0
= void fraction of packed bed = time, sec.
p
=
e
packing density, grams (dry basis)/ ml. packed volume
Subscripts .4.B. C = ion indicated by usual chemical symbol b.,f = backward and forward rate constants I = component z I = resin phase s = solution phase 0 = initial concentration I
I
13
t ISICOY,E,
7
L,
~
Figure 5. Typical curves for effluent concentrations from deep beds show the same peak for N a as in the kinetic experiments
Figure 7. With the exception of one run, experimental and computer values agree closely for deep bed data
*
Table VI.
Deep Bed Run Conditions
Ion system:
Cations: Ag+ - N a + H - ; Anion, NOS-
Resin used:
Dowex 50, 8% divinylbenzene 50- to 100-mesh Acid (H+) form
Resin capacity:
5.1730 meq./gram dry resin (Hform)
Bed heights:
0.42 to 5.52 cm. form, initial)
Reactant solutions :
Aqueous AgNOa-NaNOa Total cation normality = 0.09272-0.1048 Equivalent fractions : Ag+ 0.2515-0.7398 Na = 0.7485-0.2602
(H+
+
t
IL_CII
Figure 6. Single curves were drawn through both sets of data showing good reproducibility for deep bed runs
Solution flow rates :
0.92 to 2.66ml./sec.
Superscripts 1. 2. 3 = reaction number Acknowledgment
T h e authors thank the Massachusetts Institute of Technology Computing Center, Cambridge, Mass., and the Institute for Advanced Study Computer Project, Princeton, N. J., for free computer time. literature Cited
(1) Argersinger. W. J.. Jr., Davidson, .4. S., J . Phys. Chem. 56, 92 (1952). (2) Bonner, 0. D., .4rgersinger, FV. J., Jr., Davidson, A. UT..J . Am. Chem. SOC.74, 1044, 1047 (1952’). (3) Banner, 0 . D:, Rhett, V., J . Phys. Chem. 57, 254 (1953). (4) Dranoff, J. S., Ph.D. dissertation, Princeton University, Princeton, N. J., (19591. --, (5) Dranoff, J. S., Lapidus, L., IND. ENG. CHEM. 49, 1297 (1957). (6) Ibid., 50, 1648 (1958). (7) Gilliland, E. R., Baddour, R . F., Ibid., 42, 1120 (1950). (8) Hiester, N. K . , Vermeulen, T., Raddinp. s. B.. Nelson. R. L.. A.I.Ch.E. Journal 2, 404 (1956). (9) Juda. W., Carron, M., J . Am. Chem. sop.7. 0- ., 3295 (1948). (li)-Sujata. .4. D., Bandero, J. T., FVhite, R. R., IND.E N O .CHEM.47, 2193 (1955). \ -
~
both sets of data. showing good agreement. T h e complete data are presented elsewhere ( 4 ) . Data from integration of the differential equations were next calculated using input data corresponding to the experimental runs. However, the only successful calculated data were for five runs made on rathcr short beds. A typical result of the computer calculations is shown in Figure 7. Effluent concentrations of two ions are shown, the third ion being omitted for clarity. Because of the computer print-out cycle the results start a t t = 2 seconds. T h e upper ends of the curves are cut off because of oscillations (instability) in the numerical calculations after this point resulting from the integration technique (Milne’s method) used. Further computations in this laboratory have indicated that single-step integration methods, such as the Runge-KuttaGill, are completely stable for the present system. With the exception of one run, experimental and calculated values agree
76
INDUSTRIAL AND ENGINEERING CHEMISTRY
closely. Both exhibit the same general trends throughout and essentially the same numerical values. There is no apparent reason for the larger deviations in the data of the single run other than possible experimental error in this one case.
RECEIVED for review April 27, 1960 ACCEPTED August 29, 1960 FVork supported in part by a fellowship provided by the General Electric Charitable Educational Fund and the National Science Foundation.
Nomenclature
total solution concentration, meq. ml. k = rate constant, grams meq. sec. or ml. ’meq. sec. li = equilibrium constant L = solution flow rate, ml.’sec.-sq. cm. cross section Q = total resin capacity, meq. ‘gram of dry resin R = rate of exchange S, t = alternate independent variables I’ = volumetric flow rate, ml. ’sec. x’ = equivalent fraction of a n ion in solution phase I* = equivalent fraction of a n ion in resin phase 2 = axial distance in column measured from inlet, cm.
Co
=
Correction Statistical Methods in Fer mentation Develop ment I n this article by Jerome S. Schultz, Donald Reihard, a n d Elmer Lind [IND. ENG. CHEX 52, 827 (1960)] the first equation, column 1, page 828, is in error a n d should read:
