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Mathematical formulation of cake filtration for deformable solid particles
Shorter
Communications stoichiometric coefficient auxiliary variables
NOTATION
Di D
A,,
radius of catalyst pellet, m effective permeability, m2 effective Knudsen diffusion coefficient, m2/s effective molecular coefficient, m2/s auxiliary coefficients reaction rate constant, kmol/kg h equilibrium constant coordinate along radius of catalyst pellet, m refers to a shape of catalyst pellet, m = 2 (sphere catalyst) number of components molar flux, mole/m2 s pressure, MPa universal gas constant, J/mol K reaction rate, kmol/kg h temperature auxiliary variable mole fraction
Eh&eerhg Schce. Britnin.
ia Great
Vol.
40.
No.
4, pp.
673474,
Rz
Subscripts i refers to a number of component and a number of reaction refers to a number of component _i, k refers to the conditions at the surface of catalyst 0 pellet refers to reactions (l), (2), respectively 1, 2 1, 2, 3, 4, 5, 6 refers to CO, CO,, H,, CH,OH, H20, N2, respectively
REFERENCES
Skrzypek J., Grzesik M. and Szopa R., 1984, Theoretical analysis of two parallel and consecutive reactions in isothermal catalyst pellets using the dusty-gas model. Chem. Engng Sci. 39 515-521. Szarawara J. and Reychman K., 1980, Model kinetyczny niskociSnieniowej syntezy metanolu. Inl. Chem. Proc. 1 331-343.
Greek symbols effectiveness factor ? viscosity, Pa s p
Chemical Printed
673
1985. 0
Mathematical
formulation
ooo9-2509185 93.00+0.00 1985. P.zrgamon Press Ltd.
of cake filtration for deformable solid particles (Receiued
30 March
Mathematical formulation of cake filtration gives insight into fo-tion and structure of filter cakes. Solution of the resulting equation enables one to predict the distribution of various important variables, such as porosity and hydraulic pressure, within the filter cake. Modelling of constant applied-pressure cake filtration is first attempted by Tiller and Cooper (1960). In the derivation of the so-called “basic differential equation” the validity of the compression-permeability test cell (CPTC) is assumed. However, the ability of CPTC to simulate cake filtration is seriously questioned by several investigators (Willis et al., 1974; Wakeman, 1978; Willis and Tosun, 1980). Later Smiles (1970), Atsumi and Akiyama (1975), Wakeman (1978) and Tosun (1984) presented different formulations of cake filtration. In all of these presentations solid particles are assumed to be non-deformable. In actual practice, solid particles in the cake move and deform as a result of the drag forces associated with the flow. It is the purpose of this note to formulate cake filtration for the case of deformable solid particles. St-train relations for the solid particles are assumed to be represented by Hooke’s law and the solution is obtained for a simplified case of a onedimensional constant applied-pressure filtration. FORMULATION The development of the equations of continuity and motion for cake filtration is given by Willis and Tosun (1980) in detail using the volume averaging technique. Following the nomenclature commonly used in the filtration literature, the equations of continuity for the liquid and solid phases can be written as
1984)
a&
z+v.q=o and
Addition
-_a& +v.r=o. at of eqs (1) and (2) gives v.q=
The equation
of motion
-v.r.
or Darcy’s
q-&r=
law is
-5P. p
Taking the divergence of eq. (4) and relationship given by eq. (3) yields = ;
(4) making
use of
the
.
V (KVP).
The term r/(1 -E) in eq. (5) represents the actual solids velocity and can be related to the solid displacement vector, u. as (6) so
that eq. (5) becomes ae
-‘V.(KVP) ar-p where the volume
strain or dilatation, e=V.U.
e, is defined
by (8)
674
Shorter Communications
The condition as (Biot, 1941)
of equilibrium v.a
in a porous medium is given
=VP.
(9)
The stressstrain relationship for an isotropic, represented by Hook& law in the form u = G[VU
+ (VU)=]
+ n(V.
elastic solid is
U)I
(10)
where G and I are the Lame constants. Substitution of eq. (10) into eq. (9) gives GVZU+(l+G)Ve Divergence
= VP.
(11)
of eq. (11) results in (A + 2G)V2e
whose solution
= V2P
(12)
is (I+2G)e
= P+S
(13)
where $ is a function satisfying Laplace’s equation V’$ = 0. An analysis of cake filtration requires simultaneous solution of eqs (7) and (13).
One-dimensional case When total stress is constant, Verruijt (1969) showed that + = 0 for displacements occurring only in one direction. These conditions are met for a one-dimensional constant applied-pressure filtration. Under these circumstances, eqs (7) and (13) take the form (1+2G)e
= P
(14)
iSMAIL
(15) Substitution
NOTATION
a e f
K L P
(16) If the variation in permeability between the entrance and exit is small, then K can be considered constant. In this case eq. (16) becomes
ap alp at=as
p
at
t =O;
P=
PO(t)
at at
P = PA The solution
.
for all z
(19)
z=O;
t>o
(20)
z = L(t);
t > 0.
(21)
system for filtration.
symbols pressure drop across the cake, N/m’ porosity, dimensionless Lame constant detined by eq. (lo), N/m2 liquid viscosity, Ns/m2 effective stress tensor, N/m2 function satisfying Laplace’s equation, N/m2 REFERENCES
of eq. (17) by the Laplace
1. Coordinate
parameter defined by eq. (1 S), m2/s dilatation, defined by eq. (S), dimensionless shear modulus, defined by eq. (lo), N/m* identity tensor, dimensionless permeability, m2 cake length, m hydraulic pressure, N/m’ applied pressure, N/m’ pressure at the cake-septum interface, N/m2 superi?cial liquid velocity, m/s superficial solid particle velocity, m/s unit step function time, s solid displacement vector, m dummy variable in eq. (22). m axial coordinate, m
(18)
Considering a schematic diagram of a filter cake shown in Fig 1, the initial and boundary conditions associated with eq. (17) are PA
Greek AP,
;
(A + 2G)K
P=
U x z
: P
where a=
PO
q : t
of eq. (14) into eq. (15) gives
TOSUN
Department of Chemical Engineering Middle East Technical University Ankara, Turkey
pA
and
Fig.
Note that the pressure drop across the cake, A P,. as a function of time should be known before evaluating eq. (22). Once pressure distribution is determined, dilatation can be calculated using eq. (14).
transformation
one-dimensional
is
cake
Atsumi K. and Akiyama T., 1975, A study of cake filtrationformulation as a Stefan problem. J. Chem. Engng Japan 8 487-492. Biot M. A., 1941, General theory of three-dimensional consolidation. J. appl. Phys. 12 155-164. Smiles D. E., 1970, A theory of constant pressure filtration. Chem. Engng Sci. 25 985996. Tiller F. M. and Cooper H. R., 1960, The role of porosity in f&ration-IV. Constant pressure filtration. A.1.Ch.E. J. 6 59MOl. Tosun I., 1984, Formulation of cake filtration. Proc. Filt. Sot. submitted for publication. Verruijt A., 1969, Elastic storage of aquifers, in Flow Through Porous Media (Edited by De Weist R. J. M.), pp. 331-376. Academic Press, New York. Wakeman R. J., 1978, A numerical integration of the differential equations describing the formation of and flow in compressible filter cakes. Trans. Inst. Chem. Engrs 56 258-265. Willis M. S. and Tosun I., 1980, A rigorous cake filtration theory. Chem. Engng Sci. 35 2427-2438. Willis M. S., Shen M. and Gray K. J.. 1974, Investigation of the fundamental assumptions relating compressionpermeability data with filtration. Can. J. Chem. Engng 52 331-337.
Communications stoichiometric coefficient auxiliary variables
NOTATION
Di D
A,,
radius of catalyst pellet, m effective permeability, m2 effective Knudsen diffusion coefficient, m2/s effective molecular coefficient, m2/s auxiliary coefficients reaction rate constant, kmol/kg h equilibrium constant coordinate along radius of catalyst pellet, m refers to a shape of catalyst pellet, m = 2 (sphere catalyst) number of components molar flux, mole/m2 s pressure, MPa universal gas constant, J/mol K reaction rate, kmol/kg h temperature auxiliary variable mole fraction
Eh&eerhg Schce. Britnin.
ia Great
Vol.
40.
No.
4, pp.
673474,
Rz
Subscripts i refers to a number of component and a number of reaction refers to a number of component _i, k refers to the conditions at the surface of catalyst 0 pellet refers to reactions (l), (2), respectively 1, 2 1, 2, 3, 4, 5, 6 refers to CO, CO,, H,, CH,OH, H20, N2, respectively
REFERENCES
Skrzypek J., Grzesik M. and Szopa R., 1984, Theoretical analysis of two parallel and consecutive reactions in isothermal catalyst pellets using the dusty-gas model. Chem. Engng Sci. 39 515-521. Szarawara J. and Reychman K., 1980, Model kinetyczny niskociSnieniowej syntezy metanolu. Inl. Chem. Proc. 1 331-343.
Greek symbols effectiveness factor ? viscosity, Pa s p
Chemical Printed
673
1985. 0
Mathematical
formulation
ooo9-2509185 93.00+0.00 1985. P.zrgamon Press Ltd.
of cake filtration for deformable solid particles (Receiued
30 March
Mathematical formulation of cake filtration gives insight into fo-tion and structure of filter cakes. Solution of the resulting equation enables one to predict the distribution of various important variables, such as porosity and hydraulic pressure, within the filter cake. Modelling of constant applied-pressure cake filtration is first attempted by Tiller and Cooper (1960). In the derivation of the so-called “basic differential equation” the validity of the compression-permeability test cell (CPTC) is assumed. However, the ability of CPTC to simulate cake filtration is seriously questioned by several investigators (Willis et al., 1974; Wakeman, 1978; Willis and Tosun, 1980). Later Smiles (1970), Atsumi and Akiyama (1975), Wakeman (1978) and Tosun (1984) presented different formulations of cake filtration. In all of these presentations solid particles are assumed to be non-deformable. In actual practice, solid particles in the cake move and deform as a result of the drag forces associated with the flow. It is the purpose of this note to formulate cake filtration for the case of deformable solid particles. St-train relations for the solid particles are assumed to be represented by Hooke’s law and the solution is obtained for a simplified case of a onedimensional constant applied-pressure filtration. FORMULATION The development of the equations of continuity and motion for cake filtration is given by Willis and Tosun (1980) in detail using the volume averaging technique. Following the nomenclature commonly used in the filtration literature, the equations of continuity for the liquid and solid phases can be written as
1984)
a&
z+v.q=o and
Addition
-_a& +v.r=o. at of eqs (1) and (2) gives v.q=
The equation
of motion
-v.r.
or Darcy’s
q-&r=
law is
-5P. p
Taking the divergence of eq. (4) and relationship given by eq. (3) yields = ;
(4) making
use of
the
.
V (KVP).
The term r/(1 -E) in eq. (5) represents the actual solids velocity and can be related to the solid displacement vector, u. as (6) so
that eq. (5) becomes ae
-‘V.(KVP) ar-p where the volume
strain or dilatation, e=V.U.
e, is defined
by (8)
674
Shorter Communications
The condition as (Biot, 1941)
of equilibrium v.a
in a porous medium is given
=VP.
(9)
The stressstrain relationship for an isotropic, represented by Hook& law in the form u = G[VU
+ (VU)=]
+ n(V.
elastic solid is
U)I
(10)
where G and I are the Lame constants. Substitution of eq. (10) into eq. (9) gives GVZU+(l+G)Ve Divergence
= VP.
(11)
of eq. (11) results in (A + 2G)V2e
whose solution
= V2P
(12)
is (I+2G)e
= P+S
(13)
where $ is a function satisfying Laplace’s equation V’$ = 0. An analysis of cake filtration requires simultaneous solution of eqs (7) and (13).
One-dimensional case When total stress is constant, Verruijt (1969) showed that + = 0 for displacements occurring only in one direction. These conditions are met for a one-dimensional constant applied-pressure filtration. Under these circumstances, eqs (7) and (13) take the form (1+2G)e
= P
(14)
iSMAIL
(15) Substitution
NOTATION
a e f
K L P
(16) If the variation in permeability between the entrance and exit is small, then K can be considered constant. In this case eq. (16) becomes
ap alp at=as
p
at
t =O;
P=
PO(t)
at at
P = PA The solution
.
for all z
(19)
z=O;
t>o
(20)
z = L(t);
t > 0.
(21)
system for filtration.
symbols pressure drop across the cake, N/m’ porosity, dimensionless Lame constant detined by eq. (lo), N/m2 liquid viscosity, Ns/m2 effective stress tensor, N/m2 function satisfying Laplace’s equation, N/m2 REFERENCES
of eq. (17) by the Laplace
1. Coordinate
parameter defined by eq. (1 S), m2/s dilatation, defined by eq. (S), dimensionless shear modulus, defined by eq. (lo), N/m* identity tensor, dimensionless permeability, m2 cake length, m hydraulic pressure, N/m’ applied pressure, N/m’ pressure at the cake-septum interface, N/m2 superi?cial liquid velocity, m/s superficial solid particle velocity, m/s unit step function time, s solid displacement vector, m dummy variable in eq. (22). m axial coordinate, m
(18)
Considering a schematic diagram of a filter cake shown in Fig 1, the initial and boundary conditions associated with eq. (17) are PA
Greek AP,
;
(A + 2G)K
P=
U x z
: P
where a=
PO
q : t
of eq. (14) into eq. (15) gives
TOSUN
Department of Chemical Engineering Middle East Technical University Ankara, Turkey
pA
and
Fig.
Note that the pressure drop across the cake, A P,. as a function of time should be known before evaluating eq. (22). Once pressure distribution is determined, dilatation can be calculated using eq. (14).
transformation
one-dimensional
is
cake
Atsumi K. and Akiyama T., 1975, A study of cake filtrationformulation as a Stefan problem. J. Chem. Engng Japan 8 487-492. Biot M. A., 1941, General theory of three-dimensional consolidation. J. appl. Phys. 12 155-164. Smiles D. E., 1970, A theory of constant pressure filtration. Chem. Engng Sci. 25 985996. Tiller F. M. and Cooper H. R., 1960, The role of porosity in f&ration-IV. Constant pressure filtration. A.1.Ch.E. J. 6 59MOl. Tosun I., 1984, Formulation of cake filtration. Proc. Filt. Sot. submitted for publication. Verruijt A., 1969, Elastic storage of aquifers, in Flow Through Porous Media (Edited by De Weist R. J. M.), pp. 331-376. Academic Press, New York. Wakeman R. J., 1978, A numerical integration of the differential equations describing the formation of and flow in compressible filter cakes. Trans. Inst. Chem. Engrs 56 258-265. Willis M. S. and Tosun I., 1980, A rigorous cake filtration theory. Chem. Engng Sci. 35 2427-2438. Willis M. S., Shen M. and Gray K. J.. 1974, Investigation of the fundamental assumptions relating compressionpermeability data with filtration. Can. J. Chem. Engng 52 331-337.