I
M. v.
MENTS'
The Research Council of Israel, Jerusalem, Israel
Throughput a n d Energy Consumption for.
..
Water Desalination by Electrodialysis A clear framework of manageable expressions can be obtained from these conditions and suitable parameters
Mum
WORK HAS been done on electrodialysis (7-72), but general methods for linking influential parameters of the process d o not exist. I n the work reparted here, a framework of expressions is described for throughput and energy consumption as a function of such dimensions as resistance constants, concentrations, and stack currents.
Assumptions
A smooth homogeneous liquid stream is assumed and both dialyzate and brine streams cover each other completely in the alternate compartments. T h e electrical stream flow is homogeneous, even for concentration differences in different parts of the stack. Temperature differences are not taken into account, and a n average coulomb yield is assumed, together with a reciprocal relation between liquid concentration and resistance (72). T h e electrode potentials are small compared with total stack potential; a n average value is accepted. Apparent resistances caused by such factors as polarization effects and diffusion potentials are included in the membrane resistance. Salt and water losses by back diffusion osmosis, the presence of rinsing streams, and the fact that these streams may carry away part of the salt are accounted for by efficiency factors such as i, and k,. Validity of all assumptions was checked by actual experiment. Constant Voltage Batch Process Equations can be derived for expressing concentration as a function of location in the stack, hut because homogeneity is lacking, the results would have little meaning. I t is more relevant to calculate time dependence of the concentrations :
Rickmansworth. Equal width for both brine and dialysate compartments is assumed, together with constant high brine concentration and constant membrane resistance ( 7 , 2, 7 7 ) . These conditions can be expressed by m = 1 ; cp = m ; and R, = R,, and Equation 2 reduces to
This results in
I = T [ A In y
-
A)(i
-
I/y)]
(5)
where T h e time constant T expresses the time it would take theoretically to desalinate the dialysate container completely under the initial current conditions. Mo is the proportional contribution of the combined membrane resistance to the total stack resistance a t the outset of the run. By reshuWing Equation 2 and inserting conditions as they are a t the end of the run ( t e , C,,, 73, throughput is
When Equation 5 is differentiated,
Therefore,
2.-12. =-/
This relation shows how the currentconcentration ratio increases about 11.4 times during the run up (Table I). Using the time-current columns in Table I, graphical integration gives
f Zdt = 26,700 amp. sec. Energy consumption can be expressed by
';io
1ooo3600
Equations 3 and 4 seem to govern batch behavior. Equation 4 shows that raising the current density causes a higher consumption rate. If current density is too high or liquid velocity in the stack too low, polarization may occur, which involves greater membrane resistance iM0 and &). Also, selection of small compartment widths, b , is restricted --pumping power increases when distances between membranrs are too small. Time dependence of the three quanti'dl
-
cd2
is important.
cdi
T o illustrate the theory, the results are applied to three test runs, one a t Rickmansworth and two a t Beer Sheva.
(9)
Because voltage is constant, this results in
p = - 26,700.58 . - .
ties, C,,, i, and Present address, Smolenskin St. 14, Ahuza, Haifa, Israel.
+ (t
=
5.15 kw. hr.;
1000 gal. Using the result from Equation 9 and the definition equations for ii and T , one finds that i n = 58 and T = 38.4 min. ( F = 96,500 amp. sec./equivalent; Vd = 72.5 X 4400 cc. c d , - Cdr = 2900 p.p.m. = 5.10--5 equivalent/cc.). By constructing the tangent (Figure 1) in a concentration-time plot a t I = 0, T = 38 min. Thus, the average coulomb yield i, and this yield a t starting conditions, i,o are about equal. By inserting T = 38.4 rnin. and a value -4 = 0.50 into Equations 5, 7, and 8 for the time dependence of concentration and current, the measured results agree well with those calculated (Table I and Figure 1). When the A = 0.50 value is combined with Equation 6, M o = 0.50. Thus, thr combined membrane resistance contributes half of the total resistance a t startVOL. 52, NO. 2
*
FEBRUARY 1960
149
r
\I
35 ~
"
"5
F
55
* t(i"m,n"le*l
Figure 1. Rickmansworth test. For time dependence of concentration and current, the experimental and calculated results agree well '5@0
2500
1500
500
-Cd(wm4
ing conditions.
However, data were in-
sufficient to determine whether or not part of the liquid resistance should bc added to membrane resistance. This
Time, hlin.
membrane resistance includes extra resistances such as quotient of counter potentials a n d current, together with those dcscribed previously. By the same method used for obtaining Equation 5 , the throughput, found with the help of Equation 3, is
Volts
Table 1. Rickmansworth" Test ( 7 ) Concentration, P.P.M.b T ~ ~ ~ . , CalculatedC Amp. Concentrate Diluate "C. Y Amp. Concn.
50 50 50 50 50
0
5 10 15 20 25 30
13.6 12.3
11.5 10.5 9.7 8.6 7.5 6.5 5.7 5.0 4.3 3.9
50 50 50 50 50 50 50
35 40 45 50 53.5
10,300 10,300 10,500 10,500 10,500 10,500 11,000 10,900 10,800 10,600 10,600 10,600
3400 3000 2650 2200 1900 1600 1300 1100 830 700 590 500
43
44
1.00 1.13 1.29 1.54 1.79 2.22 2.61 3.1 3.9 4.8 5.8 6.8
13.6 12.6 11.7 10.7 9.7 8.7 7.6 6.6
5.7 4.8 4.1 3.6
3400 3000 2600 2220 1900 1600 1300 1080 840 700 580 520
Tobruk water; initial vol., 7 3 gal.; final diluate vol., 72 gal.; kw.-hr. used (by meter) 0.336; throughput, 80 gal./hr.; power consumption, 4.65 kw.-hr./1000 gal. Analysis of treated water: NaC1. 386 p.p m.; total hardness, 58 p,p.m.; power consumption (graphical) 5.05 kw.-hr./ 1000 gal. As SaCl by a salinometer. Calculated using Equations 5 and 7. (I
By inserting data, 58
G =
. 13.6 "
96,500
1
5.4 .58.5
1
0.50 In 6.8
+ 0.50 (1 - 1/6.8)
Table
-
100 cc./sec. = 82 gal./hr.
Beer Sheva, p = 1. Widths for both brine and dialysate compartments were equal as well as container volumes and initial concentration of both liquid streams. During the run, membrane resistance was constant. Thus, m = 1: C, = C,, and R, = R,. Equation 2 changes to t - In (27 -1) f 2M0 (1 - l / r ) 2 2Mo
+
7
II.
Beer Sheva Test
[ b = 0.11 cm.: k, = 1.4; S = 540 sq. cm.; n = 13; a = 100 C2-I cm.-'/equivalent/cc.; 30 volts; liquid, NaCl (sol.)] P.P.M. Calcd.b t , 11ins. 1,Amp. Cdl Cbi I Cdi
vb
3.80 3.40 2.90
2810 2450 2100 1750 1410 1100 820 540
'v = 109,000 CC. 3600 3970 4320 4670 5000 5300 5600 5870
3.70 3.65 3.60 3.50 3.35 3.20 3.00 2.60 2.20
2540 2220 1940 1670 1400 1150 920 720 530
2540 2670 2800 2930 3050 3150 3250 3340 3430
$0
4.25 4.20 4.15 4.10
0 30 60 90 120 150 180 210
4.00
0 15 30 45 60 75 90 105 120
= 1,'
(11)
By differentiating Equation 11
3.70 3.60 3.50 3.35 3.25 3.05 2.85 2.60 2.25
Now
should read T i 2.0
1 50
INDUSTRIAL AND ENGINEERING CHEMISTRY
=
1
+
1\10
y ? / ( l . T 8 y - 0 . i 8 ) f 3le'
Calculated using Equations 17 and 19.
2540 2220 1940 1670 1410 1180 950 740 540
v=
WATER D E S A L I N A T I O N This relation shows that toward the end of the run, the current-concentration ratio rises to about two or three times the original value in cases where M Q compares with unity. I n this example, throughput is
and Equation 4 for power becomes k,bioF p =(2 2Md1 l/Yd) na
+
+
-
Because the decrease in current with time was greater than it would have been had membrane resistance been constant, conductivity of the membrane was assumed to decrease proportionally to the average concentration of the surrounding liquid. Membrane resistance is
Equation 2 can be integrated and the following relations can be verified without approximations (Table 11) : In
Y
+ 2 In
(3 - I/?)
+
For the example given in Table I1 where (c = 1, the membranes were of the same type as those in the previous test (4)but with a high resistance. Resistance of the liquid a t the start of the run can be calculated :
Because total resistance a t the beginning was 7fl, Mo is about 4. Counter voltages are not taken into account; their apparent resistances are added to membrane resistance. With the help of the definition equations for 15 and 7 ij = 655% and T = 240 minutes By using this r value and Mo = 3.9 in the equations given in Table 11, the results agree with those summed u p in that table. When a tangent is constructed in a concentration-time plot a t t = 0, then T = 235 minutes. Thus, the average coulomb yield, 15, and this yield a t starting conditions, 7jo are about equal. Using the time-current columns in Table 11, graphical integration gives, f Zdt = 49,000 amp. sec. Energy consumption is P = 3.7 kw. hr./cu. m . By using Equation 14 and various data, P =
1.4.0.11(4.20/540). 96,500 0.65.100 (1
+ 3.9)
(1
watt sec. 12.5 cc.
- &)
=
1,78 =
3.5 kw. hr./cu. m.
This shows that membranes used in the Rickmansworth test would have resulted in considerably lower power consumption-down to about one third. Beer Sheva, p = 2. Brine and dialysate conipartinents were equal, dialysate container volume was half that of the brine container? inirial concentration of both streams was equal, and membrane conductivity was proportional to average conductivity of liquids on both sides of the membranes (72). This run \vas performed immediately after that where p = 1 ; therefore, .’do = 3.9 here also. However, concentrations and starting current were slightly loiver.
In
evolving
dt/(l
Equation
- P/?)
24,
a factor
appears; using Equa-
tion 22, this integral is 1/2 T In (27,
- 1)
Continuous Process T h e relations for constant current batch processes are applicable to continuous methods; however, container volumes are infinite and influent-effluent concentrations are not comparable. Thus, voltage over the stack is usually greater and power consumption is higher. Using the same conditions as for the constant current process,
E - krbio (’ + M u )In (27, l - uCo(1 - 1 / Y A and
-
1)
(26)
Comparison of Results
As in the previous case, ij, r , f l d t , P, and G can be calculated easily; the results agree well with those obtained experimentally (Table 11).
Constant Current Batch Process For constant current processes which can be described using average coulomb yield, some of the relations can be arrived at easier than those for constant voltagee.g., throughput and time dependence of concentration can be derived immediately from T because concentration difference of influent and effluent remain constant during the run. Constant current processes have no marked advantage over constant voltage processes because polarization is likely to occur during advanced desalination. However, end relations are given for one set of conditions: equal widths for both brine and dialyzate compartments, equal container volumes, equal initial concentrations of both liquid streams, and constant membrane resistance.
and
P =
For the comparative numerical study in Table 111, it is assumed that current through the stack was normalized-i.e., initial current density was equal at the beginning of all the processes. Further, it is assumed that a t the outset, the contribution of membrane resistance is equal to that of the liquid-i.e., Mo = 1. The treatment would have remained more general by inserting Mo instead of unity in the various relations; however, the value of 1 is practical. Several constants may be chosen ( 7 7) :
k,
=
F
=
Mo = n
b = 0.13 cm.; io = 1.2.10+ amp./sq. cm. 43fi-Icm. -‘/equivalent/cc. ; C, = 7.7’ 10-5 equivalent/cc. (4500 p.p.m. as NaC1) 96,500 coulomb/equivalent; .I = 0.8; S = 3500 sq. cm. 1 (identical with Rm0 = 390 sq. cm. av.) 100; E = 10volt.
= 1.5;
=
For the Rickmansworth run where Ct, = Cbo= 3Cd, and Ma is lower in a reciprocal proportion to the average concentration of the liquid, the conditions are the same except for M o = a / 4 and A = (1 3/4)-1(1 1 1 ~ ) - 1 = 3;, To shorten some of the relations, “unit amounts” may be introduced: 81 = k,io6/uCo (28) g = .InSio/FCu (29) 6 = k,bFiio/?ja (30)
+
+
In Table 111, all quanrities are simply built from the product of these unit amounts and a factor that depends on Y. only VOL. 52, NO. 2
FEBRUARY 1960
15 1
~~~
Table 111. Cdr
Rickmansworth
Beer Sheva, p = 1
1 -
2
A
1
A In y e (1 (1
+
+
Beer Sheva, Q = 2
+ 21210
2
2 2Mo In (2 y e - 1 ) 2 Mo (1 - 1/Ye)
- A) x - I/Ye)
+
Constant Current
+ 2Mo
+ 2Mo In y e + 2 In
1
-
2 (t/'T)*
2
+ 2'f~
1/n
X
1
-
+
In (2y. - I) 1 I/re 1 1 - I/?"
-
1
(3 - I / Y ~ 8) Ma+X In 4 ~ , / ( 3y e 1)
Continuous Mo) x
(1
I/?*
+
G/S
1000 500
0.92 0.70
1.10 0.87
1.03 0.81
1.29 1.13
1.29 1.13
P/@
1000
1.81 2.08
3.11 3.56
3.11 3.56
3.64 4.61
4.16 5.67
500
E (Volt)
1000 500
173 173
290 290
290 290
G (cu. m,/day)
1000 500
36 27
43 34
40 32
P (kw.-hr./cu. m.)
1000
3.5 4.0
500 a
The term
--
WCda
?n
Cdc)
~
C o m p a r a t i v e Numerical Study
5.9 6.7
290 --* 510b 290 + 820b
50 44
50 44 6.8 8.6
5.9 6.7
.
is omitted: its value, 0.2 kw. hr./cu. m.. is included in the results of the
P column.
390 460
7.8 10.5
Calculated using Equation 23
under the fictitious assumption of constant membrane resistance.
s
E
= total stack-potential, volts
E1 E1
= potential over one element, volts = unit potential/element, volts
Conclusions
F G 9
= faraday, coul./equivalent = throughput, cc./sec. = unit of throughput, cc./sec.
Using the values cited, 81 = 0.70 volt; = 450 cc./sec. = 39 cu. m./day; and 6 = 6.6 watt sec./cc. = 1.84 kw. hr./ cu. m.
-
h
T h e relations described can be used in precalculating runs and in further evaluating items such as back diffusion, osmosis, and varying yields. Also they can be used in comparing some cost aspects (3, 5, 70, 72). Table I11 shows that because of power consumption, continuous processes are less profitable than batch operations. However, whether or not larger throughput obobtained in continuous processes counterbalances these higher energy demands, should be considered, especially where fixed charges on investment are more important than cost of direct current (5).
= krksb2vdF, a help constant, cm.
Eo
= initialcurrent dens., amp. /sq. cm.
A
= effective membrane length, cm. = membrane-resistance ratio help
(1 f
dimensionless between membranes,
6
= distance
c
= concentration, equivalent/cc.
cm.
cbo
= concentration of brine container
CdO
= concentration
c dI cdz Cde
e
a t the outset, equivalent/cc. of dialyzate container a t the outset, equiv./cc. = concentration of influent dialyzate, equivalent/cc. = concentration of effluent dialyzate, equivalent/cc. = end concentration of dialyzate, equivalent/cc. = total av. electrode potential, volts
1 52
= coulomb
= current density as a function of time, amp./sq. cm. I = total current; I = Si, amp. k , = resistance factor (k, > l), dimensionless kr = stream factor ( k , < I), dimensionless m = ratio of the brine and dialyzate compartment widths, dimensionless M , = membrane resistance ratio factor ;
'T
z
n
=
P
=
6
=
R
=
Rm
=
Rmo =
s
=
t
=
te
=
vd
=
Vb
Vd
= =
c
=
y
INDUSTRIAL AND ENGINEERING CHEMISTRY
y
RmOffCdO
[
I
m2] -I,
kr dimensionless number of elements energy consumption rate, watt. sec./cc. unit of energy consumption, watt. sec.: cc. resistance, D resistance per unit of membrane area (av.), sq. cm. resistance per unit of membrane area a t the outset, R sq. cm. effective membrane area, sq. cm. time, sec. time taken until C,, = C,,, sec. dialyzate stream velocity in the stack, cm./sec. brine container volume, cc. dialyzate container volume, cc. specific conductivity, R-l cm.-';' equivalent 'cc.
= dilution ratio; y =
CdO
-,
c dI
sionless
dimensionless
ij,
iiCYEl
~
Cdo/C&
dimensionless yield a t the outset, dimensionless = time constant: defined by 'T = F V d C d ~ , sec. __ii nSi, = 1 / 6 / b d , dimensionless Cde),
hd
M, =
=
= average coulomb yield; defined
= height of membrane, cm.
Nomenclature a
re
7
dimen-
q
Subfixes 0, 1, 2, e , b , d, refer to situations a t the outset, influent, effluent, end situation, brine, dialyzate, respectively.
literature Cited (1) Chem. & Znd. 8-13 (Jan. 4, 1958). (2) Engineer (Sept. 13, 1957). (3) Ionics Inc., Bull. 3, "Fresh Water from Saline Sources," Cambridge, Mass., September 1956. (4) I N D . ENG.CHEM.49,N0.12,38A(1957). (5) Kirkham, T. A., Zbid., 63, 185-9 ( 1956). (6) Kressman, T. R. E., Tye, F. L., Trans. Faraday SOC.5 5 , 1441-50 (1959). (7) Moyers, W. H., Eng. Mining 159, 84-7 (1958). (8) National Academy of Sciences, National Research Council of the United States of America, Division of Physical
Sciences, Symposium on Saline Water Conversion, Nov. 4-6, 1957. (9) Patridge, S. M., Peers, A. M., Appl. Chem. 8,49-59 (1958). (10) Rosenberg, N. W., Tirrell, C. E., IND.ENG.CHEM.49, 780-4 (1957). (11) Water and Water Eng. (October 1957). (12) Winger, A. G., Bodamer, G. W., Kunin, R., Prizer, C. J., Harmon, G. W., IND. END. CHEM. 47, 50-60 (1955). RECEIVED for review January 27, 1958 ACCEPTED September 26, 1958 Derivations of equations are available from M. v. Ments, Smolenskin St. 14, Ahuza Haifa, Israel.
