Znd. Eng. Chem. Res. 1988,27, 2085-2092 Rasher, B. H.; Ma, Y.H. “Liquid Diffuaion in Microporous Alumina Pellets”. AIChE J. 1977, 23(3),303. Raemuson, A,; Neretnieks, I. ”Exact Solution of a Model for Diffusion in Particles and Longitudinal Dispersion in Packed Beds”. AIChE J. 1980,26(4), 686. Ruthven, D. M. Aincipks of Adsorption and Adsorption Processes; Wiley: New York, 1984. Selin, M. E.;Lavrukhin, D. S.; Kulemina, L. B. ”Adsorption by Synthetic NaA Zeolites From Alcohol Water Solutions”. Kolloid. Zh. 1964,26(4), 602. Stuchkov, G. ‘Maea Transfer during Liquid Phase Adsorption Drying of Chloromethane by Zeolites”. Khim Technol. (Kiev) 1975, 6, 40.
2085
Teo, W. K.; Ruthven, D. M. “Adsorption of Water from Aqueous Ethanol Using 3A Molecular Sieves”. Ind. Eng. Chem. Prod. Des. Dev. 1986, 25, 17. Varga, K.; Beyer, H. “Dehydration of Organic Solvents with Molecular Sieves”. Acta Chim. Acad. Sci. Hung. 1967, 52(1), 69. Wilson, E. J.; Geankoplis, C. J. “Liquid Mass Transfer at Very Low Reynolds Numbers in Packed Beds”. Ind. Eng. Chem. Fundam. 1966, 5(1),9.
Received for review May 5, 1988 Revised manuscript received August 16, 1988 Accepted August 29, 1988
A Comprehensive Technique for Equilibrium Calculations in Adsorbed Mixtures The Generalized FastIAS Method James A. O’Brien* Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-2159
Alan L. Myers Department of Chemical Engineering, The University of Pennsylvania, Philadelphia, Pennsylvania 19104
A recent paper by Moon and Tien describes a n improvement to the FastIAS technique of multicomponent adsorption equilibria prediction, originally introduced by the present authors. In this paper, we extend the original FastIAS method to some additional cases suggested by Moon and Tien. We present a generalized technique for formulating adsorption equilibria calculations by the ideal adsorbed solution theory for arbitrary input specifications and illustrate it using the latter cases. T h e calculation involves the solution of a system of N nonlinear algebraic equations, of which N - 1 are always the same and only 1, a material balance, must be altered when different equilibrium specifications are made. We further refine the original formulation of FastIAS by taking advantage of the special structure of the equations involved. Overall, the improved technique performs almost twice as fast as the Moon-Tien procedure and is much simpler to implement and extend. Excellent initial estimates of the solution are given by considering the asymptotic low-pressure case which is analytically solvable for all of the specifications considered here. We identify the condition for an infeasible specification in one case. In an earlier work (O’Brien and Myers, 1985), we presented the initial development of a technique for making rapid adsorption equilibrium calculations using the theory of ideal adsorbed solutions (IAS) as originally described by Myers and Prausnitz (1965). IAS is a useful theory since (a) it requires no mixture data and (b) it is an application of solution thermodynamics to the adsorption problem, so it is independent of the actual model of physical adsorption. We briefly review the equations involved in IAS. The basic equation of IAS is the analogue of Raoult’s law for vapor-liquid equilibrium, i.e., Pyi = Pi”(T)Xi i = 1, ...,N (1) where Pi“ is the pressure which, for the adsorption of every pure component i, yields the same spreading pressure, T , as that of the mixture. Spreading pressure is defined, in turn, by the Gibbs adsorption isotherm (Ross and Olivier (1964) and references therein)
TA
dP
i = 1, ..., N
(2)
The function n:(P) is the experimentally measured adsorption isotherm of pure i. Finally, by requiring zero area change upon mixing at constant IT and T as the condition of ideality, we obtain the total amount adsorbed, n,,as (3)
As we have explained (O’Brien and Myers, 1985),there are N + 1degrees of freedom in this thermodynamic system, and therefore, fixing any N + 1 independent variables defines a unique equilibrium state. In principle, this is a straightforward calculation. Typically, we are interested in specifying the temperature, T, N - 1independent mole fractions (a description of one of the phases, in essence) and one other quantity such as the pressure, P, or the total number of moles of material adsorbed, n,. Other specifications are possible, and we present one example below in case 3. Note that, throughout all of the following, we have assumed that the temperature is always specified. Thus, we required only N additional specifications to be made. In practice, however, there are two computational difficulties with the procedure. Firstly, the value of IT which determines Pi” is the same for all components. Thus, we really don’t need eq 2, but its inverse expression for P ~ ( T )This . requires that Pi” be found by some iterative method. Secondly, the integral in eq 2 has typically been evaluated numerically (e.g., Myers (1984))since there has been no pure-component isotherm expression for nt(P;) which both correlates experimental data and permits analytical integration. In order to facilitate the use of IAS predictions in software for the design of adsorption processes, some method needed to be found to perform this kind of calculation at very high speeds. These concerns were addressed (O’Brien and Myers, 1985) by the original FastIAS technique. I t was based on a logical series-expansion extension to the Langmuir iso-
0888-5885/ 8812627-2085$0~50/0 0 1988 American Chemical Society
2086 Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988
therm, which we derived using the theory of adsorption on heterogeneous surfaces (O’Brien and Myers, 1984; O’Brien, 1986). The isotherm which resulted from the above treatment, and which provided the original basis for FastIAS, may be expressed as
where vi is a dimensionless version of Pi“ defined by
vi
Kip: (5) (Note that, in the Moon-Tien paper, Pi” was incorrectly identified as a partial pressure.) As explained by O’Brien and Myers (1984), mi is the saturation capacity of the adsorbent, K iis directly proportional to the Henry’s constant for adsorption on a hypothetical homogeneous surface of the same mean energy as the actual surface (see eq 45), and ui is a measure of the width of the adsorption energy distribution, made dimensionless using RT. Applying the Gibbs adsorption isotherm to eq 4 above, we obtained the following explicit expression for xi*, the modified spreading pressure of component i:
spreading pressures in pairs. These were written as fi(vi) = fi+l(vi+J i = 1, -., N-1 (9) by requiring the spreading pressure of adjacently numbered components to be equal. In dimensionless form, eq 1 became KiPyi = qixi i = 1, ...,N (10) The last equation came from a material balance, requiring that the calculated values of the adsorbed-phase mole fractions, x , be summed to unity, or
3
c-KiPYi = 1 i
Vi
Thus, we expressed the problem as a system of nonlinear algebraic equations, which was written as dd = 0 (12) where gi(v) are given by i = 1, N-1 (13) gib) = f i h i ) - f i + I ( V i + J and a*.,
.
c
In the above, 9 is a vector of pressures, one for each component, Thus, one fits eq 4 to pure-gas isotherm data and determines the parameters (mi,ui, Ki). Equation 6 is then used instead of the usual numerical integration to obtain the value of the spreading pressure at any value of the pressure. I n fact, any isotherm equation having a n expression for spreading pressure which is explicit in the pressure allows the use of the FastIAS formalism described below. The other crucial requirement is that the isotherm also correlate the experimental data well. The OBrien-Myers isotherm, eq 4, is simply one such equation. Other isotherms, including the statistical model of Ruthven for adsorption in zeolites (Ruthven and Goddard, 1984), also have the property of an explicit expression for xi*. In our notation, Ruthven’s isotherm may be written as
q
I
101,1)2,
vNlT
(15)
gNIT
(16)
and g is the vector function of q g
kl,
g2,
**a)
In addition, 0 is the zero vector. We have given this introductory material in detail since the notation introduced here will be carried throughout the rest of the paper. We chose to solve this equation system using the NewtonRaphson method (Press et al., 1986), terming the complete algorithm FastIAS. Once the solution of eq 12 for 7 is known, the mole fractions, x , and the total adsorbate loading, n,, are calculated from eq 10 and 3, respectively. Thus, the entire equilibrium problem is solved. As a brief reminder, the N-variable Newton-Raphson method may be expressed as
+ $k) (17) The quantity ij(k) is the correction vector for the kth iteration. It is given by the solution to the system of linear equations 777(k+l)
where Ail = 1, and Ruthven and Goddard (1984) have given expressions for the other A,. The number of terms, M , in eq 7 is determined by the maximum number of adsorbate molecules which will fit in a typical zeolite cage or cavity. The corresponding expression for the spreading pressure is given by (Ruthven and Goddard, 1984)
a**,
(P(k)$k)
and
v(k)
=
-dv(k))
(18)
is the Jacobian matrix defined by
M
xi* =
mi In [l + x A i j v i J / j ! ] J=l
(8)
It is obvious from eq 8 above that the derivatives necessary to apply the Ruthven equation in FastIAS are easily evaluated (see below). Original FastIAS Procedure We illustrate the original FastIAS procedure (which allowed specification of pressure, P, and either adsorbedphase mole fractions, x , or gas-phase mole fraction, y ) for the case of a P, y specification. We require that the spreading pressure for each component be the same, due to the ideality condition. Adopting the notation f i ( V i ) for xi*,we may write N - 1 independent equations by equating
In the original FastIAS method, the only non-zero elements @ ( k ) were given by the following relationships:
of
4bi,p= fi’(?p) @i,i+l(k)
= -f.r + l ’( %+I
i = 1, ..., N-1 (k))
i = 1, ..., N-1
(20) (21)
Because of this, the Jacobian matrix had zeros everywhere except the main diagonal, the superdiagonal, and the last row. In all of the above, the superscript k stands for “at the kth iteration”. The expression for the derivative f/(vi)
Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2087 = dfi/dqi is given by differentiating eq 6
Table I. Summary of Specification Cases case specifications calcd 1 2a 2b
Note that, due to the Gibbs adsorption equation (21, this is just q p / q i . Initial guesses were generated from the analytical solution of the equilibrium problem at the extreme of very low pressure (Myers and Prausnitz, 1965). We will devlop this point more finely below. The convergence criterion used throughout was a relative error test of the form
In the original work, we used E = 10-lo. In order to compare with the Moon-Tien results, however, we use t = 10" in the current paper. In fact, it makes no difference to FastIAS since the convergence is very rapid-we have found the running times to be practically identical for each of the above values of e. Moon-Tien Procedure In a recent publication, Moon and Tien (1987) examined the IAS technique slightly differently, once again using the O'Brien-Myers isotherm, eq 4. Firstly, they considered two other specification types which we describe below. In addition, they reformulated the FastIAS equations in the form of a single equation for the spreading pressure, T, electing to update various quantities manually during the algorithm. The rationale for this was that "... the FastIAS procedure requires the solution of N algebraic equations, while the modified procedure requires solving a single equation." Moon and Tien go on to say that "It is reasonable to assume that the computational effort involved is a monotonically increasing function of the number of equations to be solved." We believe this to be a misleading statement. One is solving for the same quantity or quantities no matter how the problem is set up, so it seems reasonable that the various methods of attack should not differ too much in performance. We demonstrate this empirically below, by modifying the original FastIAS in a straightforward manner so that it outperforms the Moon-Tien procedure by about a factor of 2 in speed. The equations developed by Moon and Tien will not be reproduced here, as they are exceedingly complicated and would detract from an easy understanding of the material we are presenting. The reader is instead referred to Moon and Tien (1987) and references therein. On consulting the latter, it will also become apparent that each case of the Moon-Tien procedure is very different, being highly dependent upon the specifications. This much is evident from the flow charts shown. We have, however, implemented the Moon-Tien technique in order to compare it with the results of the new FastIAS procedure. Before leaving the discussion of the Moon-Tien technique, we enumerate the cases they dealt with, as it will be convenient to adopt their terminology to aid in comparisons. The original FastIAS specifications of P,y and P,x are labeled cases 1and 2a, respectively. Case 2b was defined by Moon and Tien as the specification of n,,x;it is simply the inverse problem of case 1. Finally, case 3 specifies none of the usual quantities, instead dealing with a situation which arises in the linear driving force model for adsorption rate (Hsieh et al., 1977;Wang and Tien, 1982; Larson and Tien, 1984; Tien, 1986). The specification is given by i = 1, ...,N (25) pi + ( ~ i n= i j3i
3
p,Y p, x
nt, x nt, Y
nt, x a,B
p,Y
P,Y , n,,x
X
*=
jl
X X
X X
x
x
:::
)1
Figure 1. Structure of FastIAS Jacobian matrix, non-zero entries are designated by X.
e. The only
where ai and Bi are given positive constants, pi is the partial pressure of i defined by pi Pyi (26) and ni is the amount of i adsorbed at equilibrium, defined by ni = ntxi (27) I t should be obvious that this is a reasonable specification-we add N extra equations to the problem and specify temperature and all of the ai and Bi, 2N + 1 variables in all, resulting again in a fully specified system. The various cases are synopsized for reference in Table I.
Modified FastIAS Procedure In this section, we detail the improvements which we have made to the FastIAS method. When the original FastIAS procedure was published, the gain of a factor of 25 in speed over the previous numerical integration-based method (Myers, 1984) was deemed sufficient, and no effort was expended on further improvement. However, we now feel compelled to point out that the elegance and simplicity of the original technique, along with a speed superior to the recentiy proposed method, may still be obtained even with more complicated equilibrium specifications. This must be addressed if the technique is to be useful to practicing engineers and scientists. (a) Structure of Equations. In our original formulation, we left open the question of whether the performance of FastIAS might be improved by exploiting the peculiar structure of the equation set. We have now done this, in the following manner. Instead of eq 9, write the N - 1 expressions for equality of spreading pressure as fi(qi) = ~ N ( T N ) = 1, 2, N-1 (28) now comparing each spreading pressure to that of the Nth component. No matter what the specifications imposed on the equilibrium problem, N - 1of the N equations will always be exactly the same and given by eq 28. Thus, each of the N - 1 equations will contribute two terms to the Jacobian matrix-one on the main diagonal and another in the last column. The Nth equation will, in general, give a Jacobian element for every position on the last row. Thus, we obtain, in this reformulation, a Jacobian matrix of the structure given in Figure 1-zeros everywhere except for the diagonal, the last column, and the last row. Now, we recognize that a major cost of the FastIAS method (in terms of computer time) is the solution at each iteration of the N X N linear system, eq 18. We demonstrated before (O'Brien and Myers, 1985) that approximately five iterations, independent of N, were required for convergence Consider, then, the solution of the to within < linear N X N system Ax = b (29) **a,
2088 Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 X
:j
X
*=[
..,
X
Figure 2. Structure of reduced Jacobian matrix, Ip. The only nonzero entries are designated by X.
where A has the structure shown in Figure 1. We are now able to take advantage of the fact that A is nearly upper triangular in form. First, we use the top row of the augmented system to eliminate the first element in the bottom row of A. Next, we use the second row to eliminate the second element of the bottom row and continue N - 1 times. The result is a coefficient matrix having only a main diagonal and a last column, as shown in Figure 2. This system may be solved trivially by back-substitution! The back-substitution is rendered even simpler since each row has only two entries. Furthermore, since we know that the relevant elements will be ‘keroed out”, we don’t even have to do the operation. The entire reduction of the matrix to the form shown in Figure 2 is carried out by performing 3 ( N - 1) total operations on a”, the rightmost element of the last row of the Jacobian matrix and bN,the last element of the right-hand side vector. An operation is defined here in the usual way, as multiplication or division coupled with an addition or subtraction. Thus, we solve our system in two steps. First, we reduce to the special upper-triangular sparse form of Figure 2 by putting ”(I.
-
”a
-
N-1 a N i -aiN +l aii
(30)
Table 11. Timing Results for Calculations (Times in Seconds)” modified old svstem-case FastIAS Moon-Tien FastIAS 1-1 0.05 0.06 0.06 I-2a 0.05 0.06 I-2b 0.13 N/A 1-3 0.11 0.30 N/A 11-1 0.14 0.11 0.26 11-2a 0.14 0.24 11-2b 0.29 N/A 11-3 0.34 0.26 N/A 111-1 0.18 0.28 0.67 111-2a 0.24 0.80 111-2b 0.64 N/A 111-3 0.35 0.76 N/A 111-4 0.35 0.76 N/A a t = loa; Intel 8088/8087 CPU (DEC Rainbow), 4.77-MHz clock speed, Turbo Pascal V4.0. N/A indicates not applicable-original (or old) FastIAS did not use the specifications. Blank indicates that we did not perform these calculations. See text.
(b) Closure Equations for Various Specifications. We have already mentioned that, in order to vary the thermodynamic specification of the problem, we need only change one equation, specificallyg&). Thus, the first N - 1equations are always those represented by eq 28. Here, we detail what should be the form of gN for each of the cases to be studied. In general, the Nth equation is derived from a material balance. By writing a suitable material balance, any set of specifications may be accommodated in a straightforward manner. Case 1. This has been dealt with above. However, we include it here also for completeness. Requiring that the adsorbed-phase mole fractions add up to unity (a material balance), we write
Then, we solve by back-substitution using bN
XN
=~ N N
XN-1
=
b ~ - -i a ~ -Ni
with Jacobian elements given by ~ N
(33)
aN-l N-1 XN-2
=
b ~ - 2- a ~ - N2 ~ N uN-2 N-2
XN-3
= etc.
The operation count for the back-substitution is 2N - 1. Thus, contrary to the general case of a full matrix which takes time proportional to N 3to solve (Press et al., 19861, we have the remarkable conclusion that our restructured system can actually be solved using 5N - 4 operations (3N - 3 multiplications and 2N - 1 divisions), i.e., in time proportional to N , the number of equations. That this is a significant effect may be deduced from the results in Table 11, where we compare the modified FastIAS with the original FastIAS, which used a full-matrix technique, the LU decomposition method (Press et al., 1986). For a 10component calculation of case 1 (111-1in the table), the original FastIAS takes almost 4 times as long as the modified FastIAS. The speedup obtained is not as large as would be expected from the operation scaling arguments above. We infer from this that solving the linear systems is not the sole controlling factor in the calculations; evaluating the Jacobian elements is also time consuming. The latter method is easily encoded, and we have used it in preference to Gaussian elimination in the current development. It is an integral part of each of the computer routines described later.
Case 2a. Again, this was exposed in the original paper. Now requiring that the gas-phase mole fractions add to unity, we obtain (34) This time, the Jacobian entries are
(!e)=&
(35)
Case 2b. Here, n, is specified along with x. Our material balance requirement must, therefore, be (36) following from eq 3. For the Jacobian elements, by differentiation, we arrive at (37) where the derivative above is given by differentiating eq 4 to obtain
Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2089
It is worth noting that, if the specified value of n, is too high, there is no solution-this corresponds to a thermodynamically infeasible specification. We may easily derive an expression for the maximum possible value of n,. The first step is to note that, since mi is the saturation capacity of the adsorbent for component i, n t < mi i = 1, ...,N (39) Taking the reciprocal, multiplying by xi, summing, taking the reciprocal once again, and comparing with eq 3 for n,, we obtain N
n, < l / C ( x i / m i ) i=l
n",
(40)
Thus, a specification of n, > n", yields a system of equations with no solution. For this reason, we have incorporated eq 40 above in our routine NXTOPY (see later) to screen infeasible n,,x specifications. Case 3. Here, we again require that the adsorbed-phase mole fractions, x ',sum to unity. This yields (the expression for x i was also derived by Moon and Tien)
at very low pressure. Henceforth, we refer to eq 47 as the low-pressure condition. We use the low-pressure condition as our initial estimate for 7 throughout. Specific to each case under consideration is the particular starting estimate for n, to be inserted in the low-pressure condition. We list these below. In addition, we show how the unknown variables are calculated once the values of vi have been solved for. Case 1. The low-pressure condition yields the estimate of n, as n, = PCBj*yj (48) I
With this information, we solve eq 28 and 32 for g. Finally, we calculate the unknown variables x i and n, by using KiPYi x. = (49) vi
and eq 3. Case 2a. From the definition of the selectivity factor (Myers and Prausnitz, 1965), sij, and the low-pressure condition, we obtain P
The derivative to obtain the Jacobian elements for this equation is difficult, so we introduce an approximation. We assume that each xj is approximately constant for the differentiation, an assumption whose validity is borne out a posteriori by the robustness of the algorithm. Applying the chain rule and using eq 3 for n,, we arrive at
where the quantities xi and n, are updated at each iteration by using KiPi xi vi + Kiaint
-
n,
-
l/C(xj/njo) I
(43)
The remaining derivative in eq 42 is given by eq 38. Note that, in the event that all of the ai are zero, case 3 reduces to case 1-a specification of p i (or, equivalently, P and y). (c) Initial Guess Strategy. This is arguably the most difficult part of any iterative algorithm to perfect without some sort of physical insight into the problem at hand (Press et al., 1986). However, we have the advantage of an asymptotic solution-the solution for the case of very low pressure. A t low pressure, we may write for the amount adsorbed (Myers and Prausnitz, 1965) ni = Bi*Pyi (44) where the quantity Bi* is the Henry's constant divided by R T and is given, for our isotherm, by Bi* = miKi[l + a:/2] (45) Physically, Bi* represents the limiting slope, at zero pressure, of the isotherm function, dn; Bi* =dp Now, combining eq 44 and 10 and eliminating Pyi,we may derive that
After having solved eq 28 and 34 for 7, we calculate the unknown variables n, and y i using eq 3 and Pixi
Yi =
K,P
respectively. Case 2b. Here n, is specified, so there is no need to estimate it! Having solved eq 28 and 36 for 1,we find the unknown variables P and yi as follows. First, using eq 10, we calculate the partial pressures pi to be 4iXi
p i ( I p y . )= -
'
Ki
Adding all of these together yields P, by definition of the partial pressure. Finally, we obtain each yi from Yi = Pi/P (53)
Case 3. Combining the low-pressure condition with eq 25, the specification equation for case 3, and eliminating the partial pressure, pi,we may derive the estimate (54) We then solve for 7 using eq 28 and 41. At each step in the calculation, we update x i and n, as described above. Once we have calculated the g, we obtain the unknowns P and yi as follows. First, the specification, eq 25, is written to calculate the partial pressure in terms of known quantities (55) p i = pi - aintxi Finally, using the same procedure as in eq 52 and 53 above, we sum the pi to get P and then divide each pi by P to obtain yi. Results In order to compare our results with the Moon-Tien technique, we use the same test problems they presented in their paper. Each case (except for 2a, for which they
2090 Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 Table 111. Pure-Component Isotherm Parameters Useda system mi, mol kg-' 102K,,kPa-' af 1.00 1.20 I 5.00 4.00 0.50 1.10 3.00 0.80 0.09 I1 5.00 1.00 1.20 2.00 0.60 1.10 0.80 3.00 0.09 7.00 10.0 0.90 2.00 6.00 1.00 1.00 1.20 I11 5.00 3.00 0.60 1.10 4.00 0.80 0.09 1.00 1.20 2.00 0.30 1.00 3.50 0.10 1.10 4.00 1.50 1.20 2.00 0.10 1.15 2.50 0.01 1.00 4.00 0.60 1.00 5.50
For each system, components are numbered sequentially from the top. Table IV. Specifications Used in Calculations" calcd system case conditions 1 I- 1 I y1 = 0.6, yz = 0.3, P = 300 I-2a I 2a XI= 0.6, 2 2 = 0.3, P = 1193 I-2b I 2b x1 = 0.6, x2 = 0.3, n, = 3.0 I ai = 1.0, /Ti = 5.0, i = 1-3 1-3 3 yi = 0.2, i = 1-3; y4 = 0.3; P = 300 11-1 I1 1 11-2a I1 2a xi = 0.2, i = 1-3; ~1 = 0.3; P = 11120 2b xi = 0.2, i = 1-3; xq = 0.3; n, = 3.0 11-2b I1 ai = 1.0, Bi = 5.0, i = 1-5 3 11-3 I1 1 yi = 0.1, i = 1-3, 5; yi = 0.2, i = 4, 10 111-1 I11 yi = 0.05, i = 6-9; P = 300 111-2a I11 2a xi = 0.1, i = 1-3, 5; xi = 0.2, i = 4, 10 xi = 0.05, i = 6-9; P = 23 990 I11 2b X , = 0.1, i = 1-3, 5; xi = 0.2, i = 4, 10 111-2b xi = 0.05, i = 6-9; n, = 3.0 ai = 1.0, j3i = 5.0, i = 1-10 111-3 I11 3 ai = 1.0, i = 1-3, 7, 10; a4 = 0.5; a5 = 2.0 111-4 I11 3 a6 = 4.0; as = 0.4; a g = 5.0; fli = 62 = 5.0 B3 = 2.0; B4 = 1.0; B5 = 10.0; B6 = 20.0; (3, = 3.0 = 2.50; 89 = 40.0; Blo = 2.0 "All pressures in kPa and amounts adsorbed in mol kg-'. Only the N-1 independent mole fractions for each case are given.
presented no results) was calculated for each of three different systems of components. The three systems (referred to as I, 11, and 111) have 3, 5, and 10 components, respectively, in order to highlight the effects of the number of components on the calculation time. Values of the parameters mi, Ki, and cri were chosen for each component of each of the systems. The resulting sets of parameters used are summarized in Table 111, which is effectively identical with Table I in Moon and Tien's paper. In addition, the specifications which were made for each case are listed in Table IV, which is similar to Table I1 in Moon and Tien. The values chosen for the parameters appear to be reasonable, being of the same order of magnitude as values determined by O'Brien (1986) from the data of Reich et al. (1980). The term "results" is a little misleading here. Firstly, the major part of our result is already evident-it is embodied in the statement of the improved formulation and algorithm for FastIAS which we have developed above. In addition, the actual numbers obtained from these calculations are identical no matter whose technique is used, as long as it is correctly implemented. These results are tabulated in Moon and Tien (1987) and will not be repeated here. We have also verified that, for example, applying the case 2b specification to the results of a case
Table V. Sample Specification for Moon-Tien Algorithm Failure" calcd system case conditions 111-1 I11 1 y1 = y7 = 0.1; yz = y3 = 0.01; y4 = 0.05; y5 = ye = ye = 0.2; yg = 0.02; P = 300 "All pressures in kPa.
1 calculation recovers the original specification, as it should. Table I1 summarizes the results of this study in terms of the calculation time required for each specification by each of the methods. All of the calculations were carried out by using programs written in Turbo Pascal V4.0 (Borland, 1987) on a DEC Rainbow PClOOB personal computer, a typical desk-top machine with an Intel 808818087 combination CPUInumeric data coprocessor. Timings were carried out by enclosing the entire program in a repeating loop and measuring the time taken for many passes through the loop, since the time per calculation is rather short. Care was taken to time only the calculations and not the part of the program which read in the data and wrote out the results. The Moon-Tien technique is complicated to program, so we implemented their method only for cases 1 and 3. This is a reasonable approach because, in their paper, cases 1and 3 represented the lower and upper bounds on calculation time, respectively. We compare in Table 11the modified FastIAS, the Moon-Tien, and the original FastIAS methods. (In the following, we take FastIAS to mean the modified FastIAS, unless otherwise specified.) Overall, the modified FastIAS technique is seen to perform better than the others, with a maximum advantage of more than a factor of 2 in speed. As Moon and Tien (1987) pointed out, their technique outperforms the original FastIAS. However, we have been able to improve yet further simply by a more convenient statement of the problem. One might argue that a factor of 2 is an incremental advance, although Moon and Tien published their technique on the basis of such an advantage, on average. We believe that the real advantage of our technique lies in its twin properties of simplicity and straightforward extension to any set of specifications. Actually, in the limit of very large numbers of components ( N > 40), FastIAS and the Moon-Tien procedure approach the same speed. However, in a case with many components, one expects at least some of them to be present in trace quantities, especially since the removal of trace impurities is an important application of adsorption technology (e.g., Broughton and Gembicki (1984)). Th'is can cause difficulties for the Moon-Tien method, as explained below. Some experimentation reveals that our FastIAS method is more robust than that of Moon and Tien. We performed several runs for 10-componentsystems (111) but with some of the components present in trace quantities or absent altogether. In these cases, the Moon-Tien algorithm failed to converge frequently. FastIAS, on the other hand, was robust throughout. One particular problem specification which caused the Moon-Tien method to fail is documented in Table V. The reason for the Moon-Tien algorithm's failure is their choice of initial guess for the spreading pressure, T. It is arbitrarily based on the pure-component spreading pressure of the component i with the highest value of saturation capacity, mi. However, this reference component may not even be present in a given problem specification. Modification of their initial guess is possible, but it must be on an ad hoc basis for each computational difficulty which arises. FastIAS, on the other hand, pro-
Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2091 vides a rational and robust means of starting the calculation in every case. Additional testing, using randomly generated parameter values and problem specifications in a physically reasonable range, indicates that FastIAS computation time is roughly independent of the specifications, depending mainly on the number of components, N. However, the Moon-Tien procedure’s running time depends on a given set of problem specifications as well as on N. In the original FastIAS paper, we discussed a method of ensuring that the vi remain positive during iteration always (avoiding the unphysical occurrence of negative pressures), by disallowing the newly calculated increment 6i if it caused ~i to become less than zero, instead putting vi equal to 7 i / 2 from the previous iteration. If this condition occurs often, it reduces the second-order effectiveness of the Newton method. By monitoring the invocation of this condition, we observed that the Moon-Tien algorithm for case 3 always takes at least one such unphysical step on the first iteration after the initial guess. In contrast, it is almost never necessary to be concerned with this problem for FastIAS-the only time it arose was for the first Newton step of case 2a when the specified pressure was very large. However, we include this condition in the code to ensure global robustness.
Conclusions A generalized reformulation of the FastIAS technique performs better than the most recent contending method for the calculation of multicomponent adsorption equilibria using ideal adsorbed solution theory. The technique consists of the solution of a system of N nonlinear algebraic equations for an isothermal N-component adsorbing mixture. Only one of the N equations changes when the problem specification is altered. Furthermore, we have shown how to derive the Nth “closure” equation for any problem specification by means of a material balance. The peculiar structure of the equations has enabled us to develop a solution technique for the linear Jacobian matrix system which only requires computer time proportional to N, the number of equations. Finally, consideration of the thermodynamic limit of very low pressures provides a general technique for deriving initial estimates of the variables being solved for. The FastIAS technique has not yet failed, in our experience, to converge to the solution from such an initial guess. Thus, with a high degree of confidence, failure to converge may be attributed to an infeasible thermodynamic specification. The program code for carrying out cases 1,2a, 2b, and 3 is available from the author in routines called PYTONX, PXTONY, NXTOPY, and ABTONX, respectively. Both Fortran and Pascal versions of the code are available and may be requested by contacting J.O’B. Each routine is entirely self-contained, requiring as inputs the specifications and the pure-component fitting parameters and providing the calculated quantities as outputs. Refer to Table I for the various input/output options available. Nomenclature A = surface area of adsorbent, m2 kg-l A , = parameter j for component i in isotherm eq 7 Bi* = Henry constant for component i ( I R T ) ,mol kg-’ Pa-’ fi(vi)= spreading pressure function in FastIAS, mol kg-’ Pa-’ g ( v ) = vector of g i , mol kg-’ (or dimensionless) gi(7) = functions in FastIAS method K i = parameter in isotherm eq 4, Pa-’ mi = adsorbent saturation capacity, parameter in isotherm eq 4, mol kg-’ ni = amount of pure i adsorbed from mixture, mol kg-’
n; = adsorption isotherm of pure i, mol kg-’ n, = total amount adsorbed, mol kg-’ n“, = maximum specification possible for n,, mol kg-’ P = pressure, Pa pi = partial pressure of component i, Pa Pi” = vapor pressure analogue in eq 1, Pa R = universal gas constant (4.314 J mol-’ K-I) sij = selectivity of adsorbent for i relative to j T = temperature, K x i = adsorbed-phase mole fraction of component i yi = gas-phase mole fraction of component i Greek Symbols
vector of quantities in eq 25, Pa kg mol-’ @ = vector of quantities in eq 25, Pa 6 = correction vector in FastIAS solution algorithm 7 = vector of vi vi = quantity defined in eq 5 r = spreading pressure, Pa m or J m-2 ri* = modified spreading pressure, mol kg-’ ui = parameter in isotherm eq 4 $ i , j = element of Jacobian matrix, Q,, mol kg-’ (or dimena =
sionless) Q, = Jacobian matrix for Newton-Raphson
solution in FastIAS
technique (see above) Subscript i = component i Superscripts k = at the kth iteration o = quantity at equal spreading pressure T = transpose of vector ’ = derivative with respect to vi
Literature Cited Borland Turbo Pascal V4.0; Borland International: Scotts Valley, CA, 1987. Broughton, D. B.; Gembicki, S. A. “Adsorptive Separations by Simulated Moving Bed Technology-The Sorbex Process”. In Fundamentals of Adsorption; Myers, A. L., Belfort, G., Eds.; American Institute of Chemical Engineers: New York, 1984; p 115. Hsieh, J. S. C.; Turian, R. M.; Tien, C. “Multicomponent Liquid Phase Adsorption in Fixed Bed”. AIChE J. 1977,23, 263. Larson, A. C.; Tien, C. “Multicomponent Liquid Phase Adsorption in Batch. Part I: Formulation and Development of Computation Algorithms”. Chem. Eng. Commun. 1984, 27, 339. Moon, H.; Tien, C. “Further Work on Multicomponent Adsorption Equilibria Calculations Based on the Ideal Adsorbed Solution Theory”. Ind. Eng. Chem. Res. 1987,26, 2042. Myers, A. L. “Adsorption of Pure Gases and their Mixtures on Heterogeneous Surfaces”. In Fundamentals of Adsorption; Myers, A. L., Belfort, G., Eds.; American Institute of Chemical Engineers: New York, 1984; p 365. Myers, A. L.; Prausnitz, J. M. “Thermodynamics of Mixed Gas Adsorption”. AIChE J. 1965, 11, 121. O’Brien, J. A. “Studies of Molecular Interaction Effects in Physical Adsorption on Heterogeneous Solid Surfaces”. Ph.D. Dissertation, The University of Pennsylvania, Philadelphia, 1986. O’Brien, J. A.; Myers, A. L. “Physical Adsorption of Gases on Heterogeneous Surfaces-Series Expansion of Isotherms using Central Moments of the Adsorption Energy Distribution”. J. Chem. SOC.,Faraday Trans. 1 1984,80, 1467. O’Brien, J. A.; Myers, A. L. “Rapid Calculations of Multicomponent Adsorption Equilibria from Pure Isotherm Data”. Ind. Eng. Chem. Process Des. Deu. 1985,24, 1188. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: New York, 1986; pp 19, 31-38, 241, 271. Reich, R.; Ziegler, W. T.; Rogers, K. A. “Adsorption of Methane, Ethane and Ethylene Gases and Their Binary and Ternary Mixtures and Carbon Dioxide on Activated Carbon at 212-301 K and Pressures to 35 Atmospheres”. Ind. Eng. Chem. Process Des. Deu. 1980, 19, 336. Ross, S.; Olivier, J. P. On Physical Adsorption; Wiley: New York, 1964; p 8.
I n d . Eng. Chem. Res. 1988,27, 2092-2095
2092
Ruthven, D. M.; Goddard, M. "Correlation and Analysis of Equilibrium Isotherms for Hydrocarbons on Zeolites". In Fundamentals of Adsorption; Myers, A. L., Belfort, G., Eds.; American Institute of Chemical Engineers: New York, 1984; pp 536, 537. Tien, C. "Incorporation of the IAS Theory in Multicomponent Adsorption Calculations". Chem. Eng. Commun. 1986, 40, 265.
Wang, S.-C.; Tien, C. "Further Work on Multicomponent Liquid Phase Adsorption in Fixed Beds". AZChE J. 1982, 28, 565.
Received for review February 18, 1988 Revised manuscript received June 10, 1988 Accepted July 6, 1988
Polymeric Iron Chelates for Nitric Oxide Removal from Flue Gas Streams S t e p h e n A. Bedell,* S u s a n S. Tsai, and Robert R. Grinstead+ Dow Chemical U.S.A.,Building B-250, Freeport, Texas 77541
A major problem encountered in the use of ferrous chelates for removal of nitric oxide from flue gases is the loss of the chelate in the purge stream of the built-up salt products. The use of polymeric chelating agents which can hold iron and be separated from waste streams by ultrafiltration is reported. NO absorption is most effective for the Fibrabon 35/ED3A (the reaction product of ethylenediaminetriacetic acid and an epichlorohydrin/bis(6-aminohexyl)aminecopolymer) iron chelate. Removal levels of greater than 90% have been obtained for this system. Comparison of the polymeric chelates with the ferrous chelate of ethylenediaminetetraacetate (EDTA) has been made. Scrubbing of sulfur dioxide from flue gases can be accomplished by contacting the gas with an alkaline absorbent, such as caustic or a limestone slurry. Nitric oxide removal presents some unique problems. Unlike SO2,NO is not very soluble in conventional scrubbing solutions. As developed in the 1970s in Japan, solutions containing ferrous EDTA-type chelates can be used to simultaneously remove SOz and NO from flue gas streams. The formation of a ferrous-nitrosyl complex greatly increases the solution's capacity for NO, as shown for hydrated ferrous ion in eq 1. Chelation of the iron by EDTA not only helps Fe11(H,0)6+ NO + Fe"(H,O),(NO) K,, = lo2.' (1) to keep the iron from precipitating, but also results in greater affinity for NO through ternary complex formation (eq 2). Once fixed in solution, the NO can be readily FeIIEDTA + NO + Fe"(EDTA)(NO) Keq = 106.2(2) reduced by sulfite produced in the SO2 scrubber (eq 3). Though earlier literature reports discrepancies in product Fe"(EDTA)(NO) + S032Fe'I(EDTA) + S,N products (3)
-
identification, Chang and Littlejohn (1985a,b) have determined several species of mixed sulfur-nitrogen salts along with molecular nitrogen and nitrous oxide. Distribution of these products is dependent on reaction conditions. The effectiveness of this chemistry in abating NO to low levels has been well documented (Koizumi et al., 1974; Faucett et al., 1977; Hishinuma et al., 1978). The greatest advantage of the ferrous chelate based NO, processes over many other technologies is the potential for convenient retrofitting of existing SO2 scrubbers. Though plans to install a ferrous-based NO, process in a 750-MW West German coal fired power plant were recently announced (Leimkuehler et al., 1986), a major problem hindering
* Author to whom correspondence should be addressed. +Current address: Dow Chemical U.S.A., P.O. Box 9002, Walnut Creek, CA 94596.
widespread commercialization of this type of process is the high cost of chelate blowdown losses. As wet FGD waste products (calcium sulfite or sulfate plus salts of the nitrogensulfur products) are removed in a slip stream from the process, some iron chelate will also be purged. This loss of chelate could be minimized if all the iron was in the active ferrous form. Oxygen in the flue gas will oxidize the ferrous chelate (eq 4) to form a ferric complex which has 4Fe"(EDTA)
+ 4H+ + Oz
-
4Fe"'(EDTA)
+ 2H20 (4)
little affinity for nitric oxide. Oxidation of Fe(I1) by NO can also occur (Sada et al., 1986). Though reduction of the ferric chelate by sulfite (eq 5) will compete with the Fe"'(EDTA)
+ SO3,-
-
Fe"(EDTA)
+ '/S206z- (5)
ferrous oxidation, the majority of the iron will remain in the ferric state. Since this will require more total iron to achieve NO abatement, losses during waste product removal will also increase. The key to development of this technology depends on one or a combination of two process improvements (Bedell et al., 1986): (1)efficient, cost-effective means of ferric chelate reduction; (2) full or partial recovery of the iron chelate from the waste purge. The first strategy has been shown by Tsai et al. (1987) to be technically feasible with the use of an electrochemical cell for iron reduction. The current study was undertaken to evaluate the NO removal ability of novel polynuclear iron chelates, large enough to be retained by an ultrafiltration membrane, while FGD product salts are passed through for waste disposal. The recovered iron chelate can then be returned back to the scrubbing device for reaction with additional NO. Figure 1 shows a scheme of this process. Experimental Section Preparation of Chelates. Poly(N-(carboxymethyl) ethyleneimine)ferrate(III)(Fe-CMPEI-150). PEI-150 (45.3 g) (a 33% aqueous solution of 10000 molecular weight polyethyleneimine from Virginia Chemicals) w8s added to 200 mL of water. Fifty-two grams of bromoacetic acid was added to 50.0 g of H 2 0 followed by 42.0 g of 50% KOH solution. The potassium bromoacetate solution was then 0 1988 American Chemical Society
