- Email: [email protected]
Exact uniqueness and multiplicity criteria for a first-order reversible reaction in a CSTR
O0000000000000000000000000000000000CO000000000000CO00000000000000000~O00000000
Pergamon
Chemical Enoineerin0 Science, Vol, 50, No, 13, pp. 2189 2192, 1995 Copyright ~: 1995 Elsevier Science Lid Pnnted in Great Britain. All righls resctwe,d 0009.2509/95 $9.50 + 0.00
0009-2509(95)00045-3
E x a c t uniqueness and multiplicity criteria for a first-order reversible reaction in a C S T R
(Received 25 November 1994; accepted in revised form 4 January 1995)
MODEL EQUATIONS The mass and heat balances for a single reaction of any kinetics, occurring in a nonadiabatic CSTR, are as follows:
l - X + VADaR =O
(1)
1 - y + DaftR = 0.
(2)
we finally obtain the equation from which the variable z can be determined: 4)= - I n z
The dimensionless state variables and numbers are defined in the Notation. The dimensionless rate of a first-order reversible reaction is
[
1 +Da(l
ro
exp y I -
k1(T=)CAo
e x p [ y x ( l - l/y)] K
+yflDaz=O.
Da(l + f l ) = b ;
x
Da --=c K
(1 - x)
(3}
= - (1 + bz + cz~)lnz + az = O. MATHEMATICAL
CA + Cs = Ca0.
ANALYSIS
(A) /(
/
asymptote
C . @e -
asymptote i
z
- ~3.8~-
•
.
i
(8)
The mathematical analysis of the function 4~ is carried out from the standpoint of the existence of multiple solutions to eq. (8). Due to the presence of four parameters, the highest possible codimension of a singular point can be at most four.
(4)
Upon introduction of a new dimensionless variable which defines the temperature in the reactor
"~.@e , -:D~.82
(7)
reduces the overall number of parameters in eq. (6) from five to four:
where
e.ee~
(61
The introduction of the following parameters
OaTft=a; R
+fllz+
.
Fig. I(A). 2189
,
~
"
-
"
,
C 9@. 88
Shorter Communication
2190
(B) 1.40
K
asymptote
1 , 2@ exP(3)+-z+exP(2) 1
0.60,,
I
I,
I,,
-2.00
1
s /,
I,,
I, 1.00
0.00
-1.00
d
I,,
I,
/I,, 2.00
I
c
c I ‘, 3.00
Fig. l(A). Projection of the third and fourth derivatives onto the (c, K) plane; (B) projection of the third and fourth derivatives onto the (c, K) plane (enlargement of part “d” in Fig. l(A)).
Consequently, eq. (8) cannot have more than five solutions. Determining the successive derivatives of $Jand setting them equal to zero we get: 4=0
=Z. a=
bz+cz”+llnz
(9)
Z
,$‘=O
=
b=
lnz-1
l)lnz+
--CZ”_‘[(KZ
l]
(10)
2-lnz ‘” = ’ a
’ =
(11) (K -
‘#‘“‘=O =Z. Kz=
l)Z’[(K
-
1) h
Z +
lnzz-4lnz+6 (lnz_2)lnz
23
a
1. Fourth derivative vanishes for 0 < K i 1 (Fig. l(B)), leading to z > 1, which is incompatible with the endothermal nature of the process. 2. Fourth derivative is equal to zero at a singular point + in Fig. l(A) (z E R+). Then, however, c < 0, which is again hardly justifiable from the physical point of view. 3. Third derivative equals zero (Fig. l(B)) for 0 < K < 1; thus, z > 1, which is inconsistent with the endothermal course of the reaction. The line for K < 0 in Fig. l(A) has no physical significance for either endothermal or exothermal reactions.
(12)
4”” = 0 =E. K=
-(l+,/?);
a=
12$efi-’ 5+3J3
b=2
; e-t3+J5)
lnz=3+J5;
I1nzoR;
a=
K=&-l;
K = O;
-12J5-------. 5-3fi’
a =O;
’ +fi
’
5+3J5 b=2-e 1 -d
5-3$ b =O;
c=
e,,-‘.
(J5 -
l)ez*”
5+3J3
-@+jS,.
* c=
-1. (13)
Thus, two singular points are obtained of codimension four
and coordinates as above, and also a locus of singular points of codimension four (In z E R). A graphical representation of the third and fourth derivatives is shown in Fig. l(A) and (B). The conclusions that follow depend on whether the process is exothermal or endothermal. I. Exothermal reaction (K > 1, z > 1) 1. Fourth derivative is different from zero for K > 1. 2. Third derivative is equal to zero for K > 1 in Fig. l(A). However, z is then less than one, which contradicts the fact that the reaction is exothermal. Since, additionally, c < 0, this result is also unacceptable on physical grounds. 3. Third derivative becomes zero for K > 1 (Fig. l(B)). However, in that case c < 0, which is not physically justified. II. Endothermal reaction (0 <
K <
1, 0 < z < 1)
It follows from the foregoing discussion that none of the two processes can have five steady states. Therefore, in further analysis we shall concentrate upon the case of codimension two, i.e. upon the system of eqs (9)-(11). First, it should be noted that in the case of an endothermal reaction (0 < K < 1, 0 < z < 1) the parameter c in eq. (11) becomes negative, which is physically unacceptable. Thus, an endothermal process cannot possibly have any multiple steady states, and we shall focus our attention on an exotherma1 reaction (K 2 1, z B 1). The hysteresis varieties are determined in the system of coordinates (b, c), and the results are shown in Fig. 2. It can be clearly seen that the individual curves tend asymptotically to the curve for K = 1, when In(z) = 2. The left-hand limit of this asymptotic curve is b = 0, c = ew2, and
Shorter Communication
2191
0.16- C
0.~@
0.e2
~3.0z
0.06
e.~3~
~3.10
0.',:
~.12
Fig. 2. Hysteresis variety for a first-order reversible reaction.
2o. ee ~ y
12"00i 8.0~-
',
'\\ P ,
,
.
-
.
,
,
i
. , ¸ ,
- , i
r
.
.
.
.
Fig. 3. Hysteresis variety for a first-order irreversible reaction.
the right-hand limit is given by b = e - 2, c = O. The above conclusion follows directly from eqs (10) and (11). All the hysteresis varieties, related to the individual values of x >1 1, have the same fight-hand limit at the point In z = 2,
a=4e-2 b=e-2, c=O. The extension of this limiting point yields a well-known result (Tsotsis and Schmitz, 1979; Leib and Luss, 1981) (Fig. 3): ), = 4 -
(14)
Equation (14) defines the hysteresis variety for a first-order irreversible reaction. Based on this equation it may be shown that, on a hysteresis variety, the dimensionless temperature y is given by
tion of steady state for a CSTR in which a first-order irreversible reaction takes place, the maximum number of solutions cannot exceed three. The hysteresis varieties in Fig. 2 divide the plane (b. c) into two subregions with at most three (below the line) and one steady states. All the hysteresis varieties converge at the point c = 0, b = e - 2. It is readily seen from eq. (8) that for c = 0 this point defines the hysteresis varieties for first-order irreversible reactions (Fig. 3). The steady-state equation for this case contains only two parameters (a. b) instead of three (7, fl, Da) which appear in the equations discussed by Leib and Luss (1981); this considerably simplifies the analysis. M. BEREZOWSKI A. B U R G H A R D T
1+/~ y = 2 +---"fl 2 '
(15)
It follows from the foregoing discussion that despite the presence of four (and, initially, five) parameters in the equa-
Polish Academy of Sciences Institute of Chemical Enoineering ul. Baltycka 5, 44-100 Gliwice, Poland E-mail: [email protected]
Shorter Communication
2192 cp C
Da E F Gc Ah H
NOTATION heat capacity, kJ/(kg K) concentration of reactant, kmol/m 3 Damk6hler number (V/q). [ro(T,~, x0)/CAo] activation energy, kJ/kmol area through which heat transfer occurs, m 2 mass flow rate of cooling medium, kg/s heat of reaction, kJ&mol dimensionless heat exchange parameter,
{°--I- ( L)I} qpcp k K q r R Ro T Tm U V x
reaction rate constant, I/s equilibrium constant, kl (Tm)/k2(Tm) volumetric flow rate, m3/s rate of reaction, kmol/(m 3 s) dimensionless reaction rate defined by eq. (3) gas constant, kJ/(kmol K) temperature, K mean reference temperature (To + H 7~-o)/(1 + H), K overall heat transfer coefficient, W(m2 K) volume of the reactor, m 3 dimensionless concentration, Ca/CAo
dimensionless temperature, T/T,~ dimensionless variable defined by eq. (5)
Greek letters fl ;, ~p ~b
dimensionless heat of reaction,
( -
Ah)
Cao
I-pcp(l + H)T,~] dimensionless activation energy, E~ l(Rg T,,) ratio of activation energies, E2/E~ density, kg/m 3 function defined by eqs (6) and (8)
Subscripts A B c o
component A component B cooling medium feed
REFERENCES
Leib, T. M. and Luss, D., 1981, Exact uniqueness and multiplicity criteria for an nth order reaction in a CSTR. Chem. Engng Sci. 36, 210--212. Tsotsis, T. T. and Schmitz, R. A., 1979, Exact uniqueness and multiplicity criteria for a positive-order Arrhenius reaction in a lumped system. Chem. Engng Sci. 34, 135-137.
Pergamon
Chemical Enoineerin0 Science, Vol, 50, No, 13, pp. 2189 2192, 1995 Copyright ~: 1995 Elsevier Science Lid Pnnted in Great Britain. All righls resctwe,d 0009.2509/95 $9.50 + 0.00
0009-2509(95)00045-3
E x a c t uniqueness and multiplicity criteria for a first-order reversible reaction in a C S T R
(Received 25 November 1994; accepted in revised form 4 January 1995)
MODEL EQUATIONS The mass and heat balances for a single reaction of any kinetics, occurring in a nonadiabatic CSTR, are as follows:
l - X + VADaR =O
(1)
1 - y + DaftR = 0.
(2)
we finally obtain the equation from which the variable z can be determined: 4)= - I n z
The dimensionless state variables and numbers are defined in the Notation. The dimensionless rate of a first-order reversible reaction is
[
1 +Da(l
ro
exp y I -
k1(T=)CAo
e x p [ y x ( l - l/y)] K
+yflDaz=O.
Da(l + f l ) = b ;
x
Da --=c K
(1 - x)
(3}
= - (1 + bz + cz~)lnz + az = O. MATHEMATICAL
CA + Cs = Ca0.
ANALYSIS
(A) /(
/
asymptote
C . @e -
asymptote i
z
- ~3.8~-
•
.
i
(8)
The mathematical analysis of the function 4~ is carried out from the standpoint of the existence of multiple solutions to eq. (8). Due to the presence of four parameters, the highest possible codimension of a singular point can be at most four.
(4)
Upon introduction of a new dimensionless variable which defines the temperature in the reactor
"~.@e , -:D~.82
(7)
reduces the overall number of parameters in eq. (6) from five to four:
where
e.ee~
(61
The introduction of the following parameters
OaTft=a; R
+fllz+
.
Fig. I(A). 2189
,
~
"
-
"
,
C 9@. 88
Shorter Communication
2190
(B) 1.40
K
asymptote
1 , 2@ exP(3)+-z+exP(2) 1
0.60,,
I
I,
I,,
-2.00
1
s /,
I,,
I, 1.00
0.00
-1.00
d
I,,
I,
/I,, 2.00
I
c
c I ‘, 3.00
Fig. l(A). Projection of the third and fourth derivatives onto the (c, K) plane; (B) projection of the third and fourth derivatives onto the (c, K) plane (enlargement of part “d” in Fig. l(A)).
Consequently, eq. (8) cannot have more than five solutions. Determining the successive derivatives of $Jand setting them equal to zero we get: 4=0
=Z. a=
bz+cz”+llnz
(9)
Z
,$‘=O
=
b=
lnz-1
l)lnz+
--CZ”_‘[(KZ
l]
(10)
2-lnz ‘” = ’ a
’ =
(11) (K -
‘#‘“‘=O =Z. Kz=
l)Z’[(K
-
1) h
Z +
lnzz-4lnz+6 (lnz_2)lnz
23
a
1. Fourth derivative vanishes for 0 < K i 1 (Fig. l(B)), leading to z > 1, which is incompatible with the endothermal nature of the process. 2. Fourth derivative is equal to zero at a singular point + in Fig. l(A) (z E R+). Then, however, c < 0, which is again hardly justifiable from the physical point of view. 3. Third derivative equals zero (Fig. l(B)) for 0 < K < 1; thus, z > 1, which is inconsistent with the endothermal course of the reaction. The line for K < 0 in Fig. l(A) has no physical significance for either endothermal or exothermal reactions.
(12)
4”” = 0 =E. K=
-(l+,/?);
a=
12$efi-’ 5+3J3
b=2
; e-t3+J5)
lnz=3+J5;
I1nzoR;
a=
K=&-l;
K = O;
-12J5-------. 5-3fi’
a =O;
’ +fi
’
5+3J5 b=2-e 1 -d
5-3$ b =O;
c=
e,,-‘.
(J5 -
l)ez*”
5+3J3
-@+jS,.
* c=
-1. (13)
Thus, two singular points are obtained of codimension four
and coordinates as above, and also a locus of singular points of codimension four (In z E R). A graphical representation of the third and fourth derivatives is shown in Fig. l(A) and (B). The conclusions that follow depend on whether the process is exothermal or endothermal. I. Exothermal reaction (K > 1, z > 1) 1. Fourth derivative is different from zero for K > 1. 2. Third derivative is equal to zero for K > 1 in Fig. l(A). However, z is then less than one, which contradicts the fact that the reaction is exothermal. Since, additionally, c < 0, this result is also unacceptable on physical grounds. 3. Third derivative becomes zero for K > 1 (Fig. l(B)). However, in that case c < 0, which is not physically justified. II. Endothermal reaction (0 <
K <
1, 0 < z < 1)
It follows from the foregoing discussion that none of the two processes can have five steady states. Therefore, in further analysis we shall concentrate upon the case of codimension two, i.e. upon the system of eqs (9)-(11). First, it should be noted that in the case of an endothermal reaction (0 < K < 1, 0 < z < 1) the parameter c in eq. (11) becomes negative, which is physically unacceptable. Thus, an endothermal process cannot possibly have any multiple steady states, and we shall focus our attention on an exotherma1 reaction (K 2 1, z B 1). The hysteresis varieties are determined in the system of coordinates (b, c), and the results are shown in Fig. 2. It can be clearly seen that the individual curves tend asymptotically to the curve for K = 1, when In(z) = 2. The left-hand limit of this asymptotic curve is b = 0, c = ew2, and
Shorter Communication
2191
0.16- C
0.~@
0.e2
~3.0z
0.06
e.~3~
~3.10
0.',:
~.12
Fig. 2. Hysteresis variety for a first-order reversible reaction.
2o. ee ~ y
12"00i 8.0~-
',
'\\ P ,
,
.
-
.
,
,
i
. , ¸ ,
- , i
r
.
.
.
.
Fig. 3. Hysteresis variety for a first-order irreversible reaction.
the right-hand limit is given by b = e - 2, c = O. The above conclusion follows directly from eqs (10) and (11). All the hysteresis varieties, related to the individual values of x >1 1, have the same fight-hand limit at the point In z = 2,
a=4e-2 b=e-2, c=O. The extension of this limiting point yields a well-known result (Tsotsis and Schmitz, 1979; Leib and Luss, 1981) (Fig. 3): ), = 4 -
(14)
Equation (14) defines the hysteresis variety for a first-order irreversible reaction. Based on this equation it may be shown that, on a hysteresis variety, the dimensionless temperature y is given by
tion of steady state for a CSTR in which a first-order irreversible reaction takes place, the maximum number of solutions cannot exceed three. The hysteresis varieties in Fig. 2 divide the plane (b. c) into two subregions with at most three (below the line) and one steady states. All the hysteresis varieties converge at the point c = 0, b = e - 2. It is readily seen from eq. (8) that for c = 0 this point defines the hysteresis varieties for first-order irreversible reactions (Fig. 3). The steady-state equation for this case contains only two parameters (a. b) instead of three (7, fl, Da) which appear in the equations discussed by Leib and Luss (1981); this considerably simplifies the analysis. M. BEREZOWSKI A. B U R G H A R D T
1+/~ y = 2 +---"fl 2 '
(15)
It follows from the foregoing discussion that despite the presence of four (and, initially, five) parameters in the equa-
Polish Academy of Sciences Institute of Chemical Enoineering ul. Baltycka 5, 44-100 Gliwice, Poland E-mail: [email protected]
Shorter Communication
2192 cp C
Da E F Gc Ah H
NOTATION heat capacity, kJ/(kg K) concentration of reactant, kmol/m 3 Damk6hler number (V/q). [ro(T,~, x0)/CAo] activation energy, kJ/kmol area through which heat transfer occurs, m 2 mass flow rate of cooling medium, kg/s heat of reaction, kJ&mol dimensionless heat exchange parameter,
{°--I- ( L)I} qpcp k K q r R Ro T Tm U V x
reaction rate constant, I/s equilibrium constant, kl (Tm)/k2(Tm) volumetric flow rate, m3/s rate of reaction, kmol/(m 3 s) dimensionless reaction rate defined by eq. (3) gas constant, kJ/(kmol K) temperature, K mean reference temperature (To + H 7~-o)/(1 + H), K overall heat transfer coefficient, W(m2 K) volume of the reactor, m 3 dimensionless concentration, Ca/CAo
dimensionless temperature, T/T,~ dimensionless variable defined by eq. (5)
Greek letters fl ;, ~p ~b
dimensionless heat of reaction,
( -
Ah)
Cao
I-pcp(l + H)T,~] dimensionless activation energy, E~ l(Rg T,,) ratio of activation energies, E2/E~ density, kg/m 3 function defined by eqs (6) and (8)
Subscripts A B c o
component A component B cooling medium feed
REFERENCES
Leib, T. M. and Luss, D., 1981, Exact uniqueness and multiplicity criteria for an nth order reaction in a CSTR. Chem. Engng Sci. 36, 210--212. Tsotsis, T. T. and Schmitz, R. A., 1979, Exact uniqueness and multiplicity criteria for a positive-order Arrhenius reaction in a lumped system. Chem. Engng Sci. 34, 135-137.