1530
I n d . E n g . C h e m . Res. 1992,31, 1530-1540
Effect of Intraparticle Forced Convection on Gas Desorption from Fixed Beds Containing “Large-Pore”Adsorbents Zuping Lu, Jose M. Loureiro, M. Douglas Levan,+and Alirio E. Rodriques* Laboratory of Separation and Reaction Engineering, School of Engineering, University of Porto, 4099 Porto Codex, Portugal
The importance of intraparticle convection on the desorption of gases from fixed beds with “large-pore”adsorbents is assessed on the basis of modeling studies. A complete model for isothermal fixed-bed desorption of linear and nonlinear absorbed species includes axial dispersion, intraparticle diffusion, and intraparticle forced convection. Velocity and total pressure changes in the bulk fluid phase as well as in the intraparticle fluid are considered. Several limiting cases are shown as simplifications of the complete model. In a certain range of the number of intraparticle diffusion units, intraparticle forced convection leads to a better efficiency of the desorption process. This is a consequence of the “augmented” effective diffusivity by convection inside the adsorbent particle.
Introduction The importance of intraparticle forced flow in “largepore” materials has recently received more attention in relation to the performance of chromatographic processes, e.g., high-pressure liquid chromatography (HPLC) for protein separation (Horvath, 1990; Afeyan et al., 1990; Lloyd and Warner, 1990, Rodrigues et al., 1991,1992). The concept which explains the enhanced performance of “large-pore” materials is that of “augmented” effective diffusivity by convection as already shown in the areas of reaction and bioengineering (Nir and Pismen, 1977; Rodrigues et d., 1982,1984,Cresswell, 1985; Stephanopoulos and Tsiveriotis, 1989; Prince et al., 1991). In separation engineeringintraparticle forced convection should be considered not only in liquid/solid systems, i.e., HPLC, but also in gas/solid processes, such as pressure swing adsorption (PSA). PSA is now a major industrial gas separation technique (Ruthven, 1984; Yang, 1987; Sircar, 1989). One promising innovation and development in PSA is rapid cycling with a large pressure drop in the bed (Keller and Jones, 1981). In such cases, the importance of intraparticle forced flow cannot be neglected (Rodrigues et al., 1991) when “large-pore” adsorbents are used, since the pressure gradient across or inside the adsorbent can be quite large during adsorption/desorption (Asaeda et al., 1981; Nakano et al., 1982). A number of models for fixed-bed adsorption/desorption have been developed (Zwiebel et al., 1974; Carter, 1975; Wong and Niedzwieki, 1982; Huang and Fair, 1988). Most of them are based on constant velocity and constant total concentration. Recently, Buzanowski et al. (1989) studied the effect of bed pressure drop on adsorption and desorption dynamics considering variable velocity, dilute feed (constant total concentration) and local equilibrium. The objectives of this paper are as follows: (i) One objective is to develop a model for gas desorption in fixed beds taking into account mass and momentum equations. The general model applies to variable total concentration and variable velocity situations and includes intraparticle mass transport by diffusion and convection. (ii) A second objective is to solve the model equations by the method of orthogonal collocation on finite elements. (iii) The third objective is to assess the effect of intraparticle convection on the dynamics of gas desorption in fixed beds and check the limiting cases of constant velocity
* To whom correspondence should be addressed.
Department of Chemical Engineering, University of Virginia, Charlottesville, VA 22903-2442.
and negligible mass-transfer resistance.
Mathematical Model The problem considered here is an isothermal adsorption column packed with “largepore” adsorbent materials, e.g., aluminas (slab geometry with thickness i), and the feed is a binary mixture of species A (adsorbable)and B (inert carrier). Initially the adsorbent is saturated with the adsorbable species A and the column is at steady state; i.e., at time zero there is a total pressure profile P ( z ) along the bed. However, in some simulations the bed is considered to be initially at uniform pressure as indicated later when writing the initial conditions for the model equations. At time zero, a negative step change in the concentration of the species A is made at the column inlet. Additional assumptions are 1. The flow pattern in the column can be described by the axial dispersion plug flow model. 2. The bed pressure drop is described locally by Ergun’s law (Buzanowski et al., 1989). 3. Mass transfer inside the adsorbent occulg by diffusion and convection. The intraparticle convective flow follows Darcy’s law. 4. The main resistance to mass transfer is inside the adsorbent; i.e., the external film mass-transfer resistance is negligible. 5. Ideal gas behavior is assumed for the binary mixture. 6. The adsorption equilibrium is described by a linear or a Langmuir isotherm. 7. Mole fraction and total concentration gradients in the bulk fluid are considered as approximately constant over the length of a slab adsorbent particle to set the boundary conditions for mass transfer inside the adsorbent. 8. For convenience, we assume that species A and B have identical physical properties. According to the above assumptions, the governing equations of isothermal fixed-bed desorption dynamia as well as the appropriate initial and boundary conditions can be described as follows. Mass Balances inside the Adsorbent Particle. The total mole flux of species i in the fluid inside porous pellet includes the contributions of Knudsen diffusion, molecular diffusion, and viscous flow; it is given by (Gunn and King, 1969; Allawi and Gunn, 1987)
where Dei and Dekiare the effective and Knudsen diffusivities of species i, respectively, and u is the superficial
0888-5885/92/2631-1530$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res.,Vol. 31, No. 6, 1992 1531 intraparticle convective (Darcy) velocity. From assumption 8, we get Dei = DeA = DeB and Dek = DeU = D e B
(2)
and the overall flux is then 2
C N : = N ; = -DeU
i=l
$+
uc’
(i = A and B) (3)
So, the mass balance for species A and the overall mass
balance are respectively
and up = 111is the external specific area of the slab particle. Boundary conditions are z = 0,
aYA -aax + UYA = 0; az
uc = constant (16)
(mole flux is kept constant and equal to that in steady state, after desorption is completed, at a given pressure drop AP, = P1 - Po).
z = L , -az- - 0;
c = constant
(17)
(the outlet of the column is at atmospheric pressure). The initial condition is t = 0, Y A 1; c = c o (18) where c, = Po/RT when there is no pressure drop across the bed; if there is a given pressure drop across the bed c, is obtained from u,c, = constant (u, from Ergun’s law equation) (19)
Boundary conditions are
p1 z = L,c, = ’0 (19a) RT’ RT Momentum balance for the bulk fluid phase. From Ergun’s law ap = --ac = -- 1 az RT az 1.75p0(l- e) 150p(l U + P u 2 (20) RT dp2c3 P,dpc3 z = o , c,=-*
The initial condition is
=o, y i = l ;
(8) where c 6 = Po/RT when there is no pressure drop across the bed; if there is a given pressure drop across the bed, c 6 follows the equations dc 6 c i - = constant (9) dz t
c’=ci
Adsorption Equilibrium Isotherm. klC’Yi
qA = 1
+ k2C’Y)A
1
L[
In the above equations, u is the superficial intraparticle velocity and follows Darcy’s law (eq 11))u is the superficial bulk fluid velocity and follows Ergun’s law (eq 201, and D , is the axial dispersion coefficient. The effective diffusivities DeAand DekAand the axial dispersion coefficient D , are calculated by the following equations: 1
Momentum Equation for the Fluid inside the Particle. From Darcy’s law
Dm2o Dm = P‘
Mass Balances for the Bulk Fluid Phase in the Bed.
DkA= 9.6 X 103(T/M)1/2r,cm2/s (23a) Dax = YlDm + ~ + d p (24) where Dmis the molecular diffusivity,D , is the molecular diffusivity at atmospheric pressure, and DkA is the Knudsen diffusivity. Introducing dimensionless variables
where N)Aand “are the mole fluxes of species A and total fluid transported from fluid phase to the adsorbent phase, 1.e.:
U u* = u,’
u u* = -, UO
e = - -t TO
where c, is total concentration at the column outlet (= P,/RT), u, is the bulk fluid velocity at the outlet in steady state at a given pressure drop AP,,u, = (Bp/p)(APo/L), and r0 = Jl/u,; we get the following model equations. For the adsorbent particles
1532 Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992
a (- f’
a~ b4 + f ’
~ Y I +A- -Y) -IAAoa f t ap b4 a p fp
a(v*fw-
kq* (1 + ( k - l)f
pR
= 1/2L
aP)
1
a(f’Y2)(25)
ae -
(274
The initial condition is e=o, y i = i ; f ’ = f : (29) where f 6 = 1 if there is no pressure drop across the bed; if there is a given pressure drop AfO = f l - 1 across the bed, f 6 is obtained by the following equation: f-6 + ,A&(f’J2 1 = Blp + B2 (30) b4 where B1 and B2 are constants and are obtained from the previous equation after the substitution of the following conditions:
x=1, ayA - -0; f 1 (37) ax The initial condition is e = o , YA = I , f = f o (38) where f o = 1 if there is no pressure drop across the bed; if there is a given pressure drop across the bed f o is calculated from f o = [ l + 2(b5 + b J ( l - x ) ] ’ / ~ (39) Equation 39 gives the dimensionless pressure profile in the bed in steady state; it was obtained by combining Ergun’s law with u*f = constant. The bulk fluid superficial velocity at the column outlet in steady state at a given pressure drop APo = Pl - Po (AfO = f l - 1) is -La,/Po + [(La1/P0)2+ 2La,((P,/Po)2 - 1)]1/2 u, = 2La, Ergun’s law is now
--a f
ax
b5 =
150p(l - t)2L U,; dp2c3P0
- b5u* + b$(u*)2 be =
1.75p0(l - c)L u, P0dpe3
(41) (414
In the above equations, the parameters are (Rodrigues et al., 1991) kl q* = ; k =1 k2~, 1+ k2COO (adsorption equilibrium parameters) (42)
+
Darcy’s law is now written as
(where Pe, is the Peclet number for the external fluid phase at reference conditions) For the bulk fluid phase (43b)
acu*n af 1- e +-+-N=O ax ae e
(33) (where A, is the intraparticle Peclet number at reference conditions) 6-12
.,
?do
?A2
Uoo
(454 Dmo J, (where a, is the ratio between time constant for diffusion and external convective flow; reciprocal of the number of intraparticle diffusion units) (yo=--=-70
Dmo
b4 = - (ratio between molecular and Knudsen DkA
Boundary conditions are
diffusivities) (45b)
Ind. Eng. Chem. Res., Vol. 31, No. 6,1992 1533
We will call this model the “complete”model, i.e., case
A from eqs 25 to 45b; it has four dependent variables f ,
f ’, YA, and yb, and four equations (25), (26), (32), and (33). Now we will discuss some simplified cases. Case B. No Intraparticle Convection. In this case A, = 0; from the equations above we get the following. Inside the adsorbent particles
tp
a y -1- = b4 a p 2
[
(1
+
kq* (k - l)f?’A)2
1-
WYA)
af’ 1-ep kq* a8 tp (1 + (k - i)f’y’A)2 +
k9* (1 + (k - 1 ) f Y d 2
a(u*fi --= ax k9* (1 + (k - l)fYA)2
-1 -1
a(fYA) (51)
a(fuA)
ae (52)
Boundary and initial conditions are given by eqs 36-39. Case D. Constant Total Concentration. In this case af ‘/ap = 0,aflax = 0,af yae = 0,aflae = 0,SO the SUperficial bulk fluid velocity and the intraparticle velocity are constant (i.e., a = a,, A = A, if b4 is very small). In the adsorbent particles aP
e=o,
aYA
= YA + Z y i = i
f l R
(54) (55)
In the bulk fluid phase
(47)
ae
aP2
Y’A
Boundary and initial conditions are
ae
Boundary and initial conditions are exactly the same as eqs 36-39. Case C. No Mass-Transfer Resistance inside the Adsorbent (Equilibrium Model). In the bulk fluid phase
5
= 1,
(46)
Boundary conditions are given by eqs 27 and 28 and the initial conditions are given by eqs 29-30a except that eq 30 should be changed to f 6/b4 = BlP + B2 (48) In the bulk fluid phase, the equations are exactly the game as eqs 32 and 33, but the fluxes N A and N should be changed to
a2y‘A --
P
k9* ‘p
(1 + (k - 1)y’A)’ (53)
Boundary and initial conditions are
6=0, Y A “ 1
“Complete”Model: Solution by Orthogonal Collocation on Finite Elements The orthogonal collocation method for solving partial differential equations was developed mainly by Villadsen (1970) and Finlayson (1972, 1980). The method is particularly useful for the solution of boundary value problems and, for a given accuracy of the solution, has often been found to require less computer time than standard finite difference methods (Raghavan and Ruthven, 1983,1984). It has been applied to the simulation of fixed-bed reactors (Hansen, 1971; Karanth and Hughes, 1974; Leit3o and Rodrigues, 1990) and simulation of adsorption columns (Liapis and Rippin, 1978; Raghavan and Ruthven, 1984; Morbidelli et al., 1985; Costa and Rodrigues, 1985a,b; Sereno et al., 1991). Although the commercial PDECOL package for solving partial differential equations by the method of orthogonal collocation on finite elements was developed more than a decade ago, it is not easily and directly used to solve many chemical engineering problems. In this paper, the partial differential equations (PDE’s) in two spatial dimensions p and x and in time 8 describing mass balances inside the adsorbent were first reduced to first-order partial differential equations in one spatial dimension x and in time 8 by using orthogonal collocation on finite elements on the direction p; then the resulting first-order partial differential equations together with the second-order partial differential equations describing the mass balances for the bulk fluid phase in the bed were solved with the PDECOL package. The problem of consistency between initial and boundary conditions was tackled by using an exponential time decreasing function which will be discussed later. Rearranging eqs 25-45b in such way as to isolate the time derivatives of the dependent variables, we get the following. Inside the adsorbent particle
-dY‘A -ae
1534 Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992
where
i
Table I. Values of Fluid, Particle, and System Properties Used in Numerical Simulations D,, = 0.1 c m 2 d c = 0.4 pa = 10-3 g ~ m - ~ c = 0.7 p = 10-4 g.cm-I.s-1 l != 150 cm k=land2 d, = 0.2 and 0.5 cm q* = 10 and 100 71 = 4.5 Po = 100 kPa 71 = 0.7 P1 = 120, 150, 200 kPa 72 = 0.5
bd’
plicit) spatial dimension x . The problem of these 16 PDEs at each axial position with 2 other PDEs in the bulk fluid phase was solved by the PDECOL package. Seven finite elements with two interior collocation points were used for discretization in the axial direction, so the ordinary differential equation (ODE) solver in the PDECOL package handles an initial value problem with 16 X 18 ODE’S, The PDECOL package requires compatibility between initial and boundary conditions. This problem was solved by using an exponential decreasing time function as follows:
b3
1 =A,
and
where NA and N are stated as eqs 34 and 35, and
Equation 65 is obtained from Ergun’s law (eq 41) after differentiating in order to x and solving for au*/ax. -b5
+ (b:
u*f = exp(-M,8)
(68)
(if there is no pressure drop across the bed at initial time) where the values of M,from 10 to 1O‘O were tested in order to compare the numerical results; almost no differences were found when M,>> 1. In our calculations, M, = 100 for q* = 10 and M = 10 for q* = 100 were used. The control error was set at 104-10-5. The simplified cases (case B to case D) were also solved by the method above. The computing time was between 0.5 and 2 min on an IBM RS/6000-530 computer.
In the bulk fluid phase
- 4b$ af/dX)1/2
(66) 2bd Equation 66 is obtained from Ergun’s law in dimensionless form by solving in order to u*. The set of eqs 60-66 with eqs 27-30a and eqs 36-39 defines a nonlinear boundary value problem in each position x of the adsorber involving a pair of coupled parabolic second-order PDEs in two spatial dimensions p (explicitly) and x (implicitly) and in time 8. These two equations are first reduced to first-order partial differential equations by orthogonal collocation on finite elements in the p direction. Four elements and two interior collocation points in each element are chosen inside the adsorbent. Thus, the total number of collocation points is NP=2 X 4 + 2 = 10 (including two boundary points); i.e., 2 PDEs in two spatial dimensions were reduced to 16 PDE’s in one (im-
u*
and
Simulation Results and Discussion Simulations were carried out to address the effect of adsorption equilibrium isotherm (k = 1corresponds to a linear isotherm; k = 2 corresponds to a Langmuir isotherm), adsorbent capacity (q* = 10 and q* = 1001, particle size (d, = 0.2 cm and d, = 0.5 cm), and bed pressure drop (AP,= 2 X lo4, 5 X lo4, and lo5 Pa). Values of fluid, particle, and system properties are shown in Table I. Case D. Constant Total Concentration. Let us start the analysis with case D, i.e., constant total concentration and so constant velocity. Model parameters are now q*, k (adsorption equilibrium parameter), Pe = Pe, (axial Peclet number), and a = a,, X = A, (intraparticle masstransfer parameters). Effect of X and a. The effect of intraparticle forced convection measured by h on the fixed-bed desorption is shown in Figure 1, where the mole fraction of species A at the bed outlet was plotted as a function of the reduced time for the case of linear isotherm (k = 1)at various a. If there is no intraparticle forced convection h = 0, the history of mole fraction versus 8 (Figure la) has a long tail, so desorption is inefficient because CY = 10; i.e., the time constant for intraparticle diffusion Tdo is 10 times the space time T , (low effective diffusivity). When intraparticle convection increases, Le., X increases, the desorption process becomes more efficient. The limiting case without mass-transfer resistances and without axial dispersion predicts that the mole fraction at the bed outlet is a discontinuity located at 8, * 47. Obviously at very high h we should get the result predicted by the model
Ind. Eng. Chem. Res., Vol. 31, No. 6,1992 1535
0
20
40
60
80
100
e 120 k=1
0.4 0.2
0.0 0
20
40
60
80
100
e
120
5
0
10
15
e
20
Figure 2. Effect of intraparticle convection (X = A,) on the mole fraction of adsorbable species A at the bed outlet as a function of time for linear and nonlinear equilibrium isotherm: case D with Pe = 300, q* = 10, and a = 10. (---) Equilibrium model with axial Equilibrium model without axial dispersion. (a, top) dispersion. k = 2 (Langmuir isotherm); (b, bottom) k = 1 (linear isotherm).
1.o
0.8
(.-e)
0.6
0
20
40
60
80
100
e
120
Figure 1. Effect of intraparticle convection (A = A,) on the mole fraction of adsorbable species A at the bed outlet as a function of time for various intraparticle diffusion parameters (a = a,) with linear equilibrium isotherm: case D with Pe = 300, q* = 100, and k = 1. (- - -) Equilibrium model with axial dispersion. (-) Equilibrium model without axial dispersion. (a, top) a = 10; (b, middle) a = 1; (c, bottom) a = 0.3.
including axial dispersion, i.e., the broken line shown in Figure 1. The improvement of the desorption efficiency when X increases is due to the “augmented” effective diffusivity by convection b,, i.e. (Rodrigues et al., 1982)
The effect of X is smaller when a is smaller, since lower a means that the mawtransfer resistance due to diffusion is l~ important;for very high X, the “apparent” diffusivity
is De = 0116. The desorption curves calculated for a = 10 and X = 100 should be similar to the curve calculated in the absence of intraparticle convection with a = 0.6, because of the relation 8 = af(X).
Some effluent histories for desorption in Figure l a show inflection points. This is due to intraparticle convection and will be observed whenever a = r d / r= 1 + q*, in this case for a = 11. These observations are in agreement with results reported for residence time distribution of inert tracers which show two peaks at a = 1 (Rodrigues et al., 1991). Effect of the Nature of the Adsorption Equilibrium Isotherm. Parts a and b of Figure 2 show the histories of mole fraction of species A at the bed outlet, Le., yA versus 6 for nonlinear (k = 2) and linear adsorption equilibrium, respectively. The limiting situation a t high X in the case of linear isotherm, in the absence of axial dispersion, is a discontinuity shown in Figure 2b; for nonlinear isotherm, the limiting case at high A is the dotted line in Figure 2a predicted by the following equation:
as known from the equilibrium theory. Since a Langmuir isotherm is used, desorption leads to a dispersive curve even at high A. Effect of the Adsorbent Capacity q*. Figures 2a and 3 and Figures l a and 2b show the effect of the adsorbent capacity q* for nonlinear (k = 2) and linear (12 = 1) isotherm, respectively. The reduced stoichiometric time is at Os = 6.55 for q* = 10 and at = 47 for q* = 100. Case A (Complete Case) and Case B (No Intraparticle Convection). Effect of Pressure Drop, AP,,.The pressure drop in the fixed-bed adsorber can have a strong impact on the
1536 Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992 q*=lOO k=2
5. APo=50 kPa 6. APo=lOO kPa
a=10
0.2
20
0
40
60
80
100
6
Figure 3. Effect of intraparticle convection (A = A,) on the mole fraction of adsorbable species A at the bed outlet as a function of time with nonlinear equilibrium isotherm: case D with Pe = 300,q* = 100,and a = 10. Equilibrium model without axial dispersion. (e-)
2
0
120
4
6
8
time, sec.
1 h
500-
A
~ u n
1. APo-20 kPa 2. APo=50 kPa 3. APo=lOO kPa
"
0
0
2
4
6
8
time, sec. 500
=
'-
I
Run -
400
xII
L
3
300
in
-2 V
. 200
k a
100
0
I
1
2
3
4
time, sec. Fuure 4. Effect of pressure drop on the mole fraction @J and mole flux (uyA)of adsorbable species A at the bed outlet as a function of time: case B (no intraparticle convection) with q* = 10,k = 1, d, = 0.5 cm, and Pe = 1500. 1, AP, = 20 P a ; 2,AP, = 50 kPa; 3, AP, = 100 kPa. (a, top) yA versus time; (b, bottom) U Y A versus time.
separation performance of PSA (Sundaram and Wankat, 1988, h n g and Yang,1988). The effecta of pressure drop on adsorption/desorption dynamics were studied by Buzanowski et d. (1989) using a very dilute feed. Numerical simulations showing the effect of pressure drop on the histories of mole fraction (YA) and mole flux (uyA)of species A at the bed outlet are presented in parta a and b, respectively, of Figure 4 for case B (A = 0; runs 1, 2, and 3) and in parts a and b, respectively, of Figure 5 for case A (complete case; runs 4,5, and 6). For each case the effect of increasing the pressure drop is to get an earlier history of mole fraction at the bed outlet. For case B (no intraparticle convection) the effect can be explained as follows: higher inlet pressure leads to a re-
1
2
3
4
time, sec. Figure 5. Effect of pressure drop on the mole fraction (yA)and mole flux (uyA)of adsorbable species A at the bed outlet as a function of time: case A (complete model) with q* = 10,k = 1, d, = 0.5 cm, Pe = 1500,and B, = 2 X lo-*cm2. 4,AP, = 20 P a ; 5, AP, = 50 P a ; 6, AP, = 100 kPa. (a, top) yA versus time; (b, bottom) UYA versus time. Table 11. Parameters for Numerical Simulation of Case A (Complete Model; Runs 4-6) and Case B (No Intraparticle Convection; Runs 1-3)" simul u,, r,, s cmie 4 run B,, cm2 , A, r,, s a, 0.291 m 206 0.2 1 0 7.74 0 0.171 m 351 0.5 2 0 13.2 0 0.110 = 547 1.0 3 0 20.5 0 4 2 X 10" 7.74 3.83 0.291 0.587 206 0.2 5 2X 13.2 9.58 0.171 0.235 351 0.5 6 2x 20.5 19.2 0.110 0.117 547 1.0 " d , = 0.5 cm, q* = 10, k = 1, Pe = 1500, rdo = 2.25 s. Initial conditions for total concentration are f,= f 6 = 1.
duction of the mole fraction of adsorbable species in the bulk fluid phase; because the flux of inert species at the inlet is larger, a smaller yAin the fluid phase and 80 a larger mole fraction gradient at the surface of the pellet improve desorption. However, there is an opposing effect of higher inlet pressure since the effective diffusivity is inversely proportional to the total pressure, and so desorption inside the pellet is more difficult. These two opposing factors lead to the results shown in Figure 4. When intraparticle convection is important, higher inlet pressure leads to higher intraparticle convective velocity which more than compensates the reduction of the effective diffusivity due to the pressure increase; the efficiency of desorption is better for higher pressure as shown in Figure 5. Reference parameters for the numerical simulations of case A and case B are shown in Table II. For the complete case the evolution of the intraparticle convection param-
Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992 1537 Table 111. Parameters for Numerical Simulation in Figures 8 and V simul run d,, cm Pe B,, cm2 A0 7 0.5 1500 0 0 8 0.5 1500 5.0 X lo4 0.958 9 0.5 1500 3.12 X lo4 6.0 10 0.5 1500 1.25 X lo4 24.0 lBb 0.5 1500 1.25 X lo4 24.0 11 0.5 1500 12 0.2 3750 0 0 13 0.2 3750 1.25 X lo-' 9.58 14 0.2 3750 Oq*
-
7,,
8
T,
a0
7.74 7.74 7.74 7.74 7.74
m
2.35 0.367 0.094 0.094
0.291 (206) 0.291 (206) 0.291 (206) 0.291 (206) 0.291 (206) 0.291 (206) 0.494 (121) 0.494 (121) 0.494 (121)
0.729 0.729
m
0.0376
(u& s (cm/s)
case
B A A A
A C B A C
= 10,k = 1, AfO = 0.2. Initial conditions: fo follows eq 39 and f 6 follows eq 30. bRun 18 is for k = 2. JVI
I
2.0 1
iI
".
0
2
4
a
6
t, sec. 1
--
40
1.0 4 0
2
4
6
8
time, sec. Figure 7. Reduced total pressure at the middle of the bed as a function of time for different pressure drops: case A (complete model).
N
m
2
30
I1
4
9
20
xI1
10 v
- 8 " .
0
"
2
4
6
a
4
6
8
time, sec.
8
0
2
time, sec. Figure 6. Intraparticleconvection (A) and intraparticlediffusion (a) in one adsorbent particle at the middle of the bed as a function of time for different pressure drops: case A (complete model). (a, top) X (at x = 0.5, p = 0,1688)versus time. (b, middle) X (at x = 0.5,p = 0.8312)versus time. (c, bottom) a (at x = 0.5,p = 0.1688)versus time.
eter h at two different positions inside one pellet ( p = 0.1688 and p = 0.8312) located at the middle of the bed are shown in parte a and b, respectively, of Figure 6. The intraparticle diffusion parameter a is also shown in Figure
6c for the same pellet at p = 0.1688. The peaks in these figures are due to the pressure increase in desorption since both h and a are proportional to pressure. The reduced total pressure as a function of time at the middle of the column is shown in Figure 7. Effect of the Adsorbent Particle Permeability B,. The particle permeability B, is directly related to the pore size and affects the convective velocity u, and so the intraparticle Peclet number A. The effect of the particle permeability B on the histories of mole fraction and mole flux at the b e l outlet is shown in parts a and b, respectively, of Figure 8. Reference parameters are listed in Table 111. When the particle permeability increases, the desorption history will move from a curve where no intraparticle convection occurs (curve 7 in Figure 8) to an upper limit given by the equilibrium model, case C, which corresponds to the curve of more efficient desorption (curve 11 in Figure 8). Effect of the Adsorbent Particle Size, d,. When mass-transfer resistances inside the particle are not important, it is advantageous to use a larger particle diameter since at the same velocity the bed pressure drop will be smaller. Alternatively if the mole flux at the inlet is kept constant, the velocity in the bed is higher for larger particle size and the mole fraction in the fluid phase will be smaller, so the histories of mole fraction and mole flux at the bed outlet have smaller tails. This is shown in Figure 9. In the other limit, if diffusion is the controlling step, the desorption efficiency should be better at smaller particle size as shown in Figure 9. The reference parameters are listed in Table III. The effect of the particle permeability B,, Le., of intraparticle convection, is more important for larger than smaller particle size d, if diffusion is the controlling step. Effect of the Adsorption Capacity q*. For high adsorption capacity q* = 100 the histories of mole fraction
1538 Ind. Eng. Chem. Res., Vol. 31, No. 6 , 1992
o.ll
I
/7
0.4
0.2
"."0 nn
0.0
2
time, sec.
4
6
'
0
1
2
time, sec.
4
6
600
Run -
--
6oo 500
X I1
. z
Ih
400
E
300
+,e
200
3
100 n
"0
4
6
time, sec. Figure 8. Effect of adsorbent particle permeability E , on the mole fraction (YA) and mole flux (UYA) of adsorbable species A at the bed outlet as a function of time: q* = 10, k = 1, AF', = 20 P a , Pe = 1500. Case A (complete model): run 8 (BP= 5 X lo4 cm2), run 9 (BP= 3.12 X cm2),and run 10 (Bp= 1.25 X lo-' cm2). Case B (no intraparticle convection): run 7. Case C (no intraparticle mass-transfer resistance): run 11. (a, top) yA versus time; (b, bottom) uyAversus time.
and mole flux at the bed outlet are shown in Figure 10 and reference parameters are listed in Table IV. The effect of intraparticle convection is similar to the one observed with q* = 10. The shape of yAversus time is different from the case with q* = 10 showing a sharp decrease of yAat s m d times because a high q* will cause a high pressure inside the particle which is unfavorable for desorption. Effect of Nonlinearity of the Adsorption Equilibrium Isotherm, k. The nonlinearity of the adsorption equilibrium isotherm is measured by k. For k = 1we have the linear isotherm; for k = 2 we get a favorable Langmuir type isotherm. The desorption of a favorably adsorbed species A is more difficult than the desorption of a linearly adsorbed species A for the same adsorbent capacity. The effect of intraparticle convection is just the enhancement of the desorption efficiency, but the performance is better for a linear adsorption equilibrium isotherm.
Conclusions The "complete" model describing the dynamics of gas desorption from an isothermal fixed bed with "huge-pore" adsorbent considering variable total concentration in the bulk fluid and inside the particle was derived. Some simplified cases of the model were discussed. The partial differential equations for the particle were numerically discretized by orthogonal collocation on finite elements, and the PDECOL package was used to solve the resulting discretized partial differential equations together with the bulk fluid-phase equations. The problem of
"0
2
6
t, sec. Figure 9. Effect of adsorbent particle size d,, on the mole fraction bA)and mole flux (uyA)of adsorbable species A at the bed outlet as a function of time for different adsorbent particle sizes: q* = 10, k = 1, AF', = 20 P a . Case A (complete model): runs 10 and 13 (B = 1.25 X lo-' cm2). Case B (no intraparticle convection): runs 7 an8 12. Case C (no intraparticle maas-transfer resistance): runs 11 and 14. (a, top) yA versus time; (b, bottom) UYA versus time.
Table IV. Parameters for Numerical Simulation in Figure 10" simul 7, (%), run B,, cm2 A. T,. s a, s (cm/s) case 15 0 0 = 7.74 0.291 (206) B 16 3.12 x 6.0 0.376 7.74 0.291 (206) A 17 0.291 (206) C 9* = 100,k = 1, Ai, = 0.2, d , = 0.5 cm, Pe = 1500. Initial conditions: f, follows eq 39 and f 6 follows eq 30.
consistency between the initial and the boundary conditions in the PDECOL package for many chemical engineering problems was solved by using an exponential time decreasing function. The effecta of adsorbent particle permeability B,, particle size d,, adsorbent capacity q*, and the nature of the adsorption isotherm k were discussed on the basis of numerical simulation results. An increase of the particle permeability B,, which increases the intraparticle convection, can improve the efficiency of desorption but never change the nature of the desorption transition. The effect of increasing B, is to drive histories of mole fractions between the diffusion-controlled regime and equilibrium,and is more important for lower effective diffusivities. The effect of the particle permeability and so of intraparticle convection is more important for larger than for smaller particles when diffusion is the controlling step. Larger adsorbent capacity makes desorption more difficult. The desorption performance of the linear equilibrium isotherm is better than that of the nonlinear one.
Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992 1639 1.o
de= “apparent” or “augmented”effective diffusivity, cm2/s el, ...,ea = parameters stated by eqs 60a and 61a
q*=lOO k=l
Run -
f = dimensionlesstotal concentration in the bulk fluid phase f ’= dimensionless total concentration in the fluid inside the particle f, = dimensionless total concentration in the bulk fluid phase under steady state f 6 = dimensionless total concentration in the fluid inside the particle at time zero f l = dimensionlesstotal concentration at the bed inlet under steady state 4, = dimensionlem preawe drop across the bed under steady state L = bed length, cm 1 = slab thickness, cm k = constant for the normalized isotherm, eq 42 kl, k2 = Langmuir isotherm constants N = dimensionless total mole flux from the bulk fluid to the adsorbent N A = dimensionless mole flux of species A from the bulk fluid to the adsorbent N‘ = total mole flux from the bulk fluid to the adsorbent, mol/cm2 s N’A = mole flux of species A from the bulk fluid to the adsorbent, mol/ (cm2s) N’i = mole flux of species i inside the adsorbent, mol/(cm2
time, sec.
In
3
400
200
8)
0 0
10
20
time, sec.
30
40
F m 10. Effect of adsorbent particle permeability B, on the mole fraction (yA) and mole flux (uyA)of adsorbable epeciee A at the bed outlet as a function of time for larger adsorption capacity: q* = 100, k = 1, d, = 0.5 cm, AP, = 20 P a , Pe = 1500. Case A (complete model): run 16 (BP= 3.12 X lo-* cm2). Case B (no intraparticle convection): run 15. Case C (no intraparticle mass-transfer resistance): run 17. (a, top) YA versus time; (b, bottom)cyAversus time. Acknowledgment Financial support from FUNDACAO ORIENTE, JNICT, NATO CRG 890600, and EEC JOULE 0052 is gratefully acknowledged.
Nomenclature a = external specific area of slab particle, cm2/cm3
4, B2 = constants in eq 30 bl, ..., b6 = dimensionless constants stated by eqs 41a, 43b,
45b, and 60b B, = permeability of the adsorbent, cmz c = total concentration in the bulk fluid phase, mol/cm3 c’ = total concentration in the fluid inside the adsorbent, mol/cm3 c, = total concentration in the bulk fluid phase under steady state, mol/cma c, = total concentration in the bulk fluid at the bed outlet, mol/cm3 c 6 = total concentration in the fluid inside the adsorbent under steady state, mol/cm3 d = adsorbent particle diameter, cm If, = axial dispersion coefficient, cm2/s DeA = effective diffusivity of species A, cm2/s DeB = effective diffusivity of species B, cm2/s Dei = effective diffusivity of species i, cm2/s D , = effective molecular diffusivity at atmospheric pressure, cm2/s D , = molecular diffusivity, cm2/s D,, = molecular diffusivity at atmospheric pressure, cm2/s DLA= Knudsen diffusivity of species A, cm2/s D e M = effective Knudsen diffusivity of species A, cm2/s
N ; = total mole flux inside the adsorbent, mol/(cm2 s) M, = constant in eq 67 P = pressure in the bulk fluid, Pa Po = atmospheric pressure, Pa P1 = pressure at the bed inlet, Pa AP, = pressure drop across the bed under steady state, Pa p’ = pressure in the fluid inside particle, Pa Pe = Peclet number qA = concentration in the adsorbed phase, mol/cm3 q* = constant for the normalized isotherm, eq 42 R = ideal gas constant P = average pore size, cm t = time, s T = temperature, K u = superficial velocity in the bulk fluid, cm/s u, = superficial velocity at the bed outlet under steady state, cm/s u* = dimensionless velocity in the bulk fluid u = superficial intraparticle velocity, cm/s u, = reference intraparticle velocity, cm/s u* = dimensionless intraparticle velocity x = dimemionless axial coordinate in the bed = mole fraction of species A in the bed = mole fraction of species A in the fluid inside the adsorbent z = axial coordinate in the bed, cm z’ = space coordinate in the adsorbent, cm
$:
Greek Letters CY = ratio between time constant for intraparticle diffusion and space time a0 = reference parameter defined by eq 45a 0, = constant defined by eq 31a & = ratio between the half thickness of the slab and bed length yl,y2 = constants t = bed porosity cp = adsorbent porosity { = parameter stated by eq 62 6 = time reduced by the reference space time Os = dimensionless stoichiometric time A = intraparticle Peclet number
1540 Ind. Eng. Chem. Res., Vol. 31, No. 6, 1992 A, = intraparticle Peclet number 44a) p = fluid viscosity, g/(cm s)
at reference conditions (eq
p = dimensionless space coordinate po = fluid density, g/cm3
rf = tortuosity factor for the r0 = reference space time, s T,,
rdo
in the adsorbent
particle
= reference time constant for intraparticle convection, s = reference time constant for intraparticle diffusion, s
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