-6 = shear dimensionless general variable stress at the impingement plate 7
=
Literature Cited Black, J., Hardisty, H., Sixth Thermodynamicsand Fluid Mechanics Convention, C . 73/76, 99, University of Durham, Apr 6, 1976. Burgess, B. W., Chapman, S.M., Seto. W., Pulp Paper Mag. Can., 7 3 (1l), 314 (1972a). Burgess, B. W.. et al., Pulp Paper Mag. Can., 73 (1l ) , 323 (1972b). Gauntner, J. W., Livingood, J. N. B., Hrycak, P., NASA TN D-5652 (1969). Gosman, A. D., Pun, W. M., Runchai, A. K.,Spaiding, D. B.,Wolfshtein, M., "Heat and Mass Transfer in RecirculatingFlows". Academic Press, New York, N.Y., 1969. Hardisty, H., Report No. 226 and 227, University of Bath, School of Engineering, 1973. Keebie, T. S., Aero. Res. Council, Department of Defence, Australia, ARL/ME308 Aug 1969.
Marple, V. A., Lui, B. V. H.. Whitby, K. T., J. Fluids Eng., 96, 394 (1974). Mujumdar, A. S.,Douglas, W. J. M., presented at TAPPI Meeting, New Orleans, La., Oct 3, 1972. Runchal, A. K., Spaiding, 0. B., Wolfshtein, M., High Speed Comp. Fluid Dyn., Phys. FluidSuppl., 2, 21 (1969). Scholtz, H. T., Trass, 0.. AlChEJ., 16, 90 (1970). Saad, N. R., Master of Engineering Thesis, McGili University, Montreal, 1976. van Heiningen, A. R. P., Mujumdar, A. S.,Douglas, W. J. M., J. Heat Transfer (1976a). van Heiningen, A. R. P., Mujumdar, A. S., Douglas, W. J. M., A.S.M.E. Winter Annual Meeting, New York, N.Y., Dec 5 , (1976b). van Heiningen, A. R. P., Mujumdar, A. S.,Douglas, W. J. M., Lett Heat Mass Transfer, (1976~).
Received f o r reuieu; November 15, 1976 Accepted November 15,1976
Numerical Solution of Surface Controlled Fixed-Bed Adsorption D. U. von Rosenberg;'
R. P. Chambers, and G. A. Swan
Chemical Engineering Depadrnent, Tulane University, New Orleans, Louisiana
A simple numerical method has been developed for the solution of fixed bed adsorption which can be described by equations of the form -B1 [u(dc/dx) 4- (dc/dt)] = B2(dw/dt) = f(c,w), where the function f may take various forms depending upon the mechanism controlling the adsorption process. Breakthrough curves calculated by this method show very good agreement with experimental curves for two different systems. The computation time can be decreased by a modification which allows the time increment to be increased for a fixed length increment.
Introduction The adsorption of one component from a moving fluid stream by a fixed solid in a packed bed is a common operation. In many cases transfer in the direction of flow by diffusion or dispersion is negligible with respect to convective transfer, and composition does not vary normal to the flow. Furthrmore, the adsorption process can often be described by a mass transfer or chemical reaction relationship, and variation of loading within the solid particles can be neglected. I t is this case which is treated. The method of solution developed was applied to two distinct physical problems. The first problem was the adsorption of carbon dioxide from a stream of nitrogen by a molecular sieve. The second problem treated was the adsorption of heavy metal compounds from waste water by a selective adsorbent. Governing Equations The governing equations for this model consist of two partial differential equations obtained from material balances on the fluid and solid phases. In addition, an equation describing the adsorption equilibrium or the reaction rate is needed. This equation is usually nonlinear. For the general case, the governing equations can be written as
(
-B1 u - + ax ac
dW
at
=Bzz
aw B:! - = f(c,w) at
where c is the concentration of sorbate in the fluid, w is the 1 Chemical Engineering Department, U n i v e r s i t y of Tulsa, Tulsa, Okla. 74104.
154
Ind. Eng. Chern., Fundam., Vol. 16, No. 1, 1977
concentration of sorbate on the solid, x is the length dimension in the direction of flow, t is time, u is the fluid velocity, B 1 is the capacity of the fluid for sorbate, Bz is the capacity of the solid for sorbate, and f(c,w) is a function which describes the rate of adsorption. It is this set of equations for which a simple numerical solution was developed.
Previous Methods A number of methods for the design of fixed-bed adsorbers have been developed. Vermeulen et al. (1973) developed a graphical procedure which has been used extensively for design purposes but which was not particularly adaptable to the analysis of the experimental data from the heavy metal removal. Acrivos (1956) developed a method of numerically solving the equations along the characteristics. One limitation of this method is that the sizes of the space increment and of the time increment cannot be changed once the solution has been started. The nature of adsorption problems is that small space increments must be used throughout the solution, but, as the solution proceeds, the time derivatives decrease in magnitude. Thus, it is desirable to use a method of solution in which the size of the time increments can be increased as the solution progresses. Method of Solution The numerical method to be described makes use of two types of finite difference equations. The equation used in the beginning is based on the centered difference method which has been shown to be unconditionally stable by Wendroff (1960) and which is second order correct in both length and time. The method is based on the characteristics of the system in that the ratio of the space increment to the time increment
0
x.0 n+l
n
0
o
a
0
~
i
I
0
0
0
i* I i +I
2
I
.1 c
rn n t l
0
7
o
index
Wtdex
C
Figure 1. Finite difference grid.
0
0
i
i++
w
o n i+l
i+l
Figure 2. Grid for complete equations
is set equal to the fluid velocity. The use of this value for the ratio is not necessary for stability, but it does significantly decrease truncation error. After the solution has been started with the centered equation, a backward difference equation is used for later times. Ths equation is also unconditionally stable and is second order correct in length and first order correct in time. However, there is no restriction on the relative sizes of the two increments. The grid points for the numerical solution are arranged as shown in Figure 1. The values of concentration in the fluid phase, c , are determined on the square grid points, and those of the solid phase concentration, w , are determined on the round grid points. Note that c is determined a t each end of the column and a t points equally spaced a t intervals of Ax between the ends. The concentration w is determined a t points halfway between those at which c is determined. The indexing system for length and time is shown in Figure 1.A t any given time step, all the values of c and w are known a t the old time level (denoted by n ) ,with the initial conditions supplying the values needed to start the solution. Also, the values of c are known a t x = 0 from the boundary conditions. As a result of this situation, the values of c and w can be computed from the finite difference analogs to eq 1and 2 for the first set of points downstream of x = 0. The same procedure can then be used to compute values of c and w at the next set of points, and this procedure can be repeated until the exit of the packed bed is reached. The entire procedure can then be started again a t the bed entrance for a new time step.
Procedure for the Complete Equations In Figure 2 is shown a set of points which are used in the solution of the complete equations. The values of c and w are unknown at the shaded points as shown in Figure 2. The finite difference equations are written about the center of the grid a t the point denoted by In order to minimize truncation error, the ratio of the increment sizes A/& is set equal to the fluid velocity, u . With this condition, the finite difference analog to eq 1 is
+.
-RI(c,+I
n+i
- ci,n) = B?(wi+1. n + l - w i + l . n )
0 0 I
+ 0
.it2 i
ntl
~n
C
icl
I+ I
W
Figure 3. Grid for abbreviated equations.
false position. Thus, the two unknown values of the concentrations in a set of points can be readily computed. There is one limitation to this method, and that is that the size of the time increment, A t , is limited to Axlu. Note that u is the velocity of the fluid, not the velocity with which the band of adsorbed species moves down the bed. For many cases, this may not limit the speed of the solution unduly, but in some cases, the computing time may be longer than can be tolerated. Thus, a method which does not have this limitation is desired.
Procedure for Abbreviated Equations For many cases, the concentration of the sorbate in the fluid phase does not change very rapidly with time at a given point. Thus, the time derivative of c is much smaller than the spatial derivative of c, so the time derivative can be dropped from the equation. After the solution has been started using the complete equations shown above, the abbreviated equations can be used as described below. The grid used for the abbreviated equations is shown in Figure 3. The finite difference analogs to eq 1 and 2 without the time derivative of c are written about the point a t which w is determined a t the new time level ( n 1).This point is in Figure 3. The analog to eq 1 is now the indicated by backward equation and is given as
+
+
(3)
From this equation, ~ , + l , can ~ + be ~ obtained in terms of w I +1, n + l . The centered difference equation is also written for eq 2 and is
where
The analog of eq 5 for c,+ I / > , n+ i12 is the diagonal analog. The use of this analog is an important part of this method. When the value of c,+1, n + l from eq 3 is substituted into eq 4,one single equation in w,+1, n + l is obtained. This equation is nearly always nonlinear, but it can be easily solved by the method of
As in the first method, ~ i + ~ , is ~ +obtained l in terms of w i + l , n + l from eq 7 . The analog to eq 2 is written a t the new time level (n 1)and is
+
where
As in the previous procedure, eq 8 is obtained in terms of w i + l , n + l , and it is solved by some procedure appropriate to its nonlinear character. For this method, the size of the time increment is not limited by the size of the space increment, and the time increment can be made as large as desired. I t should be noted here that the time increment must be short enough to describe the adsorption process. However, this restriction is not a result of Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
155
;I 20
40
60
80
IO0
TIME, MIN
Figure 4. CO? adsorption curve, run no. 5: -, experimental; 0,comp. Ka = 3.5 X IO+. Table I. Comparison of Mass Transfer Coefficients Run no. 2
Predicted ka 6.50 x 10V
3
4.15
5
3.23 6.50
6 7
4.15
Exptl ka 5.0 X 3.5 3.5 5.0 4.0
the numerical method but rather of the nature of the process being described. The solution can be accelerated by computing the concentrations only in that portion of the bed in which the dimensionless concentrations are significantly less than 1and greater than 0. Thus, a very rapid, efficient method of numerically solving packed bed adsorption problems has been developed. Reversible Gas Phase Adsorption The first problem for which this method was utilized was the adsorption on Linde Type 5A molecular sieve pellets of a small amount of carbon dioxide from a stream of nitrogen. It has been determined (Littman et al., 1968) that the mass transfer of the COz from the gas stream to the surface of the pellets was the controlling mechanism. A mathematical model which included intraparticle diffusion, heat effects, and heat transfer by convection and conduction was solved by numerical procedures (Schof, 1971). The results of this study confirmed that mass transfer is controlling for this system. The equations which describe this process are
-E(
u-+=ka(p-p*) aaxp aatp ) aw M -= - h a ( p - p * ) at
Ph
where p is the partial pressure of COz in the gas phase, w is the loading of COn on the solid, and p* is the partial pressure of Conin equilibrium with w .The relationship between p* and w is given by p* = awb
Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
20
60
40
Time
80
(Min)
Figure 5 . Mercuric chloride breakthrough curve.
was taken as the correct value. One such run is shown in Figure 4. For five of these runs values of the mass transfer coefficient were also predicted from a relation obtained by Eagleton and Bliss (1953) for the drying of air with silica gel. The particle size, mass velocity, and temperature for the experiments used to obtain this relation were in the same range as those for the C 0 2 experiments. The mass transfer coefficients obtained from the correlation were then corrected for the difference in diffusivities of the two systems by a method of Leva (1953). A comparison of the values predicted by the Eagleton-Bliss correlation and the values obtained from the simulation of the C 0 2 experiments is shown in Table I. The good agreement is seen. Runs 2 and 6 were for the same flow rate in two different length beds, as were runs 3 and 7. The experimental values of ka are exactly the same for runs 2 and 6 and in good agreement for the other pair. Irreversible Liquid Phase Adsorption The second system studied was the irreversible adsorption of heavy metal compounds from waste water streams by selective adsorbents. The controlling step of the adsorption is a surface reaction, the rate of which is a function of the fraction of empty sites. A complete description of the experiments used to determine the controlling step of this system may be found elsewhere (Swan e t al., 1976). The equations which describe this system are
(12)
Thus, this case fits the model of eq 1and 2. The other terms in the equations are parameters which describe the physical situation. The parameter ha is the coefficient for mass transfer of C 0 2 between the gas phase and the solid. The values of all the physical parameters except for the mass transfer coefficient were obtained from Littman et al. (1968). Also in this report were breakthrough curves for a number of isothermal adsorption experiments for this system. In order to evaluate these experiments, computer runs were made with various values of the mass transfer coefficient. The value of ka which gave the best match with the experiment 156
0
where c is the concentration of compounds in the liquid and q is the loading of compounds on the solids. The function F(q) is the fraction of unpoisoned sites available and Q-q F(q) = Q + 3vQ2(Q - q)2/:1[Ql/:3
- ( Q - q)1/3]
(15)
In this relation, Q is the total capacity of the adsorbent for
sorbate; the other terms in the equations are parameters which describe the physical situation. These physical parameters were obtained from the characteristics of the adsorbent and from laboratory studies. One set of batch experiments was made to determine the rate constant, k , and the effectiveness factor, 1.Another series of experiments was then made in a laboratory column to compare the experimentally determined breakthrough curves with those obtained by a numerical solution of the model. A comparison of the two curves for one experiment is shown in Figure 5.
q = solid phase metal concentration Q = saturated solid phase metal concentration t = time u = fluid velocity w = loading of sorbate on solid x = length At = time increment Ax = length increment t = void fraction 7 = effectiveness factor p = Thiele modulus p = gas density pb = bulk density of solid
Conclusion A simple numerical method has been developed for the solution of packed bed adsorption problems and has been successfully applied to two physical problems.
Subscripts i = lengthindex n = timeindex
Nomenclature a,b = constants in equilibrium relation for GO2 on Linde 5A molecular sieve R 1 = capacity of fluid for sorbate = capacity of solid for sorbate c = concentration of sorbate in fluid k = reaction rate constant for heavy metal compounds ha = mass transfer coefficient for COz M = molecular weight of COz p = partial pressure of COS p * = partial pressure of COS in equilibrium with solid loading of w p t = total pressure
Literature Cited Acrivos. A., lnd. Eng. Chem., 48, 703 (1956). Eagleton, L. C., Bliss, H., Chem. Eng. Prog., 49, 543 (1953). Leva, M.. "Tower Packings and Packed Tower Design", United States Stoneware co., 1953. Littman, J., et al., "A Study on the Properties of Solid Adsorbents for the Design of Regenerative CO? Removal Systems", Final Report on Contract NAS93541, Feb 1968. Schof, D.. Ph.D. Thesis, Tulane University, 1971. Swan. G. A., McElrath, K. 0.. Baricos, W. H., Cohen, W., Chambers, R. P., AlChE Symp. Ser., 71, No. 152, 96 (1976). Vermeulen, T., Hiester, N. K.. Klein, G., "Perry's Chemical Engineers Handbook", 5th ed, Sec. 16, McGraw-Hill, New York, N.Y., 1973. Wendroff, B., J. SOC.lnd. Appl. Math., 8, 549 (1960).
Received f o r reuiew August 12, 1976 Accepted November 22,1976
Freezing Point Depression; Binary Alkali Halides Alfred Carlson, Jack B. Howard, and Robert C. Reid' Department of Chemical Engineering, Massachuseffs lnsfifute of Technology, Cambridge, Massachusetts 02 739
Employing regular solution theory and experimental enthalpies of mixing, freezing points of molten binary alkali halides were estimated and good agreement was obtained with experimental data.
The use of molten salt mixtures as energy storage media is under active study by many groups; molten salts are also of value as heat transfer fluids in high temperature reactions. In this paper we illustrate a method to estimate the melting points (including the eutectic) of simple binary mixtures of alkali halides which do not form solid solutions. Such mixtures are not ideal, but it is often a good approximation to assume regular solution behavior; Le., the excess entropy of mixing is small. This assumption is discussed by Lumsden (1966). In the present study, only binaries with a common anion have been considered; the technique should, however, be applicable to other simple molten-salt systems.
Thermodynamics Consider a binary liquid mixture in equilibrium with a solid phase consisting of one of the pure components. In the binary alkali halide case of interest, the common anion is Y and we denote the smaller cation by S and the larger by L. Assume the solid phase is, for example, SY. Then
or
Rsyl - TSsyl = HSyh- Tssy6
(2)
By the use of simple thermodynamics (outlined in the Appendix), one obtains the equation -
- E s y I ( T ) I ASS^',^^ In XsuI = RT R - AHsuI(Tsu,) 1 R T with
[
$SV =
1
]
Tsyl
-$su
RT
AC,,,,[(T - T s y J - T In ( T / T s u , ) ]
(3)
(4)
Equation 3 is rigorous except for the assumption that X,,, = C,,,,,I - CpsYsis not a function of temperature. As Prausnitz (1969) suggests, $sy may usually be neglected in comparison with the other terms on the right-hand side of eq 3. We also assume that the partial molal enthalpy of mixing, L W S ~ is I , not a strong function of temperature. Finally, we assume that the excess entropy of mixing in binary alkali halides is negliInd. Ens. Chem., Fundam., Vol. 16, No. 1, 1977
157
