Brazilian Journal of Chemical Engineering
ISSN 0104-6632 Printed in Brazil
Vol. 20, No. 02, pp. 191 - 199, April- June 2003
MASS TRANSFER IN POROUS MEDIA WITH HETEROGENEOUS CHEMICAL REACTION S.M.A.G.UIson de Souza 1 and S.Whitaker2 [Departamento de Engenharia Qufmica e Engenharia de Alimentos, Universidade Federal de Santa Catarina, Phone (+55) (48) 331-9448 - R.216, Fax (+55) (48) 331-9687, Cx. P. 476, 88040-900 - Florian6polis - SC, Brazil E-mail:
[email protected] 2Department of Chemical Engineering and Material Science, University of California at Davis, Phone (+1) (916) 752-8775, Fax (+1) (916) 752-1031CA 95616, U.S.A. E-mail:
[email protected] (Received: October 5,2001; Accepted: December 2,2002)
Abstract - In this paper, the modeling of the mass transfer process in packed-bed reactors is presented and takes into account dispersion in the main fluid phase, internal diffusion of the reactant in the pores of the catalyst, and surface reaction inside the catalyst. The method of volume averaging is applied to obtain the governing equation for use On a small scale. The local mass equilibrium is assumed for obtaining the oneequation model for use on a large scale. The closure problems are developed subject to the length-scale constraints and the model of a spatially periodic porous medium. The expressions for effective diffusivity, hydrodynamic dispersion, total dispersion and the Darcy's law permeability tensors are presented. Solution of the set of final equations permits the variations of velocity and concentration of the chemical species along the packed-bed reactors to be obtained. Keywords: mass transfer, porous media, modeling.
INTRODUCTION A packed-bed reactor is illustrated in Figure 1, where the (J) region represents the porous catalyst pellets and the 11 region represents the main fluid phase.In order to develop spatially smoothed equations for the transport of chemical species in the porous medium, illustrated in Figure 1, it is necessary to use the large-scale averaging volume, Of/:,. Knowledge of the transport equations within the (J) and 11 regions is required. The (J) region contains two phases: the y phase (the fluid phase inside the (J) region) and the K phase (solid). To understand the transport process taking place within the (J) region, transport equation for the y phase must be developed
and this requires the use of the small-scale averaging volume, 0/7
SMALL-SCALE AVERAGING The boundary value problem for the concentration of a chemical species, A, in the y phase, illustrated in Figure 1, with a heterogeneous chemical reaction, can be expressed as aeAy = V.(DyVCAy)
at
in the y phase
B.C.1: -nyte.DyVCAy=kCAy
at
~
(1)
(2)
192
S.M.A.G.Ulson de Souza and S.Whitaker
2R
z
L
'll re~on
y phase
Figure 1: Mass transfer in a packed-bed reactor.
B.C.2: CAy=f(r,t) lC.: CAy = g(r)
at t = 0
at~
(3)
(4)
in which CAy is the molar concentration of chemical species under consideration, k is the heterogeneous reaction rate constant, and Dy is the y-phase molecular diffusivity of species A. Here variable ~ is used to represent the entrances and exits of the y phase at the boundary of the (J) region. Variable cd,... is used to represent the entire interfacial area within that region.
When the boundary value problem given by equations (1) through (4) is solved, the CAy concentration can be determined as a function of position and time. For design purposes, it is sufficient to determine the averaged concentration associated with the averaging volume, 0f7 The spatial averaging theorem (Howes and Whitaker, 1985) for the y - K system can be expressed as
(V\jly)=V(\jIy)+~
Brazilian Journal oj Chemical Engineering
f n"(K\jIydA A"(K
(5)
Mass Transfer in Porous Media
in which A'(K represents the area of Y-l( interface contained within O'f/: Application of the spatial averaging theorem (Whitaker, 1999) in equations (1) through (4) results in E
a
193
i
= 1,2,3
(9c)
When by and Sy are determined by equations (8) and (9), it can be shown that
Problem III:
-(CAY)Y =
Yat
'------v-------
accumulation
(lOa)
vfDy[
=
'-y V (CAY), +
~ A~
n",CA,dA
\
11-
(6)
B.C.l : -n"{K.V'1'Y = 0
at A'(K
B.C.2: '1'y(r+li)='1'y(r)
I
(lOb) i = 1,2,3
(lOc)
diffusion - aV"{Kk (CAy)'Y '---v------'
heterogeneous reaction The integral that appears in equation (6) is called the spatial deviation filter, because it acts as a filter, which allows some information to pass from the original point equation, and boundary condition, to local volume-averaged transport equation written for the intrinsic averaged concentration. To obtain a closed form of equation (6), a representation for the spatial deviation concentration must be developed.
Here Ii represents the three non-unique lattice vectors that are needed to describe a spatially periodic medium. The solution of Problem III is 'V = constant. Since this constant will not pass through the filter, the value of 'V plays no role in the closed form of the volume-averaged diffusion equation. The closed form of the volume-averaged diffusion equation can be given by
a
e -(CAY)Y = Yat
=v.[eyDy[V (CAy?
Vector by and scalar Sy are known as the closure variables and '1'y is an arbitrary function. The closure variables can be determined by the following boundary value problems:
+[J...Vy AYKf nYKbydA]SI (CAy? 11 +(11)
+v["~[ :, Ln",~+Ay)' ]The effective diffusivity tensor is defined by
Problem I:
(12) B.C.l: -n"{K.Vby = n"{K at A'(K
(8b)
i
= 1,2,3
(8c)
and the dimensionless vector associated with the chemical reaction is defined by
Problem II: (9a)
B.C.l:
- D"{K.V sy
(9b)
u = 1-
V:y
f [Dyk
sYj dA n"{K - -
(13)
A"{K
The use of these two definitions in equation (11) results in
Brazilian Journal of Chemical Engineering, Vol. 20, No. 02, pp. 191 -199, April- June 2003
194
Ey
S.M.A.O.Ulson de Souza and S.Whitaker
~t (CAy)Y = V.[ EyD~ff'V (CAY)Y] +
B.C.l: (CAY? =CAP at the ro-11 boundary (14)
+V.[EyUk(CAy?]-aV)'lC k (CAY)Y
B.C.2: -nffi1l.DpVCAl3n =
=n The second term on the right side of equation (14) represents the convective transport term that is generated by the heterogeneous reaction. This term is negligible for the case of diffusion in porous catalysts. Ryan (1983) has demonstrated that this term is equal to zero for the symmetric unit cells. Under these circumstances, equation (14) simplifies to
Ey
~t (CAy? = V.[ EyDJff'V (CAY?](15)
(17)
nffi1l.EyD~ff'V (CAY) Y
(18)
at the ro-11 boundary
(19) in the 11 region
B.C.3: CAp=dfi(r,t)
at ~e
(20a)
B.CA: (CAY? = &(r, t)
at cdroe
(20b)
-av)'lCk (CAY?
LARGE-SCALE AVERAGING The 0) and 11 regions associated with the averaging volume, ~, are shown in Figure 1. The length scales associated with the ro and 11 regions are designed by /10 and ~ and the volume fractions of these regions are identified by cP 10 and CPT]' respectively. The boundary value problem that forms the basis for large-scale averaging is given by
Ey
~t (CAY)Y = V.[ EyD~ff.v (CAY)Y](16)
-av)'lCk(CAY? in the ro region
~y'Pro ~{~
CAy)'
accumulatIon
r vLYDiCf.[ l =
lC.l: CAp=GYC'(r)
at t=O
(21a)
lC.2: (CAy? =oY(r)
at t=O
(21b)
where CAP is the ~-phase molar concentration of chemical species under consideration, Dp is the ~ phase molecular diffusivity of species A, and vp is the velocity vector of the ~ phase. The 11 region contains only the ~ phase, which is the same as the "(phase. The boundary conditions given by equations (17) and (18) are based on the idea that the local volumeaveraged concentration and flux are continuous at the 11-ro boundary. The volume-averaged form of equation (16) is given by
TJ
!{CA~}TJ
'----v-------'
accumulation
+ V'[ CJ>TJ {CA~}TJ {v~ '
conv~ctive
r]
195
given by
=
,
(CA) =_1
transport
f CAdV=
~o/£o
(24)
=CJ>ro{(CAy?f +CJ>TJ{CA~}TJ can be the proper single concentration to characterize the mass transfer process. Under the local mass equilibrium condition, the following equation can obtained:
+v{cp"D~[V{CA~}" +:" AL n"roCA~dA]] \
(E)i.(CA) + {v~ }.V (CA) = v.[ ,,~ff'V (CA)
I
diffusive Vtransport
at
'-v----'
accumulation
'-------.------ ' .
convectIve transport
,
diffusive transport
THE ONE-EQUATION MODEL
The one-equation model is based on the assumption that the mass transport process can be characterized by a single concentration. This assumption is valid when the system is in a state of local mass equilibrium (Whitaker, 1986a-b, 1991). If the size of porous particles is very small or the effective diffusivity is large, the local mass equilibrium condition can occur (Quintard and Whitaker, 1993, 1994a-e). The intrinsic spatial-averaged concentration,
To obtain a closed form of this equation, it is necessary to represent the spatial deviation concentration in the ro region and 11 region as functions of the dependent variable. As the two source terms in the closure problem are proportional to the gradient of the intrinsic spatial-averaged concentration, the following can be written: (28)
J-, (25)
where is the average porosity given by (26)
The effective diffusivity tensor for the 0)-11 system, ~eff , can be expressed as
Problem I: (30)
(31)
B.C.2 : -nroTJ.D~ VbTJ = -nroTJ.EyD ~ff .Vbro + +nroTJ. [ D~I - EyD ~ff ] at AroTJ
(32)
(29)
in which bl] and boo represent the closure variables to obtain
Brazilian Journal 01 Chemical Engineering, Vol. 20, No. 02, pp. 191 - 199, April- June 2003
(33)
196
S.M.A.O.Ulson de Souza and S.Whitaker
Periodicity: h'l(r+lj) = b11 (r) i = 1,2,3
(34a)
i = 1,2,3
(34b)
It can be proved that 'II = ; = constant, as discussed by Whitaker (1999). These constant values will have no influence on the closed form of the macroscopic equation, since they will not pass through the filter. Equations (28) and (29) can be introduced into equation (27) to obtain the following closed form:
(e)~(CA) +{v~}.V(CA)=V.[ q>ll~ff.V(CA)Jat ~, , .
'-----.r------'
accumulation
convectIve transport
v
diffusive transport
(35)
-~-V.[ 11{VpCAPrJ heterogeneous ' reaction
disp~rsive
'
DARCY'S LAW The physical process under consideration is that of a single-phase, incompressible flow in a rigid porous medium, such as the 0)-11 system illustrated in Figure 1. The boundary value problem is given by
(40) in the 11 region V.vp = 0 in the 11 regIOn
(41)
B.C.1: vp =0 at the 00-11 boundary
(42)
B.C.2: vp =f(r,t) at
~e
(43)
Here Pl3 represents the total pressure in the phase, while!J.l3 and Pl3 represent the viscosity and density of the ~ phase, respectively. ~e represents the 11 region entrances and exits. The regional average in the 11 region of Stokes's equation can be expressed as ~
transport
Here the effective diffusivity tensor for the 0)-11 system is defined by
2
0= -V {pp r + ppg +!J.p V {vp r + (44) (36)
The hydrodynamic dispersion tensor can be defined by (37)
f
+~ n 11Ol .[ -Ipp +!J.p Vvp ] dA 11 A11 0l The third term on the right side of equation (44) is known as the Brinkman correction. This term is unimportant for a flow in homogeneous porous medium as discussed by Whitaker (1986c). In order to develop the closed form of equation (44), it is necessary to solve the following boundary value problem for the closure problem:
The total dispersion tensor (Eidsath et aI., 1983; Han et aI., 1985) can be defined according to
Pl5*
= P4ff + ~
(38)
The use of equations (36) through (38) in equation (35) results in (e):t
f n11Ol.[-Ipp +!J.pVvp ] dA
11 A11 0l (46)
V.vp=O
(CA)+{V~}.V(CA)= (39)
= V.[ 11Pl5*.V (CA) ] -avYKkOl(CA)
~
(45)
B.C.l:vp =-{vpr '---v----'
source Brazilian Journal of Chemical Engineering
at~Ol
(47)
Mass Transfer in Porous Media
B.C.2: v~ = ~ at~e source Average:
(48)
{v~r =0
Here K is the permeability tensor. Equation (57) can be rearranged to give (59)
(49)
As the single source appears in this boundary value problem, a solution for the spatial deviation velocity and pressure can be proposed as (50)
and this is known as Darcy's law with the Brinkman correction. For a homogeneous porous medium, the Brinkman correction is negligible compared to the pressure term, and in this case,
(51)
(60)
Here B, b, 'II, and ~ are the closure variables. Problem I:
0=-Vb+V2B-~
197
f nTJO}"[-Ib+VB] dA
(52)
Darcy discovered experimentally this result in a one-dimensional version in 1850 (Whitaker, 1999). Whitaker (1996, 1997) has presented a theoretical development for the Forchheimer equation, given by
TJ ATJ
