- Email: [email protected]
The effect of dispersion on chemical reactor optimum residence times using perturbation expansion techniques
Chemical Engineering Science, 1968, Vol. 23. pp. 833-840.
Pergamon Press.
Printed in Great Britain.
The effect of dispersion on chemical reactor optimum residence times using perturbation expansion techniques K. L. KIPP, Jr?. and S. H. DAVIS, Jr. Department of Chemical Engineering, Rice University, Houston, Texas (First
receioed
1Augrtsf 1967; nccepred 29 January 1968)
Abstract-This work is a mathematical study of the effect of dispersion on the optimum residence times of chemical reactors. Attention is confined to the range of very small and very large dispersions so that perturbation expansions can be used to obtain asymptotic solutions to the equations for the axial dispersed plug flow tubular reactor. Only the consecutive reaction system A + B -+ C, where B is a desired intermediate product is considered. The optimum residence time is then defined as that which maximizes the yield of B in each isothermal reactor model. It is shown that the expansions for optimum residence time are excellent approximations over a wide range of operating conditions for this reactor model. An interesting maximum effect in the optimum residence time for the axial dispersed tubular reactor is discussed: INTRODUCTION
of dispersion on the optimum residence time for the axial dispersed plug flow tubular reactor is investigated in this work. The reactor residence time is chosen to maximize the yield of the desired intermediate product B in a simple consecutive reaction, A + B --* C. The mathematical technique of perturbation expansions is used to obtain solutions to the limiting cases of small and large dispersion in such reactors. In chemical reaction engineering or chemical reactor analysis perhaps the two most common mathematical models are the ideal tubular flow reactor and the continuous flow stirred tank reactor. Actual reactors are usually intermediate in their behavior between these extreme cases with partial mixing and non-uniform flow causing deviation from either simple system. Many types of models can be used to represent the effects of non-ideal flow in tubular reactors. In the axial dispersed tubular reactor non-ideal flow is characterized by a single dispersion parameter indicating the extent of fluid mixing. The reacting fluid is assumed to be a single isothermal phase with reactions taking place homogeneously. Thus, the model is applicable to packed THE
EFFECT
iPresent
bed catalytic reactors with negligible mass transfer resistance between phases. Much theoretical work has been done on dispersion in chemical reactors and the effect of dispersion on chemical conversion, but there is little literature that relates directly to the objectives or to the methods of this paper. Freeman and Houghton[6] have applied the perturbation expansion techniques of Erdelyi[S] and Carrier to the singular perturbation, concentration profile problem. However, in the formal solution procedure they introduce first order terms in a zeroth order boundary condition (thus violating the order relation) in order to bring their results into closer agreement with numerical solutions. Parish [ 121 has also applied perturbation expansions to the conversion problem with small dispersion for multiple reactions and general kinetics. Only a few articles have appeared dealing with dispersion and selectivity in chemical reactors[3,8,10,13-151. Kramers and Westerterp [lo] have numerically evaluated the axial dispersed tube reactor for the consecutive reactions A + B + C. For the same reaction system Tichacek [ 131 presents results on the decrease in intermediate yield due to dispersion in a tubular
Address: Esso Production Research Co., Houston, Texas.
$33
K. L. KIPP, JR. and S. H. DAVIS,
JR.
solution domain. One approach that can be used to treat singular perturbation problems involves the use of a coordinate stretching transformation at the end of the domain to form a ‘defect boundary layer’. In this layer a correction term added to the solution becomes sufficient in magnitude to enable the boundary condition at that end to be satisfied. Away from the boundary layer the defect boundary layer term becomes vanishingly small. The ‘defect boundary layer’ technique for ordinary differential equations as given by Goldwyn [7] will be used in this work.
reactor. He considers several sets of reaction kinetics and uses a perturbation expansion solution in one case; but to simplify the equations, he makes the reactor length semi-infinite. Thus, his results are slightly different than for the axial dispersed reactor with a finite reaction zone. In review, one can say that only Parish handled perturbation expansion solutions to dispersion problems in finite chemical reactors in a rigorously correct manner. No one to date has presented work on optimum residence times for reactors with dispersion effects using perturbation expansion techniques.
THE
AXIAL
THEORY
DISPERSED
TUBULAR
The perturbation technique is an empirical method for obtaining approximate analytical solutions to otherwise insolvable problems that are only slightly different from problems that can be solved. Basically, we desire a solution to a differential equation containing one small parameter where all other terms are assumed to be of order one. If the equation can be solved exactly when the small parameter goes to zero, then we hope that a good approximation to the exact solution of the entire differential equation can be obtained in the form of a perturbation expansion of the exact solution to the zeroth order equation. An assumed solution form, C#I = c#J,,+E& + ++..., is inserted into the differential equation and boundary conditions. The resulting sets of equations in orders of the small parameter E are solved sequentially for & starting with the zeroth order system. The various sets of equations must be satisfied identically in order for the complete solution to satisfy the differential equation and boundary conditions uniformly in the small parameter. The result is assumed to be accurate to the order of the first truncated term. This is the basic Poincare expansion procedure. One common difficulty that occurs is the highest order derivative disappears as the small parameter goes to zero. This gives a singular perturbation problem. There is difficulty in making the solution satisfy all the boundary conditions because no single asymptotic expansion is uniformly valid over the entire
PLUG
FLOW
REACTOR
Small dispersion region
This model characterizes non-ideal flow patterns in a tubular reactor by using the similarity between mixing and diffusion. In the differential component material balance, the molecular diffusivity is replaced by a dispersion coefficient that may account for several sources of mixing. This axial dispersion model is one-dimensional and is obtained from the general component material balance [ I] $+V*
VC*=DV2Ci+rC,+Si
(where D is diffusivity matrix, S is source term, V is fluid velocity) through the following assumptions: (For an excellent discussion see Bischoff [2]). There are no sources of material within the system; the reacting mixture is treated as a single phase; the mixture is of constant density; there is bulk flow in the axial direction only, and radial symmetry; the dispersion coefficients are constant; the time average velocity of the fluid is constant over the tubular cross-section and independent of axial position (plug flow); there is no variation in properties in the radial direction; the system is at steady state. The differential equation for the ith component mass balance is then
834
_~d2”‘+I,KrC*=() dz2 dz
(1)
The effect of dispersion
where D is dispersion coefficient and rcI is rate of formation of ith component/unit volume. Looking at the terms of Eq. (1) we see that the axial dispersion model describes dispersion and mixing by a diffusional mechanism, (i.e., driven by concentration gradients), superimposed on the plug flow velocity profile of the ideal tubular reactor model. The single parameter D characterizes the net effect of all dispersion phenomena which may be taking place. The Danckwerts’ boundary conditions [4] for a tubular reactor with dispersion are assumed to be applicable. dci atz = 0 c~-D-= cio CW dz Dd”i,O dz
atz=L
(2b)
-
The reactor is assumed to be isothermal so that no energy balance is required. Only reaction component A enters the reactor. Furthermore, the dispersion coefficient is assumed to be the same for both species and independent of composition. In this paper it will be assumed that the two consecutive reactions are irreversible and have first order kinetic rate expressions. AAB
r,=klcA
SO rA = - ktCA
B I; C
r2 = k2cB
rB = k,C, - k2ce.
In terms of the dimensionless
y+
variables
Lk,
g-z-
V
(y=-.
-7
k2 k,
B.C.
at
x=0
(34
= 0
&2
4A--y$$
= 1 0
at
x=7
y-@‘A dr.
=0
B.C.
at
x= 0
4B-73
=0 0
at
x=7
d’#‘B
-yz
=O. T
Note that we have reduced the number of parameters to three y, LY,and r. Thus, a perturbation expansion for Tag. will be a function of y and (Y only. The above linear system can be solved analytically. This would not be true if the reaction rates were non-linear. It will be seen that perturbation expansion solutions are simpler to obtain than exact analytical solutions in those cases where both are possible. The above set of equations forms a singular perturbation problem since as y + 0 the highest order derivatives are lost and the equations become first order. This means that it will be impossible to satisfy the boundary conditions in all orders of y with a single ordinary perturbation expansion. Fortunately we can solve the two differential equations sequentially rather than simultaneously due to the irreversibility of the reactions. Starting with the eqUatiOU for 4A, an eXpat’KiOn of the form 4A = +A0+ Y4AI+ y24As+ . . . is inserted into Eq. (3a). The solution of the resulting set of equations (determined by the condition that each coefficient of a power of y must vanish) may be solved sequentially. The first two terms in the expansion are $A0 = e-” (The solution for an ideal tubular reactor) (4) +A, = xe-” - e-” (The first order perturbation terms satisfying the inlet boundary conditions).
The system equations are
d2b db -+-&++A
(3b)
As a result of the singularity of the differential equation at y equal to zero the first order perturbation term does not satisfy the reactor outlet boundary condition. To correct this an additional term is introduced into the solution which is important near x = 7 but vanishes far away from that boundary. 835
K. L. KIPP, JR. and S. H. DAVIS,
A revised solution of the form $A = 4AAo+ Y&4*+ $JIA(rl) +. * * is assumed (p 3 0 since $A is bounded as Y ---, 0) where c#B~,,, +A1,etc. are the functions previously found. We want JiA($ + 0 as 71+ ~0.71is the boundary layer coordinate. So let r) = (r-x)Yy (this is a coordinate stretching transformation). As x + 7, r) + 0; as x + - 03, r) + ~0. Thus, 7 measures the distance away from the exit of the reactor. Inserting the assumed form for $ into Eq. (3a) we obtain: -Ye -Z - Yz(- 3e-” + xe-” )
- . . . -y2v+p+ + (-e-I)
+y(2e-“-xe-“)
+e-“+y(xe-z-ee-z)
+. . . -y”+j3
+. . .+yaJIA(~)
l-
d*JiA dr1’
The total perturbation then c#I~(~)= e-“+y
[
(x-
expansion l)e-Z-te-Te-
to order y is
1 (8)
T-X 7 +O(y’).
The singular perturbation term, $*, does not satisfy the differential equation to order y but the region in which this condition holds becomes very small as y + 0. Thus, Eq. (8) satisfies all the system equations to order y except in a boundary layer region near the reactor exit. Exactly the same procedures can be followed with Eq. (3b) to find a perturbation expansion for &. The differential equation is inhomogeneous. The solution gives & at the exit of the reactor:
drCIA drl = 0.
&3(~)= $-f(epT-e-a’)
(5)
+y [
a-(r-
I)e-T
--f+m--l)e-m]+Y--&
Note that cj~~~+Yc$~~ satisfies the differential equation identically as expected. There was difficulty with our previous solution because there were no terms in d2$,/dxz to “balance” the terms in d$,/dx and 4,., so the boundary condition at x = r could be satisfied to order y. Therefore, the defect boundary layer term JIA(r)) involving the second derivative should be of the same order in y as the lower derivative term. Then the second derivative d2~A/cIrz will not disappear as Y goes to zero, especially near the reactor exit. For this reason we pick Y so that the highest derivative terms are retained and are of the same order as lower terms. Thus, we must set 2v+p+ 1 = v+/3 so v=-l,thenq= (r-x)/Y. The equations determining +Abecome
X (e-‘-ae-“)(l)+O(y’).
(9)
Now to obtain a solution for optimum length or residence time to maximize the production of B, perturbation expansion techniques can be used to solve the nonlinear equation. Take the derivative of &(r) Eq. (9) with respect to r and set it equal to zero. This locates the T for maximizing &(T).
d&i -= dr
aeFm) + y[&eeT
-&(-e-T+ + _e (-1) a-1
(6)
_r_-e-T+-_e-” CJ? a-l
cl!% a-1
1 b> = 0
To get a perturbation expansion approximation let 7=~~-t-y7~+. . ., insert into Eq. (lo), and equate the terms in like orders of Y. We then find lncu rapt.
e=Oat
JR.
q=O.
=,-1+y
(
‘a(a-l)-l)+O(r~).(ll) a-1
Thus, we have obtained a relation for the effect To satisfy Eq. (7) to order y we must pick p = 1. of small dispersion on the optimum residence time for the maximum yield of the desired The differential equation and Eq. (6) are satisfied by ce-‘l. Equation (7) determines c = edT’. component in the consecutive reaction system: 836
The effect of dispersion
A + B + C. Figure 1 is a graph of the first order perturbation term TV,and the ratio of or to 7. a\ a function of the reaction rate constant ratio cy.
A similar procedure leads to the desired solution for $Q at the reactor outlet
MT)= (l+T);l+m)
Large dispersion region
1 Topt.
=
ro+yrt+o
(13)
1
(14)
-p
produces
=
x a2+6ad(a)
+ lOa+6~(a)
2a2+aV(cI)
+&(a)
+ 1
+o $ 0
. (15)
Figure 2 shows the first order term in Eq. (15) and the ratio of this term to T@ This expansion gives the curious result that as the amount of dispersion is reduced from infinity the optimum residence time increases since r1 is always positive. Since the maximum obtainable output concentration of component B, decreases monotonically as the dispersion increases from zero to infinity, and since the optimum residence time necessary to achieve
ik+$&&+3-* +ol (12) 10 y2’
’
0
is assumed to be valid. This procedure
gives an ordinary perturbation expansion problem since the highest derivative does not disappear as y + cc).In the course of obtaining +ik the only irregularity is that to obtain one of the constants of integration in the kth order term, the k+ 1 order set of equations must be partially solved. The asymptotic expansion for 4A is then:
1 x2 + -1+72
+‘q
The zeroth order term is the solution for a CFSTR. To find an asymptotic expansion for the optimum residence time valid near l/r = 0 the derivative of Eq. (13) is set equal to zero and an expansion of the form
Topt.
$,4(x)=
1 10
T3(2cU+a+ 1) 6((1+7)(1+(~7))~
Consider the case where the axial dispersion coefficient in the axial dispersed plug flow reactor model is infinite. In this case the reactor behaves as a continuous flow perfectly stirred tank reactor (CFSTR). We want to look at the case where the dispersion coefficient is almost infinite and find the effect of a small reduction from perfect fluid mixing. The system equations and boundary conditions are again Eqs. (3a) and (3b). This time l/r is the small parameter for the perturbation expansion. Assuming an expansion of the form
Fig. I. Semi-log plot of 71 and T,/T~vs. cxfor the axial dispersed tubular reactor with small dispersion.
837
K. L. KIPP, JR. and S. H. DAVIS, 10
I
ti
/to 0.1
0.01 001
0.1
I
ii3
100
d
Fig. 2. Log-log plot of 71and T,/T~vs. (I for the axial dispersed tubular reactor with large dispersion.
this maximum concentration is less for the reactor with no dispersion than for the perfectly mixed reactor, one would initially expect the optimum residence time to increase monotonically as the dispersion coefficient goes from zero to infinity. Yet, Eq. (15) implies that the optimum residence time as a function of the dispersion coefficient must pass through a maximum value for some finite dispersion. To verify this result, the equations of the axial dispersed plug flow model (3~) and (3b) were solved analytically and the optimum residence time was evaluated numerically. Figure 3 shows the variation of optimum residence time with dispersion over the complete range from zero to infinity for (Y= 0.1. From this curve it is clear that the effect indicated by the per-
JR.
turbation expansion solutions actually does exist. The tangents at y = 0 and y = tQ agree with the perturbation expansion solutions, Eq. (11) and (15), as shown in the same figure. For (Y= 0.1 the deviation between the numerical results and the asymptotic (tangent) curves is greater than 5 per cent only in the range O-4 c y < 10. It should be kept in mind that these are predictions of the axial dispersed plug flow reactor model. Its applicability in the region of perfect mixing is open to question, even though its behavior approaches the correct limit as y goes to infinity. The same reaction problem has been solved for a continuous flow stirred tank with an ideal tubular reactor bypass. The asymptotic expansion for the optimum residence time has been obtained for this model of large dispersion effects [9]. Since the first order perturbation term is always negative, the optimum residence time for the bypass type of model decreases monotonically as the fluid flow pattern is changed continuously from perfect mixing to plug flow with no mixing. This is in agreement with our physical feeling for the mixing process and our expectation of a monotonic transition of the component concentrations and optimum residence time. It should be noted that fluid mixing takes place in this model on a macro scale whereas mixing in the axial dispersed model is micro-mixing.
Fig. 3. Axial dispersed tubular optimum residence time vs. y/ (y + 1) or a + 1 with the asymptotic expansions for small and large dispersion maximum at y = 4.5.
838
The effect of dispersion PREDICTION
OF OPTIMUM TIMES
RESIDENCE
The axial dispersed plug flow tubular reactor has been used to model simple reactor systems and predict optimum residence times for them in the region of small dispersion effects (see Reference [9]). The systems considered were: an ideal tubular reactor with a perfectly stirred tank bypass with a small bypass flowrate, a partial bypass reactor of the same type, and an axial dispersed reactor in series with a total bypass reactor. Expressions for the predicted optimum residence times using Eq. (11) and (15) are presented in Reference [9]. Numerical evaluations indicate that for small dispersion the axial dispersed tubular reactor can successfully model and predict optimum residence times for systems where fluid mixing takes place by a different mechanism than in the axial dispersion model. EXTENSION
TO NON-LINEAR
REACTION
KINETICS
The method of perturbation expansions is perhaps most useful for those cases where an exact analytical solution of Eq. (3~) and (3b) is not possible due to non-linear kinetics. Work has been done toward developing asymptotic expansions for the optimum residence time for general reaction rate expressions (see Reference 191).
unusual phenomenon. Subsequent results indicate that this maximum is due to the choice of dimensionless parameters and the fact that the length of the reactor is optimized with the dispersion, flow velocity, and kinetic rate constants held fixed. Thus, this maximum phenomenon has significance for reactor design calculations. Although the bypass reactor model of large dispersion does not exhibit this effect, this can be explained by the fact that mixing occurs on a macro scale in this model rather than on a micro scale as in the dispersed plug flow reactor model. Acknowledgments-This work was supported by the Ethyl Corvxation and the National Science Foundation (Systems Grant GU-I 153). NOTATION Note:
Subscripts
with
these quantities
are defined where used. Capital letters A, B, C, reaction components c constants of integration D axial dispersion coefficient L length of tubular reactor Si material source term for component i V average or plug flow velocity in the axial direction of a tubular reactor Small letters Cf Cio
CONCLUSIONS
The method of perturbation expansion solutions is a simple, straightforward method for obtaining approximate analytical solutions to differential and algebraic equations in regions of limiting behavior of one parameter. Its greatest potential lies in its application to non-linear differential equations where an analytical solution is not otherwise possible. Even for linear systems, an asymptotic expansion often gives an accurate solution with much less work than a full and exact solution. The maximum in the optimum residence time of the axial dispersed plug flow reactor with the consecutive first order reaction system is an
appearing
k rAU3) ‘cl rl r2 t X
z
concentration of component i concentration of i upstream and away from the axial dispersed tubular reactor or concentration of i at the entrance to an ideal reactor linear kinetic rate constant in r rate of formation of component A(B) rate of formation of component i reaction rate for A + B reaction rate for B + C time variable dimensionless axial coordinate for tubular reactor tubular reactor axial coordinate
Greek letters Q! ratio of the reaction rate constants 839
K. L. KIPP, JR. and S. H. DAVIS,
v
parameter of the defect boundary layer stretching transformation /3 parameter of the defect boundary layer perturbation term y dimensionless axial dispersion coefficient q dimensionless residence time T,,~~. optimum dimensionless residence time
JR.
rk
optimum dimensionless residence time, kth order perturbation term 6 ACBj dimensionless concentration of A(B) c& dimensionless concentration of i, kth order solution I)~ defect boundary layer solution or singular perturbation term for the dimensionless concentration of i
REFERENCES [I] BIRD R. B., STEWART
W. E. and LIGHTFOOT
E. N., Transport Phenomena. Wiley 1960. [2] BISCHOFF K. B. and LEVENSPIEL O.,Chem. Engng Sci. 1962 17 245. [3] CARBERRY J. J. and MARTIN M. W.,A.I.Ch.E. J11963 9 129. [4] DANCKWERTS P. V., Chem. Engng Sci. 1953 2 1. [5] ERDELYI A., J. Sot. Ind. appl. Math. 1963 11 105. [6] FREEMAN L. B. and HOUGHTON G., Chem. Engng Sci. 1966 211011. [7] GOLDWYN R. M., Unpublished notes, Rice University 1966. [8] HOELSCHERH. E.,A./.Ch.E.Jl 19639569. [9] KIPP K. L., M.S. Thesis, Rice University 1967. [lo] KRAMERS H. and WESTERTERP K. R., Elements of ChemicalReactor Design and Operation. [I I] LEVENSPIEL O., Chemical Reaction Engineering. Wiley 1962. [ 121 PARISH T. D., Ph.D. Thesis, Rice University 1967. [13] TICHACEKL.J.,A.I.Ch.E.JI19639394. [ 141 TURNER J. C. R., Br. Chem. Engng I964 9 376. [15] VAN KREVELEN D. W.,Chemie-fngr-Tech. 195830553.
Academic Press 1963.
Rbumk-Voici
une etude mathtmatique de I’effet de dispersion sur les temps de residence de reacteurs chimiques. L’attention est restreinte a la gamme des dispersions tres grandes et tres petites, de sorte que les expansions de la perturbation puissent &tre utilisees pour obtenir des solutions asympotiques des equations pour un reacteur tubulaire a Ccoulement bouche avec dispersion axiale. On ne considere que le systeme consecutif de reaction A + B + C, dans lequel B est un produit intermtdiaire desire. Le temps de residence optimum est alors defini comme celui augmente au maximum la limite de B dans chaque modele de reacteur isothermique. On montre que les expansions pour un temps de residence optimum sont des approximations excellentes pour une gamme &endue de conditions de fonctionnement pour ce modele de reacteur. On discute dun effet maximum inttressant dans le temps de residence optimum pour le reacteur tubulaire a dispersion axiale. Zusammenfassung-
Dieser Artikel beschreibt eine mathematische Studie iiber den Einfluss der Streuung auf die optimalen Verweilzeiten in chemischen Reaktoren. In Betracht gezogen werden ausschliesslich sehr kleine und sehr grosse Streuungen, so dass Storungsexpansionen angewandt werden kiinnen, urn asymptotische Liisungen zu den Gleichungen fiir den PfropfstriimungsRdhrenreaktor mit Langsstreuung zu erhalten. Es wird allein das Folgereaktionssystem A + B -+C behandelt wobei B ein erwiinschtes Zwischenprodukt darstellt. Die optimale Verweilzeit wird dann als diejenige definiert, bei welcher die Hochstausbeute an B in jedem isothermen Reaktormodell erhalten wird. Es wird gezeigt, dass die Expansionen fiir optimale Verweilzeiten fur dieses Reaktormodell ausgezeichnete Annlherungen in einem weiten Bereich von Betriebsbedingungen darstellen. Es wird ein interessanter Maximaleffekt in der optimalen Verweilzeit fir den Rohrenreaktor mit Langsstreuung erortert.
840
Pergamon Press.
Printed in Great Britain.
The effect of dispersion on chemical reactor optimum residence times using perturbation expansion techniques K. L. KIPP, Jr?. and S. H. DAVIS, Jr. Department of Chemical Engineering, Rice University, Houston, Texas (First
receioed
1Augrtsf 1967; nccepred 29 January 1968)
Abstract-This work is a mathematical study of the effect of dispersion on the optimum residence times of chemical reactors. Attention is confined to the range of very small and very large dispersions so that perturbation expansions can be used to obtain asymptotic solutions to the equations for the axial dispersed plug flow tubular reactor. Only the consecutive reaction system A + B -+ C, where B is a desired intermediate product is considered. The optimum residence time is then defined as that which maximizes the yield of B in each isothermal reactor model. It is shown that the expansions for optimum residence time are excellent approximations over a wide range of operating conditions for this reactor model. An interesting maximum effect in the optimum residence time for the axial dispersed tubular reactor is discussed: INTRODUCTION
of dispersion on the optimum residence time for the axial dispersed plug flow tubular reactor is investigated in this work. The reactor residence time is chosen to maximize the yield of the desired intermediate product B in a simple consecutive reaction, A + B --* C. The mathematical technique of perturbation expansions is used to obtain solutions to the limiting cases of small and large dispersion in such reactors. In chemical reaction engineering or chemical reactor analysis perhaps the two most common mathematical models are the ideal tubular flow reactor and the continuous flow stirred tank reactor. Actual reactors are usually intermediate in their behavior between these extreme cases with partial mixing and non-uniform flow causing deviation from either simple system. Many types of models can be used to represent the effects of non-ideal flow in tubular reactors. In the axial dispersed tubular reactor non-ideal flow is characterized by a single dispersion parameter indicating the extent of fluid mixing. The reacting fluid is assumed to be a single isothermal phase with reactions taking place homogeneously. Thus, the model is applicable to packed THE
EFFECT
iPresent
bed catalytic reactors with negligible mass transfer resistance between phases. Much theoretical work has been done on dispersion in chemical reactors and the effect of dispersion on chemical conversion, but there is little literature that relates directly to the objectives or to the methods of this paper. Freeman and Houghton[6] have applied the perturbation expansion techniques of Erdelyi[S] and Carrier to the singular perturbation, concentration profile problem. However, in the formal solution procedure they introduce first order terms in a zeroth order boundary condition (thus violating the order relation) in order to bring their results into closer agreement with numerical solutions. Parish [ 121 has also applied perturbation expansions to the conversion problem with small dispersion for multiple reactions and general kinetics. Only a few articles have appeared dealing with dispersion and selectivity in chemical reactors[3,8,10,13-151. Kramers and Westerterp [lo] have numerically evaluated the axial dispersed tube reactor for the consecutive reactions A + B + C. For the same reaction system Tichacek [ 131 presents results on the decrease in intermediate yield due to dispersion in a tubular
Address: Esso Production Research Co., Houston, Texas.
$33
K. L. KIPP, JR. and S. H. DAVIS,
JR.
solution domain. One approach that can be used to treat singular perturbation problems involves the use of a coordinate stretching transformation at the end of the domain to form a ‘defect boundary layer’. In this layer a correction term added to the solution becomes sufficient in magnitude to enable the boundary condition at that end to be satisfied. Away from the boundary layer the defect boundary layer term becomes vanishingly small. The ‘defect boundary layer’ technique for ordinary differential equations as given by Goldwyn [7] will be used in this work.
reactor. He considers several sets of reaction kinetics and uses a perturbation expansion solution in one case; but to simplify the equations, he makes the reactor length semi-infinite. Thus, his results are slightly different than for the axial dispersed reactor with a finite reaction zone. In review, one can say that only Parish handled perturbation expansion solutions to dispersion problems in finite chemical reactors in a rigorously correct manner. No one to date has presented work on optimum residence times for reactors with dispersion effects using perturbation expansion techniques.
THE
AXIAL
THEORY
DISPERSED
TUBULAR
The perturbation technique is an empirical method for obtaining approximate analytical solutions to otherwise insolvable problems that are only slightly different from problems that can be solved. Basically, we desire a solution to a differential equation containing one small parameter where all other terms are assumed to be of order one. If the equation can be solved exactly when the small parameter goes to zero, then we hope that a good approximation to the exact solution of the entire differential equation can be obtained in the form of a perturbation expansion of the exact solution to the zeroth order equation. An assumed solution form, C#I = c#J,,+E& + ++..., is inserted into the differential equation and boundary conditions. The resulting sets of equations in orders of the small parameter E are solved sequentially for & starting with the zeroth order system. The various sets of equations must be satisfied identically in order for the complete solution to satisfy the differential equation and boundary conditions uniformly in the small parameter. The result is assumed to be accurate to the order of the first truncated term. This is the basic Poincare expansion procedure. One common difficulty that occurs is the highest order derivative disappears as the small parameter goes to zero. This gives a singular perturbation problem. There is difficulty in making the solution satisfy all the boundary conditions because no single asymptotic expansion is uniformly valid over the entire
PLUG
FLOW
REACTOR
Small dispersion region
This model characterizes non-ideal flow patterns in a tubular reactor by using the similarity between mixing and diffusion. In the differential component material balance, the molecular diffusivity is replaced by a dispersion coefficient that may account for several sources of mixing. This axial dispersion model is one-dimensional and is obtained from the general component material balance [ I] $+V*
VC*=DV2Ci+rC,+Si
(where D is diffusivity matrix, S is source term, V is fluid velocity) through the following assumptions: (For an excellent discussion see Bischoff [2]). There are no sources of material within the system; the reacting mixture is treated as a single phase; the mixture is of constant density; there is bulk flow in the axial direction only, and radial symmetry; the dispersion coefficients are constant; the time average velocity of the fluid is constant over the tubular cross-section and independent of axial position (plug flow); there is no variation in properties in the radial direction; the system is at steady state. The differential equation for the ith component mass balance is then
834
_~d2”‘+I,KrC*=() dz2 dz
(1)
The effect of dispersion
where D is dispersion coefficient and rcI is rate of formation of ith component/unit volume. Looking at the terms of Eq. (1) we see that the axial dispersion model describes dispersion and mixing by a diffusional mechanism, (i.e., driven by concentration gradients), superimposed on the plug flow velocity profile of the ideal tubular reactor model. The single parameter D characterizes the net effect of all dispersion phenomena which may be taking place. The Danckwerts’ boundary conditions [4] for a tubular reactor with dispersion are assumed to be applicable. dci atz = 0 c~-D-= cio CW dz Dd”i,O dz
atz=L
(2b)
-
The reactor is assumed to be isothermal so that no energy balance is required. Only reaction component A enters the reactor. Furthermore, the dispersion coefficient is assumed to be the same for both species and independent of composition. In this paper it will be assumed that the two consecutive reactions are irreversible and have first order kinetic rate expressions. AAB
r,=klcA
SO rA = - ktCA
B I; C
r2 = k2cB
rB = k,C, - k2ce.
In terms of the dimensionless
y+
variables
Lk,
g-z-
V
(y=-.
-7
k2 k,
B.C.
at
x=0
(34
= 0
&2
4A--y$$
= 1 0
at
x=7
y-@‘A dr.
=0
B.C.
at
x= 0
4B-73
=0 0
at
x=7
d’#‘B
-yz
=O. T
Note that we have reduced the number of parameters to three y, LY,and r. Thus, a perturbation expansion for Tag. will be a function of y and (Y only. The above linear system can be solved analytically. This would not be true if the reaction rates were non-linear. It will be seen that perturbation expansion solutions are simpler to obtain than exact analytical solutions in those cases where both are possible. The above set of equations forms a singular perturbation problem since as y + 0 the highest order derivatives are lost and the equations become first order. This means that it will be impossible to satisfy the boundary conditions in all orders of y with a single ordinary perturbation expansion. Fortunately we can solve the two differential equations sequentially rather than simultaneously due to the irreversibility of the reactions. Starting with the eqUatiOU for 4A, an eXpat’KiOn of the form 4A = +A0+ Y4AI+ y24As+ . . . is inserted into Eq. (3a). The solution of the resulting set of equations (determined by the condition that each coefficient of a power of y must vanish) may be solved sequentially. The first two terms in the expansion are $A0 = e-” (The solution for an ideal tubular reactor) (4) +A, = xe-” - e-” (The first order perturbation terms satisfying the inlet boundary conditions).
The system equations are
d2b db -+-&++A
(3b)
As a result of the singularity of the differential equation at y equal to zero the first order perturbation term does not satisfy the reactor outlet boundary condition. To correct this an additional term is introduced into the solution which is important near x = 7 but vanishes far away from that boundary. 835
K. L. KIPP, JR. and S. H. DAVIS,
A revised solution of the form $A = 4AAo+ Y&4*+ $JIA(rl) +. * * is assumed (p 3 0 since $A is bounded as Y ---, 0) where c#B~,,, +A1,etc. are the functions previously found. We want JiA($ + 0 as 71+ ~0.71is the boundary layer coordinate. So let r) = (r-x)Yy (this is a coordinate stretching transformation). As x + 7, r) + 0; as x + - 03, r) + ~0. Thus, 7 measures the distance away from the exit of the reactor. Inserting the assumed form for $ into Eq. (3a) we obtain: -Ye -Z - Yz(- 3e-” + xe-” )
- . . . -y2v+p+ + (-e-I)
+y(2e-“-xe-“)
+e-“+y(xe-z-ee-z)
+. . . -y”+j3
+. . .+yaJIA(~)
l-
d*JiA dr1’
The total perturbation then c#I~(~)= e-“+y
[
(x-
expansion l)e-Z-te-Te-
to order y is
1 (8)
T-X 7 +O(y’).
The singular perturbation term, $*, does not satisfy the differential equation to order y but the region in which this condition holds becomes very small as y + 0. Thus, Eq. (8) satisfies all the system equations to order y except in a boundary layer region near the reactor exit. Exactly the same procedures can be followed with Eq. (3b) to find a perturbation expansion for &. The differential equation is inhomogeneous. The solution gives & at the exit of the reactor:
drCIA drl = 0.
&3(~)= $-f(epT-e-a’)
(5)
+y [
a-(r-
I)e-T
--f+m--l)e-m]+Y--&
Note that cj~~~+Yc$~~ satisfies the differential equation identically as expected. There was difficulty with our previous solution because there were no terms in d2$,/dxz to “balance” the terms in d$,/dx and 4,., so the boundary condition at x = r could be satisfied to order y. Therefore, the defect boundary layer term JIA(r)) involving the second derivative should be of the same order in y as the lower derivative term. Then the second derivative d2~A/cIrz will not disappear as Y goes to zero, especially near the reactor exit. For this reason we pick Y so that the highest derivative terms are retained and are of the same order as lower terms. Thus, we must set 2v+p+ 1 = v+/3 so v=-l,thenq= (r-x)/Y. The equations determining +Abecome
X (e-‘-ae-“)(l)+O(y’).
(9)
Now to obtain a solution for optimum length or residence time to maximize the production of B, perturbation expansion techniques can be used to solve the nonlinear equation. Take the derivative of &(r) Eq. (9) with respect to r and set it equal to zero. This locates the T for maximizing &(T).
d&i -= dr
aeFm) + y[&eeT
-&(-e-T+ + _e (-1) a-1
(6)
_r_-e-T+-_e-” CJ? a-l
cl!% a-1
1 b> = 0
To get a perturbation expansion approximation let 7=~~-t-y7~+. . ., insert into Eq. (lo), and equate the terms in like orders of Y. We then find lncu rapt.
e=Oat
JR.
q=O.
=,-1+y
(
‘a(a-l)-l)+O(r~).(ll) a-1
Thus, we have obtained a relation for the effect To satisfy Eq. (7) to order y we must pick p = 1. of small dispersion on the optimum residence time for the maximum yield of the desired The differential equation and Eq. (6) are satisfied by ce-‘l. Equation (7) determines c = edT’. component in the consecutive reaction system: 836
The effect of dispersion
A + B + C. Figure 1 is a graph of the first order perturbation term TV,and the ratio of or to 7. a\ a function of the reaction rate constant ratio cy.
A similar procedure leads to the desired solution for $Q at the reactor outlet
MT)= (l+T);l+m)
Large dispersion region
1 Topt.
=
ro+yrt+o
(13)
1
(14)
-p
produces
=
x a2+6ad(a)
+ lOa+6~(a)
2a2+aV(cI)
+&(a)
+ 1
+o $ 0
. (15)
Figure 2 shows the first order term in Eq. (15) and the ratio of this term to T@ This expansion gives the curious result that as the amount of dispersion is reduced from infinity the optimum residence time increases since r1 is always positive. Since the maximum obtainable output concentration of component B, decreases monotonically as the dispersion increases from zero to infinity, and since the optimum residence time necessary to achieve
ik+$&&+3-* +ol (12) 10 y2’
’
0
is assumed to be valid. This procedure
gives an ordinary perturbation expansion problem since the highest derivative does not disappear as y + cc).In the course of obtaining +ik the only irregularity is that to obtain one of the constants of integration in the kth order term, the k+ 1 order set of equations must be partially solved. The asymptotic expansion for 4A is then:
1 x2 + -1+72
+‘q
The zeroth order term is the solution for a CFSTR. To find an asymptotic expansion for the optimum residence time valid near l/r = 0 the derivative of Eq. (13) is set equal to zero and an expansion of the form
Topt.
$,4(x)=
1 10
T3(2cU+a+ 1) 6((1+7)(1+(~7))~
Consider the case where the axial dispersion coefficient in the axial dispersed plug flow reactor model is infinite. In this case the reactor behaves as a continuous flow perfectly stirred tank reactor (CFSTR). We want to look at the case where the dispersion coefficient is almost infinite and find the effect of a small reduction from perfect fluid mixing. The system equations and boundary conditions are again Eqs. (3a) and (3b). This time l/r is the small parameter for the perturbation expansion. Assuming an expansion of the form
Fig. I. Semi-log plot of 71 and T,/T~vs. cxfor the axial dispersed tubular reactor with small dispersion.
837
K. L. KIPP, JR. and S. H. DAVIS, 10
I
ti
/to 0.1
0.01 001
0.1
I
ii3
100
d
Fig. 2. Log-log plot of 71and T,/T~vs. (I for the axial dispersed tubular reactor with large dispersion.
this maximum concentration is less for the reactor with no dispersion than for the perfectly mixed reactor, one would initially expect the optimum residence time to increase monotonically as the dispersion coefficient goes from zero to infinity. Yet, Eq. (15) implies that the optimum residence time as a function of the dispersion coefficient must pass through a maximum value for some finite dispersion. To verify this result, the equations of the axial dispersed plug flow model (3~) and (3b) were solved analytically and the optimum residence time was evaluated numerically. Figure 3 shows the variation of optimum residence time with dispersion over the complete range from zero to infinity for (Y= 0.1. From this curve it is clear that the effect indicated by the per-
JR.
turbation expansion solutions actually does exist. The tangents at y = 0 and y = tQ agree with the perturbation expansion solutions, Eq. (11) and (15), as shown in the same figure. For (Y= 0.1 the deviation between the numerical results and the asymptotic (tangent) curves is greater than 5 per cent only in the range O-4 c y < 10. It should be kept in mind that these are predictions of the axial dispersed plug flow reactor model. Its applicability in the region of perfect mixing is open to question, even though its behavior approaches the correct limit as y goes to infinity. The same reaction problem has been solved for a continuous flow stirred tank with an ideal tubular reactor bypass. The asymptotic expansion for the optimum residence time has been obtained for this model of large dispersion effects [9]. Since the first order perturbation term is always negative, the optimum residence time for the bypass type of model decreases monotonically as the fluid flow pattern is changed continuously from perfect mixing to plug flow with no mixing. This is in agreement with our physical feeling for the mixing process and our expectation of a monotonic transition of the component concentrations and optimum residence time. It should be noted that fluid mixing takes place in this model on a macro scale whereas mixing in the axial dispersed model is micro-mixing.
Fig. 3. Axial dispersed tubular optimum residence time vs. y/ (y + 1) or a + 1 with the asymptotic expansions for small and large dispersion maximum at y = 4.5.
838
The effect of dispersion PREDICTION
OF OPTIMUM TIMES
RESIDENCE
The axial dispersed plug flow tubular reactor has been used to model simple reactor systems and predict optimum residence times for them in the region of small dispersion effects (see Reference [9]). The systems considered were: an ideal tubular reactor with a perfectly stirred tank bypass with a small bypass flowrate, a partial bypass reactor of the same type, and an axial dispersed reactor in series with a total bypass reactor. Expressions for the predicted optimum residence times using Eq. (11) and (15) are presented in Reference [9]. Numerical evaluations indicate that for small dispersion the axial dispersed tubular reactor can successfully model and predict optimum residence times for systems where fluid mixing takes place by a different mechanism than in the axial dispersion model. EXTENSION
TO NON-LINEAR
REACTION
KINETICS
The method of perturbation expansions is perhaps most useful for those cases where an exact analytical solution of Eq. (3~) and (3b) is not possible due to non-linear kinetics. Work has been done toward developing asymptotic expansions for the optimum residence time for general reaction rate expressions (see Reference 191).
unusual phenomenon. Subsequent results indicate that this maximum is due to the choice of dimensionless parameters and the fact that the length of the reactor is optimized with the dispersion, flow velocity, and kinetic rate constants held fixed. Thus, this maximum phenomenon has significance for reactor design calculations. Although the bypass reactor model of large dispersion does not exhibit this effect, this can be explained by the fact that mixing occurs on a macro scale in this model rather than on a micro scale as in the dispersed plug flow reactor model. Acknowledgments-This work was supported by the Ethyl Corvxation and the National Science Foundation (Systems Grant GU-I 153). NOTATION Note:
Subscripts
with
these quantities
are defined where used. Capital letters A, B, C, reaction components c constants of integration D axial dispersion coefficient L length of tubular reactor Si material source term for component i V average or plug flow velocity in the axial direction of a tubular reactor Small letters Cf Cio
CONCLUSIONS
The method of perturbation expansion solutions is a simple, straightforward method for obtaining approximate analytical solutions to differential and algebraic equations in regions of limiting behavior of one parameter. Its greatest potential lies in its application to non-linear differential equations where an analytical solution is not otherwise possible. Even for linear systems, an asymptotic expansion often gives an accurate solution with much less work than a full and exact solution. The maximum in the optimum residence time of the axial dispersed plug flow reactor with the consecutive first order reaction system is an
appearing
k rAU3) ‘cl rl r2 t X
z
concentration of component i concentration of i upstream and away from the axial dispersed tubular reactor or concentration of i at the entrance to an ideal reactor linear kinetic rate constant in r rate of formation of component A(B) rate of formation of component i reaction rate for A + B reaction rate for B + C time variable dimensionless axial coordinate for tubular reactor tubular reactor axial coordinate
Greek letters Q! ratio of the reaction rate constants 839
K. L. KIPP, JR. and S. H. DAVIS,
v
parameter of the defect boundary layer stretching transformation /3 parameter of the defect boundary layer perturbation term y dimensionless axial dispersion coefficient q dimensionless residence time T,,~~. optimum dimensionless residence time
JR.
rk
optimum dimensionless residence time, kth order perturbation term 6 ACBj dimensionless concentration of A(B) c& dimensionless concentration of i, kth order solution I)~ defect boundary layer solution or singular perturbation term for the dimensionless concentration of i
REFERENCES [I] BIRD R. B., STEWART
W. E. and LIGHTFOOT
E. N., Transport Phenomena. Wiley 1960. [2] BISCHOFF K. B. and LEVENSPIEL O.,Chem. Engng Sci. 1962 17 245. [3] CARBERRY J. J. and MARTIN M. W.,A.I.Ch.E. J11963 9 129. [4] DANCKWERTS P. V., Chem. Engng Sci. 1953 2 1. [5] ERDELYI A., J. Sot. Ind. appl. Math. 1963 11 105. [6] FREEMAN L. B. and HOUGHTON G., Chem. Engng Sci. 1966 211011. [7] GOLDWYN R. M., Unpublished notes, Rice University 1966. [8] HOELSCHERH. E.,A./.Ch.E.Jl 19639569. [9] KIPP K. L., M.S. Thesis, Rice University 1967. [lo] KRAMERS H. and WESTERTERP K. R., Elements of ChemicalReactor Design and Operation. [I I] LEVENSPIEL O., Chemical Reaction Engineering. Wiley 1962. [ 121 PARISH T. D., Ph.D. Thesis, Rice University 1967. [13] TICHACEKL.J.,A.I.Ch.E.JI19639394. [ 141 TURNER J. C. R., Br. Chem. Engng I964 9 376. [15] VAN KREVELEN D. W.,Chemie-fngr-Tech. 195830553.
Academic Press 1963.
Rbumk-Voici
une etude mathtmatique de I’effet de dispersion sur les temps de residence de reacteurs chimiques. L’attention est restreinte a la gamme des dispersions tres grandes et tres petites, de sorte que les expansions de la perturbation puissent &tre utilisees pour obtenir des solutions asympotiques des equations pour un reacteur tubulaire a Ccoulement bouche avec dispersion axiale. On ne considere que le systeme consecutif de reaction A + B + C, dans lequel B est un produit intermtdiaire desire. Le temps de residence optimum est alors defini comme celui augmente au maximum la limite de B dans chaque modele de reacteur isothermique. On montre que les expansions pour un temps de residence optimum sont des approximations excellentes pour une gamme &endue de conditions de fonctionnement pour ce modele de reacteur. On discute dun effet maximum inttressant dans le temps de residence optimum pour le reacteur tubulaire a dispersion axiale. Zusammenfassung-
Dieser Artikel beschreibt eine mathematische Studie iiber den Einfluss der Streuung auf die optimalen Verweilzeiten in chemischen Reaktoren. In Betracht gezogen werden ausschliesslich sehr kleine und sehr grosse Streuungen, so dass Storungsexpansionen angewandt werden kiinnen, urn asymptotische Liisungen zu den Gleichungen fiir den PfropfstriimungsRdhrenreaktor mit Langsstreuung zu erhalten. Es wird allein das Folgereaktionssystem A + B -+C behandelt wobei B ein erwiinschtes Zwischenprodukt darstellt. Die optimale Verweilzeit wird dann als diejenige definiert, bei welcher die Hochstausbeute an B in jedem isothermen Reaktormodell erhalten wird. Es wird gezeigt, dass die Expansionen fiir optimale Verweilzeiten fur dieses Reaktormodell ausgezeichnete Annlherungen in einem weiten Bereich von Betriebsbedingungen darstellen. Es wird ein interessanter Maximaleffekt in der optimalen Verweilzeit fir den Rohrenreaktor mit Langsstreuung erortert.
840