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Comment on population balance and residence time distribution models for well mixed reactor and regenerator systems
Letters to the Editors
1820 that follows
principle
from an unthinking application to a fluidized suspension.
of Archimedes u
Department
of Chemical and Biochemical Colleye London
V
L. G. GIBILARO S. P. WALDRAM P. U. FOSCOLO Engineering
w, wb z
University Towington Place L.ondon WC1 E 7JE, U.K.
E
P PP
pressure. piezometric pressure (P = p + pgz) superficial fluid velocity particle volume effective weight of particle buoyancy force on particle vertical distance variable voidage fluid density particle density
NOTATION
particle diameter drag force on a single particle total drag force per unit cross-section acceleration due to gravity bed height
d Fd F, 9 L
Chemicd EnqineertngScience, Vol. 39, No. 12, Pnnted m Grsaf Britain.
Comment
REFERENCES Foscolo P. U., Gibilaro L. G. and Waldram S. P., Chem. Engng sci. 1983 38 1251. [2] Foscolo P. U. and Gibilaro L. G., Chem. Engng Sci. 1984 39 1667. [l]
of bed
pp. 1820-l&21.
1984.
on population balance and residence time distribution mixed reactor and regenerator systems
models
15 June 1984)
(Received
Dear Sirs, In a recent paper [I], the authors have tried to obtain activity distribution functions for a well-mixed reactorregenerator system. They have given the population balance equations for reactor and regenerator [eq. (27) and eq. (28) in Cl]]. It has been stated that this set of equations cannot be solved analytically by ordinary methods as it is a two-point boundary value problem, therefore a numerical scheme has been used. It is believed that similar difficulty has been faced in the past [2]. We would like to point out that analytical solution is possible [3] and is as given below in the forms of hypergeometric series:
where
The numerical scheme given by the authors [l] has been stated to coverage very slowly. We have checked their results
fi (4 = (-4 + C)IB AM
= D/B
where A=$[(l--p)~~:(f--.)+(m~pfl~~~p+l)a’~P(l-a) +(m-p+l)(m-p++2)(n-p+l)(n-p+f) 2!(2 -p:
a’_‘(1
.]
-a)+
~+(~-P+wm+1)+(~-P+l)(m-P+2)(n-p+l)(n-p+2) 1!(2-~)(3-P)
2-P
(--(B-l) B
u,_,(m-P+
[
1
+ . . . 2!(2 --PM3 -p)(4
lb-p+
-p)
u&.
1!(2-p)
+(m-p+l)(~--_p+2)(n-p+l)(n-_p+2)11,~~+ 2!(2 -_p)(3 -P) D=a’-P
l+(m-~+l)(n-~+l) 1!(2-p) 1
for well
~+(~-P+~)(m-P+2)(n-P+l)(n-P+2)aZ* 2!(2-P)(3
-P)
1
.
1821
Letters to the Editors eq. (33) of Cl], the first term in the denomenator a(l+b-B@)and not (l+p-B&).
with the present analytical solution and the accuracy of the numerical scheme seems to be quite good. The boundary conditions given in [1] to solve eq. (25) and eq. (26) are not necessary. However the conditions, L X(a)da= s cl
1;
should be
P. A. JOSH1 H. S. SHANKAR Department o/Chemical Engineering Indian Institute ofTechnology, Powzi Bombay 400 016, India
i= 1,2
REFERENCES
[l] WengH.S,andChenT.L.,Chem. Engng Sci. 198035915. [2] Rudd D. F., Can. J. them. Engng 1962 42 197. [3] Joshi P. A. and Shankar H. S., communicated to Chem. Engng J.
sufficient to solve the problem. In addition, there are a few minor errors; viz. Kunii and Levenspiel have derived the equations for mean activity for the ease of & = 1 and not for the case of 4 = 0; in
are
Chcmlcal Engineering Science,Vol. 39, No. 12, pp. 1821-1823. Printed in Great B&ah.
ow9-2509,84 s3m f0.U) F’ergamen Ptcss Ud.
1984.
A further study of the exact uniqueness and multiplicity lumped reaction systems (Receiwd
26 July 1984)
Dear Sirs, In a recently published study (Tsotsis et al.. 1982) the necessary and sufficient criteria for uniqueness and multiplicity for a general lumped model of chemically reacting systems were presented. The reaction rate expression used for the study was of the general form: I-@, T) =
(1)
(l+lc)’
where 0, 1+1/B,
83
-1
pi
-1
(3)
we write eq. (12) of Tsotsis et al. (1982) as: F’(Z)=
--AZ4+BZ’-CZ’+DZ-E=O
and eq. (13) of Tsotsis et al. (1982) becomes: A=l+L
B=(l+L){[B(y--2)-_p+11C1--al--3j -L[n-q(l-a)]-1
C=[l+jJ(l-a)]{(l+L)[2p+l+fi(l-a)] +2(l-qL)}-yj3(1--a)[a+2+L(l+2a)] +[p(l-a)+a][l+aL)-aqL(1 D=
-a)
(5)
-Cl+8(l-~)]ICl+B(l-~)1C~+l+(~-q)~] +2p)+yS(l-a){n[L(a+2)+2]+1}
Kckp(--E,fRBT)
[The symbols from Tsotsis ez al. (1982) are used throughout this paper, with the exception of the symbols caand Z which are defined below.] For the determination of the exact criteria for the general case it was assumed that q > p - 1 and that fi S, - 1. In the course of continuing studies of the dynamic and steady state behaviour of catalytic reaction systems, we found that an oversight was made in Tsotsis et al. (1982) in the application of Sturm’s theorem (Uspensky. 1945). In further study described here, we correct the oversight, and as a consequence, the previouscriteria are simplifiedconsiderably. We also extend the study to include thecases of q < p - 1 and fl< -1. Employing the following transformation of the dimeasionless concentration, X:
II=
criteria for a class of
(4)
E = p[l +b(l
-a)]‘-~(1
+aL.)y@(l -a).
This transformation maps the open interval a < X < 1, which is assured to have positive absolute temperatures, into the open interval 0 c Z < a). The substitution given in eq. (14) of Tsotsis et al. (1982) for Z puts eq. (15) of Tsotsis et al. (1982) in the form: -A’F’[Z(Y)]
=f(Y)=Y4+6HY2+4GY+(IA2-3H*) (6)
where G, H and I are defined in terms of A, B, C. D, E by eq. (16) ofTsotsis et al. (1982). which now uses the new forms of A, E. C, D. E given in eq. (5) above. The steady state multiplicity problem is thus reduced to the problem of ascertaining the number of real roots of the quartic equation in (6) for -B/4 < Y c CD,which corresponds to the physicallylimiteKlrangen
1820 that follows
principle
from an unthinking application to a fluidized suspension.
of Archimedes u
Department
of Chemical and Biochemical Colleye London
V
L. G. GIBILARO S. P. WALDRAM P. U. FOSCOLO Engineering
w, wb z
University Towington Place L.ondon WC1 E 7JE, U.K.
E
P PP
pressure. piezometric pressure (P = p + pgz) superficial fluid velocity particle volume effective weight of particle buoyancy force on particle vertical distance variable voidage fluid density particle density
NOTATION
particle diameter drag force on a single particle total drag force per unit cross-section acceleration due to gravity bed height
d Fd F, 9 L
Chemicd EnqineertngScience, Vol. 39, No. 12, Pnnted m Grsaf Britain.
Comment
REFERENCES Foscolo P. U., Gibilaro L. G. and Waldram S. P., Chem. Engng sci. 1983 38 1251. [2] Foscolo P. U. and Gibilaro L. G., Chem. Engng Sci. 1984 39 1667. [l]
of bed
pp. 1820-l&21.
1984.
on population balance and residence time distribution mixed reactor and regenerator systems
models
15 June 1984)
(Received
Dear Sirs, In a recent paper [I], the authors have tried to obtain activity distribution functions for a well-mixed reactorregenerator system. They have given the population balance equations for reactor and regenerator [eq. (27) and eq. (28) in Cl]]. It has been stated that this set of equations cannot be solved analytically by ordinary methods as it is a two-point boundary value problem, therefore a numerical scheme has been used. It is believed that similar difficulty has been faced in the past [2]. We would like to point out that analytical solution is possible [3] and is as given below in the forms of hypergeometric series:
where
The numerical scheme given by the authors [l] has been stated to coverage very slowly. We have checked their results
fi (4 = (-4 + C)IB AM
= D/B
where A=$[(l--p)~~:(f--.)+(m~pfl~~~p+l)a’~P(l-a) +(m-p+l)(m-p++2)(n-p+l)(n-p+f) 2!(2 -p:
a’_‘(1
.]
-a)+
~+(~-P+wm+1)+(~-P+l)(m-P+2)(n-p+l)(n-p+2) 1!(2-~)(3-P)
2-P
(--(B-l) B
u,_,(m-P+
[
1
+ . . . 2!(2 --PM3 -p)(4
lb-p+
-p)
u&.
1!(2-p)
+(m-p+l)(~--_p+2)(n-p+l)(n-_p+2)11,~~+ 2!(2 -_p)(3 -P) D=a’-P
l+(m-~+l)(n-~+l) 1!(2-p) 1
for well
~+(~-P+~)(m-P+2)(n-P+l)(n-P+2)aZ* 2!(2-P)(3
-P)
1
.
1821
Letters to the Editors eq. (33) of Cl], the first term in the denomenator a(l+b-B@)and not (l+p-B&).
with the present analytical solution and the accuracy of the numerical scheme seems to be quite good. The boundary conditions given in [1] to solve eq. (25) and eq. (26) are not necessary. However the conditions, L X(a)da= s cl
1;
should be
P. A. JOSH1 H. S. SHANKAR Department o/Chemical Engineering Indian Institute ofTechnology, Powzi Bombay 400 016, India
i= 1,2
REFERENCES
[l] WengH.S,andChenT.L.,Chem. Engng Sci. 198035915. [2] Rudd D. F., Can. J. them. Engng 1962 42 197. [3] Joshi P. A. and Shankar H. S., communicated to Chem. Engng J.
sufficient to solve the problem. In addition, there are a few minor errors; viz. Kunii and Levenspiel have derived the equations for mean activity for the ease of & = 1 and not for the case of 4 = 0; in
are
Chcmlcal Engineering Science,Vol. 39, No. 12, pp. 1821-1823. Printed in Great B&ah.
ow9-2509,84 s3m f0.U) F’ergamen Ptcss Ud.
1984.
A further study of the exact uniqueness and multiplicity lumped reaction systems (Receiwd
26 July 1984)
Dear Sirs, In a recently published study (Tsotsis et al.. 1982) the necessary and sufficient criteria for uniqueness and multiplicity for a general lumped model of chemically reacting systems were presented. The reaction rate expression used for the study was of the general form: I-@, T) =
(1)
(l+lc)’
where 0, 1+1/B,
83
-1
pi
-1
(3)
we write eq. (12) of Tsotsis et al. (1982) as: F’(Z)=
--AZ4+BZ’-CZ’+DZ-E=O
and eq. (13) of Tsotsis et al. (1982) becomes: A=l+L
B=(l+L){[B(y--2)-_p+11C1--al--3j -L[n-q(l-a)]-1
C=[l+jJ(l-a)]{(l+L)[2p+l+fi(l-a)] +2(l-qL)}-yj3(1--a)[a+2+L(l+2a)] +[p(l-a)+a][l+aL)-aqL(1 D=
-a)
(5)
-Cl+8(l-~)]ICl+B(l-~)1C~+l+(~-q)~] +2p)+yS(l-a){n[L(a+2)+2]+1}
Kckp(--E,fRBT)
[The symbols from Tsotsis ez al. (1982) are used throughout this paper, with the exception of the symbols caand Z which are defined below.] For the determination of the exact criteria for the general case it was assumed that q > p - 1 and that fi S, - 1. In the course of continuing studies of the dynamic and steady state behaviour of catalytic reaction systems, we found that an oversight was made in Tsotsis et al. (1982) in the application of Sturm’s theorem (Uspensky. 1945). In further study described here, we correct the oversight, and as a consequence, the previouscriteria are simplifiedconsiderably. We also extend the study to include thecases of q < p - 1 and fl< -1. Employing the following transformation of the dimeasionless concentration, X:
II=
criteria for a class of
(4)
E = p[l +b(l
-a)]‘-~(1
+aL.)y@(l -a).
This transformation maps the open interval a < X < 1, which is assured to have positive absolute temperatures, into the open interval 0 c Z < a). The substitution given in eq. (14) of Tsotsis et al. (1982) for Z puts eq. (15) of Tsotsis et al. (1982) in the form: -A’F’[Z(Y)]
=f(Y)=Y4+6HY2+4GY+(IA2-3H*) (6)
where G, H and I are defined in terms of A, B, C. D, E by eq. (16) ofTsotsis et al. (1982). which now uses the new forms of A, E. C, D. E given in eq. (5) above. The steady state multiplicity problem is thus reduced to the problem of ascertaining the number of real roots of the quartic equation in (6) for -B/4 < Y c CD,which corresponds to the physicallylimiteKlrangen