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42

Q = α1 A ( T1 − Ts 1 ) Q =

: QDV]\P SU]\NáDG]LH n = 3:

(4.26)

Ts 1 − Ts ( n +1 )

(4.27)

Rλ

Q = α 2 A ( Ts ( n +1 ) − T2 )

(4.28)

. 1   1 T1 − T2 = Q  + Rλ + α 2 A   α1 A

(4.29)

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α1

α2

Rk = Rα 1 + Rλ + Rα 2

(4.30)

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T1 − T2 Rk

(4.31)

5yZQDQLH WR PR*QD WH* ]DSLVDü Z SRGVWDZRZHM SRVWDFL Q = kA ( T1 − T2 )

(4.32)

MHVW ZVSyáF]\QQLNLHP SU]HQLNDQLD FLHSáD GOD ZLHORZDUVWZRZHM FLDQ\ Sáaskiej, H ZVSyáF]\QQLNLHP 3HFOHWD :VSyáF]\QQLN WHQ REOLF]D VL ]H Z]oru

gdzie k zwanym równi *

kA =

1 1 = i =n Rk 1 1 s +∑ i + α1 A i =1 λi A α 2 A

(4.33)

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Q = Ts ( n +1 ) − T2 α 2 Œd z L

(4.39)

3R GRGDQLX VWURQDPL L Z\FLJQL FLX SU]HG QDZLDV Z\UD]X Q /(πL) otrzymujemy Q 1 1 + Rλ r + ( ) = T1 − T2 ŒL α1 d w α2 dz

(4.40)

: QDZLDVLH SR]D RSRUHP SU]HZRG]HQLD FLHSáD Rλr mamy opory wnikania opisane wzorami Rα r 1 =

1 α1 d w

Rα r 2 =

1 α2 dz

(4.41)

(4.42)

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Rkr = Rα r1 + Rλ r + Rα r 2 .

(4.43)

RWU]\PDP\ Z]yU QD VWUXPLH FLHSáD SU]HQLNDMF\ SU]H] FLDQ UXURZ zwany równaniem Pecleta dla tej przegrody Q = ŒL

T1 − T2 Rkr

(4.44)

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1 = Rkr

1 i =n 1 + α1d w i =1

∑

di +1 di 1 + 2λi α2dz

.

(4.45)

ln

RWU]\PXMHP\ QDVW SXMF SRVWDü UyZQDQLD 3HFOHWD Q = kr ŒL T1 − T2

:\SURZDG]RQH UyZQDQLH VWDQRZL Z]yU QD VWUXPLH FLHSáD FLDQ UXURZ zwany równaniem Pecleta dla tej przegrody.

(4.46)

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101

Dokładny obraz rozkładu prędkości i temperatury można uzyskać z obliczeń za pomocą programu komputerowego FLUENT 6.0. Rozwiązanie układu równań przepływu, tj. równania Naviera–Stokesa oraz równania energii w programach tego typu wymaga zaawansowanych metod numerycznych, najczęściej metody objętości kontrolnej Patankara [9]. Przykładowe obliczenia wykonano dla przepływu wody w rurze poziomej o średnicy 0,05 m na odcinku długości 4 m z użyciem siatki niestrukturalnej (rys. 7.9). W przypadku dużej średniej prędkości wody (v = 0,256 m/s, rys. 7.10) liczba Reynoldsa wynosi około 11 000 i profile prędkości są wyraźnie płaskie zarówno u wlotu (rys. 7.10b), jak i na wylocie z rury (rys. 7.10a). Podobnie rozkład temperatury wody jest dość wyrównany (rys. 7.11), jednak jest ona znacznie niższa od temperatury ścianki (tść = 55 °C). Rura 1 + 2

Siatka niestrukturalna

Rys. 7.9. Siatka niestrukturalna dla rury 1+2 (L = 1 m +3 m; d = 0,05 m)

102

a) wylot rury 2: v = 0,256 m/s, tpłynu wlot. = 20 °C, tścianki = 55 °C

b) wlot rury 1, v = 0,256 m/s, tpłynu wlot. = 20 °C, Q = 0 W

Rys. 7.10. Wektory prędkości kolorowane według skali wielkości (m/s) dla średniej prędkości wody v = 0,256 m/s: a) wylot rury 2, b) wlot rury 1

103

a) wlot rury 2, v = 0,256 m/s, tpłynu wlot. = 20 °C, tścianki = 55 °C

b) wylot rury 2, v = 0,256 m/s, tpłynu wlot. = 20 °C, tścianki = 55 °C

Rys. 7.11. Kontury temperatury kolorowane według skali wielkości (K) dla średniej



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Przekrój poprzeczny przez wymiennik płaszczowo-rurowy, rurki φ 10, woda, przepływ laminarny, m1 = 0,05 kg/s, T1 = 300 K, T2 = 400 K

Rys. 7.13. Kontury prędkości wody podczas przepływu poprzecznego do rurek wymiennika φ 10 m

Przekrój poprzeczny przez wymiennik płaszczowo-rurowy, rurki φ 10, woda, przepływ laminarny, m1 = 0,05 kg/s, T1 = 300 K, T2 = 400 K

Rys. 7.14. Wektory prędkości wody podczas przepływu poprzecznego do rurek wymiennika (φ 10 mm) kolorowane według skali wielkości (m/s)

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Przekrój poprzeczny przez wymiennik płaszczowo-rurowy, rurki φ 10, woda, przepływ laminarny, m1 = 0,05 kg/s, T1 = 300 K, T2 = 400 K

Rys. 7.15. Kontury temperatury podczas przepływu poprzecznego do rurek wymiennika φ 10 mm kolorowane według skali wielkości (K) dla masowego przepływu wody m = 0,05 kg/s

Przekrój poprzeczny przez wymiennik płaszczowo- rurowy, rurki φ 10, woda, przepływ laminarny, m1 = 0,05 kg/s, T1 = 300 K, T2 = 400 K

Rys. 7.16. Kontury ciśnienia całkowitego podczas przepływu poprzecznego do rurek wymiennika φ 10 mm kolorowane według skali wielkości (Pa) dla masowego przepływu wody m = 0,05 kg/s

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b) wylot rury 2, v = 0.01025 m/s; tpłynu wlot. = 20 °C, tścianki = 55 °C

Rys. 7.19. Wektory prędkości kolorowane według skali wielkości (m/s) dla średniej prędkości wody v = 0,01025 m/s: a) wlot rury 1, b) wylot rury 2

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a) wlot rury 2, v = 0,01025 m/s, tpłynu wlot. = 20 °C, tścianki = 55 °C

b) wylot rury 2, v = 0,01025 m/s, tpłynu wlot = 20 °C, tścianki = 55 °C

Rys. 7.20. Kontury temperatury kolorowane według skali wielkości (K) dla średniej prędkości wody v = 0,01025 m/s: a) wlot rury 2, b) wylot rur



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5\V  =DOH*QRü FDáNRZLWHM HPLV\MQRFL H+ 2 RG WHPSHUDWXU\ 7 L LORF]\QX SV SRG FLQLHQLHP  EDUD 



6WRVXMH VL UyZQLH* UyZQDQLH 6WHIDQD±%ROW]PDQQD (J

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SR ZSURZDG]HQLX RGSRZLHGQLFK ZDUWRFL HPLV\MQRFL HJ GOD GDQHJR JD]X QS RGF]\ WDQ\FK ] Z\NUHVyZ U\V  L   3U]HGVWDZLRQ\ Z\NUHV GOD SDU\ ZRGQHM ]DZLHUD SRSUDZQH ZDUWRFL GOD PDá\FK VW *H SDU\ : SU]\SDGNX Z\*V]\FK VW *H SDU\ ZRGQHM RGF]\WDQH ZDUWRFL HPLV\M QRFL HJ QDOH*\ VNRU\JRZDü SU]H] SRPQR*HQLH LFK SU]H] SRSUDZN E+2 ]DOH*Q RG SU *QRFL SDU\ RUD] LORF]\QX SU *QRFL SDU\ L JUXERFL ZDUVWZ\ RGF]\WDQ ] LQQHJR Z\NUHVX U\V  

5\V  =DOH*QRü SRSUDZNL E+ 2 RG FLQLHQLD 

5\V  =DOH*QRü SRSUDZNL E&2 RG FLQLHQLD 

189

Podobnie dla CO2 QDOH*\ XZ]JO GQLü SRSUDZN βCO2 U\V  Z ]DOH*QRFL RG FLQLHQLD FDáNRZLWHJR L LORF]\QX FLQLHQLD F]VWNRZHJR &22 L JUXERFL ZDUVWZ\ 8Z]JO GQLHQLH REHFQRFL FLDQ\ ELRUFHM XG]LDá Z Z\PLDQLH FLHSáD ]QDF]QLH XWUXGQLD ]DGDQLH 'OD REOLF]H WHFKQLF]Q\FK ]DOHFD VL ]DOH*QRü qg − s

4   Tg 4  Ts   ′  = ε s C0 ε g   − ag   ,  100     100 

W/m 2

(10.46)

gdzie: qg–s – FLHSáR Z\PLHQLDQH PL G]\ JD]HP D FLDQ QD  P2 powierzchni, Tg – temperatura gazu, Ts – WHPSHUDWXUD FLDQ\ ELRUFHM XG]LDá Z Z\PLDQLH FLeSáD ε s′ = (εs + 1)/2 jest efektywn ]GROQRFL HPLV\MQ FLDQ\ εs – WHRUHW\F]Q ]GROQRFL HPiV\MQ FLDQ\ εg – ]DVW SF] ]GROQRFL HPLV\MQ JD]X ] Z\NUHVyZ Z WHPSHUDWXU]H gazu, ag – ]DVW pF] ]GROQRFL DEVRUSF\MQ JD]X Z WHPSHUDWXU]H FLDQ\ : REOLF]HQLDFK WHFKQLF]Q\FK PR*QD SU]\Mü *H ag = εg Z WHPSHUDWXU]H FLDQ\ L RGF]\WDü MH ] Z\NUHVyZ

5\V  =DOH*QRü SRSUDZNL ∆εg RG VWRVXQNX FLQLHQLD : SU]\SDGNX SURPLHQLRZDQLD PLHV]DQLQ\ JD]yZ HPLVMD SRV]F]HJyOQ\FK VNáDGQiNyZ QDNáDGD VL  ]DVW SF] ]GROQRü HPLVML L DEVRUSFML RNUHOD VL ] QDVW SXMcych ]DOH*QRFL • dla temperatury gazu

ε g = β CO2 ε g (CO2 ) + β H2 O ε g (H2 O) − ∆ε g

(10.47)

• GOD WHPSHUDWXU\ FLDQ\  Tg  ag = βCO2 ε g (CO2 )    Ts 

0 ,65

 Tg  + β H 2 O ε g ( H2 O )    Ts 

0 ,45

− ∆ε g

(10.48)

:DUWRFL SRSUDZHN ∆εg QDOH*\ RGF]\WDü ] RGSRZLHGQLFK Z\NUesów (rys. 10.16) w zaOH*QRFL RG WHPSHUDWXU\ gazu.

11. Obliczanie wymienników cieSáD .ODV\ILNDFMDZ\PLHQQLNyZFLHSáD


        
  
   

         
  
odzaje.        
  a      
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  lub zakrzywionej. Wymiana energii zachod   
  
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yn  ywa   

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ddaje     
   
 

   %      
 
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* regeneratory pieców Siemensa–Martina oraz niektóre generatory gazu.
   
   

 (    

ktorem. &   
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      ) 
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   – mie  


192

7\S\NRQVWUXNFMLZ\PLHQQLNyZFLHSáD


    * 
        

 
     
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     *ebrowana.

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*   • 
     
      
  
ików jest zgodny, • 
    
   
 
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193

•
 
*   
       
 
        
 
  

  
   o
 

 
    

  urze, przedstawiony schematycznie na rys. 11.1a. Schemat wymienników z  
  y*
     
          
  
    

 
    *
   
 
  
   
  ugim – 
*
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   *    
*             u 
  *  urze.     
 
 

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n        
   

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&


  * 
        e
 
    '  * 

•
     
        
  
     
      
  •
     czynników jest zgodny, a zwrot przeciwny, •
      


     
  o    !

      

   
               
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i   * 
   "
 (# * 
   
  odsta     
     
  
 
  *
o  
 a   
*    
   
 b (rys. 11.3), czynnik b    *
     

*    
kocowej czynnika a + *
  
 * ûta(przeciwp.) ûta(wspólp.)

194

7

7

∆9

7 7

7

/

/

% $

∆9

7

%

$

$

%

$ %

5\V  6FKHPDW ZVSyáSUGRZHJR Z\PLHQQLND FLHSáD L UR]NáDGX WHPSHUDWXU\ Sá\QyZ

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,     
 
   

  
        
 
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*  *  
  
  -  ./0         
i 700 °1      u         
   

 rnych kilkaset   *    mperatury topnienia. 2   *    
*
      

 
        
 ,    
   o* 
 
 

 
   
    iwego

 p W = mc

(11.1)

$*  *


 
      la

 
 
      Q a −b = Wa (Ta 1 − Ta 2 ) = ±Wb (Tb 1 − Tb 2 )

(11.2)

3  

      –   du. ,      "# Q a −b = m a c pa (Ta 1 − Ta 2 ) = ± m b c pb (Tb 1 − Tb 2 )

(11.3)

     

Wb T − Ta 2 = ± a1 Wa Tb 1 − Tb 2

(11.4)

4      

   
     

 
   

195

UHGQLDUy*QLFDWHPSHUDWXU\ 3U]HFLZSUGRZ\LZVSyáSUGoZ\SU]HSá\ZSá\QyZ

5 
         
 y 
  
     
   
 
   
i    k     *
 
 6    awie  *


   

        onego stru  ,  
   
      ,      Q = kAûT

 
  *  
∆T    

(11.5) 



 e  Q .

79



∆9

7

9

∆9



G9 

9

7



G

9





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/9



/9







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9



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9



∆9

9

∆9

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196

' *  
    
 
 
  * owierzchni i  

 $
    
 
   yparka, w                 ycona      *  
   6    *  
ynów ∆T = Ta – Tb   *    
      
 / i 7         
   *
  e   *  

   $

  *
           *
 
∆Tm 
 
  anie Pecleta

Q = kAûTm

(11.6)

 
       *  
   

   6  *     
  dA  

 aje w       *      dQa: dQ a = − m a c pa dTa

(11.7)

1 
     
     A, tzn. na drodze od przekroju 1 do przekroju 2 (rys. 11.5), wynosi

Ta 2

Q a = − m a

∫c

pa dTa

(11.8)

Ta 1

W równaniach tych mamy znak –   

   8   du. !    *     
   

ami:

• w odniesieniu do elementu powierzchni dA •  

dQ a −b = kdAûT

(11.9)

Q a −b = A ( kûT )m

(11.10)

     A

'  "9#   
    dA =

dQ a −b kûT

+ * 
 * 
 

  d Q a

(11.11) = dQ a −b oraz Q a = Q a −b .

Po podstawieniu wzoru (11.7) do (11.11), otrzymujemy dA = −

m a c pa dTa kûT

(11.12)

197

    Ta 2

c pa dTa

∫

A = − m a

(11.13)

kûT

Ta 1

,       
*   ":# Ta 2

Q a −b

=

− m a

c pa dTa

∫

kûT

Ta 1

( kûT )m

(11.14)

i porównaniu równania (11.14) z równaniem (11.8) otrzymujemy Ta 2

− m a

∫c

Ta 2 pa dTa

∫

= − m a

Ta 1

c pa dTa kûT

Ta 1

( kûT )m

(11.15)

     Ta 2

( kûT )m

=T

∫c

pa dTa

Ta 1 a2

∫

Ta 1

(11.16) c pa dTa kûT

  * * cp = const mamy: ( kûT )m

=

Ta 2 − Ta 1 Ta 2

∫

Ta 1

(11.17)

dTa kûT

dla k ;    ûTm =

Ta 2 − Ta 1 Ta 2

∫

Ta 1

(11.18)

dTa ûT

$*  *

 * k = const, to z równania (11.10) otrzymujemy równanie Pec-

leta w ogólnej postaci:

Q = kAûTm

(11.19)

4  
*    *  
    o  *  ∆T = f (Ta# 3*        
             
   
 
  

 $*

198

 *

    *    rzymujemy: ûTm =

∆T = aTa + b (tj. równanie y = ax + b), to po prze-

Ta 2 − Ta 1 Ta 2

∫

Ta 1

T − Ta 1 Ta 2 − Ta 1 aT − aTa 1 = T a2 = = a2 a2 1 aTa 2 + b ûT dTa dTa ln ln 2 a aT b ûT1 + a1 aTa + b ûT T

(11.20)

∫

a1

       
 b i –b mamy ûTm =

ûT2 − ûT1

ln

ûT2

(11.21)

ûT

$    
   *  

  
*  <   * 
    
   " Tb), jak np. podczas   
         

  <    e
 
  =
 
       
 
     wnaniem (11.6) lub (11.7). W s   
 
 

* "k∆T) zmienia   *   Ta lub Tb    
     a
  ( kûT )m

=

( kûT )2 − ( kûT )1 ( kûT )2 ln ( kûT )1

1   
 k   *  
∆T   
 

cjami temperatury czynnika a 5      1  ( kûT )m

=

k1 ûT2 − k2 ûT1 k1 ûT2 k2 ûT1

(11.22)


 k(11.23)

1   *

       
 
* 
 etodami. 3UG\VNU]\*RZDQHLPLHV]DQH

6  

 
  
     
 y
 3  
  
 
*
     
 >   *  
 
   
      o 
   ε. Poprawka ε  ( 
 
  ymiany  
   
      
 
*owego * 

pokazano na rys. 11.7.

199

5\V  :\PLHQQLN FLHSáD ] SU]HSá\ZHP NU]\*RZ\P %

∆9PD[

∆9$

∆9

%

∆9

∆9%

%







5\V  5R]NáDG Uy*QLF WHPSHUDWXU\ GOD ]DVW SF]HJR XNáDGX ] SU]HSá\ZHP NU]\*RZ\P

$*   *  
  
 
*  e   ûTm′ = ε ûTm = ε

ûT2 − ûT1

ln

ûT2

(11.24)

ûT1

   



  A=

Q kε ûTm

(11.25)

   *   *
 
 
 
*
    *
 
  
  w  ∆Tm. ,    *
   X i Z.

to poprawka ε

200

5\V  =DOH*QRFL ε = f (X, Z GOD Uy*Q\FK przySDGNyZ SU]HSá\ZX NU]\*RZHJR ZHZQWU] rurek i prostopDGOH GR QLFK ZHGáXJ VFKHPDWX

5\V  =DOH*QRFL ε = f (X, Z) GOD Uy*Q\FK SU]\SDGNyZ SU]eSá\ZX PLHV]DQHJR

Parametr X 
    *
 
       owej czynnika b (ogrzanie czynnika b, ∆Tb#  
  *
eratury w   ∆Tmax, parametr Z      *
 
     i    
 a "    
 a, ∆Tb#  
  *
eratury ∆Tmax:

201

X=

Tbk − Tbp Tap − Tbp

Z=

Tap − Tak Tbk − Tbp

=

ûTb ûTmax

(11.26

ûTa ûTb

(11.27)

=



 *   ε = f (X, Z   
    *  o    
  
 

 
      e  !" # $
 



  
        % lda (jedna rurka we 

  &    
   

 Hoblera [4].

2EOLF]HQLDZLHONRFLZ\PLHQQLND oparte na jego sprawQRFL


  

  '"    
  '"    

 $ 

 
 

    $    
 
  Number of Transfer Units – ()* +*  ,-  (       i
  .
  
 

  η /.
 0  0 
 n

       
   
j  &  
$
& 
$



  


  *

 
 
 

 kowicie 
  *


∆T = T – T w tym wymienniku max

η=

a1

b1

Q Q max

(11.28)

5\V  5R]NáDG WHPSHUDWXU\ GOD SU]HFLZSUGRZ\FK Z\PLHnniNyZ FLHSáD a) Wa > Wb, Wb = Wmin; b) Wb > Wa, Wa = Wmin

)    



 ∆T  *   
 

  o
 



    

 
 

 
   *  


  & 
   *

 
 
 

 e

 *
&       $ 
  +   
 


  o*

 
 W        

 +* 
 emax

min

202


& * W = W   
 -    &   


 enie *   


 
 
 b  
  $     urze wlo 
 a 1

  .

 
b

  



min

Q max = Wmin (Tap − Tbp ) = Wmin ûTmax

(11.29)

Q = Wmax (Tap − Tak ) = Wmax ûTmin

(11.30)

oraz

η=

Wa (Tap − Tak )

Wb (Tbk − Tbp )max

=

Wmax (Tap − Tak )

(11.31)

Wmin (Tap − Tbp )

+ 

 
 

&   
 - $&  

 
*enie 
  0 η=

Wb (Tbk − Tbp )

Wa (Tap − Tak )max

=

Wmax (Tbk − Tbp )

(11.32)

Wmin (Tap − Tbp )

2

*  

  
  
   Q ,  *
a 0& * 

Q = ηWmin (Tap − Tbp )

(11.33)

+* 
 
 
 

&      
 



mamy:

Q = Wb (Tb 2 − Tb 1 ) = kA

(Ta 1 − Tb 1 ) − (Ta 2 − Tb 2 )  T −T  ln  a 1 b 1   Ta 2 − Tb 2 

(11.34)

 

     0 
 

3

Z równania (11.33) otrzymujemy wy *

η=

 * 
 
& * W = W b

Wb (Tb 2 − Tb 1 )

Wmin (Ta 2 − Tb 1 )

(11.35)

min

Ta 2 = Tb 1 +

  

  
  T

b2,

to

1 (Tb 2 − Tb1 ) η

(11.36)

203

Ta 2 − Tb 2 = Tb 1 − Tb 2 +

2

*

1  1 (Tb 2 − Tb 1 ) =  − 1 (Tb 2 − Tb 1 ) η η 

Wb Ta 2 − Ta 1 = Wa Tb 2 − Tb 1

(11.37)

(11.38)

   

 



Ta 1 = Ta 2 −

Wmin (Tb 2 − Tb 1 ) Wmax

(11.39)

oraz Ta 1 − Tb 1 = Ta 2 − Tb 1 −

1  

 ,4 

 * 0 Ta 2 − Tb 1 =

  
  

 !-3 Ta 1 − Tb 1 =

Wmin (Tb 2 − Tb 1 ) Wmax

1 (Tb 2 − Tb 1 ) η

(11.40)

(11.41)

1 W  1 W (Tb 2 − Tb 1 ) − min (Tb 2 − Tb 1 ) =  − min  (Tb 2 − Tb 1 ) (11.42) η Wmax  η Wmax 

  

  *  ,5) i (11.42) podstawiamy do równania (11.34) 1 Wmin − η Wmax kA = ln 1 Wmin −1 η

 Wmin  1 −   Wmax 

(11.43)

2 $
  
  ceniu otrzymamy ostatecznie  kA  Wmin   1 − exp  − 1 −   Wmin  Wmax   η=  kA  Wmin W 1 − min exp  − 1 − Wmax  Wmin  Wmax

  

(11.44)


*

 kA/W 

$   
 
()* 6  W /W oznaczymy jako R i dla przeciwpr 
 

  

 

3 min

min

max

204

ηp =

1 − e−

NTU (1− R )

1 − Re−

(11.45)

NTU (1− R )

7   
 

   


 $  


 '& '" 
  
*

 ηw =

( 1− e 1+ R

− NTU 1+ R )

(11.46)

 *    (0  

 


  *    a 
    
 

  5R]NáDGWHPSHUDWXU\Sá\QyZLFLDQ\ 7HPSHUDWXUDSá\QyZ

# 

  

 *  
  
ienniku     

  $ 

  

 *



&  
 
 o  


 
 +       ,&  

 T = f (A) lub T = f (L), gdzie A    

& L    y 

  *
od stosu       *

     
  m      
   !  
 *

  

  
   
   $
  
 

  (z 

    otoczenia, dQ = dQ = dQ) w przeciwprdowym i 
 
  
 +* 
 
 

   
 8&  dla 
   *
 0 *


$    

  h

*

 dA): a

b

dQ = − m a c pa dTa

(11.47)

dQ = − m b c pb dTb

(11.48)

dQ = −Wa dTa

(11.49)

dQ = −Wb dTb

(11.50)

2 

 .

  *

 
  

   
 t-

szej postaci:



*

    *

 
3

i otrzymam

205

dTa = −

dQ Wa

(11.51)

dTb = −

dQ Wb

(11.52)

2 *

z

 *




3 ∆T = T – T 
 

a

b

d ( ûT ) = dTa − dTb

(11.53)

1   1 d (ûT ) = − dQ  −   Wa Wb 

(11.54)

dQ = kdA∆T

(11.55)

1   1 d ( ûT ) = − kdAûT  −   Wa Wb 

(11.56)

  
   8 8'

2 

 *

   

 2  otrzymujemy

  

 
d (ûT )

ûT

1  1 = −k  −  Wa Wb

  dA 

9 

  
   

   A 
& * k = const)

1

ln

ûT2 ûT1

= 0 do A2 oraz od ∆T1 do ∆T2

    = − kA2  −   Wa Wb 

(11.57)

  

 

1   1 − kA2  −   Wa Wb 

 (11.58) +    
*



  
  
 

  .   *



   &    

     e  *

    :  '" ûT2 = ûT1

206

ûT2 = ûT1 H− kA /W 2

(11.59)

z

+*       $   *

 
ynów w dowolnym prze 
 

    x   &  *
 $ 
0    ∆T   

 
   


 

 1

ûTx = ûT1H− kA /W x

(11.60)

z

# 0     *

   
 y*   *

    
  *

 
 

 $
0  t

 7   & 
W < W , mamy zatem: 1

*  W > W , to 1

2

1 1 1 = − Wz W1 W2

(11.61)

1 1 1 = − Wz W2 W1

(11.62)

2

  
 
 

 

   

&    
w 

 

   


Dla  * przeciwne znaki (równania (11.47)–(11.52)); dTb * y * 1 1 1 = + Wz W1 W2

(11.63)

Wz = W1

(11.64)

Wz = W2

(11.65)

+*     




  

  &   podczas skrapla

 
    $ 


  
 &  

& *  o 0   

 

 * W = ∞   
 *

 
prze
  
     

 w skraplac

 

We wzorach Hudlera (11.59)–(11.60) ∆T  *

0  

   
   
 

& 
    &  

  kszej *


mperatury. 1   
 *
$ 0 

*   
 
 1  

    
& * *  
     
 
    &  
 
T     #$ 
1

najpierw ∆T2:

b1

207

ûT2 = Ta 2 − Tb 2

(11.66)

Tb 2 = Ta 2 − ûT2

(11.67)

   T

b2

Po wyeliminowaniu temperatury na podstawie równania bilansu cieplnego (11.2) mamy: Ta 1 − Ta 2 =

Wb (Tb 1 − Tb 2 ) Wa

(11.68)

Ta 2 = Ta 1 −

Wb (Tb 1 − Tb 2 ) Wa

(11.69)

Wb W Tb 1 + b Tb 2 − ûT2 Wa Wa

(11.70)

  



 

   

 45

Po podstawieniu tego wy *

Tb 2 = Ta 1 −



 

Wb Tb 1 Wa  Wb 1 − W a 

Ta 1 − Tb 2 =

− ûT2

(11.71)

  

2

* *


∆T *
 0  *    ∆T &  
 * 0 . 
  2

1

otrzy

Tb 2 = f

(ûT1  Ta 1  Tb 1 )

(11.72)

2$

 *

 0  T lub T w dowolnym przekroju wy 
    *

  
0  

 ∆T 
 &   

  0      o  b

a

miennika: Tb = f (∆T1, Ta1, Tb1 w

Wa (Ta 1 − Ta ) = ±Wb (Tb 1 − Tb )

(11.73)

1    
   &         du. (    

 5 *
 0 
    * i 
 
   
 
 

 

208

11.6.2. TemperatuUDFLDQNL

# 

 
       

   
 $o    
  2

& * 


 a     czynnik b

Q = α a A (Ta − Tsa )

(11.74)

Q = α b A (Tsb − Tb )

(11.75)

Q = kA∆T

(11.76)

 $    

 
 

 2 

  

  
  
    Tsa = Ta −

k ∆T αa

(11.77)

Tsb = Tb +

k ∆T αb

(11.78)

oraz

+* 
 
& * T

sa

≅ Tsb ≅Ts

&    

 

Ta − Ts α b = Ts − Tb α a

  



Ts =

α a Ta + α b Tb αa + αb

(11.79)

(11.80)

)    T *
  

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*
0&    


 



   

 
   

ego. s

11.:VND]yZNLGRSURMHNWRZDQLDZ\PLHQQLNyZFLHSáD

2

 
 

  

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   *
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209

;     
 

 $ 

 
 $  
          
 8&  
3 T > T , T > T )

1.1. skupienia (w przeci  opisuje znane nam równanie

a1

a2

Q = m a c pa (Ta 1 − Ta 2 ) = m b c pb (Tb 1 − Tb 2 )

b1

b2

(11.81)


 
&  0     
a   

  
    emperatury. 1.2. Gdy po jednej stronie wymiennika zachodzi zmiana stanu skupienia czynni&      

  T > T ) lub w wyparce ogrzewanej spa  & $    
  0 w którym indeksy a i b

b2

b1

Q = m a ra = m b c pb (Tb 2 − Tb 1 )

(11.82)

Q = m a ra = m b rb

(11.83)

, +* i zmiana stanu skupienia zachodzi po obu stronach, np. w wyparce   &  
!  
 
& 

 

     

  o 


     

 
&     

  a


&  

 $   
& *3 T > T , T > T , i > i ) przyjmuje po0 a1

a2

b2

b1

a1

a2

Q = m i c pi (Ti 1 − Ti 2 ) + m i ( xa 1 ia 1 − xa 2 ia 2 ) − Kik = m b c pb (Tb 2 − Tb 1 ) (11.84)

    0 
 


 
*            & i& +<&        ' # 


 *


    
 
 (  =   
 

 
   
 *


 (  
& gdy ∆T /∆T < 2, stosu


 

  
  
&    

    

    a 
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0           
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& $ 0  


 



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0 

 
    

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   q, W/m ,   

    

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   *
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$  oblic

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  & *  0

 
 


  

e

     

     ,- <&  
 

   &8 < @ *
    

& 
          gdzie xa  suchego, K, kg/s, masowy stru

1

2

2

kr

210



 

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  y * *

 
 nstrukcyjnego wymiennika. 8 

   $  
      


 
 ( e*
  
0&   ,-A&       0 $  $

e 



 
 


&        ymiennik  
& *  $ 

   

   o 0  11.8. Obliczanie wymienników dla wybranych SU]\SDGNyZQLHXVWDORQHMZ\PLDQ\FLHSáD W praktyc 
   
 




 

 
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;        

  

  
  

$ 

   


      *   0    
  

w dwóch pod

dQ = kA (Ta − T ) dτ

(11.85)

dQ = mb c pb dT

(11.86)

oraz

9 

 8   0  
   



       &   

  
 9 

 4        

dτ od czynnika a Ta do cieczy b w zmiennej temperaturze T. Gdy Ta < T



lpii b cieczy b o masie mb w tym samym czasie dτ. Po przyrównaniu prawych stron obu

kA (Ta − T ) dτ = mb c pb dT

(11.87)

A dT dτ = mb c pb k (Ta − T )

(11.88)

 

 zy 

211


          τ po    Tp    Tk po prawej stronie

lewej stronie oraz od temperatury

Tk

τA dT =∫ mb c pb T k (Ta − T ) p

(11.89)

             *   a   
   
  !"  *                   o*      # 
   
  !" *     
    a  k.  $ 
                *          "    % Ta ≅ const; k ≅ const i       Tk

τA 1 dT 1 T − Tk = = − ln a mb c pb k T Ta − T k Ta − Tp p

∫

(11.90)

&       
          a  *    
     k jest zmienny, jak np. podczas             '  ( anie

               (   

  o   *       
  % 1 1 s 1 = + + k α a λ αb

(11.91)

                 wieniu równa !"  
  !"       

Tk

Tk

Tk

τA dT sdT dT = + + mb c pb T α a (Ta − T ) T λ (Ta − T ) T α b (Ta − T ) p p p

∫

∫

∫

(11.92)

      
   
  )

  
*

   
     e Sp            *  *      rozdz. 11.6): Ts ≈

α a Ta + α b T α a + αb

(11.93)

212

 
   

ûTa = Ts − Ta =

α a Ta + α b T T − Ta − Ta = αa α a + αb +1 αb

(11.94)

oraz ûTb = T − Ts = T −

α a Ta + α b T T − Ta = α α a + αb 1+ b αa

(11.95)

+  *    
 
     x = αb/αa i otrzymujemy: ûTa = x

T − Ta x +1

(11.96)

oraz ûTb =

T − Ta x +1

(11.97)

(            e      stronach w warunkach konwekcji naturalnej. Dla o 
 
  i-

  *   *  
*    α a = LûTa m

(11.98)

α b = NûTbn

(11.99)

oraz

+   
 !,"  !-" y  

 T − Ta  α a = Lx    x +1  m

m

(11.100)

oraz  T − Ta  αb = N    x +1 

n

(11.101)

   
      +   *     
             *  * a   *   !&"%

Zmienna x (x = αb/αa)

213 Tk

Tk

m

dT dT  x +1 1 =   α (Ta − T ) Tp  x  L (T − Ta )m (Ta − T ) Tp a

∫

∫

Tk

dT  x +1 1 = − =   x  L Tp (T − Ta )m +1 m

∫

(11.102)

 1 1  x +1 1  = −  m m  x  Lm  (Tk − Ta ) (Tp − Ta ) 

   

m

Tk

sdT s T − Tk = − ln a λ Ta − Tp − T) a

∫ λ (T

Tp

(11.103)

Tk ( x + 1) ( x + 1) dT dT = = − N α (Ta − T ) Tp N (T − Ta )n (Ta − T ) Tp b Tk

∫

n

∫

=

( x + 1)n nN

n Tk

 1 1  − n  (Tk − Ta ) (Tp − Ta )n 

dT

∫ (T − T )

n +1

Tp

=

a

(11.104)

   

.*  * Tp – Ta) = Θp oraz (Tk – Ta) = Θk  
 !&"  

       % 1  1+ x  τA =   mb c pb Lm  x 

m

 1 1  m − m Θp  Θk

 s Θ k (1 + x ) +  − ln nN λ Θ p 

n

 1 1  n − n Θp  Θk

P   
        my wówczas oznaczenia: (Ta – Tp) = Θp oraz (Ta – Tk) = Θk.

  

(11.105)

   /  e-

 &KáRG]HQLH FLHF]\ Z ]ELRUQLNX QD ZROQ\P SRZLHWU]X

*              owietrzu (rys. 11.11). Temperatura powie   Ta = const, a temperatura cieczy w           (                     
      * opis*     
 ! "  !!" ( *        m przy 
*          01  02 .*  * *  d     m = n"  
 3"  * *    ostaci

214

 1  x + 1 m ( x + 1)m τA =   + mb c pb  Lm  x  mN 

 1 1  − m m    Θ k Θp

 s Θk  − ln   λ Θp

(11.106)

+ *  '    * x=



α b NûTb m ûTa = = α a LûTa m ûTb

N  ûTa  = L  ûTb 

m +1

(11.107)

= x m +1

(11.108)

5\V  6]NLF ]ELRUQLND ] JRUF\P Sá\QHP RGGDMF\P FLHSáR GR SRZLHWU]D

4     *  ,"  *   *    

nawiasie kwadratowym równania

m ( x + 1) ( x + 1)  N 1  x +1  + = + 1 =    m Lm  x  mN mN  Lx  m

( x + 1) m  x m +1  ( x + 1) = + 1 =  mN  x m mN 

m

( x + 1) =

( x + 1)

m +1

(11.109)

mN

 * 
 cowy ( x + 1) τA = mb c pb mN

m +1

 1 1  m − m Θp  Θk

 s Θk  − ln  λ Θp

(11.110)

5  
            
  * * edbany. 11. 2JU]HZDQLH ]ELRUQLND ] FLHF] ]D SRPRF SDU\ QDV\FRQHM

+                       Ta = const, temperatura cieczy w zbiorniku wzrasta natomiast w     (
   i   αa       

215

     αb            *  
 m-

peratury: αa = f (∆Ta) oraz αb = f (∆Tb). .     
 m i n  
* 6    m = –1/4, dla konwe     n 7 01 *  y*  
 3"  

  *  x        
 
   e   *   *  * αa >> αb   x = αb/αa ≅ 0. Dla skraplania wiel * L    4, co dla m = –1/4 daje: m

1  x +1 4 x    =−   Lm  x  L  x +1

0 ,25

→0

6  
 
 3"  *  *     * przy*  * Aτ 1  1 1  s Θk =  n − n  − ln mb c pb nN  Θ k Θ p  λ Θ p

(11.111)

5 *               *   *  * –20% [4]. 2JU]HZDQLHR]L ELDQLH FLHF]\PHGLXPJU]HMQ\P FKáRG]F\P Z]PLHQQHMWHmperaturze

*   &"  
   j              z      *   Ta1 = const,       Ta2). Bilans cieplny takiego procesu opiszemy za po  a     
% •       dQ = mb c pb dTb

(11.112)

dQ = m a c pa (Ta 1 − Ta ) dτ

(11.113)

dQ = kAûTdτ

(11.114)

• dla p  •                   
 Tbp, pod     * Tbk #       ym momencie jest 
*   %

• na wlocie cieczy grzejnej ∆T1 = Ta1 – Tb, •        ∆T2 = Ta – Tb.

216

  
*      ûT1 − ûT2 (Ta 1 − Tb ) − (Ta − Tb ) T 1 − Ta ûT = = = a ûT T −T (Ta 1 − Tb ) ln 1 ln a 1 b ln ûT2 Ta − Tb (Ta − Tb )

(11.115)

5\V  0LHV]DOQLN ] FKáRG]RQ\P Sá\QHP R VWDáHM temperaturze na wlocie Po podstawieniu do równania (11.114) i przyrównaniu go do równania (11.113) otrzymujemy: m a c pa (Ta 1 − Ta ) dτ = kA

Ta 1 − Ta dτ Ta 1 − Tb ln Ta − Tb

(11.116)

        Ta 1 − Tb kA = Ta − Tb m a c pa

(11.117)

Ta 1 − Tb  = e ka/ma c pa Ta − Tb

(11.118)

ln

oraz

co daje Ta = Tb +

Ta 1 − Tb e ka/ma c pa 

(11.119)

217

Przyjmijmy oznaczenie K = e ka/ma c pa grzejnej na wylocie zapiszmy w postaci 

 
      

Ta = Tb + Po podstawieniu tej zal*  

&"   

Ta 1 − Tb K

  1"  
wnaniu

T − T    mb c pb dTb = m a c pa Ta 1 −  Tb + a 1 b   dτ = K    = m a c pa

 K −1 1   (Ta 1 − Tb ) 1 − K   dτ = m a c pa K (Ta 1 − Tb ) dτ   

(11.120) do równania

(11.121)

Rozdzielenie zmien      τ mb c pb K −1 dτ = K 0 m a c pa

∫

Tbk

dTb a 1 − Tb

∫T

Tbp

(11.122)

+        mb c pb Ta 1 − Tbk  K −1 τ ln =−  ma c pa Ta 1 − Tbp  K 

(11.123)

gdzie: Ta1 – Tbp = ∆T1p              tku procesu, Ta1 – Tbk = ∆T1k – spadkiem temperatury na wlocie cieczy grzejnej na kocu procesu.

8                

(τ 7 "  

    *  *  
 &"   awieniu Tbp lub Tbk za Tb: Tap = Tbp +

Ta 1 − Tbp K

(11.124)

oraz Tak = Tbk +

Ta 1 − Tbk K

(11.125)

(       *   zczem zastosujemy       *         
  n   
 &1"    '  
           %      Tbp – Ta1 = = ∆T1p oraz na 

 esu Tbk – Ta1 = = ∆T1k.

218

2JU]HZDQLHR]L ELDQLH FLHF]\]DSRPRF]HZQ WU]QHJR wyPLHQQLNDFLHSáDRVWDáHMWHPSHUDWXU]HPHGLXP

*G\ ]ELRUQLN PD GX* SRMHPQRü OXE JG\ Z\PLDQD FLHSáD SU]H] FLDQN MHVW XWUXGQLRQD D Z *RZQLFD QLewskazana, ]DOHFD VL VWRVRZDQLH GRGDWNRZHJR Z\PLHnQLND FLHSáD 'R UR]ZD*D SU]\MPLMP\ UR]ZL]DQLH SU]HGVWDZLRQH QD U\V  Z NWyU\P FLHF] QDOH*\ RJU]Dü GR WHPSHUDWXU\ Tbk Z ]HZQ WU]Q\P Z\PLHnQLNX FLHSáD ]D SRPRF QDV\FRQHM SDU\ ZRGQHM R WHPSHUDWXU]H Ta1 0DVRZH QDW *HQLH SU]HSá\ZX cieczy w wymienniku jest równe m b = const. 3U]\MPXMH VL  *H WHPSHUDWXUD Z SU]estrzeni zbiornika jest zmienna, ale wyrównana, równa temperaturze cieczy obiegowej na wlocie do wymiennika Tb. Temperatura cieczy na wylocie z wymiennika Tb′ jest WDN*H ]PLHQQD DOH Z\*V]D RG WHPSHUDWXU\ Tb.

Rys. 11.13. Schemat instalacji do ogrzewania mieszalnika za pomoF RVREQHJR Z\PLHQQLND FLHSáD

%LODQV FLHSOQ\ WDNLHJR SURFHVX RSLV]HP\ ]D SRPRF QDVW SXMcycK UyZQD

• dla cieczy w zbiorniku

dQ = mb c pb dTb

(11.126)

• GOD FLHF]\ Z Z\PLHQQLNX FLHSáD dQ = m b c pb (Tb′ − Tb ) dτ

(11.127)

dQ = kAûTdτ

(11.128)

• GOD SU]HQLNDQLD FLHSáD UHGQLD WHPSHUDWXUD FLHF]\ Z ]ELRUQLNX QD SRF]WNX Srocesu jest równa Tbp, pod NRQLHF RJU]HZDQLD PD ZDUWRü Tbk 6Lá QDS GRZ SURFHVX Z GDQ\P PRPHQFLH MHVW

ró*nica temperatury:

219

• na wlocie cieczy do wymiennika ∆T1 = Ta1 – Tb • na wylocie z wymiennika ∆T2 = Ta1 – Tb′  
*  temperatury wynosi zatem ûT =

ûT1 − ûT2

ln

ûT1

=

ûT2

(Ta 1 − Tb ) − (Ta 1 − Tb′ ) Tb′ − Tb = T −T (Ta 1 − Tb ) ln a 1 b ln Ta 1 − Tb′ (Ta 1 − Tb′ )

(11.129)

+    
     & "  
  do równania (11.127) m b c pb (Tb′ − Tb ) dτ = kA

Tb′ − Tb dτ Ta 1 − Tb ln Ta 1 − Tb′

(11.130)

       

ln

Ta 1 − Tb kA = Ta 1 − Tb′ m b c pb

(11.131)

   Ta 1 − Tb Ta 1 − Tb '

=e

kA m b c pb

(11.132)

co daje Tb′ = Ta 1 −

.*  *

K = eka wylocie zapiszemy w postaci

/ m b c pb

,

Ta 1 − Tb e

ka / m b c pb

(11.133)

 
       

Tb′ = Ta 1 −

Ta 1 − Tb K

(11.134)

+    *   &-"  
  
 &,"   

220

T 1 − Tb   mb c pb dTb = m b c pb Ta 1 − a − Tb  dτ K   1    = m b c pb (Ta 1 − Tb )  1 −   dτ  K   = m b c pb

(11.135)

K −1 (Ta 1 − Tb ) dτ K

Rozdzielenie zmiennych τ

K −1 m dτ = b K 0 m b

∫

Tbk

dTb a 1 − Tb

∫T

Tbp

(11.136)

    mb Ta 1 − Tbk  K −1 τ  = −  ln mb Ta 1 − Tbp  K 

(11.137)

gdzie: Ta1 – Tbp = ∆T1p jest spadkiem temperatury na wlocie cieczy do wymiennika na      Ta1 – Tbk = ∆T1k jest spadkiem temperatury na wlocie cieczy przy



   8          Tb′      τ = 0) i 

   τ"  *  *  
 12"   eniu za Tb Tbp lub Tbk Tbp′ = Ta 1 −

Ta 1 − Tbp K

(11.138)

oraz Tbk′ = Ta 1 −

Ta 1 − Tbk K

(11.139)

W przypadku ch      Ta1 wzór jest identyczny z 
 1-"            ymiennika: na po    Tbp – Ta1 = = ∆T1p   

  Tbk – Ta1 = ∆T1k.

11.9. Obliczanie regeneratRUyZFLHSáD       *     adowego, tzn. energii cieplnej gazów odlotowych (np. spalin). Regeneracyjne podgrze-

        ;          

221

     (             z                  (  
       * *       9         <  =&> ?      
 
        o   (    
  *  x"     o  * λ, a         –     * 9 * 
  * *              r *  Hausen [2] wypro     
 
*  % • dla gazów

α Aj  ∂Tg  (Ts − Tg )   =  ∂x τ m g c pg

(11.140)

α Aj  ∂Ts  (Tg − Ts )   = C  ∂τ  x

(11.141)

•   

gdzie: Ts, Tg –       
 Aj –   y               m g –   masowy gazów, cpg –       inieniem, C –   *     $       
        
 i         *     
   
  (
w      *         eneratora pochodne temperatury 
   x            a 5         :     czynnik ogrze ? 9 * 

* 
   % •      τA Q = α A A (Ts − TgA )m τ A

(11.142)

•     τB Q = α B A (TgB − Ts )m τ B

(11.143)

gdzie Q *  .0  

+    
 2&"  21"        ró*                  Q=

1 A (TgB − TgA )m 1 1 + α Aτ A α B τ B

(11.144)

222

+    
 +   *  * 
   ania

     % 1 1 1 = + k α Aτ A α Bτ B

(11.145)

 
 22"  
   * Q = kA (TgB − TgA )m = kAûTm

gdzie ∆Tm = (∆T1+∆T2"0& wylocie z regeneratora.

(11.146)

     
*       

11.10. Wybór typu konstrukcji przeponowego wymiennika FLHSáD +          *        zarówno wiedzy merytorycznej po                      ?     o*      
 
  kcyjnych, jak i  
              omii                *    * a   +  typ konstrukcji *   *               *     
   +   rodzajem konstrukcyjnym jest tzw. wymiennik wielorurkowy. Do tej grupy wymien-


    
*       *    t            " @      i  9 * 
**           aniem rur od    i          A           #        
  6       
*   wymienniki z grafitu impregnowa *                 towe, zbudowane z pakietów blach metalo  # 
             e +           *      adzona w kolektorach: wlotowym i wylotowym. Na rynku jest oferowanych wiele  
   *   
*nicowanych cenach.

@      *          A  
      * 
       zty inwestycyjne.

223

+                *        @   
 * **    
)


           -         d    6  *     *     umieni oraz             (     *     o         9        ymienni               j rur, S = L/d [4]. Kolejne za   
            
   elementów aparatury.

2]QDF]HQLD $ D E & & &
& F G ( J K ,O N / P P P ?? 1 1 3 4 I

S

"

T S S 5 5 5 U U 6 V 7 8 9

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±


± ± ± ± ± ± ± ± ± ±

SROH SU]HNURMX SRZLHU]FKQLD P G\IX]\MQRü FLHSOQD P ÂV
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