Ind. Eng. Chem. Res. 1987,26, 2042-2047
2042
Magnussen, T.; Rasmussen, P.; Fredenslund, Aa. Ind. Eng. Chem. Process Des. Deu. 1981, 20, 331. Skjold-Jorgensen, S.; Kolbe, B.; Gmehling, J.; Rasmussen, P. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 714. Ssrensen, J. M. Ph.D. Thesis, The Technical University of Denmark, 1980.
S~rensen,J. M.; Magnussen, T.; Rasmussen, P.; Fredenslund, Aa. Fluid Phase Equilib. 1979, 2, 297. Varhegyi, G.; Eon, C. H. Ind. Eng. Chem. Fundam. 1977, 16, 2.
Received for review April 18, 1986 Accepted July 7 , 1987
Further Work on Multicomponent Adsorption Equilibria Calculations Based on the Ideal Adsorbed Solution Theory Hee Moon* and Chi Tien Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13244-1240
It was found that the procedures developed previously for calculating multicomponent adsorption equilibria can be readily extended to the case where the pure component isotherm data are represented by the Langmuir expression with a first-order correction term. In contrast to the recent published work of O’Brien and Myers which considered some aspects of the same problem, the present method requires, for any N-component mixture, the solution of only one or two nonlinear algebraic equations instead of the N equations necessary with the O’Brien-Myers method. The saving in computation time becomes significant as the number (N) increases. O’Brien and Myers (1985) recently suggested a method for calculating multicomponent gas adsorption equilibria from pure component data based on the ideal adsorbed solution (IAS) theory. Their starting point is that the single-component isotherm data can be expressed by the Langmuir expression with a first-order correction, or
where the superscript o denotes the pure component state and the subscript i, the ith component. np is therefore the amount of the ith component adsorbed per unit mass of adsorbent, and m, is the maximum (saturation) value of n:. qi is defined as qi =
Kiplo
(2)
where Pp is the partial pressure of the ith component and Ki, a constant. Thus, there are three parameters (Ki, mi, and ai) to be specified when eq 1 is used to represent the pure component isotherm data. When an N-component gas mixture is in equilibrium with a particular adsorbent, the equilibrium state is defined by the concentrations in the adsorbed phase (nL,i = 1,2, ..., N) and the concentrations in the gas phase (P, = Py,, i = 1, 2, ...,N>. O’Brien and Myers developed their method (known as FASTIAS) to calculate the concentrations of one phase given the concentration of the other phase. The two cases (calculating ni with given Pi and vice versa) the researchers considered, however, do not cover all the situations of interest in adsorption calculation. As Wang and Tien (1982) and Larsen and Tien (1984) pointed out in their work on fixed-bed and batch adsorption processes, if the intraparticle diffusion is described by the homogeneous diffusion model or the lumped parameter model (also known as the linear driving force model), one
* Present
address: Department of Chemical Engineering, Chonnam National University, Kwangju 505, Korea.
is required to calculate the concentrations of both phases under the condition
Pi + Aini = Bi for i = 1, 2, ..., N (3) where Ai and Bi are constants. It should also be mentioned that several earlier studies concerned the calculation of adsorption equilibria based on the IAS theory. Tien and co-workers (Wang and Tien, 1982; Larsen and Tien, 1984; Tien, 1986) have shown that for liquid-phase adsorption, if one expresses the singlecomponent isotherm data either by the Freundlich expression or by the Freundlich expression on a piecewise basis, the required calculation is reduced to solving two nonlinear algebraic equations for any multicomponent systems. In other words, an increase in the number of components does not have a significant effect on computation. The purposes of the present work are 2-fold. First, the so-called FASTIAS procedure developed by O’Brien and Myers is extended to cover the situation described by eq 3. Second, to simplify computation, the FASTIAS procedure is reformulated according to the method developed previously by Tien and co-workers. Sample calculations illustrate how this reformulation simplifies computation.
FASTIAS Procedure The IAS theory, upon which the FASTIAS procedure is based, defines the adsorption equilibrium between a gas mixture of N components and a specific adsorbent by a system of equations: Pyi = Pprixi i = 1, 2, ..., N (4)
nt = [&/n:]-l
(6)
1
ni = ntxi i = 1, 2, ..., N (7) where x iand yiare the mole fractions of the ith component
0888-5885/87/2626-2042$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2043 in the adsorbed phase and the solution phase, respectively. Pi" is the pressure of the ith component in pure component adsorption and gives the same spreading pressure obtained in the multicomponent cases. For multicomponent adsorption, at equilibrium, the spreading pressure, ri,is the same for all components. If eq 1is used to represent the pure component adsorption isotherm data, an explicit relationship between ?ri and Pi" (or qi as defined by eq 2) can be readily obtained thus:
yi = Pi/P
(19)
Pi" = pi/Ki (22) Case 2b. Assuming that the values of x i and n, are known, the values of yi and P need to be calculated. Equation 1 2 again provides ( N - 1)equations with vi,i = 1, 2, ...,N , as unknown. Furthermore, from eq 6 and 1, an additional relationship can be obtained, Le.,
As a shorthand, one may write
1 6+ -
The basic assumption used in formulating the IAS theory requires that
= fi+l(qi+J (10) Furthermore, by applying eq 4 and the identity Cixi= 1, one has fi(d
CKiPyi/qi = 1 i
(11)
The three most common cases encountered in adsorption equilibrium calculations are (1)given Pi = Pyi, find ni = n,xi; (2) given x i and n,, find Pi = Pyi;and (3) given Pi Aini = Bi, find Pi and ni. The procedures O'Brien and Myers developed to cover the first two cases are given below. Case 1. The equations used for calculation are derived from eq 9-11: gi = fi(qi) - fi+l(qi+l)= 0 for i = 1, 2, ..., ( N - 1) (12)
]
2 (1 + q J 3 From eq 12 and 23, and with the given values of xi and n,, the values of qi can be obtained. The fluid-phase concentration can be found from eq 19-22. Case 3. This case was not covered by O'Brien and Myers. The equilibrium concentrations of the solution and adsorbed phases are related according to eq 3. By definition, one may write K.P . = K.P. = n.x. 1Yl 1 1 1 1 or
+
gN
= C(KiPyi/qi) - 1 = 0 i
ni = [Cxi/ni"]-lxi
ni = n,xi (25) If eq 24 and 25 are substituted in eq 3, xi is found to be KiBi xi = qi + KiAint
(13)
Equations 12 and 13 constitute a system of N equations. With Pi given, the N unknowns, qi, i = 1, 2, ...,N , can be readily calculated (using, for example, the NewtonRaphson method). Once the values of qi are known, the adsorbed phase concentration, ni = n,xi, can be found to be x i = PYi/Pi"
Also
(14) (15)
1
where
Case 2a. Assuming that the mole fractions of the adsorbed phase, xi, and the total pressure of the solution phase, P, are given, one may determine the concentrations of the solution phase in the following manner. With xi's and P known, eq 12 together with
constitute a system of N equations with qi,i = 1, 2, ...,N , as unknown. The solution of this system of equations yields the values of vi. The fluid-phase concentration (or the partial pressure of the various components) can be found with
The requirement that
Cixi = 1 leads to the condition
Following the procedure used by Tien and co-workers (Wang and Tien, 1982; Larsen and Tien, 1984; Tien, 1986), let
s = [Exi/ni"]-l i
(28)
Substituting the expression of n? (from eq 1)and eq 26 in eq 28, one has KiBi -1= E (29) S i ui%i(l - si) mi[ -Y (Ti + KiAiS) 1+ Vi 2(1
+
+
From eq 27 and 29, one has
I
2044 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987
Equations 30,31, and 12 constitute a system of (N + 1) equations. With A,, B,, and the pure component isotherm constants (mL,u,, and K,) known, the ( N + 1)unknowns (namely, q,, i = 1,2, ..., N , and S) can be determined. The equilibrium concentrations of the solution and adsorbed phases can be calculated according to n, = S (32) KIB, XI = (33) 71 + K,A,S n, = n,x, (34) P, = B, - A,nI (35) P = CP, (36) 1
i
ex.(
z
- Ei) - 1
=o
(44)
Equation 44 can be used to determine the spreading pressure, II. Once II is known, vi can be found from eq 39. This calculation can be done iteratively by assuming Ei as constant for each iteration. Once the values of q i are found, n, and x i can be found readily. II can be determined from eq 44 in the following manis given as ner. The (k + 1)th iterated value of nk+1= nk + 6nk (45) and
= PI/P
(37) O'Brien and Myers considered cases 1and 2a, and the equations for these two cases given in this paper are identical with those they derived. On the other hand, they did not consider cases 2b and 3. In the first three instances, the calculation is the solution of N equations. For the last case, ( N + 1) equations are required. If the Newton-Raphson method is used to solve those equations, one must initially guess the unknown to ensure rapid convergence. This point will be discussed in later sections. Y1
1-C
"i'V Di KiPi
Modified Procedure The number of equations to be solved to estimate multicomponent adsorption equilibria according to the method outlined above equals or approximately equals the number of components of the gas mixture. To predict the dynamic behavior of adsorption processes with numerical methods, one must calculate the adsorption equilibrium at every time increment. Thus, for gaseous mixtures with even a moderately large number of components (say, up to lo), the required calculation may become excessive or impractical, and the need for alternate but simple methods of calculations is obvious. The modified procedure proposed here is derived from the following consideration: Equations 1 and 8 may be rewritten as
(39) where Hi and Ei are given as
The equations used to calculate the equilibrium concentrations are given below. Case 1. Given Pi = PY,, find ni = n,xi. The mole fraction of the absorbed phase ( x i ) can be expressed as Pi KiPi x.=-=(42) Pi" vi Substituting eq 39 in eq 42, one has
Summing up xi's for all the components, one has
where F and F' are given as
exp( zz m,
\..-1
)
The derivative F'(II) was obtained by assuming Ei as constant. Case 2. Given x i and n,, find Pi = Pyi. From eq 6 and 38, one has
Substituting eq 39 in eq 49, one has
Equation 50 is similar to eq 44 of case 1 and can be used to obtain the spreading pressure, II. The iterative calculation proceeds exactly the same as that used to obtain II from eq 44. F ( I I ) and F ' ( I I ) in this case are given as
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2045
Case 3. The equilibrium concentrations of the solution and adsorbed phases are related by eq 3. By definition, Pi= P:(II)xi. From eq 39, Piis found to be
All calculations were terminated according to the following convergence criterion:
CIWVil
