Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 1188-1191
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Jensen, N. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 228. Jensen, N.; Fisher, D. G.; Shah, S. L. AIChf J., in press. Kim, Y. S.; Mc Avoy, T. J. Paper present at the ACC Conference, Arlington, VA, 1982. King, J. "Separation Processes"; Mc Graw-Hill: New York, 1980; p 393. Luyben, W. L. AIChEJ. 1970, 16, 198. Luyben, W. L. Ind. Eng. Chem. Fundam. 1975, 1 4 , 321. Mc Avoy, T. J. AIChf J. 1981, 2 7 , 613. Mc Avoy, T. J. Ind. Eng. Chem. Process Des, Dev. 1983a, 22, 42. Mc Avoy, T. J. "Interaction Analysis. Prlnciples and Applications"; Instrument Society of America: Research Triangle Park, NC, 1983b; Chapter 5, Monograph 6. Mc Avov. T. J. Ind. Eno. Chem. Process Des. Dev. 1985, 24, 229. Marinokalarraga, M.; &t Avoy, T. J.; Marlin, T. E. Paper presented at the ACC Conference, San Diego, CA, June 1984. Marindialarraga, M.; Mc Avoy, T. J.; Marlin, T. E. Paper presented at the ACC Conference, 1985.
Marino-Galarraga. M.; Mc Avoy, T. J.; Marlin, T. E. Ind. Eng. Chem. Process Des. Dev., submitted, 1985b. Marino-Galarraga, M.; Mc Avoy, T. J.; Marlin, T. E. Ind. Eng. Chem. Process Des. Dev., submitted, 1 9 8 5 ~ . Shinskey, F. G. "Process Control Systems": Mc Graw Hill: New York, 1979; p 207. Stanley, G. M. S. Thesis, University of Maryland, College Park, MD, 1985. Stanley, G.; Mc Avoy, T. J. Ind. f n g . Chem. Fundam., in press. Thurston, C. Hydrocarbon Process. 1981, 6 0 , 125. Tung, L. S.;Edgar, T. F. AIChE J. 1981, 2 7 , 690. Welschedel. K.: Mc Avoy, T. J. Ind. Eng. Chem. Fundam. 1980, 19, 379. Witcher, M.; Mc Avoy, T. J. ISA Trans. 1977, 16, 35.
Received for review July 19, 1984 Revised manuscript received March 4, 1985 Accepted March 25, 1985
Rapid Calculations of Multicomponent Adsorption Equilibria from Pure Isotherm Data James A. O'Brlen and Alan L. Myers' Chemical Englne8rlng Department, University of Pennsylvania, Philadelphia, Pennsylvania 19 I04
The method of ideal adsorbed solutions (IAS) is reformulated by means of a new adsorption isotherm equation, to allow fast computation of multicomponent adsorption equilibria. In this formulation, the I A S method for an N-component mixture is a structured system of N simultaneous, nonlinear algebraic equations. These are solved by the Newton-Raphson technique to illustrate the speed and simplicity of the procedure. The method permits the specification of the composition of either phase at equilibrium.
The prediction of the mixed-gas adsorption equilibria from pure-component adsorption isotherms has long been a goal of research on the thermodynamics of adsorption. One approach is the method of the ideal adsorbed solutions (IAS), which was developed 20 years ago (Myers and Prausnitz, 1965). Many other theories have been proposed (Lewis et al., 1950; Grant and Manes, 1966; Suwanayuen and Danner, 1980). The attractive features of IAS are that (a) it requires no mixture data and (b) it is an application of solution thermodynamics to the adsorption problem, so that it is independent of the actual model of physical adsorption. The basic equation of IAS is analogous to Raoult's law for vapor-liquid equilibrium Pyi = Pio(II)xi
(for all i)
(1)
where Pio is the pressure which, for the adsorption of pure component "in, yields the same spreading pressure, n, as that of the mixture. The spreading pressure is defined by the Gibbs adsorption isotherm
The function nio(Pio) is the experimental adsorption isotherm of pure i. The total amount adsorbed is determined by the requirement of zero area change upon mixing at constant II and T, so that (3)
In principle, this is a straightforward calculation. For an N-component adsorption equilibrium problem, there are N + 1 degrees of freedom which are normally specified 0196-4305/85/1124-1188$01.50/0
in terms of pressure ( P ) ,temperature ( T ) , and N - 1 independent vapor composition variables (yJ. There are 2N + 1 unknowns: (xi),(Pio),and II. Equations 1 and 2 plus the constraint Exi = 1 provide an equal number of equations, so that a unique solution exists. Equation 3 may be solved separately for nt because it is the only one containing this variable. In practice, however, there are two difficulties. Firstly, the value of which determines Pio is the same for all components, so that it is the inverse of eq 2 which is needed. The solution for Pio must be found by trial and error or at least by some iterative scheme. Secondly, the integral in eq 2 must be evaluated numerically, since there is currently no pure-component isotherm for n?(Pio)which both correlates experimental data and permits analytical integration. For example, the Langmuir equation can be integrated analytically but does not fit experimental data. The Toth equation (Toth, 1962) and the LUD equation (Myers, 1984) fit experimental data for type I adsorption isotherms but cannot be integrated analytically. For these reasons, IAS is too slow for equilibrium calculations in models for the dynamics of adsorption in packed columns. Theories for calculating breakthrough curves require the numerical solution of differential equations in the domain of space and time, so that subroutines for local equilibrium must be as fast as possible. Many of the models for column dynamics in use at the present time contain primitive equilibrium isotherms such as the Langmuir equation, or even linear adsorption isotherms, in order to simplify the solution. Our intention in this paper is to present a method for significantly shortening the computation time required for IAS. The method outlined below gives rise to a system of N simultaneous, nonlinear algebraic equations for an 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
N-component system. Such an explicit statement of the problem is very attractive for solving column design problems. New Isotherm Equation It is apparent from the introduction that IAS theory is essentially independent of the detailed functional form of the pure-component isotherms, nio(Pio).Thus, if an isotherm can be found which represents the data and is analytically integrable according to eq 2, then the solution can be simplified considerably. In a recent paper (OBrien and Myers, 1984), we have derived such an isotherm by considering a series expansion of the usual adsorption integral equation in terms of the central moments of the adsorption energy distribution. The Langmuir equation was used as the local isotherm. Truncation after the second term of the series yields the equation
where
= KiPiO
(5) Note that the first term in eq 4 is the Langmuir isotherm, while the second is the first-order correction for the effects of adsorbent heterogeneity. u t is the variance of the adsorption energy distribution for pure component i. After substitution of eq 4 into eq 2, the spreading pressure may be written explicitly as qi
where the gi(Fj) are given by
(i = 1, 2,
gi(Fj) = f i ( t i ) - f i + l ( t i + l )
..a,
(N - 1)) (14)
and N gN(?>
=
c(KiPyi/qi) it1
-1
(15)
Equation 13 is just a system of nonlinear algebraic equations in the variables ql, q2, ...,qN, and may be solved by any of the usual techniques. For the purpose of illustration, we will outline the solution procedure by using the Newton-Raphson method (Carnahan et al., 1969). Once the solution of eq 13 is known, the mole fractions { x i )and the total adsorbate loading may be calculated from eq 9 and 3, respectively. Thus, the entire equilibrium problem is solved. The N-variable Newton-Raphson method may be expressed as Fjk+l = f i k + $k (16) where bk, the correction vector for the kth iteration, is given by the solution to the system of linear equations Qk$k
and
= -i(Fjk)
(17)
is the Jacobian matrix defined by
For our problem, the only non-zero elements of given by the relationships For each component, the parameter set {mi,Ki,ui} is obtained from the experimental data. In the next section, it is shown how the explicit expression for the spreading pressure in eq 6 is used to simplify IAS calculations. FASTIAS Procedure Consider an N-component adsorption system (N is the number of adsorbates and does not include the solid adsorbent). At equilibrium, equality of spreading pressures IIi* must be imposed. Grouping these in pairs yields N - 1 independent equations. For simplicity and scaling purposes, in the following we define f i ( q i ) E ni*(qi)
The pairwise grouping may be written fi(qi) = fi+l(qi+J (i = 1, 2, ..., (N- 1))
(7) (8)
(for all i) The mole fraction constraint equation is
4i,i+lk= -fi+l’(ti+lk)
CKiPYi/tli = 1
i=l
(10)
The advantage of this formulation is that the problem has been reduced to a set of N equations (eq 8 and 10) in N unknowns ( t i ) . Now, when a vector of pressures is defined, one for each component
fi E h l , 72, *.., q N l t and the vector function of .fi is written
8 I [gli g2, *.*,
gNlt we can state the IAS problem in the form ia6) = 0
(11)
f [ ( q ) (E
= qik/2
(23)
whenever, at the kth iteration
+ a t IO
(24)
thereby precluding the possibility of any of the becoming zero or negative. It should be noted that if this situation occurs often, the second-orderconvergence property of the Newton-Raphson method will be lost. However, we include it to ensure the robustness of our algorithm (see flow chart in Figure 1). The convergence criterion which we used was N i=l
(13)
(20)
There is a numerical subtlety associated with the use of eq 16 in the present application. Since eq 21 and 22 contain singularities at 0 and -1, respectively, possible difficulties in the algorithm are avoided if we put
CIst/qik+’(
