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Multicomponent film-penetration theory with linearized kinetics—II. Particular cases and tests
Chemical Engineering
Science, 1974, Vol. 29, pp. 2325-2331. Pergamon Press.
Printed in Great Britain
MULTICOMPONENT FILM-PENETRATION THEORY WITH LINEARIZED KINETICS-II. PARTICULAR CASES AND TESTS S. T. LEE and G. B. DELANCEY? Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey07030, U.S.A. (Received 4 September 1973; accepted 14 June 1974) Abstract-The matrix results required for the evaluation of all interfacial flows at the film theory limit are presented for the multiple reaction system A + C + B with linear kinetics and the single reaction system A + 2B @ C with linearized kinetics. The yield of C is calculated in the former case and the enhancement factor for A is calculated in the latter case when B is nonvolatile and the reaction is irreversible. The single reaction, A + 2B + C, as well as A + B + C + D, is considered further for testing the linearized solution against the exact numerical calculations of the enhancement factor with the nonlinear kinetics. The results indicate that the linearization theory can be applied with a maximum error of 10 per cent and, in most cases, with a considerably smaller error over the complete range of gas-liquid contact times.
In a previous publication[l], the calculations required to determine the interfacial mass flows in a multicomponent gas-liquid system were presented from the viewpoint of the film-penetration theory. Isothermal conditions and an arbitrary number of liquid phase reactions were considered. Two essential features of these results will be demonstrated here: that the general results can be applied to an aribtrary absorption system via the prescribed matrix calculations alone and that nonlinear reaction rate expressions can be replaced with the proposed linear approximations without a significant loss in accuracy. The first objective will be met by summarizing the reduction of the general results necessary to calculate the interfacial flows in the film theory regime for a single reaction system, A + 2B # C, as compared to a system with two reactions, A + C @ B. The second objective will be met by comparing the numerical solutions for the enhancement factor for the single reaction with an elementary nonlinear rate expression to the results from the proposed linearized version. The comparisons will be made over the complete range of surface renewal rates for a wide range of the physical parameters. IS. T. Lee is currently in the Chemical Engineering Dept., University of Minnesota, Minneapolis, Minnesota 55455, U.S.A. Correspondence should be addressed to G. B. DeLancey.
SYSTEM 1
The single reaction components (N = 3),
(R = 1) involving
A+2B& and a nonlinear
kb
three
(1)
rate expression
* r = kfcAc:-kbcc
(2)
will be considered. When the reverse reaction is negligible, this reaction serves as an approximation to the multiple reaction sequence, A + B + C,
C + B +products,
if the second reaction is infinitely fast. The multiple reaction sequence has been studied by Brian and Beaverstock[2] and applied to the carbon dioxide-monoethanolamine and the chlorineferrous chloride systems. No thermodynamic coupling will be considered here and the diffusion coefficients of all species will be assumed equal. Equal diffusion coefficients is generally known to be a meaningful approximation for most gas absorption systems. The linearized form of the reaction rate expression in Eq. (2) can be written as (primed equation
2325
S. T. LEE and G. B. DELANCEY
2326
references refer to the general equations presented earlier [ 11)
based on the number of components and indicates that
(16’)
[jwR]=[M,]=$(kA+2ks-kc)=;Ak.
where kA = kf = kfcp,c:,
(17’)
ks = k#‘&toCso
(17’)
kc = - kb
(17’)
~+y’-YBw+YB) 13
4(1-ys3)(l
-rrSY
2 +2. (1-
[MI] has the single, positive
*
’
eigen-
,A& D
[T], in Eq. (38’) is the scalar The transformation, identity, and [A] in Eq. (39’) is identical to [MR]. The matrix [V] becomes
(l- YB4) 4 (l+ys)(l-Y83)
1 3 -3
(36’:
[Nl=[-;]
(18’)
6U + YE) 1
QB=o’
(36’:
and
Consequently, value,
and
and reaction:
YBW+ ye>
4(1-yd)(l-ys)* $+ ( -1+ys >2 1 - ys
(50’)
(18’)
A demonstration of the validity of this approximation will be given later. The matrix of stoichiometric coefficients is given by
The equivalence relationship applied to this example with -D Ak
(6’)
in Eq. (52’) can be
ks kz,
kc kA
(4) and the reaction
velocity matrix by [k] = [k,, b, kc1
(1’)
The coefficient matrix for the steady-state can then be evaluated as
ka
- ks -2ks ks
-kc -2kc kc
problem
1
(31’)
After calculation of the diagonalization matrices in Eq. (32’), the matrix [M] may be calculated:
The interfacial mass flows may now be obtained from Eq. (55’). If species B and C are nonvolatile, the equations required to determine their interfacial concentrations result from setting their interfacial flows to zero. These calculations will be considered for the irreversible case, kc = 0, when the bulk concentration of A is zero, CAL= 0. Under these conditions, the interfacial flux of species A is given by
+ &(cosh
The partitioning
shown in the previous equation is
%‘(po) CBO - W] k, -------(Cm kA + 2ke
- cm)
7
(55’)
Multicomponent film-penetration theory-II
2327
where
and
(6) flux of species B is given by
The interfacial
By dividing the interfacial flux for species A by its value for physical absorption, DGJL, the enhancement factor for species A may be obtained as E
A
,1+4-J 2
cso
2 ( c.40’ >
(7)
The condition that the interfacial flux of B vanish has been used which requires that cso be determined to satisfy
Such a reaction system has applications in the liquid phase oxidation of hydrocarbons by absorbed oxygen and the liquid phase conversion of absorbed ethylene and oxygen to acetaldehyde. In these cases the A and C components are volatile. Such a reaction system has been considered, for example, by Kuo[3] and Bridgewater [4]. It should be noted that the labeling of the B and C components, which might naturally be done in reverse order, is done so that the form indicated in Eq. (32’) will be obtained more directly. Otherwise, the transformation [P] will effectively interchange the roles of B and C, so that this labeling is convenient, not necessary. For the sake of convenience in general, all components that participate in a single reaction should be labeled among the first R components. The reaction kinetics will be assumed to take the linear forms i-1= k,jcn (10) and rz = k+c - klm
so that no linearization is The diffusion coefficients again be assumed to be stoichiometric coefficients
(11)
necessary. of all components will equal. The matrix of is given by
(6’)
[VI =
(8) Equation (8) must be solved by trial and error for the required value of CLW. The two limiting cases of physical absorption and instantaneous reaction may be noted. As the rate constant, k,, becomes small, the solution to Eq. (8) approaches CBO/C,U = q, and the enhancement factor in Eq. (7) approaches unity. As kf becomes infinite, the solution to Eq. (8) approaches cBo/cao= 0, and the enhancement factor approaches the result given by Hatta[23].
and the matrix of reaction
[kl =
In order to demonstrate the steps required for multiple reactions, the following consecutive reaction sequence involving two reactions (R = 2) and three comnonents (N = 3) will be treated:
by
[“d’_;2b k”, I
The coefficient matrix for the steady-state becomes [D]-‘[v][k]
=$
-OkIf i k If
(1’) problem
0 - bitb kw kzb - kz, 3
(31’)
After calculation of [P] and [Q] in Eq. (32’) and their inverses, [M] may be evaluated as rr
SYSTEM 2
velocities
11
1
2328
S.
Therefore,
T. LEE and G. B. DELANCEY
the matrix [I&] is given by
KI = kzrlkrf
(18)
Kz = k2b/k,,
(19) (20)
which has the eigenvalues At = k,flD,
(12)
AZ= (kzr + kzb)/D
(13)
and
The matrix [N] is given by [Nl=
11,ll
(36’)
so that
WI =
When K2 = 0, this result agrees with the one reported by Bridgewater[4] for the case of equal diffusivities and no gas phase resistance. When K2 = 0, and cAL= ccL = 0, the result also agrees with that reported by Ramachandran and Sharma[5]. The preceding examples illustrate the steps required to apply the general steady-state solution, or film theory result, reported earlier [ l] to particular cases. More complicated systems will require the same calculation procedures.
COMPARISON OF NUMERICALWITH LINEARIZEDRESULTS
W)
The nonlinear equations that film-penetration theory for the
The equivalence relationship applies to this example with r [Q’[vl
=
given in Eq. (52’)
-Dlklf -D kz, + kzb
-Dkzflh, k4 - (kzf + kzb)
I
D(1 -ka/kv) D kv - (kz, + kzb) kv + kzb
The interfacial flux expressions now follow from Eq. (55’). The results will yield the selectivity expression: J _cII= JA,,
the A +
2B & C follow directly from Eq. (20’)-(23’) with kb N = 3 and R = 1. The matrix of diffusion coefficients and the matrix of stoichiometric coefficients have been given above under System 1. The vector (r) is to be replaced with the scalar nonlinear expression given in Eq. (2). These equations have been solved by Lee[6] using an implicit finite difference technique[rl]. In each set of calculations the time and distance parameters were sequentially reduced until further reductions produced no significant changes in the results. The numerical technique was also verified by checking the numerical
results for the linear system A &B b
against the analytical solution [6]. The linearized model is given by the application of Eq. (69’) and (74’). The parameters required have been calculated above under System 1. The reaction rate (constant) vector in Eq. (74’) is given by the scalar expression
l-K2 1 - (K, + Kt) Kl - l-(K,+KS
represent reaction
&-Kz&+K,&+Kz& K,+Kz
(21)
where x’\/[(K,+ K&4 sinh
v\/[(K,
+
K2)p,l
(cod
d\/[(K + Kz)PJ cio- a) i=A,B,C
6i = -&$-JCOSh
IT = (c*o +
CEO +
ccd - (Cat +
~(/L.,)c.m-c.4L)
(16)
’
cm+ CCL)
sin;($fti,)(COsh d/(P~I)(c.to-
(17) CAL)
2329
Multicomponent film-penetration theory-II
Both the linearized model and the nonlinear model can be expressed in terms of the following dimensionless groups:
kjc4oc28oL lJA=
Dc*olL ’
(22)
ps =
2k,c,ocix,L Dcso/L ’
(23)
pc=
k~cmc&L/(Dcco/L)
(24)
po = kbcco/(k,cno c iid
(25)
K = kblkf
(26)
c kL= CAL 1%
(27)
c L = CBL /CAi
(28)
c&.= CcLlCai
(2%
The dimensionless groups denoted by pi measure the forward reaction rate with respect to the physical diffusion rate of species i. The reverse reaction rate is measured relative to the forward rate by pa. If reaction equilibrium exists at the interface, the value of p0 is unity. The enhancement factor for species A has been chosen as the basis of comparison of the approximate with thi: numerical solutions. The enhancement factor is the interfacial flux relative to that which would persist in the absence of chemical reaction given by Eq. (72’). Figure 1 summarizes the results for a range of values of the dimensionless groups given in Eq.
I
K _po--
%
[‘BL -
C'CL -
00
00
0.0
20
12 0
0.3 o-7
3-6 70 100
03 05 07
2.0 2-5 30
0 OS 008
pn zo.4. 100 0
(22)-(29). Although the variations in the equilibrium constant (K) are attached to the curves, the remaining parameters are not held constant but are changed to reflect their response to the reaction rate in a qualitative fashion. The results indicate that the least square error approach to linearizing the reaction rate gives excellent results for intermediate and low surface renewal rates defined by 1 D, 7 > 0.10. In this region the percentage error is iz J
less than 10 per cent. In fact, the error is less than 1 1 DA per cent for r ~>0*25. J For higher renewal rates, the approximation employing a constant reaction rate, namely the nonlinear one evaluated at the interface, reproduces the numerical results to within 10 per cent. The maximum error occurs as the renewal rate becomes infinite and decreases thereafter at least until _l_ 7DA = 0.10. The error is less than 5 per L J 1 Dn - < 0.10. Similar observacent when 0.02 < L J s
tions can be made concerning the comparisons presented in Figs. 2 and 3. A summary of the errors involved in the approximations is given in Fig. 4. Results have also been computed for the reaction A + B zs C + D [6]. The dimensionless groups for this system are similar to those for the preceding reaction. Figures 5 and 6 summarize the results for
p,= I-cl, /l,= 8.0
n = Numerical
SOIUtio”,
cl i CO”s+o”+ reOC+lO” rote model, 0 = lineor reaction rote model
EA
100
I.0
0 01
0 10
IO.0
Fig. 1. Comparison of approximate with numerical enhancement factors for A +2B+C at various equilibrium constants.
Fig. 2. Comparison of approximate with numerical enhancement factors for A +2B%C at various bulk concentrations. CL = 0.30; C& = 0.08; K = 0.3; p0 = 3.6; pLA= 0.4; pB = 1.0; /.L== 10.0; n = numerical solution; 0 = constant reaction rate model; q = linear reaction rate model.
2330
S. T. LEE and G. B. DELANCEV
100-O
E
A
100 --A-u__-_&3 -*--
--4-3~4-
I 0 IO
0. I
+-
I I
.o
100
P
T-5
Fig. 5. Comparison of approximate with numerical Fig. 3. Comparison of approximate with numerical enhancement factors for A +BkC +D at various enhancement factors for A +2B*C at various reaction equilibrium constants. velocities. c& = 0.0; CL = 10.0; c& = 12.0; K = 0.0; p.=O.O; pB = 1.0;pc =8.0; A = numerical solution; 0 = constant reaction rate model; q = linear reaction rate model.
1
‘OooL /--
90.0 /A
4 60-O
/
18 700 ti
J
t'
.,,k /
0 0
OIO203040-506070.60.9I-011
O-I 02
03
C-4 05
0.6 O-7
0.6 O-9 I.0 I.1 I.2 13
12131
Fig. 4. Summary of errors associated with the approximations for A +2B % C. 0 = K = 0,curve from Fig. 1; A = c L = 40, curve from Fig. 2; V = pA = 1.0, curve from Fig. 3; q = see Figs. 4 and 5 in Ref. [30]. Solid points represent calculations with constant reaction rate model; open points represent calculations with linear reaction rate model.
Fig. 6. Summary of errors associated with the approximations for A + B % C + D.O= K = 0.0, curvefromFig. 5; Cl = K = 0.3, curve from Fig. 5; A = K = 0.7, curve from Fig. 5; V = K = 1.0, curve from Fig. 5. Solid points represent calculations with the constant reaction rate model; open points represent calculations with the linear reaction rate model.
Multicomponent film-penetration theory-II
this case and indicate that observations similar to those made above can be made. Additional comparisons are available elsewhere [6]. An important observation is that the error associated with the approximation based upon the least square error does not depend appreciably upon the values of the governing dimensionless groups. The error for this approximation depends primarily upon the value of $dD/s
in all cases studied[6].
form based on the approach of least square error may be employed for calculations of the enhancement factor. In all cases the error may be expected to be not greater than approximately 10 per cent and in many cases to be considerably less. Acknowledgements-This work was partially supported with funds donated to the Chemistry and Chemical Engineering Department at Stevens Institute of Technology by the American Cyanamid Company.
CONCLUSIONS
NOTATION
error in the enhancement factor associated with the approximations based upon the least square error depends primarily upon the value of The
i
f for each of the reaction systems studied: d A+B$C; A+B+C+D. The error decreases steadily with this parameter and is nearly 10 per
cent at t
J cent beyond 1 D ?I J s.
s equal to 0.10 and is less than 1 per a value of approximately
0.30 for
The notation earlier[l] with variables:
used here follows that used the addition of the following
En enhancement factor for species A due to chemical reaction K,, Kt scaled reaction velocity constants q concentration ratio, cBLIcAL Greek
symbols Si scaled
concentration
differences
(i =
A, B, C, ‘U
For values of i
f less than 0.10, the approxiJ mation based on a constant reaction rate, namely the nonlinear one evaluated at the interface, yields an enhancement factor within approximately 10 per cent of the exact value calculated numerically. The percentage error also steadily improves with this approximation
2331
up to $/D/s
= 0.10.
The results of this study suggest that the film-penetration theory for reaction systems with nonlinear kinetics may be approximated for engineering purposes by combining the results from two linearized forms of the true rate expressions. First, a constant reaction rate equal to the nonlinear rate evaluated at the interface can be employed to calculate the enhancement factor for high to intermediate surface renewal rates. For, intermediate to low surface renewal rates, the linearized
pi p
dimensionless quantities that reflect reaction rates relative to diffusion rates ratio of reverse to forward reaction rate
Primed equation numbers equations in ref. [ 11.
refer to the associated
REFERENCES
VI DeLancey G. B., Chem. Engng Sci. (Part 1) 1974 29 2315. 121Brian P. L. T. and Beaverstock M. C., Chem. Engng Sci. 1965 20 47. 131Kuo C. H., A.1.Ch.E. JI. 1972 18 644. [41 Bridgewater J., Chem. Engng Sci. 1%7 22 185. VI Ramachandran P. A. and Sharma M. M., Chem. Engng Sci. 1970 25 1743. [61 Lee S. T., M. S. Thesis, Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, N.J. 1973. [71 Carnahan B., Luther H. A. and Wilkes J. O., Applied Numerical Methods. John Wiley, New York 1969.
Science, 1974, Vol. 29, pp. 2325-2331. Pergamon Press.
Printed in Great Britain
MULTICOMPONENT FILM-PENETRATION THEORY WITH LINEARIZED KINETICS-II. PARTICULAR CASES AND TESTS S. T. LEE and G. B. DELANCEY? Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey07030, U.S.A. (Received 4 September 1973; accepted 14 June 1974) Abstract-The matrix results required for the evaluation of all interfacial flows at the film theory limit are presented for the multiple reaction system A + C + B with linear kinetics and the single reaction system A + 2B @ C with linearized kinetics. The yield of C is calculated in the former case and the enhancement factor for A is calculated in the latter case when B is nonvolatile and the reaction is irreversible. The single reaction, A + 2B + C, as well as A + B + C + D, is considered further for testing the linearized solution against the exact numerical calculations of the enhancement factor with the nonlinear kinetics. The results indicate that the linearization theory can be applied with a maximum error of 10 per cent and, in most cases, with a considerably smaller error over the complete range of gas-liquid contact times.
In a previous publication[l], the calculations required to determine the interfacial mass flows in a multicomponent gas-liquid system were presented from the viewpoint of the film-penetration theory. Isothermal conditions and an arbitrary number of liquid phase reactions were considered. Two essential features of these results will be demonstrated here: that the general results can be applied to an aribtrary absorption system via the prescribed matrix calculations alone and that nonlinear reaction rate expressions can be replaced with the proposed linear approximations without a significant loss in accuracy. The first objective will be met by summarizing the reduction of the general results necessary to calculate the interfacial flows in the film theory regime for a single reaction system, A + 2B # C, as compared to a system with two reactions, A + C @ B. The second objective will be met by comparing the numerical solutions for the enhancement factor for the single reaction with an elementary nonlinear rate expression to the results from the proposed linearized version. The comparisons will be made over the complete range of surface renewal rates for a wide range of the physical parameters. IS. T. Lee is currently in the Chemical Engineering Dept., University of Minnesota, Minneapolis, Minnesota 55455, U.S.A. Correspondence should be addressed to G. B. DeLancey.
SYSTEM 1
The single reaction components (N = 3),
(R = 1) involving
A+2B& and a nonlinear
kb
three
(1)
rate expression
* r = kfcAc:-kbcc
(2)
will be considered. When the reverse reaction is negligible, this reaction serves as an approximation to the multiple reaction sequence, A + B + C,
C + B +products,
if the second reaction is infinitely fast. The multiple reaction sequence has been studied by Brian and Beaverstock[2] and applied to the carbon dioxide-monoethanolamine and the chlorineferrous chloride systems. No thermodynamic coupling will be considered here and the diffusion coefficients of all species will be assumed equal. Equal diffusion coefficients is generally known to be a meaningful approximation for most gas absorption systems. The linearized form of the reaction rate expression in Eq. (2) can be written as (primed equation
2325
S. T. LEE and G. B. DELANCEY
2326
references refer to the general equations presented earlier [ 11)
based on the number of components and indicates that
(16’)
[jwR]=[M,]=$(kA+2ks-kc)=;Ak.
where kA = kf = kfcp,c:,
(17’)
ks = k#‘&toCso
(17’)
kc = - kb
(17’)
~+y’-YBw+YB) 13
4(1-ys3)(l
-rrSY
2 +2. (1-
[MI] has the single, positive
*
’
eigen-
,A& D
[T], in Eq. (38’) is the scalar The transformation, identity, and [A] in Eq. (39’) is identical to [MR]. The matrix [V] becomes
(l- YB4) 4 (l+ys)(l-Y83)
1 3 -3
(36’:
[Nl=[-;]
(18’)
6U + YE) 1
QB=o’
(36’:
and
Consequently, value,
and
and reaction:
YBW+ ye>
4(1-yd)(l-ys)* $+ ( -1+ys >2 1 - ys
(50’)
(18’)
A demonstration of the validity of this approximation will be given later. The matrix of stoichiometric coefficients is given by
The equivalence relationship applied to this example with -D Ak
(6’)
in Eq. (52’) can be
ks kz,
kc kA
(4) and the reaction
velocity matrix by [k] = [k,, b, kc1
(1’)
The coefficient matrix for the steady-state can then be evaluated as
ka
- ks -2ks ks
-kc -2kc kc
problem
1
(31’)
After calculation of the diagonalization matrices in Eq. (32’), the matrix [M] may be calculated:
The interfacial mass flows may now be obtained from Eq. (55’). If species B and C are nonvolatile, the equations required to determine their interfacial concentrations result from setting their interfacial flows to zero. These calculations will be considered for the irreversible case, kc = 0, when the bulk concentration of A is zero, CAL= 0. Under these conditions, the interfacial flux of species A is given by
+ &(cosh
The partitioning
shown in the previous equation is
%‘(po) CBO - W] k, -------(Cm kA + 2ke
- cm)
7
(55’)
Multicomponent film-penetration theory-II
2327
where
and
(6) flux of species B is given by
The interfacial
By dividing the interfacial flux for species A by its value for physical absorption, DGJL, the enhancement factor for species A may be obtained as E
A
,1+4-J 2
cso
2 ( c.40’ >
(7)
The condition that the interfacial flux of B vanish has been used which requires that cso be determined to satisfy
Such a reaction system has applications in the liquid phase oxidation of hydrocarbons by absorbed oxygen and the liquid phase conversion of absorbed ethylene and oxygen to acetaldehyde. In these cases the A and C components are volatile. Such a reaction system has been considered, for example, by Kuo[3] and Bridgewater [4]. It should be noted that the labeling of the B and C components, which might naturally be done in reverse order, is done so that the form indicated in Eq. (32’) will be obtained more directly. Otherwise, the transformation [P] will effectively interchange the roles of B and C, so that this labeling is convenient, not necessary. For the sake of convenience in general, all components that participate in a single reaction should be labeled among the first R components. The reaction kinetics will be assumed to take the linear forms i-1= k,jcn (10) and rz = k+c - klm
so that no linearization is The diffusion coefficients again be assumed to be stoichiometric coefficients
(11)
necessary. of all components will equal. The matrix of is given by
(6’)
[VI =
(8) Equation (8) must be solved by trial and error for the required value of CLW. The two limiting cases of physical absorption and instantaneous reaction may be noted. As the rate constant, k,, becomes small, the solution to Eq. (8) approaches CBO/C,U = q, and the enhancement factor in Eq. (7) approaches unity. As kf becomes infinite, the solution to Eq. (8) approaches cBo/cao= 0, and the enhancement factor approaches the result given by Hatta[23].
and the matrix of reaction
[kl =
In order to demonstrate the steps required for multiple reactions, the following consecutive reaction sequence involving two reactions (R = 2) and three comnonents (N = 3) will be treated:
by
[“d’_;2b k”, I
The coefficient matrix for the steady-state becomes [D]-‘[v][k]
=$
-OkIf i k If
(1’) problem
0 - bitb kw kzb - kz, 3
(31’)
After calculation of [P] and [Q] in Eq. (32’) and their inverses, [M] may be evaluated as rr
SYSTEM 2
velocities
11
1
2328
S.
Therefore,
T. LEE and G. B. DELANCEY
the matrix [I&] is given by
KI = kzrlkrf
(18)
Kz = k2b/k,,
(19) (20)
which has the eigenvalues At = k,flD,
(12)
AZ= (kzr + kzb)/D
(13)
and
The matrix [N] is given by [Nl=
11,ll
(36’)
so that
WI =
When K2 = 0, this result agrees with the one reported by Bridgewater[4] for the case of equal diffusivities and no gas phase resistance. When K2 = 0, and cAL= ccL = 0, the result also agrees with that reported by Ramachandran and Sharma[5]. The preceding examples illustrate the steps required to apply the general steady-state solution, or film theory result, reported earlier [ l] to particular cases. More complicated systems will require the same calculation procedures.
COMPARISON OF NUMERICALWITH LINEARIZEDRESULTS
W)
The nonlinear equations that film-penetration theory for the
The equivalence relationship applies to this example with r [Q’[vl
=
given in Eq. (52’)
-Dlklf -D kz, + kzb
-Dkzflh, k4 - (kzf + kzb)
I
D(1 -ka/kv) D kv - (kz, + kzb) kv + kzb
The interfacial flux expressions now follow from Eq. (55’). The results will yield the selectivity expression: J _cII= JA,,
the A +
2B & C follow directly from Eq. (20’)-(23’) with kb N = 3 and R = 1. The matrix of diffusion coefficients and the matrix of stoichiometric coefficients have been given above under System 1. The vector (r) is to be replaced with the scalar nonlinear expression given in Eq. (2). These equations have been solved by Lee[6] using an implicit finite difference technique[rl]. In each set of calculations the time and distance parameters were sequentially reduced until further reductions produced no significant changes in the results. The numerical technique was also verified by checking the numerical
results for the linear system A &B b
against the analytical solution [6]. The linearized model is given by the application of Eq. (69’) and (74’). The parameters required have been calculated above under System 1. The reaction rate (constant) vector in Eq. (74’) is given by the scalar expression
l-K2 1 - (K, + Kt) Kl - l-(K,+KS
represent reaction
&-Kz&+K,&+Kz& K,+Kz
(21)
where x’\/[(K,+ K&4 sinh
v\/[(K,
+
K2)p,l
(cod
d\/[(K + Kz)PJ cio- a) i=A,B,C
6i = -&$-JCOSh
IT = (c*o +
CEO +
ccd - (Cat +
~(/L.,)c.m-c.4L)
(16)
’
cm+ CCL)
sin;($fti,)(COsh d/(P~I)(c.to-
(17) CAL)
2329
Multicomponent film-penetration theory-II
Both the linearized model and the nonlinear model can be expressed in terms of the following dimensionless groups:
kjc4oc28oL lJA=
Dc*olL ’
(22)
ps =
2k,c,ocix,L Dcso/L ’
(23)
pc=
k~cmc&L/(Dcco/L)
(24)
po = kbcco/(k,cno c iid
(25)
K = kblkf
(26)
c kL= CAL 1%
(27)
c L = CBL /CAi
(28)
c&.= CcLlCai
(2%
The dimensionless groups denoted by pi measure the forward reaction rate with respect to the physical diffusion rate of species i. The reverse reaction rate is measured relative to the forward rate by pa. If reaction equilibrium exists at the interface, the value of p0 is unity. The enhancement factor for species A has been chosen as the basis of comparison of the approximate with thi: numerical solutions. The enhancement factor is the interfacial flux relative to that which would persist in the absence of chemical reaction given by Eq. (72’). Figure 1 summarizes the results for a range of values of the dimensionless groups given in Eq.
I
K _po--
%
[‘BL -
C'CL -
00
00
0.0
20
12 0
0.3 o-7
3-6 70 100
03 05 07
2.0 2-5 30
0 OS 008
pn zo.4. 100 0
(22)-(29). Although the variations in the equilibrium constant (K) are attached to the curves, the remaining parameters are not held constant but are changed to reflect their response to the reaction rate in a qualitative fashion. The results indicate that the least square error approach to linearizing the reaction rate gives excellent results for intermediate and low surface renewal rates defined by 1 D, 7 > 0.10. In this region the percentage error is iz J
less than 10 per cent. In fact, the error is less than 1 1 DA per cent for r ~>0*25. J For higher renewal rates, the approximation employing a constant reaction rate, namely the nonlinear one evaluated at the interface, reproduces the numerical results to within 10 per cent. The maximum error occurs as the renewal rate becomes infinite and decreases thereafter at least until _l_ 7DA = 0.10. The error is less than 5 per L J 1 Dn - < 0.10. Similar observacent when 0.02 < L J s
tions can be made concerning the comparisons presented in Figs. 2 and 3. A summary of the errors involved in the approximations is given in Fig. 4. Results have also been computed for the reaction A + B zs C + D [6]. The dimensionless groups for this system are similar to those for the preceding reaction. Figures 5 and 6 summarize the results for
p,= I-cl, /l,= 8.0
n = Numerical
SOIUtio”,
cl i CO”s+o”+ reOC+lO” rote model, 0 = lineor reaction rote model
EA
100
I.0
0 01
0 10
IO.0
Fig. 1. Comparison of approximate with numerical enhancement factors for A +2B+C at various equilibrium constants.
Fig. 2. Comparison of approximate with numerical enhancement factors for A +2B%C at various bulk concentrations. CL = 0.30; C& = 0.08; K = 0.3; p0 = 3.6; pLA= 0.4; pB = 1.0; /.L== 10.0; n = numerical solution; 0 = constant reaction rate model; q = linear reaction rate model.
2330
S. T. LEE and G. B. DELANCEV
100-O
E
A
100 --A-u__-_&3 -*--
--4-3~4-
I 0 IO
0. I
+-
I I
.o
100
P
T-5
Fig. 5. Comparison of approximate with numerical Fig. 3. Comparison of approximate with numerical enhancement factors for A +BkC +D at various enhancement factors for A +2B*C at various reaction equilibrium constants. velocities. c& = 0.0; CL = 10.0; c& = 12.0; K = 0.0; p.=O.O; pB = 1.0;pc =8.0; A = numerical solution; 0 = constant reaction rate model; q = linear reaction rate model.
1
‘OooL /--
90.0 /A
4 60-O
/
18 700 ti
J
t'
.,,k /
0 0
OIO203040-506070.60.9I-011
O-I 02
03
C-4 05
0.6 O-7
0.6 O-9 I.0 I.1 I.2 13
12131
Fig. 4. Summary of errors associated with the approximations for A +2B % C. 0 = K = 0,curve from Fig. 1; A = c L = 40, curve from Fig. 2; V = pA = 1.0, curve from Fig. 3; q = see Figs. 4 and 5 in Ref. [30]. Solid points represent calculations with constant reaction rate model; open points represent calculations with linear reaction rate model.
Fig. 6. Summary of errors associated with the approximations for A + B % C + D.O= K = 0.0, curvefromFig. 5; Cl = K = 0.3, curve from Fig. 5; A = K = 0.7, curve from Fig. 5; V = K = 1.0, curve from Fig. 5. Solid points represent calculations with the constant reaction rate model; open points represent calculations with the linear reaction rate model.
Multicomponent film-penetration theory-II
this case and indicate that observations similar to those made above can be made. Additional comparisons are available elsewhere [6]. An important observation is that the error associated with the approximation based upon the least square error does not depend appreciably upon the values of the governing dimensionless groups. The error for this approximation depends primarily upon the value of $dD/s
in all cases studied[6].
form based on the approach of least square error may be employed for calculations of the enhancement factor. In all cases the error may be expected to be not greater than approximately 10 per cent and in many cases to be considerably less. Acknowledgements-This work was partially supported with funds donated to the Chemistry and Chemical Engineering Department at Stevens Institute of Technology by the American Cyanamid Company.
CONCLUSIONS
NOTATION
error in the enhancement factor associated with the approximations based upon the least square error depends primarily upon the value of The
i
f for each of the reaction systems studied: d A+B$C; A+B+C+D. The error decreases steadily with this parameter and is nearly 10 per
cent at t
J cent beyond 1 D ?I J s.
s equal to 0.10 and is less than 1 per a value of approximately
0.30 for
The notation earlier[l] with variables:
used here follows that used the addition of the following
En enhancement factor for species A due to chemical reaction K,, Kt scaled reaction velocity constants q concentration ratio, cBLIcAL Greek
symbols Si scaled
concentration
differences
(i =
A, B, C, ‘U
For values of i
f less than 0.10, the approxiJ mation based on a constant reaction rate, namely the nonlinear one evaluated at the interface, yields an enhancement factor within approximately 10 per cent of the exact value calculated numerically. The percentage error also steadily improves with this approximation
2331
up to $/D/s
= 0.10.
The results of this study suggest that the film-penetration theory for reaction systems with nonlinear kinetics may be approximated for engineering purposes by combining the results from two linearized forms of the true rate expressions. First, a constant reaction rate equal to the nonlinear rate evaluated at the interface can be employed to calculate the enhancement factor for high to intermediate surface renewal rates. For, intermediate to low surface renewal rates, the linearized
pi p
dimensionless quantities that reflect reaction rates relative to diffusion rates ratio of reverse to forward reaction rate
Primed equation numbers equations in ref. [ 11.
refer to the associated
REFERENCES
VI DeLancey G. B., Chem. Engng Sci. (Part 1) 1974 29 2315. 121Brian P. L. T. and Beaverstock M. C., Chem. Engng Sci. 1965 20 47. 131Kuo C. H., A.1.Ch.E. JI. 1972 18 644. [41 Bridgewater J., Chem. Engng Sci. 1%7 22 185. VI Ramachandran P. A. and Sharma M. M., Chem. Engng Sci. 1970 25 1743. [61 Lee S. T., M. S. Thesis, Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, N.J. 1973. [71 Carnahan B., Luther H. A. and Wilkes J. O., Applied Numerical Methods. John Wiley, New York 1969.