Calculations of Multicomponent Adsorption-Column Dynamics

May 20, 2000 - Although the Potential and IAS theories derive from different physical models, .... According to Amankwah and Schwarz,37 the reliabilit...
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Ind. Eng. Chem. Res. 2000, 39, 2459-2467

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Calculations of Multicomponent Adsorption-Column Dynamics Combining the Potential and Ideal Adsorbed Solution Theories Jose´ P. B. Mota* and Anto´ nio J. S. Rodrigo Departamento de Quı´mica, Centro de Quı´mica Fina e Biotecnologia, Faculdade de Cieˆ ncias e Tecnologia, Universidade Nova de Lisboa, 2825-114 Caparica, Portugal

An efficient method for combining the Adsorption Potential and Ideal Adsorbed Solution theories in simulation models of multicomponent adsorption columns is presented. The main advantage of this dual-theory approach is that it reduces the number of equilibrium measurements required to predict the breakthrough profiles. The validity of the dual-theory equilibrium procedure is illustrated by embedding it in a dynamic-column breakthrough model of adsorption of ternary mixtures of methane, ethane, and propane on activated carbon. The resulting dynamic model was found to be computationally efficient, and good agreement was obtained between the predicted and experimental breakthrough curves. It is shown that the two theories can be coupled in such a way as to provide a simple and practically useful method of correlating multicomponent adsorption equilibrium in dynamic process modeling studies. 1. Introduction Gas adsorption processes are a well-established separations tool in chemical and petrochemical industries. A large majority of such processes operate through equilibrium adsorption of the mixture, which requires reliable quantitative characterization of adsorption equilibrium in order to design them. It is therefore not surprising that many theories and adsorption equilibrium models have been developed over the years.1 One of the major theories of adsorption is the Potential theory developed by Dubinin and co-workers2-4 from ideas originally formulated by Polanyi.5,6 The potential theory is applicable to adsorption at moderate and high pressures, but it has also been applied with success at low ones,7 although at very low pressure the theory contradicts some consistency requirements.8,9 Nevertheless, for nearly a half-century it has been used extensively to describe the adsorption of vapors and gases by microporous solids.10 The first attempts to extend the Potential theory to multicomponent adsorption considered idealized adsorbates obeying Raoult’s law.11 Two examples of such extended models are the direct extension of the Dubinin-Radushkevich equation to multicomponent adsorption by Bering et al.12,13 (for further details see also Yang,14 pp 53 and 54) and the Grant-Manes model.15,16 The latter extension is based on the equipotential concept of Dubinin and is not tied to a particular form of the characteristic curve. This model was employed by Reich et al.17 to correlate the adsorption of binary and ternary gas mixtures on activated carbon. An improved method based on the Potential theory for predicting gas-mixture adsorption equilibria has been developed by Mehta and Danner.18 This method has been incorporated by Moon and Tien19 into liquidphase multicomponent adsorption calculations. A promising theory of multicomponent adsorption based on the Potential concept has been developed * To whom correspondence should be addressed. Telephone: +351 21 295 44 64 (ext. 0961). Fax: +351 21 294 83 85. E-mail: [email protected].

recently by Shapiro and Stenby.20,21 In this theory the mixture is considered as a heterogeneous substance segregated in the external field emitted by the adsorbent, and the same standard equation of state is used for segregated and bulk phases. However, the accuracy and computational efficiency of the theory in a dynamiccolumn breakthrough model has not yet been investigated. A contribution of fundamental importance to the prediction of mixed-gas adsorption is the Ideal Adsorbed Solution (IAS) theory developed by Myers and Prausnitz.22 The theory can also be extended to nonideal systems by introducing surface-phase activity coefficients.23,24 Although the Potential and IAS theories derive from different physical models, Belfort25 has shown that similarities between the two theories exist, especially when the Potential theory is formulated in the form of the Grant-Manes model.15,16 For multicomponent adsorption, the IAS theory has the advantage of being thermodynamically consistent and, as Sircar and Myers26 pointed out, the extensions of the Potential theory cited above are correct only when the surface potential of the pure adsorbates are equal at the state of saturated vapor. Greenback and Manes,27 however, point out that if adsorbate nonuniformity is incorporated in those models, the thermodynamic consistency requirement can be met. A straightforward way of extending the Potential theory to multicomponent adsorption that is thermodynamically consistent is to combine it with the IAS method. This approach was mentioned briefly by Myers,9 who considered the Potential theory in the form of the Dubinin-Radushkevich isotherm. Recently, Stoeckli and co-workers28,29 extended the method to the Dubinin-Astakhov isotherm. They were also able to predict breakthrough curves for the binary adsorption of several vapors on activated carbon, under isothermal conditions, using IAS and the Dubinin-Radushkevich isotherm.30 In this work, an efficient method for combining the two theories in simulation models of multicomponent adsorption columns is presented. Different forms of the

10.1021/ie9908478 CCC: $19.00 © 2000 American Chemical Society Published on Web 05/20/2000

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Potential theory are considered: the Dubinin-Astakhov equation, the Dubinin-Radushkevich equation, and the polynomial expansion proposed by Ozawa et al.31 The validity of the procedure is illustrated by comparing predicted and experimental breakthrough curves for adsorption of ternary mixtures of light hydrocarbons on activated carbon under nonisothermal conditions. 2. Theory The adsorption potential, φ˜ , for single-gas adsorption is defined as

f ) RT ln(Ps/P)

(1)

where P is the equilibrium pressure at temperature T and Ps(T) is the saturated vapor pressure of the adsorbate. At high pressure, however, P and Ps should be replaced by the corresponding fugacities, f and fs, to correct nonideal gas behavior. The Potential theory states that for a given gas-solid system the volume of the adsorbed phase, W, is a function of φ˜ only, i.e.

W ≡ qVa ) W(φ˜ )

(2)

where q is the amount adsorbed at equilibrium and Va(T) is the adsorbed-phase molar volume, which is temperature dependent. The functional relationship between φ˜ and W is characteristic of the particular gassolid system and is usually referred to as the characteristic curve. The Potential theory has two attractive features that make it especially useful for predicting single-component adsorption equilibrium from a limited set of experimental measurements. The first one is that according to the theory the characteristic curve is temperature independent. Therefore, only adsorption equilibrium measurements at one temperature are necessary to obtain the characteristic curve, and this is sufficient to describe the adsorption at all temperatures for the same gas-solid system.14,32 The second feature, despite being less applicable in practice, is also of great usefulness: in many cases the theory can be generalized if an affinity coefficient, β, is used as a shifting factor to bring the characteristic curves of all gases on the same adsorbent into a single curve. Hence, eq 2 can be replaced by

W ≡ qVa ) W(φ), φ ≡ φ˜ /β

(3)

which is characteristic of a given adsorbent and can be equally applicable to all adsorbates. This formulation is perhaps the best attempt to obtain a universal isotherm by which the adsorption data of one reference gas are extended to other gases.14,18 In general, under conditions below the critical temperature, Tc, of the adsorbate, Va is assumed equal to the molar volume of the saturated liquid at system temperature. Above Tc the adsorbed phase is ill-defined, and this led to different approximations for Va having been proposed. Likewise, in the supercritical region the concept of vapor pressure does not exist and Ps in eq 1 must be replaced by a pseudovapor pressure. A comprehensive summary of these methods has been given by Agarwal and Schwarz32 and by Tien.33 The work of Agarwal and Schwarz32 suggests that above Tc a suitable representation of adsorption data

by a temperature-independent characteristic curve is obtained with a constant value of β, with Ps defined as2

Ps ) (T/Tc)2Pc

(4)

where Pc is the critical pressure of the adsorbate and with Va evaluated by4

Va(T) ) Vb exp[Ω(T - Tb)], Ω ) ln(b/Vb)/(Tc - Tb) (5) In eq 5, Tb and Vb are the temperature and molar volume of the liquid adsorbate at the normal boiling point, and Ω is an estimate of the thermal expansion coefficient of the adsorbate in a superheated liquid state.31 An alternative method proposed in the literature7,34,35 is to set β equal to Va and to calculate Ps either by extrapolation of Antoine’s equation or by use of the reduced Kirchhoff equation36 interpolating the points (Pc, Tc) and (1 atm, Tb). However, the approach based on a temperature-independent β for each adsorbate, and on eqs 4 and 5, seems to give a better representation of the data in the form of a single characteristic curve.32 Furthermore, β does not frequently correlate well with Va. To improve the coalescence of data into a single characteristic curve, Greenback and Manes27 as well as Mehta and Danner18 suggested taking β ) kVs, where k is a constant determined empirically for each adsorbate and Vs is the saturated liquid molar volume of the adsorbate. According to Amankwah and Schwarz,37 the reliability of the predictions can be improved further if the exponent of 2 in eq 4 is replaced by a parameter to be determined by the adsorbent-adsorbate system. The IAS theory22 is the most successful thermodynamically consistent method for predicting multicomponent adsorption equilibrium from single-component isotherm data. Since its introduction, the method has become so popular that specialized procedures have been devised in order implement it efficiently. The most notable examples are FastIAS,38 MFastIAS,39 and GFastIAS.40 A popular approach for solving the transient behavior of distributed parameter processes is the Method of Lines,41 which uncouples the spatial and time discretization by converting the original equations into a mixed system of ordinary differential and algebraic equations (DAEs) with respect to time. Any spatial discretization scheme will do as long as it is suitable for the problem. The advantage of this procedure is that sophisticated computer programs which permit fast and accurate integration of large sets of DAEs over time can be employed (e.g., DASSL42 or DASOLV43). If such an approach is adopted for solving a dynamic adsorption model, it is our opinion that the most efficient way of embedding the IAS method in the numerical model is to consider the spreading pressure of the gas mixture, Π, as a dependent variable, just like the temperature or partial pressure of each adsorbate. In the case of batch adsorption Π ) Π(t), where t is time, whereas for a fixed bed Π ) Π(t,z), z being the distance from the column inlet. The explicit use of Π as a dependent variable has been advocated previously by Wang and Tien44 in a study on multicomponent liquidphase adsorption in fixed beds, by Tien,45 and by Moon and Tien39 in their MFastIAS method. The remaining degree of freedom due to taking Π as a dependent

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variable is eliminated by forcing Π to satisfy at every time the algebraic equation NC

xi ) 1 ∑ i)1

(6)

where xi is the ith mole fraction in the adsorbed phase derived from the values of Π, temperature, and partial pressures. The majority of codes for solving DAEs are based on implicit multistep methods, the most popular among these being the backward differentiation formulas.46 At each time step in the numerical integration, the solution is updated in a two-stage process. First, an initial guess for the solution is formed by evaluating a predictor polynomial which interpolates the solution at previous time steps. The solution which is finally accepted at the current time step is obtained by solving a corrector equation. This nonlinear system is usually solved using some variant of Newton’s method. Because the DAE solver changes the step size and the order of the method to solve the DAE system more efficiently and reliably, the initial guess of Π formed by evaluating the predictor polynomial is already close to the true solution. This ensures that Newton’s method does not experience convergence difficulties due to introducing Π as a dependent variable. For each value of Π obtained in the corrector iteration process, the other variables are computed as follows. First, the pressure P0i at which the pure-component adsorption equilibrium yields the same value of Π at temperature T, is calculated from the Gibbs adsorption equation:

∫0P q0i (P,T) d ln P

ΠA ) RT

0 i

(7)

where q0i (P,T) is the adsorption isotherm of pure component i. Then, the basic equation of IAS is employed to calculate xi:

xi )

yiP/P0i

(8)

where yi is the mole fraction of component i in the gas phase. Finally, the total amount adsorbed, q, and that of each component are computed by the following expressions:

1 q

NC

)

∑ i)1

xi

, qi ) xiq

q0i (P0i ,T)

(9)

Given that

dφ ) -(RT/β) d ln P (const T)

(10)

and taking into account eqs 3 and 7, the following expression relating ΠA and φ is obtained:

ΠA ) (β/Va)

∫φ∞W(φ*) dφ*

(11)

∫φ∞W(φ*) dφ* ) ψ(φ)

ΠA f ψi ) (Va/β)iΠA f φ0i ) φ(ψi)

(12)

Instead of calculation of P0i from eq 7, in the framework of the Potential theory, the scaled adsorption

(13)

where φ(ψ) ) ψ-1(φ) is the inverse function of ψ(φ). Likewise, the basic equation of IAS, eq 8, is rewritten as

(

xi ) exp where

φ˜ i ≡ RT ln

)

( )

yiP φ˜ 0i - φ˜ i φ˜ 0i ) exp RT Ps,i RT

( )

( )

Ps,i Ps,i , φ˜ 0i ≡ RT ln 0 ) βiφ0i yiP Pi

(14)

(15)

and Ps,i is the (pseudo) vapor pressure of pure component i at temperature T. As shown in eq 12, the advantage of rewriting the Gibbs adsorption isotherm, eq 7, as a function of φ is that all adsorbates share the same integral function ψ(φ). Therefore, if the single-component characteristic curves of the various adsorbates can be superposed using shifting factors, then their spreading pressures can be derived from a single curve also: ψ(φ). Notice that the relationship between φ and φ˜ can be extended to ψ because this variable may be viewed as the scaled value of ΠA with shifting factor β/Va. An interesting result arises when single-component isotherm data are represented as a temperatureindependent characteristic curve by eq 3 with β ) Va. In this case, for all adsorbates the scaling factor for ΠA is β/Va ) 1. Because the function ψ(φ) - ΠA has a single root, it follows from the IAS theory that the scaled adsorption potentials are equal at the standard state pressures P0i :

φ01 ) ... ) φ0NC

(16)

If Va(T) were the correct shifting factor of φ˜ for a given adsorption system, then eq 16 would greatly simplify the embedding of the IAS theory in a dynamic column breakthrough model. However, because Va(T) does not frequently correlate well with β, eq 16 is not a universal result and, as such, will not be exploited any further. Whether the steps in eq 13 are feasible to implement depends on whether the inverse function of ψ(φ) is itself feasible to compute, either analytically or numerically. Ideally, the form of W(φ) should be such that the integral in eq 12 can be evaluated and inverted analytically to give an expression for ψ that is explicit in φ and cheap to compute. The other requirement is, of course, that W(φ) correlate the pure-gas isotherm data well. Several empirical or semiempirical forms for W(φ) have been proposed in the literature. One of the most popular is the Dubinin-Astakhov (DA) equation:2

W ) Ws exp(-γφn)

or, equivalently,

ψ ≡ (Va/β)ΠA )

potential φ0i , at which the pure-component adsorption equilibrium yields the same value of Π at temperature T, is computed from eq 12, i.e.

(17)

For this particular form of W(φ), eq 12 can be integrated analytically to give

ψ(φ) )

Ws nγ1/n

Γ(1/n,γφn)

(18)

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where

Γ(a,x) ≡

∫x∞ta-1e-t dt

ing multicomponent adsorption equilibrium in dynamic process modeling studies.

(19)

is the complementary incomplete Γ function. A similar result has been derived by Lavanchy et al.28 In particular, if n ) 2, then eq 17 becomes the popular DubininRadushkevich (DR) equation, in which case eq 18 reduces to

ψ(φ) )

Ws xπ/γ erfc(xγφ) 2

(20)

where erfc(‚) is the complementary error function. This result can be traced back to Myers.9 In an attempt to correlate adsorption data obtained under wider experimental conditions by the Potential theory, Ozawa et al.31 found that an equation of the following form is preferable to eq 17: n)∞

ln(Ws/W) )

γnφn ∑ n)1

(21)

where the γn are constants characteristic of the adsorbent. The difficulty in employing eqs 18 and 20 is that they are expensive to compute and their inverses are even harder to evaluate. The case of eq 21 is worse, because its integral cannot be calculated analytically. In principle, one should expect ψ(φ) to be easier to evaluate than W(φ) because the former is a smoother function. To see why this is so, notice that the average value of W(φ) over some interval [φ1,φ2] is

W h [φ1,φ2] )

∫φφ W(φ*) dφ*

1 φ2 - φ1

2

1

(22)

which is directly tied to the definition of ψ(φ) in eq 12. Nevertheless, the difficulty in evaluating ψ(φ) for the forms of W(φ) considered in this work lies in the particular format shared by all of them, which is the exponential of a polynomial in φ. Interestingly, in many tables of mathematical functions and mathematical programming libraries, the functions in eqs 18 and 20 are also coded as the exponential of a polynomial in the argument. These facts led us to postulate that in most practical cases ψ(φ) can be approximated by the exponential of a low-order polynomial in φ. In particular, if the order of this polynomial is at most 2, i.e.,

ln(ψs/ψ) ) s1φ + s2φ2

(23)

where ψs is the value of ψ at saturation and si g 0 are constants characteristic of the adsorbent, then ψ(φ) can be easily inverted to give

φ(ψ) )

s1 [ 1 + (4s2/s12) ln(ψs/ψ) - 1] 2s2 x

(24)

This form of φ(ψ) has all that is required for a computationally efficient implementation of IAS: it is explicit in ψ and is cheap to compute. It is shown next that eq 23 is indeed a suitable representation of experimental ψ data and that the proposed dual-theory formalism provides a simple and practically useful way of correlat-

3. Results and Discussion The dual-theory equilibrium model discussed in the previous section was applied to the experimental data of Malek and Farooq47-49 for adsorption of ternary mixtures of methane, ethane, and propane on activated carbon. The overall effectiveness of the proposed model was determined by applying it to a dynamic process modeling study; all of the required data are readily available from published work: single-component isotherm data,47,48 kinetics of particle uptake,49 and breakthrough results.49 This test provides an effective measurement of the accuracy and computational efficiency of the model, which is more realistic than assessing its performance only as a regression and predictive model for adsorption equilibrium in an isolated context. Figure 1 shows different attempts of correlating the single-component adsorption equilibrium data of the three hydrocarbons by the Potential theory. Under supercritical conditions Ps and Va were evaluated using eqs 4 and 5, respectively. Nonideal gas behavior was correct by converting pressure to fugacity by means of the following equation:50

ln

Pr f ) (B0 + ωB1) P Tr

(25)

where Pr and Tr are reduced pressure and temperature, ω is Pitzer’s acentric factor, and

B0 ) 0.083 -

0.422 0.172 , B1 ) 0.139 (26) 1.6 Tr Tr4.2

Analysis of adsorption data by the Potential theory in a “universal isotherm” context can be regarded as a two-stage process. First, if there is no prior knowledge of the βi’s that convert the various isotherms into a single temperature-independent characteristic curve, they have to be determined. Next, a suitable form for W(φ) capable of correlating the characteristic curve is fitted to the experimental data. In this work the affinity factors were determined using the following procedure. Methane was arbitrarily selected as the reference adsorbate and a unit value was assigned to its affinity factor: β1 ) 1. The problem was thus reduced to determining the values of β for ethane (β2) and propane (β3) that provide the best fit of the experimental adsorption data to a single temperatureindependent characteristic curve. The affinity factor is characteristic of the adsorbate; hence, its value should be independent of the selected form for W(φ). Unbiased values of the βi’s can be obtained by employing an intermediate form for W(φ) with a sufficiently large number of degrees of freedom. The intermediate form selected for W(φ) is that given by eq 21 with n ) 4. It is used solely as a mathematical tool to quantitatively measure the scattering of the data points around a single hypothetical characteristic curve. The βi’s were calculated using a standard multivariate nonlinear minimization package. For each pair of trial values (β2, β3) supplied by the minimization package, the equation for W(φ) was fitted to the experimental isotherm data by linear regression, and the sum of squares of residuals was returned as the value of the minimization function.

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2463 Table 1. Comparison of β Values Obtained in This Study with Some Properties of the Adsorbate That Have Been Previously Employed in the Literaturea i

βi

Vb,i/Vb,1

Pe,i/Pe,1

Pm,i/Pm,1

(Pe,i/Pe,1)0.9 b

2 3

1.38 1.74

1.46 1.98

1.71 2.44

1.53 2.07

1.62 2.23

a V ) molar volume of liquid adsorbate at n.b.p. P ) molar b e polarization. Pm ) molecular parachor. b Correlation of Wood.51

Figure 1. Characteristic curve of adsorption of light hydrocarbons on activated carbon. The symbols denote the experimental data reported by Malek and Farooq47,48 for single-component adsorption equilibrium of methane (O), ethane (4), and propane (0) on activated carbon. The solid lines represent the least-squares fit of different forms of the Potential theory: DA equation (top plot), eq 21 with n ) 2 (middle plot), eq 21 with n ) 3 (bottom plot).

The values of β that best fit the experimental data are β1 ) 1.38 and β2 ) 1.74. These values were subsequently used to compare the predictive characteristics of more tractable forms for W(φ): the DA equation and two instances of eq 21, one with n ) 2 and the other with n ) 3. Note that once the βi’s have been determined these equations are easily fitted to the adsorption isotherm data using linear regression. Furthermore, they can all be inverted analytically to give an expression for φ that is explicit in W (and hence in q). As will be shown later, this is a handy feature when calculating the isosteric heats of adsorption. The fits for the three forms of the characteristic curve are plotted in Figure 1, along with the values of the parameters that provide the best fit of each model. As shown in Figure 1, the Potential theory was very successful in correlating the isotherms of the three

hydrocarbons as a single temperature-independent characteristic curve. This fact corroborates the applicability of the theory to the adsorption system under study. Although there is a general consensus about which parameters β correlates best with for vapor adsorbates,51 the same cannot be said for supercritical gases. It is not yet clear as to which property is most appropriate for correlating the isotherms of supercritical gases on activated carbon. This fact is illustrated in Table 1, which compares the obtained β values with some of the properties of the adsorbate that have been employed in the literature for vapor adsorbates: molar liquid volume, molar polarization (Pe), and molecular parachor (Pm).52 The Vb ratio (third column of Table 1) is very close to the trend observed by Mehta and Danner,18 because these authors defined β ) kVs and obtained k values close to unity for the three adsorbates. As shown in Table 1, the agreement is qualitatively correct but none of the properties examined exactly match our obtained values. Still, it is straightforward to find a suitable correlation for the system under study: for example, both Vb0.83 and Pe0.62 agree very well with the obtained β values. There are, however, no real meaningful conclusions to be drawn from these correlations because of the very small number of adsorbates considered in this study. Figure 1 shows that the DA equation provides a very close fit of the experimental data. The fit of eq 21 with n ) 2 is somewhat less satisfactory compared to the DA equation, especially near φ ) 0. It also gives an overestimated value of the specific pore volume, Ws. Equation 21 with n ) 3 is slightly superior to the DA equation in fitting the data over the entire experimental φ range. However, it has the drawback of not being a monotonically decreasing function of φ (notice the sign change in β3). In fact, the best fit of the model has an incorrect trend at very low pressure because its limit at infinite potential is limφf∞ W(φ) ) ∞ and not zero as it should be. This deficiency, which is due to the limited amount of experimental data available in the low-pressure region, can be overcome by fitting the model using an inequality-constrained linear regression algorithm.53 The least-squares problem that must be solved is

min γi

∑(Wexp - Wcal)2

1 2

s.t. γ3 g 0

(27)

This was not tried, however, because the goodness of fit of the DA equation is satisfactory for the present purposes and, therefore, employed in the rest of the work. To compute the coefficients of the approximation to ψ(φ) given by eq 23, the equation has to be fitted to tabulated values of ψ over its working range computed using the selected form for W(φ). The tabulation of ψ(φ) can be performed either analytically, using eq 18 in the case of the DA equation, or numerically, using a standard adaptive quadrature algorithm applied to eq

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Table 2. Dynamic Column Breakthrough Modela balance component material balance (i ) 1, ..., NC) particle uptake (LDF rate model) overall material balance (adsorbates plus inert carrier) energy balance

differential equation

(

)

∂ci 1 -  ∂q ji ∂ci ∂ DL + F ) - vci ∂t  p ∂t ∂z ∂z ∂q ji j i) ) ki(q/i - q ∂t j i ∂(v/T) 1 ∂T 1 -  R ∂q Fp )  P i ∂t ∂z T2 ∂t ∂q ji ∂T (1 - )Fp Cs (-∆H)i ) ∂t ∂t i vCgP ∂T 2hw ∂ ∂T λe (T - Tw) ∂z ∂z RT ∂z Rc

[ ( )

∑

]

∑

boundary conditions

vf

yf,iP ∂ci ) vci - DL RTf ∂z

v ) vf

for z ) 0

∂ci )0 ∂z

for z ) L

for z ) 0

Cgfvf ) Cgv -

λeR ∂T P ∂z

for z ) 0

∂T )0 ∂z

for z ) L

a The process is isobaric, and the ideal gas law is assumed to apply for the gas phase; subscript f refers to feed or inlet conditions; q/ i is the amount adsorbed of component i in equilibrium with the bulk gas.

Figure 2. Scaled spreading pressure, ψ ) (Va/β)ΠA, as a function of scaled adsorption potential, φ, for light hydrocarbon adsorption on activated carbon. The symbols denote the values of ψ calculated either analytically using eq 18 or numerically using an adaptive quadrature procedure applied to eq 12; the solid line is the leastsquares fit of eq 23.

12. In the latter case, the infinite range of integration [φ, ∞] has to be cut at some finite value φmax. In the present work, no loss in accuracy was observed over the working range of ψ (0-40 kJ/mol) by setting φmax ) 100 kJ/mol. Figure 2 shows the best approximation of eq 23 to the values of ψ computed using the previously fitted DA equation. It is clear that eq 23 provides an excellent fit of ψ over the entire range of adsorption potential. In fact, the agreement is so good that the fit of eq 23 is very close to interpolating the data points. The validity of approximating ψ(φ) by eq 23 has been extensively tested by the authors for the forms of W(φ) given by eqs 17 and 21 with n ) 2 or 3. Invariably, the accuracy of the approximation was always similar to that shown in Figure 2. For the DA equation, which is by far the most frequently employed form of the Potential theory, we have developed a correlation for the coefficients of eq 23 that saves the workload of tabulating ψ(φ) and subsequently fitting eq 23:

ln(γ1/nψs/Ws) ) 0.01642 - 0.89750/n + 0.88111/n1.5 (28) ln(s1/γ1/n) ) 1.91148 - 0.19853en - 1.37027/n1.5 (29) 1/2

s2 /γ

1/n

2

) -0.95514 + 1.01256n - 0.03442n

(30)

Figure 3. Comparison of values of ψ(φ) for the DA equation calculated analytically using eq 18 (symbols) and the prediction of eq 23 with coefficients computed according to eqs 28-30 (solid lines). The comparison is made for various values of the exponent n of the DA equation (see eq 17).

As demonstrated in Figure 3, the approximation to ψ(φ) using the coefficients computed according to the formulas given above remains extremely accurate and can be safely employed to compute ψ(φ) for the DA equation. The formulas given by eqs 28-30 were developed with the aim of fitting ψ(φ) as accurately as possible for values of n in the range 1 e n e 2. The validity of the formulas should be checked for values of n deviating significantly from this range. The computational efficiency of the proposed model was assessed by embedding it in a dynamic-column breakthrough model based on the adsorption system for which the equilibrium data were measured. The model is essentially a nonisothermal and variable-velocity axially dispersed plug-flow model. The linear driving force rate model is used for particle uptake, and it is assumed that there is no kinetic interaction between the three hydrocarbons for adsorption on the particle. The process is isobaric through the use of a backpressure regulator. Table 2 lists the governing equations, the details of which are described in Malek and Farooq.49 The isosteric heat of adsorption for each adsorbate, (-∆H)i, is estimated using the Clausius-Clapeyron equation as

(

(-∆H)i ) RT2

)

∂ ln yiP ∂T

q jj

(31)

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2465 Table 3. Values of the Main Parameters Employed in Multicomponent Breakthrough Simulations carrier Fp (g cm-3) dp (cm)  Cs (J g-1 K-1)

helium 0.87 0.258 0.40 0.95

L (cm) Rc (cm) hw (J cm-2 s-1 K-1) Tf (K) P (bar)

40.0 1.75 0.0025 299.15 2.583

feed mole fraction

run 1

run 2

yf,1 yf,2 yf,3

0.314 0.123 0.041

0.156 0.157 0.146

LDF coefficient (s-1)

run 1

run 2

k1 k2 k3

0.290 0.045 0.009

0.260 0.050 0.013

which when applied to the Potential theory, eq 3, gives

(

(-∆H)i ) βiφ h i + RT2

)

d ln Ps,i dT d ln Va,i β iT dT

(

)/(

Figure 4. Breakthrough curves for methane (C1), ethane (C2), and propane (C3), in activated carbon (run 1). Comparison between model predictions (lines) and experimental measurements (symbols) reported by Malek and Farooq.49 Kinetic and operating parameters are given in Table 3.

d ln W (32) dφ φ)φh i

)

where φ h i is such that W(φ h i) ) q j iVa,i. This is a straightforward calculation because the DA equation can be inverted to give an expression for φ h i that is explicit in q j i. The dynamic model was solved numerically using the finite-volume method54 for spatial discretization with flux limiting to achieve nonoscillatory solutions and a modified version of the DASSL42 code for integrating the solution over time. Details of the numerical procedure are given elsewhere.55,56 Preliminary runs were carried out in order to assess the accuracy of the computed solution. It was found that 25 equally spaced grid points give results independent of spatial resolution to within the accuracy requested (10-4 relative error tolerance). The dynamic model was solved on a low-end personal computer (200 mHz Pentium) in order to more easily determine its effectiveness in terms of computational load. Each run took less than 10 s, which demonstrates that adopting the proposed multicomponent equilibrium model incurs a relatively small computational workload. Table 3 lists the values of the main parameters employed in two multicomponent runs corresponding to different feed gas compositions. Their values are the same as those used by Malek and Farooq.49 Figures 4 and 5 compare the experimental and simulation breakthrough profiles for the two runs. It is clear that there is good agreement between experimental profiles and model predictions. The quality of the fit is nearly the same as that obtained by Malek and Farooq49 using the extended Langmuir-Freundlich isotherm model. The advantage of the dual-theory approach is that much less adsorption equilibrium measurements are required to predict the breakthrough profiles. The success of this procedure depends, however, on whether a suitable property that correlates with β for a given adsorption system is known a priori. As shown in Figure 5, the model overestimates slightly the temperature rises in the second run, where the feed gas is more concentrated in the heavier hydrocarbons. The discrepancy between the experimental temperature profile and the model prediction is

Figure 5. Breakthrough curves for methane (C1), ethane (C2), and propane (C3), in activated carbon (run 2). Comparison between model predictions (lines) and experimental measurements (symbols) reported by Malek and Farooq.49 Kinetic and operating parameters are given in Table 3.

partially due to a thermodynamic inconsistency associated with the low-pressure behavior of isotherms derived from the Potential theory. According to eq 32, the limiting value of the isosteric heat of adsorption at low pressure is limφif∞(-∆H)i ) +∞, which is clearly incorrect. This results in an overestimation of the heat of adsorption in the low-pressure region. However, because the cumulative heat of adsorption is proportional to particle loading, its overestimation is kept under reasonable bounds. Nevertheless, as shown in Figure 5, it can become noticeable when the more adsorbable species are present at sufficiently high concentration levels in the feed gas. At first sight, the thermodynamic inconsistency of the Potential theory at very low pressure, and also the fact that limPf0 φi ) +∞, prevents modeling of adsorption from a clean bed, where initially φi ) +∞. In practice this limitation is not a serious one, because the initial conditions applying for a clean bed can be replaced by those for an unclean one in which the less adsorbable component (methane in our case) is present at low concentration. The overall accuracy of the computed solution is unaffected if the trace concentration is set at a sufficiently small value.

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4. Conclusions A dual-theory multicomponent equilibrium procedure, based on the Potential and IAS theories, was presented. For the more commonly employed forms of the characteristic curve, an accurate and simple expression can be fitted to the spreading pressure data, which can be subsequently inverted to give an expression for ψ that is explicit in ψ. This feature increases the feasibility of implementing the IAS method. The applicability of the dual-theory procedure was illustrated by incorporating it in a dynamic-column breakthrough model of light hydrocarbon adsorption on activated carbon. The model provided good predictions of experimental breakthrough profiles for the test case without incurring a too large computational workload. A similar equilibrium model has been employed recently in a theoretical study of the impact of natural gas composition on its storage by adsorption.57 However, the details of the multicomponent equilibrium procedure were only briefly outlined in that reference. The cyclic operation of an on-board natural gas adsorptive storage system was modeled as a series of consecutive cycles. Each cycle is a two-step process consisting of charge with a fixed composition gas mixture, followed by discharge at a constant molar flow rate. The gas mixture studied is a six-component hydrocarbon mixture representative of natural gas. The overall computational efficiency of the dynamic model was good enough to allow 1000 cycles of operation to be simulated in a reasonable amount of time (less than 20 min) on a PC with a 400 mHz Pentium processor. Acknowledgment This work was partly supported by Fundac¸ a˜o para a Cieˆncia e Tecnologia through Program PRAXIS XXI (PCEX/C/QUI/109/96). A.J.S.R. acknowledges financial support from FCT through Grant PRAXIS XXI BD/ 18182/98. Nomenclature A ) specific surface area of adsorbent, cm2 g-1 c ) gas-phase concentration in bed, mol cm-3 Cg ) heat capacity of the gas mixture, J mol-1 K-1 Cs ) heat capacity of the adsorbent, J g-1 K-1 DL ) axial dispersion coefficient, cm2 s-1 dp ) mean particle diameter, cm hw ) wall heat-transfer coefficient, J cm-2 s-1 K-1 k ) LDF coefficient, s-1 L ) column length, cm R ) gas constant, 8.314 J mol-1 K-1 Rc ) column internal-wall radius, cm t ) time, s Tw ) wall temperature, K v ) interstitial velocity, cm s-1 z ) axial coordinate in column, cm Greek Symbols  ) bed void fraction λe ) effective axial thermal conductivity, J cm-1 s-1 K-1 Fp ) particle density, g cm-3

Subscripts f ) feed or inlet conditions i ) adsorbate component (excluding inert carrier)

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Received for review November 23, 1999 Revised manuscript received March 20, 2000 Accepted April 3, 2000 IE9908478