312
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Fixed-Bed Ion Exchange with Differing Ionic Mobilities and Nonlinear Equilibria: Ethylenediamine Dihydrochloride and Ammonium Chloride System Kenneth L. Erlckson and Howard F. Rase’ Department of Chemical Engineering, The University of Texas at Austin. Austin, Texas 78712
Simple and rapid experimental techniques are developed and demonstrated for obtaining fundamental data for use in analyzing complex industrial ion-exchange systems. A system consisting of a microreticular polymeric resin (Dowex 50Wx8) and solutions of ethylenediamine dihydrochloride and ammonium chloride was used. This typical amine wastewater stream exhibits a number of the major complexities associated with industrial systems. Equilibrium relationships, ionic diffusion coefficients in the resin, dispersion coefficients, boundary layer thicknesses, ion+xchange capacity, bed void fraction, and particle dimensions were determined experimentally. These data were then used in a model and gave good agreement with observed breakthrough curves.
The complicated nature of the majority of industrial ion-exchange operations imposes the need for carefully planned scale-up studies. These studies would be minimized if process configurations, operating conditions, and design of pilot-plant ion exchangers could be evaluated with some measure of confidence using a mathematical model based on fundamental data. Any fundamental approach, however, must ultimately deal with the realities of real situations for which certain convenient simplifying assumptions are not applicable and lengthy research programs not feasible. The study to be described addresses these issues by developing and demonstrating simple and rapid experimental techniques for obtaining fundamental data useful in analyzing complex industrial ion-exchange systems. In addition, a preliminary model employing fundamentally based parameters is proposed and tested against performance data from a fixed-bed column. The system selected for study consisted of Dowex 50Wx8, a microreticular polymeric ion-exchange resin, and aqueous solutions of ethylenediamine dihydrochloride and ammonium chloride. This system is typical of amine containing wastewater streams from which recovery of amines is important for both economic and environmental reasons. It is representative of certain complex industrial ion-exchange systems with two exchanging ions that exhibit nonlinear equilibrium relationships, significantly different ionic mobilities, and ion-exchange rates that are controlled primarily by diffusion within the resin particles, with diffusion through the boundary layers having only a small influence.
Existing Models for Fixed-Bed Ion Exchange The most common approach to modeling ion exchange is to assume that the rate is a simple function of the instantaneous bulk and equilibrium concentrations in the resin and the solution. The resulting rate equation is then combined with the equilibrium relationship and the partial differential equation representing the material balance for species A in a horizontal differential element of the column. The various models of this type summarized by Helfferich (1962d), Vermeulen (1958, 1959), and Sherwood, et al. (1975), differ primarily in the definition of the functional relation between rate and the concentrations of exchanging components in the resin and the solution and in the assumptions made relative to the equilibrium relation and the differential material balance. All of these approaches 0019-7874/79/1018-0312$01 .OO/O
require equilibrium data and an empirically determined rate constant which in the widely used method by Thomas is called a “kinetic coefficient”. More rigorous approaches have also been presented in which are solved some or all of the four equations that express, respectively, the rate of ionic diffusion within the particles, the rate of diffusion through the surrounding boundary layer in the fluid, the material balance for a horizontal differential element of the column, and the relationship between the volume-average (or bulk) and the local molar concentration of the given ionic species in the resin particles. In solving these simultaneous equations the several authors have differed somewhat on the initial and boundary conditions but all have assumed equilibrium at the fluid-particle interface. In most cases, after making appropriate simplifications, Duhamel’s theorem is applied in order to reduce some or all of the four equations to a single integro-differential equation, which eventually requires some form of numerical computation. The developments of Colwell and Dranoff (1968,1971), Cooper (1965), Fleck et al. (1973), Rosen (1952, 1954), and Tien and Thodus (1959) are recommended for study of this approach. As stated by Rosen (1952), these developments apply only when the change in solution concentration over a length equal to a particle diameter is small enough that the change can be neglected. Furthermore, the material balance for a given species, such as A, which is derived by taking the limit as the axial dimension Az of the volume element approaches zero, should be used with some caution. Helfferich (1962b) has suggested that the value of Az cannot be decreased to less than one particle diameter without the volume element losing its geometric and mechanical characteristics. The equilibrium relationships employed as boundary conditions in these developments have included irreversible equilibria; linear, Langmuir, or Freundlich isotherms; and distribution coefficients which are linear functions of concentrations. These several forms allow a fair degree of latitude in attempting to fit experimental data with an appropriate expression. For some ion-exchange problems, however, the nature of the equilibrium relationship can be critical, and it may not be possible to represent the experimental data adequately with one of these special cases. Furthermore, linear isotherms and irreversible 0 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
0
I Xu05[f
Q
w
- D a t a p o i n t s on - DIaOt aN c points u r v e on
0 I N curve
-1
t
0 .o
1 i
05
IO
XA
Figure 1. Experimental equilibrium curves.
equilibrium are not usually applicable to industrial ionexchange systems. Other distinguishing characteristics of these developments to be noted are included in the works of Colwell and Dranoff (1969,1971), who considered axial dispersion (but with the omission of a boundary layer equation) and tested their model by experiment. Also Fleck et al. (1973) applied the restriction of the constant-pattern condition. The mathematical complexity of rigorous approaches varies depending on the simplifying assumptions. Many of the complexities, however, are now less formidable when efficient numerical methods are applied. Further advances in model development will depend both on clever mathematical effort and efficient procedures for independent measurement of fundamental variables necessary for the differential equations. These variables include the equilibrium relationships required for establishing boundary conditions, ionic diffusion coefficients in the resin, dispersion coefficients, boundary layer thicknesses, ion-exchange capacity of the resin, bed void fraction, and particle dimensions. The experimental methods used in this study for obtaining values of these variables proved to be relatively simple, rapid, and accurate. When combined with values of effective diffusivities of the ions in solution and used in a model based on nonlinear equilibrium and rates that are primarily resin-diffusion controlled, good agreement was obtained with observed breakthrough curves.
Experimental Techniques and Data Equilibrium Experiments. Well-mixed 500-mL, three-necked flasks equipped with indented baffles were used to obtain equilibrium composition, co-ion intrusion, and average resin shrinkage. By using a number of flasks simultaneously, a set of equilibrium data could be generated in one experiment. Samples were removed through the short Pyrex tube which was fitted with a 100 mesh type 316 stainless-steel screen. The lower end of the long Pyrex tube was also fitted with a screen. By inverting the flask and gently forcing air through the long tube the interstitial solution could be drained through the short tube. A 24-h contacting period was found to be adequate and convenient for reaching equilibrium. Data were initially obtained by starting with the resin in the NH4+form and displacing the NH4+ ion with H3+NCH2CH2NH3+ion (favorable exchange) and then additional data were obtained using the reverse reaction (unfavorable exchange). Typical results a t 28 "C are given in Figure 1. Repeat runs were made using separate samples of resin. The results were generally accurate within fO.01 equivalent fractions in the liquid phase and f0.02 for the resin. Co-ion intrusion and water loss and shrinkage by the resin were also examined. Data are given in Table I. For
313
Table I. Parameters Obtained by Independent Emeriments at 28 'C parameter
value
dispersion coefficients 1.2 cm2/sQ G = 1.6 g/cmZs 0.75 cm2/s@ G = 0.83 g/cm2 s ionic diffusion coeffi= 1.2 x cm2/s cients in resin DB = 4.0 x cm*/s co-ion intrusion 0.023 mequiv of Cl-/mequiv total capacity 2.10 mequiv of C1-/(cm3of resin ion-exchange capacity in NH,+form) 0.036 cm3 of H,O/mequiv water loss (upon complete total capacity conversion from NH,* form t o H;NCH,CH,NHi form) 0.039 cm in NH,+ form mean radius of resin particles (broken 0.036 cm in particles not used H:NCH,CH,NH,' form in studies) bed void fraction 0.35 shrinkage 8% bulk volume decrease for conversion from NH,' to H:NCH,CH2NH: form
n~
the resin in both NH4+ and H3NCH2CH2NH3+forms, co-ion intrusion was determined by observing sorption of C1- ion upon addition of sufficient 1.5 N C1- solution to bring an initial C1- concentration of 0.05 N to about 1.0 N and by observing desorption of C1- upon dilution of the 1.0 N solutions. No significant difference in co-ion intrusion was observed between the two ionic forms of the resin. During ion exchange, water loss by the resin was, therefore, calculated from changes in C1- concentrations observed while obtaining ion-exchange equilibria. Shrinkage by the resin was determined from the change in bulk volume occurring when samples of resin were completely converted from the NH4+ t o t h e H3+NCH2CH2NH3+ form, and conversely. Void Fraction and Particle Size. The bed void fraction was estimated from the NH4+ and C1- concentrations which resulted when a known volume of distilled water was added to a known bulk volume of resin and interstitial solution of specified NH4Cl concentration. Particles having approximately uniform average radius, rK, between about 0.36 and 0.39 mm were obtained by wet sieving followed by a water elutriation technique. Dispersion Experiments. Dispersion coefficients were obtained by monitoring the effluent solution concentration in the fixed-bed column as a function of time after introducing a step change in concentration of the feed solution. The resin was used in the NH4+ form, and the initial concentration of the interstitial solution was 0.05 N NH4C1. Then a step increase to 0.1 N NH4C1was introduced. In this manner the dispersion data were free from effects of ion exchange. Concentrations were obtained by means of a flow-through electrical conductivity cell as described for the fixed-bed performance experiments. The dispersion coefficients were determined from the experimental data by selecting values which gave the best fit with calculated effluent concentration vs. time curves. The calculated curves were based on the material balance with ion-exchange terms omitted. Data are given in Table I. Ion-Exchange Capacity. The average ion-exchange capacity of the resin in the NH4+ form was determined (Table I) by converting samples of the resin from the H+ form to the NH4+form using an excess of 1.0 M NH40H and then titrating the unreacted NH40H with 1.0 M HC1.
314
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979 T v o c i Tube
-
'1
I
1I
d Nitrogen Cylinder
L
; I
2
U
L!
d E
Sample C o I I e c t o r s
Figure 2. Apparatus for evaluating effective individual ionic diffusion coefficients.
Slopcock A
lL
/I[,-TY&-.,
ABr op t! tr loet o r Disptnsinq Burette
c
1250-7 1
Gloss Bottli 15- g a l I
Figure 3. Experimental ion-exchange column and associated equipment.
cm. The actual bed heights were measured for each experiment. The maximum possible error in this determination was Test solution was first introduced into the distributor estimated at f 2 % . cone consisting of two ground-glass fittings (60/40 male Ionic Diffusion Coefficients in the Resin. Effective and 24/40 female) and two 100-mesh stainless steel coefficients for NH4+ (DB) and H3+NCH2CH2NH3+(DA) screens. This distributor was designed to produce a were determined by batch experiments at 28 "C using the uniform step change in the composition of the solution. apparatus illustrated in Figure 2. A known volume of resin After lowering the solution initially in the column to a in the NH4+form with a known volume of 1.0 N NH4Cl position midway between screens A and B, stopcock A was interstitial solution was placed in the batch contactor (flask temporarily disassembled; test solution was quickly poured B) shown in Figure 2. Then a known volume of 1.0 N through a funnel, and stopcock A was then reassembled. C1-H3+NCH2CH2NH3+C1solution was quickly introduced The layer of Pyrex beads above screen A tended to from flask A by fracturing the Saran membrane. The distribute the solution over the screen so that only one or introduction required approximately 0.3 s. Effects of two drops of the solution penetrated screen A before it was co-ion intrusion were minimized by initially contacting the completely covered with solution. Once covered with solution, the capillary effect due to the very small openings resin with 1.0 N NH&l solution. By using a large volume in screen A prevented further solution from penetrating of the C1-H3+NCH2CH2NH3+C1-solution the equivalent until either stopcock B (a 4-mm stopcock) or stopcock C fraction of this species in the solution was kept high and in the region where the equilibrium curves are close to(a 10-mm stopcock) was opened. The one or two drops gether. In this manner water loss and effects due to co-ion of test solution which penetrated screen A were distributed intrusion and to diffusion through the boundary layer were laterally by the beads between screens A and B, and no minimized. Rapid stirring also assured minimal fluid-side test solution penetrated beneath screen B. When stopcock gradients. No effect on the results was observed for stirring B or C was opened and the system was vented to the atmosphere or the system was under pressure from the speeds over the range of 500-1250 rpm. In order to obtain nitrogen cylinder, experiments with dye solutions showed solution concentration vs. time data aliquots of solution of known volume were removed at predetermined times, that initially a nearly flat surface separated the flowing test solution from the solution originally in the column and and samples were analyzed using a modified Kjeldahl technique (Erickson, 1977) for both H3+NCH2CH2NH3+ that the surface was slightly distorted as it moved down the column which was not filled with resin when per(ion A), which was sorbed, and NH4+ (ion B), which had forming experiments with dye solutions. This slight been desorbed. distortion was caused by velocity gradients near the The sorption of ion A during the batch experiments was column wall, convective currents which were perhaps due described mathematically by the simultaneous solution of to density differences between solutions, and turbulent the material balance and the diffusion equation derived disturbances near screen B. using the Nernst-Planck equations for ions A and B. After the test solution was introduced into the disValues of DA and D B were selected and the equations tributor cone and the upper fittings were reassembled, the solved numerically for the solution concentration as a height of resin above screen C was recorded, and the line function of time. The calculated data were compared with between the aspirator bottle and column was gently filled the experimental data, and the values chosen for D A and with test solution. Stopcocks B and C were then set at DBwere those which gave the best agreement. (See Table predetermined positions. During the experiments effluent I.) solution concentrations were determined by measuring the Fixed-Bed Performance. Performance data were change in electrical conductivity of the solution and obtained for this system using the equipment shown in comparing with data obtained using known standards. Figure 3. The average diameter of the Pyrex column was The conductivity cell was connected to a Wheatstone 3.98 cm and the distance between screens B and C was 62.8
bridge which was in turn connected to a strip-chart recorder. The actual experiment was initiated by simultaneously starting the chart on the recorder and opening the valve on the aspirator bottle which was under pressure from the nitrogen cylinder. From the time breakthrough was first observed until the end of the experiment, liquid samples were removed at stopcock B for determining water loss from the resin. A constant temperature of 28 f 1 "C was maintained for all tests by enclosing the equipment in a polyethylene tent. A thermistor which was placed near the ion-exchange column was connected to a controllor which actuated a small floor heater.
--. 0 6 -
~
-0
0,
3% E
-
o* o*
2- 0 4 x
"0 0 -Exoefimental Dolo *-Predicted Data
6
02-
0
do
00-
Performance Model A model was developed for this complex system and used with the independently derived parameters to predict performance at the conditions for which performance data were obtained. A material balance for species A in the Lth volume element was obtained by integrating and then combining the equations of continuity for species A in the interstitial solution and in the resin particles.
0
20
40
60
I00
80
120
I40
I60
1
ELAPSED TIME (secl
with initial conditions
XA(zl,t) = a constant
(Ib)
XA*(O) = 0 = WA(0)
(IC)
where X A * ( t ) = the volume-average solution-phase equivalent fraction of species A in the volume element a t time t , X A ( z L , t ) ,XA(zL+l,t) = the average equivalent fraction of species A over the imaginary cross-sectional surfaces at time t and coordinates zL and z ~ +respectively, ~ , Scs = cross-sectional area of bed, ED = a dispersion coefficient accounting for both molecular diffusion and macroscopic fluid-phase dispersion, NT = the total solution normality, Q = the volumetric flow rate of the solution, mA(t)= the total amount of species A in the resin particles in the volume element at time t , and e = the void fraction of the bed. Details of the complete model are given in the Appendix. Briefly, the bulk resin-phase concentration of species A was obtained from the sum of the volume integrals of the concentration profiles for ion A in each resin particle. The concentration profiles were calculated by solving the diffusion equation based on the Nernst-Planck equations for ions A and B, with one boundary condition being the assumption of equilibrium at the resin-solution interface. The solution of the diffusion equation required experimentally determined values for the effective ionic diffusion coefficients and experimentally determined ion-exchange equilibrium data. The equilibrium isotherms were highly nonlinear, and their use with the model involved a solution-side surface concentration for species A. This surface concentration was related to the bulk solution concentration by an equation analogous to that recommended by Helfferich (1962b) for diffusion through the fluid boundary layer. The boundary layer equation contained as parameters a boundary layer thickness which was determined experimentally (see -4ppendix) and an effective solution-phase diffusion coefficient which was estimated from data in the literature (see Appendix). Breakthrough curves were calculated and found to be in good agreement with the observed performance curves over a two-fold range of flow rate for both favorable and
Figure 5. Comparison of experimental and predicted column performance data, favorable exchange, volumetric flow rate: 11.5 cm3/s. l
08
o
LI
r
'
'
I
' L,
'
o*o*
f
c
J
2 04
avo
0 -
EiDerlmental Data Doto
*- P r e d i c t e d
-1 -1
Urfororable Eicilonge 0 :2 1 8 c m 3 I s e c i G = 3 64 I b l f f 2 s e c I
9
1
(D
oot 0
: ,1 , 20
40
60
80
I00
I20
I40
160
ELAPSED TIME (set1
Figure 6. Comparison of experimental and predicted column performance data, unfavorable exchange, volumetric flow rate: 21.8 cm3/s.
unfavorable exchange. See Figures 4,5, and 6. Conclusions The agreement between experimental and calculated results clearly demonstrates that mathematical models based on fundamental data and mechanism can be developed and applied to difficult industrial ion-exchange systems. Furthermore, the fundamental data required can be obtained relatively quickly and easily, either from the literature and/or by experiment. With some modifications in mathematical techniques, the basic approach described above appears to offer much promise for providing considerable insight into complex industrial ion-exchange processes while requiring only a modest but extremely useful data collection effort. Appendix: Details of Model With reference to eq 1 the term X A ( t )is approximated by the arithmetic average of XA(zL+lt)and XA(zL,t).The partial derivatives in the dispersion terms are approximated by finite difference expressions involving only values of X A at zL and zL+lin order to avoid prohibitive iterative calculations. Such approximations produced small errors in the dispersion terms, but the effect of dispersion is itself
316
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
small. Using the trapezoidal rule, eq l a is then integrated with respect to time between the limits of t and t At, and the resulting equation involves only the dependent variables X A and W A . By selecting appropriate values for Az and At, values of X A ( Z L + l , t + At) can be calculated explicitly as follows. A sufficient number of time increments are chosen, as values of L are taken sequentially, from 1to Lmm. Using iterative numerical techniques, eq 1 are then solved simultaneously with the following approximation for W A ( t At) and the necessary attendent equations
+
+
W A ( +~ At) =
WA(ZL,~
+ At) + w ~ ( Z ~ + l , +t At)
(2)
2
+
The term WA(ZL,~ + At) is the value of w ~ ( t At) that would result if at every coordinate z in AV, XA(z,t At) were equal to X A ( z L , t + At). The bulk resin-phase concentration w A ( z L L t + At) is obtained from the local particle concentrations X A K by
+
where C K = the ion-exchange capacity of the resin, m = the total number of resin particles, VK= the volume of the Kth particle, and XAK = the local equivalent fraction of species A in the Kth particle. For spherically symmetrical particles, the local equivalent fraction X M ( r , t )is given by the solution of the diffusion equation derived using the Nernst-Planck equations for ions A and B
%}) ar
(4a)
using the following boundary conditions
XM(r,O) = a constant
(4b)
a -XAK(O,t) = 0 ar
(44
X A K h , t ) = F[XAKS(~)I
(44
where D A , D B = effective diffusion coefficients for ions A and B, respectively, in the resin, F[] = the nonlinear functions which describe the ion-exchange equilibrium isotherms, rK = the radius of the Kth particle, XMs = an assumed uniform fluid-side equivalent fraction of species A a t the surface of the Kth particle, and Y A , YB = the valences of ions A and B, respectively. The surface c9ncentration X,,(t) is related to XA(zL,t + At) and W A ( z L , t + At) using the boundary layer equation recommended by Helfferich (1962b) in the form
Using a procedure similar to that discussed by Lantz (1971), the truncation error in eq 1was analyzed in order to determine values of At and Az which would minimize numerical dispersion. The analysis indicated that the dispersion resulting from terms involving XAshould be minimized by taking At = CS,SAZ/Q
Unfortunately, the analysis did not provide similar quantitative insight into the errors associated with W,+As mentioned by Helfferich (19624, the minimum realistic value of Az would be about one particle diameter. The approximation for WA(t + At) given by eq 2 involves an error which in general will not vanish as the value of Az approaches such a lower limit. A theoretical analysis of the error associated with WA(t+ At) appeared unfeasible. Instead an empirical analysis was made, and it was found that the cumulative error was minimized for values of Az/Q of about 0.14 s/cm2, which in eq 6 corresponds to values of At of about 0.6 s. Furthermore, it appeared that the cumulative error would become relatively less significant as the ratio of the length of the column to the volumetric flow rate becomes larger. For eq 1 to 5 the parameters e, rK, CK, D A , D B , and ED and the function F were evaluated as previously described. Regulation of the volumetric flow rate Q was accomplished by manipulation of the appropriate stopcocks shown in Figure 3. The value of Deffwas estimated as the average of the diffusion coefficients for the NH4+and H3+NCH2CH2NH3+ ions a t infinite dilution and 28 "C. The diffusion coefficient for the NH4+ ion was readily obtained from the literature (Newman, 1973). The diffusion coefficient for the H3+NCH2CH2NH3+ ion was assumed approximately equal to that for the ethyl ammonium ion, for which the diffusion coefficient was estimated from its limiting equivalent conductance (Dean, 1973) and the NernstEinstein equation (Newman, 1973). The value obtained for Deffwas 1.7 X cmz/s. An initial estimate for the characteristic boundary layer thickness 6 was obtained from the literature (Helfferich, 1962a). For a given value of Deff,the shape of the elongated portion of the curve occurring at small values of X A ( t 9 ) is almost entirely determined by the value of 6, while t e shape of the remaining portion of the breakthrough curve is relatively unaffected. The initially estimated values of the boundary layer thickness were improved by selecting values of 6 which produced approximately the correct form of the leading edge of the breakthrough curves. Furthermore, the values determined for 6 were reasonably insensitive to variations in the values used for Az/Q and corresponding At. For superficial mass velocities of 1.6 and 0.83 g/cm2 s, the values obtained for 6 were 0.0035 and 0.0065 cm, respectively. While this procedure for determining a boundary layer thickness involves some lack of precision, the boundary layer thickness itself is of a somewhat uncertain nature. For m uniform particles, each having surface area SK, eq 5 can be written as dwA(Zl,t
dt
where Deff = an effective solution-phase diffusion coefficient for species A, S K = the surface of the Kth particle, and 6 = a characteristic boundary layer thickness. Similar remarks apply to the term w A ( Z L + l , t + At).
(6)
At) = mSKCKD Deff -x 6 + 6t)l (7) [ X A ( Z L , ~ At) - X M S ( ~
which has the form of the much used linear-driving-force relationships. The term De,/& can be considered as a mass transfer coefficient defined by eq 7. The values of DeH/6 determined for the ion-exchange system studied in this cm/s for the mass and 2.6 X research were 4.9 X
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
velocities of 1.6 and 0.83 g/cm2 s, respectively, and were of the same order of magnitude as conventionally defined mass transfer coefficients which might be predicted from available literature (Sherwood et al., 1975). Nomenclature CK =-ion exchange capacity of the resin DA, DB = effective diffusion coefficients for ions A and B, respectively, in the resin D,ff = an effective solution-phase diffusion coefficient for species A E D = a dispersion coefficient accounting for both molecular diffusion and macroscopic fluid-phase dispersion F[] = the nonlinear functions which describe the ion-exchange equilibrium isotherms m = total number of resin particles in A V NT = the total solution normality Q = volumetric flow rate of the solution r K = radius of Kth resin particle SCS= cross-sectional area of resin bed S K = surface of Kth particle t = time V = total interior volume of the packed portion of column V, = volume of the Kth particle WA(t)= the total amount of species A in the resin particles in the volume element at time t X A = equivalent fraction of species A in a solution at equilibrium with the resin X A = resin phase concentration in equilibrium with the solution XA*(t) = volume-average solution-phase equivalent function of species A in the volume element at time t
317
X A ( Z L ,=~ average ) equivalent fraction of species A over the imaginary cross-sectional surfaces at time t and coordinate - ZL
X, = local equivalent fraction of species A in the Kth particle X M =~ an assumed uniform fluid-side equivalent fraction of species A at the surface of the Kth particle Y,a, Ye = valences of ions A and B, respectively z = axial coordinate 6 = characteristic boundary layer thickness t = solid fraction of the bed
Literature Cited Colwell, C. J., Dranoff, J. S., Ind. Eng. Chem. Fundam., 8 , 193 (1969). Colwell, C. J., Dranoff, J. S., Ind. Eng. Chem., I O , 65 (1971). Cooper, R. S . , Ind. Eng. Chem. Fundam., 4, 309 (1965). Dean, J. A., Ed., "Langes Handbook", 11th ed, McGraw-Hill, New York, 1973. Erickson, K. L., Dissertation, The University of Texas, Austin, 1977. Fleck, R. D., Jr., Kirwan, D. J., Hall, K. R., Ind. Eng. Chem. Fundam., 12, 95 (1973). Helfferich, F., "Ion Exchange", p 253, McGraw-Hill, New York, 1962a. Helfferich, F., "Ion Exchange", pp 263, 273, McGraw-Hill, New York, 1962b. Helfferich, F., ':!on Exchange", pp 449-452, McGraw-Hill, New York, 1962c. Helfferich, F., Ion Exchange, Chapter 9, McGraw-Hill, New York, 1962d. Lantz, R. E., SOC. Pet. Eng. J., 315 (Sept 1971). Newman, J. S.,"ElectrochemicalSystems", pp 228-231, RenticeHall, Englewood Cliffs, N.J., 1973. Rosen, J. E., J . Chem. Phys., 20, 387 (1952). Rosen, J. E.. Ind. Eng. Chem., 46, 1590 (1954). Sherwood, T. K., Pigford, R . L., Wilke, C. R., "Mass Transfer", Chapter 10, McGraw-Hill, New York, 1975. Thomas, H., J. Am. Chem. Soc., 66, 1664 (1944). Tien, C., Thcdos, G., AIChE J . , 5, 373 (1959). Vermeulen, T., Adv. Chem. Eng., 2, 147 (1958). Vermeulen, T., Hiester, N. K., Chem. Eng. frog. Sym. Ser., 55(24), 61 (1959).
Received for review April 17, 1978 Accepted July 27, 1979
Estimating the Jet Penetration Depth of Multiple Vertical Grid Jets Wen-Chlng Yang' and Dale L. Keairns Research and Development Center, Westinghouse Electric Corporation, Pittsburgh, Pennsylvania 75235
Literature data on jet penetration of a fluidized bed were reviewed. Available correlations for determining the jet penetration depth of a single jet were compared with the jet penetration data for multiple jets. A new correlation based on the two-phase Froude number was proposed that correlated the data to within f40%, including both single-jet and multiple-jet data obtained in both two-dimensional and three-dimensional beds.
Introduction The major ambiguity in studying the jetting phenomenon in fluidized beds is the lack of consensus in what constitutes a jet. Based on X-ray observation in a three-dimensional fluidized bed, Rowe et al. (1978) reported that the gas issuing from an orifice entered the fluidized bed primarily in the form of bubbles. They observed a permanent jet over the orifice, however, when the particles were not fluidized locally. A permanent jet was defined as a flamelike jet with a permanent void. A permanent jet was also observed and reported in the literature when a two-dimensional unit was employed or when the jet was located close to the vessel wall (Zenz, 1968; Merry, 1976; Wen, 1977). Some authors (Markhevka et al., 1971; Shakhova and Minayev, 1972) called the discharge from an orifice a "jet", even when the jet was truncated periodically into bubbles. On the basis of this concept of periodic jet, Merry (1975) proposed a correlation for estimating the jet penetration depth. The jet penetration depth was defined as the height a t which the jet 0019-7874/79/1018-0317$01 .OO/O
degenerated into a gas bubble. In a study on a recirculating fluidized bed with a draft tube, in a semicircular unit using bed materials ranging from hollow epoxy spheres (p, = 0.21 g/cm3) to sand (p, = 2.65 g/cm3), Yang and Keairns (197813) observed that a permanent jet could not be obtained above the tube supplying gas to the draft tube when sand was the bed material. When lighter materials such as polyethylene beads ( p , = 0.91 g/cm3) or hollow epoxy spheres were used as the bed material, however, a permanent jet could be readily established. Although the jet was stabilized against the flat wall of the semicircular vessel, different behavior was observed with different bed materials. It might be possible that the gas issuing from an orifice can be in the form of bubbles, a pulsating jet, (a periodic jet), or a permanent jet, depending on the relative properties of the gas and the bed materials and the operating conditions. Intuitively, it seems logical to expect the existence of a permanent jet if the gas density approaches that of the fluidized bed. At the extreme cases permanent jets have been observed with injection of gas 0 1979 American Chemical Society
