3002
Ind. Eng. Chem. Res. 1997, 36, 3002-3012
Multicomponent Transport Effects in Sorbent Particles under Pressure Swing Conditions Atanas Serbezov and Stratis V. Sotirchos* Department of Chemical Engineering, University of Rochester, Rochester, New York 14627
A comprehensive theoretical study of the problem of multicomponent adsorption-desorption in sorbent particles under pressure swing conditions is carried out in this study. A rigorous mathematical model is formulated for the process in which the dusty-gas model and the commonly used Fickian model are employed to describe the coupling of the mass fluxes and the partial pressures in the pore space, with or without viscous flow. The dusty-gas model is used as a standard against which the performance of the Fickian model is tested. Computations are carried out for binary and quaternary mixtures. It is found that the rigorous model (the one based on the dusty-gas model) predicts faster responses for the partial pressures of the components in the mixture. Faster responses are also observed when the viscous flow is accounted for in the transport model, with the effect being considerably stronger in the case of the Fickian model. The obtained results are also used to investigate the performance of the Glueckauf linear driving force approximation for multicomponent systems. It is concluded that like the Fickian diffusion model on which it is based, this approximation cannot predict, even qualitatively, the dynamic responses of the partial pressures in a multicomponent system. Introduction The use of distributed-parameter models for describing the behavior of adsorption-based separations, such as pressure swing adsorption (PSA), is becoming increasingly common as a result of the availability of better computational resources. A distributed-parameter model for PSA can be broken up into several submodels: a model for the adsorbing bed, a model for the adsorbing particle, and a model for the adsorption process at the adsorbing surface. The models for the bed and the particle are derived by applying the general conservation principles to the mass of each species and to the total energy. In general, four mechanisms of mass transport have to be considered: bulk diffusion, Knudsen diffusion, Knudsen flow, and viscous flow. In the adsorbing bed the dominant mode is typically viscous flow (Serbezov and Sotirchos, 1997) and the contributions of the other mechanisms can be neglected, but this is not the case for the sorbent particles where, depending on the operating conditions, all four mechanisms may be equally important. This is why for a mixture of many species the intraparticle mass transport model must accurately describe the multicomponent mass transport of the individual species over a wide range of conditions in order to be useful for simulations. The model of choice for describing intraparticle mass transport in the PSA literature has been the Fickian model (Yang, 1987), which assumes independent transport by diffusion of each species in the pore space. Although widely applied and accepted, this model has several drawbacks. It does not account for the viscous flow and underestimates the Knudsen flow of each species caused by the total pressure gradient, and for mixtures of more than two components, the Fickian mass transport coefficient becomes an adjustable parameter that must be obtained by fitting experimental data even for pore structures that can be represented as parallel pore bundles. The multicomponent PSA models can be significantly simplified by replacing the detailed particle model by * Author to whom correspondence should be addressed: phone, (716) 275-4626; fax, (716) 442-6686; e-mail, svs2@che. rochester.edu. S0888-5885(96)00699-9 CCC: $14.00
an approximate lumped parameter expression which accounts for the finite rate of the intraparticle mass transport. The most widely used approximation is the Glueckauf linear driving force (LDF) approximation (Glueckauf, 1955) which for PSA processes with short cycle times has been modified by Nakao and Suzuki (1983), Alpay and Scott (1992), and Carta (1993). Since it is based on the Fickian mass transport model, the LDF approximation carries all of its limitations. The occurrence of bulk diffusion, Knudsen diffusion, Knudsen flow, and viscous flow during transport of gases in porous materials is accounted in the dusty-gas model (DGM) (Jackson, 1977; Mason and Malinauskas, 1983; Sotirchos, 1989). We use this model in this study to develop a detailed multicomponent model for the simultaneous mass transport and adsorption (desorption) in sorbent particles under pressure swing conditions. Computations are carried out for binary (O2-N2 and H2-CO2) and quaternary (CO2-H2-CH4-N2) mixtures to investigate the relative importance of bulk diffusion, Knudsen diffusion, Knudsen flow, and viscous flow in a broad range of conditions. The dusty-gas model is used as standard against which the performance of the Fickian model is investigated in order to identify under what conditions the Fickian model produces inaccurate results. The obtained results are also used to examine the validity and the applicability of the linear driving force approximation in multicomponent mixtures under pressure swing conditions. Mathematical Models and Equations Mass Balance Equations. We consider a spherical particle of an isotropic porous adsorbent exposed to a multicomponent gaseous mixture of n species with constant composition, which is subjected to a cyclic change in the total pressure. The process is isothermal, and the operating conditions ensure the validity of the ideal gas law. The problem is one-dimensional in the radial direction, but a multidimensional notation will be employed since the treatment we follow is also applicable to multidimensional problems involving isotropic porous media. The set of mass balance equations for the individual species in the multicomponent mixture is written © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3003
p ∂p + ∇‚N ) Rp RgT ∂t
(1)
The partial pressures at the external surface of the particle are assumed to follow the variation of the total pressure, that is, we have the boundary conditions:
At R ) Rp
p ) PTy
y ) const
(2)
p is the porosity of the adsorbing particle, p is the vector of the partial pressures of the species, T is the temperature, Rg is the ideal gas law constant, N is the vector of the molar fluxes of the species within the particle relative to stationary coordinates, Rp is the vector of the uptake rates per unit volume of the particle, R is the radial coordinate in the particle, Rp is the radius of the particle, y is the vector of the mole fractions of the species in the bulk of the particle, and PT is the total pressure. The adsorbed and the fluid species within the pores are assumed to be locally in equilibrium with each other at all times. The vector of the local uptake rates is expressed as
Rp ) -(1 - p)
∂q ∂t
(3)
q is the vector of the equilibrium solid phase concentrations of the species which are related to the partial pressures of the individual components in the fluid phase through the adsorption isotherm. If the adsorption isotherm is explicit in the adsorbed species concentrations, i.e.
qi ) qi(p1,p2,...,pn)
(4)
∂q ∂p )Q ∂t ∂t
(5)
then
where Q is the Jacobian matrix of q with respect to p. Using eqs 3 and 5, the mass balance equation (eq 1) takes the form
[
p ∂p )I + (1 - p)Q ∂t RgT
]
(6)
(7)
n
Ni ∑ i)1
Ji ) 0 ∑ i)1
(9)
In a different representation, the molar flux of the gaseous species in the adsorbing particle relative to stationary coordinates is viewed as consisting of two additive components: a viscous flux and a diffusive flux (Jackson, 1977; Mason and Malinauskas, 1983)
Ni ) NV,i + ND,i
(10)
NV,i is the viscous flux of species i arising from the viscous flow in the particle in which the gas acts as a continuum fluid driven by total pressure gradient. The viscous fluxes of the individual species in a multicomponent mixture are independent of each other and can be expressed by D’Arcy’s law (Bird et al., 1960)
NV,i ) -
yi BePT ∇PT RgT µ
(11)
Be is the effective permeability of the adsorbing particle and µ is the viscosity of the gaseous mixture. The quantity (BePT/µ) has dimensions of transport coefficient and can be viewed as representing the effective coefficient for viscous flow in the particle. ND,i is the diffusive flux of species i resulting from the bulk (molecule-molecule interactions) and Knudsen (molecule-wall collisions) mechanisms for mass transport. The relation between the diffusive flux ND,i and the pure diffusion flux Ji can be obtained by combining eqs 7, 8, 10, and 11. The final form is
ND,i ) yiND,T + Ji
(12)
where ND,T is the total diffusive flux given by n
∇‚N
NT is the total convective flux, that is, the total molar flux resulting from the bulk motion of the fluid. It is defined as
NT )
n
-1
In order to solve eq 6 for the partial pressures of the individual components as functions of time and location in the particle, a mass transport model is needed. Its function is to provide an expression for ∇‚N as function of p, ∇p, and ∇2p. Multicomponent Mass Transport Equations. We focus on the multicomponent mass transport effects in the gas phase only, and in order to avoid unnecessary complication of the discussion, we do not consider surface diffusion in the mathematical model. The molar flux of each gaseous species in the adsorbing particle relative to stationary coordinates can be represented as the resultant of two vector quantities (Bird et al., 1960):
Ni ) yiNT + Ji
The convective flux of each component is given by the product (yiNT). Ji is the pure diffusion flux of species i, that is, the molar flux relative to a coordinate system that moves with the average molar velocity. It follows from eqs 7 and 8 that
(8)
ND,T )
ND,i ∑ i)1
(13)
There are two different approaches to model the diffusive fluxes. The first one is based on Fick’s first law of diffusion while the second one is based on the Stefan-Maxwell relations (Bird et al., 1960). Fickian Model. A constitutive equation for the pure diffusion flux Ji in eq 7 is generally known as Fick’s first law of diffusion (Bird et al., 1960)
Ji ) -
1 e D P ∇y RgT F,i T i
(14)
The Fickian model (FM) for the diffusive flux ND,i in eq 10 is formulated in analogy to eq 14
ND,i ) -
1 e D ∇p RgT F,i i
(15)
DeF,i is the effective Fickian diffusion coefficient of component i in the mixture of the remaining (n - 1) components. The values of DeF,i are commonly calculated by a formula attributed to Bosanquet (Pollard and Present, 1948)
3004 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
1 1 1 ) e + e e DF,i DF,i DK,i
(16)
DeK,i is the effective Knudsen diffusion coefficient which accounts for the molecule-wall collisions. DeF,i is the effective bulk diffusivity of component i in the mixture of the rest (n - 1) components which accounts for molecule-molecule interactions. DeF,i depends in general on the mole fractions and the molar fluxes of all species in the mixture. Because of this dependence, DeF,i cannot be determined theoretically, and its values are usually obtained by fitting experimental data. For a binary mixture DeF,i is usually taken to be equal to the binary diffusivity which can be computed from the Chapman-Enskog equation listed in Table 2. Another option is to use the Wilke approximation for the effective diffusivity of a gas in a multicomponent mixture of stagnant gases (Wilke, 1950)
( ) n
e DF,i )
∑ j)1
yj
(17)
e Di,j j*i
equations (Jackson, 1977; Mason and Malinauskas, 1983; Sotirchos, 1989)
∇pi ) -RgT
(
∑ j*i
yjND,i - yiND,j e Di,j
(
)
1 Be e Ni ) p ∇P + DF,i ∇pi RgT µ i T
(18)
(
∇pi +
Bepi
)
∇PT ) -RgT
e µDK,i
(
(19)
)
1 Be T pe + DeF ∇p N)RgT µ
Ni +
e Di,j
(20)
two-parameter Fickian model 1 ∇‚(DeF∇p) RgT
(21)
three-parameter Fickian model
) ]
1 Be T ∇‚ pe + DeF ∇p RgT µ
(22)
The corresponding forms of the mass balance equation when the Fickian model is used for the mass transport are listed in Table 1. Dusty-Gas Model. The dusty-gas model (DGM) states that the diffusive fluxes of the species in a multicomponent gaseous mixture satisfy the following set of
)
e DK,i
(24)
(25)
three-parameter dusty-gas model (26)
B and F are [n × n] matrices defined by
Bi,i ) -RgT
(
pj
∑ j*i
1 +
e Di,j
e DK,i
B i,j ) RgT
( )
(
Bepi
j*i
Fi,i ) 1 +
F i,j )
where DeF is a diagonal matrix of the Fickian diffusion coefficients and e is an n-dimensional vector with all elements equal to unity. The Fickian model is explicit in N, and ∇‚N needed in eq 6 can be obtained analytically by direct differentiation of eqs 19 and 20. We have
[(
yjNi - yiNj
two-parameter dusty-gas model
three-parameter Fickian model
∇‚N ) -
∑ j*i
In our presentation we will refer to eq 23 as the twoparameter dusty-gas model and to eq 24 as the threeparameter dusty-gas model. In vector-matrix notation eqs 23 and 24 can be written as
two-parameter Fickian model 1 De ∇p ND ) RgT F
(23)
Equations 10, 11, and 23 can be combined together in order to formulate a similar expression for the complete molar fluxes of the components
F∇p ) BN
In the discussion to follow, eq 15 will be referred to as the two-parameter Fickian model and eq 18 as the three-parameter Fickian model. Equation 18 is also referred to as the Fickian diffusion/convection model. Equations 15 and 18 can be rewritten in vector-matrix notation as
∇‚ND ) -
)
e DK,i
∇p ) BND
When eqs 11 and 15 are substituted in eq 10, the following expression for the molar fluxes is obtained:
(
Ni +
j*i
)
pj
e Di,j
)
e µDK,i
( ) Bepi
e µDK,i
(27)
(28)
(29)
(30)
For the dusty-gas model analytical expressions for ∇‚ND and ∇‚N cannot be derived since it is implicit in the fluxes. Sotirchos (1991) proposed to compute these quantities numerically at every spatial and temporal position by first differentiating eqs 25 and 26 and then solving them for ∇‚ND and ∇‚N, respectively. We have
∇‚ND ) B-1(∇2p - ∇B‚ND)
(31)
∇‚N ) B-1(∇F‚∇p + F∇2p - ∇B‚N)
(32)
∇B and ∇F are matrices whose elements are ∇Bi,j and ∇Fi,j, respectively. Equations 31 and 32 enable us to incorporate the transport equations into the mass balance equations decreasing by n the total number of equations in the overall model. A similar approach was taken by Coppens and Froment (1996) in a multicomponent pellet model describing the catalytic reforming of naphtha. The corresponding forms of the mass balance equation when the dusty-gas model is used for the mass transport are listed in Table 1. Linear Driving Force Approximation. If the adsorption isotherm is linear, i.e.
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3005
qi )
Hi p RgT i
(33)
and the two-parameter Fickian model is used as a transport model, the mass balance equation can be simplified to
[
e DF,i ∂pi ) ∇‚ ∇p ∂t p + (1 - p)Hi i
]
(34)
Using eq 33, eq 34 can be rewritten in terms of the solid species concentrations
∂qi e ) ∇‚(DS,i ∇qi) ∂t
(35)
Equation 35 is commonly referred to as the solid adsorbed phase diffusion model and the coefficient e DS,i is the solid diffusion coefficient. The relation e between the solid diffusion coefficient DS,i and the e Fickian diffusion coefficient DF,i is e ) DS,i
e DF,i p + (1 - p)Hi
(36)
The linear driving force (LDF) approximation approximates eq 35 written in terms of the average solid phase concentration q j i with a linear expression (Glueckauf, 1955):
∂q ji ) ki(qi|R)Rp - q j i) ∂t
(37)
The value of the mass transfer coefficient ki is commonly taken to be e ki ) 15DS,i /Rp2
(38)
Equation 38 has been derived for processes with dimensionless half cycle times θ1/2 greater than 0.1 with θ1/2 given by e θ1/2 ) DS,i t1/2/Rp2
(39)
and t1/2 being the real half cycle time. For instantaneous pressure changes, eq 37 has the solution
j i|R)Rp - (qi|R)Rp - q j i|t)0) exp(-kit) qi ) q
(40)
Computational Method. The mathematical models are solved numerically using a collocation scheme based on B-spline interpolation (De Boor, 1978). The spatial domain is divided into N intervals. In each interval the
spatial variables are approximated by a polynomial of order M. The separate polynomial pieces are joined at the breakpoints by matching the values (left and right) of the function and the value of its first spatial derivative. Thus, the solution is represented with a piecewise continuous polynomial (polynomial spline). The mass balance equations are discretized in space by requiring that they be satisfied at (M - 2) collocation points for each interval and the boundary conditions be valid at the end points. The discretization procedure transforms each mass balance equation from a partial differential equation to a system of N(M - 2) ordinary differential equations in time and two algebraic equations which come from the boundary conditions. For a mixture of n components the mass balance equations yield a system of nN(M - 2) ordinary differential equations and 2n algebraic equations. The integration of the resulting sets of algebraic and ordinary differential equations is carried out by a Gear-type solver (Gear, 1971). The results presented and discussed in this study are obtained using 40 intervals of equal size to discretize the particle equation, with each partial pressure profile approximated by a cubic polynomial spline over each interval. However, satisfactory results can be obtained using as many as three intervals. If the same discretization scheme is applied to the dusty-gas model equations, n[N(M - 2) + 2] more algebraic equations are obtained, and consequently the total size of the system that has to be solved by the differential-algebraic equation solver is doubled. However, we can use the differentiated forms of the dustygas model (eqs 31 and 32) and compute the divergences of the molar fluxes at every spatial and temporal position. Consequently, only the differential equations that result from the discretization of the mass balance equations and the algebraic equations obtained from the boundary conditions must be used in the integrator, thus reducing dramatically the computational time needed. Additional information on the numerical scheme is given by Sotirchos (1991). Results and Discussion We apply the mathematical models developed in the previous section to obtain the dynamic response of a single sorbent particle immersed in a multicomponent mixture of constant composition at 300 K and subjected to cyclic pressurization and depressurization. The duration times of the pressurization and the depressurization intervals are equal, and unless specified otherwise, instantaneous pressure change is employed. The initial condition for each half cycle is the final condition of the previous one. At the onset of the process the bulk phase inside the particle has an uniform composition, the same as the surrounding mixture. The
Table 1. Intraparticle Model Equations mass transport model 2-parameter dusty-gas model 3-parameter dusty-gas model
2-parameter Fickian model
3-parameter Fickian model
[ [
mass balance equation
p ∂p )I + (1 - p)Q ∂t RgT p ∂p )I + (1 - p)Q ∂t RgT
[ [
] ]
-1
B-1(∇2p - ∇B‚ND)
-1
p 1 ∂p ) I + (1 - p)Q ∂t RgT RgT p 1 ∂p ) I + (1 - p)Q ∂t RgT RgT
B-1[∇‚(F∇p) - ∇B‚N]
] ] [( -1
∇‚(DeF∇p)
-1
∇‚
) ]
Be T pe + DeF ∇p µ
3006 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 Table 2. Equations for the Effective Transport Properties coefficient
equation
( )( )
e Si,j
P ˆT T PT T ˆ
e Di,j )D ˆ i,j
D ˆ i,j ) 2.628 × 10
reference
3/2
-3
S1 Chapman-Enskog equation
xTˆ 3(Mi + Mj)/(2MiMj) 2 P ˆ Tσi,j ΩSi,j
S1 ) p/ηB e DK,i
Bird et al. (1960)
e DK,i )D ˆ K,i
D ˆ K,i ) 97rˆ
(TTˆ )
1/2
S2 Present (1958)
x
T Mi
p r S2 ) ηK rˆ BePT/µ
µ)
semiempirical formula of Wilke
yiµi
n
∑ ∑y Φ n
i)1
j
i,j
j)1
( ) [ ()( )]
1 Φi,j ) x8
-1/2
µi 1+ µj
1/2
Mj Mi
p r 2 ην 8
B)
sorbent particle is assumed spherical with radius Rp ) 10-3 m, porosity p ) 0.6, and uniform pore size, which is varied between 10-8 and 10-6 m. Three multicomponent mixtures have been considered: a binary mixture of O2 and N2 (air) with molar ratio 21:79, a binary mixture of CO2 and H2 with molar ratio 70:30, and a quaternary mixture of CO2, H2, CH4, and N2 with molar ratio 40:20:15:25. The adsorption isotherm for the O2N2 mixture is assumed linear (eq 33) with HO2 ) 5 and HN2 ) 15. An extended Langmuir isotherm is employed for the CO2-H2 and the CO2-H2-CH4-N2 mixtures, i.e.
qi ) qm,i
bipi
(41)
n
1+
∑ j)1
Bird et al. (1960)
1/4 2
Tomadakis and Sotirchos (1993) showed that this assumption cannot be justified at intermediate and high porosities. The tortuousity factors which they report for porosity p ) 0.6 are slightly smaller than the value of 3 adopted here, but this has an insignificant effect on the results of the present study. Comparison between the Dusty-Gas Model and the Fickian Model. The DGM representation of the pure diffusion fluxes and the total convective flux is obtained by combining eqs 7, 10, 11, and 23 (Jackson, 1977)
(
∑ i*j
yjJi - yiJj 1
1 +
e Di,j
bjpj
with
n
e e DK,i DK,j
qm,CO2 ) qm,H2 ) qm,CH4 ) qm,N2 ) 1.78 kmol/m3
∑ k)1
yk
e DK,k
PT
∇yi -
RgT
bCO2 ) 0.1250 MPa ; bH2 ) 0.0014 MPa -1
-1
bCH4 ) 0.0475 MPa ; bN2 ) 0.0098 MPa -1
n
-1
The parameters for the isotherms were chosen on the basis of data compiled from the literature (Yang, 1987; Kumar, 1989). The formulas used for the calculation of the transport coefficients are summarized in Table 2. In order to determine the structural parameters of the porous medium, we represent its structure as an array of long cylindrical capillaries randomly oriented in the threedimensional space. The tortuousity factors are obtained as orientationally averaged quantities with the assumption that the mass transport in the overlap regions among different capillaries affects negligibly the overall mass transport. Burganos and Sotirchos (1989) and
)
∑ j)1 NT ) -
Jj 1
∑ j)1
e DK,j
yi RgT
e DK,i
1-
∇PT (42)
yj
n
∑ j)1
( ) BePT
1
+
yj
( ) 1
e DK,j
e DK,j
n
)
-1
RgT
µ
n
∑ j)1
yj
∇PT (43)
e DK,j
These equations indicate that in a multicomponent mixture the pure diffusion fluxes are influenced by the total pressure gradient. This dependence disappears when all effective Knudsen transport coefficients are equal. This is the case for multicomponent mixtures of identical species (such as mixtures of isomers) or multicomponent mixtures of similar species (such as the binary mixture of O2 and N2 considered in this study).
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3007
Figure 1. Resistor analogue of the dusty-gas model for a mixture of identical or similar gases (e.g., binary mixture of O2 and N2) having effective Knudsen diffusivity DeK and effective binary diffusivity De.
Figure 2. Resistor analogue of the Fickian model for a mixture of identical or similar gases (e.g., binary mixture of O2 and N2) having effective Knudsen diffusivity DeK and effective binary diffusivity De.
Using De and DeK to denote the common effective bulk and Knudsen transport coefficients, respectively, eq 42 simplifies to
(Ji)DGM ) -
(
1 1 1 + RgT De De K
)
-1
PT∇yi
(44)
which has the same form as Fick’s first law of diffusion (compare eq 44 with eq 14). For the binary mixture of O2 and N2, De denotes the effective binary bulk diffusion e coefficient of O2 and N2 De ) DO and DeK stands for 2,N2 the effective Knudsen diffusion coefficient of either one e e of the species, O2 or N2 DeK ) DK,O ≈ DK,N . The DGM 2 2 representation of the total convective flux for a mixture of similar gases is obtained through simplification of eq 43. We have that
(NT)DGM ) -
(
)
e 1 B PT + DeK ∇PT RgT µ
(45)
To obtain the Fickian representation of the total convective flux for a mixture of similar gases, eq 18 is rewritten as
[ (
) ] [
BePT 1 e (Ni)FM ) yi D + ∇PT + RgT F µ -
Figure 3. Transport coefficients vs pore radius of O2 in a binary mixture of O2 and N2 at constant total pressure PT ) 0.35 MPa and constant temperature T ) 300 K.
]
1 e D P ∇y (46) RgT F T i
The second term in eq 46 stands for the pure diffusion flux of the component, Ji (eq 14). Comparing eq 46 with eq 7, we conclude that the Fickian representation of the total convective flux for a mixture of similar gases is given by
(NT)FM ) -
(
)
BePT 1 DeF + ∇PT RgT µ
(47)
To facilitate the comparison of DGM and FM flux models for mixture of similar species (such as the binary mixture of O2 and N2), we present the resistor analogues for these models in Figures 1 and 2, respectively. The fluxes correspond to currents, while the reciprocals of the transport coefficients correspond to resistances. The current through the upper branch of each circuit is driven by the composition gradient. It stands for the pure diffusion flux and is independent of the currents in the lower two branches which are driven by the total pressure gradient. In the presence of composition gradient, bulk diffusion and Knudsen diffusion occur in series and both contrib-
Figure 4. Transport coefficients vs total pressure of O2 in a binary mixture of O2 and N2 at constant pore radius r ) 10-7 m and constant temperature T ) 300 K.
ute to the pure diffusion flux of each species. The dominant mechanism is that characterized by a larger value of transport resistance (smaller value of transport coefficient). The effective transport coefficients of O2 in a binary mixture of O2 and N2 at 300 K are shown in Figures 3 and 4. The effective Knudsen coefficient of N2 is not shown because it is almost identical to that of
3008 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
O2. Figure 3 presents the transport coefficients as functions of the pore radius at constant total pressure 0.35 MPa, whereas Figure 4 presents the transport coefficients as functions of the total pressure at constant pore radius 10-7 m. As seen from these figures, the bulk diffusion coefficient is independent of pore radius and is inversely proportional to the total pressure. The Knudsen transport coefficient, on the other hand, is independent of total pressure and proportional to pore radius. Thus, the Knudsen mechanism is important in small pores and low pressures, whereas bulk diffusion prevails in big pores and high pressures. Two mechanisms contribute to the convective transport in the presence of a total pressure gradient: Knudsen flow and viscous flow. Since they occur in parallel, their relative importance is proportional to the magnitude of their transport coefficients (inversely proportional to the values of the resistances). Figures 3 and 4, as well as Table 2, show that the viscous transport coefficient is proportional to the total pressure and to the square of the pore radius. Thus, in large pores and at high pressures the contribution of the viscous flux is expected to be significant. In multicomponent mixtures of species with different molecular weights, such as CO2 and H2, the presence of total pressure gradient can also lead to significant differences between the predictions of the dusty-gas model and the Fickian model for the pure diffusion fluxes because the second term in the right hand side of eq 42 can be significant in this case. It follows from the above discussion that the Knudsen mechanism contributes to the mass transport flux under both driving forces, that is, composition gradient and total pressure gradient. Figures 1 and 2 show that for the binary mixture of O2 and N2 (or for any multicomponent mixtures of identical or similar species) the dusty-gas model and the Fickian model differ only in the predicted resistance for the Knudsen flow in the particles which is driven by a total pressure gradient. Whereas the dusty-gas model correctly predicts that this resistance is equal to 1/DeK, the corresponding resistance in the Fickian model is 1/DeF. It is seen from Figure 3 that for pores in the range (10-7-10-6) m, the value of DeK is an order of magnitude above the value of DeF. Consequently, the Fickian model is expected to underestimate significantly the dynamic response of the particle to cyclic changes in the total pressure. Figure 5 shows the dynamic response of an adsorbing particle to instantaneous pressurization from 0.1 to 0.4 MPa and subsequent instantaneous depressurization from 0.4 to 0.1 MPa for a binary mixture of CO2 and H2. Plotted in this figure is the dimensionless partial pressure of H2 at the center of the particle that is predicted by the different mass transport models. The reference pressure used to make the pressures dimensionless in this figure and everywhere else is 0.1 MPa. The half cycle time is 0.5 s, which is long enough for complete equilibration of the particle with the surrounding mixture at the end of each half cycle. The acceleration of the dynamic response of the particle caused by Knudsen flow is clearly seen by comparing the predictions of the two-parameter DGM and the twoparameter FM. Figure 6 presents composition profiles of the light component (H2) in a binary mixture of CO2 and H2 after 0.1 s following instantaneous pressurization from 0.1 to 0.4 MPa. The composition of the mixture at the external surface of the particle is maintained constant, and therefore, the mole fraction profile of H2 should be
Figure 5. Response curves for the partial pressure of H2 in a binary mixture of CO2 and H2 at the center of a particle (p ) 0.6, Rp ) 10-3 m, r ) 10-7 m) for the first cycle of a process with instantaneous pressurization and depressurization at constant temperature T ) 300 K predicted by the two- and three-parameter versions of the dusty-gas and the Fickian mass transport models.
Figure 6. Intraparticle partial pressure profiles of H2 at t ) 0.1 s for the response curves shown in Figure 5.
flat if the two species were transported in the interior of the particle at the same rate. However, H2 not only has a much larger effective Knudsen diffusivity than CO2 but also is adsorbed less strongly. This makes its effective rate of intraparticle mass transport larger than that of CO2, and as a result, pressure equilibration in the interior of the particle occurs primarily by H2 transport. This leads to higher partial pressures of H2 in the interior than at the surface. The presence of overshoot in the response of the partial pressure of H2 in the interior of the particle can be clearly seen in the response curves shown in Figure 5. Since for 10-7 m pores the effective Knudsen diffusivities (the mass transport coefficients for transport driven by the total pressure gradient in the two-parameter dusty-gas model) are much larger than the Bosanquet diffusivities (the corresponding mass transport coefficients in the twoparameter Fickian model), much larger values of intraparticle H2 partial pressures are predicted by the dustygas model.
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3009
Figure 7. Total pressure difference between the surface and the center of a particle (p ) 0.6, Rp ) 10-3 m) during instantaneous pressurization and depressurization at constant temperature T ) 300 K for two particle sizes.
Figure 8. Response curves for the partial pressure of O2 in a binary mixture of O2 and N2 at the center of a particle (p ) 0.6, Rp ) 10-3 m, r ) 10-8 m) for the first three cycles of a process with instantaneous pressurization and depressurization at constant temperature T ) 300 K predicted by the two- and threeparameter versions of the dusty-gas and the Fickian mass transport models.
Investigation of the Intraparticle Convective Effects. The driving force for the intraparticle convective transport is the gradient of the total pressure. At the beginning of each half-cycle of instantaneous pressure change, the total pressure at the outer surface of the particle changes stepwise and remains constant for the rest of the half-cycle. However, the total pressure at the center of the particle does not change stepwise because the permeability of the particle is not infinitely large. Figure 7 shows the difference between the total pressure at the surface and the total pressure at the center of the particle as a function of time for two different pore sizes (10-7 and 10-8 m) for the O2-N2 mixture. The half-cycle time is 2 s and the total pressure at the surface is varied stepwise between 0.1 and 0.4 MPa. There is a time interval at the beginning of each half-cycle during which a total pressure gradient is present in the particle, and flow driven by total pressure difference takes place. The length of this time
Figure 9. Response curves for the partial pressure of N2 in a binary mixture of O2 and N2 at the center of a particle (p ) 0.6, Rp ) 10-3 m, r ) 10-8 m) for the first three cycles of a process with instantaneous pressurization and depressurization at constant temperature T ) 300 K predicted by the two- and threeparameter versions of the dusty-gas and the Fickian mass transport models.
Figure 10. Response curves for the partial pressure of N2 in a binary mixture of O2 and N2 at the center of a particle (p ) 0.6, Rp ) 10-3 m, r ) 10-7 m) for the first cycle of a process with linear pressurization and exponential depressurization at constant temperature T ) 300 K predicted by the two- and three-parameter versions of the dusty-gas and the Fickian mass transport models.
interval depends strongly on the permeability of particle, increasing as the latter decreases. Two stages can be distinguished from the results in Figure 7 in the variation of the intraparticle total pressure difference with time during pressurization. The transition from the first stage to the second corresponds to the point at which the front of the pressure wave reaches the center of the particle and the dominant nature of the differential equation system changes from hyperbolic to parabolic. From the results of Figure 3, one finds that for pores smaller than 10-8 m the dominant mode of pressure gradient driven transport is Knudsen flow. This observation suggests that the predictions of the two-parameter models, which do not account for viscous flow, will be almost identical to those of the three-parameter
3010 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
Figure 11. Response curves for the partial pressure of N2 in a binary mixture of O2 and N2 at the center of a particle (p ) 0.6, Rp ) 10-3 m, r ) 10-7 m) for the first two cycles of a process with sinusoidal total pressure variation at constant temperature T ) 300 K predicted by the two- and three-parameter versions of the dusty-gas and the Fickian mass transport models.
models, in which the viscous transport mechanism is included, for particles having pore sizes in this range. Indeed, this is seen to be the case in Figures 8 and 9 which present dynamic responses predicted by different mass transport models for the partial pressures of O2 and N2, respectively, at the center of an adsorbing particle with pore radius 10-8 m during the first three cycles of a process involving instantaneous pressurization from 0.1 to 0.35 MPa and subsequent instantaneous depressurization from 0.35 to 0.1 MPa. For pores larger than 10-5 m, we have the other limit where the total pressure gradient causes flow primarily through the mechanism of viscous flow (Serbezov and Sotirchos, 1997). Figures 10 and 11 refer to the intermediate case (r ) 10-7 m) in which the Knudsen transport coefficient and the viscous transport coefficient are of the same order of magnitude (see Figure 3). Presented in each of these figures is the variation of the dimensionless partial pressure of N2 in a binary mixture of O2 and N2 at the center of the particle with time, as predicted by different mass transport models. Figure 10 is for linear pressurization
PT ) PT,low +
PT,high - PT,low t, t1/2
0 e t e t1/2
(48)
and exponential depressurization
PT ) PT,low + (PT,high - PT,low)e-3t,
0 e t e t1/2 (49)
which are typical conditions encountered by the adsorbing particles in commercial PSA units. Figure 11 is for sinusoidal pressure change at the surface of the particle
PT ) PT,low +
PT,high - PT,low + 2 PT,high - PT,low tcycle 2π sin t2 tcycle 4
[ (
)]
(50)
This mode of total pressure variation may be considered as being representative of rapid PSA operations where
Figure 12. Dimensionless amount of O2 adsorbed in a particle (p ) 0.6, Rp ) 10-3 m, r ) 10-7 m) from a binary mixture of O2 and N2 as function of time for the first cycle of a process with instantaneous pressurization and depressurization at constant temperature T ) 300 K predicted by the two- and three-parameter versions of the dusty-gas and the Fickian mass transport models and the Glueckauf LDF approximation.
the step changes in the total pressure at the entrance are smoothened out in the interior by the low permeability of the adsorbing bed. The duration of the cycle in both figures is 3 s, with Figure 10 showing only the first cycle and Figure 11 the first two cycles. We see from Figures 10 and 11 that the inclusion of the viscous term in the three-parameter models has a much stronger effect on the response of the partial pressure of N2 for the Fickian model. To understand the reason for this behavior, we can use the results shown in Figure 4 for the variation of the effective transport coefficients of O2-N2 mixture with the pressure for the 10-7 m pore size. It is seen there that the viscous transport coefficient, BePT/µ, is smaller than DeKsthe effective mass transport coefficient for flow driven by total pressure gradient in the dusty-gas model when viscous flow is neglectedsover the whole pressure range used in Figures 10 and 11 (0.1-0.35 MPa) but larger than DeFsthe corresponding quantity in the Fickian model (see Figures 1 and 2)sover the most part of the pressure range. Thus, the inclusion of viscous terms leads to a significant reduction in the resistance for transport driven by the total pressure gradient in the Fickian model, whereas its effect on the dusty-gas model is rather small. Investigation of the Validity of the Linear Driving Force Approximation. Since most of the LDF approximations of the general form of eq 37 are based on the two-parameter FM, it follows from the above discussion that they may produce inaccurate results for multicomponent systems. Figure 12 compares the predictions of different mass transport models with those of the LDF approximation for the binary mixture of O2 and N2. Shown in this figure is the dimensionless amount of O2 adsorbed in the particle during the first cycle. The reference quantity used to make the plotted results dimensionless is the amount of O2 that is adsorbed in the particle when the particle is equilibrated with the surrounding environment at high operating pressure. Pressurization and depressurization are assumed to be instantaneous between 0.1 and 0.35 MPa. The duration of the half-cycle is 1.5 s, which in dimen-
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3011
Figure 13. Dimensionless amounts of CO2, H2, CH4, and N2 adsorbed in a particle (p ) 0.6, Rp ) 10-3 m, r ) 10-7 m) from a quaternary mixture of these gases as function of time for the first cycle of a process with instantaneous pressurization and depressurization at constant temperature T ) 300 K predicted by the three-parameter dusty-gas model.
sionless units (based on N2 at 0.35 MPa and 300 K) is 0.228. The LDF parameter ki is obtained from eq 38. The predictions of the various models for the O2-N2 system in Figure 12 are qualitatively similar to those of Figure 5 for the CO2-H2 system. Similar responses has been observed experimentally by Hu et al. (1993) and Asaeda et al. (1981). The overshoot in the response of H2 in Figure 5 is caused by both the faster transport of this species in the interior of the particle (by Knudsen flow and diffusion) and its lower adsorption equilibrium constant. For the binary system of O2 and N2, on the other hand, the effective Knudsen transport coefficients of the two species have similar values. As a result, the overshoot in the solid phase concentration of O2 is caused by its smaller adsorption constant which makes its intraparticle mass transport to be effectively faster than that of N2. Because of its faster transport, O2 is the main contributor to total pressure equilibrationsas is the case with H2 in Figure 5sbut once this happens, diffusion becomes the primary mode of transport with O2 diffusing outward and N2 inward. Thus, the average uptake of O2 goes through a maximum and then decreases to its equilibrium value. It is clear from the results of Figure 12 that the Glueckauf LDF approximation fails to predict the behavior of O2 not only quantitatively but also qualitatively. The same turns out to be the case for the behavior of H2 in Figure 5. The two-parameter FM and, hence, the LDF approximation cannot predict the overshoot in the concentration response of the fast moving/weakly adsorbed species because they treat the transport of each species in the mixture as being independent of the transport of the other components in the mixture. A generalized LDF approximation which accounts for the coupling of the mass fluxes of the species in a multicomponent system has been developed by Serbezov and Sotirchos (1997). For mixtures having more than two components it is possible to observe overshoot in the response curves of more than one species. Figure 13 presents dimensionless amount adsorbed vs time curves predicted by the three-parameter DGM for instantaneous pressurization and depressurization of a quaternary mixture of CO2,
H2, CH4, and N2 between 0.1 and 0.4 MPa with 1 s halfcycle time for 10-7 m pore size. The amounts adsorbed were rendered dimensionless by dividing by the equilibrium amount for each species at the high-pressure limit. It is seen that three of the components of the mixture exhibit overshoots in their response curves, H2, CH4, and N2. Because of its having the largest effective Knudsen mass transport coefficient and the smallest adsorption equilibrium constant, H2 presents the highest overshoot. CH4 is transported faster than N2, but since it is adsorbed more, its effective rate of transport in the particle is smaller than that of N2 and its overshoot considerably smaller. The Glueckauf LDF response curves are not shown in the figure because in order to construct such curves, one must first identify a value for DeF,i. We mentioned earlier that DeF,i is a function of the mole fraction and the molar fluxes of all species in the mixture. It must be noted that even if a concentration-dependent DeF,i is employed in eq 16, the two-parameter Fickian model given by eq 15 can predict only monotonically increasing variation of each species with timesprovided that DeF,i does not take negative values for some mole fraction combinations. Summary and Concluding Remarks A general dynamic mathematical model was formulated for describing mass transport and adsorption of multicomponent gaseous mixtures in porous particles under pressure swing conditions. A state of local equilibrium was assumed to exist between the gaseous species and the adsorbed species in the intraparticle space. Adsorption isotherms that are explicit in the adsorbed phase concentrationssas is commonly the caseswere employed in the computations, but the computational scheme can be adapted easily to deal with those rare situations where the isotherms do not conform to this assumption. The coupling of mass transport fluxes, partial pressures, and partial pressure gradients in the intraparticle space was described using the dusty-gas model, and both of its versionsstwoparameter (without viscous fluxes) and three-parameter (with viscous fluxes)swere used for computations. In order to elucidate the effects of the multicomponent mass transport on the behavior of the pressure swing adsorption process, computations were also carried out using the three- and two-parameter versions (with and without viscous fluxes, respectively) of the Fickian flux model in which the diffusive transport of each species in the mixture is assumed to be independent of those of the other species. D’Arcy’s law was used to account for the intraparticle viscous flow in both the dusty-gas model and the Fickian model. Results were obtained for O2-N2 and CO2-H2 binary mixtures and a quaternary mixture of CO2, H2, CH4, and N2. Since they allow for viscous flow in the particles, the three-parameter models predicted faster responses for all the species in the mixture than the corresponding two-parameter models in all cases. Allowing for viscous flow had much stronger influence in the case of the Fickian model. The main reason for this effect is that the two-parameter Fickian model not only does not account for the interaction of the transport rates of different species but also underestimates the mass transport rate of each species by flow driven by the total pressure gradient in the absence of viscous flow. In agreement with the observations of past experimental studies, the three-parameter dusty-gas model
3012 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
was found to predict overshoot in the responses of the partial pressures (local or average) of the species that exhibited larger effective rates of mass transport by the total pressure gradient. These were not necessarily the species with the larger effective Knudsen mass transport coefficients, since the effective mass transport rate is also influenced by the adsorption equilibrium constant, decreasing as the latter increases. Overshoots, but smaller, were also predicted by the two-parameter dusty-gas model and the three-parameter Fickian model. However, this was not the case with the results obtained by the two-parameter Fickian model, in which the transport of each species is determined only by its own partial pressure gradient. Monotonically increasing (during pressurization) or decreasing (during depressurization) behavior was also observed in the average amount adsorbed vs time results that were obtained using the linear driving force (LDF) approximation. This is hardly surprising considering that the derivation of this approximation is based on the two-parameter Fickian model. Notation Be: permeability of the adsorbing particle, m2 B: matrix defined in eqs 27 and 28 DF: Fickian diffusion coefficient, m2/s DK: Knudsen transport coefficient, m2/s DS: solid diffusion coefficient, m2/s Di,j: binary diffusion coefficient for gases i and j, m2/s DF: Fickian bulk diffusion coefficient, m2/s F: matrix defined in eqs 29 and 30 Hi: adsorption equilibrium constant in eq 33 ki: LDF mass transfer coefficient, s-1 Ji: molar flux of species i relative to moving coordinates, kmol/(m2‚s) M: molecular mass (used in Table 2) n: number of species in the gaseous mixture Ni: molar flux of species i relative to stationary coordinates, kmol/(m2‚s) PT: total pressure, Pa pi: partial pressure of species i, Pa p: vector of the partial pressures, Pa qi: solid phase concentration of species i, kmol/m3 q: vector of the solid phase concentrations, kmol/m3 Q: Jacobian matrix of q with respect to p r: pore radius, m Rg: ideal gas law constant, J/(kmol K) Rp: radius of the adsorbing particles, m R: vector of the molar rates of sorption, kmol/(m3‚s) t1/2: half cycle time, s T: temperature, K yi: mole fraction of species i Greek Letters p: porosity of the adsorbing particles η: tortuousity factor θ1/2: dimensionless half cycle time µ: viscosity of the gaseous mixture, kg/(m‚s) Subscripts D: quantities referring to the diffusive flux F: quantities referring to the Fickian model K: quantities referring to the Knudsen transport mechanism T: total quantities V: quantities referring to the viscous flow Superscripts e: effective quantities
ˆ : reference quantities -: averaged quantity Others DGM: dusty-gas model FM: Fickian model PSA: pressure swing adsorption LDF: linear driving force
Literature Cited Alpay, E.; Scott, D. M. The Linear Driving Force Model for FastCycle Adsorption and Desorption in a Spherical Particle. Chem. Eng. Sci. 1992, 47, 499. Asaeda, M.; Watanabe, J.; Kitamoto, M. Effect of Total Pressure Gradient on Unsteady-State Diffusion Rates of Gases in Porous Media. J. Chem. Eng. Jpn. 1981, 14, 129. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. Burganos, V. N.; Sotirchos, S. V. Knudsen Diffusion in Parallel, Multidimensional or Randomly Oriented Capillary Structures. Chem. Eng. Sci. 1989, 44, 2451. Carta, G. The Linear Driving Force Approximation for Cyclic Mass Transfer in Spherical Particles. Chem. Eng. Sci. 1993, 48, 622. Coppens, M.-O.; Froment, G. F. Fractal Aspects in the Catalytic Reforming of Naphtha. Chem. Eng. Sci. 1996, 51, 2283. De Boor, C. A. A Practical Guide to Splines; Springer-Verlag: New York, 1978. Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations; Prentice Hall: Englewood Cliffs, NJ, 1971. Glueckauf, E Theory of Chromatography. Part 10. Formulae for Diffusion into Spheres and Their Application to Chromatography. Trans. Faraday Soc. 1955, 51, 1540. Hu, X.; Rao, G. N.; Do, D. D. Multicomponent Sorption Kinetics of Ethane and Propane in Activated Carbon: Simultaneous Adsorption. Gas. Sep. Purif. 1993, 7, 39. Jackson, R. Transport in Porous Catalysts; Elsevier: New York, 1977. Kumar, R. Adsorption Column Blowdown: Adiabatic Equilibrium Model for Bulk Binary Gas Mixtures. Ind. Eng. Chem. Res. 1989, 28, 1677. Mason, E. A.; Malinauskas, A. P. Gas Transport in Porous Media: The Dusty-Gas Model; Elsevier: New York, 1983. Nakao, S.; Suzuki, M. Mass Transfer Coefficient in Cyclic Adsorption and Desorption. J. Chem. Eng. Jpn. 1983, 16, 114. Present, R. D. The Kinetic Theory of Gases; McGraw-Hill: New York, 1958. Pollard, W. G.; Present, R. D. On Gaseous Self-Diffusion in Long Capillary Tubes. Phys. Rev. 1948, 73, 762. Serbezov, A.; Sotirchos, S. V. Mathematical Modeling of the Adsorptive Separation of Multicomponent Gaseous Mixtures. Chem. Eng. Sci. 1997, 52, 79. Serbezov, A.; Sotirchos, S. V. Generalized Linear Driving Force Approximation for Modeling Multicomponent Adsorption-Based Separations. Submitted for publication. Sotirchos, S. V. Multicomponent Diffusion and Convection in Capillary Structures. AIChE J. 1989, 35, 1953. Sotirchos, S. V. Dynamic Modeling of Chemical Vapor Infiltration. AIChE J. 1991, 38, 1365. Tomadakis, M. M.; Sotirchos, S. V. Ordinary, Transition, and Knudsen Regime Diffusion in Random Capillary Structures. Chem. Eng. Sci. 1993, 48, 3323. Wilke, C. R. Diffusional Properties of Multicomponent Gases. Chem. Eng. Prog. 1950, 46, 95. Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: Boston, MA, 1987.
Received for review November 4, 1996 Revised manuscript received February 6, 1997 Accepted February 12, 1997X IE960699A
X Abstract published in Advance ACS Abstracts, June 15, 1997.
