Electrochemical Water Processing

and (4.15)

Equations (4.15) relate to a constant flow through a constant inter-electrode spacing cell with a length, L, as shown in Figure 4.7. 4.3.2.3

Water Flow Rate in Processed Chamber

Another important relationship to define in establishing the parametric analysis is the fluid speed, v, through the cell. A cell with a fixed dc voltage impressed across its electrodes will conduct more current at higher water flow rates because the emergent and the average water TDS will be higher than those of slower flow rates. The water speed in terms of distance per unit time may be expressed in terms of volume of water flow rate and cell geometry.

116

ELECTROCHEMICAL WATER PROCESSING

Specific relationships can be derived to express the volumetric flow in terms of fluid speed. One form of this relationship in the English system of units, which will be employed here, is described as follows. Because 1 gallon has a volume of about 231 cubic inches, the speed of a unit volume of water flowing through a cell can be determined by v =F

231 (W-p)

in/min

(4.16)

where W = width of the cell, and p = spacing between membranes as shown in Figure 4.7 4.3.2.4

Solute Concentration Along the Length of the Cell

Now, we can estimate the TDS, cx, within the processed chambers at any point, x, along the flow path. The equation is derived as follows. In Figure 4.8, we show the volume of water that is passing through the cell between the cation and anion membranes. The direction of water flow is in the x-direction, and electric (ionic) current flow is perpendicular to the x direction as shown. Figure 4.8 illustrates that the net change in solute concentration along the length of a cell is due to the extraction by electro-dialytic migration from both faces all along the length of the processed water cell section. Solute ion extraction

Flow in

Figure 4.8 Section of processed water compartment.

Flow out

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

117

Let us now examine the manner in which the quantity of solutes changes as the stream of solution progresses down the length of the cell. The amount of solute, Qf, that passes into (enters) any section, p-W-dx, from the more concentrated upstream regions of the cell whose cross section is W-p, and at some distance, x, from the entrance, and over a time interval, dt, can be expressed as Q f =W-p-v-/?-c x -if

(4.17)

The multiplier, ß, in Equation 4.17 is the density of the solution flowing through the cell. Even though ß will change with either time or distance down the cell length, it will be treated as constant in these analyses. Low concentrations of solutes encountered in most water processing tasks make the densities of the solutions about equal to that of pure water. When making more accurate performance estimates for seawater desalinators, one may wish to account for the higher densities and for the change in density along the cell length. However, it would appear to be an unnecessary complication in the mathematics for a first performance approximation. The quantity of solute change, dQ f , or lost from the flow stream is now dQ f =W-p-v-/?-dc x -d£,

(4.18)

or, more appropriately, (à&) =W-p-v-ß-dcx \ dt Jt

in2- —-^-ppm3 min in

rr

(4.19)

If we stay with the English system, the terms above are expressed dimensionally in units of p v c

in in/min ppm

118

ELECTROCHEMICAL WATER PROCESSING

W Q

in in 3 -ppm/min

Now, we must examine the rate of which the solutes are leaving that same region or volume of cell by mechanism of electrodialysis. Equation (4.19) is now related to the rate of which the dissolved solids are leaving the stream via dialysis through the membrane walls. The current flow, Ix, through the element of area W • dx is I x = i x - W - d x —¿--in-in (4 20) in If a constant, X, that is a conversion factor for electric charge to quantity of solids transported is introduced then Equation 4.3.8 may be rewritten later in the form of charge transfer. And, the rate with which dissolved solids, (dQ/dt) d , are removed by the electrodialysis process from the region volume W-p-dx, whose length in the flow direction is dx, and whose area is W • dx, is represented by — 1 =i x -77-W-dx-A g m / u n i t t i m e , V dt Jd

(4.21)

The constant, X, is in units of grams per unit time per amp. The term, n, is the coulombic efficiency of solute transfer through the membranes. Referring again to the Faraday equivalent, and using the average gram molecular weight, GMW, of the solids that we identified earlier as 60 gm, we can assign a numerical value to X. About 60 grams of solute should be displaced by 26.8 amp-hr or 1 amp hr should produce about 2.3 gms at 100% coulombic efficiency. Dividing the 2.3 g m / h r figure by 60 to convert from hours to minutes, the conversion factor becomes A= 0.038gms/amp - min

(4.22)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

119

In order to rapidly arrive at quantitative solutions to the mathematical relationships of this first approximation, some major assumptions made so far are: 1. Non turbulent flow 2. Equivalent conductance of solutes remain essentially constant with concentration 3. Equivalent weights of solutes is averaged at 60 The equations can be kept more general and in shorter form by retaining the symbols instead of numerical substitutions for some quantities such (p and X. Returning to the two rates of change of quantity of solutes as represented by Equations (4.19) and (4.21), we must now equate the removal rate of dissolved substances that enter the region in question to the rate of which the solutes are entering the region, p-W-dx, in order for steady state conditions to be established. To solve for cx in the cell, Equation (4.19) must be set equal to the negative of Equation (4.21).

ÉQ)=-\*Q.\ dt i,

(4.23)

\ dt

In order to equate and evaluate the two expressions, it is necessary to put them into the same units. If in Equation (4.19) ß is in units of grams/in 3 , and if the right side of Equation (4.18) is multiplied by 10~6to convert concentration from ppm to fractional ratio, the two relationships in question can be equated. Equating, we obtain W-p-v-dcx-ß-10-6=ix-i]-W-dx-A where cx continues to be expressed in ppm.

(4.24)

120

ELECTROCHEMICAL WATER PROCESSING

We now want to put the variable, i , in terms of the variable, cx, which we wish to solve. If the applied voltage, E, is constant all along the cell length the following expression becomes simple to solve exactly. Substituting for ix in the above net expression, as was shown in Equations (4.15), we have Ec

(4.25)

and

X

p-v-dc x -ß-l(r 6 =

77-A-dx.

(4.26)

Transposing terms, and solving the differential equation, dc x

-E • r¡- X

v

. i

ln(c x ) =

(4.27)

(pp2v-ß

cx

EriÀx

=

+ K(const), or

2

(p-p -v-ß

(4.28)

cx =K-e" M x , and the constant, K, is evaluated as co from the limiting condition that at x = 0, c = c . Then c finally becomes '

X

O

J

X

c„e

E-ifA-x p2vß

(4.29)

The total current, I, passing t h r o u g h the cell is 1= W- \ix- dx. Substituting from Equations 4.25 a n d 4.28 for i a n d cx a n d performing the integration, w e get l

1=

(P-P

■\e

E-1J-2.-X

^Vßdx

(4.30)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

121

Equation (4.30) has the form L

I = A-¡e~Bxdx,

(4.31)

0

and when the integration is carried out, has the final form I = -~[e-BL-l]

(4.32)

By re-substituting the polynomials for A and B, the expression for I is W-v-

ß-p-c0 TJÀ,

(4.33)

For many initial estimations of performance, we will let ß = 16.4 gms/in 3 for dilute solution density, and we will use the previously estimated value for À, of 0.038 gms/amp-min. 4.3.2.5

Figure of Merit

Introducing the idea of a coefficient of performance—or figure of merit—facilitates performance comparisons of operating ED systems. Among the characteristics of an ED system such as size, life and cost, perhaps the most important parameter is that of the "effectiveness" of operation in terms of amount of power required to remove a given quantity of dissolved solids per unit time from the input water stream. Let us define a figure of merit, FM, simply as ™ . Quantity removed unit time FM = Power Input _ gallons per minute x ppm change input watts = F-(C0-CL) = F-(C0-CL) power I•E

(4 34)

122

ELECTROCHEMICAL WATER PROCESSING

Such a relationship enables us to compare various design trade-off in an even more meaningful fashion. 4.3.3

Numerical Evaluation Program

Numerous and popular personal computer software programs can find the solution of the above equations in a simultaneous manner. A convenient means of solving a series of simultaneous equations is afforded by a basic program called FORMULA ONE published by Alloy Computer Products, Inc., and MathCad by Mathsoft, Inc. Computations made by means of these programs have been rapid and quite straightforward. All of the relationships developed so far are sufficient to perform a first approximation of the performance of a flow-through type of ED water processor. The system of simultaneous relationships have been collected together and conveniently listed below as Equations 4.35. r

*

cx

= 26-10 4 -^-lE-]-x-2.8-W-3/p2-v)

y

V

= c0-e = F-231/W-p

h

= E-c o /26-10 4 -p

h

=

I

»

E-cL/26-104-p c0-W-v-ß-p

C

L

h

= ia-L-W

Q Q

= 10-/.-/

= F-(c0-cL) power = I-E

1-

e

ErjXL (f>-v2vß

(4.35)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

123

For a series of examples and the implementation of numerical computations based upon the set of equations in (4.35), refer to Chapter 4, Section 4.4 and Appendix G. 4.3.3.1

Second Approximation, Part II

In the preceding analysis, a number of assumptions were made that do not entirely represent actual operating conditions or take into account the fact that both current density and coulombic efficiency vary along the length of the cells in the direction of water flow. Examples of the manner in which the current density varies with distance along the fluid flow path are shown in Figures 4.9 through 4.14. These are graphs showing the manner in which both the concentration, TDS, of solutes and electric current density vary along the length of a cell as flow rate, F, and voltage, E, are changed. Since the current density is directly

Figure 4.9 Single cell performance factors. For F = 0.10 gal/min, E = 10 volts.

124

ELECTROCHEMICAL WATER PROCESSING

proportional to the TDS concentration, it is seen to vary in shape in the same manner as concentration. Figures 4.9 and 4.10 clearly show how both current density and TDS are non-linear with distance at 10 volts as cell potential. If the cell potential is reduced to 2 volts, the curves become more straight lines. Similarly, Figures 4.11 and 4.12 show how the shape of the curves for current density and concentration versus distance along the cell length is less non-linear at the higher flow rate of 0.50 gal/min, and becomes even more linear at the lower cell potential of 2 volts. These charts show the differences in the two curves if the flow rate is increased from 0.10 to 0.50 gal/min at the two voltage values. The last two graphs, Figures 4.13 and 4.14, show the relative effects of increasing cell potential to 20 volts at 0.50 gal/min, and changing to an even higher flow rate of 1.0 gal/min at 10 volts potential.

Figure 4.10 Single cell performance factors. For F = 0.10 gal/min, E = 2 volts.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

125

The graphs show that as impressed voltage is increased or asflowrate is decreased for any particular cell configuration, the curves become more pronouncedly non-linear.

Figure 4.11 Single cell performance factors. For F = 0.50 gal/min, E = 10 volts.

Figure 4.12 Single cell performance factors. For F = 0.50 gal/min, E = 2 volts.

126

ELECTROCHEMICAL WATER PROCESSING

Figure 4.13 Single cell performance factors. For 0.5 gal/min, E = 20 volts.

Figure 4.14 Single cell performance factors. For 1 gal/min, E = 10 volts.

Averaging the entrance and exit current densities are not reasonable approximations.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

4.3.4

127

Multiple Cells in Parallel

Different characteristics of performance can be achieved by placing more than one cell in either series or parallel connection hydraulically. We will now proceed to evaluate the advantages and limitations of such design variations. 4.3.5

General Characteristics

The amount of solids removed from a stream of water per unit time flowing through a cell is solely a function of the following parameter: 4.3.6

Total Electric Current through the Electrodes and Membranes

For constant voltage operation, the change in concentration in the water stream between input and output values is also dependent upon the flow rate through the cell. At constant voltage, if the flow rate is lower, the net change in ppm is greater because the water spends more time in the processing chamber. The total quantity removed per unit of time is constant and directly proportional to the total electric current. Let us now examine some of the trade-off factors between design parameters in the construction of a single cell unit. The design goal is to provide for the needed dissolved solids removal rate at minimum capital cost and minimum power cost, while maintaining an electric current density below some prescribed level. Referring to the set of Equations (4.35) developed so far, we can now proceed with an exploration of the parameter interdependencies that lead to design optimization. We need to view both the flexibility and constraints associated with the basic cell operation. A specific set of design criteria

128

ELECTROCHEMICAL WATER PROCESSING

serves as an initial example to demonstrate a performance analysis approach. • The quantity of dissolved solids to be removed per unit of time determines the total electric current required in the cell, regardless of the length and width of a cell and the flow rate of water. • The procedure in establishing the design parameters is as follows; Let us define a set of performance requirements. For example: a. Total volumetric flow rate = 0.50 gal./min. b. Input TDS = 200 ppm c. Desired output TDS = 2.0 ppm This defines the set of criteria that must be met by the unit. For further details on computer formatting, see Appendix G. However, there are numerous other considerations to be taken into account before hardware design can be finalized. Some of these considerations are: • • • •

Maximum allowable current density Materials and component costs Aspect ratio of packaging Volt-amp ratio

A file (see Appendix G) has been set up to compute the results of various design inputs. If we set the above criteria as input values and set some maximum value of current density to between 0.05 and 0.20 amp/in 2 , along with a cell width of 6 inches, we can evaluate the voltage, length, and power requirements for a single cell. We let F be such that the cell will produce water at 2 ppm from 200 ppm entrance at 0.50 gal/min flow volume. The removal rate of solutes at the 0.5 gal/min and 198 ppm change is 99 gal-ppm/min.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

129

The Table 4.4 below illustrates the range of dependent variable values obtained. As the current density is raised from 0.05 to 0.20 amps/in 2 the cell potential must be increased accordingly and the length of cell is less to achieve the removal of 198 ppm from the water flow-through. We see that the only changes are the length of the cell and the voltage needed to extract solutes at the 99 gal-ppm/ min rate. As the input current is increased, the power level increases accordingly because the cell length had been reduced proportionately with the consequent increase in cell internal resistance, necessitating a rise in voltage to maintain the ppm difference at input and output. Note that the total current, I, is constant. Thus, it is evident that in order to increase "power efficiency" for any removal rate, keeping all other parameters constant, current density should be reduced with the consequent increase in cell length. Now, let us examine a multiple cell arrangement with the same general parameters of spacing, cell widths, 0.5 gal/min flow, and input-output concentrations. Another computer file described in Appendix G addresses the mathematics relating to a stacked series of n cells. Again, for purposes of illustration, let us set the number of cells in the stack at an arbitrary figure of n=10. The Table 4.4 Sample computation for a single cell. E

L

I

Power

0.05 A/in 2

13 volts

92 in.

9.9 amps

128 watts

0.10

26

45.9

9.9

256

0.20

52

23

9.9

513

i

0

130

ELECTROCHEMICAL WATER PROCESSING

Table 4.5 Sample computation for a 10 cell array. i

E

L

I

Power

16.3 volts

15.3 in.

0.99 amps

160 watts

o

0.05 A/in

2

0.10

32.5

7.7

0.99

320

0.20

65

3.8

0.99

641

data calculated by the parametric analysis are given in the Table 4.5 above. As in the case of the single cell, the only parameters that have changed in moving to higher current densities are the stack voltage and stack cell length. Some power consumption sacrifice is made in stacking the cells, but a gain is achieved in compactness of design. The question now is why does one wish to stack up these ED cells at all? The resulting multi-cell configuration has little effect on performance. The main benefits of a multiple cell design over a single cell is more practical module dimensions of length and width, and the avoidance of very high currents and low voltages if the cells are connected in series electrically. Lower current power supplies along with more practical configurations and system packaging will also tend to lower capital costs of manufacture. The configuration of a stacked module versus a thin and rather long single cell is undoubtedly a more practical design for most applications. 4.3.7

Coulombic Efficiency Variation

So far, coulombic efficiency has been treated as constant throughout the range of the operation of ED cells and at all TDS values and current densities. In actuality, the coulombic efficiency, r\c, does, indeed, depend upon many factors such as concentration availability of the specific ions, current density, pH range, membrane properties and even the characteristics of the ions.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

131

The probability that any one ion in solution with other ionic species will be the charge carrier across a membrane is proportional to its mobility and availability (or concentration), as compared to that of other similarly charged ionic species. Another factor influencing the selectivity of one ion over another in the transfer process is their relative mobility. We can show what general form the relationships would probably take. Taking the transport process for cations, and more specifically, for sodium ions as an example of how the development of a relationship between population density of ions and coulombic efficiency may proceed, we will define the following variables. QN QH V /i+ t+

= = = =

quantity of Na + ions quantity of H + ions volume of electrolyte solution ion mobility of cation transport number ratio for cation through membrance

The rate at which a specific cation is leaving an electrolyte volume, V, is directly proportional to the concentration ratio of that ion to those of all other available (+) ions in the volume. The dependency is further modified by the relative values of the mobility of each ion and their relative transport numbers through the membrane. To abbreviate the necessary mathematical characters that will be manipulated, the constants, a and b, are given as composites, or products of the mobility and transport number. In other words, a = ßjN-tN b = fjH-tH

and (4.36)

where the constants, jo, and t, are mobilities and transport numbers for sodium and hydrogen ions, respectively.

132

ELECTROCHEMICAL WATER PROCESSING

Based upon the probability of transfer of specific ions, the rates can be represented as follows: aQ ^^-i-A-, " , , and dt (aQN+bQH) bQ ^-=-i-A-, " x dt (aQN+bQH)

(4.37)

Achieving the solution of these equations can be made simple for the sake of illustration by assuming that the concentration of hydrogen ions in solution is constant. This assumption is not entirely unreasonable because the H + ion concentration is very low due to ionization of mid pH range water, and reasonably constant because they are being replenished by continuous ionization of water as dialysis proceeds. Treating Q H as constant, we solve the first differential equation for Q N by direct integration in the manner shown below. aQ

N

^

+

bQ

N

^=-i-AQ

N

,

or multiplying through by dt, and dividing by b • Q H • dQ N we get dQw bQ H

+

¿ Q N

=

QN

.jA.d bQ H

t

(4.38)

Integration, and re-arrangement of terms yields •Q N + l n Q N = - ^ ¿ — t + K,or bQ H ^ ^ bQ H i-At

QN

b H

QN-e °

=Ke"

bQH

(4.39)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

133

Boundary conditions are such that when t=0, Q N is some maximum value, N. And K = N • e1*2", thus the equation for Q N becomes N-Q N

(4.40)

Ut

Q N =N-e b Q « -e"bQH Differentiating the last expression, with respect to t to obtain the rate of change of sodium ions with time, N-Q N

J_C[QN_ = gbQ H

N

dt

U-t

.g bQH dQ f dt bQ H +

iX bQ H

moving terms to solve for dQ N /dt .-

dQ N

dt

. .. 1

N

N-Q N

1A

bQH

e

bQfl

,

.

l

bQH

e

Ut

• e bQH

N-Q N Ut b< 2 H . e bQ H

(4.41)

We are primarily interested at this juncture in seeing how the coulombic efficiency depends upon the variables at any time, t. The coulombic efficiency, r¡c, is defined as the ratio of the rate of equivalent ion transfer to electric current flow over, and may be represented as dQ > 7 77c

a

dt

(4.42)

Without pursuing the mathematical manipulations further, one can see that the form taken by the expression for r/c is that of an exponential with Q N appearing in the exponent.

134

ELECTROCHEMICAL WATER PROCESSING

As an approximation, and as a first compensation for the variation in r¡c, we will consider only the effect of ion concentration. In general, as the concentration or availability of specific ions to be transported across membranes becomes lower, the coulombic efficiency for transporting that specific ion will diminish. The decrease in 77c is simply due to the fact that the competition of H + and OH" ions becomes greater because their availability as charge carriers becomes proportionately greater than the less populous ions. Hence, the functional dependence of r¡c upon concentration, cx, at some point x within a cell must be determined. A simplified, but useful, form of the functional dependence that could be taken would be the exponential relationship i% = (l-e^),

(4.43)

where \]/ has a value such that the coulombic efficiency is nearly unity when cx was at some level where the competition for transport by other ions was negligibly small. Such values of cx can be determined experimentally. To define the function completely, at least one empirical data point is needed. Some simple measurements should be made for each of the membranes to be employed at more than one concentration of dissolved substances. We have graphed Equation (4.43) to show how coulombic efficiency varies with concentration for different values of the constant, \\f. Empirical data provides the information needed to quantify \\f. To properly account for the dependency of r¡c upon cx it is necessary to insert the expression for r¡c into the preceding formulas for cx and ix as seen in Equations (4.31) and (4.35). Because of the additional variable terms, the resultant expression becomes more complex mathematically when accounting for a non-constant r¡c. For example, the solute concentration takes the form

ANALYSIS «fe MODELING ELECTRODIALYSIS SYSTEMS

pvdcx/? =

Ec

X

r¡- À- dx

135

(4.44)

substituting for r/c from the equation rç = ( l - e ^ ) ,

(4.45)

E-c,

(4.46)

and we get p-v-dcx-ß:

(l-e-^)-A-dx,


Rearranging Equation (4.46), and putting it into a form for solution of c , dc v cx(l-e^)

-El •dx
(4.47)

A graphical approach to an approximation of solute concentration may be taken by simply assuming values of \j/, and multiplying the concentration step-by-step along the cell length by the adjusted values of r¡c. A program (anal-1. wk3) was devised to accomplish this series of calculations to explore the effect upon cell performance by a varying coulombic efficiency. The next four graphs show the dependence of r¡c upon TDS for two different values, 0.01 and 0.10, of the coefficient \|/, along with their effect upon the variation of cx along the length of the cell. Figure 4.15 and Figure 4.16 show both coulombic efficiency and TDS data for \j/=0.01. Figure 4.16 shows that the manner in which the TDS varies with distance is more pronounced by assuming a constant (r¡c, represented by the letter, J, in these LOTUS plotted graphs) than it does with a variable r¡c and similarly Figures 4.17 and 4.18 for y=0.10. The TDS change calculations with distance along the cell with an assumption of either variable or constant r¡c.

136

ELECTROCHEMICAL WATER PROCESSING

Figure 4.15 Coulombic efficiency versus solute concentration. y/= 0.01.

Figure 4.16 TDS versus distance.

y/=0M.

The value of y must be determined prior to more accurate evaluation of coulombic efficiencies at lower TDS. This can be simply accomplished by performing a series of measurements of sodium ion transport numbers with the specific

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS Exponential approximation (1-exp(-cx*p)) for P = 0.1

1.2

s

137

1

x

g- 0 - 8 £ 0.6 | 0.4 ° 0.2 0

0

50

100

150

200

250

Concentration PPM Figure 4.17 Coulombic efficiency versus solute concentration. \f/= 0.10.

Figure 4.18 TDS versus distance. \¡/=Q.\Q.

cation membrane needed. These measurements should be performed at different concentrations of sodium salt on the transfer source side. Experiments to make these measurements are straightforward to implement.

138

ELECTROCHEMICAL WATER PROCESSING

4.3.8

Further Considerations

All of the preceding analyses have been performed with many assumptions in mind, and by ignoring some important factors. In our continuing development of design mathematics, we will need to account for many of these factors. These include: a. b. c. d. e. f. g.

Membrane resistance Resistance of the waste water compartments Electrode potentials and polarization effects Concentration potentials across membranes Variable coulombic efficiencies Heat dissipation and effects on cell resistance Electrical losses by conduction through common water manifolds

In the next section we will treat those systems with either static solutions in waste and processing compartments, or with a limited volume of fluids in continuous re-circulation.

4.4

Flow-Through Design Exercises

In this section, we will go through a few design variations with the mathematical developments of Section 4.3. All of the following exercises are for a system in which the water passes through the electrode assembly only once. First, let us quickly review the overall status of the analysis at this point in its development. The design types based upon these assumptions and equations will be called Designs-A. These are the basic assumptions made so far. 1. Uniform, non-turbulent flow of processed water through cells 2. One pass through cells for total processing 3. Constant equivalent conductance of solutions—hence resistance is directly proportional to concentration

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

139

4. No losses due to diffusion or fluid interconnections 5. No opposing electrical potentials 6. All electrical resistance is due only to processed water solutions 7. Coulombic efficiency, n, is constant The last assumption is maintained here even though some thought was devoted to exploring variable, current density dependent functions for n. Since all ED extraction of dissolved solids takes place in one flow-through operation, the equations employed at this point for a design exercise are those developed in Section 4.3. The salient parameters for this design approach are listed below. Through our approach to the design exercise, we will determine which parameters are independent and which are treated as dependent variables. For additional detailed performance estimates, see Appendix G. The variables and constants employed alomg with their abbreviations are defined in Table 4.6. Some results of these computations are given below as examples of the type of performance expected from a cell operating within the envelope of the above assumptions. The important issue to resolve at this time is the establishment of a methodology to arrive at the specifics of design for any particular application. In other words, how does one go about designing an assembly of electrodes and membranes along with the needed power supply requirements to meet the needs of an application, and in a reasonably optimized fashion? The first step is to define the performance needs in terms of input TDS, desired output TDS, and flow rate of water to be processed—or the quantity of processed water output required per unit time. Immediately after that determination, it is necessary to define certain operating limits of the cell components, such as maximum allowable current density, practical maximum length of the electrodes, number of

140

ELECTROCHEMICAL WATER PROCESSING Table 4.6 Abbreviations for the various parameters. Definition

Dimensions

io

entrance current density

amps/in 2

iL

exit current density

amps/in 2

ri

coulombic efficiency

%/100

P

cell spacing

in

c

o

entering TDS

ppm

E

cell potential

volts

V

flow speed

in /min

F

volume now rate

gal/min

C

exit TDS

ppm

L

cell length

in

W

electrode width

in

I

total current

amps

power

power input

watts

Quantity

L

cells in a series stack, and acceptable physical dimensions of the stack module(s) for a specific application. In general, the practical constraints placed upon the hardware design must be clearly identified. For illustrative purposes, we assign the somewhat unrealistic, but convenient value of 100% for r|, along with values for some of the other cell parameters. Now, we can proceed with a sample design. 4.4.1

Exercise #1

Let us take a single cell structure, and calculate the cell parameters necessary to produce a water treatment capacity of 0.50 gallons per minute flow rate (see Appendix G).

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

141

We also wish to have a 10:1 reduction ratio between the input and output water TDS. The maximum sustained current density for extended life of electrodes is about 0.10 amps/sq. in. The first step in estimating the design configuration is the determination of the cell area needed to accomplish this task. When we insert the values of input TDS, co, and output TDS, cL, to solve for cell width and length, these physical dimensions of the cell will determine the ratio cjch at a specific flow rate independent of the magnitude of the concentration of salts in the input water, if the voltage applied to the cell is kept constant. Only the power input requirement will be affected. A more practical restriction for cell operation is to limit the current density. If the maximum current density, i , is made constant at a specific value of perhaps 0.10 amp/sq, then the ratio cJcL is linearly dependent upon the total cell area. If we let the input TDS = 200 ppm, then the output TDS is 20 ppm. The data computes to the following values for cell operation: co = 200 ppm cL = 20 ppm Ae = 730 sq in

Input Power = 320 watts F = 0.50 gal/min I = 10 amps

To determine the voltage necessary from an external power supply to the cell, it is necessary to decide upon cell spacing, p, (distance from membrane to membrane and membrane to electrode). We will assume all spacing to be the same, and for this exercise, we will let p = 0.25 in. Then the dc power supply potential, E, is 32 volts, and the power is 320 watts. Further examination of some of these parameters shows that if either the spacing or maximum input current were to be changed, the power required and the power supply voltage would also change. For example, if the spacing were reduced to 0.12 in., the power would decrease to 160 watts, and the driving cell potential would then need to

142

ELECTROCHEMICAL WATER PROCESSING

be only 16 volts instead of the 32 volts to provide 10 amps through the cell. These changes all occur because of the lower cell internal resistance at the smaller spacing. However, the engineering trade-off in this last instance is the greater difficulty in fabricating and controlling uniform water flow as cell spacing is made smaller. Lower operating cost for the cell in terms of electric energy per processed gallon may also entail a lower degree of reliability of cell operation. Another interesting trade-off is between current density, operating cost and equipment size. If the maximum current density, io, is lowered, for example, to 0.05 a m p / s q in, the power is reduced from 320 watts to 160 watts for the 0.25 in. spacing cell. In essence, the energy efficiency is doubled, and the operating life of electrodes is extended, but the cell becomes twice the size and twice the cost of the materials. One must decide what set of values provides the best overall cell performance and lowest total cost. To further illustrate the various combinations of parameter values in the design and operation of single cells, the following additional short examples are offered. Again, the following are some results obtainable from the system of Equations 4.35, when other values are selected for cell geometry, current density and flow rates. There is a wide range of choices in assigning dependent and independent variables. 4.4.2

Exercise #2

For example, if we select cell dimensions and electric current densities as the independent values 4.4.2.1 Predetermined Independent Variables L = 100 in. W = 10in. i o = 0.10 amp/in 2

r¡ = 1.0

p = 0.25 co = 200ppm F = 1.0 gal./min.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

143

Then, we may solve for these variables as dependent ones, and their values become, I = 19.8 amps E = 32.5 volts

Power = 644 watts cL = 41 ppm

Another calculated term can be introduced here that we will refer to as the Figure of Merit, FM. It is defined as FM = F/(co/cL). FM is a measure of the quantity of material removed from the water stream per unit time for the power input required. FM is in units of gal-ppm/watt-min., and it enables us to conveniently assess and compare the effectiveness of one design to another. The figure of Merit for the case above is FM = 0.246 gal-ppm/watt-min Cell voltage, E, and entrance current density, io, can be interchanged as dependent-independent variables. If it is desirable to establish a maximum current density because of electrode life considerations, then io is chosen as independent. Note: The value 0.10 amp/in 2 was selected from the data developed in laboratory studies with carbon electrode structures snowing carbon electrode erosion rate versus current density in dilute solutions. If the current density exceeded 0.10 amps/in 2 for most of the carbon types employed in these tests, the life of electrodes began to diminish rapidly. If, on the other hand, one limits the operating voltages to some maximum value for safety reasons, then E may become the independent selection. Similarly for all the other parameters of the cell, one may select and make any combination of the parameter independent as long as the conditions of sufficient independent equations matches the number of parameters left as dependent ones (see Appendix G). In the above solutions, we see that the potential necessary to provide a maximum current density of 0.10 amp/in 2

144

ELECTROCHEMICAL WATER PROCESSING

is 32.5 volts, and the emerging solute concentration in the processed water chamber is 20 or more ppm. If the ppm level of the input water were to be either increased or decreased, only the operating voltage would be affected either upward or downward, respectively, and the ratio of the ppm of the input water to that of the output water would remain constant. 4.4.3

Exercise #3

If we wish to change the ratio of input to output ppm, it is necessary either to change the maximum permissible value of current density, io, or the cell geometry. As another example, if the length of the cell is shortened to 50 inches, and all other independent parameters of the above circumstances are kept the same, these are the operating results. Calculated Values iL = 0.021 amp/in 2 v = 46.2 in/min E = 32.5 volts I = 9.9 amps

r x = 445 ohm-in. cx = 146 gms c* = 41.4 ppm power = 322 watts

Even though the emerging processed water from the 50 inch long cell is at a measurably higher TDS level, the power level, total current and operating voltage are all essentially unaltered. There is no sacrifice in the figure of merit with the shortened cell length. The current density at a point 50 inches down the length of a 100 inch long cell is 0.021 amps/in 2 , the same as the current density, cL, at the exit of the 50 inch long cell. In other words, the processed water has no knowledge at any point in its path of the total length of the cell. Valuable design exercises such as the one above can be performed rapidly to explore the engineering, cost and performance trade-off possibilities.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

145

We will see later that the most effective way to increase the FM of operation is by the reduction of internal cell resistance. Due to overcoming the ohmic loss through a cell, the only manner available to increase the FM at this point in the presentation is by the reduction of electrical path length through the cell, or smaller spacing, p. Reducing the spacing will result in improvements in FM because less voltage will be required to conduct the same current through the cell. 4.4.4

TDS Removal Rate Capacity

A useful quantity describing the overall performance of an ED unit is the rate with which the dissolved solids are removed as a function of other independent variables. A meaningful numerical dimension for this rate, C, is in terms of gallons-ppm per minute. To obtain the total weight removed per unit time, multiply C by the density of the water solution. With any given set of fixed design factors, such as cell dimensions and operating current, the quantity of dissolved solids removed per unit time will be constant. However, as the flow rate is increased or decreased through the unit, the change in ppm, (or TDS), of the incoming to outgoing water will vary. Thus, the importance of C as a factor that characterizes the unit under all flow conditions. The removal rate, C, is defined simply as C- F-(c0-cL)

gallons-ppm/min.

(4.48)

If we look at the computations performed above, the values for removal rates are For L = 100 in., For L = 50 in.,

C = 96 gal-ppm/min, and C = 79 gal-ppm/min

146

4.4.5

ELECTROCHEMICAL WATER PROCESSING

Stacked Cell Configuration

We can now take the next analytical step to easily determine performance for units (modules) in which a number of cells are stacked in an array. The most meaningful array configuration is a number of cells through which the water flow is in parallel, and that are connected electrically in series. This has been developed in more detail as Case-A.wk3 in Appendix G.

4.4.6

Expanded Analysis

The expanded analysis shows that the number of cells, n, in series modified the final performance figures. Total voltage, for example, across the stack of n cells in parallel is nE. If the total water flow through the stack is F, then the removal capacity is still C. The terms employed in these expanded equations are listed in Table 4.7. Table 4.7 Operating parameters for designs-A. Dimensions

Symbol

Description

V

in/min

Water Speed

F

gal/min

Total Flow Rate

W

in

Cell Width

P

in

Cell Spacing

i

a m p / s q in

Exit Current Density

E

volts

Cell Volts

c

ppm

TDS at distance x


260000 ohm-in/ppm

Resistivity factor

X

X

Coulombic Efficiency

il X

in

Distance from entrance along cell

c

ppm

Input TDS

0

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

147

Table 4.7 (cont.) Operating parameters for designs-A. Dimensions

Symbol

Description

I

amps

Total Stack Current

i

amps/sq in

Input Current Density

watts

Total DC Electrical Power Input

0

Power

Number of Cells

n A A

sqin e

Area per Electrode Area per Membrane

m

Size

cu in

Module Volume

nE

volts

Stack or Module Volts

Thickness

in

Stack Thickness

CompCost

$

Electrode & Membrane Cost

FM

gal/watt-min

Figure of Merit Coefficient of Performance

COP EnergyGal

wh/gal

Energy per Processed Gallon

We can now pursue another example of performance based upon the general conditions stipulated above for single and multiple cells. Costs of electrode and membrane components are treated in Appendix G. Maintaining the same values as in the first example, a stack of n cells in series electrically, but with parallel water flow through all the cells, will produce n-times as much processed water for the same current densities as a single cell of the same length, L. Taking the cell design of Exercise #1, n such cells will perform as follows. co = 200 ppm cL = 20 ppm Ae = 730 sq in

Input Power = nE x I watts nF = 0.50 x n gal/min I = 10 amps

148

ELECTROCHEMICAL WATER PROCESSING

Thus, the total voltage impressed across the stack is nE, and the power is greater proportionately. If the number, n, of cells is 10, then the quantity of processed water flowing through the stack would be 10 x 0.50 = 5 gallons per minute; the stack voltage is 320 volts and the power supplied is 3200 watts. The Figure of Merit would be unaltered. The same performance in terms of 5 gallons per minute can be achieved by placing 10 such cells in parallel electrically. Then the power supply would have to provide the same 3200 watts in the form of 100 amps at 32 volts. The trade-off in such design considerations is strictly a matter of the economics of electrical characteristics most convenient for the user in terms of higher voltages and lower current versus the opposite. The economic considerations concern dc power supply costs for different ranges of volt-amp relationships, as well as the increase in electrode costs for a parallel electrical construction. In both single cell, and in a series array of cells, the electrode area (2 per stack) is the same. If cells are in parallel electrically, there would need to be (n+1) electrodes for the necessary electrical connections. With electrically parallel stacks, there are a correspondingly greater number of waste water compartments, (n+1), in which electrolysis is taking place versus just 2 for a series stack. The larger number of waste water compartments would also give rise to more hydrogen and oxygen generation by electrolysis, accompanied by a correspondingly greater stack voltage requirement. There are many possible combinations of series and parallel designs, such as placing two or more series stacks in parallel hydraulically to lessen the hydraulic impedance to water flow. In general, it is desirable to make the cell spacing and width large and the cell length small to reduce the pressure needed for water flow. However, in doing so, the necessary stack becomes larger, and the power required is increased, consequently lowering of the Figure of Merit.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

149

4.5 Batch Process Analysis: Re-Circulating or Static Water Processing System In some applications, it is desirable to have a water purification system not function in a "once through" manner. Examples of such requirements are those cases where only a small or limited quantity of processed water is required over an extended period of time, and the need for immediate and continuous peak quantities of water delivery are absent. In these instances, a much more economical design may be that of a re-circulated water system with a processed water storage tank, or reservoir. In these cases, the water processor unit, the ED stack, can be made much smaller in size and with significantly lower costs in parts and materials. Figure 4.19 is a schematic illustration of such a configuration in which both the waste and processed water are circulated around through the ED module and back into the reservoirs. An analysis of the performance of such a design can be made on the basis of a simplifying assumption that the water circulates many times in any one processing cycle,

Waste water reservoir Volume = V„

Processed water reservoir Module

0

Pump

Figure 4.19 Constant volume system batch processing.

Volume = Vn

_© Pump

150

ELECTROCHEMICAL WATER PROCESSING

and that complete mixing of the water takes place continuously in the reservoirs. In that manner, the TDS of each of the two quantities of water throughout the module and reservoirs is uniform and constant at any moment in time. In a fashion similar to the preceding computations, we will first identify the system parameters and the system constants, and then establish their mathematical relationship. The total resistance, R, of a stack of n cells is (ignoring the membrane resistance for the present), R = R(waste water) + R(processed water) = R w + Rp

(4.49)

If the TDS is the same all along the length of the cells at any time, t, the rate with which the ionic species are being transported across the membranes is represented by ^=i-n-A-W-L-77 dt

(4.50)

where the terms are identified as, i = current density, amps/in 2 , rj - coulombic efficiency, W = width of cells, in., L = length of cells, in., À = 0.038 gm/amp-min., conversion factor, and n = number of cells in series electrically In order to compute the dependence of Q upon time, the current density, i, must be represented in terms of Q. Electric current density, i, in a stack is i

=

* = ^ = -**-. R n-R R w +R p

(4.51)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

151

As a means of simplifying the analyses at this time, we will let the thickness of the waste water and processed water compartments be the same. Also, we will approximate the ratio of the number of processed water to waste water chambers to be unity, i.e., there are the equal numbers of compartment types. In actuality, there are (n+1) waste water compartments, if n = number of stacked cells. The resistance terms in the last equation must now be replaced as functions of the solute concentrations, C w andC . p

From our earlier assessments we found that the resistivities, r, are; r

=26xl04-£-,and

rp = 26xl0 4 -£- ohm-in 2 ,

(4.52)

p

where C w and C are the concentrations in units of ppm of the waste water and processed water volumes respectively. In general, the relationship between TDS in ppm and the quantity, (mass), of solutes is A Q „ i n 3J xlO = A p p m = AC, V where V is the volume in liters, and Q is in grams. It is also undertood that

jf*-*2

(4.53)

152

ELECTROCHEMICAL WATER PROCESSING

Substituting the expressions for the resistances into the equation for i, we obtain, i =


|


Cp

E„ C

W

C

(4.54)


d

QJXTQ.103

p

(4.55)

v„

The (+) sign in the Q w equation accounts for the fact that solutes are being transported into the waste water region, pWdx, by the flowing water stream, and the (-) sign in the Q equation is due to the fact that an equivalent amount is being removed by electrical transport from the processed water during operation of the ED stack. Now, the above may be substituted in the differential equation for Q so that there is only one dependent variable, Q, and one independent variable, t. Qwo+Q i =

10

3 , Qpo-Q

v„

V. (PP

1Q3

(Q po -Q)(Q wo +Q) VV p

w

,or 6

10

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

V,.

-+-


v„

Q Dpo0 -Q

■ io3

153

(4.56)

Inserting this expression (4.56) for i into the equation (4.50) for d Q / d t , we obtain, dQ , ECWL V,. v + —-- =n-A-77-—Q w o +Q Q p o - Q dt (pp n-EcWLA^ cpp

•10 3 ,or

Vw(Qpo-Q)+Vp(Qwo-Q) (Q wo +Q)(Q po -Q)

■103

(4.57) Rearranging terms to solve the differential Equation (4.57) becomes dQ

V,.

-+-

Qwo+Q

V Q P o-Q

above,

n - A - 7 7 - E c ' W ' L - 1 0 3 - d t . (4.58) (PP

If the coulombic efficiency is treated as constant throughout the range of TDS concentrations, then we can integrate and solve the above directly and simply as follows: Vw-ln(Qwo+Q)-Vp-ln(Qpo-Q) = n-A-7?-Ec'W'L-103-t + K
(4.59)

K is a constant to be evaluated from boundary conditions. These limits are: when t=0, Q=0. Thus, the constant, K, is found as K = V w -lnQ w o -V p .lnQ p o

(4.60)

154

ELECTROCHEMICAL WATER PROCESSING

It is now possible to plot the change in TDS of both the waste and processed water volumes as a function of elapsed time. However, before performing the evaluation, it is necessary to decide upon the electrical mode of operation—constant voltage or constant current. If constant voltage is chosen, one must exercise some precautions in the design of the power supply. In order to make the removal rate of solutes reasonable toward the end of a cycle, the impressed voltage will have to be quite high. That same cell potential at the beginning of a processing cycle can produce extremely high currents if not limited in some manner. Thus, it would seem that some control on the maximum current available from the power supply would be necessary to avoid inordinately high power dissipation at the beginning phase of a constant voltage design system. One can eliminate the problem by limiting the system current or self-regulating, regardless of the solute concentration at any time of operation. 4.5.1

Coulombic Efficiency

An improvement in accuracy of process description can be achieved by modifying the preceding analysis to account for non-constant coulombic efficiency. Changes in coulombic efficiency during system operation as a function of the concentration changes of solutes in the processed water volume decreases during system operation are taken into account. As before, r\ may have the form ^(l-e^")

(4.61)

and it may be substituted for the constant efficiency term in the expression for Q versus t. Quite obviously that would lead to significantly increased complications and greater difficulties in arriving at mathematical solutions in closed form. Another approximate approach to the problem of compensating for variable coulombic efficiency is to adopt a linear dependence of n upon C for the region where rj begins to change quickly. That relationship can be as simple as

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

155

(4.62)

v=r-cv

where y is a proportionality coefficient applicable to the region of concentration where r\ is changing rapidly. That would necessitate solving two separate equations for Q. Each equation would cover different ranges of values of C . A bit of manipulation would be needed to determine where these equation boundaries lie. 4.5.2

Single Cell Analysis

The simplest configuration of the system to fabricate is a single cell. Such a reduced configuration eliminates the necessity of making a manifolding arrangement, and simplifies the stacking problems. In order to account for different chamber thicknesses, p and p , and the fact that a single cell has two waste water chambers and one processed water chamber, the analysis is modified as follows. The resistances become: r w =2ç>-^-, and P

(4.63)

vc

Because the current density, i, is merely the cell voltage divided by the total resistance per unit area, we have

E„

>[2p w C p +C wPp ]

(p

2pwCp+CwPF

c c

(4.64)

156

ELECTROCHEMICAL WATER PROCESSING

Substituting the expressions for concentrations in terms of quantities of solutes, as before, we obtain Q w o + Q Qpo-Q

V,.

y

Q P o-Q ^

IO3+PF

i=-


2p v

v

V

P

1Q6

Qwo+Q •IO3

j V

v

w

2pwvw(Qpo-Q)+Ppvp(Qwo+Q)
A

• IO 3

(Q W O +Q)-(Q P O -Q)

2pwVw , P P V P Q W o+Q Q PO DO -Q

■ io3

(4.65)

Now, from the expression for the rate of change of solute concentration in the processed w a t e r side, w e obtain this as the modified, single cell, differential equation:

dQ-

2p w V w , P P V P Qwo+Q

Qppo0 -Q

=A-W-L-77-^-103-dt

(4.66)

Still treating the coulombic efficiency as constant, the above expression is integrated w i t h respect to Q a n d t. 2-Pw-Vw-ln(Qwo+Q) - Pp-Vp-ln(Qpo-Q) (4.67)

The constant, K, is evaluated as before b y the fact that at t=0 a n d Q=0. Thus, K = 2-Pw-Vw-ln(Qwo)

-pp-Vp-ln(Qpo)

(4.68)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

4.5.3

157

Single Cell - Special Case

In the generalized situations where the waste and processed water volumes are not the same, and where the cell spacing is not necessarily the same, the mathematics becomes a bit complex and more difficult to evaluate. For example, explicit evaluation of Q versus time, as expressed above, is not easily accomplished directly. A simultaneous equation solving program, such as MathCad or Formula-One, enables such computations to be made, but with some difficulty. If we assign equal values to the two volumes (waste water and processed water, Vw and V ), and cell compartment spacings, p w and p , the mathematics is significantly simplified without great sacrifice to actual hardware design parameters or performance evaluation. Hence, we will proceed on the basis of Pp=Pw=P>and V w =V p =V

(4.69)

The reservoir volumes are presumably adjusted to give equal total fluid volumes. Now we may solve for Q explicitly as follows: ln(Q w o + Q) - l n ( Q p o - Q ) = A-77-Ec'W'L-103-t + K p-p-V

(4.70)

Raising both sides to the exponents of the natural logarithm, we obtain this resultant equation of the form

Qpo-Q

=KeMt

(4.71)

158

ELECTROCHEMICAL WATER PROCESSING

Transposing terms, and solving for Q, N-2wo+^

Q-po DO -Q

TX „Mt =K • e Mt , becomes

Q w o + Q= (Q p o - Q ) • K • e M t , a n d the final expression is

QpoK-e--Qwo U

K-eMt+l

^/Z)

Where the terms, M, and K are ..

.

ECWL

M = À-TJ-— ç-p-V

in3

10

K=lnQ w o -lnQ p o

(4.73)

By substituting Equations (4.73) for M and K into Equation (4.72) for Q, we can solve for Q explicitly. This relationship can then easily be placed into a spreadsheet type of format to be numerically evaluated. Graphical representations can also be plotted for different independent variables to quickly and conveniently observe how Q changes with time as any one or more of other parameters such as L, V, W, p, etc. are changed. 4.5.3.1

Ohmic Energy loss and Water Temperature Rise

A straightforward method of assessing the highest temperature rise in both the processed and waste-water volumes is outlined as follows. The worst-case situation, or maximum temperature attained upon sustained operation, is the main interest. This worst-case situation would occur if the following conditions were true: a. The process is adiabatic (perfect thermal insulation to the outside)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

159

b. All electrical input power is dissipated as ohmic heating (very little voltage drop due to ion transport across concentration gradients). That assumes that there are no voltage gradients across membranes, and that no energy is required for the oxidation/reduction reactions at the electrodes. c. Heating of all water compartments is equal, or thermal energy distribution throughout the internal structure is immediate and uniform). The total volume, VT, of a water processing system operating in a batch mode is simply Vw+Vp=VT

(4.74)

If the volumes are given in terms of Liters, then the mass, M, of the water in grams is 1000xV r Because 1 joule = 1 amp-sec =1/4.18 calorie, the temperature rise, AT, of the volume of water is watt-sec input - — = AT • M • C, where 4.18 C = specific heat of water = 1 deg C/gram-calorie (4.75) The energy input over some operating period of time, t, from the electrical power supply is t

Energy (watt-minute input) = j i • L • W • E c dt

(4.76)

Thus, the final expression for AT is A T =

Energy-60 MC-4.18

These values are easily implemented also by numerical integration in a LOTUS spreadsheet. The temperature

160

ELECTROCHEMICAL WATER PROCESSING

rise can also be computed by insertion of the analytical expression for i as given in Equation (4.51) into (4.77) for the total energy input to the systems. Direct integration of Equation (4.77) for the energy simply gives Q = n-A-W-L-77-Ji-di

(4.78)

Substituting for ¡i ■ dt into Equation 4.79, we obtain for the electrical energy input based upon our assumptions of constant voltage and total conversion of power input to the system as heat Energy = - ^ - ^ -

(4.79)

n-À-rj

Substituting again into Equation 4.77, we are able to estimate the highest possible temperature attainable by operating the system over any time period. 4.6

D e s i g n Exercises for Water Re-Circulation S y s t e m s

The following are some examples of design trade-off calculations for an electrodialysis system in which the processed water is re-circulated from a reservoir many times through the electrolytic module. These are based upon the analysis and series of mathematical relationships developed in Chapter 4, Section 4.5. As was done in Section 4.4, we will select some operating and design characteristics that typify actual application performance requirements. Let us suppose that we want to process 500 gallons in a period of one day, (24 hours). Also assume that the input water supply is quite hard at 400 ppm concentration, and that we wish to low that level to at least 40 ppm at the processed water output from the ED processing

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

161

system. By doing a series of number substitutions in the set of equations developed in Section 4.5, it is possible to arrive at a system configuration with electrical inputs that are acceptable on the basis of practical considerations. There are two methods of electrically controlling the processor cell, or module. They are either by constant current or by constant voltage dc sources. 4.6.1

Exercise #1

A constant current mode of operation is our first example. Let us take a configuration with the following dimensions and input/output goals, and limit the current to 0.05 amp/sq. in. to afford extended electrode life. L=10in. W = 20 in.,

co=400ppm c° = 40 ppm

The total current is then established as 10 amps constant. Table 4.8 gives some values of reservoir ppm, c , at various times after startup of the equipment. In the first single cell, the case where n = 1, the water is not processed to the 50 ppm level within the 24 hours with the current limitation at 0.05 a m p / s q in, and with a Table 4.8 Constant current. n=l

Time, hours

E

n=5 C

E

C

p

p

0

9

400

45

400

1

9.2

387

52.5

336

5

10.5

3366

200

83

10

12.7

273

815

20

20

23

147

162

ELECTROCHEMICAL WATER PROCESSING

total cell area of L x W = 200 sq in. Hence, it is necessary to either increase the cell area or the current density. The former would generally be more practical In the case of the five stack array, n = 5, the water ppm is achieved well within the 24 hours, but the voltages and power requirements are much too high to be practical. A more practical arrangement would be to have a near-constant current power source in which the current is maintained constant up to some maximum output voltage. After that maximum potential is reached, the current diminishes as the cell stack resistance goes up with lowering TDS. A current limiting power supply would be appropriate for this type of operation. 4.6.2

Exercise #2

A second manner in which the cells or array of cells can be operated is on a constant voltage basis. In general, that sort of operation more closely matches the characteristics of a simple power supply circuit. If we take the dimensions of the cell in exercise #1 as an illustration of such a system, the conditions of operation are listed in the table below. We have listed four different modes of operation, and one of which is for a five-cell stack shown in Table 4.9. Table 4.9 Constant voltage. Time Hours

E=10 volts n=l

I

C p

E=50 n=5

E=25

I

C p

E=50 n=l

I

C p

I

C p

0

400

11.2

400

11.1

400

28

400

56

1

385

10.8

329

9.4

362

25.6

329

47

5

329

9.4

151

4.5

246

17.8

151

22

10

271

7.8

57

1.7

151

11.2

20

183

5.4

8

0.25

57

4.3

8.6 8

57 1.2

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

163

As shown in Tables 4.8 and 4.9, single cells, or multiple cells operated at a constant voltages do not process the water to the low values of TDS within the same amount of time. Again, there are innumerable variations on designs and types of power inputs to the units that can be devised to achieve the same end performance of reducing a quantity of water time from one TDS level to another within a specific period of time. For more details about the sets of equations, list of variables, and methods of making the computations see Appendix G.

4.7 Cell Potential and Membrane Resistance Contributions In the development of the previous equations for an operating dialysis cell, we did not take into account the contribution of membrane resistance and back-emf factors. Even though they were identified as parameters, they were ignored in the mathematical development. For those wishing to be somewhat more precise in their estimates of cell behavior, these terms should be taken into account. The importance of these factors as contributions to the overall potential required from a d.c. power supply will depend largely upon these issues: • Membrane-electrode and inter-membrane spacing • Membrane type and its resistivity • Average concentration of water solutions being processed • Type of electrodes employed and their surface properties As spacing between the electrodes and membranes, and between membranes, is made larger, the contribution of

164

ELECTROCHEMICAL WATER PROCESSING

electrolyte (water solutions) becomes larger, and becomes a greater fraction of the total cell resistance. The lowest resistance of electrolyte in any situation would be encountered in a desalination unit at the very start of its operation when all compartments are filled with sea water. This salt concentrated condition would give a specific conductivity of the solution in the range of 0.10 mho-cm -1 , or about 10 ohm-cm. 4.7.1

Membranes

Let us examine the above as the condition where cell membranes may have their largest contribution. Using relatively small spacing between cell components of 1/8 inch, or about 0.32 cm, the resistance per square centimeter area of a three-compartment cell filled with seawater is about 10 ohm-cm x 0.32 cm x 3/1 cm 2 = 9.6 ohms.

(4.80)

The highest membrane specific resistivity that we have employed in ED cells is in the range of 5+ ohm-cm 2 . These values would indicate that a large portion of cell resistance is due to the membranes in such a limiting situation. In those cases where salt concentration is less than seawater by a factor of a hundred or greater, as in the case or water softeners and demineralizers, where the TDS of the input water is in the range of 200 to 1,000 ppm, solution resistances are about 100 to 200 or more ohms for a one square centimeter of cell area. That is considerably higher than the contributions of even the high resistance membranes above, which may have specific resistivities of 10+ ohm-cm 2 at these lower TDS ranges. 4.7.2

Electrodes

Many different electronically conductive materials have been extensively studied and evaluated for application to water electrochemistry. Metals lend themselves to mechanical handling and shape configurations better than carbon.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

165

They are generally much more malleable and electrical connections are more easily made to external circuitry. However, those few metals, such as platinum as rubidium, that will withstand chemical attack as electrodes— especially when made the negative terminal—are very costly and tend to be in scarce supply. Their high costs make them rather impractical as electrode components for application in moderately priced consumer products. In many instances, other "base' metals with thin deposited coatings of platinum or ruthenium have been used for such purposes. The coatings withstand attack and erosion for a prolonged period of use. However, a base metal such as titanium is usually employed so that when the surfacing is consumed and the base conductor is exposed, it will not be attacked very rapidly. The exposed titanium is not directly useful as a positive electrode, where oxidation takes place because its oxidized surface ceases to conduct electrical current. Hence, we resort to carbon as the principal electrode material. Composite structures of polymer (plastic) bonded carbon particles have been attempted with little success. The carbon particles are rather rapidly eroded away at the positive electrode surface by the gasses generated within the structures, leaving behind only the supporting, nonconductive polymer binder, and the electrode eventually acquires too high a resistance for practical use. Fused carbon plates give the most satisfactory results because as they erode, they present the same surface to the electrolyte. Their relatively low cost enables us to tolerate the gradual loss of carbon from the plate electrodes. Low conduction of these plates as compared to metals is of little concern because in most applications the ionic concentration is low, giving rise to generally high resistivity electrolyte. This subject is covered a bit more, later in this text. Electronic resistance of electrodes as current carriers is completely negligible because of the vast difference in specific resistivity of metallic conductors (carbon included) and

166

ELECTROCHEMICAL WATER PROCESSING

those of electrolytes. In most instances, the bulk resistivity ratio between metallic resistivities, pM, and electrolytic resistivities, pe, for conduction through the materials is in the order of ^<10"4 Pe

(4.81)

However, the interface resistance encountered between the electrode surfaces and the electrolyte can be significant. In a single cell, that contribution to total resistance is greatest because there are two such interfaces per cell. In multiple cell arrays, that resistance is amortized over the number of cells. There are only two electrodes in a series stack per n number of cells. Typical interface resistance for carbon composite electrodes of the type employed in most of the hardware designs are in the range of 0.05 and 0.10 ohm-cm 2 for two electrodes. With desalinators operating with entering seawater, the contributions of both electrode and membrane resistances are not negligible. Even though the electrode interface resistances are not constant over the entire range of TDS from 40,000 to 100 ppm, the variations are not very large. Referring to Equation (4.11), RT=2R,+Rœ

+ RD + R _+K + ,

the total resistance of an array of n-cells becomes RT in) = 2R¿ +(n +1) R +nR+nR

_+nR

+

(4.82)

As n becomes greater, the contribution of R, interface resistance becomes smaller. Equation (4.82) should be substituted for (4.11) to be more rigorous in the analyses. However, the approximations made in the previous sections are certainly valid for dialysis systems operating as demineralizers with ten or more cells in series.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

167

To see how the various elements contribute to the total array resistance, some values may be inserted into Equation (4.82). If we take n = 10 for a desalinator array, cell spacing of 0.60 cm, sea water in all cells, and electrode-water interface resistance as 0.05 ohms, the total resistance per square centimeter of the area of such an array is Rt(n) = 2- (0.05) + (2w + l)-10-0.60 + 2n(5) = 0.10 + 126 + 100 = 226.1 ohms

(4.83)

The resistance for the processed and waste water sides (initial condition at the start of operation) are equal. Almost half of the total resistance is due to membranes in this case. If the array was being used to demineralize water from about 200 ppm of salt to less than 10 ppm, then the ratios would change drastically. For example, the same 10 cells would have a resistivity per square centimeter of area in the order of the following: R. = 0.10 ohms R1 = R = 200 x 0.60 ohms w

p

m

m

R = R + = 10 ohms and the total resistance becomes R((n) = 2-(0.10) + (2n + l)-200-0.60 + 2n-10 = 0.20+ 2100+ 200 = 2300.2 ohms

(4.84)

Now, the contribution of the membranes to the total resistance is quite small. In actuality, the interface and membrane resistance are functions of solute concentrations in the water, but introducing such terms would further complicate the mathematics of analysis. If we return to the analyses in Section 4.3, we can insert these resistance terms easily as constants into the overall relationships to account for their contributions. Equation (4.83) assumed that the waste water compartment resistances were very small compared with the far

168

ELECTROCHEMICAL WATER PROCESSING

more dilute processed water compartments in a cell or array of cells. And, it neglected the contributions of the membranes and electrode interfaces. Instead of employing the simplified expression for r , we can take into account all of the terms above and have as the expression for an array equation (4.82). In those cases where the ppm of the processed water is low, membrane and interface resistance can be ignored. When a unit has been operating for a period of time sufficient to increase the level of TDS in the waste water side to 10 times or greater that of the processed water chambers, the contribution of the waste water to total resistance is also negligible and may be ignored. Then Equation (4.83) is a reasonable expression to use. However, if these conditions are not true, then use the full expression in (4.82). In general, however, it is reasonable to discard the interface term, R, from the expressions because of its very small magnitude. In fact, the anion and cation membrane electrical properties are sufficiently close together to enable us to conveniently lump them together, and the net expression for cell resistance can be Rt=Rp + 2Rm

(4.85)

after steady state is achieved wherein RW({R ■ Substituting this expression for the total resistance into the subsequent relationships of section (4.7.2), we have rt=rp

+

2rm=££

+ 2rm

(4.86)

And, the current density becomes . _£_

n

E ^

+

2r.

Ec, ç-p + 2rm/cx

(4.87)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

169

Returning to equation (4.24), we can substitute Equation (4.85) for ix, and we get -E-e, rj-A-dx (p-p + 2rm/cx

p-v-dcx ■ ß-10-6

(4.88)

Again, transposing terms and solving the differential Equation (4.83), we obtain

r

c,-
*w;

-ErjA, p-v-ß-lQ-

-dx

(4.89)

Integrating Equation (4.87), the expression for the concentration of solutes in the processed water side at some point x is found as [çj-pln

C

—TC

_2

T=

-E-TJÀX

p-v-ß-10'6

(4.90)

where the boundary conditions are the same as in Section 4.3, proceeding with the solution of Equation (4.88)
-EÎ]ÀX

p-v-ß-lQ-6

■x+ç-p\nc0-rmcf

(4.91)

The remainder of the mathematics follows the same guidelines as in the earlier section to arrive at expressions for the total electric current, while taking into account the membrane resistance contributions. 4.7.3

Opposing Voltages

There are two sources of voltages opposing that of the external power supply driving electric current through a series of cells. They are: concentration potentials and

170

ELECTROCHEMICAL WATER PROCESSING

decomposition potentials at the electrodes where electrolysis is taking place. The half-cell potentials at the respective electrodes are primarily due to the decomposition of water at the negative electrode, and, depending upon the materials in solution in the waste water compartments, the production of elemental chlorine a n d / o r oxygen at the positive electrode. The solution in the (-) side will become alkaline with the production of H 2 and OH" The principle reaction is 2H20 + 2e~ -> H 2 + 20H~

(4.92)

At the positive electrode the reactions are variously 2Cl~ -» Cl2 + 2e~ @ -1.36 volts, and 2H20 -+02 + 4H + + Ae~ @ -1.23 volts

(4.93)

The potential resulting from electrolysis that is opposing the input voltage from a dc power source is in the order of 3 volts. Concentration potentials across membranes all cancel out across the array of cells in series because they have alternatively opposite polarities, due to the differences in solute concentrations between waste and processed water. The expressions 4.15 and 4.25 in Section 4.3 for the current density at some point x along the cell length, should be replaced by the following modified equation V

ej_^

( 4 9 4 )

where Ee is the opposing emf due to electrolysis at the electrode surfaces. One can see how the mathematical relationships for cx and ix grow more complex as we account for more variable parameters.

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

171

4.8 Diffusion Losses of Ions and Molecules Across Membranes Many secondary transport processes are taking place during the operation of an ED cell. Among these are: • • • •

Osmotic transport Water molecules of hydration transport Molecular (thermal) diffusion of solutes Negative ions back transfer through cation membranes • Positive ion back transfer through anion membranes • Transport of hydrogen ions into waste sides • Transport of hydroxyl ions into waste sides

Some of these effects are most pronounced at the latter stages of processing, where the processed water is very dilute with solutes, and the concentration difference is greatest between waste and processing chambers. Of all the above secondary processes, the diffusion of solutes back into the dilute chamber of the cell has, perhaps, the largest negative effect on overall ED performance. Osmotic pressure differences result in water loss only, and do not actually detract much from cell performance. Water molecules being transported as attached to ionic clusters from one compartment to another is also not a significant factor except for some slight water loss. Depending upon the ionic transport number ratio for membranes that are employed, the back transfer of ions through these membranes can also be quite small. This last degradation in performance can easily be handled analytically by slight modification in the mathematics as either additional transport terms, or more easily within the efficiency factor, n. For the present analyses, we will ignore these factors except for the thermal diffusion of solutes as undissociated components back into the processed water cell chamber.

172

ELECTROCHEMICAL WATER PROCESSING

In the continuous flow through the configuration of an ED processor, the gradient of concentration of processed water along the length of the cells results in a changing rate of back diffusion of species through the membrane. This rate of diffusion is not constant along the length of the cell. The following is a modification of the expressions for concentration of exiting processed water from an operating cell. We must return to the mathematical developments earlier in this chapter to account for this back diffusion and modify the expressions accordingly. Equations 4.19 and 4.23 need to include an additional term for the diffusion factor. As was proposed in Section 4.3, the loss rate, (dQ/dt)d, of dissolved ionic species in the water stream passing through the processed water chamber is due to the electro-dialytic effect of electric current passing through the membranes. This was described as (see Equation 4.21): ^ =ix.^W.dx-A^/ dt )A min

(4.95)

It is now necessary to add a new term due to molecular diffusion of the solute-bearing water back through the membranes to the processed water chamber because of concentration difference of solutes. That new term can be represented as (dQ/dt)m, the quantity diffusing back into the dilute side through membrane area of 2Wdx. Because the processed water chamber has two membrane walls, the area term must be multiplied by two. The amount diffusing out of the two walls of an element of cell whose area is Wdx, according to Fick's Law, is fê5") V ut Jw dx

=

*" ' A C ' a r m = kmic0-cxyW-dx

(4.96)

ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

173

where km = molecular diffusion constant peculiar to the membrane co = input as well as the waste water concentration Then the relationship for the net transport of solutes from the processed water chambers to the waste water sides in steady state conditions is Input rate of solutes from water flow + molecular diffusion into the pWdx space = electrical removal rate of ions and can be expressed as dQ^ _ dt 'w,dx

(dQA^dQ^ dt v dt

(4.97)

for an area of Wdx. Equation (4.97) then would essentially replace Equation (4.21). The ensuing mathematics must be modified now for the additional diffusion factor. We now have the resultant equation to manipulate where W-p-v-dcx-ß-10~6 = -ix-rj-W-dx-A

+ km-(c0-cx)-W-dx-ß-10-6

(4.98)

Then the mathematics can be developed to account for the losses by thermal diffusion. We can modify the relationship as follows: pvdcx/M0 E c x r]-A-dx+k (c -c )-ß-dx m 0 x (P-P

(4.99)

The equations retain the same format, but become a bit more complex in the sense that more terms are in the

174

ELECTROCHEMICAL WATER PROCESSING

expressions. If one wishes to compensate for the thermal diffusion back into the purified water, these modifications must be followed through in the same fashion as they were in 4.8.3. In a very similar manner the expression for cx becomes, dcv

p-v-ß-10'

■ = dx, or

~^ft^+ka(c0-cxy (P-P dc cx\km-ß-W-6

ß-io-^

+ V-Ä~ + kmc0ß-W W

dx pvß-10~ (4.100)

Direct integration to solve for cx as a function of distance, x, yields 1 kJ-W

-In

kj-l^

+ rjX
pvß-lCT

+ nk— +fcmco/M0-6
■ + constant

(4.101)

Applying the boundary conditions of cx - co when x = 0, the constant is Const. 1

kj-w^

+ v^-

-ln co\kmß-\0-6 + VÄ—

+kmc0ß-10-6


ANALYSIS & MODELING ELECTRODIALYSIS SYSTEMS

175

w e find the final expression containing cx becomes

1

-ln cx\kmßlO-6

6

V

kmß-W- + VÄ-

+ VX—

W)

+kmcoß-10-


1 Íb

. ., E

fcm/MO- +77A pvß-lQ-

-In

kmß.lO-6 + r1X— +fc m c o /M0- 6 V

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

5 System Design Exercises & Examples 5.1 Electrolytic Generation of Bromine and Chlorine: Design Procedures Design procedures of electrolysis devices for the production of free bromine are relatively straightforward, and easily lend themselves to implementation as mathematical programs. The electrolytic processes that take place in the "brominator" electrode array, as shown in Figure 3.17, are described in Chapter 3. In this section, we will address the rates of the processes and the analytical development of the methods to determine appropriate electrode areas, inter-electrode spacing and numbers of electrodes in an array. Voltages and currents needed to achieve the type of performance desired are also considered in the following. There are a number of factors that influence the behavior of an electrolytic bromine generator. In order to design an 177

178

ELECTROCHEMICAL WATER PROCESSING

operating system, it is necessary to identify, and quantify each parameter. These are: Coulombic efficiency for Br2 production Driving electrode potential Electrolyte (water) specific resistivity

r| E p

With these three basic materials parameters, it is possible to calculate the electrode areas, spacing and electric power input to the electrodes needed to produce a given amount of bromine per unit time. However, the coulombic efficiency must be explored in a bit more detail before we are able to proceed with some degree of rigor. The efficiency of the production of free bromine at the (+) electrode surface is dependent upon the concentration of bromide ions at the electrode surface, the nature of the electrode surface and the electric current density (per unit area of electrode). The competing process is the generation of oxygen from the available OH' ions. Thus, ri = (/>{CBr_,i,)

(5.1)

some function, (j), of current density, i, and bromide ion, CBr_, is the concentration for a particular electrode structure. In general, n will decrease with increasing i and decreasing CBr_. As was stated before, the maximum value of r| is 0.50 due to the recombination of bromine and NaOH to reform NaBr, the original salt that is decomposed to produce free bromine at the beginning of the reactions. Let us proceed with setting up a series of equations to establish a design process for electrode configurations and electric power supply specifications to produce bromine at a given rate. The amount of molecular bromine, B, produced per unit time per unit area of electrode by electrolysis is simply d

ydt

= i-î]-A-y/

(5.2)

SYSTEM DESIGN EXERCISES & EXAMPLES

179

where dB/dt = rate of production of elemental bromine = gm/sec = 8.56-10-4-z'v4-77 A = total working area of a single cell, square inch, *F= conversion constant = 1.07xl0~5 gm-equiv-wt/amp-sec = 8.56x1o-4 gms/amp-sec i = amp/in 2 The voltage, E, needed to produce the necessary total current for a single cell is given as (5.3)

E = Ec + i-A-Rc

where Ec = electrochemical cell potential for the couple 2Br + 2H20 -> 2 0 H ' + H 2 + Br2 P' S

Rr = cell resistance = —— A and s = cell spacing (distance between electrodes)

(5.4)

Cell voltages (back, or opposing emf), encountered with chloride and bromide salts are in the order of 2+ volts. It is difficult to be exact because so much depends upon the flow rate of solution passing over the electrode, polarization effects, concentration of solute, etc. For purposes of illustration, we will use a value of 2.0 volts. Let us now perform a calculation to estimate the power input, current requirements, and electrode size for producing a given amount of bromine per unit time. For example, we will select a body of water of 20,000 gallons (a small swimming pool volume) to be provided with 1 ppm of free bromine per hour. This is a strictly arbitrary situation, but it is within the realm of actual circumstances.

180

ELECTROCHEMICAL WATER PROCESSING

The equivalent conductance, A+, and A of the separate ions, Na+, and Br, are 50 and 78 cm^equiv^-ohm" 1 respectively (see Section 4.2). A typical concentration of salt, NaBr, in a pool that would be well below the taste threshold and metal corrosion level is in the order of 1,000 ppm. The equivalent conductance, A, for the salt in dilute solutions is A = A + + A - = 128,

(5.5)

and the specific resistivity is given as 1000

« ^

. (5.6) NA A convenient conversion factor for putting normality of NaBr solution in terms of ppm is as follows. Normality, N, is 1N=GEW/1000 gm, and since the GEW of NaBr is 103 gm, the conversion factor for this salt is p= F

ppm = 103xl0 3 N

(5.7)

Given these basic facts, and assuming, for the moment, a coulombic efficiency, n, of 0.50, we can proceed to estimate the size of the hardware and the electric power input. The drawing in Figure 5.1 shows the dimensions of an electrode array to produce the necessary rate of bromine. If the total working area of a single cell is A, regardless of whether that cell is comprised of two electrodes or n-number of electrodes connected in parallel electrically, and if the spacing between the electrodes is s, as shown in Figure 5.1, the electrolytic resistance of the cell is R =

S

nA

=

1000 nA NA S

(5.8)

We can now set some values to the independent parameters to calculate the electrode area needed to generate the 1 ppm of bromine per hour of operation. In order to do so, we

SYSTEM DESIGN EXERCISES & EXAMPLES

181

^ 2

Figure 5.1 Brominator electrode array.

must decide an acceptable current density at the electrodes. Again, we will be somewhat arbitrary and set that value at 0.10 amp/in 2 . The units implicit in the above equations are in metric values, or cgs. Hence, s and A must be in centimeters. Generation of 1 p p m / h r in a volume of 20,000 gallons, or about 160,000 lbs of water is 0.16 lbs, or 72.6 gm of bromine per hour. Because 8.56xl0"4 gm of Br2 are produced per second at one amp current, (or about 3.08 g m / h r at one amp), a current of 47.2 amps is necessary for that generation rate of bromine with a coulombic efficiency = 0.50. Returning to the remainder of the design problem, we see that an area of 472 square inches of electrodes is necessary to keep the current density at the 0.10 amp/in 2 , and if the spacing, s, is set at a mechanically and hydraulically convenient value of 0.25 inches, the voltages are: E = E +23.6

0.25-2.54 l-236-(2.54) 2

1000 pptn •128 103xl0 3

(5.9)

Carrying out the arithmetic, and putting in the value of 2.0 volts for Ec, and 1000 for the NaBr ppm, we obtain

182

ELECTROCHEMICAL WATER PROCESSING

the following for the dc potential needed from an external power supply: E = 2.0+ 15.6 volts.

,cim (5.10)

It is now straightforward to establish a system of equations to assess and optimize design parameters in terms of cost, current density, physical size and shape, energy consumption, etc. We can solve the following system of simultaneous equations by substituting various values for independent parameters, and one can even decide which parameters will be dependent and independent. As in many computations before, solution of these relationships can be implemented within a number of computer software programs, such as MathCad, and Formula One. Because LOTUS and EXCEL do not permit simultaneous solutions, these equations must be rearranged so that their solutions are accomplished in a sequential fashion. There are many engineering trade-off possibilities that affect capital equipment operating life, cost, efficiency and energy consumption, as well as practical considerations such as geometry. The relationships and definition of terms are listed below. t

*B/,. = i-rj-A-y/= grams/hr E = Ec + i-A-Rc = volts

1000 p= """" = o ohm-cm ppm =

3; l03xWN

Rrc = —— = ohms A The units of the constants are: y/-1.07xl0~5gew/amp-sec, = 3.08 gms/hr-amp

and for bromine

A = 128 cm2 - equiv : - ohm"1, for bromine

(5.11) (5.12)

SYSTEM DESIGN EXERCISES & EXAMPLES

183

In addition to these relationships that have been described earlier, we may wish to identify the following in our optimization calculations: Power Input = P = i- A- E

(5.13)

And defining a measure of bromine output against input power as COP, we have Coefficient of Performance = COP =

1

âVi

P àt

=

(Ri)

P

(5.14)

Costs and specifics of device and system design will be discussed in some detail in a later volume on the same subject, but dealing with empirical data and fabrication methods. Costs will vary with design. For example, if the current density is lowered, then the electrode area must be increased for the same bromine yield. However, resistive losses are lower and the power supply cost will be reduced, thus compensating for the greater electrode array size. In addition, the electrode life will be extended at the lower current densities. An increase in coulombic efficiency due to lower electrode starvation and lower polarization effects will also be realized at lower current densities. Similarly, the amount of salt in solution will influence the power requirements by determining solution resistance, as well as coulombic efficiency. Some examples of design trade-off opportunities independent of cost changes are presented here by employing the above set of equations as simultaneous relationships. In these computations, n is treated as having a constant value, even though in reality it will depend upon current density and salt concentration. If the current density is reduced, the required power supply voltage decreases as well as the power input level, and the COP is increased. Examples are listed below in Table 5.1.

184

ELECTROCHEMICAL WATER PROCESSING

Table 5.1 cm2

volts

watts

Electrode Area

Cell Potential

Power Input

0.05

9433

4.5

212

0.34

0.01

4716

7.0

332

0.22

0.15

3144

9.5

450

0.16

0.20

2358

570

0.13

amps/cm 2

5.1.1

12

COP

D e s i g n Geometry Comments

Practical designs of the this type of oxidizer-producing electrode stack may need to be installed in high pressure water systems such as those in residential and commercial water lines. Pumped circulation systems for pool filtration are usually at fairly high pressure as well. Pressures encountered can be as high as 100 psi. A cylindrical or near spherical housing configuration would be best as pressure vessel containers for electrode assemblies. If cylindrical housing is selected, for example, one would want to achieve maximum space utilization of the inside of the cylinder for the electrode array as a design goal. A short analysis is given below of such a design, where a stack of parallel plate electrodes is inserted into a cylindrical container as the outside housing. Figure 5.2 shows an end view of a stack of electrodes in a cylinder with a set of dimensions as indicated. • • • • • • •

Number of electrodes in parallel = n Width of stack = b Stack thickness = a Electrode thickness = u Interelectrode spacing = s Cylinder inside radius = R Stack length = L

SYSTEM DESIGN EXERCISES & EXAMPLES

185

Figure 5.2 Electrode stack.

We are now ready to estimate the optimum design for any given total area, A, needed for proper electrochemical operation as determined by the preceding set of equations. The total electrode area of the stack above is A = La(n-l)

(5.15)

If the electrodes are to be sized such that the end ones are in contact with the cylinder wall as shown, the relationships between the stack parameters and the cylinder radius are b-n(u + s)-s

HííiS

(5.16)

Figures 5.3 and 5.4 are photographs of an electrode stack inserted in a cylindrical housing in process of fabrication. The electrical wires and connections to the carbon plates are through an epoxy casting surrounding the edges of the plates as well as encapsulating the electrical connections

186

ELECTROCHEMICAL WATER PROCESSING

Figure 5.3

Figure 5.4

themselves. Water flow is parallel to the electrodes and goes from one end of the cylinder to the other via entrance and exit pipes in the end-caps. The computations for A, or any three of the parameters in the above three equations (5.15 to 5.17) may be performed after deciding upon the other variables. For example, if we wish to solve for L, a and n while keeping all other

SYSTEM DESIGN EXERCISES & EXAMPLES

187

parameters fixed at some pre-determined value, the results are obtainable as a variety of choices. Maximum utilization of space within the cylinder can be derived by maximizing A in the above system of equations to determine the optimum relationship between R, a and b as a function of the rest of the parameters of s, u and n. The optimum expression is found by setting the derivative of A with respect to either a or b to zero. Substituting for a and n in equation in Equation (5.15), we obtain: b+s u+s

A=L

[4R2-b2f2, or

(u + s)A = Lb(AR2 -b2)Á

-Lu(4R2 -b2)A

(5.18)

Differentiating A with respect to b, we get

(u + sy — = db

L(AR2-b2)y2-Lb2(AR2-b2yV2

+ Lub(4R2-b2yV2

(5.19)

Setting dA/db = 0 to find the maximum of A, and multiplying Equation (5.19) through by UR2 -b2\2 /(u + s), t n e result is 0 = L(4R2 -b2)-Lb2 2b2-ub-4R2

=0

+ hub, or more simply (5.20)

Calculations can now proceed with that these equations to arrive at the best value of a, b, and n to obtain maximum area, A of an electrode stack.

188

ELECTROCHEMICAL WATER PROCESSING

5.1.1.1

Example

Our reference are Equations (5.3). Calculations can now proceed with these equations to arrive at the best value of a, b, and n to obtain maximum area, A of an electrode stack. We can now assume some operating criteria, and based upon them, find out how the trade-off factors are interrelated quantitatively, and perhaps find some optimum combinations of size, initial cost, and electrical power requirements. It is known that the life of an electrode assembly is directly related to the electrical current density. However, it is important to also be aware of the improvement realized in long term operating costs of an electrode array by reducing its replacement frequency. For equation setup and computation procedures, refer to Appendix H. An interesting investigation made during 1983 to 1988 was the design and implementation of a number of complete systems for field-testing in residential and commercial pools. The photographs in Figures 5.5 and 5.6 are of some of these experimental systems. They show the programmed power supply and connections to the pool pump

Figure 5.5

SYSTEM DESIGN EXERCISES & EXAMPLES

189

Figure 5.6

and reservoir. These chlorinator/brominator units were operated continuously, in some installations for over three years. A Chlorine or bromine salt, i.e., NaCl or NaBr, was added to the pool water to bring the concentration up to between 600 to 1000 parts per million. The devices demonstrated their ability to maintain clear water conditions without increasing hardness or requiring the addition of other chemical agents.

5.2 Simple Estimate of Capital Equipment and Operating Cost of Electrochemical Desalination Apparatus The following is a preliminary cost estimation of a sea water/potable water system. The cost of membranes is set at the high value of $5/ft 2 and electrodes are also $5 to $10 per ft2. Input water is taken as 30,000 ppm (3% by weight salt) and the output is set at 300 ppm, a reasonable level for general potability.

190

ELECTROCHEMICAL WATER PROCESSING

Two stages of the water processing system are employed. Each stage is capable of reducing the TDS to 10% of the input at a flow rate of 10 gallons per minute. The first stage brings the sea water down to 3,000 ppm, and the second stage to 300 ppm. The accompanying data and calculation sheets show the dependence of power consumption and initial capital equipment investment versus the number of stages of cells in the stack. Let us take 60 cells as the figure for a good compromise between power consumption and capital cost. This gives the following figures for the cost of the system and cost of operation on a per gallon basis: Stage 1 Equipment cost = $5177 (materials) Power consumption = 48 KW Power cost @ $0.05/KWH = $2.40/hr = $0.004/gal. Stage 2 Equipment cost = $5177 (materials) Power consumption = 5 KW Power cost @ $0.05/KWH = $0.25/hr = $0.0004/gal As one can see, there is a trade-off between energy efficiency and initial capital equipment cost. There are many engineering design parameters that can be varied to further optimize the system performance, such as spacing and construction materials. Now, let us look at the amortization of capital equipment costs. If the apparatus lasts only one year of continuous operation without significant repair or replacements of parts, then we have 365 days x 24 hrs/day x 600 gal/hr = 5.256 x 106 gal/year

SYSTEM DESIGN EXERCISES & EXAMPLES

191

processed by the system. At an initial cost of $10,354 for the two stage equipment, the cost per gallon is $10,354 / 5,256,000 gal, or $.002/gal of processed water. Our guess at this time is that the equipment should have a life between 5 and 10 years of continuous use. Hence, the capital equipment costs would be about 10% of the total cost of processed seawater. It appears that the cost of producing potable water from seawater is in the range of 0.5 cents per gallon with the ED approach. When we are able to employ lower cost membranes and electrodes in the near future, the total system cost will be markedly reduced. Membranes that cost $1.00/ft2 would reduce the equipment cost from the $5177 figure to $1180 in terms of materials alone. With additional engineering development and testing, the system seems more than competitive with RO or distillation.

5.3

Cost Estimates Outline for an Electrodialysis De-ionizing System

The following is a brief summary of cost factors in the design and manufacture of an electrochemical de-ionizing unit recently configured at TRL. Costs of a system are essentially proportional to the quantity of dissolved matter removed per unit time from a water stream (flow through) or a reservoir of water (recirculation system). The quantity of dissolved substance removed per unit time from the water, d Q / d t , is directly proportional to the electric current flow, or R = d Q / d t = ni (rate of dissolved matter removal), where r) = coulombic efficiency.

192

ELECTROCHEMICAL WATER PROCESSING

Convenient units for R are: • • • •

ppm-gallons/min. grains/min. grams/min. lbs/min.

Coulombic efficiencies are generally between 70% and 90% in a practical system. A 70% figure is quite conservative. Values of n = 70% to 90% have been regularly obtained at "higher" current densities. Generally, when the TDS drops down to low values, i.e. 1 to 20 ppm, the efficiency, r\, falls off to 70% or less. Some useful conversion figures are provided below. 1 gal = 3.79 liters 1 gram = 15.4 grains 1 gm/liter = 8.35 x 10"3 lb/gal 100 ppm = 0.1 gm/liter = 0.379 gm/gal. = 1.54 grain/liter = 5.85 grain/gal. At 100% coulombic efficiency, approximately 26 amp-hours remove one gram equivalent weight of dissolved substances. If we set 1 equiv. wt. = 60, then 1 amp-hour is equivalent to about 60/26 gram of dissolved solids. Thus, the passage of 1 amp-minute is equivalent to 1/26 gram, or about 0.04 gm of DS. Normalizing the above to grains per minute, or ppm-gal/min, we obtain for a single cell: • 1 amp removes 0.62 grains per min. • 1 amp removes 10 ppm-gal per min. Now, we must relate this to electrode and membrane area, device size, and power supply specifications to estimate manufacturing materials costs.

SYSTEM DESIGN EXERCISES & EXAMPLES

193

Continuous current densities in the order of 0.04 to 0.05 amps/in 2 are acceptable with present electrodes and result in reasonably low erosion rates. Hence, taking i = 0.05 amp/in 2 as an operating current density, we find that 20 in2 area are required for removal rate of 10 ppm gal/min. Current densities at peak water flow of 0.10 amp/in 2 or more are acceptable, as well. If a system were to be designed that was capable of the peak performance, below, as a once through flow, or • 350 ppm input water - to - 50 ppm output water • 10 gal/min. maximum flow rate, then the effective area for the system would be: Total area = 2 in 2 /ppm-gal/min x 300 ppm x 10 gal/min = 6,000 in2 = 40 ft2 This design is equivalent to a removal rate of 1800 grains per minute. Ion exchange membranes employed presently cost about $5/ft 2 in moderate quantities. Electrodes presently utilized are also about $5 /ft2. Future costs may come down to $l/ft 2 for membranes and $0.20 to $0.50/ft2 for electrodes with some further development. The ratio of electrode area to membrane area will range between 1:2 to 1:20 depending upon series/ parallel design specifics. The volume of the system described above would be about 5 to 6 ft3 with 1/4 in. inter-electrode spacing, or about 2.5 to 3.0 ft3 with 1/8 in. spacing. The plastic encapsulation volume with manifolding, etc., would probably be about an additional 0.4 to 0.5 ft3 in volume. External housing, if any, to withstand the 100+ psi line pressures is not estimated here. At an average voltage of 3 volts/cell, and a total of 3000 amps-cell, a power supply with about 10 KW

194

ELECTROCHEMICAL WATER PROCESSING

output is required. Power supply is a simple, non-regulated, non-filtered dc source with no special requirements other than overload protection. The following is a cost summary for the full-scale system, capable of producing 10 gal of water per minute with a TDS change of 300 ppm from input to output. Table 5.2 Factory cost summary for 1800 grain/min. system. Present

Future Production

Cost

Size

Cost

$200

40 ft2

$40 to $80

Electrodes

$5 to $10

5 ft2

$1 to $5

Structure (Plastic)

$10+

4 ft3

$5 to $10

?

5 to 7 ft3

?

Power Supply

$300+

10 KW Peak

$200

Total Cost

$515+

Membranes

Housing

$245 to $295

It should be noted that the above figures are merely estimates. They do not account for possible volume production cost reductions, direct labor or overhead. A system of 1/10 the above size with a holding tank of perhaps 50 gallons may be a more practical design for residential applications, where water usage is intermittent and peak flow has usually very short duration. In that case, virtually all the above costs would be reduced by a factor of 10, but the cost of the tank reservoir plus any necessary controls would need to be added.

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

6 Applications Discussion 6.1

Demineralizer: Electrodialysis

ED water processing systems can be produced in a wide range of sizes. The process lends itself readily to designs and product configurations ranging from small, hand held, portable units to large-scale systems for industrial and municipal uses. Units can be stacked easily to handle large quantities of water, or to increase purity levels of output. Electrodialysis can be used to convert salt water (seawater) to drinkable water by removing salts from input water to potable concentration levels. The system will also sanitize (disinfect) the water if configured to do so. Electrical injection of strong oxidizers generated from the sea water at the start of processing would destroy any micro-organisms that might be present. Seawater normally contains salt at a concentration of 3 to 3.5% by weight, or about 35,000 ppm. Water composition from drinking wells is often at 200 to 400 ppm levels. Water from city or municipal reservoirs is usually between 80 to 150 ppm, and is considered soft for washing purposes. 195

196

ELECTROCHEMICAL WATER PROCESSING

Depending upon the length of operating time (duration of processing), the purity of the water emerging from an ED system may exceed that of reservoir water. 6.1.1

Advantages of Electrodialysis

The method uses an entirely electrical process (electrochemical) that requires only a source of a.c. or d.c. power to operate. Some of the attendant advantages are outlined below. 6.1.1.1 General Characteristics Silent operation No mechanically moving parts Operates at standard temperature and pressure Low power consumption, efficient Self regulating, needs no on-off switching Made entirely of plastic - no corrosion or maintenance No vibration, sturdy and abuse resistant Small waste water ratio Not affected by impurities usually found in water Operates over wide range of power inputs No maintenance service required Additional features: • Units can be portable to sites such as camps, motor homes, etc. • Modular construction enables multiple unit systems 6.1.2

Desalination System - Module Specifications

The drawing below shows the simple system with its essential parts. The power converter is an integral component of the module.

APPLICATIONS DISCUSSION

197

To 120 volts

Figure 6.1 Electrolyte desalt system.

A basic unit consists of the following parts: a. b. c. d. e.

ED module (Desalter) Power limited & water condition indicator Flexible plastic lines for water input and outputs Two circulation pumps Line cord adapter for operation from 12 volts d.c.

An example of a practical design is outlined below. The desalt module has these specifications: Weight Size Maximum output rate Maximum input power Internal water capacity

30 lbs 7" x 8" x 22" 1 gallon per hour 500 watts 0.7 gallons (2.6 liters)

The photograph Figure 6.2 is a laboratory size module showing a direct current power supply and the pump

198

ELECTROCHEMICAL WATER PROCESSING

Figure 6.2 Module.

needed for water circulation. Depending upon the hydraulic configuration that is set up, it may be employed as a water softener, a demineralizers, or even a desalination system. Figure 6.3 graphically shows the performance of the above module as a demineralizer when operated at about 120 volts d.c. over a two hour period. The electric current falls off quite rapidly as the salt (mineral) concentration in the processed water compartments is reduced by the electrolytic transport of ions to the seawater side of the system. A simple version of such a system would involve only connection to a 120 volt a.c, or a 12 volt d.c. source, and connection of the hydraulic lines, as shown in the above drawing for its operation. Water purification rate would be dependent upon the applied voltage and consequent electric current.

APPLICATIONS DISCUSSION

199

• c

»,

a» o

-£

«

§

0.

Figure 6.3 DeSalt module performance. (Same as shown in Figure 6.6 when starting as desalina tor).

The waste water line is connected either to the primary water source tank, or directly into the ocean by a submersible pump. Processed water lines connect to a body of water, usually a reservoir containing an initial quantity of sea water to be desalted. After the unit has been operating for the specified design time, a conductivity meter could provide the output signal to indicate that the processed water is sufficiently desalted to be usable for cooking, drinking or washing purposes. The length of time required to process the water depends upon the amount of water initially in the reservoir and power input level. 6.1.3

Performance Characteristics

The curves below show the type of performance one can expect from this size unit. The graphs in Figures 6.4, 6.5 and 6.6 are plots of power level and salt level in the processed water reservoirs as a function of time. The data is presented in three graphs because they are divided inrto three ranges of initial salt

200

ELECTROCHEMICAL WATER PROCESSING

Figure 6.4 DeSalt module performance. (High salt levels).

Figure 6.5 DeSalt module performance. (Medium salt levels).

concentrations. Actually, the process is continuous, but it is shown in the form of three separate processes in order to change scales to make the data more easily visible, especially at the low end of salt concentrations.

APPLICATIONS DISCUSSION

201

Figure 6.6 DeSalt module performance. (Low salt levels).

Output ppm and power consumption versus operating time are plotted for a 2 gallon-size reservoir of processed water when the module is powered from a rectified 120 volt a.c. line. Electrical power from a current limited supply, (5 amps), increases to a maximum of 500 watts as salt content in the processed water diminishes. Then, as the processed water becomes more purified, power consumption decreases to low wattage. If the fresh water (processed water) reservoir has a 10 gallon capacity, the processing time would be 15 hours instead of the 3 hours shown in the graphs for 2 gallons. Between 15 and 20 gallons can be processed from seawater per day of operation. 6.1.4

Cost Factors

Initial equipment cost of the system is continually being assessed. A factory cost is anticipated for the system of between $500 and $700, depending upon production rate and features embodied in the system. Operating costs

202

ELECTROCHEMICAL WATER PROCESSING

for electrical energy are estimated based on $0.10 per kilowatt-hour. Such a small unit requires about 500 watthours to produce 2 gallons of desalted water. Electrical energy costs are about $0.02 to $0.03 per gallon of potable water.

6.2 6.2.1

R e s i d e n t i a l Water S o f t e n e r Product D e s i g n Description

To illustrate the possibilities of developing and manufacturing small water softening systems employing electrodialysis principles, a design exercise is offered in the following pages of this section. The design goal is a system to non-selectively remove all ionized and dissolved substances from the input water stream into a home or any small establishment where the peak water flow demand is 10 gallons per minute or less. Input water could originate from any primary source, such as wells or reservoirs. Water entering the unit is "softened" and "de-ionized" in the sense that the total dissolved solids, TDS, is lowered as it passes through the system. Electric current passing across ion-selective membranes within the device separates all electrically charged substances from the main water stream and into a waste water outlet. This waste water contains minerals, etc. removed from the main stream in concentrated form, and is discarded or drained back into the soil (its original source). Thus, water leaving the unit would normally have 90% or more of its original dissolved materials removed. If water of 500 ppm TDS (quite hard) is the source concentration, for example, the system would deliver between 5 and 50 ppm TDS to the home, depending upon the demand rate for water by the user.

APPLICATIONS DISCUSSION

203

Some of the types of materials, (mostly inorganic compounds), which would be removed by an ED system are listed below. Cations

Anions

Others

Sodium Potassium Calcium Magnesium Manganese Lead Copper Iron Mercury

Carbonates Chlorides Bromides Sulfates Nitrates Sulfides Phosphates Borates Citrates Acetates

Acids Alkalis Hydrogen Sulfide

However, virtually no organic substances are removed from the processed stream. Un-ionized materials such as sugar, alcohols and solvents are unaffected. In order to remove or decrease the concentration of such non-ionic materials, filters a n d / o r adsorbent charcoal beds would be required. 6.2.2

Physical Description of the System

Figure 6.7 shows the general configuration of a proposed system for residential application. Dimensions and hydraulic connections are schematically outlined. The component parts are identified below. 1. Control Unit is a valve device that switches on the flow of waste water through the processor when the mainstream water flow is initiated by house demands from faucets. 2. Power Supply is a D.C. source at 120 volts, and a maximum power level of 4 KW to process up to 1,000 ppm input water. A timer circuit

204

ELECTROCHEMICAL WATER PROCESSING Processed water output

Power line input

Figure 6.7 ED Residential Water Processing System.

switches electrical polarity periodically, and briefly, to clear scale build-up from the negative electrode. 3. Electrochemical Processor is an assembly of cells (membrane/electrode stack) connected hydraulically in parallel, and electrically in series. Current flow causes dissolved solids to be continuously removed from the main water stream during usage. When the water flow is stopped, the electric current quickly diminishes to zero because of the high electrical resistance of the main body de-ionized water. Hence, the power supply and module are self regulating, eliminating the need for switching during times of zero or low water flow. The module is shown as a plastic cylinder of about 50 inches in length, and 8 inches in diameter. The structure is designed to withstand 300 psi. Pressures as high as 100 psi may be encountered in some water input lines.

APPLICATIONS DISCUSSION

6.2.3

205

Operation

Water from the primary source, (wells, etc.), enters the system via the flow controller. When a water demand is made, the flow controller permits flow of water through the waste channels. Waste water ratio is less than 5% of the total. The power supply is continuously on-line and supplies current when there is some water flow. Electric power is self-regulatory, and will supply current as needed, continuously and smoothly varying in accordance with flow rates and input TDS. No external switching is necessary. 6.2.4

D e s i g n Example

An ED system can be sized for residential application in a very direct manner. Cost, simplicity and physical dimensions are among the critical factors that determine its practicality. A geometry and size was selected for this design exercise that seems to be in the range of economic and performance criteria for applicability to typical home water supply needs. Cylindrical configuration for outer case structures is generally more suitable as pressure containers for on-line installations. This aspect of construction has been discussed to some extent in Chapter 5. The dimensions and operating conditions selected as a proposed initial design approximation are as follows. First, it is important to try to minimize the quantity of materials and number of components in this first design draft. If a plastic cylinder is selected as the pressure bearing outer shell for the stack of ED cells, then we should seek the smallest diameter possible to minimize cost of the cylinder. As the cylinder diameter increases, its walls will need to become thicker to sustain the pressure differential, and generally cost increases non-linearly with diameter of pipes and cylinders because of increased manufacturing difficulties.

206

ELECTROCHEMICAL WATER PROCESSING

Also, we need to bear in mind that there is a maximum practical length to the module, or cell stack, especially if it is to be placed indoors and perhaps installed with its long axis in a vertical position. Four to five feet in length seems to be about the upper limit for a unit to be readily installed in most residences. There are some instances where much longer units could be used installed horizontally, but there are other problems of fabrication that are encountered when electrodes and membranes become very narrow and very long. With these considerations in mind, let us select a cylinder with inside diameter of about 12 inches and a working length of 50 inches, and then explore the performance and costs of such a configuration. The following graphs show the performance that may be expected from the design described in Figure 6.8. In Figure 6.8, the power input requirements are plotted against the flow rate for different Input Water TDS Values. Particulars of the design, in addition the container diameter and length, are: Inter-membrane spacing Cell width Cell length Number of cell Total stack potential

0.10 inch 8 inches 50 inches 40 120 volts

The estimated cost of components and materials on limited production basis for this module design is between $300 and $500 depending upon pricing of membranes and production levels. Coulombic efficiency is assumed to be 100%, not a realistic figure, but it will serve as an initial and convenient starting point. It will be noted in Figure 6.8 that the power level is much lower at the lower flow rates water TDS because the average resistance of the processed water is higher.

APPLICATIONS DISCUSSION

207

Figure 6.8 For different flow rates and input water TDS. Power input requirements.

Since the unit is operated at constant voltage, electric current drawn from the power supply is considerably less at low TDS values. Figure 6.9 shows the change in output TDS as flow rate is increased with three different input TDS values of 1000, 500 and 200 ppm. This graph is correlated with Figure 6.8. Most problem residential groundwater sources that are not from municipal reservoirs have TDS values ranging from 200 ppm to 500 ppm. Higher values than 500 ppm are not rare, but probably much smaller in number. Hence, it would seem that the design center of operation for a "water softener" system intended for this application would be in the regions shown in both of the above graphs. Also, most water discharge rates for home use are between 2 and 4 gallons per minute or less. Rarely are flows of 10 gallons per minute encountered in normal, single residence buildings. Figures 6.8, 6.9 and 6.10 are performance data for electrode lengths of 50 inches.

208

ELECTROCHEMICAL WATER PROCESSING

Figure 6.9 For different flow rates and input water TDS. Output water TDS.

Figure 6.10 For different flow rates and input water TDS. Energy per gallon output.

If these are indeed the circumstances, then we see that the unit described above has been somewhat over-designed for this application. The TDS output is much lower than necessary at the peak flow rates of 2 to 6 gallons per minute.

APPLICATIONS DISCUSSION

209

A level of 50 or more ppm is generally acceptable for most potable use. There are two directions that a design modification can take to raise the output TDS level. They are: decrease voltage and current from the power supply, or reduce the physical size of the module. The latter is the more attractive direction because power consumption costs per gallon of output water, as shown in Figure 6.10 are very small. In fact, they are almost negligible, and would be even less for smaller sized units operated under the same 120 volts. If the length of the module is reduced to 25 inches, half the power is needed, but much more importantly, the cost is reduced by a factor almost two, and the output TDS of processed water in the flow rate and input TDS range commonly encountered will still be acceptable, good quality water. Figure 6.11 and Figure 6.12 show the dependence of Power requirements and output TDS upon flow rates with the shorter length unit.

Figure 6.11 For different flow rates and input water TDS. Power input requirements.

210

ELECTROCHEMICAL WATER PROCESSING

Figure 6.12 For different flow rates and input water TDS. Output water TDS.

Life expectancy of hardware should be between five and ten years. Amortizing capital equipment costs of $500 over a five-year period would be less than $10 per month for hardware replacement. The average household uses over 10,000 gallons per month. Dividing the numbers gives a cost per gallon of less than $0.001 per gallon of processed (softened) water. It would seem that application of ED to commercial and consumer water softening problems could be a very attractive and a sound subject for serious investigation and development. Among the many benefits that ED offers is the absence of regeneration cycles and the elimination of need to introduce salt to the systems as is done presently for cation resin bed softeners. 6.2.5

Competitive Methods

Conventional water softeners, employing cation resins with automatic features for regeneration, cost between $1,000 and $2,000. Salt (sodium chloride) consumption costs will vary depending upon hardness, efficiency of operation and

APPLICATIONS DISCUSSION

211

the price of salt. The utilization factor for the salt in terms of cation displacement is generally very poor. Usually salt cost per gallon of water, if the initial hardness is in the range of 500 ppm, will be greater than $0.001 to $0.02. Reverse osmosis systems are too slow in operation and too costly to be employed as practical, in-line water demineralizers. An RO unit capable of delivering 4 to 6 gallons per minute peak demand without storage tanks would be quite large and costly.

6.3

Electrical Water Processor Portable Design

Many areas of the U.S. and other countries have water with unacceptable levels of dissolved substances, salts, acids, alkalis, etc. The potability of such water is very limited. Water is either not drinkable or is at least very distasteful. There are only a limited number of ways in which to treat the problem on a small, decentralized scale. Other than introducing improved large-scale, municipal water treatment systems, some alternative options are listed below. 6.3.1

Present Solutions

1. Bottled water - distilled or de-ionized. Cost is high and convenience factor low because of necessity to transport large bulk. 2. Water softener (home type) - replaces dissolved solids with unwanted sodium salts, and requires purchase and handing of salt. 3. De-ionization - Expensive two resin bed systems that are generally not available because of cost and hazard problems associated with regeneration chemicals. 4. Reverse osmosis - High quality water output, but requires high-pressure pumps. Maintenance is significant, equipment cost is high and portability is low.

212

ELECTROCHEMICAL WATER PROCESSING

Another possibility, electrodialysis processing, is an alternative. Inexpensive, table-top batch processing can be achieved with self-contained reservoirs for both waste water and processed water. Two chambers configured to electrically concentrate the dissolved solids into one set of compartments while the other, central chamber, is freed of dissolved materials to a wide range of desired levels. The water in the waste chambers is discarded. No chemicals other than the initial water are needed. The unit does not have to be monitored during operation since the process is self-limiting. The device works well with almost any level of dissolved solids. Heavy concentrations of solutes will not block or contaminate the system. In necessary cases, the unit could be employed as a desalinator (seawater to fresh water converter). 6.3.2

Operation of an ED System

1. 2. 3. 4.

Check to close both drains Fill all chambers from the top Connect unit into wall outlet or (power supply) When indicator shows very low current drain processed water into container. 5. Discard water from the waste water chamber 6. Repeat steps 1 through 5. 6.3.3

D e s i g n Prototype

On the basis of the above operational description, we will briefly outline an early version of a portable processor designed to treat a small quantity of water for personal use. Purification of small quantities of water in a batch process manner for drinking, cooking, washing or other generalpurpose use, such as refilling storage batteries, steam irons, etc., is desirable on frequent occasions. No chemical agents are employed; only electric power is required.

APPLICATIONS DISCUSSION

213

Figure 6.13 shows a small portable unit with a connector mounted at the top, an opening on the top lid for pouring water into the processing chambers, and a release valve at the bottom of the unit from which the processed water is drained into a separate container for later use. At the end of a processing cycle, and after the processed water has been drained out, the waste water is removed from the unit by pouring out of the top opening. Water processed by this device is essentially de-ionized. The device will lower the total dissolved solids (TDS) of the input water to a few parts per million, (1 to 10 ppm). The resultant output will have the properties of relatively high quality distilled or de-ionized water, which is frequently purchased at stores as bottled potable water.

Figure 6.13 Portable demineralizer with static fluids.

214

ELECTROCHEMICAL WATER PROCESSING

Water with dissolved solids as high as 1,000 to 10,000 ppm can be treated with the apparatus above. 6.3.4

Description

The device can have a variety of configurations depending upon the use for which it is intended. As a counter top product for home use, a simple two component system would probably be most appropriate. 1. Power supply (probably wall-plug mounted type) 2. Water processor housing Capacity: 1/2 to 1 gal per processing Processing: 20 minutes to 1 hour depending upon hardness of water and level of purification desired Configuration: Rectangular container with filler holes/ pouring spouts; compact and placed directly on counter. Maintenance: Other than periodic cleaning and washing, there are no other special maintenance requirements that are known at this time. General Characteristics Voltage Current Size Power Input Useful Life Manufacturing Costs Materials:

between 12 and 40 volts d.c. depending upon design 0.25 to 1 amp d.c. depending upon size of unit 1 pint to 2 quarts 10 to 50 watts 3 to 5 years minimum continuous use (very approximate, & dependent upon size) $5 to $12

APPLICATIONS DISCUSSION

Parts: Labor:

215

$3 to $5 $2 to $5

There are no known difficult manufacturing operations involved in the production of the apparatus, nor are there any expensive, rare or strategic materials of construction. • No heating during normal operation because the system is not a thermal process, distillation apparatus. • No moving parts • Water pH can also be controlled • Compact versions of the apparatus make it available to traveling and military applications. • Low salt aspects of the output water make it attractive for health reasons. • Almost maintenance free operation • Safe handling, and low voltage permits easy UL approval.

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

Appendix A: Some Physical Constants and Conversion Factors Useful conversions from one system of units to another are listed below. 1 gram = 15.432 grains 1 once = 28.35 gm. 1 newton = 0.224 lb. 1 pound = 445,000 dynes 1 atmosphere = 14.7 l b / in2 1 joule = 107 ergs 1 calorie = 4.186 joules 1 in2 = 6.45 sq. cm. 1 sq. ft. = 929 sq. cm. 1 cu. in. = 16.39 cu. cm. 1 gallon = 3.785 liters 1 gallon = 231 cu. in. 1 pound = 453.6 gm. 1 kilogram = 2.205 lb. 217

218

ELECTROCHEMICAL WATER PROCESSING

1 slug = 14.6 kgm. 1 foot-pound = 1.355 joules 1 B.T.U. = 252 calories 1 B.T.U. = 778 ft-lb. 1 horse power = 746 watts Some pertinent Physical-Chemical constants are presented for convenience below. Avogadro constant

N

Gas Constant

R

Boltzmann constant

k=R/N

Electronic charge

e=F/N

Absolute Temperature T of the 0° C point o Coulomb q Faraday constant F

6.0238 x 1023 molecules/ mole 1.987 cal/deg-mole 82.06 cm3 atm/deg-mole 0.08205 1 atm /deg-mole 8.31 joules/deg-mole 1.38 xlO 1 6 erg/ deg-molecule 1.602 x 1019 coulombs 273.2° K 1 amp-sec. 96,493 coulombs/ equivalent 26.8 amp-hours

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

Appendix B: Conductance and Solubility The following tables provide the reader with some of the more common and readily available materials properties of specific ions and compounds. A commonly used inorganic salt for electrolytic experiments and in conjunction with reference electrodes is potassium chloride, KC1. Some data for this compound in relatively dilute solutions are presented below, where the equilibrium constant for salt dissociation, K, is defined as

K+

B.l

cr

KC1 Ionization Constants Normality N 0.0001 0.001 0.01

K 0.0075 0.035 0.132 219

220

ELECTROCHEMICAL WATER PROCESSING

Normality, N 0.10 1.0

K 0.495 2.22

As a helpful guide to calculating conductance and corresponding resistivity, the Table B.l below provides some additional related data for the salt KC1. To further provide the reader with typical properties data for some of the more common solutes, Table B.2 below lists some equivalent conductance at different concentrations. Limiting ion mobilities are significant factors for making assessments of conductance and ionization coefficients. The term limiting mobility refers to the value at infinite dilution, i.e., conditions in which interaction with other ionic components is minimum. Some values of the more common ions are listed in Table B.3 below. An extremely useful table to have available for calculating equivalent weights, etc., is a copy of the International Atomic Weights of the Elements (Table B.4).

Table B.l Equiv. per Liter

Dilution ml per Equiv.

Specific Conductance

Equivalent Conductance

1

103

0.1119

111.9

0.1

104

0.0129

128.9

0.01

105

0.00141

141.3

0.001

106

0.000147

146.9

0.0000149

148.9

0.0001

7

10

APPENDIX B: CONDUCTANCE AND SOLUBILITY

221

Table B.2 Equivalent conductance at 298 Deg. K.

c

NaCl

KC1

NaOH

KN03

HC1

HN03

1/2H 2 S0 4

0.0005

125

148

246

143

423

416

413

0.0010

124

147

245

142

421

415

400

0.0020

123

146

141

419

413

390

0.0050

121

144

241

139

416

409

365

0.010

119

141

238

136

412

405

336

0.020

116

138

227

132

407

401

308

0.050

111

134

221

126

399

393

273

0.100

107

129

120

391

384

251

0.200

102

124

113

380

374

234

0.500

93

117

101

359

357

223

333

333

1.00

112

Table B.3 Limiting ion mobilities cm2 /volt-sec. Ion

/ixl0s

Li+

40

ci-

Na+

52

K+

76

H+

363

Ion

//xlO 5

Ion

/ixlO5

79

Ca ++

62

Br

81

Ba++

66

r

80

so4=

83

205

NO-

74

OH

121.8

9 209 10.8 79.9 112.4 40.1 12 140.1 132.9

51

18

33

56

4

82

5

35

48

20

6

58

55

Sb

A

AS

Ba

Be

Bi

B

Br

Cd

Ca

C

Ce

Cs

Antimony

Argon

Arsenic

Barium

Berylium

Bismuth

Boron

Bromine

Cadmium

Calcium

Carbon

Cerium

Cesium

137.4

74.9

40

26.97

13

Al

Aluminum

Atomic Wt.

Atomic No.

Element

Symbol

Table B.4 Atomic weights and numbers.

Mo Nd

Molybdenum Neodymium

Radium

Protactinium

Praseodymium

Potassium

Platinum

Phosphorous

Palladium

195.2 39.1

78 19

Pt K

Ra

Pa

88

91

226

231

140.9

30.98

15 P

59

106.7

46 Pd

Pr

16

8

O

Oxygen

190.2

76

Os

14

7

N

Nitrogen Osmium

58.7

28

Ni

Nickel

144.3

60

20.18

96

42

10

Atomic Wt.

Atomic No.

Ne

Neon

Symbol

Element

o

h-1

on

!» Tl sa O n a

>

H tu

n >

a

n

O

n

M

r

m 1

178.6 4.00

72

2

Hf

He

Helium

Tellurium

Tantalum

Te

Ta

S

Hafnium

197.2

Sulfur

79

Au

Gold

162.5

63.6

92.9

58.9

52

72

16

127.6

180.9

32

87 38

Sr

Strontium

72.6

32

Ge

Germanium

23

11 Na

Sodium

69.7

31

Ga

Galium

107.9

47 Ag

Silver

156.9

64

Gd

Gadolinium

28

14

Si

Silicon

19

9

F

Fluorine

79

34

Se

Selenium

152

63

Eu

Europium

45.1

21

Sc

Scandium

167.2

68

Er

Erbium

150.4

62

Sm

Samarium

66

Dy

Dysprosium

44

Ru

Ruthenium

29

Cu

101.7

Copper

85.5

37

Rb

Rubidium

41

Cb

Columbium

102.9

45

Rh

Rhodium

27

Co

Cobalt

186.3

75

Re

Rhenium

52

24

Cr

Chromium

222

86

Rn

Radon

35.5

17

Cl

Chlorine

z

H

t->

> a (X) o

M

d

a o Z n

Z

o

n

ça

X

o

M

>

Thallium Thorium

Uranium vanadium

1.01 114.8 126.9 193.1 55.9 83.7 138.9 207.2 6.9 174.9 24.32 54.9 200.6

1

49

53

77

26

36

57

82

3

71

12

25

80

H

In

I

Ir

Fe

Kr

La

Pb

Li

Lu

Mg

Mn

Hg

Hydrogen

Indium

Iodine

Iridium

Iron

Krypton

Lanthanum

Lead

Litium

Lutecium

Magnesium

Manganese

Mercury

Zirconium

Zinc

Yttrium

Ytterbium

Xenon

Tungsten

Titanium

Tin

Thullium

Terbium

164.9

67

Holmium

Ho

Element

Atomic Wt.

Symbol

Element

Atomic No.

Table B.4 (cont.) Atomic weights and numbers.

232.1 169.4 118.7 47.9

90 69 50 22

Th Tm Sn

131.3

54

Zr

40

30

39

Y Zn

70

Yb

Xe

51

23

V

91.2

65.4

88.9

173

238.1

92

u

183.9 74

W

Ti

204.4

81

Tl

159.2

65

Tb

Atomic Wt.

Atomic No.

Symbol

o

z

on

ta

o n

ta TI

5

>

ñ

g

M

n

O

!«

H

n

M

r1

m

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

Appendix C: Feeder Tube and Common Manifolding Losses In designing an array of cells, there are many additional factors to consider for the purposes of optimization. Among these are such considerations as the shape and size, manner in which cost changes with geometry, electrical power supply requirements and electrical current losses encountered in the common manifolding of cells that are supplied with water from a common source. Cells are interconnected by common manifolds that are so large in cross-sectional area that they present little electrical resistance to short circuiting currents. The only significant resistances to the common connections are those in the smaller diameter, longer length feeder tubes. In electrodialysis systems, there are four such circuits in parallel electrically. Two manifolds and corresponding feeder tubes for the entering water, and two manifolds for the separate exiting water processed and waste water 225

226

ELECTROCHEMICAL WATER PROCESSING

channels. This situation is true regardless of the specifics of the design geometry and manufacturing methods of module construction. Figure C.l shows the general configuration with only two of the four alternate manifolds illustrated. Manifold cross sectional area is much larger than the individual feeder tubes for the individual cells. The following is the development of the equations describing the common loop current losses, (parasitic currents), that an ED array experiences. Figure C.2 shows the simple mesh current model that represents the leakage paths through the parallel hydraulic circuits model employed in arriving at the expressions and consequent estimates of coulombic losses. In other words, electrical current that does not contribute to the separation of dissolved substances from the processed water stream. Each cell is represented as a voltage source, E, and the electrical resistance path due to the feeder tubes or channels from each cell to the common manifold is represented as the constant, R. If all the fluid supply channels are the Exit manifold

Electrodes Membranes

Feeder tubes

Entrance manifold Figure C.l Manifold configuration.

FEEDER TUBE AND COMMON MANIFOLDING LOSSES

R:

R -I E

^

E

227

R:

Hh E

-I E

Figure C.2 Equivalent circuit for common manifold.

E

E

E

Figure C.3 Single manifold - Equivalent manifold.

same in construction the resistances, R, are all equal and constant. There are a number of ways in which the analysis of the electrical current losses can be determined. Simple application of Kirchoffs Law enables us to calculate the dissipative current through any cell in an array. Figure C.3 shows the loop currents assumed in the series of circuits of the array of cells. Each small loop is treated as an independent circuit except for the interconnectivity due to currents flowing in the adjacent resistances from other loops. After establishing the basic relationships, we can construct a series of simultaneous, linear equations to be solved for calculating the currents. The form of these equations is as follows. For any one loop in the circuit array the expression has the form of equation C.l. Tracing the potential around the nth loop results in the sum of all current producing voltages passing through the respective feeder tube resistances equals the voltage of that particular cell produced by current flow through the cell. The cell voltage, E, is that expressed in Equations (4.25) and (4.87).

228

ELECTROCHEMICAL WATER PROCESSING

2i n -R = E + i n . 1 -R + i n+1 -R

(C.1)

where n =1 to N, if there are N number of cells in the array. The current passing through the n t n cell due to the common manifold circuit is i n . A sample calculation of the above solution can be made by taking a 10 cell array, and solving for the currents in each cell. The equations are: 2irR = E + i2-R 2i2-R = E + irR + i3-R (C.2) 2i9R = E + i 8 R + i10R 2i10R = E + i 9 R If we assign some values to both R and E that are typically in the range of operating ED systems, we can solve for the currents to obtain an idea of the dissipation current distribution. Also, in order to see what the shape and the magnitudes of the dissipation currents are we must settle upon values for the geometry of the feeder channels from the manifold to the cells. Let E = 1 volt, and the electrolyte resistivity be in the order of 200 ohm-in., and the length, 1, and diameters, d, of the feeder channels can be about 0.10 in. and 1 in., respectively. The solution of the equations above give the calculated values for the currents through cells one through ten as follows. For p=200 ohm-in.

For p=2 ohm-in.

ii = 1.96x10""4 amps \ = 3.53 i3 = 4.7 \ = 5-5 \ = 5.89

0.0196 amps 0.0353 0.047 0.055 0.0589

FEEDER TUBE AND COMMON MANIFOLDING LOSSES

For p=200 ohm-in.

For p=2 ohm-in.

\ = 5.89 \ = 5.5 i = 4.7 \ = 3.53 i10 = 1.96

0.0589 0.55 0.47 0.353 0.0196

229

Dissipation currents are highest in the middle cells, and diminish toward either end cell. The solution concentration in the waste water channels can be between 10 and 100 times greater than in the processed water side. In fact, if the ratio of waste water to processed water is to be as high as practical, these could very well be the range of concentrations encountered in any operating ED systems. The values above are, for most water demineralizing applications, quite high. Significant coulombic losses can certainly be encountered in desalination ED systems where the salt water conductivities are high. The two entering water concentrations will be the same, and at the level of the initial input water source. However, one of the manifold systems at the exit will be at the concentration of the waste water, and will be the major contributor of any parasitic current losses. Figure C.4 is a plot of the two values of currents from the above table. As is readily seen, the most severe dissipative current, or coulombic losses are in the inner most, or middle cells of an array. Another current calculation that results in the same answers as derived by the preceding calculations is shown in Figure C.4. Here, the loop currents are taken differently. Instead of independent for each small circuit in the series, the currents are assumed as all originating from the first cell and circulating through each of the succeeding circuits in turn. Thus, the first current, iv passes through only the first cell, and the second current, i , passes from the first resistance, R, and then through around through the third resistor, R, and then through the two cell sources, E.

230

ELECTROCHEMICAL WATER PROCESSING Ten cell array 0.07 0.06

E

0.05

< 0.04 "'<

' ■

0.03 O

/

!

0.02 0.01 n

4

5 6 7 Cell number 200 ohm-in 2 ohm-in

10

11

Feeder tube diameter = 0.10 in. Figure C.4 Dissipative current per cell in an array. Consecutive and cumulative current loop analysis

E

E

E

E

E

Figure C.5 Consecutive and cumulative current loop analysis.

The relationships that describe the above current flows again are a series of N simultaneous equations having the form: N

X(inR) + inR = nE

(C.3)

The dissipative current, i n , passing through the n*" cell is given as N

in = X i n ' R

(C.4)

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

Appendix D: Variable Current Density D.l

Current Density Variation

The equations for the non-linear change in current density along the length of an ED cell with continuous and constant water flow were developed in Chapter 4. However, another, somewhat more detailed derivation is present below. For a better estimation of cell performance, we will treat the variation of current density along the flow path in closer approximation of reality. In Figure D.l the area current density is shown as the current conducted through an area of unit width and length, dx. Because the current density is constant in any direction normal to the cell length, or the x-direction, the current element taken for integration is ixW. The electric current density, ix, at any point at distance, x, from the entrance is (see Figure D.l)

231

232

ELECTROCHEMICAL WATER PROCESSING Current element, 1x Water flow direction

Figure D.l Area current density.

i x =E-c x /0-p

or

E-TJX-X

i

2 ° , Pe ^P ' -v-,

=

(D.l)

0-P

Because the current density is assumed constant along any path normal to the water flow, we can represent the electric current element, px, of width, W, as (D.2)

Px =i x -W

To calculate the total current flow across the electrode area, WL, we must integrate the above expression, or J px ■ dx = I,

the total current

(D.3)

By substitution, we obtain the complete expression for the current density as

px = w ^ ^ d x
WEcc 0-p

•e-Mxdx

(D.4)

This has the following general form, and for the sake of convenient shorthand we will temporarily substitute the symbols M and N for the respective polynomials. j"oLpx • dx = j*LNe"Mxdx = I

(D.5)

APPENDIX D: VARIABLE CURRENT DENSITY

233

Integrating directly for I, between the limits L and 0,

2i- e -M-x

NÍ L e" M x dx

M

Jo

+ constant

(D.6)

Boundary conditions are when x=0, 1=0, Thus the constant is evaluated, and the solution is

I=[l-e"ML] 21 M

(D.7)

Replacing terms into the equation for I for quantitative evaluation, we obtain E-77-A

l_e
Wc0pvyg

(D.8)

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

Appendix E: Mathematical Analysis: Water pH Control Cell and Ion Exchange Resin Regeneration The purpose of presenting the following analyses and proposed hardware configurations is to stimulate further thought in these directions, and promote development of practical systems for the many application needs that exist in the industrial and commercial world. To facilitate the design of an electrochemical system for the regeneration of resin stacks in water treatment devices, the following analysis has been performed. This mathematical treatment is independent of the analyses to follow in the other appendices, and it is specifically directed at the issue of generating hydroxyl and hydrogen ions for pH control purposes. This study of the processes involved in the operation of a pH control cell will be of assistance in the selection 235

236

ELECTROCHEMICAL WATER PROCESSING

of materials combinations, components and cell stack geometry for multiple cell arrays. The types of considerations entering into the design engineering of a total, self-regenerating system include these items: 1. Ion transfer membrane thickness and type of material including such parameters as ion diffusion, water transfer, and electrolytic conductivity 2. Inter-electrode spacing for optimum performance 3. Resin properties and size of beads 4. Electrode area 5. Electric current densities. The analysis that ensues was conducted on the basis of minimum initial assumptions in deriving the first series of expressions. This text is directed toward developing a series of equations along with their solutions that results in a quantitative relationship for the concentration of H + (hydrogen) ions a n d / o r OH" (hydroxide) ions in either the resins or in the water as functions of time. These concentrations are given in terms of the various physical parameters of the operating system in question. An important assumption made in these first derivations is that the generation rates by electrolysis of H + and OH" ions are constant, i.e. independent of specific ion concentrations at any point in time- In other words, the supply of appropriate ions is essentially unaltered during the cell operation, and there are no competing electrode processes such as deposition of metals at the (-) electrode, or generation of halogens at the (+) electrode. Figure E.l shows the general configuration employed in which the regeneration electrodes are an integral part of the deionization stacks. The resin beads are stacked between the membrane and their respective electrode. The cation exchange resin is contained in the "positive" compartment,

APPENDIX E: MATHEMATICAL ANALYSIS

Cation Resin

Cation membrane

Input water

237

Anion resin

Anion membrane

Deionized water output

Figure E.l Three chambered - Continously operating cell basics.

and the anion exchange resin in the "negative" compartment of a cell. The membrane is held rigidly in the center of the cell by suitable plastic screen spacers. During operation, water (electrolyte) flows into the positive, (+), side of the cell, and thence out and into the negative, (-), side. The exiting fluid is the de-ionized water. During regeneration, water is stagnant in the cell and provision is made for venting the H 2 and the 0 2 gas by product. Cell volumes and final salt levels are then estimated to determine the practicality of regeneration with a non-flowing water configuration. Non-flowing-fluid situation during regeneration would simplify the valving and switching requirements. Flow through during regeneration would necessitate the introduction of additional hydraulic circuits to convert from one configuration to another when operating modes are changed. Flow during operation of the system as a deionizer is serial. If flow were to be needed during regeneration, a parallel path would have to be provided through the two cell compartments. Some tests were run with a two-cell configuration in which the resins were integral with the electrode/membrane

238

ELECTROCHEMICAL WATER PROCESSING

assembly. Among the critical factors in the assembly design are: • • • •

Coulombic efficiency Resin capacity Resin void volume ratio time allowed for regeneration

For example, if we set the coulombic efficiency at 50% and the resin capacity at 12,000 grains per cubic foot volume of resin, resin void volume percentage at 30%, and regeneration time at 5 hours, the second set of numbers in Table I gives the current densities needed as the electrode spacing is increased. Current densities range from 0.011 to 0.18 amps/sq in, and the max concentration is constant at 1.43 normal. Equations and derivations for table E.l are given in Appendix A.

E.l

Analytic Approach

Figure E.2 is an illustration of the basic processes in a regeneration cell that contains no resins. This cell would function Hydrogen & hydroxyl ion injection system - symmetrical Low pH input

Effluent @ high pH

H+ Ions

+

a

Main water supply input Figure E.2 p H control cell.

OH" Ions

l

K

.so:

No+

.OH"

H+

-o'

APPENDIX E: MATHEMATICAL ANALYSIS

239

strictly as a pH control device in which acid and base are produced from salts initially present in the water. As shown in Figure E.2, the various ionic species and the rate processes associated with them are listed below. It must be remembered that all densities, concentrations, etc. are normalized on the basis of per square inch of electrode area. Table E.l provides a listing of these processes and the ions present in the two sides of the cell. Now, let us take each one of the above rates and ion specie concentrations and describe them in terms of the system Table E.l Rate process. Symbol

General Description on a per Square Inch Basis

RH(G)

Rate of H + ion generation via electrolysis

ROH(G)

Rate of OH" ion generation via electrolysis

RH(TD)

Rate of H + ion trans-barrier thermal diffusion

ROH(TD)

Rate of OH" trans-barrier thermal diffusion

RH(ED)

Rate of H + transfer via electric current

ROH(ED)

Rate of OH" transfer via electric current

RC(TD)

Rate of cation, Na+,Ca++, etc thermal diff.

RC(ED)

Rate of cation transfer in electric field

RA(TD)

Rate of Anion, Cl", S0 4 = , etc thermal diff.

RA(ED)

Rate of Anion transfer in electric field

If ion exchange resins are present, These are additional terms must be taken into account RR(H)

Rate of Cation Resin H + concentration change

RR(C)

Rate of Cation Resin cation concentration, change

RR(OH)

Rate of Anion Resin OH" concentration change

RR(A)

Rate of Anion Resin anion concentration change

240

ELECTROCHEMICAL WATER PROCESSING

Table E.l (cont.) Rate process. Quantity Symbol

General Description

Qc

Cation resin cation concentration, GEW per cu. in.

QA

Anion Resin anion concentration

QH

Cation Resin H + concentration.

QOH

Anion Resin OH" concentration.

q+c

Cation concentration, in Positive (+) Side

qH

H + concentration, in (+) side

q+A

Anion concentration, in (+) side

q.c

Cation concentration, in Negative (-) Side

qOH

OH" concentration, in (-) side

q.A

Anion concentration, in (-) side

Q0

Initial resin cation concentration., full regeneration.

q0

Initial salt concentration in water

operating parameters. It is possible now to relate these various parameters to each other in a quantitative manner. Taking each of the rate quantities and relating them to the electric current density and the relevant ion concentrations, we obtain the following. Rate of generation of H+ ions is directly proportional to the electric current. Since all computations are performed per unit area of electrode, the relationship is RH(G) = ia (at the negative electrode) where the constant, a, is the faraday equivalent, F, times the coulombic efficiency. The units are: i = amps/in 2 F = 26 amp-hours/gm equiv. wt. RH(G) = gm equiv. wt./in 2 -hour

APPENDIX E: MATHEMATICAL ANALYSIS

241

(The above also assumes negligible metallic plating on the negative electrode) Rate of hydroxide ion generation is similarly defined, if there are no competing reactions in the electrochemistry at the positive electrode. If we assume no generation of free halogens, essentially the only reaction possible is e + H20==OH- + l / 2 0 2 , and the expression is identical to that for RH(G). Thus ROH(G) = RH(G) = ai RH(TD) is the net rate of H + ion migration across the barrier membrane from one cell compartment to another, this diffusion is due to thermal phenomena only and is independent of any electric current flow and charge carrier transport process. The rate is given as RH(TD) = k ^ gm equiv. wt./in 2 -hr where qH is in gm equiv wt per in3 There are no H + ions on the negative side. ROH(TD) is described in the same fashion as RH(TD), and is ROH(TD) = k2qOH There are no OH" ions in the positive side. RH(ED) is the rate with which H + ions are transported from the (+) to the (-) side as part of the charge carriers in the conduction of electric current. This is a loss factor of the generated H + on the (+) side. The rate is proportional to the product of the electric current flow times the H + concentration, and is represented as RH(ED) = ibqH

242

ELECTROCHEMICAL WATER PROCESSING

R„„(ED) is the rate of electric field induced flow of hydroxide ions across the membrane, and has the same form as all "(ED)" processes. Since there are no OH" ions in the (+) side, the net expression is ROH(ED) = icqOH RC(TD) is similar functionally to RH(TD). It is the diffusion due to the concentration difference in the (+) and (-) sides of the cell. Generally, the cations will be produced in the (+) side due to the displacement of cations in the (+) side resin by the H + ions in solutions and generated electrochemically. Diffusion is from the (+) to the (-) side. RC(TD) = k3q+c - k4q.c RC(ED) is the rate due to the conduction of electric current through the cell, and since cations have a (+) charge, the migration is toward the negative side, and expressed as RC(ED) = irq+c RA(TD) = k5q_A-k6q+A Flow usually from the (-) to the (+) side. RA(ED) = isq.A In the case where there are ion exchange resins present in the cell compartments, these additional rates are involved; RR(H) is proportional to the quantity per unit volume, or concentration of cations in the (+) side resin multiplied by the concentration of H + ions in the surrounding water, or RR(H) = eq+cQHW2v Similarly, RR(C) = fqHQcW2v

APPENDIX E: MATHEMATICAL ANALYSIS

243

RR(OH) = gq.AQOHWV RR(A) = hqOHQAW2v The next step is the establishing of equations for solving. The purpose is a final expression which will give the concentration H + or OH" or cations or anions in either the water or in the resin stack as a function of time, and in terms of the various system and materials constants. The equations are as follows. Table E.l shows the various rate process directions. The total cations in the system is Q o + 2qo (if the volumes of water and resin on the (+) and (-) sides are equal.

Qo + 2qo = qc-q-c + Qc

<E.D

The balance of ions on the (+) side, q+c + qH = qA (no OH" in (+) side, pH>7)

(E.2)

The balance of ions on the (-) side, q.c = qOH + q.A (no H + in (-) side, pH<7)

(E.3)

The cation resin ion balance, QC = Q O - Q H

(E.4)

Total anion balance in the system, Qo + 2q0 = qA + q.A+ Q A

Œ-s)

Anion resin ion balance, QA = Q0-QOH

(E.6)

The rate, dq H /dt, with which the concentration of H + ions is changing in the ('+) side is the sum of some of the above rates, and is given below.

244

ELECTROCHEMICAL WATER PROCESSING

d q H / d t = RH(G) - RH(TD) - RH(ED) - ROH(TD) -R OH (ED) + R R (H)-R R (C)

(E.7)

Similarly, the rate of change of OH" ion concentration in the (-) side is, dq O H /dt = ROH(G) - RH(TD) - RH(ED) - ROH(TD) - ROH(ED) + RR(OH) - RR(A)

(E.8)

Other relationships that can be written are the necessary charge balance of total ion migration across the membrane. Thermal diffusion of ions across the membrane must result in zero net charge transfer. Hence, we have RH(TD) - ROH(TD) + R(TD) - RA(TD) = 0

(E.9)

Electric field gradient induced diffusion, (charge transport for electric current flow support), must result in the faraday equivalent of charge transport across the membrane corresponding to the net electric current density, i. RH(ED) + ROH(ED) + R(ED) + RA(ED) = iF

(E.10)

where F = faraday equivalent. The above rates in equation (E.10) are all "loss terms" with respect to their side of origination, and hence are additive as shown. Rate of loss and gain of ionic species within the two ion exchange resins are expressed below. As a convenient shorthand notation for the first derivative of a variable we will use the "pounds". symbol, or # = d/dt. In the cation resin, W#Q c = dQ c /dt = -dQ H /dt = RR(C) - RR(H)

APPENDIX E: MATHEMATICAL ANALYSIS

W#Q c = (fqHQc -eq_cQH)W2v

245

(E.ll)

In the anion resin, W#Q A = d Q A / d t = -dQ O H /dt = RR(A) - RR(OH) W#Q A = (hqOHQA - gq A Q OH )W^

(E.12)

Remember, all q terms are concentrations, and we are performing all calculations on the basis of 1 in2 of electrode surface area. Some additional relationships that will be needed to solve the total transport problem are discussed below. Since there are no sources or sinks for anions in the (+) side, the only gain or loss mechanism in that region for anions is the diffusion from the (-) side, or #q+A = dq + A /dt = isq.A

(E.13)

Also, since there are no sources or sinks for cations in the (-) side, changes in cation concentration in the (-) side can be due only to trans-compartment diffusion, or #q_c = dq_ c /dt = irq+c

(E.14)

NOTE: The thermal diffusion rates are assumed, for the sake of greater simplicity, to be very much smaller than those resulting from current flow. Henceforth, in this analysis, the (TD) terms will be dropped. To further simplify the nomenclature employed here, and to make it more readily usable in computer math programs, we will substitute more abbreviated symbols for the system variables. Taking inventory of the various parameters and equations also enables us to assess the status of our analysis.

246

ELECTROCHEMICAL WATER PROCESSING

Let us substitute the following for the various ionic quantities identified earlier. A

=qH B=q. c C=q+C D=Qc I =QoH

E = q+A L = q-A G= q O H H = QH

J

= QA

There are ten variables, the solution of which requires ten independent equations. Also, we let Q o = Q, and qo = q. The equations now become, Q + 2q = B + C +D

(E.l)

A+C=E

(E.2)

B=G +L

(E.3).

D=Q-H

(E.4)

Q + 2q = E + L + J

(E.5)

J = Q-I

(E.6)

#A = ia - ibA - icG + eCH - fAD = irC + isL + #Qc (E.7) #G = ia -ibA - icG + gLI - hGJ = irC + isL + #QA (E.8) F = bA + rC + cG + sL

(E.9)

#Q c = eCH - fAD

(E.10)

#QA = gLI-hGJ

(E.11)

#E = isL

(E.12)

APPENDIX E: MATHEMATICAL ANALYSIS

#B = irC

247

(E.13)

If we choose to solve for qH as a function of time, equations (E.l) through (E.7) and equation (E.9) provide eight of the ten needed relationships. The two additional equations may be obtained as follows. Differentiating equation (E.3), we obtain, #B = #G + #L Substituting for #B via equation (E.13), for #G via equation (E.8), and for #L via the derivative of equation (E.5), 0 = #E + #L + #J The ensuing operations are shown below. dq_ c /dt = dq O H /dt + dq_ A /dt = dq O H /dt + (-dq +A /dt - dQ A /dt) #B = #G + #L = #G + ( #E - #J) Substituting irq+c = (ia - icqOH -ibqH + QA) + (-isq.A - QA) the above becomes ir

q +c + isq.A = ig - ic q OH - i b q H

or rC + sL = a -cG -bA

(E.14)

The tenth equation, shown at the bottom of page 244, may be obtained by substituting from equations (E.l2) and (E.13) into equation (E.9), and solving for dq H /dt. F = bA + r(E - A) + cG + s(B - G)

248

ELECTROCHEMICAL WATER PROCESSING

differentiating, we obtain 0 = b#A + r(#E - #A) + c#G + s(#B - #G) = #A(b - r) + #Er + #G(c - s) + s#B = #A(b - r) + isrL + #G(c - s) + irsC 0 = (ia - icG - ibA + #Qc)(c - s) + #A(b - r) + isrL + irsC

(E.15)

E.2 Special Case Evaluation - No Resins Present in System A first approximate solution to a special configuration is of interest here. The simple analysis that follows is for a cell that contains no resins on either side of the membrane. It will operate strictly as a pH control or acid/base-generating cell. Some fraction of the salts initially present in the cell water will be converted to acids and alkalis. The following assumptions are made: • Constant electric current density • Only H 2 and 0 2 are evolved at the electrodes. Now equations (E.7) and (E.8) become, dq O H /dt = ia - icqOH - ibqH, and d q H / d t = ia - icqOH - ibqH.

(E.16) (E.17)

Obviously dq O H /dt = dq H /dt. Also, since there are no sinks or sources for H+ and OH ions on the (-) and (+) sides, respectively, of the membrane other than their diffusion across the membrane, and also because diffusion of H + into the (-) side and diffusion of OH" into the positive side cancels out with the the formation of water,

APPENDIX E: MATHEMATICAL ANALYSIS

249

The diffusion coefficients, b and c, do not necessarily have the same value. The resultant equation for #qH, (dq H /dt), is #qH = ia - iqH(b +c)

(E.18)

Transposing terms to solve for this first-degree differential equation dqH

i a - i q H ( b + c)

-=dt

vu \ J t -i(b+c)dq H or, -i(b +c)dt = -^Lia-iqH(b+c) Direct integration yields ln(ia - iqH(b + c)) = -i(b + c)t + const. ia -iqH(b + c) = K^xpHO) + c)t) + K2

(E.19)

To evaluate the constants the following limits are employed. When t = 0; When t = infinity. When t = 0

qH = 0 dqH/dt = 0 d q H / d t = ig

The constants are evaluated as, ia - i(0) = Kjexp (0) - K2, , „ u . .„ v -Kiexp(-it(b + c)) - K2 - ig and #qH = ia - i(b + c) x ^~ 4 i(b + c)

250

ELECTROCHEMICAL WATER PROCESSING

The constants are, K1 = ia, K2 = 0 And the final expression is, qH =

a - a.exp(-it(b + c))

*7d

m

_m

(E 20)

'

NOTE: All of the above assumes that the supply of cations on the (+) side and the supply of anions on the (-) side are inexhaustible for the duration of a regeneration run. Hence, we find that all generation rates for ions are such that the increase in the ions is a constant term minus an exponential. Another important consideration in these relationships is the need to distinguish between quantities of materials transported or in solution and concentration of the same material. In all of the above equations involving rate processes the terms involving rates are quantities per unit time per unit area, because all computations are performed on the basis of normalized electrode and membrane area of one in2 Hence, the foregoing equations must be modified to reflect this issue as follows. Returning to equations (E.16) and (E.17), we have, dq O H /dt = 1/w (ia - icq OH - ibqH) = d q H / d t

(E.21)

The term, w, is the thickness of the cell compartments, i.e. the distance from electrode to membrane, and is assumed the same on both sides of the cell. Since the rate term is in dimensions of GEW/in 2 of electrode, dividing by cell half width only is necessary for the specific volume correction. E.2.1

Non-Constant Electrochemical Generation Rates for H + and OH

If we introduce the fact that the availability of cation and anion species will become increasing limited as they

APPENDIX E: MATHEMATICAL ANALYSIS

251

are depleted in their respective cell compartments, and that coulombic efficiency will diminish accordingly, the following modifications in the equations are made. The constant generation term, ia, on page 241 for the rates RH(G) and ROH(G) is replaced by trans-barrier diffusion terms. Since the production and net increase of a H + ion in the (+) side must be accompanied a loss of a cation or gain of an anion in that side, the generation rate may be more appropriately represented by RH(G) = irq+c + isq.A. Similarly, the net gain of a OH" ion on the (-) side is due to either the loss of an anion or gain of a cation to the (-) side. Hence ROH(G) = irq+c + isq.A, and RH(G) = ROH(G). With these new generation terms equations (E.7) and (E.8) become, d q H / d t = #qH = irC + isL - ibA - icG + eCH - fAD

(E.7a)

dq O H /dt = #qOH = irC + isL - ibA - icG +gLI -hGJ

(E.8a)

As before, we may now proceed to find the needed ninth and tenth equations. The first eight equations are (E.l) through (E.6) and (E.7a) and (E.9). Differentiation of equation (E.3) gives #B = #G + #L

(E.22)

252

ELECTROCHEMICAL WATER PROCESSING

Equation (E.22) says dq c / d t = dq O H /dt _ dq A / d t = dq O H /dt + (dq + A /dt - dQ A /dt) Because #B = irC, and #G = irC + isL - ib A - icG + QA and #L - - #E - #J from the derivative of equation (E.5), we have by substitution into equation (E.22), irC = (irC + isL - ibA - icG +QA) -isL - QA The above reduces to, bA = sG.

(E.14a)

The tenth equation may be extracted from equation (E.9) and its derivative. Substituting equation (E.14a) into (E.9), F = 2bA + rC + sL.

(E.23)

Differentiating equation (E.9) gives, 0 = 2b#A + r#C + s#L.

(E.24)

In order to substitute for #C and #L in the above equation, we must return to equations (E.7a) and (E.8a) as well as (E.2) and (E.3). Differentiating (E.2) and (E.3), we obtain

#L = #B - #G, and

(E.25)

#C = #E - #A.

(E.26)

equations (E.12) and (E.13), #L = irC - #G, and

(E.27)

#C = isL - #A.

(E.28)

APPENDIX E: MATHEMATICAL ANALYSIS

253

Substitution of values for #G and #A from equations (E.7a) and (E.8a), we find #L = ibA + icG - isL - QA, and

(E.29)

#C = ibA + icG - irC - Qc.

(E.30)

Now we can return to the derivative of equation (E.9) again and complete the algebraic manipulations. 0 = 2b#A + r(ibA + icG - irC - #Qc) + s(ibA + icG -isL should be isL

(E.31)

It is further necessary now to substitute for #Q c and #Q A in the above expression. 0 = 2b#A + irbA + ircG -ir2C - r(eCH - fAD) + isbA + iscG - is2L -s(gLI - hGJ).

(E.15a)

Solution of the preceding resultant equations is obviously quite laborious. They are complex expressions with many terms. Ultimate solution of the time dependence relationships for qH, Q H , etc. necessitates the solution of the net differential equation which one extracts from the above and which contains only the constants and is entirely in terms of the one dependent variable and time, t. The above has been performed with the solution of A, or qH, in terms of time in mind. The equations can, of course be manipulated such that the final, resultant expression is in terms of any one of the other dependent variables versus time. Solution of these equations would be greatly facilitated in the sense of reduction of the number of terms if numerical values for the constants can be substituted. Even with that advantage the expressions are cumbersome enough to suggest the use of a symbolic mathematics program to assist with the algebra and substitutions. A program known

254

ELECTROCHEMICAL WATER PROCESSING

as MuMath (Microsoft) has been employed to significant advantage. Continuation of these calculations should be done with computer assistance. E.2.2

D i m e n s i o n s and Units

Diffusion processes are generally described by the Fick's Law relationship that has the form

jp=-kfA-

(E.32)

dt dx The quantity of material, Q, which diffuses or migrates through a medium is directly proportional to the product of the concentration gradient of that material in the medium and the cross sectional area of the path. The dimensions of the terms are; Q C x A k

= quantity of material, grams, moles, etc. = quantity of material per unit volume, gm/in 2 ,... = length, cm, in.,... = area, cm2, in 2 ,... = reciprocal (time)(length), cm-1 sec 1 ,...

In setting up the equations for this electrochemical/ ion-exchange process, the units of the q quantities are "normals/in 2 ", or "gram-equivalent-weights per square inch" of electrode area. Thus the constant terms appearing in the equations would have to contain the specific volume values of the cell components or additional provision must be made to normalize the expressions dimensionally. Reviewing the equations and normalizing for specific volumes, we have: Cell volume/in 2 = 2W, since W is the distance from the membrane to either electrode, (equal spacing in (+) and (-) side. The void volume ratio for the resins is v, and vW gives the void volume available for water.

APPENDIX E: MATHEMATICAL ANALYSIS

255

The previous equations are modified accordingly and listed below. Qo + 2q0 = qc + q-c + Q c

OE-D

Q + 2q = B + C + D same A + C = E

(E.2)

same B = G + L

(E.3)

same D = Q - H

(E.4)

Qo + 2q0 = q+A + q . A + Q c

<E-5)

Q + 2q = E + L + J same J = Q -1

(E.6)

vWdq H /dt = vWia - ibA - icG + (eCH - fAD)W2v

(E.7)

vWdq H /dt = irC + isL - ibA - icG + (eCH - fAD)W2v

(E.7a)

vWdq O H /dt = vWia - ibA - icG +(eCH - fAD)W2v

(E.8)

vWdq O H /dt = irC + isL -ibA - icG + (gLI - hGJ)W2v

(E.8a)

vWF = bA + rC +cG + sL

(E.9)

rC + sL = avW - cG - bA

(E.14)

samebA=sG

(E.14a)

0 = (iavW - icG - ibA + #QcvW)(c - s) + #A(b - r)

(E.15)

0 = 2b#A + irbA + ircG - ir2C - (eCH - fAD)rvW + isbA + iscG -is2L - (gLI - hGJ)svW

(E.15a)

256

E.2.3

ELECTROCHEMICAL WATER PROCESSING

Variable Electric Current Densities

Unless constant current supplies are employed in the electrochemical regeneration mode, a more realistic set of electrical circumstances would involve current density dependence upon dissolved solids level in the water. As regeneration proceeds, the concentration of acid and dissolved salts will increase in the (+) side, and alkali and dissolved solids will increase in the (-) side. With this increase in solute content the electrical conduction rises. If a constant voltage supply were employed then the d.c. current through the cell would increase directly as the specific conductivity of the water rose. The following is another simplification of probably real circumstances, but it more closely approaches actual conditions than the previous assumption of constant current throughout the regeneration process. The salt concentration will increase from some few hundreds parts per million level to perhaps tens of thousands of parts per million toward the end of regeneration. That is quite a large spread in concentration and conductivity. The conductivity, P, of the water may be expressed as P = P(q+C + q H ) / W ' or P = p(q.A + q OH )/W, or P = p(q +A )/W, or P = p(q_c)/W, if the concentrations on either side is at about the same level, and if the spacings on either side are equal. P is in units of ohm 1 in"1, and p is a proportionality factor having the dimensions of ohm 1 in2 GEW. If there is a great disparity in total ionic concentrations between the (+) and (-) sides, then an expression of the form below might be employed as an averaged value of conduction. p = p(q+c + qH + q-A + qoH)/2w.

Œ.33)

APPENDIX E: MATHEMATICAL ANALYSIS

257

The electric current density, i, is then represented at anytime as, i = EP

(E.34)

where E is the applied electric potential in volts d.c.

E.3 Estimation of Resin Constants The constant coefficients, e, f, g, and h, which determine the rates with which the various ions are displaced at the resin sites must be inserted in the balance equations. These coefficients may be approximated for the present from general information regarding resin bed regeneration via standard acid and alkali washes. There is no question that the time constants or reaction rates are dependent upon the size of resin beads and the state of regeneration of the resin as well as the flow pattern or "film depth" of the water medium surrounding the beads. A concentration of 1 normal HC1 or H 2 S0 4 will regenerate a resin bed within 1 hour, and the solution undergoes a depression in concentration of about 50%. Thus we may represent, as an averaged expression, the decay in ionic concentration as the following. The diffusion relationship has the form; vWdQ c /dt = (eCH - fAD)vW2 = K[CH - AD]vW2

(E.35)

Integrating, and letting e = f = K, indicating that the resin has no greater or lesser preference for H + than for other cations, we obtain; «Q= . . * _ dQc vW(eq+cQH - fqHQc) K[q+cQH - qHQc]vW

258

ELECTROCHEMICAL WATER PROCESSING

Now, during the period of regeneration we will set the following conditions as a trial; Qc goes from Q o to 0.05Qo qH = constant qc = constant

qVqH = lN QH = Q O - Q C

But, we must first solve the above equation to evaluate K. The above equation becomes, after substituting for Q H , vWdQ c /dt = K[qc(Qo - Qc) - qHQc]vW = -K[Qc(qH + q c )-q c Q o ]vW

(E.37)

Transposing terms and factoring for integration,

= "K d t ( ^ + qH)vW

(E.38)

ln[(qc + qH)Qc - qcQo] = -Kt(qc + qH)vW + Kl

(E.39)

[(qc + qH)Qc - qcQo] = Kl exp(-Kt(qc +qH)vW)

(E.40)

(qC + dQc u[(qc + q HVn ) Q c - ni qcQ0]

Integrating, we get;

Or,

To solve for the value of the coefficient, Kl, the constant of integration we set the limits, when t - 0, Qc = Q o , which describe the initial conditions of the resin at the start of the regeneration process. (This initial condition should reflect more accurately the level or extent of regeneration of the resin beds due to the electrochemical process).

APPENDIX E: MATHEMATICAL ANALYSIS

259

Substitution of the above conditions yield Kl = q H Q 0 .

(E.41)

Inserting that value of Kl into the main equation gives Kqc + qH)Qc - qcQ0] = qHQD ex P (-Kt(q c + qH)vW)

(E.42)

Based upon general data regarding performance and chemical regeneration of resins in present systems, it seems that these are reasonable conditions to impose for the evaluation of K. Apparently a cation resin bed can be regenerated within one hour by passing a 1 normal solution of HCl over the bed. The concentration of the emergent acid is in the range of 0.7 to 0.6 N after removal of the cations, and the remainder is 0.3 to 0.4 N cation solution. The regenerated resin is evidently 90%, or greater, regenerated in the process. Thus it is reasonable to set these as additional boundary conditions in assessing the material constant, K. When t = 1 hour, Qc = 0.05Qo, from an initial value of Q o . Substitution again gives 0.05qH - 0.95qc = qH exp(-K(qc +qH)vW)

(E.43)

Qo cancels out of both sides of the equation. Now for the various constants of the materials and system! If an ion bed has the capability of removing 17,000 grains per cubic foot, then this corresponds to 1130 grams per cu. ft., or 0.65 grams/in 3 . At an average of 50 grams per gram equivalent weight of most inorganic dissolved substances, the capacity of the resin then is 1.31xl0-2 GEW per in3 of bulk resin volume. It is interesting to note here that the values of K exist over a limited range for the conditions imposed in the above. If e = f = g = h = K, we see that K has values not greatly dependent upon qc, qH, or percentage of regeneration. Examination

260

ELECTROCHEMICAL WATER PROCESSING

of the table below shows the variation in K over a range of values of concentrations of emergent acid and cations. NOTE: The q values are given in terms of GEW per in3. 0.05qH - 0.95qc = qH exp(-K(qH + qc)vW)

qH 0.016 GEW/in 3 0.0155 0.0154 0.0153

KvW

—

0 0.00048 0.00064 0.00072

187 in 3 /Hr-GEW 242 285 328

0.10qH -0.90qc = qH exp(-K(qH + qc)vW) 0.0152 0.0147 0.0146

0.0008 0.0013 0.00144

0.0024

0.0131

0.00288

0.004 0.0046

(E.46)

177 232

0.30qH - 0.70qc = qH exp(-K(qH + qc)vW) 0.012 0.0114

(E.45)

184 239 281

0.20qH - 0.80qc = qH exp(-K(qH + q > W ) 0.0136

(E.44)

(E.47)

169 266

It can be seen that the value of K ranges from about 160 as a minimum to 330 as a maximum over the allowable spread of qH and qc concentrations. If the acid concentration is too low the resin cannot be regenerated to the preset percentage of full capacity.

APPENDIX E: MATHEMATICAL ANALYSIS

261

E.4 Electrolytic Resistance of the System Water When evaluating the expressions for quantitative relationships it becomes necessary to specify the constants for the transport terms due to electric current flow. The RA(ED), RC(ED), RH(ED) and ROH(ED) terms are electric current dependent. Equations E.7(a), E.8(a), E.14(a) and E.15(a) on pages 197 and 198 all contain R(ED), or current terms. The constants b, c, r and s are equivalent resistance (reciprocal equivalent conductance) terms. Appendix B gives a table of some specific ion values of conductance. These resistance values are inversely proportional to the ion mobilities for that specific ion. If the equivalent conductance of a substance is M, and its solution concentration is S, in GEW per liter, then: Conductance = SM/1000, or Resistivity, R = 1000/SM ohm-cm. Equivalent Conductance, M, is given as mho/cm-equiv. Converting to the English system, R = 1000/2.54SM = 400/SM ohm-in Now, the electric current is V/R, where V is the voltage drop along the path of flow of the specific, subject ion. The units would be amperes. To convert to equivalent weights transported per unit time, we need to change dimensions. Since 26 amp-hr corresponds to about 1 gram equivalent weight of substance, The above equation becomes d q / d t = V/26R GEW per hour.

(E.48)

If we are interested in the quantity, d q / d t , transported per hour per square inch of electrode area in a particular cell, the inter-electrode or electrode-membrane spacing, W, must be taken into account, and d q / d t becomes V/26RW.

(E.49)

262

ELECTROCHEMICAL WATER PROCESSING

Since the q terms in the system expressions are in units of GEW per in3, it is necessary to normalize them in terms of GEW per liter. Hence, the expression d q / d t = V/26RW = V/26(400/SM)W = VSM/10400W,

(E.50)

becomes d q / d t = 60VqM/10400W = VqM/173W.

(E.51)

Typical values of M at low concentrations are: MH+ 340 mho/cm-equiv. MOH.190 M Na+ 44 M r 60 Ca M

C1-

6 5

MS04 75 Now, we may insert numerical values for the constants, b, c, r, and s as follows. b = M H+ /173W = 2/W + mho/equiv.

(E.52)

c = M OH ./173W = 1.1/W

(E.53)

r = M c+ /173W = 0.29/W +

(E.54)

s = MA_/173W = 0.40/W

(E.55)

A potential problem exists in determining the value of voltage to use in the rate equations since the division of the cell potential from the membrane to the electrodes will depend upon the ratio of the resistances. For example, if the resistance on the positive side is R+ and that of the negative side is R, then the potential drop, V , (driving voltage) on the (-) side of the cell is

APPENDIX E: MATHEMATICAL ANALYSIS

V = VR./(R + R+),

263

(E.56)

and similarly for the positive side voltage, V+ = VR + /(R + R+).

(E.57)

The R's are obviously obtained by taking the reciprocal of the sum of all the conductances due to the various ionic species in either side of the cell. For example, R=(l/ROH. + l / R A + l / R / = ([SM]OH_/400W + [SMIA/400W_ + [SM]_C/400W)"1 (E.58) and similarly for the positive side, R+ = (l/R H +

+

l/R+A+l/R+c)-i

= ([SM]H+/400W+ + [SM]+A/400W+ + [SM]+C/400W+)-1 (E.59) A reasonable first approximation which may be employed in the computations associated with the rate processes is to let the total resistance on either side of the membrane be equal then the driving potential for the specific ions will be 1/2V. In most designs the spacing in the (-) and (+) cell compartments will be about equal also, or W = W+ = W.

E.5 Solution of the Simultaneous System Equations The series of equations that were described earlier now assume the forms given below. These are the modified forms of equations (E.l), (E.2), (E.3), (E.4), (E.5), (E.6), (E.7a), (E.8a), E.9, (E.14a) and (E.15a) in light of the further definitions provided the coefficients. The q and Q terms have been substituted in the following equations as shown on page 9 for the sake of brevity

264

ELECTROCHEMICAL WATER PROCESSING

of use of symbols, and in order to make the nomenclature more adaptable to MuMath for symbolic manipulations, FORMULA ONE and LOTUS for numerical evaluations. Q + 2q = B + C + D

(E.60)

A+C =E

(E.61)

B=G +L

(E.62)

D = Q-H

(E.63)

Q + 2q = E + L + J

(E.64)

J = Q-I

(E.65)

v W d A / d t =: V+rC + V sL - V+bA - V cG + (eCH - fAD)W2v

(E.66)

v W d G / d t == VrC + VsL - V bA +

- V cG + (gLI - hGJ)W2v EvWF/Rj. := V+bA + V+rC + V cG + V sL VbA=VsG

(E.67) (E.68) (E.69)

0 = 2V + bdA/dt + V+r(V+bA + VcG - V+rC - (eCH - fAD)W2v) + V s(V+bA + V cG - VsL - (gLI - hGJ)W2v)

(E.70)

NOTE: The quantity, Rp, appearing in equation (E.68) is the total electrical resistance of the cell, or Rj. = R + R+ Since H + ions do not exist in the negative cell side, and because OH" ions do not survive in the positive side, current is carried by these ions only in their respective sides of origination. Hence the driving potential for these ions is that of the half-cell side, or V+ for the H + ions, and V for the OH" ions.

APPENDIX E: MATHEMATICAL ANALYSIS

265

The conduction process is due to all ion migration on both sides of the membrane. Hence, the total resistance is found as follows. The conductance on the positive side is the sum of all charge carriers, or l/R + = b A + r C + sE

(E.71)

l / R = c G + rB + sL

(E.72)

Since Rj. = R+ + R, the final expression for the total resistance becomes; RT = 1

bA + rC + sE

+

cG + rB + sL

(E.73)

The above value for the total resistance must then be substituted into equation (E.68) for Rj, Equation (E.70) has the form above only if Rj, is treated as constant in equation (E.68). Since it is in fact not constant, equation (E.70) must be modified accordingly to reflect this fact. Differentiating equation (E.68) as before is now a very complex operation. There are some approximations that are allowed while still preserving the basic relationships of interest. For example, the resistance on both sides of the membrane will probably be about equal throughout the operation of the system. Thus, one may let the total resistance be equal to twice that of either side, or R =2R+ = T

+

bA + rC + sE

(E.74)

Substitution of (E.74) into (E.68) gives, VvWF[bA +rC +sE]/2 = VbA/2 +VrC +VcG/2 +VsL (E.75)

266

ELECTROCHEMICAL WATER PROCESSING

Eliminating both C and L from equation (E.75) by substituting from (E.61) and (E.62) gives VvWF(bA +r(E-A)+ sE)/2 = VbA/2 + Vr(E - A)/2 +VcG/2 + Vs(B - G)/2

(E.76)

Differentiating equation (E.76) with respect to time yields vWF(bdA/dt + r(dE/dt - d A / d t ) + sdE/dt) = b d A / d t + r(dE/dt -dA/dt) + cdG/dt + s(dB/dt - dG/dt)

(E.77)

Now we can substitute again for the derivatives d E / d t and d G / d t and dB/dt from equations (E.12), (E.8a) and (E.13), respectively. vWF(bdA/dt + r(VsL/2 - dA/dt) +Vs2L/2) = bdA/dt + r(VsL/2 - dA/dt) + cdG/dt + s(VrC/2 - VrC/2 - VsL/2 + VbA/2 + VcG/2 -dQ A /dt) (E.78) dA/dt(bvWF - rvWF - b + r) + vVWFrsL/2 + vVWFs 2 L/2 = VrsL/2 + sVrC/2 - rsVC/2 - Vs 2 L/2 + VsbA/2 + VscG/2 - sdQ A /dt + VrcC/2 + VscL/2 - VbcA/2 + Vc 2 G/2 - cdQ A /dt Equation (E.78) becomes the new version of equation (E.70), and equation (E.68) has the form of equation (E.75). Thus, the new series of relationships are equations (E.60), (E.61), (E.62), (E.63), (E.64), (E.65), (E.66), (E.67), (E.75), (E.69) and (E.78). We may now proceed with the solution of these equations in generalized form and then in terms of specific values of the system constants.

APPENDIX E: MATHEMATICAL ANALYSIS

267

Equations (E.60) through (E.65) and equation (E.75) are solved as a series of linear, simultaneous algebraic relations. Their solutions for the variables B, C, D, E, G, H and I in terms of the variables A, J and L are listed below. B = (AvWsF + AvWbF - ab + 2vqWrF + 2vqWsF + vHWrF + vHWsF -2qr - Ls + Lc - H r ) / (-r + c + vWrF + vWsF)

(E.79)

C = (-AvWsF - AvWbF + Ab + 2qc + Ls - Lc +Hc)/ (-r + c + vWrF + vWsF) D = Q-H

(E.80) (E.81)

E = (AvWrF - AvWbF - Ar + Ab + Ac + 2qc + Ls - Lc + He)/ (-r + c + vWrF + vWsF)

(E.82)

I = (AvWrF - AvWbF - Ar + Ab + Ac - 2vqWrF - 2vqWsF + vLWrF + vLWsF + 2qr - Lr + Ls + He)/ (-r + c + vWrF + vWsF)

(E.83)

J = (-AvWrf + AvWbF + Ar - Ab - Ac + vQWrF + 2vqWrF + 2vqWsF - vLWrF - vLWsF - Qr + Qc - 2qr + Lr - Ls -He)/ (-r + c + vWrF + vWsF)

(E.84)

G = (AvWscA2F + AvWbc A 2F -Abc A 2 - 2vqWrc A 2F + 2vqsWc A 2F - vLWrcA2F - vLWscA2F + vHWrc A 2F + vHWsc A 2F - 2qrc A 2 + Lrc A 2 - LscA2 - Hrc A 2)/ (vWrcA2F + vWsc A 2F - rc A 2 + cA3)

(E.85)

268

ELECTROCHEMICAL WATER PROCESSING

These are employed in conjunction with equations (E.66), (E.75) and (E.78) to totally evaluate A, (as it is set up at present), as a function of time. To accomplish this end the first order differential equation, d A / d t , must be solved. The relationships rapidly become extremely complex and lengthy. Hence, it serves simplicity to insert numerical values wherever possible and as early as practical to shorten the lengths of these expressions. E.6

S a m p l e S o l u t i o n of O p e r a t i n g S y s t e m

It seems that the initially most interesting relationships from which to obtain quantitative information and to graph are the following: • Time dependence of qH and Q H for different values of V, W, and perhaps v. • Maximum level of regeneration achievable in the resin for a particular set of conditions • Coulombic efficiency versus time, V, W, and v. • Different resin properties will affect the performance as well Let us now insert the first series of numerical values for the system constants as a test for the type of analytic information afforded here. The values for e,f,g and h are set to K = 200/vW in 3 / hr-GEW The values of the remainder are: b = 2 / W mho/equiv c = l.l/W r = 0.29/W s = 0.40/W V = about 10 volts, but is one of the variables

APPENDIX E: MATHEMATICAL ANALYSIS

269

W = W+ = W_ = about 1 inch, but is one of the variables Q o = Q = 1.3xl0 2 GEW/in 3 for 15,000 grain/ft 3 resins qo = q = 1.3x10-* GEW/in 3 for 400 ppm input water If we substitute the above values for the constants, the equations are: B = 2.7A + 2.137xl0 4 + 0.113L +0.822H C = -2.7A + 0.436xl0"4 - 0.113L + 0.177H D = 1.3xl0- 2 -H E = -1.7A + 0.436xl0 4 - 0.113L + 0.177H I = -1.7A - 2.137xl0 4 + 0.886L + 0.177H J = 1.7A + 1.3xl0-2 + 2.137xl0"4 - 0.886L - 0.177H G = 2.7A + 2.137x1o4 - 0.886L + 0.822H and 0.3dA/dt = - 20A + 2.9C - 11G + 4L -200AD + 200CH 2A = 0.4G 10.764L + 11.628dA/dt = -7A + 1.595C + 8.25G + 1.98L +300GJ -300LI If we leave in the parameters W and V as indeterminate, the expressions are; B = (-0.00007 - 2A + 0.7L - 0.29H + 18.72AW + 0.00139W + 5.382HW)/(0.81 +5.382W) C = (0.0002 + 2A- 0.7L + 1.1H - 18.72AW)/(0.81 + 5.382W)

270

ELECTROCHEMICAL WATER PROCESSING

D-0.013-H E = (0.0002 + 2A- 0.7L +1.1H - 13.338AW)/(o.81 + 5.382W) I = (0.00007 + 2.81 A + O.llL + I.IH - 0.00139W - 13.338AW + 5.382LW)/(0.81 + 5.382W) J = (0.01045 - 2.81 A - O.llL - I.IH + 0.07136W + 13.338AW - 5.382LW)/(0.81 + 5.382W) G = (-0.00009 - 2.42A - 0.1331L - 0.3509H + 0.00169W + 22.6512AW - 6.51222LW + 6.51222HW)/ (0.9801 + 6.51222W) 0.3dA/dt = -200ADW -AV + 200CHW + 0.29CV -1.1GV +0.4LV 2A = 0.4G -1.71dA/dt + 1.0764LWV + 13.338WdA/dt = -0.7AV + 0.1595CV + 300GJ +0.825GV -300LI + 0.198LV The above equations can now be solved for A as f(t) for different voltage ranges and cell widths.

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

Appendix H: Mathematics for Simple Electrochemical Biociding In Chapter 5, Section 5.1 we outlined the salient factors in the design of an electrochemical bromine generator. The assembly of electrodes is designed to accomplish a particular bromination task, and hence require electrodes of sufficient size along with a dc power supply for the necessary current flow. The first of the following two sets of equations quantitatively relates the geometry of an array of electrodes, and voltages necessary to production of a given amount of bromine per unit time. This set of equations is named Figure H.l. The parameters in the above equations are identified and defined as follows. Bt Ec

Bromine generation rate, g m / h r Cell potential, volts 303

304

ELECTROCHEMICAL WATER PROCESSING Bt= i■ tj ■ Ae-3.08 Ec= Er+ i ■ Ae- Rc

p=1000/(/V-JJ

ppm = 103- 10 3 -/V R^s-pKN-AJ COP=Bt/P ppmBr= Bt- 106/(454 ■ V ■ 8) l=i-Ae P=l-Ec

Figure H.l Bromine generator array.

E r A

e

Î1

ppmBr

ppm N R c p n P s COP V

J

Electrode reaction potential, volts Working electrode area, sq in Coulombic efficiency Bromine concentration rate, p p m / m i n ppm of NaBr salt NaBr solution normality Cell resistance, ohms Power Input (electrical), watts Number of cells (electrodes) in parallel Specific resistivity, ohm-in Inter-electrode spacing, in. Coefficient of performance, gm of Br/ hr-watt Gallons of water being brominated, Equivalent conductance, 130 cm 2 / equiv-ohm

The second set of equations, Figure H.2, concerns the geometrical arrangement for housing an array of electrodes of a brominator. The simplest, least costly and most effective manner of housing an array to withstand high hydraulic pressures normally encountered in most application settings, is a cylinder. As was described in Section 5.1, the relationship is between the dimensions of electrodes and cylinder diameter to give the largest utilization of available space within the cylinder to the greatest total electrode area.

MATHEMATICS FOR SIMPLE ELECTROCHEMICAL BIOCIDING

305

The short set of such relationships is £>= n2 ■ u ■ sAe=L-W-n-L-W

s

4 f i 2 = W2+b2 2b2- u ■ b -4R2=0

Figure H.2 Cylindrical configuration.

The above relates the maximum area, A , and the inside radius of the cylindrical housing. The parameters are identified as follows: b n Ae L W R u s

Stack thickness, in. Number of plates (electrodes) Operating electrode area, sq in Plate length, in Plate width, in Cylinder radius, in. Plate thickness Inter-electrode spacing, in.

To illustrate the usefulness of establishing these equations into a suitable mathematics program, the following is offered as a typical computation for a convenient size of "reactor module", or array of cells needed to supply sufficient bromine for a body of water with a volume of 40,000 gallons - a large swimming pool. Strictly as a matter of illustration, let us specify that about 1 ppm of bromine must be provided to this body of water when electrolyzing a water solution of about 1000 ppm of sodium bromide. That concentration of NaBr is a 0.0097 normal solution. If the above equations are entered into either a MathCad type or spreadsheet program and solved for various lengths of electrodes and diameter of cylindrical housings, we can manipulate the variables to arrive at a maximally acceptable configuration of electrode array.

306

ELECTROCHEMICAL WATER PROCESSING

Let us see what the physical dimensions and electrical power requirements estimates are for the following set of operating conditions. These electrodes are assumed to be all connected electrically in parallel making a single cell with a multiplicity of electrodes. uirements: Single cell r, = l ppm = 1000 ppm ppm-Br = 1 p p m / h r

E =3 s i = 0.10 amp/in 2 = 0.016 amp/cm 2 s = 0.25 in = 0.625 cm V = 40,000 gal.

From equations in Figure H.l, the values of the dependent variables are found to be, Bt = 145 gm/hr, A = 2950 cm2 = 461 in2 E = 11.2 volts, I = 47 amps, and P = 530 watts regardless of the number of cells or specific geometry of the unit as long as the electrode spacing, s, is kept the same. Now, in order to decide upon the actual number of cells and size and shape of the unit, we make use of the set of equations in Figure H.2. The only actual choice we have in placing an electrode assembly into a cylinder is the inside radius of the housing. The equations are set up to provide the values of L and W, and n for the maximum area in the shortest stack length, L. We will select two values of R to complete the illustrative computations, and to provide the reader with an idea of the size and power required to produce electrolytically 1 p p m / h r of bromine in 40,000 gallons of water at a coulombic efficiency of 100%.

MATHEMATICS FOR SIMPLE ELECTROCHEMICAL BIOCIDING

For R = 4 in: n = 17 plates

L = 5.1 in

W = 5.6 in For R = 2 in: n = 9 plates W = 2.8 in.

L = 21 in

307

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

Appendix F: Industrial Chlorination and Bromination Equipment Cost Estimates The processes and cost information described here are based upon current practices and available information. Equipment cost figures was estimated from engineering experience and cost figures was estimated from 1992 dollars using the Engineering News Record Construction and Cost Indexes (CCI). This feasibility study has produced an order-of-magnitude investment and operating cost estimate for a proposed bromination circuit. Both the chlorination and bromination water treatment processes have been sized to treat a three million gallon industrial water supply in an open, re-circulating system. It has been assumed that a recirculation rate of 100,000 gallons per minute (gpm) occurs in an induced-draft cooling tower.

271

272

ELECTROCHEMICAL WATER PROCESSING

Table E l summarizes the design and operating conditions chosen for this analysis and systems comparison. No forced blow-down in the bromination process design was assumed because of the water softening capabilities of the unit. Though no specific calculations or field tests have been conducted on this assumption, the experience with operating these types of devices supports this contention. Zero forced blow-down also minimizes bromide salt make-up requirements. Chemical storage facilities for chlorine gas and bromide salt have been provided to accommodate one tanker or truckload delivery of the gas or solid, respectively. No building storage or shelter, other than that indicated on the equipment list, was considered. Chlorine demand Table El Water treatment design assumptions and operating conditions. Chlorination

Bromination

Recirculation Rate - gpm

100,000

100,000

Turnover Rate - min

30

30

Tower Load - F

20

20

Evaporation - gpm

2,000

2,000

Windage Loss - gpm

100

100

Forced Blowdown - gpm

400

-

Total Blowdown - gpm

500

100

Total Makeup - gpm

2,500

2,100

Cycles of Concentration

5

21

Chemical Disinfectant

chlorine (g)

bromine (s)

Chemical Demand - ppm

3

2

Chemical Demand - lb / d a y

6,000

1,200*

*Expressed as sodium bromide.

EQUIPMENT COST ESTIMATES

273

was calculated based on average ranges reported and the industry's published "rules-of-thumb" for processes with no product contamination. 3 Bromine demand was assumed based on experience at TRL, Inc. The brominator module and power supply were sized assuming a bromine loss rate of 1 ppm per hour, requiring the generation of 25 lbs per hour of bromine. The efficiency (coulombic) of bromine generation was assumed at 50%, and the current density applied was 0.10 amps per square inch of cell area. All cells were connected in series at 5 volts per cell. The process demand is approximately 75 to 80% of the equipment design capacity. The following is a major equipment list for a chlorination circuit, which will supply 6,000 lbs/day of chlorine to an open, re-circulating cooling water system with a recirculation rate of 100,000 gallons per minute. Table F.2 is an Table F.2 Chlorination system equipment list. Equipment

Specification

Materials of Construction

Cost-1992

Cl2 Gas Storage Tank & Shield

4,800 gallons

MS

$25,000 E

Weigh Platform

30 ton

mf g std

23,900

Storage Tank Eductor

2 in. air oper.

mf g std

1,000

Air • • •

50 SCFM at 125 psi 50 hp

mfg std

32,200

Flexible Tank Connections 2 required

10 ft. L

Monel

2,000

Padding System air compressor air dryer aftercoolers receiver • air filter • moisture indicator

274

ELECTROCHEMICAL WATER PROCESSING

Table F.2 (cont.) Chlorination system equipment list. Equipment

Specification

Materials of Construction

Cost -1992

Expansion Tanks

20% header volume

MS

600

Injector

25 in. Hg vacuum

titanium

Chlorinator • inlet reducing valve • retameter • oriface meter • regulating valve • relief valve

8,000 lb/day

mfg std

Injector Pump

150 gpm 150 ft. 15 hp

Iron

2,100

Evaporator • temperature alarms • level & pressure switch

8,000 lb/day 12 KW

mfg std

57,900 E.

Residual Controller/ Analyzer / Monitor

-

mfg std

5,000

Safety Equipment • Cl 2 Leak Detector • Breathing Apparatus • Container Kits • Floor Level Fans (2)

10 hp @ 2500 acfm

mfg std frp

7,000

Total Erected E q u i p m e n t

$82, 900

Total Non-erected E q u i p m e n t

$86,600

EQUIPMENT COST ESTIMATES

275

equipment, materials listing along with their estimated costs. Below is a glossary of abbreviations found in the equipment list. dia frp hP lb gpm rpm sqft MS RL mfg std E El

diameter fiber-reinforced polyester horsepower pound, avoirdupois gallons per minute revolutions per minute square feet mild steel rubber lined manufacturers standard field erected

B r o m i n a t i o n E q u i p m e n t List

The following table is a major equipment list for a bromination circuit which will supply 1,200 lb/day of sodium bromide salt to the cooling tower system, and maintain a bromine concentration of 2 ppm to effect slime, algae and bacteriological control. The system has been specified Table F.3 Bromination system. Equipment

Specification

Materials of Construction

Cost -1992

NaBr Feed Bin w / load cell

45,000 lb

MS,RL

$24,600

NaBr Vibrating Feeder

8,000 lb/hr, 1/4 hp

MS, RL

2,900

NaBr Discharge

-

MS,RL

800

NaBr Solution Tank

4,000 gallon, 9 ft dia, 10 ft deep

MS,RL

17,000

276

ELECTROCHEMICAL WATER PROCESSING

Table F.3 (cont.) Bromination system. Equipment

Specification

Materials of Construction

Cost -1992

NaBr Mixing Agitator

32 in dia impeller 5 hp,125 rpm

MS,RL

5,500

Feed Water Pump

100gpm-30 ft 4hp

RL

7,400

Solution Feed Pump

variable gpm50ft,l/2hp

mfg std

2,800

Pneumatic Conveyor System

5 ton/hr capacity

mfg std

14,700

Bromination Channel Tank

3 ft x 2 1/2 ft xl9ft

frp

6,100

Bromination Modules • 12 modules/10 cells per • 1200 sq ft plates

5 sq ft per plate

carbon plates copper bus

7,500

Residual Analyzer/ Recorder /Monitor pH Analyzer Power Supply

5,000 " 150 amps 240 volts 36 KW

mfg std

Total Erected E q u i p m e n t

$24,600

Total Non-erected E q u i p m e n t

$79,700

10,000

EQUIPMENT COST ESTIMATES

277

for an open, re-circulating cooling water system with a recirculation rate of 100,000 gpm. F.2

Capital Cost A n a l y s i s

Capital costs were developed from the major equipment lists and preliminary capital cost estimating techniques and factors. Installation labor costs were estimated based upon purchased equipment costs. Installed instrumentation and control costs were based upon equipment and process type; 30% of the purchased equipment costs for the chlorination system and 20% of the purchased equipment costs for the bromination system. This cost difference is required when the safety precautions necessary in the chlorine supply, handling and metering system are examined. Because the chlorine is handled as a liquid and gas in duplicate header systems in the storage area, and as a gas in the metering system, the piping costs associated with the chlorination process are greater; 40% of purchased equipment costs, versus 30% for the bromination circuit. Electrical costs are estimated at 10% of equipment costs for the bromination circuit, and 5% of equipment costs for the chlorination costs. Because of the complexity of the power supply and circuitry associated with brominator modules, it is estimated that the electrical installation costs for the bromination circuit would be a significant cost. Services and platform construction is necessary for both processes, and include lighting, site preparation and access platforms to the storage facilities. Both costs are approximately 30% of the purchased equipment costs. All of the above costs, plus the equipment costs, make up the total construction costs'4-5'. As a check against estimating techniques developed in this treatment, the construction costs figures published by the US EPA in January 1978 for chlorine storage and feed systems were referenced. Escalated to 1992 dollars, the construction cost for an 8,000 lb / d a y capacity chlorine system

278

ELECTROCHEMICAL WATER PROCESSING

was approximately $332,0006. The total construction costs estimated for this exercise is approximately $363,000. Because the proposed bromination system is a new treatment method for industrial water systems, there are no construction data available for comparison. Indirect costs, such as engineering and supervision, construction expenses and profit, administration and legal fees, and contingency were taken as percentages of the total construction costs usually associated with water treatment facilities6. In the bromination circuit, an initial charge of bromide salt was included in the indirect costs as a one-time fee. The interest charges were calculated based on estimated construction times and an 8% interest rate. For the chlorination system, the fixed-capital investment is estimated at $607,500. The bromination system is estimated to cost $375,000, approximately 60% of the chlorine circuit. The following tables (F.4 and F.5) list the individual cost components and charges for each system.

Table F.4 Chlorination system fixed capital summary. Total Non-erected Equipment

$86,000

Total Erected Equipment

82,900

Installation Labor

13,100

Instrumentation & Controls (installed)

50,700

Freight & Site Handling

3,400

Piping (installed)

67,600

Electrical

8,400

Service, Platforms

50,700

Total Construction Costs (TIC)

362,800

Contingency @15% TCC

54,400

EQUIPMENT COST ESTIMATES

Table F.4 (cont.) Chiorination system fixed capital summary. Contractor Overhead & Profit @ 20% TCC

43,500

Engineering @ 13% TCC

47,200

Legal, Fiscal & Administrative @ 20% TCC

72,600

Subtotal

580,500

Interest During Construction @ 8%

27,000

Fixed Capital Investment

607,500

Table F.5 Bromination system fixed-capital summary. Total Non-Erected Equipment

$79,700

Total Erected equipment

24,600

Installation Labor

13,000

Instrumentation & Controls (installed)

20,900

Freight & Site Handling

2,000

Piping (installed)

33,000

Electrical

10,600

Service, Platforms

33,000

Total Construction Costs (TCC)

216,800

Contingency @ 15% TCC

32,500

Contractor Overhead & Profit @ 12% TCC

26,000

Engineering @ 13% TCC

28,200

Legal, Fiscal & Administrative @ 20% TCC

43,400

Sodium Bromide Initial Charge, delivered 27,000 lb

16,200

Subtotal

363,100

Interest During Construction @ 8%

12,000

Fixed-Capital Investment

375,100

279

280

F.3

ELECTROCHEMICAL WATER PROCESSING

O p e r a t i n g Cost A n a l y s i s

Operating costs were calculated based on factored estimates delivered from the fixed capital investment. Percentages typical of water treatment facilities were utilized for plant life and depreciation, maintenance and insurance. Raw material costs were restricted to bromide and chloride salts for each respective process. Corrosion inhibitors, chemical cleaners and flocculants were not included. Connect hp and make-up water requirements from evaporation, windage and forced blow-down estimates were used to calculate utility charges. Administrative charges include office and laboratory labor, and operation charges include labor and engineering. The following operating cost summary in Table F.6 indicates no appreciable operating cost difference between chlorination or bromination under the design basis and assumptions outlined earlier. Because of safety considerations and maintenance and insurance increases possibly Table F.6 System o Derating cost comparison. Bromination Usage/yr Depreciation @ 10% TFI

$/yr

-

60,700

1050 tons

210,000

Chlorination Usage/yr

$/yr

-

37,500

210 tons

252,000

Maintenance @ 4% TFI Insurance @ 2% TFI Chemicals (delivered) • Chlorine® $0.10/lb • Sodium Bromide @ $0.60/lb

EQUIPMENT COST ESTIMATES

281

Table F.6 (cont.) System operating cost comparison. Bromination Usage/yr

$/yr

Utilities • Cooling 1,260 MM Gal 88,200 water @ $0.07/ 1,000 gal • Electricity® 0.30 MM 15,000 KWH $0.05/KWH Labor • Administrative • Operation Total Operating Cost

-

37,800 109,200

Chlorination Usage/yr

$/yr

1,058 MM Gal 74,100

0.36 MM KWH

-

$557,300

18,200

37,800 89,600 $531,700

associated with the storage and handling of chlorine, a possible operating cost increase in the estimate of $35,000 to $40,000 would not be unreasonable. However, this would not be a significant difference in the operating cost comparison for this order of magnitude estimate.

F.4 Conclusions and Comments The preliminary capital investment and operating cost estimates have indicated that the bromination system for disinfection and control of micro-organisms, algae, and slime growths in re-circulating water systems is a cost effective alternative to chlorination. A capital savings of approximately 40% is realized with the installation of the bromination process versus the chlorination equivalent. The assumptions for the design of the brominator and power supply were chosen to provide a severe duty estimate for equipment selection. Because of the water-softening action, iron, manganese and other dissolved metal removal

282

ELECTROCHEMICAL WATER PROCESSING

achieved by the brominator unit, it is assumed no forced blow-down will be required in this system. This assumption significantly decreases make-up bromide salt requirements, but needs to be verified in an actual operating system. When demineralized water is used for tower make-up, the tower cycles are limited only by windage losses, and the assumption of zero forced blow-down would be completely valid for both operations. There are many significant advantages to bromination over chlorination in water treatment applications. In the past, use of bromine biocides were limited to uses where chlorine disinfection was unacceptable because of higher chemical costs of bromine, and the oxidizing agents needed to produce the bromine residuals. With an electrolytic bromination system, based upon electrochemical "activation" of the bromide salts, the higher operating costs of bromination have essentially been eliminated. Because bromine is supplied in relatively inert salt form as sodium bromide, the storage and handling hazards associated with elemental halogens do not exist in this system. Operator safety training, control and safety equipment, system maintenance, leak detection apparatus are not required in the storage of bromide salts. The use of this system should also favorably impact insurance costs because of reduced safety hazards to operating personnel. Producers of bromine chemicals, such as l-bromo-3chloro-5, 5 dimethylhydantoin (Aquabrome), are actively pursuing industrial cooling water applications of bromine disinfection. References in the following describe the results of field tests and operations in several water-cooling water facilities. Another bromine biocide, 2.2-dibromo-3-nitrilo proprionamide (DBNPA), found to be an effective disinfectant, with no environmental toxicity because of it accelerated decomposition upon the application of heat or the increase in pH (78) . The data taken with these compounds have shown the effectiveness and advantages of bromine disinfection in open, re-circulating water systems.

EQUIPMENT C O S T ESTIMATES

283

Table F.7 Industrial re-circulating water rates. Industry

Recirculation Rate

200 MW Power Station

120,000 gpm

900 MW Power Station

405,000

200,000 MTPY Lead Production Hydrometallurgical Process Ammonia Production Facility (Capacity Unknown) Aquitance Refinery (Capacity Unknown)

5,300 36,000 292,000

Due to environmental regulations concerning toxicity and thermal pollution, cooling systems are being built by industry in an effort to meet government standards. In the power industry, two thirds of the fuel supplied must be dissipated as waste heat, requiring large cooling systems, and many of which have recirculation rates over 200,000 gpm. Cooling water requirements and quantities vary greatly, depending upon the industry. Table F.7 outlines several industrial systems cooling tower recirculation rates and production capacities.

References 1. White, George C , Handbook of Chlorination, page 712, Van Nostrand Reinhold Co., New York, 1972 2. Colturi, T. F, and Kozelski, K.J., "Corrosion and Biofouling Control in a Cooling Tower System", Material Performance, Vol. 23, No. 8, pages 43-47, August 1984. 3. White, pages 557-558. 4. White, pages 711-712. 5. Peters, Max S., and Timmerhaus, Klaus D., Plant Design and Economics for Chemical Engineers, pages 100 -141.

284

ELECTROCHEMICAL WATER PROCESSING

6. Montgomery, J.J., Water Treatment Principles and Design, pages 656-673. 7. Colyuri, pages 43-47. 8. Cappeline, G.A., and Carroll, J.G., "Enhance Your Cooling Systems's Performance Through Proper Use of Microbiocides," Power, Vol. 121, No. 10, pages 56-61, October 1977.

Electrochemical Water Processing by Ralph Zito Copyright © 2011 Scrivener Publishing LLC.

Appendix G: Design Mathematics in Computer Format The groups of mathematical equations developed in Chapter 4 are listed here, along with their corresponding lists of variables (basic parameters), for the convenience of those readers that wish to perform design estimates of operating ED systems. The information contained here is also useful is their implementation into computer files and programs. Because all the mathematical derivations and assumptions upon which they are based are to be found in the text, the reader can modify these expressions to reflect other analytical approaches or to allow other assumptions to be made. Three basic analyses will be given in this appendix. They are based upon these types of configurations: Case-A Once through, multiple cell array Case-B Re-circulation array with constant current Case-B Re-circulation array with constant voltage 285

286

ELECTROCHEMICAL WATER PROCESSING

In the first case, A, the idea of either constant current or constant voltage is not relevant in the analysis since the composition of the water flowing through the cells remains constant after steady state conditions are met. The dc power supply will merely provide whatever the current demand is at the impressed cell or stack voltage - after steady state is achieved. The analysis is set up such that the cell voltage or cell current at the input end can be set to a predetermined value. For the derivation of the following listed equations for Case-A refer to Chapter 4, Sections 4.2 and 4.3. The set of relationships employed here are direct results of those developed earlier and listed in Equations (4.35). In the re-circulating configurations, B and C, the mathematical representation is quite different, and the composition, TDS, of the re-circulating water changes with time. Hence, one must decide whether to establish a constant current, constant voltage, constant current or voltage limited, etc. system before the analysis and numerical computations can begin.

G.l

Case A

A continuous water flow-through configuration through a module establishes a set of mathematical expressions that were developed in Section 4.3, and are listed below. These relationships in the form given in Table G.l are set up for simultaneous solution in programs with the capability of performing such computations. The variables and constants employed throughout are identified and defined below in Table G.2. Let us examine the set of variables and constants in Table G.2. The last two columns in the table list the typical options of either initially setting a value for a parameter such as width of membranes or stack voltage, or setting some other parameter and solving for that particular one.

APPENDIX G: DESIGN MATHEMATICS IN COMPUTER FORMAT

287

Table G.l Equation group for case-A. Once flow-through design. v = 23\-F/W-p-n x =Vc;e/V-P

{

i-Ec-ri-ix/
x~co'e

i0 = co-Ec/
l-e

cp-p2-v-ß-10~6

Power = I ■ Ec ■ n Ae = W-x A-m = \ ' 2 ' n Size = W-x-p-n/l728 Compnts = 5 • \2Ae + Am J A 44 Thick = p ■ (l + 2nj FM = F ■ Í c0 - cx \j Power COP =

(c0-cxyc0

Energy = Power/F ■ 60 Es = n-Ec

Ignoring the last nine items beginning with Power in Table G.2, there are basically nine variables that can be handled as either independent or dependent ones. Almost invariably the single most important parameter we wish to either solve for or control is the TDS, cL, of the emerging water from a cell or a stack of cells in an array (module). The variables that we would usually most wish

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ELECTROCHEMICAL WATER PROCESSING

Table G.2 Parameter

Definition total current

I

Units Dimensions amps 2

Value

LOTUS Symbol

cale

I

input

io

entrance current density

amps /in

i

exit current density

amps/in 2

cale

ix


conversion factor

ohm-in-ppm

26X10A

D

ß

water density

gm/in 3

16.4

B

X

conversion factor

gm/amp-min.

0.038

A

TI

coulombic efficiency

%/100

input

J

P

cell spacing

in

input

P

c

entering TDS

ppm

input

CO

E

cell potential

volts

cale

Ec

V

flow speed

in/min

cale

V

F

volume flow rate

gal/min

input

F

W

electrode width

in

cale or input

W

n

number of stacked cells

input

n

X

distance from entrance

in

select

x

r

process side resistivity

ohm/in2

cale

rx

TDS at point x, or at exit

ppm

cale

ex

i

o

X

o c

c

X

APPENDIX G: DESIGN MATHEMATICS IN COMPUTER FORMAT

289

Table G.2 (cont.) Parameter

Definition

Units Dimensions

Value

LOTUS Symbol

E

stack voltage

volts

cale or input

Es

power

power input

watts

cale

Power

COP

coefficient performance

cale

COP

FM

figure of merit

cale

FM

Energy

energy per gallon out

wh/gallon

cale

Energy

Thick

stack thickness

in.

cale

Thick

Compnts

elect. & memb cost

$

cale

Compnts

Size

module volume

ft3

cale

Size

A

electrode area

in2

cale

A

2

cale

A

s

A

e m

membrane area

in

e m

to control are the dimensions of a stack and the number of cells as well as the electrical input. Of these nine variables, only six independent equations are available. Hence, three of the nine variables must have their values predetermined in order to solve for the remaining six dependent variables. Which variables are established as independent and dependent is strictly optional. These nine variables are ix, p, F, cx, n, W, n, x, and Es. Of the above parameters, (f> and r| are constants. Usually the TDS, co, of the entering water is known, or is determined by outside conditions over which we have little control. Either the flow speed, v, or the volume flow rate, F, is also usually a given condition or requirement for a specific

290

ELECTROCHEMICAL WATER PROCESSING

application. That then leaves us with the module configuration to manipulate in terms of membrane area and cell spacing. The number of cells in a stack is relevant only to the resultant volt-amp characteristics of an operating array. The amount of processing that is accomplished is strictly determined by the total electric current through the unit or the current density times the membrane area, if the area is taken as the total of all cells in series. To further illustrate the position of these various parameters the variables have been shaded in Table G.2 to indicate their normal placement. The lightly shaded areas identify those parameters that are usually given some initial values (becoming independent variables), and the more heavily shaded ones are the variables (dependent ones) that are of most interest and for which the program provides solutions. The parameters that are left unshaded are also dependent variables and are solved as part of the process. The cost outlines are based upon $5 per square foot of membrane and electrode components, and can be changed at any time to new, more realistic values. Cell spacing is almost always set at some minimum practical value that will permit ease of fabrication and minimal chance of obstruction of cell passages, while still affording a low cell electrical resistance. Most of our experience and design estimates have been with spacing, p, of between 0.10 to 0.20 inches. For example, one may wish to determine the membrane area necessary for processing a given quantity of water, F, per unit time through a module of cells with a given spacing, p, and a maximum allowable current density, i o , at the input to the module. If the input TDS, co, is stipulated as one of the conditions for that particular application, then it becomes quite simple to calculate the total membrane area needed to process that flow of input water to some specified output TDS, cx.

APPENDIX G: DESIGN MATHEMATICS IN COMPUTER FORMAT

291

Performing the computations in the manner of a series of simultaneous equations offers a better opportunity for optimization of parameters to satisfy any specific set of requirements. Because of the manner in which the set of equations is structured, it is necessary to set a convenient value of W as a trial computation, and then pick a value of x, the length of the cell for a preliminary calculation to be performed. The number of cells, n, can be arbitrarily set at any value, usually n=l for a start since n only determines the voltage needed and the length and thickness of the stack. For example, if we let W = 10 inches, F = 1 gal/min, p = 0.20 inch, io.= 0.10 amp/in 2 , and an input TDS of 1,000 ppm, we can then proceed to see what the exit TDS would be for different lengths of the cell, or what length of cell is needed to attain a predetermined exit TDS. If the value of io is specified, then the cell voltage, Ec, must not be fixed. It will be calculated for the value of voltage necessary under the other fixed parameters. The equation solving for the above conditions gives the following output information. Tables G.3 and G.4 list the values for the independent variables with n = 1, and n = 5, respectively. The stack voltage is given, in the case of n = 1, E =E. c

s

We note that the only difference between the two configurations is that the impressed voltage is higher for the Table G.3 W = 10 in, F = 1 gal/min, p = 0.20 in, io.= 0.10 amp/in 2 , co = 1000 ppm, ri = 1.0 for n =1. c TDS

E Volts

I Amps

Power, Watts

10

904

5.2

9.5

49

20

818

18

94

50

605

39

204

100

366

63

328

200

134

86

450

X

X

S

292

ELECTROCHEMICAL WATER PROCESSING

Table G.4 For n = 5. 10

605

20 40

7.9

204

366

12.6

328

134

17

450

26

stacked five cells. Power, efficiency, etc. are all unaltered in going from a single cell to a multiple cell array as long as the total area is kept constant. The only way in which the COP (coefficient of performance), or FM (figure of merit), can be improved is to reduce the cell spacing - unfortunately with the attendant increase in mechanical design and reliability problems. An example of the manner in which these equations can be arranged in a spreadsheet program to afford ease of computation and plotting various variables against each other for comparison and optimization purposes is show in the graph, Figure G.I. As the title indicates, this is a plot of exiting TDS and power input as a function of the length of the cell stack. Power increases and TDS decreases as the length becomes greater. This graph was done for fixed values of p, F, io, co, W, n, tj. The values can be changed at will within a spreadsheet to produce the corresponding graph. In fact, the basic equations can be rearranged to enable one to calculate the value of x, or W, etc. that will result in a predetermined value of cx. That would entail, for example, reformatting the equation for cx by taking the log of each side and then solving for perhaps x or W for any given set value of cx. The worksheet (spreadsheet) from which graphs can be plotted have standard forms containing mathematical expressions (formulas) for the evaluation of the dependent variables. An example of such a worksheet is shown below

APPENDIX G: DESIGN MATHEMATICS IN COMPUTER FORMAT

293

as Table G.5. In this table the values of the p a r a m e t e r s are those found represented in Figure G.I.

Output TDS and input power versus cell stack length 1200 1000 IB

Q. CL CO

Q

s

800

0)

600

I

¡400

a

n

200 10

20 30 40 Stack length, Inches - Exit TDS

50

60

■ Power

W = 10, n = 5, F = 1, p = 0.2, co = 1000, ¡o = 0.1, J = 1 Figure G.l Flow-through configuration case-A.

Table G.5 Spreadsheet form for Lotus or Excel. CASE-A output TDS and power input change with cell length. Independent Variables with Pre-set Values io = 0.04

D = 260000.00 B = 16.40

p = 0.20

co = 200.00 F = 0.10

A = 0.04

W =6

J = 1.00

n=l

v = 19.25

Ec = 10.4

294

ELECTROCHEMICAL WATER PROCESSING

Table G.5 (cont.) Spreadsheet form for Lotus or Excel. CASE-A output TDS and power input change with cell length.

Calculated Values of the Dependent Variables Formulas TDS@ ex

i amps

FM

0.00

200.00

0.00

2.00

157.21

4.00

COP

E

#DIV/0!

0.00

volts 10.40

0.00

0.43

0.96

0.21

10.40

4.44

123.57

0.76

0.96

0.38

10.40

7.92

6.00

97.14

1.03

0.96

0.51

10.40

10.67

8.00

76.35

1.23

0.96

0.62

10.40

12.82

10.00

60.02

1.40

0.96

0.70

10.40

14.51

12.00

47.18

1.52

0.96

0.76

10.40

15.85

14.00

37.08

1.62

0.96

0.81

10.40

16.89

16.00

29.15

1.70

0.96

0.85

10.40

17.71

18.00

22.91

1.77

0.96

0.89

10.40

18.36

20.00

18.01

1.81

0.96

0.91

10.40

18.87

25.00

9.87

1.90

0.96

0.95

10.40

19.71

30.00

5.40

1.94

0.96

0.97

10.40

20.18

35.00

2.96

1.96

0.96

0.99

10.40

20.43

40.00

1.62

1.98

0.96

0.99

10.40

20.57

45.00

0.89

1.99

0.96

1.00

10.40

20.64

50.00

0.49

1.99

0.96

1.00

10.40

20.69

X

G.2

s

Power watts

Case B

The second operating configuration, labeled Case-B, for a water processor in which the processed water is re-circulated many times before the entire quantity of water

APPENDIX G: DESIGN MATHEMATICS IN COMPUTER FORMAT

295

is processed to the desired level of TDS. This design is a batch process system in which a quantity of the processed water is stored in a reservoir or tank. The waste water may also be so stored in a separate reservoir, or made to pass slowly through the waste water compartments only once. For the derivation of the mathematics refer to Chapter 4, Section 4.5. The additional quantities included in the set of expressions below provide some preliminary means of assessing costs of components and polymeric case. Somewhat arbitrarily establishing case thicknesses and borders determines the dimensions and volumes of the case. These values are set, for illustrative purposes only, at Top and bottom case thickness = 1.0 in. Vertical edge case thickness = 0.50 in. Case face thickness = 0.25 in. Electrode thickness is 0.25 inch, and cell spacing in both the waste water and processed water compartments are the same. The electrode cost is set at $10 per square foot, and polymer cost at $0.12 per cubic inch. Membrane cost is estimated at about $8 per square foot of area. Also, a means for approximating temperature rise of the processed water from joule heating is provided. Hence, we arrive at the additional quantities as shown in Table G.5. As before, two constants appearing in the equations are: X = 0.038 gm/amp-minute ç - 26.104 phm-in-ppm The set of equations applicable to this situation wherein the electric current is kept constant is as follows.

296

ELECTROCHEMICAL WATER PROCESSING

Table G.6 Term

Description

Calories

Thermal energy generated by ohmic losses

Temprise

Rise in processed water temperature

V

Volume of processed water in its reservoir p

ModVol

Volume of module inner components

MembCost

Membrane cost- $/sq ft

ElectCost

Cost of electrodes - $/sq ft

EpoxyCost

Cost of polymer - $/cu in

EpoxyVol

Volume of the outer polymer case or shell - cu in

CaseCost

Cost of polymer case

ModuleCost

Total Cost of basic module materials

ModWt

Total weight of the Hence the additional termsmodule

ElectWt

Weight of the two end electrodes

T

Depth, or thickness of the module

EpoxyWt

Weight of the outer polymer shell

In a fashion similar to the one for Case A, the mathematics may be set up in either a spreadsheet form, or other math program to actually perform the computations and plot graphs of the change in values for the dependent parameters as a selected independent one is varied. An example of a simple computation for Case B is given below. To illustrate the behavior of a constant current, re-circulating water system we will choose some values for the necessary parameters, and solve for the ones we will make independent here employing the relationships shown in Table G.7. We may choose, for example, values for the independent variables as shown in Table G.8.

APPENDIX G: DESIGN MATHEMATICS IN COMPUTER FORMAT

297

Table G.7 Equation group for case B. Batch process - constant current. E

c

RW

= i R

[ w

=

+ R

p)

(

P-P/CW

Rp =
=

i-r¡-W-L-n-A-t

I = i-WL Power = i2 ■ (RW +

Rp)-n-L-W

Calories = Power ■ 60/4.1 Temprise = Calories/V„ • 10 ModVol = L-W -n-p/60 ts = n ■ tc MembCost = 8 ■ L • W ■ 2n/l44 ElectCost = 2- L-W -10/144 EpoxyCost = 0.12 EpoxyVo/ = 2-(W + l)-(T + l)-(0.25 + l) + 2 - ( r + l) ■(L + l)-(0.25 + l) + 2-(W-L) •0.50 CaseCost = EpoxyCost ■ EpoxyVol T = n-(p + p) + p + 2-0.25 ModuleCost = CaseCost + ElectCost + MembCost ModWt = 16 • 2 • (L • W ■ 0.25) ■ 2/454 +1.1 EpoxyVol-16/454 ElectWt = 16 ■ 2 • (L • W ■ 0.25) • 2/454 EpoxyWt = 1.1 • EpoxyVol ■ 16/454

And, then we obtain the results shown in Table G.9 for the dependent variables from the equations of state listed in Table G.7.

298

ELECTROCHEMICAL WATER PROCESSING

Table G.8 Independent parameters. i = 0.10 amps

V =50 liters

p = 0.20 in.

W = 10 in

cw = 10,000 ppm

L = 20 in.

c o = 1000 ppm

n=l

p

Table G.9 De rendent & calculated parameters versus time. t, Min.

E , Volts c'

C ,TDS

Power Watts

I, Amps 20

p'

0

5.72

1000

114

10

6.56

848

133

20

8

696

159

30

10

544

201

40

13.8

392

276

50

22

240

444

60

59.6

88

1192

Total current, I, is a constant 20 amps. As can readily be seen, the power and cell voltage begin increasing very rapidly as the TDS of the processed water becomes low. At some point the power supply will be required to deliver impractically high powers. Thus, a current limited power supply would most likely be employed in such configurations wherein the voltage would reach some maximum level and remain there as current begins to diminish with increasing water resistance as the processing goes on. This latter stage of operation would place the system into a "constant voltage" mode of operation, or into Case C situation. A further example of the computational possibilities afforded by the mathematical relationships when employed

APPENDIX G: DESIGN MATHEMATICS IN COMPUTER FORMAT

299

2500

2000

s

Q. 0. 1500 (A

I I

a g. 1000

500 <» 0 0 0 0 <

10

20 30 40 Elapsed time - Minutes -Output TDS

50

60

■ Input power

co = 1000, p = 0.2, W = 5, L = 40,Vp = 40, n = 1, i = 0.10 Figure G.2 Case-B constant current batch process.

in a spreadsheet program is offered in the graph, Figure G.2. In this instance the fixed parameters are L, W, n, r|=l, I, and co. The value assigned to cw is ten times that of the initial c = co because the processed water ratio to waste water volumes is at least 10 to 1.

G.3 Case C Operating an electrodialysis system from a constant voltage power supply is probably more practical, and would be the form most taken in the future. The distinct disadvantage of such a dc source is the need for limiting current because initial water resistance can be quite low for high TDS sources. The set of equations for this third version, Case C constant voltage configuration is presented below (Table G.10). The relationships are different from the previous two cases, and need to be handled accordingly. For derivations and design exercises of constant voltage systems refer to Chapter 4, Sections 4.5 and 4.6.

300

ELECTROCHEMICAL WATER PROCESSING

Table G.10 Equation group for case-C. Batch process - constant voltage. i = E C / (Rw + Rp ) c

Rw =
w

Rp = (p-V p c