The Rate of Ion Exchange E. R. GILLILAND AND R. F. BADDOUR .Massachusetts Institute of Technology, Cambridge, Mass.
I
N VIEW of the importance of ion exchange processes and the
where the rate constant, k , is
sparsity of information on the rate of ion exchange, experimental work on the subject has been performed. Available data on the effect of changing particle size, flow rate, and concentration either show wide discrepancy or are not well defined. For example, rate constants have been reported t o vary with velocity to the zero, the 0.7, the one third, and the first power (3-6, 1 2 ) . Since these variables are basic t o the operation of an ion exchange bed, it was decided to study them quantitatively, and as some investigators report an effect of bed height on the rate constant, this variable was studied also. Bed diameter was varied for the sake of completeness.
The group a / k L represents the liquid-side resistance and k~ contains those factors which determine the rate a t which ions are tranrported from the bodv of the liquid to the liquid-solid interface. Similarly, co/KkR is the solid-side resistance, and k g contains the factors which determine the rate a t which ions are transported from the body of the particle t o the solid-liquid interface. The sum of these two represents the total resistance, and, by inspection, the difference between the products of concentrations represents the net driving force for the exchange. The fact that Equation 3 fits the rate data will be taken as sufficient evidence that the assumptions made in its derivation are permissible. The first step, then, was to check Equation 3 in the most direct manner practicable to see if it is a satisfactory expression for the rate of exchange of Na + for H + using Dowex 50. This was done in batch experiments, using the integrated form of Equation 3, which may be written
CHOICE O F MATERIALS
The ion pair selected was Na+-H+ because of its simplicity and because of the ease of analysis for the hydrogen ion. Dowex 50 was selected as the exchanger because (a)this material contains only one type of active anionic group-namely, the -SO-s group; ( b ) in the hydrogen form the material behaves like a strong acid and may be easily analyzed; and (c) the material is available in the form of very regular spheres. RATE EQUATION
Derivation. For later use in the differential equation for ion exchange in a packed bed, an equation is necessary which expresses the rate of ion exchange in terms of average concentrations in the main body of the solution and in the ion exchange particles. I n the derivation of this equation some important assumptions were made and these are summarized here:
where
E =a g x a - I
1. The factors limiting the rate of ion exchange are those which limit the rate a t which ions are transported to the interface between the ion exchange particle and the liquid, by whatever mechanism this transport is accomplished. 2 . For either phase, this transfer may be described by a rate equation of the form:
XC
and x = concentration of H + in solution, meq./cc. 2, = concentration of H + in solution a t equilibrium, meq./ cc. a = initial concentration of K a + in solution, meq./cc. b = meq. of H + in resin initially per cc. of solution t = time, sec.
Rate = k’A (c - c i ) where k’ = mass transfer coefficient; A = area; c = concentration in bulk of phase of material being transferred; and ci = concentration a t interphase of material being transferred. 3. Equilibrium exists a t the interface.
This work was performed by Andonian ( 1 ) and by Sjenitzer (12). Their technique was to introduce salt solution into a beaker in which Dowex 50 in the hydrogen form and water were agitated with a stirrer. Sodium ions in solution exchanged with hydrogen ions in the resin, thereby changing the p H of the solution. This change of p H with time was followed using a pH meter. The results of these readings were used to calculate values of the group on the left side of Equation 5 , and these values were plotted versus 1. The procedure was as follows:
For the exchange of sodium ions in solution for hydrogen ions in the resin phase, the process may be represented by the exchange equation
Nat
+ H i + N a g + Ht
(1)
the subscripts L and R refer to liquid and resin phases, respectively. For this process, it has been shown (8) that an equilibrium constant defined as
1. Place a predetermined amount of the hydrogen form of Dowes 50 in a beaker of water. 2. Into the water insert glass and calomel electrodes connected to a direct-reading p H meter, agitate the water with a stirrer (allowing sufficient time for the particles t o become saturated with water) and adjust the p H meter. 3. From a weighing bottle quickly introduce an amount of concentrated sodium chloride solution predetermined to give approximately the desired concentration in the beaker, and start the timer. 4. Take readings of p H a t various time intervals.
(where concentrations refer to equilibrium concentrations) remains practically constant for a given total solution normality, co, although the mole fraction of N a i is varied ovkr a wide range. Using these assumptions, an equation for the rate of exchange has been derived (2),and may be expressed in the form
2
= k
[c ( l - E )
- &co
- c)
1
If Equation 3 is a valid expression for the rate of this process a plot of In (1 - E z / z e ) / (1 - z/ze) vs. t should be a straight line’
(3)
330
du,v )
IO0
=f
Bapu.
(u,u )
These values off are presented in Figure 2. According to Equation 6 the slope of the elutriation curve at the halfway point (C/CO = 0.5) is d (c/co)i/z = (8) dY 4v + a
k o
. x
I 0
10
30
20
40
50
60
70
t , SEC Figure 1. Na+-H+ Exchange in a B a t c h System
The conclusions from these and the other results of Andonian and of Sjenitzer is that Equation 3 satisfactorily represents the exchange of Na+ for H+, for solution concentrations between 0.001 N a n d 0.1 N . THE EQUATION OF ION EXCHANGE IN A FIXED BED
Using rate Equation 3 and a differential material balance on a bed packed with ion exchange particles, the equation for the concentration of h'a+ in the solution leaving the bed divided by the concentration entering has been presented by Thomas (161, and may be rewritten in the form 1
c = co
K--l
l+Ge where u =
'y,dimensionless
v =
k32 m, dimensionless
(Kv
-
= flow rate through bed = total exchange capacity
FIXED-BED RUNS
Preparation of Material. For the fixed-bed runs, ion exchange material was prepared which consisted of whole, uncracked spheres with a small size distribution. Particle size was determined by preparing two samples from any one size and measuring 75 particles in each sample using a microscope. Particle size measurements were made with the material in the sodium form and in the presence of distilled water.
~
TABLE I. PARTICLE SIZE MEASUREMENTS (1 sosle division = 0.01344 mm.) Number of Particlea
Scale Divisions
u)
Slide
Total 3 24 34 55 22 11 0 1 48.21
45.5 46.5 47.5 48.5 49.5 50.5 51.5 52.5
Av. DSV. = 0.0643 om.
y = volume of solution through bed
V
where u is a term which is zero under the conditions for which Equation 7 is applicable. Equation 8 presents a method, then, for relating the rate constant to a quantity which may be determined readily from experimental elutriation curves. For cases in which the limiting form applies, the halfway point occurs at the point of inflection of the forward elutri8tion curve and in the general case i t occurs near the point of inflection, so that the slope at this point may be measured quite accurately with little effort.
No. 1 1
'
8 16 26 18 6
...
4s: i 3
Slide No. 2 2 16 18 29 4 5 0 1 47.99
of bed in meq.
The function, G, approaches one for large values of o if K is greater than one. For this case, Equation 6 has the limiting form
10
f
(7)
0 09
K--Uk where k = (___ K Evaluation of g(u, o)-Approximation Formulas. For large values of u, an asymptotic expansion of the +function by Onsager (IO) has been converted t o the g-function, rearranged, and somewhat simplified. It is
where H and H' are the prpbability function and derivative of the probability function, respectively (16).
w Figure 2.
Factors f o r Calculating g-Function
,
INDUSTRIAL AND ENGINEERING CHEMISTRY
332
D = 0.02217
D = 0.03644
D = 0.0446
D
D = 0.0522
Figure 3.
Vol. 45, No. 2
=
0.1005
Photographs of Particles Used in Fixed-Bed Runs
The measurements for one size are shown in Table I. For a particle size listed as 45.5 scale divisions, this means that the diameter was between 46 and 46 scale divisions. Two separate samples were taken because, statistically, the agreement between the samples is a measure of how representative the samples are of the whole group from which they were taken. The measurements show good agreement between the two samples. Also, for all the sizes most of the particles in any group have a diameter within 5% of the average, and the masimum deviation from the average is less than 10%. Photographs of samples from all the particles used are shown in Figure 3. Data on Swelling. I n order to estimate the diameter of the particles in other forms, measurements were made xvhich show the diameter of the particles under various conditions relative to the diameter in the sodium form in the presence of water. ThesP values are listed in Table 11. Equilibrium Constants. Equilibrium constants, defined by Equation 2, were determined a t three different concentrations. I n each case equilibrium was approached from both directions. I n the forward determinations, resin in the hydrogen form was
allowed to come to equilibrium with a solution of sodium chloride. In the reverse determinations, resin in the sodium form was allowed t o come t o equilibrium with a solution of hydrochloric acid. I n all cases, the relative quantities of resin and solution were adjusted to give a t equilibrium a Polutiori equinornial with respect to the ions. The results are shown in Table 111. Preparation of Fixed Bed. Runs were made using beds of different heights and of two different diameters-15 and 30 mm. -4diagram of the 15-mm. column is shown in Figure 4. The bed was supported on a piece of 200-mesh stainless steel screen clamped between a ball and socket. Rubber cement was used between the joints to hold the screen and seal the joint.
TABLE 11. Sodium Form in Presence of
Water 0 . 0 1 N,UaC1 0.1 N hac1 1 N NaCl
REL.4TIYE DIAMETERS OF TEE RESIN
Relative Diameter 1 0.999 0.998 0.984
Hydrogen Form in Presence of Wat e 1’ 0 . 0 1 A’HCI 0.l.VHC1 1 N HC1
Relative Diameter 1.025 1.025 1.022 1.010
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1953
side-arm hose was released in order to flush out the distilled water in the system above the glass packing. The system was ready for a run. The flow was started, and a timer was switched on simultaneously. Approximately the first 100 cc. was taken in an Erlenmeyer flask weighed to 10.05gram. This sample was later analyzed for acid and chloride ion, the latter value being used to determine the voids volume. From this point on, samples were taken according to a predetermined schedule. For most runs these samples were about 10 cc. in volume and were taken in small vials weighed to 3t0.005 gram. Depending on the conditions, the time was read when a sample was switched or samples were taken for a predetermined period of time. Sample volumes were determined by weighing, and the entire sample was titrated with 0.1 N sodium hydroxide using phenolphthalein as an indicator. The flow rate wm determined by dividing the volume of solution necessary for the main part
TABLE 111. EQUILIBRIUM CONSTANTS Forward Determinations Solution Normality K
1.561 1.601 1.494 1,510 1.418 1,418
0.01018 0.01026 0.1000 0.1000
0.819 0.792
Reverse Determinations Solution Normality K
1.596 1.556 1.494 1.508 1,429 1.424
0.01052 0.01038 0.1050 0.1050 0,782 0.803
Nominal Solution Normality
Value of K Used in Caloulations
0.01 0.1 1
1.58 1.50 1.42
333
After the column was filled with a water slurry of particles, another bed of glass spheres was placed in the upper section. This was found to be necessary, and sufficient, to give slug flow through the expansion joint. Air bubbles were removed and the column carried through four complete cycles so that the acTABLE IV. FIXED-BEDRATEDATA-0.1 N SOLUTIONS companying swelling and shrinking would bring the (All data are for nominal solution concentrations of 0.1 N,room temperature (ca. 25' C.) particles to a stable packing arrangement. The and 15 mm. diameter columns except for run numbers marked b y an asterisk, which give bed was then ready for a run. * data for columns 30 mm. in diameter)
Procedure. Most of the runs were made by passing sodium chloride solution through a bed of particles in the hydrogen form (called forward runs).' I n addition three runs were made in which hydrochloric acid was passed through a bed of particles in the sodium form (reverse runs). For the forward runs the bed was converted to the hydrogen form by passing an excess of 1 N hydrochloric acid through the bed very slowly (dropwise). All solutions were introduced by siphoning' from '5-gallon bottles located about 6 feet above the column. The acid line was then disconnected, and the distilled water line was connected to the same inlet. Water was run through the column until the washings tested acid free. The flow rate was adjusted approximately using distilled water, and then the upper stopcock was closed. With a finger over the exit of the column, the two-way stopcock was turned to admit the salt solution, and the clamp on the O i S T i L L EO
EL ECTROL YTlC SOLU TlON
WATER
n n
/5 MM TUBE
T. 200 MESH
5,s. SCREEN
/ON EXCHANGE PARTiCL ES
V
Run No.
Figure 4.
Column Used for FixedBed Runs
3c0/4V*
St/a
1/x
1/&1.05
a/kLD2
1/kD2
D = 0.02217om.
0.174 0.180 0.329 0.543 0.863
40 37 39 42 43 45
0.0521 0.258 0.535 0.108 0.285 1.41 0.568 2 81 1.20 5.94 2.04 10.1
3740 2310 1440 1160 922 852
75 70 66 74 64 63 72 68 76
0.0261 0.0319 0.0553
4690 4050 3050 2050 1990 1380 1160 928 939
15 28 8* 29 7* 6 27 9 26 11* 30 25 12 24
0 0207 0 0587
0.100
0.119 0.280 0.564 1.04 1.11
0 697 0 0 0 0 0 0 0
1 1 2 2
111 138 292 309 539 616 901 19 39 06 32
57 73 67 50 54 65 52 62 53 71 59 69
0.0206 0.0270 0.0470 0.0471 0.122 0.124 0.279 0.284 0.561 0.592 1.08
1.12
D 0,0208 0.0214 0.0196 0.0149 0.0129 0.00913 1.86 2.79 0.00701 2.91 0.00693
-
9.29 3.78 3.07 1.58 1.64 0.86 1.02 0.75 0.69
11.77 11.22 6.63 5.42 3.96 2.68 2.42 2.35
l/E-1.50 I10.27 9.72 5.13 3.92 2.46 1.18 0.92 0.85
5600 5340 3160 2580 1890 1280 1150 1120
0.0594 0.0615 0.112 0.185 0.294 0.634 0.948 0.990
61
10.34 4.83 4.12 2.63 2.69 1.91 2.07 1:80 1.74
4890 4630 2440 1870 1170 562 438 405
32 38 33 44 34 35 51 36
49
0.0262 0.0258 0.0216 0.0182 0.0169 0.0126 0.0114 0,00839 0.00856
691 508 467
56 48
0.133 0.289 0.406 0.753 0.792 1.50 1.54 2.26 2.41
60
0.0264 0.0260 0,0219 0.0186 0.0173 0.0132 0.0120 0.00965 0.00938
7000 3270 2790 1780 1820 1290 1400 1220 1180
0,0540 0.118 0.165 0.306 0.322 0.608 0.624 0.919 0.981
58 46 55 47
200 MESH
S.S. SCREEN
DVvp/p
0.02644cm.
6290 2560 2080 1070 1110 588
D = 0.0446om. l/2-4.2 0.0129 0.0126 22.41 18.21 3040 0.0102 0.00982 1610 13.87 9.67 0.00661 0.00598 735 8.61 4.41 458 0.00449 0.00372 6.94 2.75 222 0.00306 0.00221 5.53 1.33 0.00227 0.00141 5.11 0.91 152 D = 0.0522om. 1/%6.0 0.151 0.0151 38.4 32.4 0.185 0.0143 33.2 27.2 0.320 0.0113 25.0 19.0 0.579 0.00942 16.8 10.8 0.689 0,00840 16.3 10.3 1.62 0.00565 11.29 5.29 3.27 0.00380 9.52 3.52 6.02 0.00293 7.60 1.60 6.43 0.00278 7.69 1.69 D = 0.0643em. 1/sE-9.5 0.00723 0.148 0.0113 0.0112 67.6 58.1 4680 0.419 0.00782 2010 34.5 25.0 0.498 0.00174 0.00172 1690 30.5 21.0 1320 0.792 0.00514 0.00511 25.9 16.4 0.985 0.00136 0.00125 21.6 12.1 974 2.08 0.00318 0.00296 596 16.9 7.4 2.21 0.00289 0.00280 596 16.9 7.4 3.85 0.00232 0.00201 320 13.47 3.98 4.40 0.00268 0.00181 294 13.15 3.65 6.43 0.000434 0.000354 12.34 2.84 229 8.50 0.00133 0.00115 10.76 1.26 101 9.92 0.00125 0.00106 9.92 0.42 33.8 14.7 0.00108 0.000703 10.87 1.37 110 70.8 16.6 0.00131 0.000608 10.38 0 . 8 8 D = 0.1005om. I/>-21 0.230 0.00795 0.00739 95.6 74.6 2460 0.301 0.00729 0.00668 81.2 60.2 1990 0.524 0.00561 0,00489 63.4 42.4 1400 0.526 0.00542 0.00470 1480 65.7 44.7 1.36 0.00363 0.00270 44.2 23.2 766 1.38 0,00353 0.00272 43.1 22.1 729 3.11 0,00227 0.00145 36.0 15.0 495 3.17 0.00241 0.00157 32.7 11.7 386 6.26 0,00162 0.000890 29.16 8.16 269 6.61 0,00166 0.000908 27.08 6.08 201 82.8 12.0 0,00126 0.000573 23.51 2:51 71.3 1215 0.00125 0.000562 23.16 2.16 '
5440 2780 2460 2080 1740 1360 1360 1080 1060 993 866 799 875 836 3150 2680 2090 2170 1460 1420 1190 1080 962 894 776 764
INDUSTRIAL AND ENGINEERING CHEMISTRY
334
I O -
0 8
06
C -
CO 0 4
-
0 2
0 I00
200
4 00
J 00
500
600
1.0
08
0 6
C C0 0 4
02
0
I
I
j
!
I
I
!
I
I
of the break-through curve by the time required for this volume t o flow through the column. For each particle size, runs were made at various flow rates in B 15-mm. diameter column using 0.1 N sodium chloride solutions. For one particle size, 0.0643 cm. in diameter, runs were made at various flow rates with ( a ) one column 30 cm. high and 30 mm. in diameter, ( b ) at concentrations of sodium chloride of 0.01 N , 0.1 N , and 1 N , and (c) with column heights of approximately 12 cm., 30 cm., and 50 cm., all 15 mm. in diameter. For this particle size three reverse runs were made also. RESULTS OF FIXED-BED RUNS
I n order t o compare the curves calculated from Equations 6 and 7 with experimentally determined elutriation data, values of t h e rate constant were determined from the experimental data using Equation 8. These values were inserted into Equations 6 a n d 7 to construct the curves of C / C O versus y. The results are given in Figure 5 . Figure 5a is for a low flow rate and small particles. T h e complete equation and the limiting form give equivalent results for this case, and both agree well with the experimental data. Figure 5b represents medium flow rate and medium size particles. The curves calculated from the two equations differ significantly, and the fit is adequate but is not so good as for the previous example. Figure 5c shows the results for the extreme case of large particles and high flow rate. The two forms of the equation give widely different results, and the fit with the complete equation is much superior. Rate Constants for 0.1 N Solutions. Table IV gives the experimental rate data for the various particle sizes versus the superficial velocities. The slope a t the halfway point (Sl/z)was corrected by the amount, u, to give kc0/4V* (see Equation 8) from which the values of were determined. It should be noted
Vol. 45, No. 2
that the preliminary calculated results are presented in terms of IC, the rate constant IC multiplied by the group ( K - 1)/K which is constant throughout all the runs and therefore should cause no confusion. It might be point,ed out that the group ( K - 1 ) / K is a good index of selectivity. For no selectivity ( K = l), this group has the value zero; for an infinite value of K this gives a selectivity of unity. Effect of Column Dimensions. I n Figure 6E, l/%is plotted for a given part,icle size and various column dimensions (upper points). These results show t h a t there is no detectable effect on the rate constant of changing the column height fourfold or the column diameter twofold in this range. It is important to note that the smaller column has a diameter more than twenty times that of the particles. Boyd et al. (6) report that, qualitatively, they observed no effect of column diameter so long as i t was at least twenty times the particle diameter. Treatment of Rate Constants. On the basis of other data on heat and mass transfer, i t is expected that the liquid-side resistance, a / k ~ should , decrease with increasing flow rate, varying inversely with the velocity to some power between 0.5 and 1. The solid-side resistance, c o / K k ~should, , however, be independent of flow rate. This presents a method of separating the two resistances then. If the total mass-transfer resistance is plotted versus velocity on log-log paper for a range of values of fluid velocity and the constant value of the solid-side resistance subtracted from each value of the t’ot,alresistance, the values which remain, representing the liquid-side resistances, should b e on a straight line. The slope of this line is the exponent of the velocity term. Figure 6 gives the results of the application of this method t o the data for the various particle sizes. The technique employed was t o estimate the value of tho solid-side resistance, subtract this from the values of the total resistances, and plot’ the points. If the resulting points did not lie on a straight line, the initial estimate was corrected by an appropriate amount and the process repeated. Once the proper value of the solid-side resistance for each size was determined, columns of liquid-side resistances were prepared as shown in Table IV. An alternate technique would be to estimate the exponent of the velocity term and on ordinary graph paper to plot the values of l / k versus velocity to the assumed power. This would be equivalent t o the Wilson plot (9) used in heat transfer problems. The technique shown in Figure 5 is considered a faster and more accurate one for determining the value of the exponent. Once this value has been established, however, the Wilson plot might be equally good or better for determining the value of the solidside resistance, especially if only a few data points are available. From measurements of the slopes of the lines in Figure 6, the value of the exponent for the velocity was determined to be 0.83. The results for the largest particle size, Figure BF, are somewhat anomalous in that the slope of the straight line is significantly different from that of all the others. Furthermore, the points drop off at the end. Each value was rechecked and, as may be seen from the plot, the agreement is close enough. It, is believed t h a t this effect is caused by the fact that the column diameter is only fifteen times that of the particles for this case, as previously discussed. Correlation of Liquid-Side Coefficients for Different Particle Sizes. If it is assumed that ( a ) mass tramfer in the liquid occurs by diffusion from the surface of a particle t,hrough an effective distance Ar, at which point the solution concentration is that of the main stream and ( 6 ) ”that the area term in the equation for mass transfer may be taken as the geometric mean of 4 ~ r i and 4 4 r 0 (precise for steady diffusion between two concentric spheres), then the result is
+
(9)
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1953
On the basis of previous work on mass transfer it is expected that A D / D should be a function of the Reynolds number, D V p / p and the Schmidt number, p / p D ~ . Since the terms involved in the Schmidt number remained practically constant throughout these experiments A D / D should be a function of the Reynolds number only. Solving Equation 9 for A D / D in terms of the liquid-side resistance, a/kL, gives
4e D
1)
%)(&J
= (Y)(l+
335
versus Reynolds number. This is shown in Figure 7 . The equation of the correlating line is
AD (k2) (1 + 5 )
= 1010 Re-O.g4
(11)
From Equations 10 and 11
(10)
(12)
Since A D / D should be small compared to 1, a plot of a/kLD2 versus the Reynolds number should correlate the data for different particle sizes. When the data were plotted in this way a straight line resulted, the equation of which was a/kLD2 = 980 Re-0.83. Using this correlation, the equation for A D / D was determined and a corrected plot was made of
Correlation of Solid-Side Coefficients for Different Particle Sizes. If the equation is determined for the rate of diffusion in a single spherical particle with the concentration of the solution a t the surface maintained constant, an infinite series of exponentials is obtained. .If only the first term is retained the result yields
I
l0
IO
10
/O
If a n alternate derivation is made with the assumption of a linear concentration gradient within the particle, a similar expression is obtained with
b E
x*
0.1
0 1
I O
v , CM
1 0
/SEC IW
IO
2 E
These expressions indicate that the solid-side resistances should be proportional to the diameter squared. In order to check this for the different particle sizes, the solid-side resistances, listed in Table v, were each divided by the appropriate value of 0 2 . The agreement among the different values is good, the deviation from the average being &5%. The average value of the solid-side resistance is
IO
2
(1/D2) ( ~ o / K k z=z) 720
(15)
B
10
TO I O
0 1
v, CM
v, CM/SEC
100
I00
IC
IO
$
E
B
I O
O /
/SEt
Using this value and Equations 13 and 14, DE is calculated to be 2.5 X 10-6 and 2.0 X 10-6, respectively. These values are in the range of those obtained for coefficients determined in an independent manner but using a different ion exchange resin ( I S , I 4 j , Total Resistance. Combining Equations 11, 12, and 15 gives 1 __ kD2
+ 0.049 -I- 720
'O1'
(16)
2
I O " I
v'
CM/scC
Figure 6.
I
. 0
-
10 TOTAL RESISTANCE TOTAL /)Es/STANCE
..
n - ,,
1"
v,
CM/SEC
- SOLID-S1DERESlSTANCE
Determination of Solid-Side Resistance
Equation 16 summariaesall the data on rate constants determined from the exchange of Na + in 0.1 N sodium chloride solutions for H + in deep beds of sized spheres of Dowex 50 for a range of particle diameters, column heights, and two different column diameters. I n Figure 8, Equa-
336
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 2
APPLICATION OF TECHNIQUE TO OTHER SYSTEMS
The assumptions made in the derivation of the rate equation for ion exchange are sufficiently general that the results should be applicable, with proper modifications, to other problems, such as heat transfer in packed columns or exchange adsorption. CONCLUSIONS
The assumption that the rate of ion exchange is limited by the rate of mass transfer permits the derivation of a rate equation R hick correlates the data over the range of conditions covered arid which offers a method of estimating results for other conditions from a knowledge of fundamental properties of the ionic solution and of the exchange material. Over the range of variables tested, the rate constant for ion exchange in a packed bed is not a function of bed height or of bed diameter (except as this affects solution flow rate) so long as the column diameter is sufficiently greater than particle diameter-approximately twenty times as great. Both the resistance in the liquid phase and the resistance in the solid phas'e are important in determining the total resistance to ion exchange in the range of concentrations, particle diameters, and flow rates tested. The concentration of ions in the exchanging solution affects the value of the rate constant, and this effect can be predicted with a moderate degree of success. The elutriation curves for the exchange of Hfin solution for N a + on the resin can be predicted successfully from the rate data taken for the exchange of Na + in solution for H + in the resin.
TABLE V.
CONPARISOX OF SOLID-SIDE RESISTANCES
D 0.02217 0.02644 0.0446 0.0522 0.0643
0.lOO~
CQ/K~R 1.05 1.50 4.2 6.0 9.5 21
(1/D2) (co/K~~R) 710 713 700 733 763 697
ACKNOWLEDGMENT
The resin used in this investigation vias donated by the Dow Chemical Co. The data presented here were taken with the assistance of Ruth E. Leffler. NOMENCLATURE
ci
= volumetric capacity of resin, meq./cc. particle = concentration of N a + in bulk of solution, meq./cc. = concentration of &-a+ in liquid a t solid-liquid interface,
co
= solution normality, meq./cc.
C
= average mole fraction of iYa+ in particle = mole fraction of X a + in the resin phase a t solid-liquid
u
tion 16 is compared with the data points.
t o be quite good. Results at Different
The correlation seems
Concentrations. Using Equations 4, 9, 12, and the appropriate values of D , DL, and a, values of l/k were predicted for 1 N and 0 01 N solutions. These were compared with the experimental data, listed in Tables VI and VII. The results are shown in Figure 9. The predictions are reasonably good considering the range of concentrations covered. Reverse Runs. Three reverse runs were made with a particle diameter of 0.0643 cm., tvio in which 0.1 N hydrochloric acid was passed through the bed in the sodium form, and one in which 1 N hydrochloric acid was used. Theoretically, the rate data obtained for the forward runs should be directly applicable to the reverse runs. Thus, if the value of the forward rate constant for the condition of the test is divided by the value of the forward equilibrium constant, this should yield the value of the reverse rate constant. If this constant is used in Equation 6, the resulting expression should describe the reverse elutriation curve. I n order to check this, for the conditions under which the three reverse runs were made, elutriation curves were calculated using rate constants determined from forward runs. I n Figure 10 these curves are compared with the data points. The agreement is considered to be quite good, and it is concluded t h a t elutriation curves for reverse runs can be determined satisfactorily using rate data obtained from forward
rune.
c
5,
meq./cc.
interface
D = particle diameter, cm. AD = 2Ar (see Ar) D L = diffusivity of N a + - H + in resin particle, sq. cm./sec. g (u, v) = function involved in fixed-bed equation
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1953 100
I
~
I
I
- .
!
I i l l l I
!
1 1 1 1 1 1
I
1
:
’
337
I
TABLE VII.
FIXED-BEDRATEDATA-1 N SOLUTIONS
( D = 0.0643 om., column diameter = 30 mm.)
8
RunNo.
V
DVP/P
22 16 10
0.0276 0.0747 0.147 0.259 0.843
0.192 0.518 1.02 1.80 5.85
I8
- 0 01 N @UNS
20
rc0/4v* 0.00733 0.00419 0.00291 0.00153 0.00535
Sll8
0.00837 0,00522 0.00324 0.00266 0.00140
l/k 54.7 34.9 25.5 27.6 24.2
IO
I
7f
1 0
O B
e C co
O b
0 4
1 0 0 2
0
0 4 10
01
/O I O
A DV f
Figure 9.
O B
Comparison of Prediction a t Different Concentrations with Data
c CO
TABLE VI.
FIXED-BEDRATED A T A - O . ~ N ~ SOLUTIONS
(D = 0.0643 om., column diameter = 15 mm.) Run No. V DVp/r Si/a. Ica/4V* 31 21 23 19
0.121 0.308 0.838 2.01
0.871 2.22 6.03 14.5
0.000798 0.000464 0.000274 0.000200
0.000748 0.000451 0.000252 0.000175
0 6
0 4
0 2
-
COLU“
D / l = / 5 MAf
1 ”
0
l/k 6.08 3.95 2.61 1.56
I
0
400
8 00
1zw
/6W
y,cc 10
OB
0 4
C
10 II k kt
k~
E
K
I
I
ro
AT
, /K) = function involved in fixed-bed equation, g ( K O u 1 - B (v, u ) = modified Bessel function of first kind, of zero order =)
first order modified Bessel function of first kind
= rate constant, cc./meq.-sec. = liquid-side coefficient, l/sec. = solid-side coefficient, l/sec. = (K 1) k / K , cc./meq.-sec. = equilibrium constant
= particle radius, cm. = distance from surface of particle into liquid over which a
concentration gradient is assumed to exist, em. S1/2= slope of plot of C/CO us. y a t C/CO = 0.5 t = time, sec. u = dimensionless group, k c ~ y / V v = dimensionless group, h%/KV V = superficial velocity of solution through bed, cm./sec. V* = volumetric flow rate of solution through bed, cc./sec. 5 = meq. resin in bed y = cc. solution through bed = viscosity in poises, grams/cm.-sec. p = density of solution, grams/cc. Q = mathematical group involved in slope of elutriation curve
LITERATURE CITED
(1) Andonian, M. D., S.B. thesis, chem. eng., M.I.T. (1950). (2) Baddour, R. F., Sc.D. thesis, chem. eng., M.I.T. (1951). (3) Brtuman, W. C., and Eichorn, J., J . Am. Chem. SOC.,69, 2830 (1947). (4) Beaton, R. H., and Furnas, C. C., IND.ENG.CHEM.,33, 1500 (1941)
.
c co
0 4
O.?
0 0
400
aoo
Figure 10. Comparison of Calculated Curves and Experimental Points for Reverse
Runs ( 5 ) Boyd, G. E., Adamson, A. W., and Myers, L. S., Jr., J . Am. Chem. SOC.,69, 2837 (1947). (6) Boyd, G. E., Myers, L. S., and Adamson, A. W., Ibid., 69, 2849 (1947). (7) Brinkley, 8. R., Jr., and Brinkley, Ruth F., unpublished table
(1950).
(8) Duncan, J. F., and Lister, B. A. J., J . Chem. SOC.,1949, p. 3285. (9) McAdams, W. H., “Heat Transmission,’’ 2nd ed., p. 272, New York, McGraw-Hill Book Co., lnc., 1942. (IO) Onaager, L., from Nachod, F. C., “Ion Exchange,” 1 s t ed., p. 42,
New York, Academic Press, Inc., 1949. (11) Selke, W. A., and Bliss, H., Chem. Eng. Progress, 46, 509 (1950). (12) Sjeniteer, F., unpublished work done under Foreign Students Summer Project a t M.I.T. (June to September 1950). (13) Spiegler, K. S.,excerpt from L. N. S. E. Quarterly Progress Report Nuclear Inorganic Group (Nov. 30, 1951-Feb. 28, 1952). (14) Spiegler, K. S., and Coryell, C. D., J . Phys. Chem., 56, 106 (1952). (15) “Tables of Probability Functions,” prepared under sponsorship of National Bureau of Standards (1941). (16) Thomas, H. C., J . Am. Chem. SOC.,66, 1664 (1944). RECEIVED for review June 14, 1952. ACCBIPTEDOctober 29, 1952. Presented before the Division of Industria1 and Engineering Chemistry, 122nd Meeting, AMERICAN CHEMICAL SOCIETY,Atlantic City, N. J.
