Everyday Heat Transfer Problems: Sensitivities to Governing Variables

Author: M. Kemal Atesmen

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EVERYDAY HEAT TRANSFER PROBLEMS Sensitivities To Governing Variables

by M. Kemal Atesmen

© 2009 by ASME, Three Park Avenue, New York, NY 10016, USA (www.asme.org) All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. INFORMATION CONTAINED IN THIS WORK HAS BEEN OBTAINED BY THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS FROM SOURCES BELIEVED TO BE RELIABLE. HOWEVER, NEITHER ASME NOR ITS AUTHORS OR EDITORS GUARANTEE THE ACCURACY OR COMPLETENESS OF ANY INFORMATION PUBLISHED IN THIS WORK. NEITHER ASME NOR ITS AUTHORS AND EDITORS SHALL BE RESPONSIBLE FOR ANY ERRORS, OMISSIONS, OR DAMAGES ARISING OUT OF THE USE OF THIS INFORMATION. THE WORK IS PUBLISHED WITH THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARE SUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER ENGINEERING OR OTHER PROFESSIONAL SERVICES. IF SUCH ENGINEERING OR PROFESSIONAL SERVICES ARE REQUIRED, THE ASSISTANCE OF AN APPROPRIATE PROFESSIONAL SHOULD BE SOUGHT. ASME shall not be responsible for statements or opinions advanced in papers or ... printed in its publications (B7.1.3). Statement from the Bylaws. For authorization to photocopy material for internal or personal use under those circumstances not falling within the fair use provisions of the Copyright Act, contact the Copyright Clearance Center (CCC), 222 Rosewood Drive, Danvers, MA 01923, tel: 978-750-8400, www.copyright.com.

Library of Congress Cataloging-in-Publication Data Atesmen, M. Kemal. Everyday heat transfer problems : sensitivities to governing variables / by M. Kemal Atesmen. p. cm. Includes bibliographical references and index. ISBN 978-0-7918-0283-0 1. Heat–Transmission–Problems, exercises, etc. 2. Materials–Thermal properties– Problems, exercises, etc. 3. Thermal conductivity–Problems, exercises, etc. 4. Engineering mathematics–Problems, exercises, etc. I. Title. TA418.54.A47 2009 621.402’2–dc22

2008047423

TABLE OF CONTENTS Introduction ................................................................................................... 1 Chapter 1 Heat Loss from Walls in a Typical House ............................. 5 Chapter 2 Conduction Heat Transfer in a Printed Circuit Board .... 13 Chapter 3 Heat Transfer from Combustion Chamber Walls.............. 25 Chapter 4 Heat Transfer from a Human Body During Solar Tanning ............................................................................ 33 Chapter 5 Efficiency of Rectangular Fins .............................................. 41 Chapter 6 Heat Transfer from a Hot Drawn Bar .................................. 51 Chapter 7 Maximum Current in an Open-Air Electrical Wire .......... 65 Chapter 8 Evaporation of Liquid Nitrogen in a Cryogenic Bottle ...................................................................... 77 Chapter 9 Thermal Stress in a Pipe ........................................................ 85 Chapter 10 Heat Transfer in a Pipe with Uniform Heat Generation in its Walls ......................................................... 93 Chapter 11 Heat Transfer in an Active Infrared Sensor .................. 103 Chapter 12 Cooling of a Chip ................................................................. 113

iii

Everyday Heat Transfer Problems Chapter 13 Cooling of a Chip Utilizing a Heat Sink with Rectangular Fins ......................................................... 121 Chapter 14 Heat Transfer Analysis for Cooking in a Pot ................ 131 Chapter 15 Insulating a Water Pipe from Freezing ........................... 139 Chapter 16 Quenching of Steel Balls in Air Flow .............................. 147 Chapter 17 Quenching of Steel Balls in Oil......................................... 155 Chapter 18 Cooking Time for Turkey in an Oven .............................. 161 Chapter 19 Heat Generated in Pipe Flows due to Friction ............. 169 Chapter 20 Sizing an Active Solar Collector for a Pool ................... 179 Chapter 21 Heat Transfer in a Heat Exchanger ................................. 195 Chapter 22 Ice Formation on a Lake .................................................... 203 Chapter 23 Solidification in a Casting Mold ....................................... 213 Chapter 24 Average Temperature Rise in Sliding Surfaces in Contact .............................................................................. 221 References .................................................................................................... 233 Index .............................................................................................................. 235

iv

I NTRODUCTION

Everyday engineering problems in heat transfer can be very complicated and may require solutions using ﬁnite element or ﬁnite difference techniques in transient mode and in multiple dimensions. These engineering problems might cover conduction, convection and radiation energy transfer mechanisms. The thermophysical properties that govern a particular heat transfer problem can be challenging to discover, to say the least. Some of the standard thermophysical properties needed to solve a heat transfer problem are density, speciﬁc heat at constant pressure, thermal conductivity, viscosity, volumetric thermal expansion coefﬁcient, heat of vaporization, surface tension, emissivity, absorptivity, and transmissivity. These thermophysical properties can be strong functions of temperature, pressure, surface roughness, wavelength and other properties. in the region of interest. Once a heat transfer problem's assumptions are made, equations set up and boundary conditions determined, one should investigate the sensitivities of desired outputs to all the governing independent variables. Since these sensitivities are mostly non-linear, one should

1

Everyday Heat Transfer Problems

analyze them in the region of interest. The results of such sensitivity analyses will provide important information as to which independent variables should be researched thoroughly, determined accurately, and focused on. The sensitivity analysis will also provide insight into uncertainty analysis for the dependent variable, (Reference S. J. Kline and F. A. McClintock [9]). If the dependent variable y is deﬁned as a function of independent variables x1, x2, x3, … xn as follows: y = f(x1, x2, x3, … xn) then the uncertainty U for the dependent variable can be written as: U = [(∂y/∂x1 u1)2 + (∂y/∂x2 u2)2 + (∂y/∂x3 u3)2 + … + (∂y/∂xn un)2]0.5 where ∂y/∂x1, ∂y/∂x2, ∂y/∂x3, …, ∂y/∂xn are the sensitivities of the dependent variable to each independent variable and u1, u2, u3, …, un are the uncertainties in each independent variable for a desired conﬁdence limit. In this book, I will provide sensitivity analyses to well-known everyday heat transfer problems, determining ∂y/∂x1, ∂y/∂x2, ∂y/∂x3, …, ∂y/∂xn for each case. The analysis for each problem will narrow the ﬁeld of independent variables that should be focused on during the design process. Since most heat transfer problems are non-linear, the results presented here would be applicable only in the region of values assumed for independent variables. For the uncertainties of independent variables—for example, experimental measurements of thermophysical properties—the reader can ﬁnd the appropriate uncertainty value for a desired conﬁdence limit within existing literature on the topic. Each chapter will analyze a different one-dimensional heat transfer problem. These problems will vary from determining the maximum allowable current in an open-air electrical wire to cooking a turkey in a convection oven. The equations and boundary conditions for each problem will be provided, but the focus will be on the sensitivity of the governing dependant variable on the changing independent

2

Introduction

variables. For the derivation of the fundamental heat transfer equations and for insight into the appropriate boundary conditions, the reader should refer to the heat transfer fundamentals books listed in the references. Problems in Chapters 1 through 6 deal with steady-state and one-dimensional heat transfer mechanisms in rectangular coordinates. Chapters 7 through 10 deal with steady-state and one-dimensional heat transfer mechanisms in cylindrical coordinates. Unsteady-state problems in one-dimensional rectangular coordinates will be tackled in Chapters 11 through 14, cylindrical coordinates in Chapter 15, and spherical coordinates in Chapters 16 through 18. The following six chapters are allocated to special heat transfer problems. Chapters 19 and 20 deal with momentum, mass and heat transfer analogies used to solve the problems. Chapter 21 analyzes a counterﬂow heat exchanger using the log mean temperature difference method. Chapters 22 and 23 solve heat transfer problems of ice formation and solidiﬁcation with moving boundary conditions. Chapter 24 analyzes the problem of frictional heating of materials in contact with moving sources of heat. I would like to thank my engineering colleagues G. W. Hodge, A. Z. Basbuyuk, E. O. Atesmen, and S. S. Tukel for reviewing some of the chapters. I would also like to dedicate this book to my excellent teachers and mentors in heat transfer at several universities and organizations. Some of the names at the top of a long list are Prof. W. M. Kays, Prof. A. L. London, Prof. R. D. Haberstroh, Prof. L. V. Baldwin, and Prof. T. N. Veziroglu. M. Kemal Atesmen Ph. D. Mechanical Engineering Santa Barbara, California

3

CHAPTER

HEAT LOSS

1

FROM WALLS IN A TYPICAL HOUSE

H

eat loss from the vertical walls of a house is analyzed under steady-state conditions. Walls are assumed to be large and built in a planar fashion, so that one-dimensional heat transfer rate equations in rectangular coordinates may be used, and only conduction and convection heat transfer mechanisms are considered. In this analysis, radiation heat transfer effects are neglected. No air leakage through the wall was assumed. Also, the wall material thermal conductivities are assumed to be independent of temperature in the region of operation. Assuming winter conditions—the temperature inside the house is higher than the temperature outside the house—the convection heat transferred from the inside of the house to the inner surface of the inner wall is: Q/A = hin (Tin – Tinner wall inside surface)

(1-1)

Most walls are constructed from three types of materials: inner wall board, insulation and outer wall board. The heat transfer from these wall layers will occur by conduction, and is presented by the following rate Eqs., (1-2) through (1-4):

5

Everyday Heat Transfer Problems

Q/A = (kinner wall/tinner wall) (Tinner wall inside surface – Tinner wall outside surface) (1-2) Q/A = (kinsulation/tinsulation) (Tinner wall outside surface – Touter wall inside surface) (1-3) Q/A = (kouter wall/touter wall) (Touter wall inside surface – Touter wall outside surface) (1-4) The heat transfer from the outer surface of the outer wall to the atmosphere is by convection and can be expressed by the following rate Eq. (1-5): Q/A = hout (Tout – Touter wall outer surface)

(1-5)

Eliminating all the wall temperatures from Eqs. (1-1) through (1-5), the heat loss from a house wall can be rewritten as: Q/A = (Tin – Tout)/[(1/hin) + (tinner wall/kinner wall) + (tinsulation/kinsulation) (1-6) + (touter wall/kouter wall) + (1/hout)] The denominator in Eq. (1-6) represents all the thermal resistances between the inside of the house and the atmosphere, and they are in series. In the construction industry, wall materials are rated with their R-value, namely the thermal conduction resistance of one-inch material. R-value dimensions are given as (hr-ft2-F/BTU)(1/in). The sensitivity analysis will be done in the English system of units rather than the International System (SI units). The governing Eq. (1-6) for heat loss from a house wall can be rewritten in terms of R-values as follows: Q/A = (Tin – Tout)/[(1/hin) + Rinner wall tinner wall + Rinsulation tinsulation + Router wall touter wall + (1/hout)]

(1-7)

where the deﬁnitions of the variables with their assumed nominal values for the present sensitivity analysis are given as: Q/A = heat loss through the wall due to convection and conduction in Btu/hr-ft2

6

Heat Loss From Walls In A Typical House

Tin = 68ºF (inside temperature) Tout = 32ºF (outside temperature) hin = 5 BTU/hr-ft2-F (inside convection heat transfer coefﬁcient) Rinner wall = 0.85 hr-ft2-F/BTU-in (wall inside board R-value) tinner wall = 1 in (wall inside board thickness) Rinsulation = 3.5 hr-ft2-F/BTU-in (insulation layer R-value) tinsulation = 4 in (insulation layer thickness) Router wall = 5 hr-ft2-F/BTU-in (wall outside board R-value) touter wall = 1 in (wall outside board thickness) hout = 10 BTU/hr-ft2-F (outside convection heat transfer coefﬁcient). The heat loss through a wall due to changes in convection heat transfer is presented in Figures 1-1 and 1-2. Changes in the convection heat transfer coefﬁcient affect the heat loss mainly in the natural convection regime. As the convection heat transfer coefﬁcient increases into the forced convection regime, heat loss value asymptotes. Resistances from both inside and outside convection heat transfer are too small to cause any change in heat loss through the wall. The heat loss through a wall due to changes in insulation material R-value is presented in Figures 1-3 and 1-4. Higher R-value insulation

1.8

Q/A, Btu/hr-ft2

1.79 1.78 1.77 1.76 1.75

5

10

15

20

25

Outside Convection Heat Transfer Coefficient, Btu/hr-ft2-F

Figure 1-1 Wall heat loss versus outside convection heat transfer coefﬁcient

7

Everyday Heat Transfer Problems

Q/A, Btu/hr-ft2

1.8 1.78 1.76 1.74 1.72

1

3

5

7

9

Inside Convection Heat Transfer Coefficient, Btu/hr-ft2-F

Figure 1-2 Wall heat loss versus inside convection heat transfer coefﬁcient

material is deﬁnitely the way to go, depending upon the cost and beneﬁt analysis results. The thickness of the insulation material is also very crucial. Thicker insulation material is deﬁnitely the best choice, depending upon the cost and beneﬁt analysis results.

Q/A, Btu/hr-ft2

2 1.8 1.6 1.4 1.2

3

3.5

4

Insulation "R" Value,

4.5 hr-ft2-F/Btu-in

Figure 1-3 Wall heat loss versus insulation R-value

8

5

Heat Loss From Walls In A Typical House

Q/A, Btu/hr-ft2

6

4

2

0

0

2

4 6 8 Insulation Thickness, in

10

12

Figure 1-4 Wall heat loss versus insulation thickness

The effects on heat loss of inner and outer wall board R-values and thicknesses are similar to the effects of insulation R-value and thickness, but to a lesser extent. Sensitivities of heat loss to all the governing variables around the nominal values given above will be analyzed later. Sensitivity of heat loss to the outside convection heat transfer coefﬁcient can be determined in a closed form by differentiating the heat loss Eq. (1-7) with respect to hout: ∂(Q/A)/∂hout = (Tin – Tout)/{h2out [(1/hin) + Rinner wall tinner wall + Rinsulation tinsulation + Router wall touter wall + (1/hout)]2}

(1-8)

Sensitivity of heat loss to the outside convection heat transfer coefﬁcient is given in Figure 1-5. Similar sensitivity is experienced for the inside convection heat transfer coefﬁcient. The sensitivity of heat loss to the convection heat transfer coefﬁcient is high in the natural convection regime, and it diminishes in the forced convection regime. Sensitivities of heat loss to insulation material R-value and insulation thickness are given in Figures 1-6 and 1-7 respectively.

9

Everyday Heat Transfer Problems

∂ (Q/A) / ∂hout, F

0.09

0.06

0.03

0

10 15 5 Outside Convection Heat Transfer Coefficient, Btu/hr-ft2-F

0

20

Figure 1-5 Sensitivity of house wall heat loss per unit area to the outside

convection heat transfer coefﬁcient

These two sensitivities are similar, as can be expected, since the linear product of insulation material R-value and insulation thickness affects the heat loss, as shown in the governing heat loss Eq. (1-7). Absolute sensitivity values are high at the low values of insulation

∂(Q/A) / ∂ Rinsulation, (BTU/hr-ft2)2(in/F)

0 −0.5 −1 −1.5 −2 −2.5

0

1

2

3

4

5

6

Insulation Material R-Value, hr-ft2-F/BTU-in

Figure 1-6 Sensitivity of house wall heat loss per unit area to insulation

material R-value

10

∂ (Q/A) / ∂t insulation, BTU/hr-ft2-in

Heat Loss From Walls In A Typical House

0

−0.5

−1

−1.5

0

2

4 8 6 Insulation Thickness, in

10

12

Figure 1-7 Sensitivity of house wall heat loss per unit area to insulation

thickness

material R-value and insulation thickness. Sensitivities approach zero as insulation material R-value and insulation thickness values increase. A ten-percent variation in independent variables around the nominal values given above produces the sensitivity results given in Table 1-1 The sensitivity results are given in a descending order and they are applicable only in the region of assigned nominal values, due to their non-linear effect to heat loss. The one exception is temperature potential, (Tin − Tout), which will always be ±10% due to its linear effect on heat loss. Material R-value and its thickness change have the same sensitivity, since their linear product affects the governing heat loss equation. Heat loss through the wall is most sensitive to the temperature potential between the inside and outside of the house. Changes in wall insulation R-value and thickness affect heat loss as much as the temperature potential. Continuing in order of sensitivity, wall outer board R-value and thickness changes affect heat loss the most, followed by wall inside board R-value and thickness. Wall heat loss is

11

Everyday Heat Transfer Problems

Table 1-1 House wall heat loss change per unit area due to a 10% change in variables nominal values

Variable

Nominal Value

House Wall Heat Loss Change Due To A 10% Decrease In Nominal Value

Tin − Tout

36°F

−10%

+10%

Rinsulation

3.5 hr-ft -F/BTU-in

+7.467%

−6.497%

2

House Wall Heat Loss Change Due To A 10% Increase In Nominal Value

tinsulation

4 in

+7.467%

−6.497%

Router wall

5 hr-ft2-F/BTU-in

+2.545%

−2.545%

touter wall

1 in

+2.545%

−2.545%

Rinner wall

0.85 hr-ft -F/BTU-in

+0.424%

−0.424%

tinner wall

1 in

+0.424%

−0.424%

2

hin

5 BTU/hr-ft -F

−0.110%

+0.090%

hout

10 BTU/hr-ft -F

−0.055%

+0.045%

2

2

least sensitive to both the inside and outside heat transfer coefﬁcient changes. Wall heat loss sensitivity to both the inside and outside heat transfer coefﬁcient changes is an order of magnitude less than sensitivity to temperature potential changes.

12

CHAPTER

CONDUCTION

HEAT TRANSFER IN A PRINTED CIRCUIT BOARD

2

C

onduction heat transfer in printed circuit boards (PCBs) has been studied extensively in literature i.e., B. Guenin [4]. The layered structure of a printed circuit board is treated using two different thermal conductivities; one is in-plane thermal conductivity and the other is through-thickness thermal conductivity. One-dimensional conduction heat transfers in in-plane direction and through-thickness direction are treated independently. Since the signiﬁcant portion of the conduction heat transfer in a PCB occurs in the in-plane direction in the conductor layers, this is a valid assumption. Under steady-state conditions and with constant thermophysical properties, the in-plane (i-p) conduction heat transfer equation for a PCB can be written as: Qin-plane = Q1i-p + Q2i-p + … + Qni-p

(2-1)

where the subscript refers to the layers of the PCB. Using the conduction rate equation in rectangular coordinates for a PCB with a width of W, a length of L, layer thicknesses ti and layer thermal conductivities ki, Eq. (2-1) can be rewritten as:

13

Everyday Heat Transfer Problems

W Σti kin-plane (TL=0 – TL=L)/L = W t1 k1 (TL=0 – TL=L)/L + W t2 k2 (TL=0 – TL=L)/L + W tn kn (TL=0 – TL=L)/L

(2-2)

In-plane conduction heat transfer in a PCB represents a parallel thermal resistance circuit which can be written as: (1/Rin-plane) = (1/R1) + (1/R2) + … + (1/Rn) where Ri = L/(kitiW) (2-3) where kin-plane = Σ(kiti)/Σti

(2-4)

Through-thickness (t-t) conduction heat transfer in a PCB represents a series thermal resistance circuit, and the through-thickness conduction heat transfer equation for a PCB can be written as: Qthrough-thickness = Q1t-t = Q2t-t = … = Qnt-t

(2-5)

which can be expanded into following equations: W L kthrough-thickness (Tt=0 – Tt=Σti)/Σti = W L k1 (Tt=0 − Tt=t1)/t1 = W L k2 (Tt=t1 − Tt=t2)/t2 = … = W L kn (Tt=tn-1 − Tt=tn)/tn

(2-6)

Inter-layer temperatures can be eliminated from Eqs. (2-6), and a series thermal resistance equation extracted as follows: Rthrough-thickness = R1 + R2 + … + Rn where Ri = ti/ki

(2-7)

kthrough-thickness = Σti/Σ(ti/ki).

(2-8)

where

A printed circuit board is commonly built as layers of conductors separated by layers of insulators. The conductors are mostly alloys of copper, silver or gold, while the insulators are mostly a variety of

14

Conduction Heat Transfer In A Printed Circuit Board

epoxy resins. Therefore the in-plane thermal conductivity Eq. (2-4) for a PCB can be rewritten as: kin-plane = [kconductor Σtconductor + kinsulator (ttotal − Σtconductor)]/ttotal (2-9) Similarly, the through-thickness thermal conductivity Eq. (2-8) for a PCB can be rewritten as: kthrough-thickness = ttotal/[(Σtconductor/kconductor) + (ttotal − Σtconductor)/kinsulator] (2-10) The sensitivities of these two PCB thermal conductivities are analyzed for a 500 µm-thick printed circuit board, with the assumed nominal values for thermal conductivity of the conductor and insulator layers given below: kconductor = 377 W/m-C for copper conductor layers and kinsulator = 0.3 W/m-C for glass reinforced polymer layers. In-plane thermal conductivity versus percent of conductor layers to total printed circuit board thickness is given in Figure 2-1. In-plane thermal conductivity starts at the all-insulator thermal conductivity value of 0.3 W/m-C, and increases linearly to conductor thermal conductivity at no insulator layers. Sensitivities of in-plane thermal conductivity to changes in kconductor and kinsulator are represented in Figure 2-2. As you can see, the two sensitivities are opposite. The sensitivity of in-plane thermal conductivity to conductor thickness is a constant, 0.75 W/m-C-µm. The sensitivity of in-plane thermal conductivity to insulator thickness is also a constant, and it is the opposite of sensitivity to conductor thickness, namely –0.75 W/m-C-µm. A ten percent variation in variables around the nominal values given above produce the sensitivity results in Table 2-1 for in-plane thermal conductivity. For these nominal values, in-plane thermal

15

Everyday Heat Transfer Problems

kin-plane, W/m-C

400 300 200 100 0

0

20

40 60 80 % Conductor Layers Thickness

100

Figure 2-1 In-plane thermal conductivity versus percent of conductor layers

∂ kin-plane / ∂kconductor & ∂ kin-plane / ∂ kinsulator

thickness to total printed circuit board thickness

1

∂kin-plane / ∂kinsulator

0.8 0.6 0.4 0.2 0

∂kin-plane / ∂kinsulator 0

20 40 60 80 % Conductor Layers Thickness

100

Figure 2-2 Sensitivity of in-plane thermal conductivity to kconductor and

to kinsulator

16

Conduction Heat Transfer In A Printed Circuit Board

Table 2-1 In-plane thermal conductivity change due to a 10% change in variables around nominal values for a 500 micron thick PCB

Nominal Value

In-Plane Thermal Conductivity Change Due To A 10% Decrease In Nominal Value

In-Plane Thermal Conductivity Change Due To A 10% Increase In Nominal Value

kconductor

377 W/m-C

−9.99%

+9.99%

Σtconductor

250 µm

−9.98%

+9.98%

Σtinsulator

250 µm

+9.98%

−9.98%

kinsulator

0.3 W/m-C

−0.01%

+0.01%

Variable

conductivity is 188.65 W/m-C. Insulator thermal conductivity is the least effective independent variable in this case due to its low value. Variations in the sum of conductor thicknesses and the sum of insulator thicknesses affect in-plane thermal conductivity in opposing directions, but with the same magnitude. Through-thickness thermal conductivity versus percent of conductor thickness has a non-linear behavior, and it is given for a 500-micron PCB in Figure 2-3. Through-thickness thermal conductivity is similar to insulator layer thermal conductivity for up to 80% conductor layer thickness of the total printed circuit board, and therefore is not a good conduction heat transfer path for printed circuit boards. Sensitivities of through-thickness thermal conductivity to kconductor and to kinsulator are given in Figure 2-4. The sensitivity of through-thickness thermal conductivity to changes in conductor thermal conductivity is negligible throughout the percent conductor layer thickness. The sensitivity of through-thickness thermal conductivity to changes in insulator thermal conductivity increases and becomes signiﬁcant as the thickness percentage of the insulator layers decreases.

17

Everyday Heat Transfer Problems

kthrough-thickness, W/m-C

20 15 10 5 0

0

20

40 60 80 % Conductor Layers Thickness

100

Figure 2-3 Through-thickness thermal conductivity versus percent

∂kthrough-thickness / ∂ kconductor & ∂ kthrough-thickness / ∂ kinsulator

conductor layer thickness

20

∂ kthrough-thickness / ∂kinsulator

16 12 8 4 0

0

50 % Conductor Layers Thickness

100

∂ kthrough-thickness / ∂ kconductor

Figure 2-4 Sensitivity of through-thickness thermal conductivity to

conductor and insulator thermal conductivities versus percent conductor layer thickness

18

∂kthrough-thickness / ∂ kconductor & ∂ kthrough-thickness / ∂kinsulator, W/m-C-um

Conduction Heat Transfer In A Printed Circuit Board

0.05 ∂ kthrough-thickness / ∂ kconductor

0 −0.05 −0.1 −0.15 −0.2

0

20 40 60 80 % Conductor Layers Thickness

100

∂ kthrough-thickness / ∂ kinsulator

Figure 2-5 Sensitivity of through-thickness thermal conductivity to

conductor and insulator thickness

Sensitivities of through-thickness thermal conductivity to conductor and insulator thickness are given in Figure 2-5. The sensitivities become signiﬁcant as conductor thickness approaches 100%. A ten-percent variation in independent variables around the nominal values given above produce the sensitivity results given in Table 2-2 for through-thickness thermal conductivity, which has a nominal value of 0.6 W/m-C. In this case, through-thickness thermal conductivity is very resistant to conductor thermal conductivity variations in the region of interest. A second and a similar analysis can be performed for a plated or sputtered thinner circuit. A 10-µm thick circuit is considered with the assumed nominal thermal conductivities below: kconductor = 377 W/m-C for copper conductor layers and kinsulator = 36 W/m-C for aluminum oxide insulating layers. In-plane thermal conductivity versus percent of conductor layers to total thickness is given in Figure 2-6. In-plane thermal conductivity

19

Everyday Heat Transfer Problems

Table 2-2 Through-thickness thermal conductivity change due to a 10% change in variables around nominal values for a 500 micron thick PCB Through-Thickness Through-Thickness Thermal Conductivity Thermal Conductivity Change Due To Change Due To Nominal A 10% Decrease In A 10% Increase In Value Nominal Value Nominal Value

Variable Σtinsulator

250 µm

+11.09%

−9.08%

kinsulator

0.3 W/m-C

−9.99%

+9.99%

Σtconductor

250 µm

−9.08%

+11.09%

kconductor

377 W/m-C

−0.01%

+0.01%

kin-plane, W/m-C

400 300 200 100 0 0

20

40 60 80 % Conductor Layers Thickness

100

Figure 2-6 In-plane thermal conductivity versus percent of conductor layers

to total thickness

starts at the all insulator thermal conductivity value of 36 W/m-C and increases linearly to a conductor thermal conductivity of 377 W/m-C. Sensitivites of in-plane thermal conductivity to kconductor and kinsulator are given in Figure 2-7. As you can see, the two sensitivities are opposite.

20

Conduction Heat Transfer In A Printed Circuit Board

∂kin-plane / ∂kconductor & ∂ kin-plane / ∂ kinsulator

1

∂ kin-plane / ∂ kconductor

0.8 0.6 0.4 0.2 0

∂ kin-plane / ∂ kinsulator 0

20 40 60 80 % Conductor Layers Thickness

100

Figure 2-7 Sensitivity of in-plane thermal conductivity to kconductor and

to kinsulator

The sensitivity of in-plane thermal conductivity to conductor thickness is a constant at 34.1 W/m-C-µm. The sensitivity of in-plane thermal conductivity to insulator thickness is a constant, and it is the opposite of sensitivity to in-plane thermal conductivity, namely –34.1 W/m-C-µm. A ten percent variation in variables around the nominal values given above produce the sensitivity results in Table 2-3 for in-plane thermal conductivity, which has a nominal value of 87.15 W/m-C. Insulator thickness is the dominant independent variable in this region of interest. Through-thickness thermal conductivity versus percent of conductor thickness has a non-linear behavior, and it is given in Figure 2-8. The percentage of the conductor layers thickness to total circuit thickness affects the through-thickness thermal conductivity at all conductor layer thicknesses. Through-thickness conduction heat transfer is much more prominent in thin-plated or sputtered circuits. Sensitivities of through-thickness thermal conductivity to kconductor and to kinsulator are given in Figure 2-9. The sensitivity of throughthickness thermal conductivity to changes in conductor thermal conductivity is negligible at low percentages of conductor layer thickness. On the other hand, the sensitivity of through-thickness

21

Everyday Heat Transfer Problems

Table 2-3 In-plane thermal conductivity change due to a 10% change in variables around nominal values for a 10-micron thick circuit

Nominal Value

In-Plane Thermal Conductivity Change Due To A 10% Decrease In Nominal Value

In-Plane Thermal Conductivity Change Due To A 10% Increase In Nominal Value

Σtinsulator

8.5 µm

+33.3%

−33.3%

kconductor

377 W/m-C

−6.5%

+6.5%

Σtconductor

1.5 µm

−5.9%

+5.9%

36 W/m-C

−3.5%

+3.5

Variable

kinsulator

kthrough-thickness, W/m-C

thermal conductivity to changes in insulator thermal conductivity is signiﬁcant; it increases to a maximum at around 90% conductor layer thickness, and ﬁnally decreases sharply as conductor thermal conductivity starts to dominate.

400 300 200 100 0

0

20

40

60

80

100

% Conductor Layers Thickness

Figure 2-8 Through-thickness conduction heat transfer coefﬁcient versus

percentage of conductor layer thickness

22

∂ kthrough-thickness / ∂ kconductor & ∂ kthrough-thickness / ∂kinsulator

Conduction Heat Transfer In A Printed Circuit Board

3.5 3

∂kthrough-thickness / ∂ kinsulator

2.5 2 1.5 1 0.5 0

0

20 40 60 80 % Conductor Layers Thickness

100

∂ kthrough-thickness / ∂ kconductor

Figure 2-9 Sensitivity of through-thickness thermal conductivity to

∂ kthrough-thickness / ∂kinsulator & ∂kthrough-thickness / ∂kconductor, W/m-C-µm

conductor and insulator thermal conductivities versus percentage of conductor layer thickness

10

∂ kthrough-thickness / ∂kconductor

0 −10 −20 −30 −40 −50

0

20 60 40 80 % Conductor Layers Thickness

∂ kthrough-thickness / ∂kinsulator 100

Figure 2-10 Sensitivity of through-thickness thermal conductivity to

conductor thickness and to insulator thickness

23

Everyday Heat Transfer Problems

Table 2-4 Through-thickness thermal conductivity change due to a 10% change in variables around nominal values for a 10-micron thick circuit Through-Thickness Through-Thickness Thermal Conductivity Thermal Conductivity Change Due To A 10% Change Due To A 10% Decrease In Increase In Nominal Value Nominal Value

Variable

Nominal Value

kinsulator

36 W/m-C

−9.85%

+9.18%

Σtinsulator

8.5 µm

+9.76%

−8.17%

Σtconductor

1.5 µm

−1.55%

+1.59%

kconductor

377 W/m-C

−0.18%

+0.15%

Sensitivities of the through-thickness heat transfer coefﬁcient to conductor and insulator thickness are given in Figure 2-10. The sensitivity to insulator thickness becomes signiﬁcant as the percent of conductor thickness approaches 100%. A ten percent variation in independent variables around the nominal values given above produce the sensitivity results in Table 2-4 for through-thickness thermal conductivity, which has a nominal value of 41.65 W/m-C. In this case, through-thickness thermal conductivity is most sensitive to insulator thermal conductivity and insulator thickness variations.

24

CHAPTER

3 FROM COMBUSTION HEAT TRANSFER

CHAMBER WALLS

C

ooling the walls of a combustion chamber containing gases at high temperatures, i.e., 1000°C, results in parallel modes of heat transfer, convection and radiation. This problem can be approached by assuming one-dimensional steady-state heat transfer in rectangular coordinates and with constant thermophysical properties. Convection heat transfer per unit area, from hot gases to the hot side of a wall that separates the cold medium and the hot gases, can be written as: (Q/A)convection = hcg (Tg – Twh)

(3-1)

Radiation heat transfer per unit area from hot gases which are assumed to behave as gray bodies to the hot side of a wall can be written as: (Q/A)radiation = hrg (Tg – Twh) = = εg σ(Tg4 – Twh4)

(3-2)

And so the total heat transfer from the hot gases to a combustion chamber wall is:

25

Everyday Heat Transfer Problems

(Q/A)total = (Q/A)convection + (Q/A)radiation = (hcg + hrg) (Tg – Twh) (3-3) These two heat transfer mechanisms act as parallel thermal resistances, namely: (1/Rtotal) = (1/Rconvection) + (1/Rradiation)

(3-4)

where hcg is the convection heat transfer coefﬁcient between gas and the hot side of a wall, hrg is the radiation heat transfer coefﬁcient between gas and the hot side of a wall, Tg is average gas temperature and Twh is average hot side wall temperature. The radiation heat transfer coefﬁcient hrg is deﬁned as: hrg = εg σ(Tg4 – Twh4)/(Tg – Twh)

(3-5)

where εg is emissivity of gas and σ is the Stefan Boltzmann constant. Heat transfer occurs through the wall by conduction and is deﬁned as: (Q/A)total = (kwall/L)(Twh – Twc)

(3-6)

where kwall is wall material thermal conductivity, L is thickness of the wall and Twc is the wall temperature at the cold medium side of the wall. Heat transfer between the cold medium side of the wall and the cold medium occurs by convection and is deﬁned as: (Q/A)total = hc (Twc – Tc)

(3-7)

where hc is the convection heat transfer coefﬁcient between the cold medium side of the wall and the cold medium, and Tc is the average temperature of cold medium. In this example, Twh is going to be the dependent variable, and it will be solved by iterating a function using a combination of above Eqs. (3-3), (3-5), (3-6) and (3-7), as follows: (Tg – Tc)/{[1/(hcg + hrg)] + (L/kwall) + (1/hc)} − (hcg + hrg)(Tg – Twh) = 0

26

(3-8)

Heat Transfer From Combustion Chamber Walls

The above governing equation can be rewritten as an iteration function K as follows: K = −(Tg – Tc) + {[1/(hcg + hrg)] + (L/kwall) + (1/hc)}(hcg + hrg)(Tg – Twh)

(3-9)

During iteration to determine Twh, all temperatures should be in degrees Kelvin because of the fourth power behavior of radiation heat transfer. Also, all thermophysical properties are assumed to be constants. Nominal values of these variables for the sensitivity analysis are assumed to be as follows: Tg = 1000ºC (1273 K) Tc = 100ºC (373 K) kwall = 20 W/m-K L = 0.01m hcg = 100 W/m2-K hc = 50 W/m2-K εg = 0.2 σ = 5.67×10−8 W/m2-K4 For these nominal variables, the iteration function crosses zero at 1075.93 K as shown in Figure 3-1, and 42.5% of the total heat transfer from hot gases to the hot side of the wall comes from radiation mode; the rest, 57.5%, comes from convection mode. The effects of hot gas temperature and cold medium temperature to hot side wall temperature are shown respectively in Figures 3-2 and 3-3. Hot gas temperature affects hot side wall temperature almost one-to-one, namely a slope of 0.925. However, cold side medium temperature affects hot side wall temperature almost ﬁve-to-one, namely a slope of 0.221. The effects of wall parameters—thermal conductivity and wall thickness—on hot side wall temperature are shown in Figures 3-4 and 3-5. Changes in wall thermal conductivities below 10 W/m-K are more effective on hot side wall temperature. Hot side wall temperature sensitivity to wall thickness is pretty much a constant, 3.5 C/cm.

27

Everyday Heat Transfer Problems

Iteration Function, K

2000

1000

0

−1000 400

600

800

1000

1200

1400

Twh, K

Figure 3-1 Iteration function versus Twh

Hot Side Wall Temperature, °C

The convection heat transfer coefﬁcients on both sides of the wall have opposite effects on hot side wall temperature, as shown in Figures 3-6 and 3-7. As the hot gas side convection heat transfer coefﬁcient increases, hot side wall temperature

2000 1600 1200 800 400 0

0

500

1000 1500 Gas Temperature, °C

Figure 3-2 Hot side wall temperature versus gas temperature

28

2000

Hot Side Wall Temperature, °C

Heat Transfer From Combustion Chamber Walls

1000

900

800

700

0

200

400 600 800 Cold Medium Temperature, °C

1000

Figure 3-3 Hot side wall temperature versus cold medium temperature

Hot Side Wall Temperature, °C

increases as well. Sensitivity of hot side wall temperature to variations in the hot gas side convection heat transfer coefﬁcient is more prominent at lower convection heat transfer coefﬁcient values.

860

840

820

800

0

10

20 30 40 Wall Thermal Conductivity, W/m-K

50

Figure 3-4 Hot side wall temperature versus wall thermal conductivity

29

Hot Side Wall Temperature, °C

Everyday Heat Transfer Problems

840 830 820 810 800

0

0.02

0.04 0.06 Wall Thickness, m

0.08

0.1

Figure 3-5 Hot side wall temperature versus wall thickness

Hot Side Wall Temperature, °C

The variation of hot side wall temperature at different hot gas emissivities is given in Figure 3-8. Hot side wall temperature is more sensitive to changes in lower values of hot gas emissivity.

1000 900 800 700 600

0

100

200

300

400

Hot Gas Side Convection Heat Transfer Coefficient,

500 W/m2-K

Figure 3-6 Hot side wall temperature versus hot gas side convection heat

transfer coefﬁcient

30

Hot Side Wall Temperature, °C

Heat Transfer From Combustion Chamber Walls

1000 800 600 400 200

0

100

200

300

400

500

Cold Medium Side Convection Heat Transfer Coefficient, W/m2-K

Figure 3-7 Hot side wall temperature versus cold medium side convection

heat transfer coefﬁcient

Hot Side Wall Temperature, °C

When the nominal values of the variables given above are varied ±10%, the results shown in Table 3-1 are obtained. Hot side wall temperature sensitivities to a ±10% change in the governing variables are given in descending order of importance,

950 900 850 800 750 700

0

0.2

0.4

0.6

0.8

1

Hot Gas Emissivity

Figure 3-8 Hot side wall temperature versus hot gas emissivity

31

Everyday Heat Transfer Problems

Table 3-1 Effects of ±10% change in nominal values of variables to hot side wall temperature

Nominal Value

Change In Hot Side Wall Temperature For A 10% Decrease In Nominal Value

Change In Hot Side Wall Temperature For A 10% Increase In Nominal Value

Tg

1273 K

−11.493%

+11.981%

hc

50 W/m2-K

+2.070%

−1.991%

hcg

100 W/m2-K

−1.262%

+1.136%

εg

0.2

−0.912%

+0.860%

Tc

373 K

−0.295%

+0.295%

20 W/m-K

+0.056%

−0.046%

0.01 m

−0.051%

+0.051%

Variable

kwall L

and they are applicable around the nominal values assumed for this study. Hot side wall temperature is most sensitive to variations in hot gas temperature. Next in order of sensitivity are the convection heat transfer coefﬁcients on both sides of the wall. Changes to emissivity of hot gases affect the dependent variable at the same level as the convection heat transfer coefﬁcients. Next in order of sensitivity is the cold medium temperature. Hot side wall temperature is least sensitive to variations in wall thermal conductivity and wall thickness. This variable order of sensitivity is applicable around the nominal values assumed for this case, due to the nonlinear relationship between the dependent variable and the independent variables.

32

CHAPTER

HEAT TRANSFER

FROM A HUMAN BODY DURING SOLAR TANNING

4

T

he solar tanning of a human body was analyzed under steady-state conditions with one-dimensional rate equations in rectangular coordinates, and using temperature-independent thermophysical properties. Human skin that is exposed to direct solar radiation is considered to be in an energy balance. Energy goes into the skin from both direct solar radiation and solar radiation scattered throughout the atmosphere. Energy leaves the skin through a variety of means and routes: by convection heat transfer and radiation heat transfer (into the atmosphere), by conduction (to the inner portions of the body), by perspiration, and by body basal metabolism. Other energy gains and losses, such as those due to terrestrial radiation, breathing and urination, are negligible. Energy balance at the human skin gives the following heat transfer equation: Qsolar radiation absorbed + Qatmospheric radiation absorbed – Qconvection – Qconduction to body – Qradiation emitted – Qperspiration − Qbasal metabolism = 0 (4-1)

33

Everyday Heat Transfer Problems

For the present sensitivity analysis, the heat transfer rate equations that will be used, and the nominal values that will be assumed for energies outlined in Eq. (4-1), are given below. Qsolar radiation absorbed = 851 W/m2

(4-2)

which assumes a gray body skin with an absorptivity, α = ε, of 0.8, on a clear summer day at noon, with full sun exposure Qatmospheric radiation absorbed = 85 W/m2

(4-3)

which is assumed to be about 10% of Qsolar radiation absorbed. Qconvection = h(Tskin – Tenvironment)

(4-4)

where h is the heat transfer coefﬁcient between the skin surface that is being tanned and the environment. In the present analysis, h is assumed to be 28.4 W/m2-K and Tenvironment is 30°C. Qconduction to body = (kbody/tbody)(Tskin – Tbody)

(4-5)

where kbody = 0.2 W/m-K, tbody = 0.1 m, and Tbody = 37°C. Qradiation emitted = εσ(T4skin – T4environment)

(4-6)

where emissivity of skin surface ε = 0.8 and σ = 5.67 × 10−8 W/m2-K4. Qperspiration = 337.5 W/m2

(4-7)

which corresponds to a 1 liter/hr perspiration rate for a human body with a perspiration area of 2 m2. Qbasal metabolism = 45 W/m2

(4-8)

which represents a 30-year-old male at rest. There are ten independent variables that govern the dependent variable Tskin in this heat transfer problem. Sensitivities to these ten

34

Heat Transfer From A Human Body During Solar Tanning

variables are analyzed in the region of the nominal values given above. The governing Eq. (4-1) takes the following form, and can be solved for Tskin by trial and error. C1T4skin + C2Tskin = C3

(4-9)

where C1 = εσ, C2 = h + kbody/tbody, and C3 = Qsolar radiation absorbed + Qatmospheric radiation absorbed – Qperspiration − Qbasal metabolism + εσT4environment + hTenvironment + kbody/tbodyTbody All the calculations are performed in degrees Kelvin for temperature, since the governing equation is non-linear in temperature. These sensitivities are presented in Table 4-1 below in the order of their signiﬁcance. The most effective variable on the skin temperature is Tenvironment and the least effective is the thermal conductivity of human tissue. The skin temperature is an order of magnitude less sensitive to changes in the thermal conductivity of human tissue— skin-to-body conduction heat transfer length, heat transfer due to basal metabolism, body temperature, atmospheric radiation absorbed, and emissivity of skin surface—than changes in the temperature of the environment, solar radiation absorbed, convection heat transfer coefﬁcient, and heat lost due to perspiration. Changes in some variables, such as the environmental temperature and heat transfer due to perspiration, behave linearly in the region of interest, and give equal percentage changes to the dependent variable on both sides of the variable's nominal value. It is important to remind the reader that the order shown in Table 4-1 is only useful in this region of the application due to non-linear behavior of the sensitivities. The non-linear affects of variables such as the convection heat transfer coefﬁcient are given in Figure 4-1 for Qsolar radiation absorbed = 851 W/m2 and for three different perspiration rates. The sensitivity of skin temperature to the convection heat transfer coefﬁcient is signiﬁcant up to 50 W/m2-K. The heat transfer coefﬁcient

35

Everyday Heat Transfer Problems

Table 4-1 Effects of ±10% change in nominal values of variables to skin temperature

Variable

Nominal Value

Skin Temperature, Tskin, Change Due To A 10% Decrease In Nominal Value

Tenvironment

30°C

−6.03%

+6.03%

−5.11%

+5.13%

Qsolar radiation absorbed

851 W/m

2

Skin Temperature, Tskin, Change Due To A 10% Increase In Nominal Value

h

28.4 W/m -K

+2.93%

−2.51%

Qperspiration

337.5 W/m

+2.03%

−2.03%

0.8

+0.53

−0.51

85 W/m2

−0.51%

+0.51%

37°C

−0.45%

+0.45%

+0.27%

−0.27%

Emissivity of skin surface, ε Qatmospheric radiation absorbed Tbody

2

2

Qbasal metabolism

45 W/m

Skin-to-body conduction length, tbody

0.1 m

−0.12%

+0.10%

0.2 W/m-K

+0.11%

−0.11%

ktissue

2

from the skin surface to the environment can be determined from appropriate empirical relationships found in References [6] and [10]. The heat transfer coefﬁcient in the natural convection regime is around 5 W/m2-K. If the wind picks up to, say, 8.9 m/s (20 mph), then the heat transfer coefﬁcient is in the turbulent ﬂow regime, and it increases to 20 W/m2-K. As the perspiration rate goes down, this sensitivity increases. The sensitivity curves are given in Figure 4-2. Similar results are obtained for an afternoon solar radiation by assuming half the noon solar radiation, i.e., Qsolar radiation absorbed = 425.5 W/m2, and they are given in Figures 3-3 and 3-4. As the solar radiation goes down, the skin temperature and its sensitivity to the convection heat transfer coefﬁcient decreases.

36

Heat Transfer From A Human Body During Solar Tanning

Tenvironment=30°C, Tbody=37°C, Qsolar radiation absorbed=851 W/m2 Skin Temperature, °C

70 65 Perspiration=0.5 liters/hr Perspiration=1.0 liter/hr Perspiration=1.5 liter/hr

60 55 50 45 40 35 30

0

100

50

150

200

Convection Heat Transfer Coefficient, W/m2-K

Figure 4-1 Skin temperature versus convection heat transfer coefﬁcient

for Qsolar radiation absorbed = 851 W/m2 and for three different perspiration rates

∂ Tskin / ∂h, m2-K2/W

0 −0.2

Perspiration=0.5 liters/hr Perspiration=1.0 liters/hr Perspiration=1.5 liters/hr

−0.4 −0.6 −0.8 −1

0

50

100

150

200

Convection Heat Transfer Coefficient, W/m2-K

Figure 4-2 Skin temperature sensitivity to convection heat transfer

coefﬁcient versus convection heat transfer coefﬁcient for Qsolar radiation absorbed = 851 W/m2 and for three different perspiration rates

37

Everyday Heat Transfer Problems

Tenvironment=30°C, Tbody=37°C, Qsolar radiation absorbed=425.5 W/m2 Skin Temperature, °C

45 Perspiration=0.5 liters/hr Perspiration=1.0 liter/hr Perspiration=1.5 liter/hr

40 35 30 25

0

50

100

150

200

Convective Heat Transfer Coefficient, W/m2-K

Figure 4-3 Skin temperature versus convection heat transfer coefﬁcient

for Qsolar radiation absorbed = 425.5 W/m2 and for three different perspiration rates

∂ Tskin / ∂h, m2-K2/W

0.2 0.1 0

Perspiration=0.5 liters/hr Perspiration=1.0 liters/hr Perspiration=1.5 liters/hr

−0.1 −0.2 −0.3 −0.4 −0.5

0

50

100

150

200

Convection Heat Transfer Coefficient, W/m2-K

Figure 4-4 Skin temperature sensitivity to convection heat transfer

coefﬁcient versus convection heat transfer coefﬁcient for Qsolar radiation absorbed = 425.5 W/m2 and for three different perspiration rates

38

Heat Transfer From A Human Body During Solar Tanning

60

Tskin, C

55 50 45 40 35 30

20

24

28 32 Tenvironment, C

36

40

Figure 4-5 Skin temperature versus environment temperature

In high perspiration rates, the skin temperature is below the environmental temperature, and it approaches the environmental temperature as the heat transfer coefﬁcient increases.

Tskin, C

50

45

40

35 0.5

0.6 0.7 0.8 0.9 Emissivity Of Human Skin Surface, ε

1

Figure 4-6 Skin temperature versus emissivity of human skin surface

39

Everyday Heat Transfer Problems

∂ Tskin / ∂ε, C

30

28

26

24 0.5

0.6

0.7

0.8

0.9 Emissivity Of Human Skin Surface, ε

1

Figure 4-7 Skin temperature sensitivity to emissivity of human skin surface

versus emissivity of human skin surface

Temperature of the skin behaves as shown in Figure 4-5 with a constant sensitivity to the changes in the temperature of the environment, in the domain of interest. The sensitivity behavior is almost one-to-one; namely, ∂Tskin/∂Tenvironment equals 0.92. Another variable that is analyzed in detail in the region of interest is the emissivity of human skin surface. Skin temperature increases with an increase in emissivity as shown in Figure 4-6. Figure 4-7 provides the skin temperature sensitivity to emissivity of human skin surface. The sensitivity decreases in a linear fashion as the emissivity increases.

40

CHAPTER

EFFICIENCY OF

RECTANGULAR FINS

5

H

eat transfer from a surface can be enhanced by using ﬁns. Heat transfer from surfaces with different types of ﬁns has been studied extensively, as seen in References by Incropera, F. P. and D. P. DeWitt [6] and by F. Kreith [10]. The present sensitivity analysis represents rectangular ﬁns under steady-state, one-dimensional, constant thermophysical property conditions without radiation heat transfer. Energy balance to a cross-sectional element of a rectangular ﬁn gives the following second order and linear differential equation for the temperature distribution along the length of the ﬁn. d2T/dx2 – (hP/kA) (T – Tenvironment) = 0

(5-1)

where h is the convection heat transfer coefﬁcient between the surface of the ﬁn and the environment in W/m2-C, k is the thermal conductivity of the ﬁn material in W/m-C, P is the ﬁn cross-sectional perimeter in meters, and A is the ﬁn cross-sectional area in m2. There can be different solutions to Eq. (5-1) depending upon the boundary condition that is used at the tip of the ﬁn. If the heat loss

41

Everyday Heat Transfer Problems

to environment from the tip of the ﬁn is neglected, the following boundary conditions can be used: T = Tbase at x = 0 and (dT/dx) = 0 at x = L

(5-2)

The solution to Eqs. (5-1) and (5-2) can be written as (T – Tenvironment) = (Tbase – Tenvironment) [cosh m(L-x)/cosh (mL)] (5-3) where L is the length of the rectangular ﬁn in meters and m = (hP/kA)0.5 in 1/m. The heat transfer from the rectangular ﬁn can be determined from Eq. (5-3) by ﬁnding the temperature slope at the base of the ﬁn, namely Qﬁn = −kA(dT/dx) at x = 0 or

(5-4)

Qﬁn = (Tbase – Tenvironment) sqrt(hkPA) tanh(mL)

(5-5)

Here the sensitivities of variables that affect the efﬁciency of a rectangular ﬁn will be analyzed. Fin efﬁciency is generally deﬁned by comparing the ﬁn heat transfer to the environment with a maximum heat transfer case to the environment, where the whole ﬁn is at the ﬁn base temperature, namely η = Qﬁn/Qmax where Qmax = hAﬁn(Tbase − Tenvironment). For a rectangular ﬁn, the ﬁn heat transfer efﬁciency is approximated by using Eq. (5-5), and by adding a corrected ﬁn length, Lc, for the heat lost from the tip of the ﬁn. η = tanh(mLc)/(mLc)

(5-6)

where m = [h 2(w + t)/k wt ]0.5 and Lc = L + 0.5t. For cases where the ﬁn width, w, is much greater than its thickness, t, m becomes m = (2h/kt)1/2

42

(5-7)

Efﬁciency Of Rectangular Fins

There are four independent variables that affect the rectangular ﬁn heat transfer efﬁciency. These are the convection heat transfer coefﬁcient, h; the thermal conductivity of the ﬁn material, k; length of the ﬁn, l; and thickness of the ﬁn, t. The sensitivity of efﬁciency to these four independent variables can be obtained in closed forms by differentiating the efﬁciency equation with respect to the desired independent variable. For example: ∂η/∂h = (0.5/h)[(1/cosh2(mLc)) – (tanh(mLc)/(mLc))]

(5-8)

Fin efﬁciency as a function of the convection heat transfer coefﬁcient for two different thermal conductivities—aluminum and copper—is given in Figure 5-1. The sensitivity of rectangular ﬁn efﬁciency with respect to the convection heat transfer coefﬁcient is given in Figure 5-2. Fin efﬁciency is good in the natural convection regime and degrades as high forced convection regimes are used. Sensitivity of ﬁn efﬁciency to the convection heat transfer coefﬁcient is high in the

1 Fin Efficiency

0.9 kcu=377.2 W/m-C

0.8 0.7

kal=206 W/m-C

0.6 0.5 0.4

0

100

200

300

Convection Heat Transfer Coefficient,

400

L=0.0508 m t=0.002 m

W/m2-C

Figure 5-1 Rectangular ﬁn efﬁciency versus convection heat transfer

coefﬁcient for two different ﬁn materials with L = 0.0508 m and t = 0.002 m

43

Everyday Heat Transfer Problems

0

∂h/∂ h, m2-C/W

−0.001

kcu=377.2 W/m-C kal=206 W/m-C

−0.002 −0.003

L=0.0508 m t=0.002 m

−0.004 −0.005

0 100 200 300 400 Convection Heat Transfer Coefficient, W/m2-C

Figure 5-2 Sensitivity of rectangular ﬁn efﬁciency to convection heat

transfer coefﬁcient versus convection heat transfer coefﬁcient for two different ﬁn materials with L = 0.0508 m and t = 0.002 m

natural convection regime and decreases as the forced convection heat transfer coefﬁcient increases. Fin efﬁciency as a function of ﬁn material thermal conductivity for two different convection heat transfer coefﬁcients—natural convection regime and forced convection regime—is given in Figure 5-3. The sensitivity of rectangular ﬁn efﬁciency with respect to ﬁn material thermal conductivity is given in Figure 5-4. Fin material thermal conductivity does not affect ﬁn efﬁciency in the natural convection regime except in the region of low thermal conductivity materials. However, in the forced convection regime the behavior is quite different. Fin material thermal conductivity affects ﬁn efﬁciency, and high thermal conductivity materials have to be used in order to achieve high ﬁn efﬁciency. The sensitivity of ﬁn efﬁciency to ﬁn material thermal conductivity is high for low thermal conductivities. The sensitivity diminishes as high ﬁn material thermal conductivities are utilized.

44

Efﬁciency Of Rectangular Fins

Fin Efficiency, η

1 h=5 W/m2-C

0.8

h=100 W/m2-C

0.6

0.4

0

100 200 300 400 Fin Thermal Conductivity, W/m-C

500

L=0.0508 m t=0.002 m

Figure 5-3 Rectangular ﬁn efﬁciency versus ﬁn material thermal

conductivity for two different convection heat transfer coefﬁcients with L = 0.0508 m and t = 0.002 m

∂ η/∂ k, m-C/W

0.0084 0.0063

h=5 W/m2-C

0.0042

h=100 W/m2-C

0.0021 0

L=0.0508 m t=0.002 m 0

100

200

300

400

500

Fin Thermal Conductivity, W/m-C

Figure 5-4 Sensitivity of rectangular ﬁn efﬁciency to ﬁn material thermal

conductivity versus ﬁn material thermal conductivity for two different convection heat transfer coefﬁcients with L = 0.0508 m and t = 0.002 m

45

Everyday Heat Transfer Problems

Fin Efficiency, η

1 0.9

h=5 W/m2-C

0.8

h=100 W/m2-C

0.7 0.6

t=0.002 m k=377.2 W/m-C 0

0.02

0.04

0.06

0.08

0.1

Fin Length, m

Figure 5-5 Rectangular ﬁn efﬁciency versus ﬁn length for two different

convection heat transfer coefﬁcients with t = 0.002 m and k = 377.2 W/m-C

Fin efﬁciency as a function of ﬁn length for two different convection heat transfer coefﬁcients—natural convection regime and forced convection regime—is given in Figure 5-5. The sensitivity of rectangular ﬁn efﬁciency with respect to ﬁn length is given in Figure 5-6. Figure 5-6 shows sensitivities for combinations of two different convection heat transfer coefﬁcients and two different thermal conductivities. Figure 5-5 shows that ﬁn efﬁciency is a weak function ﬁn length in the natural convection regime, but this weakness becomes a strong function of ﬁn length in the forced convection regime. These results can also be seen in Figure 5-6. In the natural convection regime, sensitivity of ﬁn efﬁciency to ﬁn length is low, but increases as the ﬁn length increases. In the forced convection regime, sensitivity of ﬁn efﬁciency to ﬁn length starts low, goes through a maximum as the ﬁn length increases, and decreases as the ﬁn length increases further. Fin efﬁciency as a function of ﬁn thickness for two different convection heat transfer coefﬁcients—natural convection regime and forced convection regime—is given in Figure 5-7. The sensitivity of rectangular ﬁn efﬁciency with respect to ﬁn thickness is given in Figure 5-8. Figure 5-8 shows sensitivities for combinations of two

46

Efﬁciency Of Rectangular Fins

8 h=5 W/m2-C & @ k=377.2 W/m-C h=100 W/m2-C & @ k=377.2 W/m-C h=5 W/m2-C & @ k=206 W/m-C h=100 W/m2-C & @ k=206 W/m-C

7 ∂ η/∂ L, 1/m

6 5 4 3 2 1 0

t=0.002 m 0

0.05 Fin Length, m

0.1

Figure 5-6 Sensitivity of rectangular ﬁn efﬁciency to ﬁn length versus ﬁn

length for combinations of two different convection heat transfer coefﬁcients and two different thermal conductivities with t = 0.002 m

1

Fin Efficiency, η

0.9 h=5 W/m2-C h=100 W/m2-C

0.8 0.7 0.6

k=377.2 W/m-C L=0.0508 m

0.5 0.4

0

0.001

0.002 0.003 Fin Thickness, m

0.004

0.005

Figure 5-7 Rectangular ﬁn efﬁciency versus ﬁn thickness for two different

convection heat transfer coefﬁcients with k = 377.2 W/m-C and L = 0.0508 m

47

Everyday Heat Transfer Problems

9 8

∂η/∂ t, 1/m

7

h=5 W/m2-C & @ k=377.2 W/m-C h=100 W/m2-C & @ k=377.2 W/m-C h=5 W/m2-C & k=206 W/m-C h=100 W/m2-C & @ k=206 W/m-C

6 5 4 3 2 1 0

L=0.0508 m 0

0.001

0.002

0.003

0.004

0.005

Fin Thickness, m

Figure 5-8 Sensitivity of rectangular ﬁn efﬁciency to ﬁn thickness versus

ﬁn thickness for combinations of two different convection heat transfer coefﬁcients and two different thermal conductivities with L = 0.0508 m

different convection heat transfer coefﬁcients and two different thermal conductivities. Figure 5-7 shows that ﬁn efﬁciency is a weak function of ﬁn thickness in the natural convection regime, but this weakness becomes a strong function of ﬁn thickness in the forced convection regime. In Figure 5-8, in the natural convection regime, sensitivity of ﬁn efﬁciency to ﬁn thickness starts high at low ﬁn thickness values, but decreases as the ﬁn thickness increases. In the forced convection regime, sensitivity of ﬁn efﬁciency to ﬁn thickness starts low, goes through a maximum as the ﬁn thickness increases, and decreases as the ﬁn thickness increases further. A ten percent variation in independent variables around the nominal values produces the following sensitivity results (Table 5-1) for ﬁn efﬁciency. The results are given for the natural convection regime in descending order, from the most sensitive variable to the least. Table 5-2 gives similar results for the forced convection regime.

48

Efﬁciency Of Rectangular Fins

Table 5-1 Rectangular ﬁn efﬁciency change due to a 10% change in variables nominal values for the natural convection regime

Variable

Nominal Value

Rectangular Fin Efﬁciency Change Due To A 10% Decrease In Nominal Value

L

0.0508 m

+0.218%

−0.239%

k

377.2 W/m-C

−0.129%

+0.106%

t

0.002 m

−0.124%

+0.102%

h

5 W/m -C

+0.117%

−0.116%

2

Rectangular Fin Efﬁciency Change Due To A 10% Increase In Nominal Value

The order of signiﬁcance in ﬁn efﬁciency change follows the same pattern with respect to variables in both the natural convection and forced convection regimes. However, in the forced convection regime, ﬁn efﬁciency changes are an order of magnitude higher than the natural convection regime.

Table 5-2 Rectangular ﬁn efﬁciency change due to a 10% change in variables nominal values for the forced convection regime

Variable

Nominal Value

Rectangular Fin Efﬁciency Change Due To A 10% Decrease In Nominal Value

Rectangular Fin Efﬁciency Change Due To A 10% Increase In Nominal Value

L

0.0508 m

+3.436%

−3.478%

k

377.2 W/m-C

−1.916%

+1.639%

t

0.002 m

−1.844%

+1.574%

h

100 W/m -C

+1.806%

−1.729%

2

49

CHAPTER

HEAT

TRANSFER FROM A HOT DRAWN BAR

6

A

hot drawn bar, assumed to be moving at a constant velocity out of a die at constant temperature, will be treated as a one-dimensional heat transfer problem. The Biot numberfor the bar, htD/2k, will be assumed to be less than 0.1, to assure no radial variation of temperature in the bar. Here ht is the total heat transfer coefﬁcient from the bar surface in W/m2-K, the sum of the convection heat transfer coefﬁcient and radiation heat transfer coefﬁcient, D is the bar diameter, and k is the bar thermal conductivity in W/m-K. Conduction, convection, and radiation heat transfer mechanisms affect the temperature of the drawn bar. Energy balance can be applied to a small element of the bar with a width of dx: Conduction heat transfer into the element – Conduction heat transfer out of the element – Convection heat transfer out of the element to the environment – Radiation heat transfer out of the element to the environment = Rate of change of internal energy of the element

51

Everyday Heat Transfer Problems

The energy balance on the element can be written as follows: Qconduction at x – Qconduction at x+dx – Qconvection from dx – Qradiation from dx = ρcpAdx(dT/dθ) (6-1) where ρ is the density of the bar in kg/m3, cp is the speciﬁc heat of the bar at constant pressure in J-kg/K, A is the bar cross-sectional area in m2, T is the temperature of the bar element in K, and dx/dθ is the drawn bar velocity in m/s. Assuming that all the bar thermophysical and geometrical properties are constants, the following one-dimensional, second-order and non-linear differential equation is obtained: d2T/dx2 – (hP/kA)(T-Tenvironment) – (σεP/kA)(T4 – T4environment) = (ρcpU/k)dT/dx

(6-2)

where P is the bar perimeter in m, σ is Stefan Boltzmann constant, 5.67×10-8 W/m2-K4, ε is the bar surface emissivity, and U=dx/dθ the speed of the hot drawn bar. The differential equation (6-2) reduces to steady-state heat transfer from ﬁns with a uniform cross-sectional area; if the radiation heat transfer and the rate of change of internal energy are neglected, see References by F. Kreith [10] and by Incropera, F. P. and D. P. DeWitt [6]. The boundary condition for this heat transfer problem can be speciﬁed as follows: T = Tx=0 (temperature of the drawn bar at the die location) at x = 0 (6-3) and T = Tenvironment as x goes to 4

(6-4)

The governing differential equation (6-2), along with boundary conditions (6-3) and (6-4), can be solved by ﬁnite difference methods and iteration, in order to determine the temperature at the i'th location along the bar.

52

Heat Transfer From A Hot Drawn Bar

Another method to solve this non-linear heat transfer problem is to deﬁne a radiation heat transfer coefﬁcient utilizing the temperature of the previous bar element, i-1, as follows: hradiation = σε(T4i-1 – T4environment)/(Ti-1 – Tenvironment)

(6-5)

and write the differential equation for location i along the bar as d2Ti/dx2 – [(hconvection + hradiation) P/kA](Ti -Tenvironment) = (ρcpU/k)dTi/dx

(6-6)

The solution reached by linear differential equation (6-6) for location i along the bar is valid for small x increments along the bar, i.e. < 0.01 m, since the radiation heat transfer coefﬁcient is calculated using the temperature of the previous element i-1. The solution to the above second order linear differential equation (6-6) which satisﬁes both boundary conditions, (6-3) and (6-4), is: (Ti – Tenvironment)/(Tx=0 – Tenvironment) = exp{[(U/2α) – sqrt((U/2α)2 + m2)]x}

(6-7)

where α = k/ρcp is the thermal diffusivity of the bar in m2/s and m2 = (hconvection + hradiation)P/kA in 1/m2.

(6-8)

This temperature distribution solution reduces to steady-state heat transfer from rectangular ﬁns with a uniform cross-sectional area and the above applied boundary conditions, (6-3) and (6-4), if the radiation heat transfer and the hot drawn bar velocity are neglected, i.e., hradiation=0 and U=0 (see References by F. Kreith [10] and by Incropera, F. P. and D. P. DeWitt [6]). Sensitivity to governing variables is analyzed by ﬁxing the drawn bar temperature at the die, i.e., Tx=0=1273 K. There are eight independent variables that affect the temperature distribution of the hot drawn bar. The sensitivities of bar temperature to these variables are analyzed by assuming the following nominal values for a special steel bar:

53

Everyday Heat Transfer Problems

D = 0.01 m ρ = 8000 kg/m3 cp = 450 J/kg-K k = 40 W/m-K ε = 0.5 U = 0.02 m/s Tenvironment = 298 K The last independent variable is the convention heat transfer coefﬁcient between the surface of the bar and the environment. The convection heat transfer coefﬁcient can be determined from a drawn bar temperature requirement at a distance from the die. In the present analysis, Tx=10 is speciﬁed to be 373 K. At approximately x=10 m the radiation heat transfer contribution almost diminishes. The convection heat transfer coefﬁcient that meets the Tx=10 = 373 K requirement is determined from the above solution to be: hconvection = 46.17 W/m2-K which is in the turbulent region of forced cooling air over the cylindrical bar, i.e., ReD = 1083 where ReD= VairD/νair. The empirical relationship for the convection heat transfer coefﬁcient for air ﬂowing over cylinders is given in Reference [10] by F. Kreith as: hconvectionD/kair = 0.615 (VairD/νair)0.466 for 40 < ReD < 4000

(6-9)

where Vair is mean air speed over the cylinder (2.18 m/s in this case), kair is air thermal conductivity, and νair is air kinematic viscosity. Air thermophysical properties are calculated at ﬁlm temperature, namely the average of bar surface temperature and environmental temperature. A comparison of convection and radiation heat transfer coefﬁcients as a function of distance from the die is given in Figure 6-1. Heat transfer due to radiation is at the same order of magnitude around the die. As the bar travels away from the die, radiation heat transfer diminishes rapidly.

54

Heat Transfer Coeffieicents W/m2-K

Heat Transfer From A Hot Drawn Bar

50 hconvection

40 30 20 10 0

hradiation 0

2

4 6 8 Distance From Die, m

10

Figure 6-1 Radiation and convection heat transfer coefﬁcients as a function

of distance from die

Hot drawn bar temperature distributions, both with and without radiation heat transfer, are shown in Figure 6-2. Radiation heat transfer effects on bar temperature cannot be neglected below x=4 meters from the die. In the initial sensitivity analysis, radiation heat transfer effects will be neglected, namely hradiation=0. Hot drawn bar temperatures as a function of distance from the die for different convection heat transfer coefﬁcients are given in Figure 6-3. Temperatures are very sensitive to low convection heat transfer coefﬁcients. The sensitivity of bar temperature at x=10 m to the convection heat transfer coefﬁcient is given in Figure 6-4. Bar temperature sensitivity is high at natural convection and at low forced-convection heat transfer regions. As the forced-convection heat transfer coefﬁcient increases, bar temperature sensitivity to the convection heat transfer coefﬁcient decreases. Bar temperature at x=10 m as a function of the convection heat transfer coefﬁcient is shown in Figure 6-5. Increasing the convection heat transfer coefﬁcient reduces its effects on bar temperature.

55

Everyday Heat Transfer Problems

Temperature, K

1400 1200 1000

Without Radiation With Radiation

800 600 400 200 0

0

2 6 4 8 Distance From Die, m

10

Figure 6-2 Hot drawn bar temperature with and without radiation heat

transfer effects

1400 hconvection=5 W/m2-K hconvection=20 W/m2-K hconvection=40 W/m2-K hconvection=60 W/m2-K

Temperature, K

1200 1000 800 600 400 200

0

2

4

8 6 Distance From Die, m

10

Figure 6-3 Hot drawn bar temperature for different convection heat trans-

fer coefﬁcients with hradiation=0

56

∂T/∂ hconvection, m2-K2/W

Heat Transfer From A Hot Drawn Bar

0 −10 −20 −30 −40 −50

0

10

20

30

40

50

hconvection, W/m2-K

Figure 6-4 Bar temperature sensitivity @ x=10 m to convection heat trans-

Bar Temperature @ x=10 m, K

fer coefﬁcient

1000 800 600 400 200

0

10

20 30 40 hconvection, W/m2-K

50

60

Figure 6-5 Bar temperature @ x=10 m as a function of convection heat

transfer coefﬁcient

57

Bar Temperature @ x=10 m, K

Everyday Heat Transfer Problems

400

380

360

340 7000

7500

8000 Bar Density,

8500

9000

kg/m3

Figure 6-6 Bar temperature @ x=10 m versus bar density

Bar temperature at x=10 m varies close to a linear behavior with bar density, as shown in Figure 6-6. ?T/?ρ is 0.0239 K-m3/kg in the 7000 to 8000 kg/m3 bar density range. Bar temperature at x=10 m also varies close to a linear behavior with bar-speciﬁc heat at constant pressure, as shown in Figure 6-7. ?T/?cp is 0.4254 K2-kg/J in the 400 to 500 J/kg-K bar speciﬁc heat range. Bar temperature at x=10 m varies linearly with bar thermal conductivity, as shown in Figure 6-8. Bar temperature is a weak function of bar thermal conductivity in this problem. ?T/?k is 0.0007 K2-m/W in the 20 to 60 W/m-K bar thermal conductivity range. Bar temperature at x=10 meters versus bar velocity is given in Figure 6-9. Hot drawn bar velocity does not affect bar temperature at low velocities, i.e., U<0.01 m/s. As bar velocity increases above 0.01 m/s, bar temperature increases rapidly. Bar temperature at x=10 meters has a sensitivity of about 10,000 K-s/m to changes in bar velocity around 0.03 m/s. As bar velocity increases further, bar temperature sensitivity at x=10 meters starts to decrease.

58

Bar Temperature @ x=10 m, K

Heat Transfer From A Hot Drawn Bar

400

380

360

340 400

420

440 460 Bar Specific Heat, J/kg-K

480

500

Figure 6-7 Bar temperature @ x=10 m versus bar speciﬁc heat at constant

Bar Temperature @ x=10 m, K

pressure

373.04 373.03 373.02 373.01 373

20

30

40

50

60

Bar Thermal Conductivity, W/m-K

Figure 6-8 Bar temperature @ x=10 m versus bar thermal conductivity

Bar temperature at x=10 meters versus bar diameter is given in Figure 6-10. Bar temperature at x=10 meters is not sensitive to bar diameter changes in small diameter values, i.e., D<0.005 m. As bar diameter increases, bar temperature sensitivity goes through a

59

Bar Temperature @ x=10 m, K

Everyday Heat Transfer Problems

700 600 500 400 300 200

0

0.01

0.02 0.03 Bar Velocity, m/s

0.04

0.05

Bar Temperature @ x=10 m, K

Figure 6-9 Bar temperature @ x=10 m versus bar velocity

1000 800 600 400 200

0

0.01

0.02 0.03 Bar Diameter, m

0.04

0.05

Figure 6-10 Bar temperature @ x=10 m versus bar diameter

maximum of about 19,500 K/m at around D=0.02 meters, and starts to decrease as the bar diameter increases. Bar temperature at x=10 meters versus environmental temperature is given in Figure 6-11. The relationship is linear as expected.

60

Bar Temperature @ x=10 m, K

Heat Transfer From A Hot Drawn Bar

380 375 370 365 360

15

20 25 Environment Temperature, C

30

Figure 6-11 Bar temperature @ x=10 m versus environment temperature

The slope of the curve is 0.9231 K/C under the assumed nominal conditions. When the nominal values of the independent variables given above are varied +-10%, the results shown in Table 6-1 are obtained. The sensitivity analysis is conducted by neglecting radiation heat transfer at x=10 meters. The convection heat transfer coefﬁcient, bar diameter, bar velocity, bar density and bar-speciﬁc heat at constant pressure have the same order of magnitude sensitivity on bar temperature at x=10 m. Changes in the temperature of the environment affect bar temperature at x=10 m, at an order of magnitude less. Changes to the thermal conductivity of the bar have the least effect on bar temperature at x=10 m. The sensitivity magnitudes and order that are shown in Table 6-1 are only valid around the nominal values that are assumed for the independent variables for this analysis. Bar velocity, bar density and bar-speciﬁc heat at constant pressure have the same sensitivity effects on the temperature of the bar as can be seen in Eq. (6-7). Another interesting sensitivity analysis can be performed around x=0.5 m, where both the convection and the radiation heat transfers are in effect.

61

Everyday Heat Transfer Problems

Table 6-1 Effects of ±10% change in nominal values of variables to bar temperature @ x=10 m

Nominal Value

Bar Temperature @ x=10 m Change Due To A 10% Decrease In Nominal Value

Bar Temperature @ x=10 m Change Due To A 10% Increase In Nominal Value

46.17 W/m2-K

+5.879%

−4.549%

D

0.01 m

−4.986%

+5.280%

U

0.02 m/s

−4.986%

+5.279%

ρ

8000 kg/m

3

−4.986%

+5.279%

cp

450 J/kg-K

−4.986%

+5.279%

25ºC (298 K)

−0.619%

+0.619%

40 W/m-K

−0.0007%

+0.0007%

Variable hconvection

Tenvironment k

Table 6-2 Effects of ±10% change in nominal values of variables to bar temperature @ x=0.5 m, including radiation heat transfer

Nominal Value

Bar Temperature @ x=0.5 m Change Due To A 10% Decrease In Nominal Value

Bar Temperature @ x=0.5 m Change Due To A 10% Increase In Nominal Value

D

0.01 m

−1.386%

+1.183%

U

0.02 m/s

Variable

−1.386%

+1.183%

ρ

3

8000 kg/m

−1.386%

+1.183%

cp

450 J/kg-K

−1.386%

+1.183%

46.17 W/m -K

+0.788%

−0.783%

hconvection ε Tenvironment k

62

2

0.5

+0.500%

−0.482%

25ºC (298 K)

−0.046%

+0.046%

40 W/m-K

−0.0003%

+0.0003%

Heat Transfer From A Hot Drawn Bar

Sensitivities to a +-10% change in independent variables are given in descending order of importance in Table 6-2. All the sensitivities to governing independent variables at x=0.5 meters are at the same order of magnitude, except Tenvironment and k. Variations in radiation and convection heat transfer losses from the bar have similar effects on bar temperature close to the die.

63

CHAPTER

MAXIMUM

CURRENT IN AN OPEN-AIR ELECTRICAL WIRE

7

M

aximum current in an open-air single electrical wire (not in a bundle), will be analyzed under steady-state and one-dimensional cylindrical coordinates, with constant property conditions. The wire is assumed to be a cylindrical conductor with a certain diameter and with certain material characteristics; i.e., resistivity. The wire conductor is insulated with concentric layers of insulation material that can stand up to a certain wire conductor temperature, Tc, which will be the temperature rating of the wire. Heat generated in the wire conductor is I2R and the temperature within the wire conductor is assumed to be uniform—the conductor temperature does not vary radially from the center to the outer radius of the wire conductor. Heat is transferred by conduction through the wire insulator and by convection from the surface of the wire insulator to the environment. Radiation heat transfer from the surface of the wire insulator is neglected. The conduction heat transfer from the conductor to the outer radius of the wire insulator can be written by the rate equation in cylindrical coordinates: Q = 2πLkins (Tc – Tinsulation outer radius)/ln(rw/rc)

(7-1)

65

Everyday Heat Transfer Problems

The convection heat transfer from the outer surface of the wire insulator to the environment can be written by the rate equation in cylindrical coordinates: Q = 2πrwLh (Tinsulation outer radius – Tenv)

(7-2)

The heat transfer mechanisms in Eqs. (7-1) and (7-2) are in a series thermal resistance path. The energy balance for this heat transfer problem, heat generated by the conductor equals heat lost to the environment, can be written as follows, by eliminating Tinsulation outer radius from Eqs. (7-1) and (7-2): I2R = (Tc − Tenv)/{[ln(rw/rc)/(2πLkins)] + (1/2πrwLh)}

(7-3)

where the ﬁrst term in the denominator is the conduction heat transfer resistance in the wire insulation, and the second term is the convection heat transfer resistance at the insulated wire's outer surface. Deﬁnitions of variables: I = Current through the conductor in amps R = Resistance of the wire can also be written as ρL/πrc2 where ρ = Resistivity of the conductor material in Ω-m L = Length of the conductor in meters rc = Conductor radius in meters Tc = Temperature rating of the wire in C Tenv = Temperature of the environment in C rw = Radius of insulated wire in meters kins = Thermal conductivity of wire insulation in W/m-C h = Convection heat transfer coefﬁcient in W/m2-C The maximum current that a wire can stand can be written from Eq. (7-3) by replacing resistance of the wire with resistivity of the conductor material: Imax = πrc {(2/ρ)(Tc − Tenv)/[ln(rw/rc)/kins + (1/rwh)]}1/2

66

(7-4)

Maximum Current In An Open-Air Electrical Wire

There are six independent variables that govern the maximum current allowed in the wire. These are the conductor radius, rc, the conductor resistivity, ρ, the temperature potential (the temperature rating of the wire minus the temperature of the environment), Tc˛− Tenv, the radius of the insulated wire, rw, the thermal conductivity of the insulating material, kins, and the open air convention heat transfer coefﬁcient, h. The sensitivities of maximum current allowed with respect to these six variables will be analyzed as a function of wire conductor diameter. Since wire gauges are used in the industry, the sensitivity results are presented for different wire gauges. Table 7-1 provides the standard wire gauges and their conductor diameters. In this problem it is fortunate that the sensitivities can be obtained in a closed form by differentiating the above maximum current

Table 7-1 Wire conductor diameter versus AWG Wire Conductor Diameter, mm

American Wire gauge (AWG)

0.8128

20

1.0236

18

1.2900

16

1.6280

14

2.0520

12

2.5900

10

3.2640

8

4.1150

6

5.1890

4

6.5430

2

7.3406

1

8.2550

1/0

9.2710

2/0

10.4140

3/0

11.6840

4/0

67

Everyday Heat Transfer Problems

∂Imax/∂ (2rc), A/mm

40

30

20 AWG20 10

0

2

AWG6 4

AWG2 6

AWG1/0 AWG3/0 8

10

12

Wire Conductor Diameter, mm

Figure 7-1 Maximum wire current to wire conductor diameter sensitivity as

a function of wire diameter

equation with respect to a desired variable while holding the other variables constant, namely ∂Imax/∂(2rc), ∂Imax/∂Tenv, ∂Imax/∂ρ, ∂Imax/∂rw, ∂Imax/∂kins, and ∂Imax/∂h. For example, ∂Imax/∂Tenv = πrc {(2/ρ)(Tc − Tenv)/[ln(rw/rc)/kins + (1/rwh)]}−1/2 {−(1/ρ)/[ln(rw/rc)/kins + (1/rwh)]}

(7-5)

The sensitivity of maximum wire current to conductor radius is given in Figure 7-1. Change in maximum wire current with respect to wire conductor diameter is high at small diameters, and decreases as the wire diameter increases. Figure 7-1 was obtained by using the following values for the other variables: ρ = 1.60E-08 Ω-m Tc = 60ºC Tenv = 30ºC

68

Maximum Current In An Open-Air Electrical Wire

Maximum Current Allowed, A

400 NEC Current Allowed, A

300 200

Maximum Current Allowed, A

100 0 0

2

4

6

8

10

12

Wire Conductor Diameter, mm

Figure 7-2 Maximum wire current allowed versus wire conductor diameter

@ a wire temperature rating of 60ºC

rw = 3rc kins = 0.3 W/m-C h = 9 W/m2-C Under the above conditions the maximum wire current calculated is compared to National Electric Code (NEC) recommendations in Figure 7-2. The calculated maximum wire currents are very much in line with the NEC-recommended values up to AWG6. The variable assumptions given above start to predict higher maximum current values for large diameter wires, and therefore have to be adjusted for large diameter wire applications. The sensitivity of maximum wire current to environmental temperature, ∂Imax/∂Tenv, is given in Figure 7-3. Sensitivities in Figure 7-3 are for a wire temperature rating of 60ºC. As the environmental temperature approaches the wire rating temperature, the maximum wire current allowed becomes more sensitive to environmental temperature changes.

69

Everyday Heat Transfer Problems

∂ Imax/∂ Tenv, A/C

0 –2 –4 Tenv=0°C Tenv=20°C Tenv=40°C Tenv=50°C

–6 –8 –10 –12

0

2

4

6

8

10

12

Wire Conductor Diameter, mm

Figure 7-3 Maximum wire current to environment temperature sensitivity as

a function of wire conductor diameter at different environmental temperatures for a wire temperature rating of 60°C

The sensitivities of Maximum wire current allowed to wire insulation thickness, rw − rc, are given in Figures 7-4 and 7-5. For a given insulation thickness, the change in maximum allowable wire current reaches its peak at a certain wire conductor diameter and then starts to decrease, as seen in Figure 7-4. As the insulation thickness increases, the maximum sensitivity point moves towards smaller diameter conductors. Another way to look at the maximum allowable wire current sensitivity to insulation thickness is shown in Figure 7-5, where the sensitivities decrease as the wire insulation thickness increases for different gauge wires. The sensitivities of maximum wire current allowed to wire insulation thermal conductivity, ∂Imax/∂kins, are given in Figure 7-6. Maximum wire current is more sensitive to wire insulation thermal conductivity variations at lower thermal conductivity values and in large gauge wires. Maximum wire current is almost insensitive to wire insulation thermal conductivity variations below 3 mm diameter (AWG8) conductors.

70

Maximum Current In An Open-Air Electrical Wire

∂ Imax/∂ (rw-rc), A/m

10000 8000 6000 4000 2000 0 0.000 0.002 0.004 0.006 0.008 0.010 0.012

Ins Thickness= 0.5*Wire Dia Ins Thickness= 0.75*Wire Dia Ins Thickness= 1.0*Wire Dia Ins Thickness= 1.25*Wire Dia Ins Thickness= 1.5*Wire Dia

Wire Conductor Diameter, m

Figure 7-4 Maximum wire current to insulation thickness sensitivity as

a function of wire conductor diameter for different insulation thicknesses

∂ Imax/∂ (rw-rc), Amp/m

8000 AWG20 AWG16 AWG12 AWG8 AWG4 AWG1 AWG2/0 AWG4/0

7000 6000 5000 4000 3000 2000 1000 0.5*d

0.75*d

d

1.25*d

1.5*d

Insulation Thickness

Figure 7-5 Maximum wire current to insulation thickness sensitivity as a

function of insulation thickness for different wire diameters

71

Everyday Heat Transfer Problems

∂ Imax/∂ kins, A-m-K/W

300 AWG20 AWG16 AWG12 AWG8 AWG4 AWG1 AWG2/0 AWG4/0

250 200 150 100 50 0

0.2

0.25

0.3

0.35

0.4

Thermal Conductivity Of Wire Insulation, W/m-K

Figure 7-6 Maximum wire current to thermal conductivity of wire insulation

sensitivity for different wire gauges

∂ Imax/∂ h, A-m2-K/W

500 AWG20 AWG16 AWG12 AWG8 AWG4 AWG1 AWG2/0 AWG4/0

400 300 200 100 0

8

12

16

20 2

Convection Heat Transfer Coefficient, W/m -K

Figure 7-7 Maximum wire current to convection heat transfer coefﬁcient

sensitivity for different wire gauges

72

Maximum Current In An Open-Air Electrical Wire

∂ Imax/∂ h, A-m2-K/W

600 500 h=7 W/m2-K

400

h=8 W/m2-K

300

h=9 W/m2-K h=12 W/m2-K

200

h=15 W/m2-K h=20 W/m2-K

100 0 0

0.002 0.004 0.006 0.008

0.01

0.012

Wire Diameter, m

Figure 7-8 Maximum wire current to convection heat transfer coefﬁcient

sensitivity as a function of wire diameter for different convection heat transfer coefﬁcients

The sensitivities of maximum wire current allowed to convection heat transfer coefﬁcient, ∂Imax/∂h, are given in Figures 7-7 and 7-8. Maximum wire current is more sensitive to convection heat transfer coefﬁcient variations at high convection heat transfer coefﬁcient values and in large gauge wires. As the wire surface area increases in large gauge wires, changes in the heat transfer coefﬁcient gain importance. The sensitivities of maximum wire current allowed to wire conductor resistivity, ∂Imax/∂ρ, are given in Figure 7-9. The value of ∂Imax/∂ρ is always negative; when wire resistivity increases maximum wire current decreases, and vice versa. Maximum wire current is more sensitive to wire conductor resistivity variations at low wire conductor resistivity values and in large gauge wires. The above sensitivity graphs show that maximum wire current sensitivity to all the governing variables behaves non-linearly. Sometimes it is more appropriate to analyze these sensitivities in the region of interest, and rank them according to their effects on maximum wire current.

73

∂ Imax/∂ ρ x 10-8, A/Ohm-m

Everyday Heat Transfer Problems

0 AWG 20 AWG 16 AWG 12 AWG 8 AWG 4 AWG 1 AWG 2/0 AWG 4/0

–50 –100 –150 –200 1.00E-08

1.50E-08

2.00E-08

2.50E-08

3.00E-08

Wire Resistivity, ρ, Ohm-m

Figure 7-9 Maximum wire current to wire conductor resistivity sensitivity as

a function of wire conductor resistivity for different wire gauges

For example, maximum wire current sensitivities are analyzed for a 4.115 mm diameter, AWG6, conductor with a temperature rating of 60ºC for the following nominal values of the variables: Conductor diameter: 2rc = 4.115 ± 0.4115 mm Conductor resistivity: ρ = 1.60E-08 ± 0.16E-08 Ω-m Temperature Rating Of Wire − Temperature of the environment: Tc − Tenv = 30 ± 3ºC Wire insulation thickness: rw − rc = 2rc ± 0.2rc = 4.115 ± 0.4115 mm Wire insulation thermal conductivity: kins = 0.3 ± 0.03 W/m-C Convection heat transfer coefﬁcient: h = 9 ± 0.9 W/m2-C The nominal values are varied ±10%, and the effects of these variations are presented in Table 7-2. The sensitivity effects to maximum wire current shown in Table 7-2 are given in descending order. The most effective variable is the wire conductor diameter, and the least effective is the thermal conductivity of the wire insulation material. There is an order of magnitude difference in the effects of these two variables. In this region of interest, it would be more advisable to focus on the accuracy

74

Maximum Current In An Open-Air Electrical Wire

Table 7-2 Effects of ±10% change in nominal values of variables to maximum wire current for a 4.115 mm diameter, AWG6, conductor with a temperature rating of 60ºC

Variable

Maximum Wire Maximum Wire Current Change Current Change Due To A 10% Due To A 10% Nominal Decrease In Increase In Value Nominal Value Nominal Value

Conductor Diameter, 2rc

4.115 mm

−13.9%

+14.5%

Conductor Resistivity, ρ

1.60E-08 Ω-m

+5.4%

−4.7%

Temperature Rating Of Wire Tc – Temperature Of Environment, Tenv

30ºC

−5.1%

+4.9%

Convection Heat Transfer Coefﬁcient, h

9 W/m2-C

−4.32%

+4.01%

Insulation Thickness, rw − rc

2rc

−2.35%

+2.17%

Thermal Conductivity Of Wire insulation, kins

0.3 W/m-C

−0.93%

+0.78%

of the wire conductor diameter rather than the accuracy of the wire insulation thermal conductivity. It is important to remember that the maximum wire current change order shown in Table 7-2 is good only in this region of the application, due to non-linear behavior of the sensitivities.

75

CHAPTER

EVAPORATION

8

OF LIQUID NITROGEN IN A CRYOGENIC BOTTLE H

eat transfer in cryogenic bottles involves conduction, convection and radiation modes. Dewar invented the vacuum ﬂask at the beginning of the twentieth century to minimize heat transfer and contain low or high-temperature ﬂuids in it. In this example, a cryogenic bottle with a stainless steel inner tube, a vacuum gap and an outer insulation layer will be utilized to store liquid nitrogen. The bottle will have a venting system to release the evaporating nitrogen. Heat transfer from the sides of the cryogenic bottle will be considered. The top and the bottom surfaces of the bottle are assumed to be well-insulated. The temperature of the inner wall of the inner tube is assumed to be that of liquid nitrogen, namely a negligible convection heat transfer resistance between the liquid nitrogen and the inner wall of the inner tube. Heat transfer occurs from the environment to the nitrogen under steady-state conditions and in one-dimensional cylindrical coordinates. The heat transfer from the sides of the tube can be calculated by using the following series circuit: Q = (Tenvironment – Tliquid nitrogen)/Σ Rij

(8-1)

77

Everyday Heat Transfer Problems

where Q is the steady-state heat transferred from the environment to the liquid nitrogen in Watts, T is the temperature in K, and Rij represents the thermal resistances on the thermal path in series between Tenvironment and Tliquid nitrogen. R21 is the conduction heat transfer thermal resistance between the inner surface of the inner tube, radial location (1), to the outer surface of the inner tube, radial location (2); in other words, thermal resistance through the thickness of the inner tube. R21 = ln(r2/r1)/(2πLkss)

(8-2)

where ln is natural logarithm of argument (r2/r1), r1 = inner radius of inner tube in m, r2 = outer radius of inner tube in m, L = height of the tube in m, and kss = thermal conductivity of stainless steel inner tube in W/m-K. R32 is the radiation heat transfer thermal resistance between the inner surface of the outer insulation tube, radial location (3), and the outer surface of the inner tube, radial location (2); in other words, through the vacuum gap. R32 = 1/(2πr3Lhr)

(8-3)

where r3 = inner radius of outer insulation tube in m, hr = εσ(T34 – T24)/(T3 – T2) in W/m2-K, ε = emissivity of inner surface of outer insulation tube, σ = Stefan-Boltzmann constant, namely 5.67 × 10−8 W/m2-K4 T3 = temperature of inner surface of outer insulation tube in K, T2 = temperature of outer surface of inner tube in K R43 is the conduction heat transfer thermal resistance between the outer surface of the outer insulation tube, radial location (4) and the inner surface of the outer insulation tube, radial location (3).

78

Evaporation Of Liquid Nitrogen In A Cryogenic Bottle

R43 = ln(r4/r3)/(2πLkinsulation)

(8-4)

where r4 = outer radius of outer insulation tube in m and kinsulation = thermal conductivity of insulation material of the outer tube in W/m-K. Renvironment-4 is the convection heat transfer thermal resistance between the environment and the outer surface of outer insulation tube, radial location (4). Radiation heat transfer between the environment and the outer surface of outer insulation tube is assumed to be negligible. Renvironment-4 = 1/(2πr4ho)

(8-5)

where ho = convection heat transfer coefﬁcient between the environment and the outer surface of the outer tube in W/m2-K. Since hr depends on unknown temperatures T3 and T2, an iterative method is used to calculate these temperatures, and thereafter hr (i.e., the radiation heat transfer coefﬁcient between the inner surface of the outer insulation tube and the outer surface of the inner tube, or through the vacuum gap). All calculations are done using the Kelvin temperature scale due to fourth power calculations in the radiation heat transfer coefﬁcient. The thermophysical properties used in these calculations are assumed to be constants. Nominal values of the variables used in this problem are as follows: Tenvironment = 293 K Tliquid nitrogen = 77 K r1 = 0.1 m r2 = 0.106 m, namely inner tube stainless steel wall thickness is 6 mm r3 = 0.116 m, namely vacuum gap is 0.01 m r4 = 0.166 m, namely outer tube insulation thickness is 0.05 m L = 0.8 m kss = 10 W/m-K

79

Everyday Heat Transfer Problems

kinsulation = 0.002 W/m-K ho = 10 W/m2-K, namely in natural convection regime ε = 0.5, for inner surface of outer insulation tube Hfg = 2 × 105 J/kg, latent heat of evaporation for liquid nitrogen ρliquid nitrogen = 800 kg/m3 It is assumed that the initial volume of liquid nitrogen in the bottle is 5 liters, or that it contains 4 kg, Mliquid nitrogen, of liquid nitrogen. The dependent variable in the calculations is the time, t in seconds, that it takes to completely evaporate the liquid nitrogen. Heat losses, Q, from the cryogenic bottle, times time, t, should equal the total latent heat of vaporization for the liquid nitrogen, Mliquid nitrogen Hfg, for complete evaporation. t = Mliquid nitrogen Hfg/Q

(8-6)

It takes 198 hours, or 8.25 days, for 5 liters of liquid nitrogen to evaporate under the above nominal conditions. Unknown temperature T3, temperature of the inner surface of the outer insulation tube, is obtained by iteration, as shown in Figure 8-1.

Temperature, K

180 160 140 120 100 21

19

17

15

13

11

9

7

5

3

1

80 Number Of Iterations

Figure 8-1 Convergence of temperature, T3, at the inner surface of an outer

insulation tube

80

Evaporation Of Liquid Nitrogen In A Cryogenic Bottle

Temperature calculations converge to a ﬁnal value after ten iterations, with less than 1% error. Liquid nitrogen evaporation time is very sensitive to the cryogenic bottle's insulation layer thermal conductivity, kinsulation. Five liters of liquid nitrogen evaporation time versus insulation layer thermal conductivity is given in Figure 8-2. The sensitivity of evaporation time increases rapidly as thermal conductivity decreases below 0.001 W/m-K, as shown in Figure 8-3. The second dominant heat transfer mechanism is the radiation heat transfer R32. The sensitivity of evaporation time to the emissivity, ε, of the inner surface of the outer insulation tube, is shown in Figure 8-4. Evaporation time increases as emissivity of the insulation inner surface, and therefore the radiation heat transfer, decreases. When the nominal values of the variables given above are varied ±10%, the results shown in Table 8-1 are obtained. Sensitivities of the time that it takes for ﬁve liters of liquid nitrogen to evaporate to a ±10% change in independent variables are given in descending order

Evaporation Time, hr

1500

1000

500

0 0

0.002

0.004

0.006

0.008

0.01

Insulation Layer Thermal Conductivity, kinsulation, W/m-K

Figure 8-2 All-liquid nitrogen evaporation time versus bottle insulation

layer thermal conductivity

81

Everyday Heat Transfer Problems

∂ t/∂ kinsulation, hr-m-K/W

0.0E+00

–1.0E+06

–2.0E+06

–3.0E+06 0

0.001

0.002

0.003

0.004

0.005

Insulation Layer Thermal Conductivity, kinsulation, W/m–K

Figure 8-3 Sensitivity of all-liquid nitrogen evaporation time to bottle

insulation layer thermal conductivity

0

∂ t/∂ ε

–100

–200

–300

–400

0

0.2

0.4

0.6

0.8

1

Emissivity, ε, Of Inner Surface Of Outer Insulation Tube

Figure 8-4 Sensitivity of all liquid nitrogen evaporation time to emissivity of

the inner surface of an outer insulation tube

82

Evaporation Of Liquid Nitrogen In A Cryogenic Bottle

Table 8-1 Effects of ±10% change in the nominal values of variables to time for 5 liters of liquid nitrogen to evaporate

Variable

Nominal Value

Change in Time Change in Time For 5 Liters Of For 5 Liters Of Liquid Nitrogen Liquid Nitrogen To Evaporate For To Evaporate For A 10% Decrease A 10% Increase In Nominal Value In Nominal Value

0.8 m

+11.111%

−9.091%

0.002 W/m-K

+9.323%

−7.694%

r1

0.1 m

+8.062%

−6.931%

r4 − r3 (outer tube insulation thickness)

0.05 m

−7.212%

+6.947%

0.5

+1.671%

−1.434%

Tenvironment

293 K

+1.064%

−1.043%

r3 − r2 (vacuum gap)

0.01 m

+0.751%

−0.740%

r2 − r1 (stainless steel inner tube wall thickness)

0.006 m

+0.449%

−0.445%

ho

10 W/m2-K

+0.0315%

−0.0258%

kss

10 W/m-K

+0.00006%

−0.000049%

L kinsulation

ε

of importance, and they are applicable only around the nominal values assumed for this study. The time required for ﬁve liters of liquid nitrogen to evaporate is most sensitive to cryogenic tube height, insulation layer thermal conductivity, inside radius of the inner tube, and outer tube insulation thickness. The second tier of sensitivities are an order of magnitude less; emissivity of inner surface of the outer insulation tube, temperature of environment, vacuum gap thickness, and stainless steel inner tube wall thickness. Time for ﬁve liters of liquid nitrogen to

83

Everyday Heat Transfer Problems

evaporate is least sensitive to the convection heat transfer coefﬁcient between the environment and the outer surface of the outer tube, and the thermal conductivity of the stainless steel inner tube. R43, conduction heat transfer thermal resistance between the outer surface of the outer insulation tube and the inner surface of the outer insulation tube, dominates the results in this case. The second dominant heat transfer mechanism is R32, radiation heat transfer thermal resistance between the inner surface of the outer insulation tube and the outer surface of the inner tube; in other words, the vacuum gap. It is important to remember that these calculated sensitivities are good only around the nominal values assumed for this case. Changing these nominal values will change the magnitudes and orders of these sensitivities, a result of the non-linear form of governing heat transfer equations.

84

CHAPTER

9 STRESS IN A PIPE THERMAL

T

hermal stresses generated by temperature variations in the wall of a pipe have been studied extensively in Reference by Timoshenko, S. and J. N. Goodier [17]. The stress, strain, radial displacement relationships in cylindrical coordinates are treated in detail in Reference [17]. To calculate the thermal stresses in a pipe wall, the temperature distribution in the pipe wall has to be known. The temperature distribution in the radial direction, R, can be obtained from a steady-state, one-dimensional heat conduction equation in cylindrical coordinates. By assuming constant thermophysical properties and no heat sources in the pipe wall, the heat conduction equation for the temperature distribution, T, is: d2T/dR2 + (1/R)dT/dR = 0

(9-1)

If the temperatures at the inner surface, Ti, and the outer surface, To, of the pipe wall are known, Eq. (9-1) can be solved by using the following boundary conditions: T = Ti at R = Ri

(9-2)

85

Everyday Heat Transfer Problems

and T = To at R = Ro

(9-3)

Equation (9-1) can be integrated and the following temperature distribution in the pipe wall can be obtained by applying the boundary conditions in Eqs. (9-2) and (9-3): (T − To) = (Ti – To) [ln(Ro /R) / ln(Ro /Ri)]

(9-4)

Derivation of thermal stress integral equations for σR, σθ, σZ, thermal stresses in three cylindrical coordinates, are detailed in Reference [17]. The thermal stress integral equations for σR, σθ and σZ were developed by applying three conditions to the stress-strain relationships. One of the conditions results from the fact that the strain along the length of a long pipe is zero. The integration constants are determined from two boundary conditions. These boundary conditions come from the radial stress, σR, being zero at the inner and outer surfaces of the pipe wall. The thermal stress integral equations are: σR = [αE/(1 – ν)](1/R2){[(R2 – Ri2)/(Ro2 – Ri2)] }

(9-5)

σθ = [αE/(1 – ν)](1/R2){[(R2 + Ri2)/(Ro2 – Ri2)] – TR2} σZ = [αE/(1 – ν)]{[(2/(Ro2 – Ri2)]

(9-6) -T}

(9-7)

where σR, σθ, and σZ are the thermal stress distributions at the pipe wall in MPa (Mega Pascals), α is the coefﬁcient of thermal expansion of the pipe wall material in m/m-C, E is modulus of elasticity for the pipe wall material in MPa, and ν is Poisson's ratio for pipe wall material.

86

Thermal Stress In A Pipe

The temperature distribution in Eq. (9-4) is substituted into Eqs. (9-5) through (9-7) to determine thermal stress distributions in three directions in cylindrical coordinates. If the heat is ﬂowing from inside the pipe to the environment, maximum compressive stresses occur at the inside radius of the pipe wall, and the maximum tensile stresses occur at the outside radius of the pipe wall. In the present sensitivity analysis, the maximum tensile stress condition that occurs at the outside radius of the pipe wall will be investigated. In the heat transfer analysis, only convection and conduction heat transfer mechanisms are considered. Radiation heat transfer from the outer surface of the pipe to the environment is neglected. The convection heat transfer per unit length from the ﬂuid in the pipe to the inner surface of the pipe wall is: Q = 2πRi hi (Tﬂuid – Ti)

(9-8)

The conduction heat transfer per unit length through the pipe wall can be written as: Q = 2πk (Ti – To)/ln(Ro/Ri)

(9-9)

The convection heat transfer per unit length from the outer surface of the pipe wall to the environment is: Q = 2πRo ho (To – Tenvironment)

(9-10)

The heat is transferred in a series thermal circuit. Combining Eqs. (9-8) through (9-10) demonstrates that heat transferred from the ﬂuid in the pipe to the environment is: Q = (Tﬂuid – Tenvironment)/[(1/2πRihi) + ln(Ro/Ri)/2πk + (1/2πRoho)] (9-11) where Q = heat transfer through pipe wall in W/m Tﬂuid = Mean temperature of ﬂuid in the pipe in C

87

Everyday Heat Transfer Problems

Tenvironment = Environment temperature outside the pipe in C Ri = Pipe inside radius in meters hi = Heat transfer coefﬁcient between inner surface of pipe wall and ﬂuid in pipe in W/m2-C Ro = Pipe outside radius in meters ho = Heat transfer coefﬁcient between outer surface of pipe wall and environment in W/m2-C k = Thermal conductivity of pipe wall material in W/m-C Maximum tensile stress due to the temperature distribution given by Eq. (9-4) occurs at the outer radius of the pipe. At the outer radius Ro, the maximum axial stress is the same as in the circumferential direction, σZ = σθ. Eqs. (9-6) and (9-7) produce the same results at R = Ri and at R = Ro. Also, it is important to remember from the boundary conditions that σR is zero at R = Ri and at R = Ro. Eq. (9-11) can be combined with Eq. (9-6) to provide the thermal stress at R = Ro. σθ at R=Ro = {α E Q/[4π(1 − ν)k]}{1 – [2Ri2/(Ro2 − Ri2)]ln(Ro/Ri)} (9-12) Assumed nominal values for the independent variables for the sensitivity analysis in Eqs. (9-11) and (9-12) are as follows: Tﬂuid = 50ºC Tenvironment = −50ºC Ri = 0.2 m Ro = 0.3 m hi = 500 W/m2-C ho = 100 W/m2-C α = 2 × 10−5 m/m-C E = 210000 MPa ν = 0.3 k = 20 W/m-C Thermal stress at the outside radius of the pipe wall is a linear function of heat transferred from the ﬂuid to the environment, or (Tﬂuid − Tenvironment), as shown in Figure 9-1. The slope of this curve is 0.828 MPa/C.

88

Thermal Stress In A Pipe

Thermal Stress At Ro, MPa

120 100 80 60

Tenvironment= –50°C

40 20 0

0

20

40

60

80

Temperature Of Fluid Flowing In Pipe, °C

Figure 9-1 Thermal stress at outside radius of pipe versus temperature of

ﬂuid ﬂowing in pipe

As the pipe outside radius, Ro, increases for a constant wall thickness of 0.1 m, the heat transfer increases, and therefore the thermal stress increases, as shown in Figure 9-2. Thermal stress sensitivity to pipe outside radius decreases as the pipe gets larger. The effects of pipe wall thickness on thermal stress are studied for a constant outside radius pipe, where Ro = 0.3 m, in Figure 9-3. Thermal stress increases as pipe wall thickness increases, reaching a maximum at around 0.15 to 0.2 meters of wall thickness. Thermal stress then starts to decrease as pipe wall thickness further increases, due to decreasing heat transfer. The inside and outside heat transfer coefﬁcients, hi and ho, affect thermal stress similarly, as shown in Figures 9-4 and 9-5. Thermal stress sensitivity to changes in the heat transfer coefﬁcient are signiﬁcant at smaller values of hi and ho. When the nominal values of the variables given above are varied ±10%, the results shown in Table 9-1 are obtained. Maximum thermal stress at Ro sensitivities to a ±10% change in the governing independent variables are given in descending order of importance, and they are applicable only around the nominal values assumed for this study.

89

Thermal Stress At Ro, MPa

Everyday Heat Transfer Problems

86 84 82 80 78 76 0.2

0.25

0.3

0.35

0.4

0.45

0.5

Pipe Outside Radius, Ro, m

Figure 9-2 Thermal stress at Ro versus pipe outside radius for a constant

wall thickness of 0.1 meters

Thermal Stress At Ro, MPa

120

80

40

0 0

0.05

0.1

0.15

0.2

0.25

0.3

Pipe Wall Thickness, m

Figure 9-3 Thermal stress at Ro versus pipe wall thickness for a pipe outside

radius of 0.3 meters

90

Thermal Stress At Ro, MPa

Thermal Stress In A Pipe

100 80 60 40 20 0

0

200

400

600

800

1000

Heat Transfer Coefficient, hi, W/m2-C

Figure 9-4 Thermal stress at Ro versus the heat transfer coefﬁcient at inside

surface of pipe wall

Thermal Stress At Ro, MPa

160 120 80 40 0

0

200

400

600

800

1000

Heat Transfer Coefficient, ho, W/m2-C

Figure 9-5 Thermal stress at Ro versus the heat transfer coefﬁcient at

outside surface of pipe wall

91

Everyday Heat Transfer Problems

Table 9-1 Effects of ±10% change in nominal values of variables to maximum thermal stress at Ro

Variable Pipe Wall Thickness, Ro-Ri

Nominal Value

Change In Change In Thermal Stress Thermal Stress At Ro For A 10% At Ro For A 10% Decrease In Increase In Nominal Value Nominal Value

0.1 m

+10.049%

−13.011%

α

2 × 10−5 m/m-C

−10%

+10%

E

210000 MPa

−10%

+10%

100ºC

−10%

+10%

k

20 W/m-C

+7.311%

−6.378%

ho

100 W/m2-C

−5.502%

+5.002%

ν

0.3

−4.110%

+4.478%

hi

500 W/m -C

−1.717%

+1.450%

Ro

0.3 m

+0.920%

−1.248%

Tﬂuid-Tenvironment

2

Thermal stress at the pipe outside radius is most sensitive to changes in pipe wall thickness, the coefﬁcient of thermal expansion, modulus of elasticity, temperature potential between the ﬂuid and the environment, and thermal conductivity of pipe material. The second tier of independent variables affecting thermal stress at the pipe outside radius are the heat transfer coefﬁcient between outside surface of pipe wall and environment, and Poisson's ratio of pipe material. Thermal stress at the pipe outside radius is least affected by changes to the heat transfer coefﬁcient between inside surface of pipe wall and ﬂuid in pipe, and outer radius of pipe. This variable sensitivity order is applicable only around the nominal values assumed for this case.

92

CHAPTER

10

HEAT

TRANSFER IN A PIPE WITH UNIFORM HEAT GENERATION IN ITS WALLS U

nder steady-state conditions, constant thermophysical properties and uniform heat generation in the walls, the one-dimensional conduction heat transfer equation in radial direction of a pipe can be written as (see Reference by Carslaw, H. S. and J. C. Jaeger [17]): d2T/dR2 + (1/R)dT/dR + Q/k = 0

(10-1)

where the radial heat ﬂux is positive in the negative radial direction (towards the center of the pipe), Q is uniform heat generation in the pipe walls in W/m3, and k is pipe wall thermal conductivity in W/m-C. This differential Eq. (10-1) can be solved for the radial temperature distribution in the pipe wall by specifying the pipe wall temperatures with the inner and outer wall radii as boundary conditions. T = Ti at R = Ri

(10-2)

T = To at R = Ro

(10-3)

and

93

Everyday Heat Transfer Problems

The radial temperature distribution in the pipe wall which satisﬁes boundary condition Eqs (10-2) and (10-3) is: T(R) = To + (QRo2/4k)[1 − (R2/Ro2)] − {(To − Ti) + (QRo2/4k) (10-4) [1 − (Ri2/Ro2)]} [ln(Ro/R)/ln(Ro/Ri)] In this sensitivity study, convection heat transfer and conduction heat transfer equivalence at the pipe wall surfaces at R = Ri and at R = Ro will be used to determine the thermal energy generation requirements in the pipe wall. Energy balances at the inside and outside surfaces of the pipe wall provide: (2πRiL) k dT/dR = (2πRiL) hi (Tﬂuid − Ti) at R = Ri

(10-5)

and (2πRoL) k dT/dR = (2πRoL) ho (To − Tenvironment) at R = Ro (10-6) where hi is the convection heat transfer coefﬁcient at inside surface of the pipe in W/m2-C, Tﬂuid is mean ﬂuid temperature in C in the pipe, ho is the convection heat transfer coefﬁcient at outside surface of the pipe in W/m2-C, and Tenvironment is environmental temperature in C. L is the length of the pipe in meters and it cancels out from the energy balance equations. Obtaining dT/dR at R = Ri from Eq. (10-4), the governing energy balance Eq. (10-5) at R = Ri becomes: hi (Tﬂuid − Ti) = 0.5 Q Ri − [k/Ro ln(Ro/Ri)][(To − Ti) + (Q/4k)(Ro2 − Ri2)] (10-7) Obtaining dT/dR at R = Ro from Eq. (10-4), the governing energy balance Eq. (10-6) at R = Ro becomes: ho (To − Tenvironment) = 0.5 Q Ro − [k/Ro ln(Ro /Ri)][(To − Ti) + (Q/4k)(Ro2 − Ri2)] (10-8)

94

Heat Transfer In A Pipe With Uniform Heat Generation

The uniform thermal energy generation required per unit length in the pipe wall, Q, is treated as the dependent variable, and its sensitivity to other independent variables is analyzed for a speciﬁed inner surface temperature, Ti. Eqs. (10-7) and (10-8) can be combined to eliminate To and obtain a relationship for thermal energy requirement as a function of eight independent variables. Nominal values of independent variables for the present sensitivity analysis are as follows: Ri = 0.10 m Ro = 0.11 m k = 15 W/m-C hi = 30 W/m2-C Tﬂuid = 50°C ho = 10 W/m2-C Tenvironment = −10°C Ti = 40°C With these nominal values, the uniform thermal energy generation required is 23,530 W/m3. Uniform thermal energy generation requirements behave non-linearly with pipe inner radius variations, as shown in Figure 10-1. The uniform thermal energy generation requirement increases with decreasing pipe wall thickness. As the wall thickness decreases, the conduction heat transfer resistance decreases, and it gets difﬁcult to keep the inner radius pipe wall temperature at a constant 40°C. Uniform thermal energy generation requirements increase as pipe outer radius increases, as shown in Figure 10-2. Uniform thermal energy generation requirements decrease with increasing pipe wall thickness. As wall thickness increases, conduction heat transfer resistance increases, and it becomes easier to keep the inner radius pipe wall temperature at a constant 40°C. Thermal energy requirement versus pipe wall thermal conductivity is shown in Figure 10-3. Thermal energy generation requirement is mostly a constant for thermal conductivities above 20 W/m-C, and it begins rapidly to decrease as the wall material becomes an insulator.

95

Energy Generation, W/m3

Everyday Heat Transfer Problems

250000 200000 150000 100000 50000 0 0.09

0.095

0.1

0.105

0.11

Pipe Inner Radius, m

Figure 10-1 Thermal energy generation requirement versus pipe inner

Energy Generated, W/m3

radius

200000 150000 100000 50000 0 0.1

0.105

0.11

0.115

0.12

Pipe Outer Radius, m

Figure 10-2 Thermal energy generation requirement versus pipe outer

radius

96

Energy Generated, W/m3

Heat Transfer In A Pipe With Uniform Heat Generation

24000 23000 22000 21000 20000

0

20

40

60

80

100

Pipe Wall Thermal Conductivity, W/m-C

Figure 10-3 Thermal energy generation requirement versus pipe wall

thermal conductivity

Thermal energy generation requirement decreases as the convection heat transfer coefﬁcient at pipe inner surface increases, as shown in Figure 10-4. The slope of the curve is −955.8 C/m, and no thermal energy generation is required at hi = 54.6 W/m2-C. At higher convection heat transfer coefﬁcients at the pipe inner surface, thermal energy has to be taken out of the pipe walls in order to keep the pipe inner walls at 40°C. Thermal energy generation requirement increases as the convection heat transfer coefﬁcient at pipe wall outside surface increases, as shown in Figure 10-5. The slope of the curve is 5062.1 C/m, and no thermal energy generation is required at ho = 5.27 W/m2-C, namely in the natural convection region. At lower convection heat transfer coefﬁcients at the pipe wall outside surface, thermal energy has to be taken out of the pipe walls in order to be able to keep the pipe inner walls at 40°C. The present case assumes a fully developed ﬂow inside the pipe. Thermal energy generation requirement versus mean ﬂuid temperature inside the pipe is shown in Figure 10-6. Thermal energy

97

Energy Generated, W/m3

Everyday Heat Transfer Problems

60000 40000 20000 0 –20000 –40000

0

20

40

60

80

100

Convection Heat Transfer Coefficient At Pipe Wall Inside Surface, W/m2-C

Figure 10-4 Thermal energy generation requirement versus convection heat

Energy Generated, W/m3

transfer coefﬁcient at pipe wall inside surface

500000 400000 300000 200000 100000 0 –100000

0

100 20 40 60 80 Convection Heat Transfer Coefficient At Pipe Wall Outside Surface, W/m2-C

Figure 10-5 Thermal energy generation requirement versus convection heat

transfer coefﬁcient at pipe wall outside surface

98

Heat Transfer In A Pipe With Uniform Heat Generation

Energy Generated, W/m3

150000 100000 50000 0 –50000 –100000 –150000

0

20 40 60 80 Mean Fluid Temperature Inside The Pipe, °C

100

Figure 10-6 Thermal energy generation requirement versus mean ﬂuid

temperature inside the pipe

generation requirement decreases as the mean ﬂuid temperature increases. The slope of the curve is −2867.4 W/m3-C, and no thermal energy generation is required at Tﬂuid = 58.2°C. For higher mean ﬂuid temperatures inside the pipe, thermal energy has to be taken out from the pipe walls in order to be able to keep the pipe inner walls at 40°C. Thermal energy generation requirement versus environmental temperature is shown in Figure 10-7. Thermal energy generation requirement increases as the environment temperature decreases. The slope of the curve is −1044.1 W/m3-C, and no thermal energy generation is required at Tenvironment = 12.5°C. Pipe walls have to be cooled above the 12.5°C environmental temperature in order to keep Ti at 40°C. Thermal energy generation requirement versus pipe inside surface temperature, Ti, is shown in Figure 10-8. Thermal energy generation requirement increases as the pipe inner surface temperature requirement increases. The slope of the curve is 3911.5 W/m3-C, and no thermal energy generation is required at Ti = 34°C. When the nominal values of the independent variables given above are varied ±10%, the results shown in Table 10-1 are obtained.

99

Energy Generated, W/m3

Everyday Heat Transfer Problems

60000 40000 20000 0 –20000 –40000 –40

–20

0

20

40

60

Environment Temperature, °C

Figure 10-7 Thermal energy generation requirement versus environmental

temperature

Energy Generated, W/m3

250000 200000 150000 100000 50000 0 –50000 –100000 –150000

0

30 60 Pipe Inner Surface Temperature, °C

90

Figure 10-8 Thermal energy generation requirement versus pipe inner

surface temperature

100

Heat Transfer In A Pipe With Uniform Heat Generation

Table 10-1 Effects of a ±10% change in nominal values of variables to thermal energy generation requirement

Nominal Value

Change In Required Thermal Energy Generation For A 10% Decrease In Nominal Value

Change In Required Thermal Energy Generation For A 10% Increase In Nominal Value

Ti

40°C

−66.49%

+66.49%

Tﬂuid

50°C

+60.93%

−60.93%

Ro

0.11 m

+104.35% @ 5% decrease in nominal value

−45.38%

Ri

0.10 m

−41.75%

+84.65% @ 5% increase in nominal value

ho

10 W/m2-C

−22.08%

+22.06%

30 W/m -C

+12.19%

−12.19%

−10°C

+4.44%

−4.44%

15 W/m-C

−0.13%

+0.11%

Variable

hi Tenvironment k

2

Thermal energy requirement sensitivities to a ±10% change in the governing variables are given in descending order of importance, and they are applicable only around the nominal values assumed for this study. Thermal energy generation required is most sensitive to pipe inner surface temperature, mean ﬂuid temperature inside the pipe, pipe outer radius and pipe inner radius. Convection heat transfer coefﬁcients at the outside and inside surfaces of the pipe wall, and environmental temperature, are the next independent variables in order of sensitivity. Thermal energy generation requirement has the lowest sensitivity to pipe wall thermal conductivity around the assigned nominal values for this case.

101

CHAPTER

HEAT

11

TRANSFER IN AN ACTIVE INFRARED SENSOR I

n an active infrared sensor, the surface temperature is held constant during the measurement process by providing controlled energy to the sensor's surface. The sensor surface receives radiation heat transfer energy from the surface of the object being measured. The sensor also loses thermal energy to its environment. The temperature at the sensor's surface can be analyzed by using unsteady-state and one-dimensional heat transfer rate equations in rectangular coordinates. Unsteady-state heat transfer in an active infrared sensor is detailed in the Reference by J. Fraden [3]. The energy balance of a sensor element can be written as follows: Change in internal energy of the sensor with respect to time = Control energy supplied to regulate the surface temperature of the sensor − Energy lost from the sensor to the environment by conduction and by convection heat transfer + Net radiation heat transferred from the object being measured to the sensor Net radiation heat transfer between the object and the sensor is assumed to occur between two gray bodies that are opaque to

103

Everyday Heat Transfer Problems

radiation. If that is true, then the emissivity, ε, and reflectivity, ρ, characteristics of a gray surface have the following relationship: ε+ρ=1

(11-1)

In this analysis, it is assumed that the emissivity and reflectivity of the object and the sensor surfaces are constants in the infrared region of the electromagnetic radiation spectrum. The radiation emitted by the object to the sensor can be written as: Qradiation emitted by object = Aσεobject T4object

(11-2)

Some of the irradiation reaching the surface of the sensor is reflected due to the reflectivity of the sensor surface. The reflected portion of the radiation emitted by the object can be written as: Qradiation reflected by sensor = ρsensor (Aσεobject T4object)

(11-3)

Eqs. (11-1) through (11-3) can be combined to get the net radiation emitted by the object to the sensor: Qnet radiation emitted by object = Aσεsensor εobject T4object

(11-4)

In a similar fashion, the net radiation emitted by the sensor to the object can be obtained: Qnet radiation emitted by sensor = Aσεobject εsensor T4sensor

(11-5)

The net radiation heat transfer between the object and the sensor is determined by combining Eqs. (11-4) and (11-5): Qnet radiation = Aσεsensor εobject (T4object − T4sensor)

(11-6)

Control energy supplied to regulate the surface temperature of the sensor is assumed to be in the form of I2R. Energy lost from the sensor to the environment by conduction and convection heat

104

Heat Transfer In An Active Infrared Sensor

transfer can be written as a rate equation, where the two heat transfer mechanisms act in series: Qconduction + convection = (1/RT) (Tsensor − Tenvironment)

(11-7)

Energy balance for the sensor can be written as a first order and non-linear differential equation: ρcpV(dTsensor/dθ) = I2R − (Tsensor − Tenvironment)/Rtotal + Aσεobjectεsensor (T4object − T4sensor) (11-8) where ρcpV is the sensor thermal capacitance in W-s/K dTsensor/dθ is the time rate of change of sensor temperature in K/s I2R is control power supplied to regulate the surface temperature of the sensor in W Tenvironment is the environment temperature in K Rtotal is the total heat transfer resistance between the sensor and the environment due to conduction and convection heat transfer in K/W A is the sensor area in m2 σ = 5.67×10-8 W/m2-K4 is the Stefan-Boltzmann constant εobject is the surface emissivity of the object being measured εsensor is the surface emissivity of the sensor Tobject is the temperature of the object being measured in K Tsensor is the temperature of the sensor surface in K For the steady-state sensor temperature, Eq. (11-8) can be rewritten by eliminating the left-hand side of the equation, which means that the change in internal energy of the sensor with respect to time becomes negligible. Tsensor is calculated from the following quartic equation, by trial and error: T4sensor + C1 Tsensor = C2

(11-9)

C1 = 1/(RT Aσεobjectεsensor)

(11-10)

where

105

Everyday Heat Transfer Problems

and C2 = (I2R/Aσεobjectεsensor) + (Tenvironment/Rtotal Aσεobjectεsensor) + T4object (11-11) The transient solutions to Eq. (11-8) are obtained from the following explicit finite difference equation, using small time intervals Δθ: ρcpV(Tsensor @ (i+1) − Tsensor @ i /Δθ) = I2R − (Tsensor @ i − Tenvironment)/Rtotal (11-12) + Aσεobjectεsensor (T4object − T4sensor @ i) For the present sensitivity analysis, nominal values of the independent variables are assumed to be as follows: ρcpV = 0.014 W-s/K I2R = 0.1 W Tenvironment = 20°C (293 K) Rtotal = 100 K/W A = 0.0001 m2 (a 1 cm × 1 cm sensor surface area) εobject = 0.9 εsensor = 0.8 The sensitivities are analyzed for three different object temperatures, namely 100°C (373 K), 500°C (773 K) and 1000°C (1273 K). Sensor temperature as a function of time for three different object temperatures is given in Figures 11-1, 11-2, and 11-3. As the temperature of the object increases, the sensor time constant, or the time the sensor reaches 63.2% of its steady-state temperature, decreases. Also as the sensor emissivity increases, the sensor time constant decreases. As the object temperature approaches the environmental temperature, the sensor temperature deviates from the object temperature because fixed nominal values are used for the analysis of this heat transfer problem. The control circuit of the sensor has to respond and change control energy, I2R, in order to achieve accurate results. The details of the control circuit are explained in the Reference by J. Fraden [3].

106

Heat Transfer In An Active Infrared Sensor

200

Tsensor, C

150

Emissivity Sensor=0.9 Emissivity Sensor=0.8 Emissivity Sensor=0.7

100 50 0 0

1

2 Time, s

3

4

Figure 11-1 Sensor temperature versus time for different sensor emissivities

and for an object at 1000°C

200

Tsensor, C

150

Emissivity Sensor=0.9 Emissivity Sensor=0.8 Emissivity Sensor=0.7

100 50 0 0

1

2

3

4

Time, s

Figure 11-2 Sensor temperature versus time for different sensor emissivities

and for an object at 500°C

107

Everyday Heat Transfer Problems

Tsensor, C

40 35 30 25 20 15 10 5 0

Emissivity Sensor=0.9 Emissivity Sensor=0.8 Emissivity Sensor=0.7 0

1

2 Time, s

3

4

Figure 11-3 Sensor temperature versus time for different sensor emissivities

and for an object at 100°C

Figure 11-4 Sensor temperature versus time for different total heat transfer

resistances between the sensor and the environment, due to conduction and convection heat transfer for an object at 1000°C temperature

108

Heat Transfer In An Active Infrared Sensor

Table 11-1 Thermal time constant change due to a 10% change in variables around the nominal values for an object temperature of 1000°C

Variable Sensor Thermal Capacitance, ρcpV

Nominal Value

Thermal Time Thermal Time Constant Change Constant Change Due To A 10% Due To A 10% Decrease In Increase In Nominal Value Nominal Value

0.014 W-s/K

−20.39%

+10.38%

Tobject

1000°C (1273 K)

+16.73%

−15.38%

A

0.0001 m2

+8.12%

−7.44%

εobject

0.9

+8.12%

−7.44%

εsensor

0.8

+8.12%

−7.44%

Rtotal

100 K/W

−3.04%

+2.31%

Tenvironment

20°C (293 K)

−0.15%

+0.15%

0.1 W

+0.049%

−0.049%

I 2R

Another variable affecting the performance of the sensor is the heat loss from the sensor to the environment due to conduction and convection heat transfer. The sensitivity of the sensor response to total heat transfer resistance between the sensor and the environment, due to conduction and convection heat transfer, is given in Figure 11-4 for an object at 1000°C temperature. As the total thermal resistance between the sensor and the environment increases, as seen in Figure 11-4, the sensor is better insulated for losses due to conduction and convection heat transfer. In such cases, the sensor temperature approaches the object temperature. A ten percent variation in variables around the nominal values given above produces the sensitivity results given in Tables 11-1 and 11-2, for thermal time constant and for steady-state temperature, respectively, for an object that is at 1000°C.

109

Everyday Heat Transfer Problems

Table 11-2 Steady-state sensor temperature change due to a 10% change in variables around the nominal values for an object temperature of 1000°C

Variable

Nominal Value

Steady-State Steady-State Sensor Sensor Temperature Temperature Change Due To Change Due To A 10% Decrease A 10% Increase In Nominal Value In Nominal Value

Tobject

1000°C (1273 K)

−18.33%

+18.22%

Rtotal

100 K/W

−4.13%

+3.48%

0.0001 m

−4.07%

+3.44%

εobject

0.9

−4.07%

+3.44%

εsensor

0.8

−4.07%

+3.44%

20°C (293 K)

−0.11%

+0.11%

0.1 W

−0.054%

+0.054%

0.014 W-s/K

0%

0%

A

Tenvironment I 2R Sensor Thermal Capacitance, ρcpV

2

A change in sensor thermal capacitance has the most significant effect on the thermal time constant in this region of operation. The thermal time constant experiencees non-linear sensitivity behaviors from all the variables in the present region of operation, with the exception of the environmental temperature and the I2R power input to the sensor. The thermal time constant experiencees similar magnitude sensitivity behaviors from the sensor area, sensor emissivity, and object emissivity variables, as expected. A change in the object temperature has the most significant effect on the steady-state sensor temperature in this region of operation. The next set of independent variables that most affect the sensor temperature are the total heat transfer resistance between the sensor

110

Heat Transfer In An Active Infrared Sensor

Table 11-3 Thermal time constant change due to a 10% change in variables around the nominal values for an object temperature of 100°C

Variable Tenvironment Sensor Thermal Capacitance, ρcpV RT

Thermal Time Constant Change Due To A 10% Decrease In Nominal Value

Thermal Time Constant Change Due To A 10% Increase In Nominal Value

20°C (293 K)

−14.97%

+13.16%

0.014 W-s/K

−10.33%

+10.39%

100 K/W

−9.90%

+9.83

Nominal Value

2

0.0001 m

−0.45%

+0.45%

εobject

0.9

−0.45%

+0.45%

εsensor

0.8

−0.45%

+0.45%

0.1 W

−0.028%

+0.028%

100°C (373 K)

+0.023%

−0.023%

A

2

IR Tobject

and the environment, the sensor area, and surface emissivities of the sensor and the object. Sensor thermal capacitance does not affect the steady-state sensor temperature, as expected. The steady-state sensor temperature experiences non-linear sensitivity behaviors from all the variables except the environmental temperature and the I2R power input to the sensor, in the present region of operation. The steady-state sensor temperature experiences similar magnitude sensitivity behaviors from the sensor area, sensor emissivity, and object emissivity variables, as expected. A similar sensitivity analysis is performed for a low object temperature case, namely Tobject = 100°C, and the results are given in Tables 11-3 and 11-4.

111

Everyday Heat Transfer Problems

Table 11-4 Steady-state sensor temperature change due to a 10% change in variables around the nominal values for an object temperature of 100°C

Variable

Nominal Value

Steady-State Sensor Temperature Change Due To A 10% Decrease In Nominal Value

Steady-State Sensor Temperature Change Due To A 10% Increase In Nominal Value

Tenvironment

20°C (293 K)

−5.58%

+5.57%

RT

100 K/W

−3.99%

+3.95%

2

0.1 W

−2.79%

+2.79%

100°C (373 K)

−2.27%

+2.46%

0.0001 m2

+1.18%

−1.19%

IR Tobject A εobject

0.9

+1.18%

−1.19%

εsensor

0.8

+1.18%

−1.19%

0.014 W-s/K

0%

0%

Sensor Thermal Capacitance, ρcpV

Temperature of the environment and thermal resistance to heat loss from the sensor by conduction and convection to the environment become the prominent variables in this low object-temperature application. Both the thermal time constant and the steadystate temperature of the sensor are very sensitive to variations in the temperature of the environment and the total thermal resistance. It is important to remember that these calculated sensitivities are good only around the nominal values assumed for this case. Changing these nominal values will change magnitudes and orders of these sensitivities, resulting from the non-linear form of governing heat transfer equations.

112

CHAPTER

COOLING

OF A CHIP

12

P

ower dissipation in electronic chips is a challenging heat transfer phenomenon, as the chips get smaller and smaller. Most chips or chip sets use copper or aluminum heat sinks to enhance the heat transferred out. These heat sinks are attached in a variety of ways to the chips in order to minimize the thermal resistance between the chip and the heat sink. In this analysis, a chip in the shape of a rectangular box, 10 mm × 10 mm × 0.1 mm, is considered. The chip is attached to its copper heat sink, also in the shape of a rectangular box, 10 mm × 10 mm × 10 mm, by a thermally conductive epoxy of 10 μm thickness. Transient heat transfer which occurs during the cooling of a chip can generally be solved by using the same energy balance equation as in Chapter 11, without the radiation heat transfer effects. The temperature of a chip can be investigated by using unsteady-state and one-dimensional heat transfer rate equations in rectangular coordinates. Energy balance for a chip can be written as follows: Change in internal energy of the chip with respect to time = Power generated by the chip − Energy lost from the chip to its heat sink and to the environment by conduction and by convection heat transfer. 113

Everyday Heat Transfer Problems

This energy balance can be written as a first order and linear differential equation: ρcpV(dTchip/dθ) = P – (Tchip – Tenvironment)/RTotal

(12-1)

where ρcpV is the chip thermal capacitance in W-s/C dTchip/dθ is the time rate of change of chip temperature in C/s P is the energy generated in the chip in W Tenvironment is the environment temperature in C RTotal is the total heat transfer resistance between the chip, its heat sink and the environment due to conduction and convection heat transfer in C/W Tchip is the temperature of the chip in C. Since the governing equation is linear in temperature, centigrade dimension is used instead of Kelvin. Total heat transfer resistance between the chip, its heat sink and the environment has two parallel components. One component is convection heat transfer resistance between the chip and the environment. The other is the heat loss in series circuit from the chip through the adhesion layer by conduction, from the adhesion layer through the heat sink by conduction, and from the heat sink to the environment by convection. The heat transfer resistance between the surface of the chip and the environment can be written as: Rchip to environment = 1/(hAchip for convection)

(12-2)

The series circuit heat transfer resistance between the chip and the environment going through the adhesive and the heat sink can be written as: Rchip through heat sink = [Ladhesive/(kadhesiveAadhesive for conduction)] + [Lheat sink/(kheat sinkAheat sink for conduction)] (12-3) + [1/(hAheat sink for convection)]

114

Cooling Of A Chip

The total heat transfer resistance from the chip is a parallel combination of Eqs. (12-2) and (12-3): (1/RTotal) = (1/Rchip to environment) + (1/Rchip through heat sink)

(12-4)

The solution to the governing first order differential Eq. (12-1) can be written as follows by using the initial condition of Tchip = Tenvironment: Tchip = Tenvironment + PRTotal [1 – exp(−Θ/ρcpVRTotal)]

(12-5)

where ρcpVRTotal is the thermal time constant for the chip, which is the product of its thermal capacitance and its total thermal resistance. Under steady-state conditions, the solution in Eq. (12-5) provides the allowable chip power dissipation as follows: P = (Tchip − Tenvironment)/RTotal

(12-6)

For the present sensitivity analysis, the nominal values of the above independent variables are assumed to be as follows: ρcpV = 0.0197 W-s/C (assuming a silicon dioxide chip with 10 mm × 10 mm × 0.1 mm dimensions) h = 200 W/m2-C (convection heat transfer coefficient at the chip and heat sink surfaces) Tenvironment = 30ºC Tchip = 90ºC Achip for convection = 0.000104 m2 Ladhesive = 0.00001 m Aadhesive for conduction = Aheat sink for conduction = 0.0001 m2 Lheat sink = 0.01 m Aheat sink for convection = 0.0005 m2 (assuming a copper alloy heat sink with 10 mm × 10 mm × 10 mm dimensions) kadhesive = 10 W/m-C (assuming silver epoxy adhesive) kheat sink = 400 W/m-C

115

Everyday Heat Transfer Problems

200

RTotal, C/W

150 100 50 0 0

100

200

300

400

Convection Heat Transfer Coefficient, W/m2-C

Figure 12-1 Total chip heat transfer resistance versus the convection heat

transfer coefficient

Chip Power Disipation, W

Total heat transfer resistance between the chip, its sink and the environment is given as a function of the convection heat transfer coefficient in Figure 12-1. As the convection mechanism goes into forced convection, and especially forced convection in liquids,

35 30 25

with heat sink

20 15

without heat sink

10 5 0 0

200

400

600

Convection Heat Transfer Coefficient,

800

1000

W/m2-C

Figure 12-2 Chip power dissipation versus the convection heat transfer

coefficient with and without a heat sink

116

Cooling Of A Chip

total heat transfer resistance decreases dramatically. This decrease in total heat transfer resistance allows the chip to be operated at higher power dissipation, as shown in Figure 12-2. Figure 12-2 also shows the effects of using a heat sink. The slope of the chip's power dissipation with a heat sink versus the heat transfer coefficient, (∂P/∂h), is 0.033 m2-C. The slope of the chip's power dissipation without a heat sink versus the heat transfer coefficient is much lower, ∂P/∂h = 0.012 m2-C. A ten-percent difference in variables around the nominal values given above produce the sensitivity results given in Table 12-1 for the power dissipation capability of the chip. The results are given in descending order from the most sensitive variable to the least.

Table 12-1 Steady-state chip power dissipation capability change due to a 10% change in variables around the nominal values

Variable Tchip − Tenvironment h Aheat sink for convection Achip for convection

Nominal Value

Steady-State Chip Power Dissipation Capability Change Due To A 10% Decrease In Nominal Value

Steady-State Chip Power Dissipation Capability Change Due To A 10% Increase In Nominal Value

60ºC

−10%

+10%

−9.81%

+9.77%

−8.05%

+8.01%

−1.76%

+1.76%

200 W/m-C 2

0.0005 m

0.000104 m

2

Aadhesive for conduction = Aheat sink for conduction

0.0001 m2

−0.23%

+0.19%

kheat sink

400 W/m-C

−0.22%

+0.18%

Lheat sink

0.01 m

+0.20%

−0.20%

kadhesive

10 W/m-C

−0.009%

+0.007%

Ladhesive

0.00001 m

+0.008%

−0.008%

117

Chip Time Constant, s

Everyday Heat Transfer Problems

3.5 3 2.5 2 1.5 1 0.5 0 0

100

200

300

400

Convection Heat Transfer Coefficient, W/m2-C

Figure 12-3 Chip thermal time constant versus the convection heat transfer

coefficient

Changes in chip temperature potential, convection heat transfer coefficient and heat sink surface area for convection heat transfer affect the chip power dissipation capability the most. Chip surface area for convection heat transfer comes next in the order of sensitivity. Changes in independent variables affecting the conduction heat transfers, Aadhesive for conduction, Aheat sink for conduction, kheat sink Lheat sink, kadhesive, and Ladhesive, contribute the least to sensitivities in chip power dissipation. These sensitivity results shown in Table 12-1 are valid only in the region of analysis for the governing independent variables. The thermal time constant, ρcpVRTotal , is an important parameter in chip design and testing. Thermal time constant, which is the product of thermal capacitance and total thermal resistance, is analyzed as a function of the convection heat transfer coefficient, and is given in Figure 12-3. As the convection mechanism goes into forced convection, and especially forced convection in liquids, the thermal time constant decreases dramatically. A ten-percent difference in variables around the nominal values given above produces the sensitivity results given in Table 12-2 for the

118

Cooling Of A Chip

Table 12-2 Chip thermal time constant change due to a 10% change in variables around the nominal values

Variable h ρcpV

Nominal Value

Chip Thermal Time Constant Change Due To A 10% Decrease In Nominal Value

Chip Thermal Time Constant Change Due To A 10% Increase In Nominal Value

200 W/m-C

+10.88%

−8.90%

0.0197 W-s/C

−10%

+10%

2

0.0005 m

+8.76%

−7.42%

0.000104 m2

+1.79%

−1.79%

Aadhesive for conduction = Aheat sink for conduction

0.0001 m2

+0.233%

−0.190%

kheat sink

400 W/m-C

+0.223%

−0.183%

Lheat sink

0.01 m

−0.201%

+0.201%

kadhesive

10 W/m-C

+0.009%

−0.007%

Ladhesive

0.00001 m

−0.008%

+0.008%

Aheat sink for convection Achip for convection

thermal time constant of the chip. The results are given in descending order, from the most sensitive variable to the least. Changes in the convection heat transfer coefficient, chip thermal capacitance and heat sink surface area for convection heat transfer affect the chip thermal time constant the most. Sensitivities of the chip thermal time constant to the rest of the independent variables follow the same order as in Table 12-1. Changes in independent variables affecting the conduction heat transfer contribute the least to chip thermal time constant sensitivities.

119

CHAPTER

13 A CHIP UTILIZING COOLING OF

A HEAT SINK WITH RECTANGULAR FINS H

eat transfer from a surface can be enhanced by using fins. This chapter combines ideas from Chapters 5 and 12, and analyzes the cooling of a chip utilizing a heat sink with rectangular fins. The temperature of a chip can be analyzed using unsteady-state and one-dimensional heat transfer rate equations in rectangular coordinates. Energy balance for transient heat transfer during the cooling of a chip can be written as follows: Change in internal energy of the chip with respect to time = Power generated by the chip − Energy lost from the chip to its heat sink and to the environment by conduction and convection heat transfer

This energy balance can be written as a first order and linear differential equation: ρcpV(dTchip /dθ) = P − (Tchip − Tenvironment)/RTotal

(13-1)

121

Everyday Heat Transfer Problems

where ρcpV is the chip thermal capacitance in W-s/C dTchip /dθ is the time rate of change of chip temperature in C/s P is the heat generated in the chip in W Tenvironment is the environment temperature in C RTotal is the total heat transfer resistance between the chip, its heat sink with fins and the environment, due to conduction and convection heat transfer in C/W Tchip is the temperature of the chip in C Since the governing equation is linear in temperature, centigrade dimension is used instead of Kelvin. Total heat transfer resistance between the chip, its sink and the environment has two parallel paths. One path is the direct convection heat transfer between the chip and the environment. The other is the heat loss in series from the chip to the adhesion layer by conduction, from the adhesion layer to the heat sink body by conduction, and from the heat sink and its fins to the environment by convection. The convection heat transfer from the heat sink and its fins to the environment has two parallel components; one is the convection heat transfer from the sink base surface to the environment, and the other is the convection heat transfer from the fins’ surfaces to the environment. The convection heat transfer resistance between the heat sink’s un-finned surfaces and the environment can be written as follows: Rheat sink convection from un-finned surfaces = 1/(hAheat sink convection un-finned surfaces) (13-2) The convection heat transfer between heat sink fin surfaces and the environment can be written by combining Eqs. (5-5) and (5-6): Qfin total = NηhAfin (Tfin base − Tenvironment)

(13-3)

The convection heat transfer resistance between the heat sink’s fin surfaces and the environment can be written as follows: Rheat sink convection from fin surfaces = 1/(NηhAfin)

122

(13-4)

Cooling Of A Chip Utilizing A Heat Sink With Rectangular Fins

N is the number of rectangular fins and η is the rectangular fin heat transfer efficiency as defined in Chapter 5. “h” represents the convection heat transfer coefficient at the chip surfaces, the heat sink fin surfaces and the un-finned surfaces. It is assumed that the convection heat transfer coefficient does not vary with a change in the number of fins and fin lengths. The series circuit heat transfer between the chip and the environment that goes through the adhesive layer and the heat sink as conduction mechanisms, and from the heat sink to the environment as parallel convection mechanisms between the un-finned and finned heat sink surfaces, can be written as follows: Rchip through heat sink = [Ladhesive /(kadhesiveAadhesive for conduction)] + [Lheat sink /(kheat sinkAheat sink for conduction)] + [1/(NηhAfin + hAheat sink convection un-finned surfaces)] (13-5) The heat transfer resistance between the surface of the chip and the environment can be written as: Rchip to environment = 1/(hAchip for convection)

(13-6)

The total heat transfer resistance between the chip, its heat sink with fins and the environment can be written by combining Eqs. (13-5) and (13-6) in a parallel thermal circuit: RTotal = {1/[(1/Rchip to environment) + (1/Rchip through heat sink)]}

(13-7)

The solution to the governing first order differential Eq. (13-1) can be written as follows, using the initial condition of Tchip = Tenvironment and assuming temperature-independent thermophysical properties: Tchip = Tenvironment + PRT [1 − exp(−Θ/ρcpVRTotal)]

(13-8)

where ρcpVRTotal is the thermal time constant for the chip, which is the product of its thermal capacitance and its total thermal resistance.

123

Everyday Heat Transfer Problems

For the present sensitivity analysis, the nominal values of the above independent variables are assumed to be as follows: ρcpV = 2.52 W-s/C (assuming a silicon dioxide chip in the shape of a rectangular box with 0.03 m × 0.03 m × 0.002 m dimensions) h = 50 W/m2-C Tenvironment = 30°C Tchip = 90°C Achip for convection = 0.00114 m2 which can be detailed as [0.03 × 0.03 + 2 × (0.03 + 0.03) × 0.002] Ladhesive = 0.002 m Aadhesive for conduction = Aheat sink for conduction = 0.0009 m2 which can be detailed as (0.03 × 0.03) Lheat sink = 0.005 m N = 7 (seven rectangular fins which are nominally 0.002 m thick and 0.03 m in length) Aheat sink convection un-finned surfaces = 0.00108 m2 which can be detailed as [with 0.002 m thick seven fins, namely 0.03 × (0.03 − 7 × 0.002) + 2 × (0.03 + 0.03) × 0.005] Afin = 0.00186 m2 which can be detailed as [for a 0.002 m thick, 0.03 m long, and 0.03 m wide fin, namely 2 × 0.03 × (0.03 + 0.5 × 0.002)] kadhesive = 100 W/m-C (assuming silver epoxy adhesive) kheat sink = 300 W/m-C It is assumed that the heat sink and the fins are machined from the same block material, and the minimum machinable spacing between the fins is 0.002 m. For seven fins, the resulting fin thickness is 0.002 m. The heat transfer efficiency for a rectangular fin can be calculated from Eq. (5-6) given in Chapter 5, η = tanh(mLc)/(mLc) where m = [2h(w + t)/kwt ]1/2, Lc = L + 0.5t. For cases where the fin width, w, is much greater than its thickness t, m becomes m = ( 2h /kt )1/2. The rectangular fin efficiency for this nominal case is 0.95. Using the above nominal values, and changing the fin length and the number of fins on the heat sink, the chip heat dissipation characteristics given in Figure 13-1 are obtained. As fin length increases, so does chip power dissipation. The chip power dissipation

124

Chip Power Dissipation, W

Cooling Of A Chip Utilizing A Heat Sink With Rectangular Fins

100 80

Lfin=0.01 m Lfin=0.03 m Lfin=0.05 m Lfin=0.07 m Lfin=0.09 m Lfin=0.15 m

60 40 20 0 0

2 4 6 Number Of Rectangular Fins

8

Figure 13-1 Chip power dissipation versus number of fins for different fin

lengths

asymptotes around Lfin = 0.15 meters for an eight-fin heat sink. With increasing fin length, the efficiency of the fin decreases, even if the convection heat transfer area increases. Adding the maximum amount of fins (eight in this case due to machinability constraints), with lengths of up to 0.15 m per fin, can enhance the chip heat dissipation by as much as 12-fold, as compared to a non-finned heat sink. The sensitivity of chip heat dissipation to fin length is given in Figure 13-2. The sensitivity approaches zero as the fin length increases. By increasing the fin length, the chip power dissipation improves less and less, and choosing the right fin length becomes a cost-benefit issue. For example, using Figure 13-1, if the desired chip power dissipation is 40 W, choosing a four-fin heat sink design, with a fin length of 0.05 m, will suffice. Chip power dissipation versus the convection heat transfer coefficient is shown in Figure 13-3. Cases with seven fins and with no fins are compared for a fixed fin length of 0.03 meters. Chip power dissipation varies linearly with the convection heat transfer coefficient, since conduction heat transfer resistances through the

125

Everyday Heat Transfer Problems

1400 ∂ P/∂ Lfin, W/m

1200 1000

N=4 N=5 N=6 N=7 N=8

800 600 400 200 0 0.01

0.04

0.07

0.1

0.13

Rectangular Fin Length, m

Chip Power Dissipation, W

Figure 13-2 Chip heat dissipation sensitivity to fin length versus fin length

160 140 120 100

N=7 N=0

80 60 40 20 0 0

50

100

150

200

h, W/m2-C

Figure 13-3 Chip power dissipation versus convection heat transfer coefficient

with no fins and with seven fins for fin length = 0.03 m

126

Thermal Time Constant, s

Cooling Of A Chip Utilizing A Heat Sink With Rectangular Fins

40 30 20 10 0 0

50 100 150 Convection Heat transfer Coefficient, W/m2-C

200

Figure 13-4 Chip thermal time constant versus convection heat transfer

coefficient

adhesive and through the heat sink are negligible for this case study. The slope of the chip power dissipation versus the convection heat transfer coefficient line for a seven-fin design, (∂P/∂h), is 0.71 m2-C. The slope of the chip power dissipation versus the convection heat transfer coefficient line for the finless design is 0.13 m2-C. It is apparent that the present nominal heat sink design, with seven fins, enhances the chip power dissipation by over five-fold. The thermal time constant for the chip is the product of thermal capacitance and total thermal resistance, ρcpVRT, and it is given as a function of the convection heat transfer coefficient in Figure 13-4. The thermal time constant for the chip is a strong function of the convection heat transfer coefficient at low values of forced convection heat transfer regime; i.e., h < 50 W/m2-C. In this finned heat sink design, the heat sink base and the fins are assumed to be the same material. The thermal conductivity of the heat sink and fin material also starts to affect the chip power dissipation at lower values of kheat sink , i.e., kheat sink < 100 W/m-C. The chip power dissipation versus heat sink and fin material thermal conductivity is given in Figure 13-5 for a seven-fin design with a fin

127

Everyday Heat Transfer Problems

Power Dissipation, W

45 40 35 30 25 20 0

100 200 300 Heat Sink & Fin Thermal Conductivity, W/m-C

400

Figure 13-5 Chip power dissipation versus heat sink and fin thermal

conductivity for Lfin = 0.03 m and N = 7

0.7 ∂ P/∂ ksink, m-C

0.6 0.5 0.4 0.3 0.2 0.1 0 0

100 200 300 Heat Sink & Fin Thermal Conductivity, W/m-C

400

Figure 13-6 Chip power dissipation sensitivity to heat sink and fin thermal

conductivity versus heat sink and fin thermal conductivity for Lfin = 0.03 m and N = 7

128

Cooling Of A Chip Utilizing A Heat Sink With Rectangular Fins

Table 13-1 Steady-state chip power dissipation capability change due to a 10% change in variables around the nominal values for Lfin = 0.03 m and N = 7

Variable Tchip − Tenvironment

Nominal Value

Steady-State Chip Power Dissipation Capability Change Due To A 10% Decrease In Nominal Value

Steady-State Chip Power Dissipation Capability Change Due To A 10% Increase In Nominal Value

60°C

−10%

+10%

h

50 W/m-C

−9.41%

+9.29%

kheat sink

300 W/m-C

−0.574%

+0.476%

Lheat sink

0.005 m

−0.289%

+0.287%

kadhesive

100 W/m-C

−0.148%

+0.122%

Ladhesive

0.002 m

+0.134%

−0.134%

length of 0.03 meters. The sensitivity of chip power dissipation to heat sink and fin thermal conductivity is shown in Figure 13-6, again for a seven-fin design with a fin length of 0.03 meters. The sensitivity increases fast for low values of heat sink and fin material thermal conductivity. A ten-percent difference in independent variables around the nominal values given above produces the sensitivity results given in Table 13-1, shown in descending order for the power dissipation capability of the chip. The chip’s physical dimensions are assumed to be constants. This sensitivity analysis is performed for a seven-fin heat sink design with a fin length of 0.03 meters. Changes in chip-to-environment temperature potential and the convection heat transfer coefficient affect the chip power dissipation capability the most. Changes in conduction heat transfer variables about the nominal values chosen for this analysis have the least affect on chip power dissipation capability.

129

CHAPTER

HEAT

14

TRANSFER ANALYSIS FOR COOKING IN A POT F

ood cooking in a pot placed on a gas burner can be a very complicated heat transfer problem, if multi-dimensional transient heat transfer for the pot and for the food in the pot are considered. The heat transfer mechanisms can get challenging, if boiling heat transfer regime is treated. The best way to approach such a heat transfer problem is to make simplifying assumptions and create a simple model, then verify the results of the model by reliable experiments. In the present heat transfer model, to simulate cooking in a pot, an unsteady-state and a coupled one-dimensional heat transfer analysis will be utilized with the following assumptions. A cylindrical pot gets its cooking energy from a gas burner at a constant rate. The pot is assumed to be made out of copper and have a uniform temperature throughout, namely negligible internal conduction resistance. The pot conduction heat transfer resistance, Lc/k, is small as compared to the pot surface's convection heat transfer resistance, 1/h, or Biot Number = hLc/k < 0.1, where Lc is a characteristic length for the pot, which is the volume of copper divided by the pot's outer surface area.

131

Everyday Heat Transfer Problems

The pot loses energy to food from its bottom and its side by natural convection heat transfer. The natural convection heat transfer mechanism between the pot and the food is analyzed up to a food temperature of 105°C at sea level atmospheric conditions; boiling heat transfer regime is not considered. The pot also loses energy to the environment from its bottom and its side by natural convection and radiation. The food receives energy from the bottom and the side of the pot by natural convection. Temperature gradients in the food are neglected, and the natural convection heat transfer coefficient between the pot and the food is calculated using an average food temperature. Conduction and radiation heat transfer mechanisms between the pot and the food are neglected. Also, the natural convection and radiation heat transfers from the top of the food to the environment are neglected. Neglecting these secondary heat transfer mechanisms introduces an initial error of about 7% in the net heat transferred to the food, but this error diminishes fast to zero as cooking time increases. Food is assumed to have the same thermophysical properties as water. Temperature-dependent variations for all of the thermophysical properties are considered. The thermophysical properties are calculated at a film temperature, which is the average of the surface and the medium temperatures. Energy balance for the food can be written as follows: Change in internal energy of the food with respect to time = Energy input by convection from the bottom of the pot to the food + Energy input by convection from the sides of the pot to the food Energy balance for the pot can be written as follows: Change in internal energy of the pot with respect to time = Energy input from the heater to the bottom of the pot − Energy output by convection from the bottom of the pot to the food − Energy output by convection from the sides of the pot to

132

Heat Transfer Analysis For Cooking In A Pot

the food − Energy output by convection from the bottom of the pot to the environment − Energy output by convection from the sides of the pot to the environment − Energy output by radiation from the bottom and sides of the pot to the environment The governing coupled and transient first-order differential equations for the food and the pot are detailed as follows: (ρcpV)food(dTfood/dΘ) = hpot bottom to food natural convection Apot bottom (Tpot − Tfood) + hpot sides to food natural convection Apot sides (Tpot − Tfood)

(14-1)

(ρcpV)pot(dTpot/dΘ) = Qin − hpot bottom to food natural convection Apot bottom (Tpot − Tfood) − hpot sides to food natural convection Apot sides (Tpot − Tfood) − hpot bottom to environment natural convection Apot bottom (Tpot − Tenvironment) − hpot sides to environment natural convection Apot sides (Tpot − Tenvironment) − hpot to environment radiation (Apot bottom + Apot sides) (Tpot − Tenvironment) (14-2) All the natural convection heat transfer coefficients can be obtained from empirical relationships in literature for the appropriate geometries; for example, for natural convection heat transfer from heated vertical plates and horizontal plates, see References [6] and [10]. The natural convection heat transfer coefficient from the bottom of the pot to the food is: (hL/k) = 0.5 RaL0.25

for 104 < RaL < 107

(14-3)

(hL/k) = 0.15 RaL0.33

for 107 < RaL < 1011

(14-4)

The natural convection heat transfer coefficients from the sides of the pot to the food and the environment are obtained from the empirical relationship: (hL/k) = {0.825 + 0.387RaL /[1 + (0.492/Pr)9/16]8/27}2

(14-5)

133

Everyday Heat Transfer Problems

The natural convection heat transfer coefficient from the bottom of the pot to the environment is obtained from the empirical relationship: (hL/k) = 0.27 RaL0.25

for 105 < RaL < 1010

(14-6)

where the Rayleigh number, RaL, is the product of the Grashof number and the Prandtl number, RaL = gβ(Tpot − Tfood)L3/(να). The Grashof number represents the ratio of buoyancy forces to viscous forces in a natural convection heat transfer system. The Prandtl number is the ratio of momentum diffusivity to thermal diffusivity, namely Pr = ν/α. Tfood in the RaL equation becomes Tenvironment when the natural convection heat transfer coefficient considered is between the pot and the environment. Thermophysical properties, variables and constants in the above Eqs. (14-1) to (14-6) are defined as: (ρcpV) = thermal capacitance, namely product of density, specific heat at constant pressure and volume, in W-hr/C T = Temperature in C or in K when radiation calculations are used Θ = Time in hr h = heat transfer coefficient for natural convection or radiation heat transfer mechanisms in W/m2-C A = Heat transfer area in m2 Qin = Heat input from the gas burner to the pot in W k = Thermal conductivity in W/m-C L = Characteristic length for the surface area, where natural convection heat transfer occurs and is defined as the ratio of heat transfer surface area to its perimeter in m g = Gravitational acceleration in m/s2 β = Volumetric thermal expansion coefficient in 1/K ν = Kinematic viscosity in m2/s α = (k/ρcp) thermal diffusivity in m2/s Governing Eqs. (14-1) and (14-2) are solved using the explicit finite difference method with a ten-second time interval. Temperature-dependent thermophysical properties of water and

134

Heat Transfer Analysis For Cooking In A Pot

air are obtained from References [6] and [10]. The radiation heat transfer coefficient is obtained from the following equation: hradiation = σε(T4pot − T4environment)/(Tpot − Tenvironment)

(14-7)

where σ is the Stefan-Boltzmann constant, 5.67 × 10−8 W/m2-K4, and ε is the emissivity of the copper pot outer surface. The heat input from the gas burner to the pot is 750 W. Dimensions of the pot and the food height are as follows: Pot inside diameter = 0.127 m Pot height = 0.076 m Pot side thickness = 0.0018 m Pot bottom thickness = 0.0063 m Food height in the pot = 0.051 m Food and pot temperatures are given as functions of time in Figure 14-1. Food temperature is calculated up to 105°C, where the applicability of the natural convection heat transfer coefficient equations end and the boiling heat transfer regime starts. In addition, water heating experiments are performed in a copper pot with the above geometrical dimensions to verify the present model. The results of these experiments are also given in Figure 14-1. The results of the present simple heat transfer model match the experimental results closely. Deviations between the model and the experiments are seen at initial cooking times; i.e., less than one minute, and as the water temperature approaches the boiling temperature. The present heat transfer model can be improved by including the temperature gradients in the water, by using a more appropriate empirical relationship for the natural convection heat transfer coefficient as the water temperature approaches its boiling temperature, and by taking smaller time increments during calculations. The natural convection heat transfer coefficients from the bottom of the pot and the sides of the pot to the food are given in Figure 14-2. The heat transfer coefficients increase as the pot and the food temperatures increase, mainly due to the temperature dependent properties of water. As the food temperature increases,

135

Food and Pot Temperature, C

Everyday Heat Transfer Problems

120 100 80

Tfood, C Tpot, C Experiment, C

60 40 20 0

2

4

6

Time, min

Figure 14-1 Food and pot modeling temperatures, and food experiment

Natural Convection Heat Transfer Coefficient, W/m2-C

temperatures, as functions of time

1600 1200 800

hpot bottom to food, W/m2-C

400

hpot sides to food, W/m2-C

0 0

2

4

6

Time, min

Figure 14-2 Pot bottom to food and pot sides to food natural convection

heat transfer coefficients

136

Heat Transfer Analysis For Cooking In A Pot

Table 14-1 Changes in time for the average food temperature in the pot to reach 105°C due to a 10% change in variables around the nominal values

Nominal Value

Change in Time for the Food in the Pot to Reach 105°C Due To A 10% Decrease In Nominal Value

Change in Time for the Food in the Pot to Reach 105°C Due To A 10% Increase In Nominal Value

750 W

+11.16%

−9.09%

Thermal Capacity of Food

Varies with time, W-hr/C

−8.36%

+8.36%

Apot bottom

0.01267 m2

−2.71%

+2.84%

Thermal Capacity of Pot

0.1286 W-hr/C

−1.65%

+1.65%

0.02027 m2

−0.84%

+0.84%

Heat Loss from Pot to Environment

Varies with time, W

−0.30%

+0.30%

hpot bottom to food

Varies with time, W/m2-C

+0.17%

−0.15%

Varies with time, W/m2-C

+0.13%

−0.12%

Variable Qin

Apot sides

natural convection

hpot sides to food natural convection

its volumetric thermal expansion coefficient increases and its kinematic viscosity decreases. The natural convection heat transfer between the bottom of the pot and the food is more dominant than the one between the side of the pot and the food, mainly due to the difference in characteristic lengths that affect the Rayleigh number. A ten-percent variation around the nominal values of independent variables given above produces the sensitivity results given in Table 14-1, for the time that the average food temperature in the pot

137

Everyday Heat Transfer Problems

reaches 105°C. The sensitivity results are given in descending order, and they are applicable only in the region of assigned nominal values due to their non-linear effects. The time it takes for the food to reach 105°C is most sensitive to heat input from the gas burner into the pot, and to the thermal capacity of the food. The sensitivity order continues with the pot geometry and the pot thermal capacity. The time it takes for the food to reach 105°C is an order of magnitude less sensitive to changes in energy loss to the environment, and to changes in the natural convection heat transfer coefficients between the pot surfaces and the food. If all of the 750 W heat input from the gas burner goes into the food, the time for the food to reach 105°C is 5 minutes. All the heat losses from the pot to the environment, as well as the thermal capacity of the pot, lower the efficiency of the food cooking system. It takes 6.23 minutes for the food to reach 105°C under nominal conditions.

138

CHAPTER

HEAT

TRANSFER AND INSULATING A WATER PIPE

15

P

rotecting water pipes from freezing requires transient heat transfer calculations and extensive knowledge of the thermo physical properties of insulating materials. Under severe environmental conditions, insulating the water pipe alone might not be sufficient to prevent it from freezing. Other protections, such as running water in the pipe or heating tape around the pipe, might be necessary. In this heat transfer study, a long pipe with an insulation layer wrapped around it is considered. The interface between the pipe’s outer surface and the insulation inner surface is assumed to be in good contact, with no air gaps or other imperfections to cause any contact resistance. The pipe is filled with stationary water, and is only exposed to the environment where there is convection heat transfer between the outer surface of the insulation and the environment. The time that it takes the water in the pipe to start freezing is analyzed, and its sensitivities to governing independent variables are investigated. Solutions to the one-dimensional form of the transient heat conduction equation in cylindrical coordinates, for a long cylinder and with appropriate boundary conditions, govern this heat transfer

139

Everyday Heat Transfer Problems

problem and are given in the Reference by P. J. Schneider [16] as follows: ∂2T/∂r2 + (1/r)∂T/∂r = (1/α)∂T/∂θ

(15-1)

where T is the temperature in centigrade along the radial position, r, of the insulated pipe in meters, θ is time in seconds, and α is thermal diffusivity of pipe or insulation material, k/ρcp, in meter squared per second. In literature, Eq. (15-1) has closed form solutions as a series of Bessel functions for a variety of boundary conditions (see Reference [1]). In the present analysis, Eq. [15-1] is solved using the explicit finite difference method, (see References [6] and [16]), with the appropriate boundary conditions. Eq. (15-1) can be written in a finite difference form, and a radial node temperature at a new time step is determined by known temperatures at the surrounding nodes from previous time step calculations, as follows: Trθ+1 =[1 − (2+Δr/r)Fo] Trθ + (1+Δr/r)Fo Tr+1θ + Fo Tr−1θ

(15-2)

where Fo is the Fourier number defined as αΔθ/(Δr)2, Δθ is the time interval in seconds used in calculations, and Δr is the radial node interval in meters used in calculations. In order to achieve stable step-by-step solutions in time to Eq. (15-2), the coefficient of Trθ has to be positive (see References [6] and [16]). In the present calculations, appropriate time and spatial intervals are used so that the Fourier number is always less than 1/(2 + Δr/r). The initial condition and the boundary conditions to solve Eq. (15-2) are as follows—initial condition, θ = 0, for the finite difference calculations is: Tr = Twater

(15-3)

Boundary conditions for the outer pipe-insulation interface and the pipe inner radial nodes are obtained from applying an energy balance to a control volume around that particular node. At the inner-most node of the pipe, it is assumed that the water and half the pipe wall

140

Heat Transfer And Insulating A Water Pipe

thickness have the same temperature. This assumption is good for small-diameter pipes and for pipe wall materials with high thermal conductivities. The pipe inner node energy balance per meter of pipe is given below. The inner node starts at the center of the pipe and goes out half way in wall thickness of the copper pipe: Energy lost due to conduction out from the radial element = Change in stored thermal energy in nodal volume. [This nodal volume starts at r = 0 and goes to 0.5(r2 − r1), which includes the water in the pipe and half the pipe wall thickness.] or kcopper 2π (Tr1θ − Tr2θ)/ln(r2/r1) = {ρwatercp waterπr12 + ρcoppercp copper2π[r1 + 0.25(r2 − r1)] 0.5(r2 − r1)} (15-4) (Tr1θ+1 − Tr1θ)/Δθ where r1 = inner radius of copper pipe in meters r2 = outer radius of copper pipe in meters kcopper = thermal conductivity of copper pipe in W/m-C ρcopper = density of copper pipe in kg/m3 cp copper = specific heat of copper pipe in J/kg-C ρwater = density of water in kg/m3 cp water = specific heat of water in J/kg-C It should be noted that thermophysical properties are assumed to be constant during the present calculations. The interface node energy balance per meter of pipe for the node at the interface between pipe outer surface and insulation layer inner surface is: Energy gained due to conduction into radial element − Energy lost due to conduction out from the radial element = Change in stored thermal energy in nodal volume. [This nodal volume starts at the middle of the pipe wall thickness, 0.5(r2 − r1), and goes out to the middle of the first insulation layer increment Δr, (r2 + 0.5Δr).]

141

Everyday Heat Transfer Problems

or kcopper 2π (Tr1θ − Tr2θ)/ln(r2/r1) − kinsulation 2π (Tr2θ − Tr2+Δrθ)/ln(r2 + Δr/r2) = {ρcoppercp copper2π[r2 − 0.25(r2 − r1)] 0.5(r2 − r1) + ρinsulationcp insulation2π[r2 + 0.25(r2 + Δr)] 0.5Δr} (Tr2θ+1 − Tr2θ)/Δθ (15-5) where kinsulation = thermal conductivity of insulation material in W/m-C ρinsulation = density of insulation material in kg/m3 cp insulation = specific heat of insulation material in J/kg-C The outer node energy balance per meter of pipe—the node at the interface between the insulation layer outer surface and the environment is: Energy gained due to conduction into the radial element − Energy lost to environment due to convection out from the radial element = Change in stored thermal energy in nodal volume [This nodal volume starts at −0.5Δr from the outer radius of the insulated pipe and goes out to the outer radius, r3.] or kinsulation 2π (Tr3-Δrθ − Tr3θ)/ln(r3/r3 − Δr) − ho 2πr3(Tr3θ − Tenvironment) (15-6) = ρinsulationcp insulation2πr3 0.5Δr (Tr3θ+1 − Tr3θ)/Δθ where ho = Convection heat transfer coefficient between the outer surface of insulation and the environment in W/m2-C r3 = Outer radius of insulated pipe in meters Nominal values of independent variables used in the present sensitivity analysis are as follows: r1 = 0.023 m r2 = 0.025 m r3 = 0.035 m

142

Heat Transfer And Insulating A Water Pipe

Twater = 12°C Tenvironment = −20°C ρwater = 1000 kg/m3 cp water = 4200 J/kg-C kcopper = 50 W/m-C (Note: water pipe is assumed to be copper alloy) ρcopper = 8800 kg/m3 cp copper = 400 J/kg-C kinsulation = 0.04 W/m-C ρinsulation = 26 kg/m3 cp insulation = 835 J/kg-C ho = 10 W/m2−C Δrcopper pipe = 0.001 m Δrinsulation 0.001 m Δθ = 0.2 s The dependent variable in the calculations is the time it takes the water in the pipe to reach 0°C. After water reaches 0°C at the inner radius of the copper pipe, it will take extra time for it to freeze completely. This freezing time can be calculated from the energy required to compensate for the total latent heat of fusion, Hfg, required for the water in the pipe: Hfg water = 334800 W-s/kg After the temperature of the inner node reaches 0°C, the time required for complete freezing is as follows: θfor complete freezing = ρwaterπr12 Hfg /[kcopper2π(0 − Tr2θ=time @ r1=0°C)/ln(r2/r1)]

(15-7)

From Eq. (15-7), the flow rate required to prevent complete freezing also can be calculated: Flow rate to prevent complete freezing in kg/s = [kcopper2π(0 − Tr2θ=T @ r1=0°C)/ln(r2/r1)] L /Hfg

(15-8)

143

Time To Start Freezing, hr

Everyday Heat Transfer Problems

10 8 k insulation=0.02 W/m-C k insulation=0.04 W/m-C k insulation=0.06 W/m-C

6 4 2 0 0

10 20 30 Insulation Thickness, mm

40

Figure 15-1 Time for water to start freezing versus insulation thickness for

three different insulation thermal conductivities

Time For Water To Start Freezing, hr

12 10 8 6 4 2 0 –50

–40

–30

–20

–10

Environment Temperature, C

Figure 15-2 Time for water to start freezing versus environmental

temperature

144

0

Heat Transfer And Insulating A Water Pipe

Table 15-1 Effects of ±10% change in nominal values of variables to time for water to start freezing

Nominal Value

Change In Time For Water To Start Freezing For A 10% Decrease In Nominal Value

Change In Time For Water To Start Freezing For A 10% Increase In Nominal Value

ρwater

1000 kg/m3

−8.645%

+8.645%

cp water

4200 J/kg-C

−8.645%

+8.645%

kinsulation

0.04 W/m-C

+8.282%

−6.774%

Twater

12°C

−8.115%

+7.816%

Tenvironment

−20°C

−7.363%

+8.669%

Insulation thickness, r3 − r2

0.01 m

−5.766%

+5.640%

ho

10 W/m2-C

+2.829%

−2.314%

ρcopper

8800 kg/m3

−1.315%

+1.315%

cp copper

400 J/kg-C

−1.315%

+1.315%

Copper pipe thickness, r2 − r1

0.002 m

+0.243%

−0.240%

ρinsulation

26 kg/m3

−0.041%

+0.041%

cp insulation

835 J/kg-C

−0.041%

+0.041%

kcopper

50 W/m-C

+0.003%

−0.003%

Variable

where Tr2 is the temperature at r2 at the time when r1 reaches 0°C and L is the length of the insulated pipe in meters. Under the above given conditions, water reaches 0°C in 3.31 hours and complete freezing will occur in an additional 14.94 hours. Time for the water to reach 0°C is calculated for different insulation thicknesses and for three different insulation thermal conductivities. The results are given in Figure 15-1. At lower insulation thermal conductivities, time for the water to reach 0°C becomes more sensitive to insulation thickness.

145

Everyday Heat Transfer Problems

Another significant variable affecting the time for water to start freezing is the environmental temperature. Figure 15-2 shows the effects of environmental temperature on time for water to start freezing under the nominal conditions. Time sensitivity to environmental temperature increases rapidly above −10°C. When the nominal values of the independent variables given above are varied ±10%, the results shown in Table 15-1 are obtained. Sensitivities to a ±10% change in the governing independent variables of time for water to start freezing are given in descending order of importance, and these results are applicable only around the nominal values assumed for this study. The changes in variables most affecting the time for water to start freezing are water thermal capacitance, insulation material thermal conductivity, and water and environmental temperatures. Insulation thickness, the convection heat transfer coefficient at the outer surface of insulation layer, and the copper pipe thermal capacitance properties compose the middle of the pack in order of sensitivity. The time for water to start freezing is least sensitive to changes in copper pipe thickness, insulation material thermal capacitance, and copper pipe thermal conductivity.

146

CHAPTER

QUENCHING

OF STEEL BALLS IN AIR FLOW

16

Q

uenching, or rapid and controlled cooling, has been used for centuries to harden steel and increase the toughness of metal alloys. Quenching poses a difﬁcult heat transfer problem, as it deals with high-temperature materials being cooled in a controlled way, in mediums such as air, water, oil, liquid nitrogen, or some other special quenching ﬂuid. The material properties and the ﬂuid properties change fast, and during the quenching process, heat transfer between the material body and the medium can go through several regimes. The medium surrounding the hot body can have ﬁlm boiling, transition from ﬁlm boiling to nucleate boiling, nucleate boiling, and natural convection as quenching time progresses. In this chapter, the quenching of a small spherical ball made out of steel is analyzed in a forced convection air ﬂow. The conduction heat transfer resistance within the ball is assumed to be much less than the heat transfer resistance between the surface of the ball and the quenching medium—the Biot number that is deﬁned in Eq. (16-2) is less than 0.1. This assumption allows the use of the following energy balance in the present unsteady-state heat transfer application (see Reference by J. P. Holman [5]).

147

Everyday Heat Transfer Problems

Rate of decrease of internal energy in spherical steel ball = Heat lost to surrounding air from the surface of the spherical ball Since the air properties, convection heat transfer coefﬁcient, and radiation heat transfer coefﬁcient change by time, the ﬁnite difference form of the energy balance can be written as follows: (ρVcp) (Tsphere (i+1) − Tsphere (i))/dt = −hi A (Tsphere (i) − Tair)

(16-1)

Tsphere (i+1) is temperature of the ball at time (i + 1) in C, Tsphere (i) is temperature of the ball at time (i) in C, Tair is temperature of air away from the ball in C, h is the total heat transfer coefﬁcient in W/m2-C, between the surface of the steel ball and air, which is the sum of the convection and radiation heat transfer mechanisms, and varies with time. It is always calculated at the i’th time in order to be able to determine the temperature at i + 1’th time. A is surface area of the steel ball in m2, dt is time interval used in calculations in seconds, ρ is density of the steel ball in kg/m3, cp is speciﬁc heat of the steel ball in J/kg-C, and V is volume of the steel ball in m3. ρcpV/hA is deﬁned as the thermal response time for the quenching process in seconds. Tsphere initial is initial temperature of the ball in C when the quenching process starts. The Biot number is deﬁned as Biot number = hmax R/ksteel

(16-2)

where hmax is the maximum heat transfer coefﬁcient, the sum of convection and radiation heat transfer mechanisms between the surface of the steel ball and the quenching medium encountered during the quenching process, R is the radius of the sphere ball in meters, and ksteel is thermal conductivity of the steel ball in W/m-C.

148

Quenching of Steel Balls in Air Flow

The convection heat transfer coefﬁcient is calculated from an empirical relationship, given in literature, for forced convection heat transfer from the surface of a sphere to air (see Reference [5]). hforced convection = 0.37 (kair /D) (UmD/νair)0.6 for 17 < ReD < 70000 (16-3) where kair is the temperature-dependent thermal conductivity of air in W/m-C, D is diameter of sphere ball in m, Um is mean velocity of air away from the ball in m/s, νair is temperature dependent kinematic viscosity of air in m2/s, and ReD is the Reynolds number deﬁned as (UmD/νair). The radiation heat transfer coefﬁcient is calculated from hradiation = ε σ (T4sphere − T4air)/(Tsphere − Tair)

(16-4)

where ε is emissivity of steel ball surface and σ is Stefan-Boltzmann constant, namely 5.67 × 10−8 W/m2-K4. During the calculations, instantaneous thermophysical properties of air, kair and νair, are calculated at mean temperatures of Tsphere and Tair, namely at (Tsphere + Tair)/2. Steel ball properties ρ, cp, ε, and D are assumed to be constants. The nominal values of governing independent variables for the present sensitivity calculations are as follows: D = 0.01 m, ρ = 7800 kg/m3 cp = 430 J/kg-C ksteel = 40 W/m-C ε = 0.2 Um = 5 m/s

149

Everyday Heat Transfer Problems

Tair = 27°C Tsphere initial = 800°C dt = 1 s The present case is analyzed below using different time intervals. As the time interval is reduced to around one second, the resulting temperatures-time proﬁles are overlaid, and therefore dt = 1 second is used for the results presented in the present sensitivity calculations. For the nominal case, the Biot number is 0.022 for a hmax of 174 W/m2-C. Steel ball temperature versus time is given in Figure 16-1. For the nominal case, the steel ball reaches 36.8% of (Tsphere initial − Tair), the thermal response time for the quenching process in 57.0 s, and it reaches 50°C in 153.4 s. Initially radiation heat transfer is 18% of the total heat transfer. Radiation heat transfer effect diminishes fast and convection heat transfer dominates the cooling process, as shown in Figure 16-2. The effect of mean air velocity is analyzed without violating the convection heat transfer coefﬁcient’s range of application in Eq. (16-3), 17 < ReD < 70000, and the Biot number being less than 0.1. The time for the steel ball to reach 50°C and the time for it to reach 36.8% of (Tsphere initial − Tair) are shown as a function of mean

Temperature, °C

800 600 400 200 0

0

50

100 Time, s

Figure 16-1 Steel ball temperature versus time

150

150

200

Quenching of Steel Balls in Air Flow

h, W/m2-C

200 150 h convection h radiation

100 50 0

0

50

100 150 Time, s

200

250

Figure 16-2 Convection and radiation heat transfer coefﬁcients versus time

air velocity in Figure 16-3. Changes in mean air velocity below 5 m/s affect the quenching process time signiﬁcantly. Figure 16-4 shows that for the present analyses, the internal conduction resistance for the steel ball is much less than the

400 Time To Reach 50°C, s

Time, s

300

Time To Reach 36.8% of (Tsphere initial-Tair), s

200 100 0

0

5 10 15 Mean Air Velocity, m/s

20

Figure 16-3 Time for the steel ball to reach 50°C and to reach 36.8% of

(Tsphere initial − Tair) versus mean air velocity

151

Everyday Heat Transfer Problems

Biot Number

0.06

0.04

0.02

0

0

10

5

15

20

Mean Air Velocity, m/s

Figure 16-4 Biot number versus mean air velocity

Table 16-1 Effects of ±10% change in nominal values of independent variables to time for the steel ball temperature to reach 50°C, t50°C

Nominal Value

t50°C Change For A 10% Decrease In Nominal Value

t50°C Change For A 10% Increase In Nominal Value

D

0.01 m

−14.25%

+14.79%

kair

varies with temperature

+11.07%

−9.18%

ρ

7800 kg/m3

−10.51%

+10.44%

cp

430 J/kg-C

−10.51%

+10.44%

Um

5 m/s

+6.53%

−5.66%

νair

varies with temperature

−6.17%

+5.90%

773°C

−3.70%

+3.02%

0.2

+0.47%

−0.47%

Variable

Tsphere initial − Tair ε

152

Quenching of Steel Balls in Air Flow

convection heat transfer resistance between the surface of the steel ball and the environment, for the range of mean air velocities considered in the present calculations, i.e., Biot number < 0.1. When the nominal values of the independent variables given above are varied ±10%, the results shown in Table 16-1 are obtained. Time for the steel ball to reach 50°C, t50°C and sensitivities to a ±10% change in the governing independent variables are given in descending order of importance, and they are applicable only around the nominal values assumed for this study. Cooling time to 50°C is most sensitive to independent variables such as ball diameter, density, and speciﬁc heat which make up the steel ball’s thermal capacitance and the medium’s thermal conductivity. Next in the order of sensitivity is the mean air velocity and the air kinematic viscosity. Changes in the initial temperature potential, (Tsphere initial − Tair), affect t50°C about ± 3% in this case study. Steel ball surface emissivity is the least effective of independent variables to t50°C.

153

CHAPTER

QUENCHING 17

OF STEEL BALLS IN OIL

I

n Chapter 16, the quenching medium analyzed was air. When water, oil and other similar mediums are used for quenching, heat transfer gets more complicated because the quenching medium goes into different heat transfer regimes, such as ﬁlm boiling, transition between ﬁlm boiling and nucleate boiling, nucleate boiling, and free convection, as the surface temperature of the material being quenched decreases. Heat transfer in the boiling regime depends on material surface, quenching medium and material surface combination, and on the quenching medium's saturated liquid and vapor thermophysical characteristics. The best way to achieve heat transfer coefﬁcients for boiling heat transfer in a quenching process is by experimental measurements (see Reference by Lee, W. J., Kim, Y. and, E. D. Case [12]). In order to assume a uniform temperature in the steel ball, namely Biot number < 0.1, (see Reference by J. P. Holman [5]), oil quenching instead of water quenching is considered. Oil quenching heat transfer coefﬁcients are in the order of 1000 W/m2-C

155

Everyday Heat Transfer Problems

If water is used as a quenching medium, the heat transfer coefﬁcients can easily exceed 10,000 W/m2-C. To satisfy the Biot number criteria, a steel ball with a diameter of 0.0006 m or less has to be used. In small surface areas, boiling heat transfer gets more complicated as the ratio of buoyant and capillary forces play signiﬁcant roles in boiling characteristics. For Biot number > 0.1 and for time varying heat transfer coefﬁcients, ﬁnite difference or ﬁnite element methods in multi-dimensional and unsteady-state heat transfer have to be utilized. The energy balance for the steel ball in oil quenching medium can be written as: Rate of decrease of internal energy in spherical steel ball = Heat lost to surrounding oil from the surface of the spherical ball Since the heat transfer coefﬁcient will change by time, the ﬁnite difference form of the energy equation is the same as Eq. (16-1): (ρVcp) (Tsphere (i+1) − Tsphere (i) )/dt = − hi A (Tsphere (i) − Toil) (17-1) Tsphere (i+1) is temperature of the ball at time (i+1) in C, Tsphere (i) is temperature of the ball at time (i) in C, Toil is temperature of oil in C, hi is the experimentally obtained total heat transfer coefﬁcient in W/m2-C, between the surface of the steel ball and oil, which is the sum of the convection and radiation heat transfer mechanisms, and varies with time. A is surface area of the steel ball in m2, dt is time interval used in calculations in seconds, ρ is density of the steel ball in kg/m3, cp is speciﬁc heat of the steel ball in J/kg-C, and V is volume of the steel ball in m3. In the present calculations, experimentally obtained heat transfer coefﬁcients for oil quenching are used (see Reference [12]). These are assumed to be total heat transfer coefﬁcients; namely, they account

156

Quenching of Steel Balls in Oil

htotal, W/m2-C

3000 nucleate boiling

2000

B

free convection

1000

C A

0

0

transition

200

400 600 Steel Ball Temperature, °C

film boiling (vapor blanket) 800

Figure 17-1 Total heat transfer coefﬁcient versus steel ball temperature for

oil quenching

for both boiling and radiation heat transfer mechanisms. For oil quenching, the total heat transfer coefﬁcient as a function of steel ball surface temperature is shown in Figure 17-1. In temperatures above 750°C, point C in Figure 17-1, the steel ball encounters ﬁlm boiling heat transfer and is covered with a vapor blanket. The temperatures between points B and C, 600°C to 750°C, constitute the transition region from boiling heat transfer to ﬁlm boiling heat transfer. The temperatures between points A and B, 200°C to 600°C, are in the boiling heat transfer regime. Temperatures below 200°C are considered to be in the free convection heat transfer regime. Steel ball properties ρ, cp, ksteel, and D are assumed to be constants during the oil quenching process. The nominal values of governing independent variables for the present sensitivity calculations are as follows: D = 0.012 m, ρ = 7800 kg/m3 cp = 430 J/kg-C

157

Everyday Heat Transfer Problems

ksteel = 60 W/m-C Toil = 70°C Tsphere initial = 900°C dt = 0.1 s For the present calculations, a time interval of 0.2 seconds is used because the thermal time constant for quenching in oil has dropped an order of magnitude as compared to quenching in air (see Chapter 16). Calculations are also repeated with a 0.1 second time interval, which improves the results by only one percent. The Biot number for this case, hR/ksteel, is 0.1 for an average heat transfer coefﬁcient of 1000 W/m2-C, where R = D/2. Steel ball temperatures versus time are shown in Figure 17-2 for nominal heat transfer coefﬁcients, and for a ±10% variation about the nominal heat transfer coefﬁcient values given in Figure 17-1. Figure 17.2 shows the initial seconds of the quenching process to emphasize the effects of heat transfer coefﬁcient variations. Boiling heat transfer phases occur during the initial seconds of the oil quenching process. For the nominal case, after 8.6 seconds, the natural convection heat transfer regime starts.

Temperature, °C

800 Nominal h (Figure 17-1)

600

10% higher h than nominal

400 200

10% lower h than nominal

0

2

4

6

8

10

Time, s

Figure 17-2 Steel ball temperature versus time for oil quenching for three

different heat transfer coefﬁcient distributions

158

Quenching of Steel Balls in Oil

Table 17-1 Effects of a ±10% change in nominal values, of heat transfer coefﬁcients given in ﬁgure 17-1, to time for the steel ball to reach 100°C, and to time for it to reach 36.8% of (Tsphere initial − Toil)

Variable h

Variable h

Nominal Value

Change in Time to Reach 100°C For A 10% Decrease In h Nominal Value

Change in Time to Reach 100°C For A 10% Increase In h Nominal Value

Figure 17-1

+11.3%

−9.6%

Nominal Value

Change in Time to Reach 36.8% of (Tsphere initial − Toil) For A 10% Decrease In h Nominal Value

Change in Time to Reach 36.8% of (Tsphere initial − Toil) For A 10% Increase In h Nominal Value

Figure 17-1

+12.3%

−9.6%

Sensitivities of time for the steel ball to reach a temperature of 100°C and time for the steel ball to reach 36.8% of (Tsphere initial − Toil), thermal response time for the quenching process, are analyzed as the dependent variables to the changing heat transfer coefﬁcients. The results are provided in Table 17-1. Oil quenching time sensitivities to changes in heat transfer coefﬁcients are signiﬁcant, and they are one-to-one. These results emphasize the importance of process controls during quenching.

159

CHAPTER

COOKING

TIME FOR TURKEY IN AN OVEN

118 8

E

very year when Thanksgiving comes around, the question of a turkey’s cooking time in an oven looms. This is an unsteady-state conduction heat transfer problem in spherical coordinates, whose solution can be found in References [1], [6], [11], and [16]. The turkey is assumed to be stuffed and in a spherical shape in which the spatial variations of temperature are only in the radial direction. Turkey also has constant thermophysical and physical properties in space and in time. Energy balance in the radial direction, r, for a small increment of time, t, gives the following partial differential equation for the temperature-time distribution for a turkey: ∂2Tturkey /∂r2 + (2/r)∂Tturkey /∂r = (1/α)∂Tturkey /∂t

(18-1)

A closed form solution to Eq. (18-1) can be obtained by applying the following initial condition: Tturkey = Tturkey initial at time t = 0

(18-2)

and the following two boundary conditions, one at the center

161

Everyday Heat Transfer Problems

at radius r = 0, r(∂Tturkey /∂r) = 0

(18-3)

the other at the outer surface at radius r = R, − k(∂Tturkey /∂r) = h(Tturkey − Toven)

(18-4)

where α = (k/ρcp) is the thermal diffusivity of turkey in m2/s, k is the thermal conductivity of turkey in W/m-C, ρ is the density of turkey in kg/m3, cp is the specific heat of turkey at constant pressure in J/kg-C, and h is the convection heat transfer coefficient between the surface of the turkey and the oven environment in W/m2-C. The solution to Eqs. (18-1) through (18-4) can be found in References [1], [6], [11], and [16], and it is as follows: (Toven − Tturkey @ r)/(Toven − Tturkey initial) = (18-5)

where δn is the n’th positive root of the following transcendental equation: 1 − δn cot(δn) = Biot number = hR/k

(18-6)

The center temperature of the turkey is of particular interest for cooking. The USDA recommends that for a cooked turkey, the center of the stuffing should reach a temperature of 74°C. In the present sensitivity analysis, the temperature time history at the center of the bird is analyzed. Eq. (18-5) reduces to the following relationship at the center, r = 0, of the turkey:

162

Cooking Time For Turkey In An Oven

(Toven − Tturkey @ r=0)/(Toven − Tturkey initial) = (18-7)

There are eight independent variables that govern the present sensitivity analysis. They are: W = Weight of turkey in kg ρ = Density of turkey in kg/m3 k = Thermal conductivity of turkey in W/m-C cp = Specific heat of turkey at constant pressure in kJ/kg-C Tturkey initial = Initial temperature of turkey in C Tturkey final = Final temperature desired at the center, r = 0, of turkey in C Toven = Oven temperature in C h = Convection heat transfer coefficient in W/m2-C The radius, R, for the turkey is obtained from its weight and its spherical assumption, using R = (3W/4πρ)1/3. There are three non-dimensional variables that can capture all the temperature time distributions resulting from Eq. (18-7): The dependant variable is a non-dimensional temperature, Θ, namely Θ = (Toven − Tturkey @ r=0)/(Toven − Tturkey initial).

(18-8)

The independent variable is the dimensionless time, namely the Fourier number, Fo = αt/R2.

(18-9)

The dimensionless parameter is the Biot number, Bi = hR/k,

(18-10)

which is the ratio of internal to external thermal resistance for the turkey. Most solutions to unsteady-state heat transfer problems

163

Everyday Heat Transfer Problems

shown in literature, i.e., References [10] and [16], are shown in graphical form by using the non-dimensional variables given in Eqs. (18-8), (18-9) and (18-10). To find these solutions, five terms of Eq. (18-5) are utilized: Θ = C1 exp(−δ12Fo) + C2 exp(−δ22Fo) + C3 exp(−δ32Fo) (18-11) + C4 exp(−δ42Fo) + C5 exp(−δ52Fo) + … The five-term expansion solution in Eq (18-11) gives very accurate results. The fifth term only contributes at very small Fo numbers, i.e., contribution to Θ for Fo = 0.05 is less than 10−5. First the δ1, δ2, δ3, δ4, and δ5 have to be determined from the positive roots of the transcendental Eq. (18-6) for different Biot numbers. The ovens that will be considered in the present sensitivity analysis will have a convection heat transfer coefficient range of 1 to 60 W/m2-K. The turkeys will have a weight range of 2 to 20 kg. The turkey’s thermophysical properties are assumed to be ρ = 1000 kg/m3, cp = 3.33 kJ/kg-K, and k = 0.5 W/m-C. With these assumptions, the diameter of turkey ranges from 16 cm to 34 cm. The Biot number range will be 0.13 to 16. The temperature time history of interest for cooking the turkey falls in a time range between one and ten hours. For this cooking time range, the Fourier number range is 0.1 to 0.2. The temperature time history for the turkey can be presented by using the three non-dimensional variables in Eqs. (18-8), (18-9) and (18-10), as shown in Figure 18-1. A typical turkey is cooked at an oven temperature of 190°C, with an initial turkey temperature of 5°C and a final desired temperature at the center of the cooked turkey to be 80°C. These temperature values provide a region of interest for the dimensionless temperature Θ to be at 0.43. The present sensitivity analysis is for a 7.3 kg stuffed turkey with a 24 cm diameter. There are two variables that characterize the oven that is being used, oven temperature and the convection heat transfer coefficient on the surface of the turkey inside the oven. For a fixed oven temperature, Toven = 190.5°C , the cooking times can be obtained

164

Dimensionless Temperature At Center Of Turkey

Cooking Time For Turkey In An Oven

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1

Bi=0.13 Bi=0.5 Bi=1 Bi=2 Bi=3 Bi=5 Bi=7 Bi=10 0.2 0.3 0.4 Fourier Number, Fo

0.5

Bi=16

Figure 18-1 Dimensionless temperature at the center of a stuffed turkey

versus Fourier number for different Biot numbers in the region of interest for cooking a turkey

from Figure 18-1 for different convection heat transfer coefficients or Biot numbers. The cooking times versus the convection heat transfer coefficient for three different size turkeys are given in Figure 18-2. The results in Figure 18-2 are for a final desired temperature of 73.9°C at the center of the turkey. The USDA recommended cooking times for different size turkeys are also shown in Figure 18-2. USDA recommended cooking times require a good convection oven with a convection heat transfer coefficient ranging from 15 to 20 W/m2-C. As the convection heat transfer coefficient increases cooking time decreases, and this decrease in cooking time is very sensitive to changes in the convection heat transfer coefficients, especially in low values. The behaviors of these curves for different stuffed turkey weights are similar. The slopes of the curves in Figure 18-2 are given in Figure 18-3. Figure 18-3 shows that ∂(Cooking Time)/∂h decreases sharply as the forced convection heat transfer coefficient values approach natural convection heat transfer coefficient values. The other variable that is governed by the oven is oven temperature. The sensitivities

165

Cooking Time, hr

Everyday Heat Transfer Problems

4.5 kg (10 lb) stuffed turkey 7.3 kg (16 lb) stuffed turkey 10 kg (22 lb) stuffed turkey

15 14 13 12 11 10 9 8 7 6 5 4 3 2

USDA recommended cooking times: for a 4.5 kg (10 lb) stuffed turkey 3-3.5 hrs, for a 7.3 kg (16 lb) 0 5 10 15 20 stuffed turkey 4-4.5 hrs for a 10 kg (22 lb) Convection Heat Transfer Coefficient, W/m2-K stuffed turkey 4.75-5.25 hrs

Figure 18-2 Stuffed turkey cooking time as a function of the convection

∂(Cooking Time) / ∂h, hr-m2-C/W

heat transfer coefficient for stuffed turkeys of different weights for Toven = 190.5°C and for Tturkey final = 73.9°C

0 −0.1

W=4.5 kg (10 lb) W=7.3 kg (16 lb) W=10 kg (22 lb)

−0.2 −0.3 −0.4 −0.5 0

10

Convection Heat Transfer

20

30

Coefficient,W/m2-C

Figure 18-3 Stuffed turkey cooking time to convection heat transfer

coefficient sensitivity as a function of the convection heat transfer coefficient for stuffed turkeys of different weights, for Toven = 190.5°C, and for Tturkey final = 73.9°C

166

∂ (Cooking Time) / ∂ (Toven), min/C

Cooking Time For Turkey In An Oven

−0.8 −1

Turkey Center Final Temperature=73.9°C

−1.2

Turkey Center Final Temperature=79.4°C

−1.4

Turkey Center Final Temperature=85.0°C

−1.6 −1.8 180

190 200 210 Oven Temperature, °C

220

Figure 18-4 A 7.3 kg stuffed turkey cooking time to oven temperature

sensitivity versus oven temperature for different turkey center final temperatures

of the cooking time to oven temperature for different stuffed turkey centerline temperatures are given in Figure 18-4. The absolute value of the cooking time sensitivity to oven temperature decreases as the oven temperature increases. Sensitivity curves are similar in this application region for different final turkey center temperatures. The above sensitivity graphs show that the cooking time sensitivity to governing independent variables behave non-linearly. Therefore, it is more appropriate to analyze these sensitivities in the region of interest and rank them according to their effects on the cooking time. The results in Table 18-1 are obtained from Figure 18-1 by interpolating between appropriate non-dimensional temperatures, Biot numbers and Fourier numbers. The effects shown in Table 18-1 to cooking time are given in descending order. The most effective variable is the specific heat of the stuffed turkey at constant pressure. The cooking time sensitivities to all the independent variables are at the same order of magnitude.

167

Everyday Heat Transfer Problems

Table 18-1 Effects of a ±10% change in nominal values of variables to cooking time

Nominal Value

Cooking Time Change Due To A 10% Decrease In Nominal Value

Cooking Time Change Due To A 10% Decrease In Nominal Value

3.33 kJ/kg-C

−10%

+10%

Tturkey final @ r=0

73.9°C

−8.12%

+8.12%

Toven − Tturkey initial

186.1°C

+8.10%

−6.41%

Thermal Conductivity, k

0.5 W/m-C

+7.10%

−6.14%

7.3 kg

−6.02%

+6.02%

Convection Heat Transfer Coefficient, h

10 W/m2-C

+5.13%

−4.42%

Density, ρ

1000 kg/m3

−4.32%

+4.63%

Variable Specific Heat @ Constant Pressure, cp

Weight, W

Some variables, such as the turkey’s specific heat at constant pressure, final temperature desired at the center of turkey, and weight of the turkey behave linearly in the region of interest, and give the same magnitude cooking time change percentages on both sides of their nominal values. It is important to remember that the ranking order shown in Table 18-1 is good only in the region of nominal values for the present calculations, due to non-linear behavior of the sensitivities.

168

CHAPTER

HEAT

GENERATED IN PIPE FLOWS DUE TO FRICTION

19

H

eat generated in pipes or in oriﬁces due to ﬂuid friction in high-viscosity ﬂuids can be substantial. In the present analysis, the heat generated in steady-state and fully developed pipe ﬂows is investigated for ﬂuids of different viscosities. The pressure drop in a pipe due to ﬂuid friction, ∆P, is generally deﬁned as the product of the ﬂuid friction factor, f, non-dimensional length of the pipe, L/D, and kinetic energy of the ﬂuid, ρVm2/2, ﬂowing in it. ∆P = f (L/D) (ρVm2/2)

(19-1)

Eq. (19-1) has been determined by dimensional analysis (see Reference [15]). The friction factor is a function of Reynolds number, ReD = ρVmD/µ, and pipe inner surface roughness, e/D. The friction factors, f, for fully developed pipe ﬂows have been obtained experimentally for different Reynolds numbers and pipe inner surface roughness conditions, and they are given in References [6] and [10] as graphs, called Moody Diagrams, or as empirical equations.

169

Everyday Heat Transfer Problems

The pressure loss in the pipe is related to the heat generated in the pipe by the ﬁrst law of thermodynamics: Q = (VmA) ∆P

(19-2)

where ∆P is the pressure drop in pipe in N/m2, L is the length of pipe in m, D is the internal diameter of pipe in m, ρ is the density of ﬂuid in kg/m3, Vm is the average ﬂuid velocity in pipe in m/s, Q is heat generated due to friction in W, and A is cross sectional area of pipe in m2. The ﬁrst example analyzes a high-viscosity engine oil, and ﬂow rates are in the laminar ﬂow region; i.e., ReD < 2000. The friction factor for laminar ﬂow is given as follows, (see Reference [10]), and it is independent of pipe surface roughness: f = 64/ReD

(19-3)

Then the heat generated due to friction, after combining Eqs. (19-1), (19-2) and (19-3), becomes: Q = (128/π) µL (VmA)2/D4

(19-4)

Nominal values of the independent variables for this example are assumed to be as follows; the density of oil cancels out of the governing Eq. (19-4). L = 400 m, D = 0.1 m, VmA = 1000 liters/min (0.0167 m3/s), and µ = 0.486 N-s/m2 @ 27°C. Temperature effects on the kinematic viscosity of engine oil, µ/ρ, from 5°C to 55°C, are given in Figure 19-1. The kinematic viscosity of

170

Kinematic Viscosity, m2/s

Heat Generated In Pipe Flows Due To Friction

0.003

0.002

0.001

0.000

5

15

25 35 45 Mean Oil Temperature, °C

55

Figure 19-1 Kinematic viscosity of oil as a function of mean oil

temperature oil decreases as its temperature increases. Similar behavior is seen in Figure 19-2 for the heat generated due to friction. The mean temperature increase in oil behaves the same way as shown in Figure 19-3. ∆Tmean is calculated from Q/(ρVmAcp), assuming that all

Heat Generated, kW

150

100

50

0

5

15

25

35

45

55

Mean Oil Temperature, °C

Figure 19-2 Heat generated by friction for oil ﬂowing in a pipe

171

Temperature Increase, °C

Everyday Heat Transfer Problems

5 4 3 2 1 0

5

15

25 35 Mean Oil Temperature, °C

45

55

Figure 19-3 Temperature increase in oil due to friction

the heat generated due to ﬂuid friction goes to increase the temperature of oil. cp is the speciﬁc heat of oil at constant pressure, in J/kg-C. Sensitivities of heat, generated by oil ﬂowing in a pipe in the laminar ﬂow region, to the governing independent variables can be obtained by differentiating Eq. (19-4): ∂Q/∂L = (128/π) µ (VmA)2/D4

(19-5)

∂Q/∂µ = (128/π) L (VmA)2/D4

(19-6)

∂Q/∂D = (−512/π) µ L (VmA)2/D5

(19-7)

∂Q/∂(VmA) = (256/π) µ L (VmA)/D4

(19-8)

The sensitivity of heat generated to pipe internal diameter, Eq. (19-7), is shown in Figure 19-4. The sensitivity is prominent at low pipe internal diameters, and approaches zero fast as the pipe diameter increases for this case. The sensitivity of heat generated to pipe ﬂow rate, Eq. (19-8), is given in Figure 19.5. The sensitivity,

172

Heat Generated In Pipe Flows Due To Friction

∂ Q/∂ D, W/m

0.E+ 00

−1.E+ 07

−2.E+ 07

−3.E+ 07 0.05

0.07

0.09

0.11

0.13

0.15

Pipe Inside Diameter, m

Figure 19-4 Sensitivity of heat generated due to friction to pipe internal

diameter for fully developed laminar ﬂows

∂Q/∂(VmA) increases linearly with the increasing ﬂow rate. These sensitivities are calculated at a mean oil temperature of 27°C. When the nominal values of the independent variables given above are varied ±10%, the results in Table 19-1 are obtained. These heat generation sensitivities are given in descending order of importance, and they are applicable only around the nominal values assumed for this case. Heat generated due to friction is most sensitive to pipe internal diameter and to ﬂow rate. Heat generated due to friction has a one-to-one sensitivity relation to the length of the pipe and to ﬂuid viscosity. Similar sensitivity calculations are done for water, which has a three order of magnitude lower viscosity than the engine oil analyzed previously. Nominal values of the independent variables for this example are assumed to be as follows: L = 400 m, D = 0.1 m, VmA = 1000 liters/min (0.0167 m3/s), and µ = 0.855 × 10−3 N-s/m2 @ 27°C. ρ = 997 kg/m3 @ 27°C

173

Everyday Heat Transfer Problems

∂ Q/∂ (VmA), W-s/m3

1.5E+07

1.0E+07

5.0E+06

0.0E+00 0.01

0.03

0.05 Flow Rate, m3/s

0.07

0.09

Figure 19-5 Sensitivity of heat generated due to friction to pipe ﬂow rate for

fully developed laminar ﬂows

In this case friction factors are lower, ReD is higher in the order of 105, and this case falls into a steady-state and fully developed turbulent ﬂow region in a pipe. For the friction coefﬁcient in smooth pipes in the turbulent ﬂow region, the

Table 19-1 Effects of ±10% change in nominal values of variables to heat generated due to friction for steady-state and fully developed laminar ﬂow of engine oil in a pipe

Variable D VmA L µ @ 27°C

174

Nominal Value

Heat Generated Due To Friction Change For A 10% Decrease In Nominal Value

Heat Generated Due To Friction Change For A 10% Increase In Nominal Value

0.1 m

+52.42%

−31.70%

1000 L/min

−19%

+21%

400 m

−10%

+10%

−10%

+10%

0.486 N-s/m

2

Heat Generated In Pipe Flows Due To Friction

following empirical equation is used (see Reference [6]): f = (0.790 ln(ReD) − 1.64)−2 for 3000 < ReD < 5 × 106

(19-9)

Kinematic Viscosity, m2/s

The heat generated due to friction for fully developed turbulent ﬂow in a smooth pipe can be obtained by combining Eqs. (19-1), (19-2) and (19-9). Heat generated by water ﬂow in a pipe is analyzed between the mean water temperatures of 5°C and 55°C. The kinematic viscosity of water is three orders of magnitude less than the engine oil kinematic viscosity, as shown in Figure 19-6. Heat generated due to friction, and therefore the mean temperature increase for water, is two orders of magnitude less than the previous oil ﬂow case, as shown in Figures 19-7 and 19-8. When the nominal values of the variables given above are varied ±10%, the results in Table 19-2 are obtained. These heat generation sensitivities are given in descending order of importance, and they are applicable only around the nominal values assumed for this case. Heat generated due to friction is most sensitive to pipe internal diameter and then to the ﬂow rate, as in the previous case. A change in the

1.50E– 06 1.30E– 06 1.10E– 06 9.00E– 07 7.00E– 07 5.00E– 07

5

15

25

35

45

55

Mean Water Temperature, °C

Figure 19-6 Kinematic viscosity of water as a function of mean water

temperature

175

Everyday Heat Transfer Problems

Heat Generated, kW

2.6

2.4

2.2

2

5

15

25

35

45

55

Mean Water Temperature, °C

Figure 19-7 Heat generated by friction for fully developed and turbulent

water ﬂow in a pipe

Temperature Increase, °C

viscosity of water is the least effective independent variable to heat generated due to friction. Changes in pipe inner surface roughness also effect the heat generated due to friction in a pipe. The effects of pipe inner surface

0.038 0.036 0.034 0.032 0.030 0.028

5

15

25

35

45

55

Mean Water Temperature, °C

Figure 19-8 Temperature increase in mean water temperature due to friction

176

Heat Generated In Pipe Flows Due To Friction

Table 19-2 Effects of ±10% change in nominal values of variables to heat generated due to friction for a steady-state and fully developed turbulent ﬂow of water in a smooth pipe Heat Generated Heat Generated Due To Friction Due To Friction Change For A 10% Change For A 10% Decrease In Increase In Nominal Value Nominal Value

Nominal Value

Variable D

0.1 m

+65.95%

−36.75%

1000 L/min

−25.60%

+30.67%

L

400 m

−10%

+10%

ρ

997 kg/m

−8.15%

+7.99%

−2.02%

+1.86%

VmA

3

µ @ 27°C 0.855 × 10 N-s/m

% Increase in Heat Generated

-3

2

160 120 80 40 0

0

200 400 600 800 Pipe Inner Surface Roughness, µm

1000

Figure 19-9 Percent increase in heat generated due to friction as a

function of pipe inner surface roughness for water ﬂowing in a pipe, ReD = 2.47 × 10+5

177

Everyday Heat Transfer Problems

roughness are analyzed for ReD = 2.47 × 10+5 using the Moody Diagram (see Reference [10]). The results are given in Figure 19-9 for steady-state, fully developed, and turbulent water ﬂowing in a pipe. The effect of pipe inner surface roughness to heat generated due to friction increases linearly above e = 100 µm. A pipe with an inner surface roughness of 100 µm generates 38% more heat than a smooth one.

178

CHAPTER

20 ACTIVE SOLAR SIZING AN

COLLECTOR FOR A POOL S

olar collectors used to heat water have been improving steadily in solar energy conversion efﬁciency as the demand to go to greener energy sources increases. To be able to design a pool solar collector system, a good knowledge of year-around environmental conditions has to be gathered. Average daily incident solar radiation, average daily daytime and nighttime environment temperatures, and average daily daytime and nighttime relative humidity must be known. Sizing an active (directly circulating the pool water) solar collector to heat a pool's water requires knowledge of the average solar insolation at the location of the collector, and the heat transfer efﬁciency of the solar collector. A pool's physical and thermophysical properties must be known to deal with radiation heat transfer, convection heat transfer, and evaporative cooling from the water's surface during the day and night. Conduction heat transfer from the pool's structure to the earth also has to be treated. A plastic pool surface cover will be considered in order to minimize heat losses to the environment when the pool is not in use. Therefore, a plastic cover's physical and thermophysical properties must also be known.

179

Everyday Heat Transfer Problems

The present heat transfer model for the pool solar collector system neglects temperature gradients in pool water. The model is for unsteady-state heat transfer in one-dimensional rectangular coordinates. There are 25 independent variables that govern the dependent variable, size of the solar collector, and are considered for sensitivity analysis. The present heat transfer model has the following assumptions: The pool is covered at night for 14 hours and open for swimming during the day for ten hours. Energy lost from the water in the solar collector at night and energy lost in piping between the pool and the solar collector are considered to be lumped into the solar collector efﬁciency. Average heat transfer properties are used during both day and night hours. When a pool cover is used at night, there is no evaporation from the pool's surface. The following energy balances can be constructed for the water in the pool for the day-time and night-time hours. Energy balance during the day for the pool: Qin solar collector + Qin convection from environment during day − Qout evaporation during day + Qin net radiation on pool surface − Qout conduction to earth (20-1) = mw cpw (dTpool/dθ) Energy balance during the night for the pool with a cover: − Qout convection to environment at night − Qout conduction to earth − Qout radiation from pool cover surface = mw cpw (dTpool/dθ)

(20-2)

Energy balance during the night for the pool without a cover: − Qout convection to environment at night − Qout conduction to earth − Qout radiation from pool surface − Qout evaporation at night = mw cpw (dTpool/dθ)

(20-3)

where mw is weight of water in the pool in kg, cpw is speciﬁc heat of water at constant pressure in W-hr/kg-C, Tpool is average pool temperature in C, and θ is time in hours.

180

Sizing An Active Solar Collector For A Pool

Each term in energy balance Eqs. (20-1), (20-2) and (20-3) is deﬁned as follows. Net solar energy the water receives while circulating in the solar collector is: Qin solar collector = η q Ac

(20-4)

where η is the solar collector heat transfer efﬁciency, or the percentage of solar insolation that can be converted to heating the pool water by the collector and its connecting pipes, which can vary between 0.4 and 0.7. q is the average solar insolation on the collector in W/m2. Ac is the area of the solar collector in m2. Convection heat transferred from the environment to the water surface during the day is: Qin convection from environment during day = hday Apool water surface (Tenvironment during day − Tpool)

(20-5)

where hday is the average convection heat transfer coefﬁcient between pool water surface and environment during the day, Apool water surface is pool water surface area (pool length times pool width in m2) and Tenvironment during day is average environmental temperature during ten hours of day in C. Heat lost from the water due to evaporation to the environment during the day is: Qout evaporation during day = M Hfg

(20-6)

where M is evaporation rate in kg/hr-m2 and Hfg is latent heat of evaporation for water in air in W-hr/kg. The mass transfer rate equation can be deﬁned similar to the convection heat transfer rate equation. By applying the perfect gas law, the mass transfer rate equation can be written as follows: M = (hdiffusion day/R Tpool )(pw − pa)

(20-7)

181

Everyday Heat Transfer Problems

By assuming thermal diffusivity of air, α in m2/hr, being equal to the diffusion coefﬁcient of water in air, D in m2/hr, and by assuming the water in air concentration gradient at the water-air interface to be the same as the temperature gradient—Nusselt number (hday L/ka) = Sherwood number (hdiffusion day L/D), the mass transfer coefﬁcient can be obtained from the heat transfer coefﬁcient (see Reference [2]): hdiffusion day = hday/ρa cpa

(20-8)

hdiffusion day is the mass transfer coefﬁcient between pool water surface and air in m/hr, R is the universal gas constant which is equal to 0.08205 m3-atm/kmol-K, pw is the saturation pressure of water vapor on pool water surface at pool temperature, pa is partial pressure of water vapor in air at average environmental temperature (calculated from pa = φpsaturation, where φ is relative humidity in the air and psaturation is the saturation pressure of water vapor in air at average environmental temperature), L is a characteristic length of pool surface, namely pool length, ka is thermal conductivity of air in W/m-K, ρa is density of air in kg/m3, and cpa is speciﬁc heat of air at constant pressure in W-hr/kg-C. The net solar radiation that is absorbed by water is: Qin net radiation on pool surface = αw G Apool water surface

(20-9)

where αw is average solar radiation absorptivity at water surface and G is average solar insolation on pool water surface in W/m2, that includes both direct and diffused solar energy. Conduction heat transfer from the walls of the pool to earth is: Qout conduction to earth = (kpool wall Apool-earth surfaces/tpool wall)(Tpool − Tearth)

(20-10)

where kpool wall is thermal conductivity of pool walls in W/m-K, Apool-earth surfaces are surface areas between the pool and earth in m2, deﬁned as [(2 × pool length + 2 × pool width) × pool water height + pool length × pool width], tpool wall is average pool wall thickness in m, and Tearth is average earth temperature.

182

Sizing An Active Solar Collector For A Pool

Heat lost during the night from the pool with a cover to the environment can be written as a series thermal circuit of convection and conduction heat transfer: Qout convection to environment at night = Apool water surface (Tpool − Tenvironment during night)/[(1/hnight ) + (tcover /kcover)] (20-11) where Tenvironment during night is average environmental temperature at night in C, hnight is the average convection heat transfer coefﬁcient at night integrated over the length of the pool, given by the following empirical relationship (see Reference [6]), in W/m2-K: (hnight L/ka) = (0.037 ReL0.8 − 871) Pr0.333 for 0.6 < Pr < 60, 5 × 105 < ReL < 1 × 108, and Recritical = 5 × 105 (20-12) where ReL = VL/ν is Reynolds number, with V being average air speed in m/hr over the length of the pool L in m and ν being kinematic viscosity of air in m2/hr at ﬁlm temperature, the average of water surface and environmental temperature. Pr = ν/αt is the Prandtl number, where αt is thermal diffusivity of air in m2/hr at ﬁlm temperature, the average of surface and environmental temperature. Recritical is the transitional Reynolds number from laminar to turbulent ﬂow on a ﬂat plate. tcover is thickness of pool cover in meters and kcover is thermal conductivity of pool cover at pool temperature. Heat radiated from the pool cover surface to the night sky is: Qout radiation from pool cover surface = εpool cover surface σT4pool cover surface

(20-13)

where εpool cover surface is emissivity of pool cover surface and σ the is Stefan-Boltzmann constant, namely 5.67 × 10−8 W/m2-K4. Energy balance during the night for the pool without a cover has convection and radiation heat transfers deﬁned as follows:

183

Everyday Heat Transfer Problems

Qout convection to environment at night = hnight Apool water surface (Tenvironment during night − Tpool) (20-14) Qout radiation from pool surface = εpool water surface σT4pool

(20-15)

where εpool water surface is emissivity of pool water surface. Qout evaporation at night is calculated the same way as Qout evaporation during day except that hdiffusion night is calculated from the heat transfer coefﬁcient integrated over the length of the pool; Eq. (20-12), using an analogy between heat and mass transfer. (hdiffusion night L/D) = (0.037 ReL0.8 − 871) Sc0.333 for 0.6 < Sc < 3000, 5 × 105 < ReL < 1 × 108, and Recritical = 5 × 105 (20-16) where D is the diffusion coefﬁcient for water in air in m2/hr and Sc is the Schmidt number deﬁned as ν/D. Governing Eqs. (20-1) and (20-2) are solved by explicit ﬁnite difference method and by iteration, to determine the required solar collector area that will heat the pool water to a minimum of 25°C in two days, speciﬁcally one ten-hour day followed by a 14-hour night, followed by another ten-hour day. Nominal conditions for the independent variables are assumed to be as follows: Solar collector variables: η = 0.7 q = 500 W/m2 Pool variables: L = 50 m (pool length) W = 25 m (pool width) H = 2 m (pool water height) G = 350 W/m2 αw = 0.96 εpool water surface = 0.96 ρw = 1000 kg/m3

184

Sizing An Active Solar Collector For A Pool

cpw = 1.162 W-hr/kg-K kpool wall = 0.7 W/m-K tpool wall = 0.1 m (average wall thickness) kcover = 0.04 W/m-K tcover = 0.02 m εpool cover surface = 0.1 Tpool initial = 20°C (pool water temperature at θ = 0) Environmental variables: Tenvironment during day = 30°C (average environmental temperature during the ten-hour day) Tenvironment during night = 15°C (average environmental temperature during the fourteen-hour night) Tearth = 20°C (average earth temperature around pool walls) Other properties during the day: hday = 2 W/m2-K ρa = 1.1614 kg/m3 cpa = 0.28 W-hr/kg-K φ = 40% (relative humidity of air during the day) pw = saturation pressure of water vapor on pool water surface at pool temperature is obtained from dry saturated steam temperature tables (see Reference [8]). psaturation = saturation pressure of water vapor in air at average environmental temperature is obtained from dry saturated steam temperature tables (see Reference [8]). Other properties at night: D = 0.0936 m2/hr ν = 0.0572 m2/hr Pr = 0.707 ka = 0.0263 W/m-K V = 3600 m/hr φ = 40% (relative humidity of air during the night) pw = saturation pressure of water vapor on pool water surface at pool temperature is obtained from dry saturated steam temperature tables (see Reference [8]).

185

Everyday Heat Transfer Problems

psaturation = saturation pressure of water vapor in air at average environmental temperature is obtained from dry saturated steam temperature tables (see Reference [8]). Under these nominal conditions, and with a pool cover at night, a solar collector of 1192 m2 (95% of the pool surface area) is required to heat the pool water from the initial temperature of 20°C to 25°C in two days. Without a pool cover at night, the solar collector area requirement almost doubles to 2054.5 m2. Radiation heat loss to night sky and evaporation heat loss to environment cause the required collector size to double without a pool cover at night. Solar collector efﬁciency and average solar insolation are the two dominant variables that affect similarly the required solar collector area. Required solar collector area is very sensitive to the product of solar collector efﬁciency and the average solar insolation. Solar collector area versus its efﬁciency is shown in Figure 20-1. The sensitivity of solar collector area to its efﬁciency is shown in Figure 20-2. Sensitivity of required solar collector area to these variables increases fast as collector efﬁciency and solar insolation values decrease.

Collector Area, m2

8000 6000 4000 2000 0

0

0.2

0.4 0.6 Collector Efficiency

0.8

1

Figure 20-1 Solar collector area versus solar collector efﬁciency at average

solar insolation of 500 W/m2

186

∂ (Collector Area) / ∂ η, m2

Sizing An Active Solar Collector For A Pool

0

−10000

−20000

−30000 0.2

0.4

0.6

0.8

1

Collector Efficiency, η

Figure 20-2 Sensitivity of solar collector area to collector efﬁciency at

average solar insolation of 500 W/m2

Initial pool water temperature, average environmental temperature during the day, and the earth's temperature under the pool have signiﬁcant effects on sizing the solar collector area. These effects are shown in Figures 20-3, 20-4 and 20-5, respectively, and their behaviors

Collector Area, m2

3000

2000

1000

0

15

17

19

21

23

Pool Water Initial Temperature, °C

Figure 20-3 Solar collector area versus pool water initial temperature

187

Everyday Heat Transfer Problems

Collector Area, m2

1250

1220

1190

1160

25 27 29 31 33 35 Average Environment Temperature During The Day, °C

Figure 20-4 Solar collector area versus average environmental temperature

during the day

Collector Area, m2

2000

1500

1000

500

5

10

15

20

25

Average Earth Temperature Under The Pool, °C

Figure 20-5 Solar collector area versus average earth temperature under

the pool

188

Sizing An Active Solar Collector For A Pool

Collector Area, m2

1500 1300 1100 900 700

30

35

40

45 50 Pool Length, m

55

60

Figure 20-6 Solar collector area versus pool length

are almost linear. Required solar collector area decreases 384 m2 per degree centigrade increase in the initial pool water temperature. Similarly, required solar collector area decreases at a much smaller rate, −7.2 m2, per degree centigrade increase in average environmental

Collector Area, m2

2000 1600 1200 800 400

10

15

20

25

30

35

40

Pool Width, m

Figure 20-7 Solar collector area versus pool width

189

Everyday Heat Transfer Problems

Collector Area, m2

2000 1600 1200 800 400 0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Pool Water Height, m

Figure 20-8 Solar collector area versus pool water height

temperature during the day. Required solar collector area sensitivity to average earth temperature under the pool is −52.7 m2/C. Another set of variables signiﬁcantly affecting required solar collector area are the pool's physical dimensions. Sensitivities for pool length, width and water height are shown in Figures 20-6, 20-7 and 20-8, respectively. These sensitivities are constants. A 23.7 m2 increase in solar collector area is required for every meter of increase in pool length, a 47 m2 increase in solar collector area is required for every meter of increase in pool width, and a 1049 m2 increase in solar collector area is required for every meter of increase in pool water height. Solar insolation on the pool water surface during the day also affects required solar collector area signiﬁcantly. The effects on required solar collector area of average solar insolation on pool water surface, and of water absorptivity, are depicted in Figures 20-9 and 20-10, respectively. Required solar collector area decreases with increasing average solar insolation on pool water surface by 3.4 m4/W. Required solar collector area decreases 12.5 m2 per 1% increase in water surface absorptivity of incoming solar radiation. Relative humidity also has an effect on sizing the solar collector. This effect is compared in Figure 20-11 for a pool with and without

190

Sizing An Active Solar Collector For A Pool

Collector Area, m2

2000

1500

1000

500 200

250

300

350

400

450

500

Average Solar Insolation On Pool Surface, W/m2

Figure 20-9 Solar collector area versus average solar insolation on pool

surface

a cover at night. Required solar collector area decreases 4.8 m2 for every percent increase in relative humidity for a pool that is covered at night. The sensitivity for a pool without a cover at night is a 6.5 m2 collector area decrease per 1% increase in relative humidity. This

Collector Area, m2

1800 1600 1400 1200 1000 0.5

0.6 0.7 0.8 0.9 Absorptivity Of Pool Water Surface

1

Figure 20-10 Solar collector area versus absorptivity of pool water surface

191

Everyday Heat Transfer Problems

Collector Area, m2

2800 Without Cover At Night

2300 1800

With Cover At Night

1300 800

0

0.2

0.4

0.6

0.8

1

Relative Humidity

Figure 20-11 Solar collector area versus relative humidity

higher sensitivity comes from higher evaporation heat loss to the cold night atmosphere. If a region with a relative humidity of 10% is considered to build a solar heated pool, as compared to a region with a relative humidity of 60%, with everything else being the same and considering a pool cover at night, the dry region requires 240 m2 more solar collector area because more heat is lost due to evaporation to the dry atmosphere. There are over 25 independent variables that govern this heat transfer problem. When the nominal values of the variables given above are varied ±10%, the results shown in Table 20-1 are obtained for a pool that is covered at night. Required solar collector area sensitivities to a ±10% change in the governing variables are given in descending order of importance, and they are applicable only around the nominal values assumed for this study. Required solar collector area is most sensitive to changes in initial pool water temperature, followed by pool water height and pool water speciﬁc heat at constant pressure. The next set of variables in order of sensitivity are solar collector efﬁciency, average solar insolation on collector, average solar insolation on pool water surface, average solar

192

Sizing An Active Solar Collector For A Pool

Table 20-1 Effects of a ±10% change in nominal values of variables to required solar collector area for a pool covered at night (Note: Effects less than 1% are not included in Table 20.1)

Variable

Nominal Value

Change In Required Solar Collector Area For A 10% Decrease In Nominal Value

Tpool initial

20°C

+64.35%

−64.35%

H, pool water height

2m

−17.53%

+17.62%

1.162 W-hr/kg-K

−17.37%

+17.45%

η, solar collector efﬁciency

0.7

+11.16%

−9.06%

q, average solar insolation on collector

500 W/m2

+11.16%

−9.06%

G, average solar insolation on pool water surface

350 W/m2

+10.07%

−10.07%

cpw

Change In Required Solar Collector Area For A 10% Increase In Nominal Value

αw, average solar radiation absorptivity at water surface

0.96

+10.07%

−4.19% with only a possible 4.2% increase to 1

L, pool length

50 m

−9.90%

+9.98%

W, pool width

25 m

−9.82%

+9.90%

Tearth

20°C

+8.89%

−8.81%

Tenvironment during day

30°C

+1.85%

−1.76%

φ, relative humidity

0.40

+1.60%

−1.59%

tpool wall

0.1 m

+1.26%

−1.01%

kpool wall

0.7 W/m-K

−1.09%

+1.17%

193

Everyday Heat Transfer Problems

radiation absorptivity at water surface, pool length, pool width, and average earth temperature around pool walls. The sensitivities then drop an order of magnitude to less than 2%. The average environmental temperature during the day continues in the order of sensitivity, followed by relative humidity, average pool wall thickness, and pool wall thermal conductivity. The rest of the independent variables have less than 1% sensitivities on the required solar collector area. These variables, in order of descending sensitivity, are the emissivity of the pool cover, temperature of the environment at night, the convection heat transfer coefﬁcient during the day, thermal conductivity of pool cover, thickness of pool cover, mean air speed over the pool at night, and thermophysical properties of air.

194

CHAPTER

HEAT

TRANSFER IN A HEAT EXCHANGER

21

T

he design of a heat exchanger can be a good challenge for engineers. The design methods cover a vast variety of engineering disciplines, such as heat transfer, ﬂuid mechanics, stress analysis, corrosion, materials, economics, etc. There are two popular heat transfer design methods covered in literature; see References [5], [6], [10] and [15]. One heat transfer design method is called the log mean temperature difference method. If the hot and cold ﬂuid inlet and/or outlet temperatures are speciﬁed, the heat exchanger design can be performed by using this method. On the other hand, if the hot and cold ﬂuid inlet and/or outlet temperatures are not speciﬁed, and a comparison between various types of heat exchangers is required, then the effectiveness method is preferred. In this chapter, the sensitivity of a heat exchanger design to governing independent variables is analyzed. Heat transfer in a counterﬂow concentric pipe liquid-to-liquid heat exchanger is considered using the log mean temperature difference design method. Hot engine oil ﬂows in the inner pipe and the cold water ﬂows in the outer pipe. Under steady-state conditions, by neglecting heat

195

Everyday Heat Transfer Problems

transfer to the environment, and assuming constant thermophysical properties, the heat transferred through a small element of the heat exchanger from the hot oil equals the heat received by the cold water. dQ = −ρhVmhAh cph dTh = ρcVmcAc cpc dTc

(21-1)

Heat transferred through a small element of the heat exchanger in Eq. (21-1) can also be expressed in terms of an overall heat transfer coefﬁcient between the hot ﬂuid and the cold ﬂuid, through the separating wall: dQ = U (Th − Tc) dAwall

(21-2)

Eqs. (21-1) and (21-2) can be combined to eliminate dQ, and then integrated between the inlet and outlet temperatures of the heat exchanger to give the following log mean temperature difference design method equations for a counterﬂow heat exchanger:

Q=

Q = ρhVmhAh cph (Th in − Th out)

(21-3)

Q = ρcVmcAc cpc (Tc in − Tc out)

(21-4)

UAwall [(Th out − Tc in) − (Th in − Tc out)] ln[(Th out − Tc in)/(Th in − Tc out)]

(21-5)

where UAwall = UhAwall hot ﬂuid side = UcAwall cold ﬂuid side

(21-6)

and Awall hot ﬂuid side represents inside surface area of the inner tube and Awall cold ﬂuid side represents outside surface area of the inner tube. The overall heat transfer coefﬁcient from the hot ﬂuid side is: Uh = 1/[(1/hh) + Rh foul + (Dh ln(Dc/Dh)/2kss) + (Awall hot ﬂuid side Rc foul/Awall cold ﬂuid side) + (Awall hot ﬂuid side/Awall cold ﬂuid sidehc)]

196

(21-7)

Heat Transfer In A Heat Exchanger

All the variables in these governing Eqs., (21-1) through (21-7), are deﬁned as follows: ρh = Hot oil density in kg/m3 VmhAh = Hot oil ﬂow rate in liters/minute cph = Hot oil speciﬁc heat at constant pressure in J-kg/C Th in = Hot oil inlet temperature to the heat exchanger in C Th out = Hot oil outlet temperature from the heat exchanger in C ρc = Cold water density in kg/m3 VmcAc = Cold water ﬂow rate in liters/minute cpc = Cold water speciﬁc heat at constant pressure in J-kg/C Tc in = Cold water inlet temperature to the heat exchanger in C Tc out = Cold water outlet temperature from the heat exchanger in C Uh = Overall heat transfer coefﬁcient based on the inside surface of the inner tube in W/m2-C Awall hot ﬂuid side = Inside surface area of the inner tube, πDhL, in m2 Uc = Overall heat transfer coefﬁcient based on the outside surface of the inner tube in W/m2-C Awall cold ﬂuid side = Outside surface area of the inner tube, π(Dh + 2t)L = πDcL, in m2 Dh = Inside diameter of the inner tube in m Dc = Outside diameter of the inner tube, Dh + 2t, in m, and where t is the thickness of the inner tube kss = Wall tube material thermal conductivity in W/m-C hh = Convection heat transfer coefﬁcient between the hot oil and the inner tube inside surface in W/m2-C hc = Convection heat transfer coefﬁcient between the cold water and the inner tube outside surface in W/m2-C Rh foul = Fouling resistance for the inside surface of the inner tube in m2-C/W Rc foul = Fouling resistance for the outside surface of the inner tube in m2-C/W In a heat exchanger, fouling resistances affecting heat transfer on surfaces of walls are caused by corrosion or by foreign material deposited over time, and they are determined by experimental methods during heat exchanger life tests.

197

Everyday Heat Transfer Problems

In order to be able to determine the convection heat transfer coefﬁcients, ﬁrst the Reynolds numbers for inner and outer tubes have to be determined. For the inner tube the Reynolds number deﬁnition is given by ReDinner tube = ρhVmhDh/µh , where µh is the viscosity of hot oil in N-s/m2. For the outer tube the Reynolds number deﬁnition is given by ReDouter tube = ρcVmcDc-hydraulic/µc , where µc is the viscosity of cold water in N-s/m2. Dc-hydraulic is the hydraulic diameter of the outer tube given as Dc-hydraulic = 4 ﬂow cross-sectional area/ﬂow wetted perimeter, or Dc-hydraulic = Dinside diameter of outer tube − Dc in meters. The nominal values for the independent variables of the present heat exchanger sensitivity analysis are assumed to be as follows: For hot oil variables: VmhAh = 5 liters/minute Th in = 70°C Th out = 40°C ρh = 870 kg/m3 cph = 2000 J-kg/C µh = 0.1 N-s/m2 kh = 0.14 W/m-C For cold water variables: VmcAc = 25 liters/minute Tc in = 20°C ρc = 1000 kg/m3 cpc = 41.8 J-kg/C µc = 0.00096 N-s/m2 kc = 0.6 W/m-C For geometry and other variables: Dh = 0.02 m t = 0.001 m Douter tube = 0.025 m kss = 15 W/m-C Rh foul = 0.0008 m2-C/W Rc foul = 0.0002 m2-C/W

198

Heat Transfer In A Heat Exchanger

ReDinner tube is calculated to be 46, which is less than 2000, and therefore the hot oil ﬂows in the laminar region. It is assumed that the laminar ﬂow is fully developed and has constant heat ﬂux at the wall. Therefore the Nusselt number, NuDh, is a constant (see Reference [6]): NuDh = (hhDh)/kh = 4.36

(21-8)

ReDouter tube is calculated to be 11,758, which is greater than 4000, and therefore the cold water ﬂows in the turbulent ﬂow region. It is assumed that the turbulent ﬂow is fully developed and the Nusselt number, NuDc, is determined from an empirical relationship (see Reference [6]): NuDc = (hcDc-hydraulic)/kc = 0.023 (ReDouter tube)0.8 Prc0.4

(21-9)

Heat Exchanger Length, m

where Prc is the Prandtl number, the ratio of momentum to thermal energy diffusion, deﬁned as (µc/ρc)/(kc/ρccpc) or (µccpc/kc). µc is the viscosity of cold ﬂuid and kc is the thermal conductivity of cold ﬂuid. The counterﬂow heat exchanger length is considered as the main dependent variable. Heat exchanger length versus hot oil ﬂow rate is given in Figure 21-1. The behavior is linear since the hot oil is ﬂowing

300

200

100

0

0

5

10

15

Hot Oil Flow Rate, L/min

Figure 21-1 Heat exchanger length versus hot oil ﬂow rate

199

Length of Counterflow Heat Exchanger, m

Everyday Heat Transfer Problems

160 120 80 40 0

30

35

40

45

50

Hot Oil Outlet Temperature, °C

Figure 21-2 Heat exchanger length versus hot oil outlet temperature for

Length of Counterflow Heat Exchanger, m

Th in = 70°C and Tc in = 20°C

300

200

100

0

0

20 30 10 Cold Water Inlet Temperature, °C

40

Figure 21-3 Heat exchanger length versus cold water inlet temperature for

Th in = 70°C and Th out = 40°C

200

Heat Transfer In A Heat Exchanger

Table 21-1 Effects of a ±10% change in nominal values of independent variables to heat exchanger length for a concentric tube counterﬂow heat exchanger

Variable Th out

Nominal Value

Heat Exchanger Length Change For A 10% Decrease In Nominal Value

Heat Exchanger Length Change For A 10% Increase In Nominal Value

40°C

+24.98%

−20.27%

70°C

−16.78%

+14.65%

0.14 W/m-C

+10.74%

−8.79%

5 liters/minute

−10.27%

+10.33%

ρh

870 kg/m

−10.27%

+10.33%

cph

2000 J-kg/C

−10.27%

+10.33%

Tc in

20°C

−6.25%

+7.21%

Dh

0.02 m

+0.448%

−0.377%

25 liters/minute

+0.347%

−0.282%

ρc

1000 kg/m

3

+0.347%

−0.282%

cpc

41.8 J-kg/C

+0.341%

−0.277%

0.0008 m -C/W

−0.235%

+0.236%

0.025 m

−0.116%

+0.126%

0.0002 m -C/W

−0.049%

+0.049%

15 W/m-C

+0.040%

−0.033%

t

0.001 m

+0.013%

−0.014%

kc

0.6 W/m-C

+0.0090%

−0.0077%

µc

0.00096 N-s/m

−0.0057%

+0.0054%

µh

0.1 N-s/m2

0%

0%

Th in kh VmhAh

VmcAc

Rh foul Douter tube Rc foul kss

3

2

2

2

201

Everyday Heat Transfer Problems

in the laminar ﬂow region and the convection heat transfer coefﬁcient is independent of ReDinner tube. Hot oil inlet and outlet temperatures, and cold water inlet temperature, have dominant effects on the heat exchanger length. Heat exchanger length as a function of hot oil outlet temperature is given in Figure 21-2. The heat exchanger length decreases logarithmically as the requirement for the hot oil outlet temperature increases. Heat exchanger length is also a strong function of cold water inlet temperature, and is shown in Figure 21-3. The heat exchanger length increases as the cold water inlet temperature increases, and the length increase behaves exponentially as the required hot oil outlet temperature is approached. When the nominal values of the independent variables given above are varied ±10%, the results shown in Table 21-1 are obtained. These heat exchanger length sensitivities are given in descending order of importance, and they are applicable only around the nominal values assumed for this case study. Heat exchanger length is most sensitive to hot oil outlet and inlet temperature requirements. Hot oil ﬂow rate and hot oil thermophysical properties also affect the heat exchanger length at the same order of magnitude as the hot oil outlet and inlet temperatures. From the cold water variables, the heat exchanger length is most sensitive to the cold water inlet temperature. The sensitivities of the heat exchanger length to the rest of the independent variables diminish fast in the present region of application. The heat exchanger length is least sensitive to changes in fouling resistances, inside diameter of the outer tube, thickness of the inner tube, thermal conductivity of the tube wall material, viscosity of water, and thermal conductivity of water.

202

CHAPTER

I CE

FORMATION ON A LAKE

22

H

eat transfer during ice formation has been studied in detail, in papers by Lin, S. and Z. Jiang (Reference [13]), and by London, A. L. and R. A. Seban (Reference [14]). In the present sensitivity study, planar ice formation is considered. References [13] and [14] also treat ice formation in cylindrical and spherical shapes. In the present heat transfer analysis, thermal capacitance of ice is neglected; in other words, a linear temperature proﬁle through the ice thickness is assumed. Ice thermophysical properties are assumed to be constants and are considered at sea level conditions. Lake water is assumed to be fresh. For proper ice formation heat transfer modeling, lake water temperature is assumed to be greater than zero degrees centigrade, and the atmosphere’s temperature is assumed to be less than zero degrees centigrade: Tlake water > Tfreezing > Tatmosphere . Heat transfer from the lake water, through the ice layer, to the atmosphere is assumed to be in an unsteady state and in

203

Everyday Heat Transfer Problems

one-dimension rectangular coordinates. Governing heat transfer rate equations are as follows: Convection heat transfer from the lake water to the growing ice surface is: Qlake water to growing ice layer = hlake water-growing ice surface (Tlake water − Tfreezing) (22-1) Heat transfer from the growing ice layer to the atmosphere is: Qgrowing ice layer to atmosphere = (Tfreezing − Tatmosphere)/[(1/hice upper surface-atmosphere) + (x/kice)] (22-2) Energy balance at the growing ice layer for the latent heat of fusion required per unit area of ice layer formation is: ρiceHice (dx/dt) = Qgrowing ice layer to atmosphere − Qlake water to growing ice layer

(22-3)

T represents temperature in C. h represents the convection heat transfer coefﬁcient in W/m2-C. kice is the thermal conductivity of ice in W/m-C. ρice is the density of ice in kg/m3. Hice is the latent heat of fusion for water in W-hr/kg. x is the ice thickness in meters at time, t, in hours. The governing differential Eq. (22-3) can be rewritten after separating the time and the space variables in the following non-dimensional form: dt* = (1 + x*)dx*/[1 − Z(1 + x*)]

(22-4)

where the non-dimensional time is t* = [h2ice upper surface-atmosphere (Tfreezing − Tatmosphere) t/(ρice Hice kice)] (22-5) where the non-dimensional ice thickness is x* = hice upper surface-atmosphere x/kice

204

(22-6)

Ice Formation On A Lake

and where Z is a non-dimensional parameter which is the ratio of two convection heat transfer mechanisms that govern this heat transfer problem: Z=

[hlake water-growing ice surface × (Tlake water − Tfreezing)] [hice upper surface-atmosphere × (Tfreezing − Tatmosphere)]

(22-7)

Eq. (22-4) is integrated using the initial condition x* = 0 at t* = 0

(22-8)

to obtain the following ice formation rate equation: t* = (1/Z2) ln[(Z − 1)/(Z(x* + 1) − 1)] − (x*/Z)

(22-9)

Nominal values of the independent variables used in the present sensitivity analysis are as follows: hice upper surface-atmosphere = 30 W/m2-C hlake water-growing ice surface = 10 W/m2-C Tlake water = 6°C Tatmosphere = −20°C Tfreezing = 0°C ρice = 920 kg/m3 kice = 1.88 W/m-C Hice = 93 W-hr/kg The time to form a certain thickness of ice under the nominal conditions given above is shown in Figure 22-1. The ice thickness limit under the above given nominal conditions that will support the latent heat of fusion required for ice formation is 0.564 m. Maximum ice thickness versus the convection heat transfer coefﬁcient between ice upper surface and atmosphere is given in Figure 22-2. This condition occurs when the convection heat transfer from the upper surface of the ice to the atmosphere starts to reach the same level as the convection heat transfer from the lake water to

205

Everyday Heat Transfer Problems

Ice Thickness Limit = 0.564 m Under Given Nominal Conditions Time To Form Ice, hr

2000 1500 1000 500 0

0

0.1

0.2 0.3 0.4 Thickness Of Ice, m

0.5

0.6

Figure 22-1 Time to form ice versus thickness

Maximum Ice Thickness, m

the growing ice surface. Then there is no more energy left to support the latent heat of fusion of water to keep forming the ice layer. Under this condition, the natural logarithm term in the denominator of the ice formation rate Eq. (22-9) approaches zero, or: 0.6

0.4

0.2

0

0

10

20

30

40

50

Heat Transfer Coefficient, hice upper surface-atmosphere, W/m2-C

Figure 22-2 Maximum ice thickness versus heat transfer coefﬁcient

between ice upper surface and atmosphere

206

Ice Formation On A Lake

x* = (1/Z) − 1

(22-10)

∂ t0.2 m / ∂ Tatmosphere, hr/C

Maximum ice thickness is very sensitive to variations in the convection heat transfer coefﬁcient between ice upper surface and atmosphere at low values. The time to achieve a certain thickness of ice depends on the values of seven independent variables. In this study, sensitivities of time to form 0.2 meters of ice thickness are investigated. Sensitivities of time to form 0.2 meters of ice thickness to temperatures of atmosphere and of lake water are shown in Figures 22-3 and 22-4, respectively. In Figure 22-3, as the temperature of the atmosphere increases, the sensitivity of time to form 0.2 meters thick of ice increases. Under the present conditions, the temperature of the atmosphere has a limit of −8.4°C for the ice to be able to reach a thickness of 0.2 meters and support the energy requirement for freezing. Similarly, in Figure 22-4, as the temperature of the lake water increases, the sensitivity of time to form 0.2 meters thick of ice increases. Under the present conditions, the temperature of the lake water has a limit of 14.3°C for the ice to be able to reach a thickness of 0.2 meters.

300 250 200 150 100 50 0 –50

–40

–30

–20

–10

Temperature Of Atmosphere, °C

Figure 22-3 Sensitivity of time to form 0.2 m thick of ice to temperature of

atmosphere

207

Everyday Heat Transfer Problems

∂ t0.2 m / ∂ Tlakewater, hr/C

80 60 40 20 0

2

4

6 8 10 Temperature Of Lake Water, °C

12

14

Figure 22-4 Sensitivity of time to form 0.2 m thick of ice to temperature of

lake water

Sensitivities of time to form 0.2 meters thick of ice to hice upper surface-atmosphere and hlake water-growing ice surface are shown in Figures 22-5 and 22-6, respectively. The convection heat transfer coefﬁcients on both sides of the ice have opposite effects to the growth of ice. In Figure 22-5, as the convection heat transfer coefﬁcient between the ice upper surface and the atmosphere decreases, the absolute value of ice formation time sensitivity increases, or the time for formation of 0.2 meters of ice increases. On the other hand, in Figure 22-6, as the convection heat transfer coefﬁcient between the lake water and the growing ice surface increases, the ice formation time and its sensitivity increase. The ice formation time sensitivities to ice density and to the latent heat of fusion of water are constants. The ice formation time sensitivity to ice density is positive, 0.115 hr-m3/kg. The ice formation time sensitivity to latent heat of fusion of water is also positive, 1.14 kg/W. The ice formation time sensitivity to thermal conductivity of ice is non-linear, and is given in Figure 22-7. As the thermal conductivity of ice increases, the time to form 0.2 meters thick of ice decreases, as does the absolute value of its sensitivity.

208

∂ t0.2 m / ∂ hice upper surface–atmosphere, hr-m2-C/W

Ice Formation On A Lake

0 –5 –10 –15 –20 –25

0

100

20 40 60 80 hice upper surface–atmosphere, W/m2-C

Figure 22-5 Sensitivity of time to form 0.2 m thick of ice to the convection

∂ t0.2 m / ∂ hlake water-growing ice surface, hr-m2-C/W

heat transfer coefﬁcient between ice upper surface and atmosphere

30

20

10

0

5

10

15

20 hlake water-growing ice surface, W/m2-C

25

Figure 22-6 Sensitivity of time to form 0.2 m thick of ice to the convection

heat transfer coefﬁcient between lake water and growing ice surface

209

Everyday Heat Transfer Problems

∂ t0.2 m / ∂ kice, hr-m-C/W

0 –50 –100 –150 –200 –250 1

1.5

2

2.5

3

Thermal Conductivity Of Ice, W/m-C

Figure 22-7 Sensitivity of time to form 0.2 m thick of ice to thermal

conductivity of ice

Table 22-1 Effects of a ±10% change in nominal values of variables to time for ice thickness to reach 0.2 m

Variable

Nominal Value

Tatmosphere

−20°C

Change in Change in Time For Ice Time For Ice Thickness to Thickness to Reach 0.2 m For Reach 0.2 m For A 10% Decrease A 10% Increase In Nominal In Nominal Value Value −12.67%

+17.05%

kice

1.88 W/m-C

+10.87%

−8.28%

ρice

920 kg/m

−10%

+10%

Hice

93 W-hr/kg

−10%

+10%

hice upper surface-atmosphere

30 W/m -C

+5.80%

−4.61%

hlake water-growing ice surface

10 W/m -C

−4.31%

+4.78%

6°C

−4.31%

+4.78%

Tlake water

210

3

2 2

Ice Formation On A Lake

When the nominal values of the independent variables given above are varied ±10%, the results shown in Table 22-1 are obtained. Sensitivities of time for the ice thickness to reach 0.2 meters to a ±10% change in the governing independent variables are given in descending order of importance, and they are applicable only around the nominal values assumed for this study. Ice thickness formation time is most sensitive to the temperature of the atmosphere. Ice thermophysical properties are next in the order of sensitivity. Convection heat transfer coefﬁcients on both sides of the ice surfaces, and the lake water temperature, round up the ice formation time sensitivity. Ice formation time sensitivities to all the independent variables are non-linear, except for ice density and latent heat of fusion for water. All the ice formation time sensitivities for the nominal independent variables values assumed for the present analysis are at the same order of magnitude.

211

CHAPTER

SOLIDIFICATION 23

IN A CASTING MOLD

E

xact temperature distribution solutions to unsteady-state onedimensional heat conduction equations in rectangular, cylindrical, and spherical coordinates, for initial condition and different boundary conditions, have been provided in literature (see References [1], [6], [10] and [16]). The reference by Carslaw, H. S. and J. C. Jaeger [1] extended these exact solutions to moving boundary conditions with phase change. In this chapter, sensitivity analysis for the solidiﬁcation front of a semi-inﬁnite liquid in a semi-inﬁnite mold is considered. The unsteady-state one-dimensional heat conduction equation in rectangular coordinates for the mold region, solidiﬁed cast material region and liquid cast material region, with constant property assumptions, is given by: ∂2T/∂x2 = (1/α) ∂T/∂θ

(23-1)

where temperature T varies with space x and with time θ. α is thermal diffusivity, k/ρcp, of the region in m2/s, where k is the thermal

213

Everyday Heat Transfer Problems

conductivity of the region in W/m-C, ρ is the density of the region in kg/m3, and cp is the speciﬁc heat of the region at constant pressure in J/kg-C. Initial and boundary conditions for each region are as follows: Minus inﬁnity to x = 0 is assumed to be the mold region. In this case, it is assumed that the mold is made out of sand. x > 0 to plus inﬁnity is the cast material’s region, which is initially ﬁlled with liquid silver. The solidiﬁcation front for the liquid silver starts initially at x = 0 and grows in the +x direction. Eq. (23-1) is applied to three different regions, in order to obtain the temperature distributions in each region with the following initial and boundary conditions: At zero time, all the mold region from x = 0 to x = −∞ is at the initial temperature of the sand, Tsand: T(−x, 0) = Tsand

(23-2)

The temperature of the mold region far away from x = 0 always stays at the initial temperature of the sand, Tsand: T(−∞, θ) = Tsand

(23-3)

The interface between the mold region and the solidiﬁed material at x = 0 has the following energy balance boundary condition: ksand ∂Tsand/∂x = ksolid ∂Tsolid/∂x

(23-4)

The solidiﬁed region initial condition is: L = 0 at θ = 0

(23-5)

where L is the solidiﬁcation front in meters. The solidiﬁcation region boundary at x = L is always at the solidiﬁcation temperature of the liquid cast material: T(L, θ) = Tsolidiﬁcation

214

(23-6)

Solidiﬁcation In A Casting Mold

At zero time, all the liquid cast material region from x = 0 to x = +∞ is at the initial temperature of the liquid cast material, Tliquid: T(+x, 0) = Tliquid

(23-7)

The temperature of the liquid cast material region far away from x = 0 always stays at the initial temperature of the liquid cast material, Tliquid: T(+∞, θ) = Tliquid

(23-8)

The interface between the liquid cast material region and the solidiﬁed cast material region at x=L has the following boundary condition, which is the result of an energy balance at the solidiﬁcation front that supports the latent heat of fusion for the liquid cast material: kliquid ∂Tliquid/∂x − ksolid ∂Tsolid/(x = ρsolid H∂dL/dθ)

(23-9)

where H is the latent heat of fusion in W-s/kg. Exact solutions to Eq. (23-1) have been obtained using the conditions in Eq. (23-2) through Eq. (23-9) for three regions, and they can be found in Reference [1]. In the present sensitivity analysis, propagation of the solidiﬁcation front is analyzed instead of temperature distributions for each region. Reference [1] shows that in order to satisfy the temperature continuity at the solidiﬁcation front all the time, the solidiﬁcation front L has to move proportionately to the square root of time, namely: L = aθ0.5

(23-10)

where “a” is the proportionality constant, has the dimensions of m/s0.5, and can be obtained by satisfying the energy balance, Eq. (23-9), at the interface of the solidiﬁed cast material and liquid cast material (see Reference [1]):

215

Everyday Heat Transfer Problems

exp(−a2/4αsolid)/{[(ρsolidcp solidksolid)/(ρsandcp sandksand)]0.5 + erf(a/2α0.5solid)} − [(ρliquidcp liquidkliquid)/(ρsolidcp solidksolid)]0.5 [(Tliquid − Tsolidiﬁcation)/(Tsolidiﬁcation −Tsand)] exp[−a2(ρsolid/ρliquid)2/4αliquid]/erfc[a(ρsolid/ρliquid)/2α0.5liquid] − aH(πρsolid/ksolidcp solid)0.5/2(Tsolidiﬁcation − Tsand) = 0

(23-11)

where the error function “erf” is deﬁned as z

erf(z) = (2/π0.5) ∫0 exp(−u2) du

(23-12)

and the complementary error function “erfc” is deﬁned as erfc(z) = 1 − erf(z)

(23-13)

Eq. (23-11) is solved for “a” by trial and error. For the present analysis, liquid silver is considered as the cast material, and sand is considered as the mold material. For solidiﬁcation to occur there are two conditions that have to be met. One condition is that the liquid cast material’s temperature cannot exc eed the melting temperature of the mold material: Tliquid < Tsand melting point

(23-14)

The second condition is that the liquid cast material has to have enough heat loss initially through the mold by conduction heat transfer to be able to start solidifying. Eq. (23-11) provides this condition by setting a equal to zero: (Tliquid − Tsolidiﬁcation)/(Tsolidiﬁcation − Tsand) < [(ρsandcp sandksand)/(ρliquidcp liquidkliquid)]0.5 (23-15) For the present sensitivity analysis, the following nominal values are assumed for the independent variables that also satisfy Eqs. (23-14) and (23-15): Sand values for the mold region:

216

Solidiﬁcation In A Casting Mold

Tsand = 100°C Tsand melting point = 1430°C ρsand = 2330 kg/m3 cp sand = 712 J/kg-C ksand = 42 W/m-C Solidiﬁed silver values for the solidiﬁed cast region: Tsolidiﬁcation = 960°C ρsolid = 10500 kg/m3 cp solid = 235 J/kg-C ksolid = 400 W/m-C

Solidification Front L, m

Liquid silver values for the liquid cast region: Tliquid = 1000°C which is less than Tsand melting point of 1430°C to satisfy Eq. (23-14), and is less than 1180°C, which is the “a” equals zero condition in Eq. (23-15) to start solidiﬁcation. ρliquid = 9300 kg/m3 cp liquid = 318 J/kg-C kliquid = 360 W/m-C H = 104400 W-s/kg

0.4 0.3 0.2 0.1 0

0

5

10

15 Time, min

20

25

30

Figure 23-1 Cast solidiﬁcation front L versus time

217

Everyday Heat Transfer Problems

“a” is obtained by iteration for these nominal values, and is 0.008137 meters/second0.5. The solidiﬁcation front versus time for the nominal case is given in Figure 23-1. When the nominal values of the variables given above are varied ±10%, the results shown in Table 23-1 are obtained. “a” sensitivities to a ±10% change in the governing variables are given in descending order of importance, and they are applicable only around the nominal values assumed for this study. The variable Tliquid is varied on the negative side −4% in order not to fall below Tsolidiﬁcation. The variable Tsolidiﬁcation is varied on the positive side +4.2% in order not to exceed Tliquid.

Table 23-1 Effects of a ±10% change in nominal values of variables to “a” in solidiﬁcation front Eq. (23-10), L = aθ0.5

Nominal Value

Change in “a” For A 10% Decrease In Nominal Value

Change in “a” For A 10% Increase In Nominal Value

Tliquid

1000° C

−20.738% for −4% change

+69.208%

Tsolidiﬁcation

960° C

+58.119%

−17.677% for −4.2% change

104400W-s/kg

+9.745%

−8.206%

ρsolid

10500 kg/m

+9.641%

−8.062%

ρsand

3

2330 kg/m

−3.192%

+2.933%

cp sand

712 J/kg-C

−3.192%

+2.933%

ksand

42 W/m-C

−3.192%

+2.933%

cp liquid

318 J/kg-C

−1.520%

+1.517%

ksolid

400 W/m-C

−1.324%

+1.143%

Variable

H

3

Tsand

100° C

+0.773%

−0.777%

ρliquid

9300 kg/m3

−0.717%

+0.686%

kliquid

360 W/m-C

−0.717%

+0.686%

cp solid

235 J/kg-C

+0.673%

−0.652%

218

Solidiﬁcation In A Casting Mold

Solidiﬁcation front propagation parameter “a” is most sensitive to the temperature of the liquid cast material and the solidiﬁcation temperature of the cast material. The second tier of “a” sensitivity belongs to the latent heat of fusion and to density of solid region. In order of sensitivity, the next set of independent variables are mold region thermophysical properties, namely ρsand, cp sand, and ksand. “a” is least sensitive to variations in independent variables cp liquid , ksolid , Tsand , ρliquid , kliquid and cp solid. Changes to these least sensitive independent variables affect “a” two orders of magnitude less than changes to temperature of the liquid cast material and changes of solidiﬁcation temperature of the cast material.

219

CHAPTER

AVERAGE

TEMPERATURE RISE IN SLIDING SURFACES IN CONTACT

24

F

rictional heating of materials in contact has been studied thoroughly in heat transfer literature. In this case, an approximate method developed by J. C. Jaeger (see Reference [7]) for frictional temperature rise on a sliding square contact area will be analyzed for sensitivity. In order to determine the steady-state average temperature at contacting surfaces, Jaeger [7] used the temperature solution resulting from an instantaneous point source in an inﬁnite solid for the following unsteady-state and three-dimensional conduction heat transfer differential equation, in rectangular coordinates (see Reference [1]): ∂2T/∂x2 + ∂2T/∂y2 + ∂2T/∂z2 = (1/α)(∂T/∂θ)

(24-1)

The temperature distribution and history resulting from an instantaneous point source at (x1, y1, z1) of strength Q that satisﬁes Eq. (24-1) can be expressed as: T = [Q/8(παθ)0.5] exp{[(x − x1)2 + (y − y1)2 + (z − z1)2]/4αθ}

(24-2)

221

Everyday Heat Transfer Problems

T in Eq. (24-2) represents the temperature distribution and history in an inﬁnite medium at location (x, y, z) and at time θ. α is the constant thermal diffusivity of the medium deﬁned as (k/ρcp), where k is the medium’s thermal conductivity, ρ is medium’s density, and cp is medium’s speciﬁc heat at constant pressure. Eq. (24-2) can be integrated to obtain steady-state temperatures for moving and stationary line and square contact areas, where heat transfer was restricted only to contact area (see References [1] and [7]). In the present sensitivity analysis, a square contact area, 2L × 2L, that is sliding with a constant velocity, V, over a semi-inﬁnite body is considered. For LV/2α > 5, the average temperature at the surface of the semi-inﬁnite medium that is under the sliding square contact area, designated as area “1,” is approximated as follows (see Reference [7]): Taverage at surface of contacted semi-inﬁnite medium = (1.064 Q/k1) (α1L/V)0.5 (24-3) The sliding square contact area also has a conducting semi-inﬁnite body behind it, and its temperature rises due to the heat generated during contact. The average temperature of the square sliding contact area designated as “2” is (see Reference [7]): Taverage for sliding square contact area = 0.946 L Q/k2

(24-4)

Eq. (24-3) assumes that all the heat generated during the contact goes to the semi-inﬁnite medium under the sliding contact area, but in reality this is not the case. A portion, mQ, of the heat generated during the sliding contact goes into the semi-inﬁnite body and the rest, (1-m)Q, goes into the square sliding contact area. Equating the average temperatures in Eqs. (24-3) and (24-4), the proportionality constant “m” can be determined: m = k1 (LV)0.5/[1.125 k2 α10.5 + k1 (LV)0.5]

(24-5)

The heat ﬂow proportionality constant, Eq. (24-5), is inserted into Eq. (24-3) to determine the sliding contact area average temperature:

222

Average Temperature Rise In Sliding Surfaces In Contact

Tsliding contact area average temperature = (1.064 Q L α10.5)/[1.125 k2 α10.5 + k1 (LV)0.5]

(24-6)

The heat generated during the contact comes from the mechanical energy that is dissipated during the contact: Q = fd W g V/(4 L2 J)

(24-7)

where fd is the dynamic coefﬁcient of friction during the contact, W is the load at the contact area in kg, g is the gravitational constant, 9.8 m/s2, V is the relative velocity of the sliding contact area over the semi-inﬁnite medium in m/s, L is the half width of square sliding contact area in m, and J = 1 kg-m2/s3-W, which is a proportionality constant that equates mechanical work to heat from the ﬁrst law of thermodynamics. Combining Eqs. (24-6) and (24-7) gives the ﬁnal form of the sliding contact area average temperature for (LV)/(2 α1) > 5: Taverage = (0.266 fd W g V α10.5)/{L J [1.125 k2 α10.5 + k1 (LV)0.5]}

(24-8)

where α1 = thermal diffusivity of semi-inﬁnite medium in m2/s, k1 = thermal conductivity of semi-inﬁnite medium in W/m-C, k2 = thermal conductivity of the square sliding contact area in W/m-C. The following nominal values for the independent variables are assumed for the present sensitivity analysis. Also both the semi-inﬁnite medium and the square sliding contact area are assumed to be stainless steel. fd = 0.5, W = 1000 kg, V = 10 m/s, α1 = 4 × 10−6 m2/s, k1 = 15 W/m-C, k2 = 15 W/m-C, L = 0.01 m

223

Everyday Heat Transfer Problems

Sliding Contact Area Average Temperature, °C

With these nominal values, sliding contact area average temperature rise is 546°C. Average temperature rises linearly with load at the sliding contact area and with the dynamic coefﬁcient of friction. Average temperature sensitivity to load at the sliding contact area is 0.546 C/kg, and to the dynamic coefﬁcient of friction is 109.1°C per 0.1 change in fd. Sliding contact area average temperature rise versus relative velocity of two bodies is shown in Figure 24-1. Average temperature increases with increasing relative velocity as a V0.5 function. Sliding contact area average temperature rise sensitivity to relative velocity of two bodies is shown in Figure 24-2. Average temperature sensitivity decreases as V-0.5 as relative velocity increases. Sliding contact area average temperature versus thermal diffusivity of semi-inﬁnite body under sliding contact area is shown in Figure 24-3; average temperature increases as α10.5. Average temperature sensitivity to α1 is given in Figure 24-4, and it behaves as α1−0.5. Sliding contact area average temperature versus thermal conductivity of semi-inﬁnite body under the sliding contact area is shown in Figure 24-5; average temperature decreases as k1−1.

1000 800 600 400 200 0

0

10

5

15

20

25

30

Relative Velocity Of Bodies, m/s

Figure 24-1 Sliding contact area average temperature versus relative

velocity of bodies

224

Average Temperature Rise In Sliding Surfaces In Contact

∂ Taverage / ∂ V, C-s/m

400 300 200 100 0

0

5

10

15

20

25

30

Relative Velocity Of Bodies, m/s

Figure 24-2 Sliding contact area average temperature sensitivity to relative

Sliding Contact Area Average Temperature, °C

velocity of bodies

2000 1600 1200 800 400 0 0.E+00

1.E−05

2.E−05

3.E−05

4.E−05

5.E−05

Thermal Diffusivity, α1, Of Semi-Infinite Body Under Sliding Contact Area, m2/s

Figure 24-3 Sliding contact area average temperature versus thermal

diffusivity, α1, of semi-inﬁnite body under sliding contact area

225

Everyday Heat Transfer Problems

∂ Taverage / ∂α1, C-s/m2

1.2E+08 1.0E+08 8.0E+07 6.0E+07 4.0E+07 2.0E+07 0.E+00

1.E−05

2.E−05

3.E−05

4.E−05

5.E−05

Thermal Diffusivity, α1, Of Semi-Infinite Body Under Sliding Contact Area, m2/s

Figure 24-4 Sliding contact area average temperature sensitivity to thermal

Sliding Contact Area Average Temperature, °C

diffusivity, α1, of semi-inﬁnite body under sliding contact area

1600 1200 800 400 0

0

50

100

150

200

250

300

Thermal Conductivity, k1, W/m-C

Figure 24-5 Slider contact area average temperature versus thermal

conductivity of semi-inﬁnite body under sliding contact area

226

Average Temperature Rise In Sliding Surfaces In Contact

Average temperature sensitivity to thermal conductivity of semi-inﬁnite body under the sliding contact area is given in Figure 24-6, and it behaves as k1−2. Sliding contact area average temperature versus thermal conductivity, k2, of sliding square contacting body is shown in Figure 24-7; average temperature decreases slightly, and almost linearly, with increasing thermal conductivity of the sliding square contacting body. Average temperature sensitivity to k2 is given in Figure 24-8, and it can be considered a constant at −0.25 C2-m/W. Under the given nominal conditions, slider contact area average temperature is most sensitive to the size of the square contact area. Sliding contact area average temperature versus half width, L, of square contact area is depicted in Figure 24-9. Average temperature decreases as L−1.5 as side of contact area increases. Average temperature sensitivity to half width of square contact area is given in Figure 24-10, and it behaves as L−2.5. When the nominal values of the independent variables given above are varied ±10%, the results shown in Table 24-1 are obtained. Sliding contact area average temperature sensitivities to a ±10% change in

∂ Taverage / ∂ k1, C2-m/W

10 –30 –70 –110 –150

0

50

100 150 200 250 Thermal Conductivity, k1, W/m-C

300

Figure 24-6 Sliding contact area average temperature sensitivity to thermal

conductivity of semi-inﬁnite body under sliding contact area

227

Sliding Contact Area Average Temperature, °C

Everyday Heat Transfer Problems

550

540

530

520

100 0 20 40 60 80 Thermal Conductivity Of Sliding Square Contacting body, k2, W/m-C

Figure 24-7 Sliding contact area average temperature versus thermal

∂ Taverage / ∂ k2, C2-m/W

conductivity of sliding square contacting body

–0.2

–0.25

–0.3

0

20

40

60

80

100

Thermal Conductivity Of Sliding Square Contacting Body, k2, W/m-C

Figure 24-8 Sliding contact area average temperature sensitivity to thermal

conductivity of sliding square contacting body

228

Average Temperature Rise In Sliding Surfaces In Contact

Sliding Contact Area Average Temperature, °C

3000 2500 2000 1500 1000 500 0

0

0.02

0.04

0.06

0.08

0.1

Half Width, L, Of Square Contact Area, m

Figure 24-9 Sliding contact area average temperature versus half width, L,

of square contact area

∂ Taverage / ∂ L, C/m

0 –50000 –100000 –150000 –200000 –250000 –300000

0

0.02

0.04

0.06

0.08

0.1

Half Width, L, Of Square Contact Area, m

Figure 24-10 Sliding contact area average temperature sensitivity to half

width, L, of square contact area

229

Everyday Heat Transfer Problems

Table 24-1 Effects of a ±10% change in nominal values of independent variables to sliding contact area average temperature

Nominal Value

Change In Sliding Contact Area Average Temperature For A 10% Decrease In Nominal Value

Change In Sliding Contact Area Average Temperature For A 10% Increase In Nominal Value

0.01 m

+17.08%

−13.30%

15 W/m-C

+11.02%

−9.03%

−10%

+10%

W, load at the sliding contact area

1000 kg

−10%

+10%

V, relative velocity of sliding square contact area and the semi-inﬁnite body

10 m/s

−5.17%

+4.92%

α1, thermal diffusivity of semi-inﬁnite body

4 × 10−6 m2/s

−5.10%

+4.84%

15 W/m-C

+0.07%

−0.07%

Variable L, half width of sliding square contact area k1, thermal conductivity of semi-inﬁnite body fd, dynamic coefﬁcient of friction0.5

k2, thermal conductivity of sliding square contact area

230

Average Temperature Rise In Sliding Surfaces In Contact

the governing independent variables are given in descending order of importance, and they are applicable only around the nominal values assumed for this analysis. Sliding contact area average temperature is most sensitive to the size of the contact area. Thermal conductivity of semi-inﬁnite body under the sliding contact area, dynamic coefﬁcient of friction and contact area load are next in the order of sensitivity. However, these sensitivities have the same order of magnitude as sensitivity to the size of the contact area. Relative velocity of two bodies and thermal diffusivity of semi-inﬁnite body under the sliding contact area come next in the average temperature order of sensitivity. Sliding contact area average temperature sensitivities to these two independent variables are signiﬁcant, and they have half the magnitude as the sensitivities to previous tier’s independent variables. Sliding contact area average temperature has its lowest sensitivity to thermal conductivity of the sliding square contacting area. This sensitivity is two orders of magnitude lower than the sensitivity to the size of the contact area.

231

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234

Kreith, F., “Principles Of Heat Transfer,” Second Edition, International Textbook Company, Scranton, Pennsylvania, 1965. Kutz, M. “Heat Transfer Calculations,” Mc-Graw Hill, New York, 2005. Lee, W. J., Kim, Y., and Case, E. D., “The Effect Of Quenching Media On The Heat Transfer Coefﬁcient Of Polycrystalline Alumina,” Journal Of Materials Science, Volume 28, pp. 2079–2083, 1993. Lin, S. and Jiang, Z., “ An Improved Quasi-Steady Analysis For Solving Freezing Problems In A Plate, A Cylinder And A Sphere,” ASME Journal Of Heat Transfer, Volume 125, pp. 1123–1128, 2003. London, A. L. and Seban, R. A., “Rate Of Ice Formation,” Transactions Of The ASME, Volume 65, pp. 771–778, 1943. Roshenow, W. M. and Choi, H. Y., “Heat, Mass, And Momentum Transfer,” Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1961. Schneider, P. J., “Conduction Heat Transfer,” Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1957. Timoshenko, S and Goodier, J. N., “Theory Of Elasticity,” Second Edition, McGraw-Hill Book Company, Inc., New York, 1951.