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Residence time distribution and fractional conversion for a multistage reactor with backmixing between real stages
Chemical Engineering Science, 1968, Vol. 23, pp. 139-145.
Pergamon Press.
Printed in Great Britain.
Residence time distribution and fractional conversion for a multistage reactor with backmixing between real stages M. REHAKOVA and Z. NOVOSAD Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Prague-Suchdol (Received 7 August 1967)
Abstract-Amathematical model is suggested for expressing the residence time distribution in a multistage system with backmixing between stages and non-ideal mixing in each stage. This non-ideal behaviour is taken into account by considering each stage to be composed of a series of vessels with backmixing between the vessels. The residence time distribution curves are obtained by numerical solution of a set of mass balance differential equations. It is shown how the ordinate and abscissa of the maximum point of the residence time distribution curve can be used to evaluate the parameters of experimental curves. An analytical expression is derived for fractional conversion calculation for a homogeneous reactor based on the above model and the relative importance of individual model parameters is indicated. 1. INTRODUCTION
FOR THE mathematical
description of residence time distribution curves in multistage equipment it is possible to employ either the dispersion or the tanks in series model. Particularly for extraction equipment, where usually a large number of stages is used, either model can be successfully applied. It has been shown[ l] that multistage column reactors on the other hand need not have a large number of stages and the tanks in series model is therefore more realistic than the dispersion model. In a multistage column reactor, backmixing between stages is an important variable which influences the residence time distribution behaviour of the system. In establishing mathematical models for a multistage system with backmixing between stages it has hitherto been always assumed that each stage behaves as an ideally mixed vessel (vessel with backmix flow)[2-7]. In this work a model is suggested which considers the general case of non-ideal mixing in each stage and backmixing between stages. Non-ideal mixing in one stage has been mathematically described by several authors [S- 151. The models used (mixed, circulation and dispersion models) are all rather involved and in combination with backmixing between stages would yield very cumbersome expressions. The model
for expressing this non-ideality suggested in our work uses a relatively simple concept which can be easily combined with backmixing between stages. Besides the mathematical formulation of the model and interpretation of the physical aspects of individual parameters, we have included also a method of evaluation of experimental curves and a calculation of fractional conversion for a first order reaction. These results are useful for estimating the relative importance of individual model parameters. 2. DESCRIPTION
OF MODEL
It is assumed that each stage of the reactor consist of a series of ideally mixed vessels with backmixing between adjacent vessels as shown on Fig. 1. Let S be the number of such vessels in each stage and let b be the coefficient of backmixing between these vessels. There is also backmixing between the individual stages of the reactor. Let the number of stages be N and let the coefficient of backmixing between stages be r. The non-ideal mixing within a stage is thus expressed by means of the parameters S and b. The mass balance equations used for describing the process inside a stage are thus formally equivalent to the equations used to describe the process as a whole. This fact is of great impor-
139
and z. NOV~SAD
M. REHAK~VA
The mass balance of tracer in dimensionless form for the first vessel of each stage is vfhbh,, ------
--$$=
(l+r)c,,_,,+bc,-(l+b+r)c,l (la)
where 8 is a dimensionless time 8 = t/SW and t is the mean residence time in one vessel. For an intermediate vessel
i=2,3...(S-1)
(lb)
j=l,2...N. The mass balance for the last vessel of each stage is &2=
(l+b)c,,_,V-trclo+l,-(l+r+b)c~~. (lc)
I
"'SN
Fig. 1. The general mathematical model and notation for mass balance equations
tance since it considerably simplifies the solution of resulting equations. The mathematical model has three parameters: the number of ideally mixed vessels S, the coefficient of backmixing between the vessels b and the coefficient of backmixing between stages r. The total number of stages N is not considered to be a parameter for it is directly given by the number of actual stages. Calculation of the response curve to an instantaneous or pulse tracer injection corresponding to a given set of parameters is performed by setting up mass balance equations for individual vessels. On Fig. 1 the concentrations of tracer leaving a vessel are denoted by a double subscript ij. The first letter indicates the number of the vessel inside a stage and the second 1eLter is the stage number.
Setting up the mass balance equations for all vessels in the system results in N.S linear differential equations. Their solution with the initial condition that c, = 0 when t d 0 and boundary conditions determined by the mass balance for the first stage of the system with a delta function input
1 dell N.S~=
6+&n--
(l+bh
(14
and the last stage of the system
1 &iv myjg-=
(~+~)(cG-I~N-csN)
(W
yields directly the impulse residence time distribution function E(8). For the general case the system of equations cannot be solved analytically. It was therefore solved numerically on the NE 803 computer by using a Runge-KuttaMerson procedure [ 161. 3. PHYSICAL
r
140
INTERPRETATION PARAMETERS
OF MODEL
The coefficient of backmixing between stages is the fraction of the feed which is recirculated
Residence time distribution and fractional conversion for a multistage reactor
between adjacent stages. It therefore has the same significance as in the tanks in series model with backmixing between adjacent tanks. The extent of mixing inside individual stages is expressed by a series of S ideally mixed vessels and a coefficient of backmixing b. It has been verified experimentally [ 161 that the parameter S depends only on the geometry of the system. For a mechanically agitaged multistage column with stage height to dia. ratio h/d = 1, the best fit to experimental curves was obtained for a value of S = 3. It was further shown [ 161 that the coefficient of backmixing b between vessels in a stage depends on the number of revolutions according to the well known expression for the circulation ratio in circulation models
[ 131. Our parameter S seems to be an analogy of van de Vusse’s parameter n, which expresses the diffusivity inside the circulation loop by the number of ideally mixed vessels in series. It also depends only on the geometry of the system and does not depend on the other variables. Our parameter b can be compared with van de Vusse’s circulation ratio r,. In order to substantiate these analogies we calculated several residence time distribution curves according to both models for equal variances. Van de Vusse shows that the variance of his model is
The variance of a series of vessels with backmixing was calculated by Miyauchi and Vermeulen [2] as
b = const. $. In view of the above relations we can consider the model bf a series of vessels with backmixing b,tween vessels which describes the mixing inside a stage to be analogous to the circulation model as shown on Fig. 2. Let us now compare
2_ 1+2b CT2--- S
2b(l+b)
{l-(&r}.
S2
(4)
Equating the variances we obtain the following relation for van de Vusse’s circulation ratio r,,
S(S-n,-22n&)+2n,(b+b2)
(
r, =
l-exp
(
-Sin
9 (3
Sn,(1+2b-S)-2n,(b+b2)(l-exp(-Slny)) the parameters of this model with one of the more recent circulation models of van de Vusse
vb
A value of S = 3, giving the best fit of experimental data[ 161, and a value of n, = 4 as recommended by van de Vusse, was substituted into Eq. (5). For several values of b the corresponding values of r, from this equation were then calculated. On Fig. 3 are shown the curves obtained by calculation from these parameters. The similarity of the general trend of these curves supports the foregoing discussion.
v(l ‘+bl
Fig. 2. Analogy with the circulation model.
’
4. EVALUATION
OF EXPERIMENTAL
DATA
In order to determine the best parameter corresponding to an experimental values residence time distribution curve it is necessary to compare this curve with a theoretical curve calculated from known parameters. As a criterion of agreement we may choose equal variances or 141
M. REHAKOVA
and Z. NOVOSAD
1c E(e)
Fig. 3. Comparison with model of van de Vusse. l-van de Vusse, n,,= 4, r, = 4.357; 2-our model, S = 3, b = 5; 3-van de Vusse, n, = 4, r, = 0.928; 4-our model, S = 3, b = 1.
equal coordinate values of characteristic points on the curve. Using equal variances is not recommended since for its determination great accuracy of the tail part of the residence time distribution curve is required. It is preferable to use the method of characteristic points. A suitable point frequently used is the maximum of the curve. In practical evaluation of curve parameters it is convenient to prepare charts for each number of stages of the measured equipment by numerical solution of Eq. (1). As an example on Fig. 4 is shown the relation between the ordinate of the maximum point E,,,,, and the abscissa 19,~~ for various parameter values b and r (S = 3) for a threestage column which is being used in research work for this is the least number of stages which includes both end stages and an intermediate stage as well. By reading off the coordinates of the maximum point of the experimental residence time distribution curve, the parameters r and b can be directly determined from this figure.
5. FRACTIONAL
CONVERSION ORDER
FOR
A FIRST
REACTION
In order to determine the relative importance of the parameters of the model we shall derive an expression for the fractional conversion of a first order homogeneous reaction as a function of these parameters. For this case Danckwerts [ 171 showed that X= LmE(t)(l-e-KL)
dt
= JbmE(t) dt -jre-ktE(t) = 1 -E(k)
dt (6)
where k is the reaction velocity constant and j?(k) is the Laplace transform of the residence time distribution curve. This can be obtained by solving the set of differential equations (1) by means of the Laplace transformation. This will be shown for a value of S = 3 and any number of stages N. The solution is presented in dimensional time t = BfNS.
142
Residence time distribution and fractional conversion for a multistage reactor
095
090
085
OHJ
Fig. 4. Determination of curve parameters r and b for a three-stage column and S = 3 from the coordinates of the maximum point ,?Z,- and O,,,,.
By a step. by-step elimination of concentration terms we obtain a relation between the Laplace transforms of the tracer concentration in any three subsequent stages (j- l), (~7 and (j+ 1) (see Fig. 1) C33(1+1)-
~q,+Ji;,(,-l,
=
0
where
(7)
where
K=b(l+b)(FG-(l+r)rG+2TGF -Fb(l+b)-TTb(l+II)) -G2T(P-r(l+r)) L=@(b(l+b)-GT) T= (l+fk+b) M=
(~+!J)~(FG--b(l+b)).
I=b(l+b)(3PG-r(l+r)-%(l+b))+G2F(r(l+r)-F) rb2(b(l+b)-GF) J=
(l+b)2(l+r) rb2
F=
(l-t-fk+b+r)
G = (l+fk+2b). Equation (7) is a difference equation which (3) The third boundary condition is the mass must be solved with the following boundary balance for the last stage in Laplace transforms conditions: Q&9 - PC33(~__1) = 0 (8~) (1) at t = 0 cfj = 0 for all i, j. @a) (2) The second boundary condition is the mass where balance for the first stage in Laplace transforms Q= TGF-b(l+b)(t+F) KC331_Lii33+M=
0
(8b) 143
P=
(l+b)2(l+r).
M. REHAK~vA
NOVOSAD
mdz.
Solving Eq. (7) with these boundary conditions we obtain an expression for the Laplace transform of the concentration from anyj-th stage
between stages complicates the mathematics somewhat but this complication is not too serious since the solution has to be performed numerical-
A4(uzJ ul+l (Q-ju1)-ua,ju2N-’ h=
(Q
u~~-Z’U~~--‘)
(Ku~-Lu~~)-
where u1,2are roots of characteristic U 1,2 =
z+A@--4J) 2
(Ku~--Lu~~)
Eq. (7)
*
Substituting j = N into Eq. (9) we obtain the Laplace transform of the tracer concentration from the last stage or the Laplace transform of the residence time distribution curve
E’(k)=
t& =
(Q-j&)
~ ’
(Q~$‘-f’a,~-‘)
ly in any case. To show the relative importance of the parameters of the model we have calculated for several cases the fractional conversion for a homogeneous reaction of first order according to Eq. (6) and (10). The results are presented graphically on Fig. 5 for values of r = 0, 1,5 and reaction numbers 4 = k(fN.9) = 0*9,1+$, 3.6. From the curves it is apparent that for,b > 10
M(u~~u N-‘(Q-Nul)-u,Nu2N-1 (Q-Nu,)) (Kul--~a,~)(Qu,~-Pu,~-')' (QulN-PulN-’ )(Ku~-L~~)-
The fractional conversion can then be calculated by substituting for g(k) from Eq. (10) into Eq. (6).
(9)
(10)
the stages can be considered as being ideally mixed. For lower values of b the non-ideal behaviour in a stage must be taken into &count and the effect of the value of this parameter increases with increasing r. It is further interesting to note that this kind of non-ideality improves reactor performance so that under certain conditions it may be a desirable effect. NOTATION
b
I
0
I
I
lo
20
I b
30
Fig. 5. Fractional conversion X for various reaction numbers
q and parameter values r vs. coefficient of backmixing b for a three-stage column and S = 3. 6. THE RELATIVE
IMPORTANCE
OF MODEL
PARAMETERS
The inclusion of non-ideal mixing within a stage into the multistage model with backmixing 144
coefficient of backmixing between vessels, B/v B flow rate of backmixing between vessels in a stage L3P c dimensionless concentration of tracer E Laplace transform of c of tracer in CU dimensionless concentration the i-th vessel of the j-th stage d stage dia. L time distribution E(@ impulse residence function Laplace transform of E(8) ordinate of the maximum point on the E(8) curve stage height L reaction rate constant for first order reaction t-l
,
Residence time distribution and fractional conversion for a multistage reactor
t f
time tmean residence time in one ideally mixed vessel t V feed flow rate L3t-’ X fractional conversion 6 delta function 6 dimensionless time t/SNf e max abscissa of the maximum point on the E(O) curve
number of revolutions per unit time t-l diffusion parameter of van de Vusse n, N number of stages 4 reaction number, k(tNS) r coefficient of backmixing between stages, R/v R flow rate of backmixing between stages II
LV-’
ru
s
circulation ratio of van de Vusse number of ideally mixed vessels in a stage
REFERENCES [l] NOVOSAD Z., Colln Czech. them. Commun. 1967 32 3842. [2] MIYAUCHI T. and VERMEULEN T., Ind. Engng Chem. Fundls 1963 2 304. [3] PROCHAiXA J. and LANDAU J., Colln Czech. them. Commun. 1963 28 1927. [4] STAINTHORP F. P. and SUDALL N., Trans. Instn them. Engrs. 1964 42T198. [5] RETALLICK W. B., Ind. Engng Chem. Fundls 1965 4 88. [6] BELL R. L. and BABB A. L., Chem. Engng Sci. 1965 24llOOl. [7] MI&A J., Colln Czech. them. Commun. 1967 32 1518. [8] GUTOFF E. B.,A. I. Ch. E. Jll960 6 347. 191 HOVORKA R. B., Ph. D. Thesis, Case Institute of Technology, Ohio 1961. [lo] MARR G. R. and JOHNSON E. F., Chem. Engng Progress; Symp. Ser. 196157 36,109. [ 111 SINCLAIR G. G.,A. I. Ch. E. Jll9617 709. [12] LEVENSPIEL O., Chemical Reaction Engineering, John Wiley 1962. [13] van de VUSSE J. G., Chem. Engng Sci. 1962 17 507. [14] WOLF D. and RESNICK W., Ind. Engng Chem Fundls 1963 2 287. [15] VONCKEN R. M., ROTTE J. W., and HOUTEN A. T., A. 1. Ch. E.-Inst. Chem. Engng Joint Meeting, London 13-17June 1965. [16] REHAKOVA M., Ph.D. Thesis, Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Prague 1966. [17] DANCKWERTS P. V., Chem. Engng Sci. 1953 2 1.
R&u&l On sug&re un modele mathematique pour exprimer la distribution du temps de rksidence dans un systtme a stages multiples avec melange posterieur entre les stages et un mtlangenon-ideal a chaque stage. On tient compte de ce comportement non-ideal en considerant chaque stage comme &ant compose dune s&e de recipients avec mtlange posterieur entre ceux-ci. Les courbes de distribution du temps de residence sont obtenues par une solution nurmbique dun ensemble d’equations differentielles d’tquilibre de masse. On montre comment l’ordonnee et l’abscisse du point maximum de la courbe de distribution du temps de residence peut &tre utilisb pour tvaluer les parametres des courbes experimentales. Une expression analutique est tiree du calcul de la conversion fractionnelle pour un reacteur homogene base sur le modele ci-dessus et l’importance relative des parametres d’un modele individuel est indiquee. Zusammenfassung-Ein mathematisches Model1 wird zur Erklarung der Verweilzeitverteilung in einem Vielstufensystem mit Rllckmischung zwischen den Stufen und nicht-idealer Mischung in jeder Stufe vorgeschlagen. Das nicht ideale Verhalten wird bertlcksichtigt, indem man jede Stufe als eine Reihe von Gefassen mit Riickmischung zwischen ihnen betrachtet. Die Kurven der Verweilzeitverteihmg werden durch numerische Liisung einer Reihe von Massengleichgewicht-DifIerentialgleichungen erhalten. Es wird aufgezeigt, wie die Ordinate und Abzisse am Maximalpunkt der Verweilzeitverteilungskurve dazu verwendet werden kiinnen, die Parameter der experimentellen Kurven auszuwerten. Ein analytischer Ausdruck ftir fraktionelle Umwandlungsrechnung wird tiir einen homogenen Reaktor auf der Grundlage des oben angeflihrten Modells abgeleitet, und die relative Bedeutung der individuellen Modell-Parameter wird eriirtert.
145
C.E.S.-D
Pergamon Press.
Printed in Great Britain.
Residence time distribution and fractional conversion for a multistage reactor with backmixing between real stages M. REHAKOVA and Z. NOVOSAD Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Prague-Suchdol (Received 7 August 1967)
Abstract-Amathematical model is suggested for expressing the residence time distribution in a multistage system with backmixing between stages and non-ideal mixing in each stage. This non-ideal behaviour is taken into account by considering each stage to be composed of a series of vessels with backmixing between the vessels. The residence time distribution curves are obtained by numerical solution of a set of mass balance differential equations. It is shown how the ordinate and abscissa of the maximum point of the residence time distribution curve can be used to evaluate the parameters of experimental curves. An analytical expression is derived for fractional conversion calculation for a homogeneous reactor based on the above model and the relative importance of individual model parameters is indicated. 1. INTRODUCTION
FOR THE mathematical
description of residence time distribution curves in multistage equipment it is possible to employ either the dispersion or the tanks in series model. Particularly for extraction equipment, where usually a large number of stages is used, either model can be successfully applied. It has been shown[ l] that multistage column reactors on the other hand need not have a large number of stages and the tanks in series model is therefore more realistic than the dispersion model. In a multistage column reactor, backmixing between stages is an important variable which influences the residence time distribution behaviour of the system. In establishing mathematical models for a multistage system with backmixing between stages it has hitherto been always assumed that each stage behaves as an ideally mixed vessel (vessel with backmix flow)[2-7]. In this work a model is suggested which considers the general case of non-ideal mixing in each stage and backmixing between stages. Non-ideal mixing in one stage has been mathematically described by several authors [S- 151. The models used (mixed, circulation and dispersion models) are all rather involved and in combination with backmixing between stages would yield very cumbersome expressions. The model
for expressing this non-ideality suggested in our work uses a relatively simple concept which can be easily combined with backmixing between stages. Besides the mathematical formulation of the model and interpretation of the physical aspects of individual parameters, we have included also a method of evaluation of experimental curves and a calculation of fractional conversion for a first order reaction. These results are useful for estimating the relative importance of individual model parameters. 2. DESCRIPTION
OF MODEL
It is assumed that each stage of the reactor consist of a series of ideally mixed vessels with backmixing between adjacent vessels as shown on Fig. 1. Let S be the number of such vessels in each stage and let b be the coefficient of backmixing between these vessels. There is also backmixing between the individual stages of the reactor. Let the number of stages be N and let the coefficient of backmixing between stages be r. The non-ideal mixing within a stage is thus expressed by means of the parameters S and b. The mass balance equations used for describing the process inside a stage are thus formally equivalent to the equations used to describe the process as a whole. This fact is of great impor-
139
and z. NOV~SAD
M. REHAK~VA
The mass balance of tracer in dimensionless form for the first vessel of each stage is vfhbh,, ------
--$$=
(l+r)c,,_,,+bc,-(l+b+r)c,l (la)
where 8 is a dimensionless time 8 = t/SW and t is the mean residence time in one vessel. For an intermediate vessel
i=2,3...(S-1)
(lb)
j=l,2...N. The mass balance for the last vessel of each stage is &2=
(l+b)c,,_,V-trclo+l,-(l+r+b)c~~. (lc)
I
"'SN
Fig. 1. The general mathematical model and notation for mass balance equations
tance since it considerably simplifies the solution of resulting equations. The mathematical model has three parameters: the number of ideally mixed vessels S, the coefficient of backmixing between the vessels b and the coefficient of backmixing between stages r. The total number of stages N is not considered to be a parameter for it is directly given by the number of actual stages. Calculation of the response curve to an instantaneous or pulse tracer injection corresponding to a given set of parameters is performed by setting up mass balance equations for individual vessels. On Fig. 1 the concentrations of tracer leaving a vessel are denoted by a double subscript ij. The first letter indicates the number of the vessel inside a stage and the second 1eLter is the stage number.
Setting up the mass balance equations for all vessels in the system results in N.S linear differential equations. Their solution with the initial condition that c, = 0 when t d 0 and boundary conditions determined by the mass balance for the first stage of the system with a delta function input
1 dell N.S~=
6+&n--
(l+bh
(14
and the last stage of the system
1 &iv myjg-=
(~+~)(cG-I~N-csN)
(W
yields directly the impulse residence time distribution function E(8). For the general case the system of equations cannot be solved analytically. It was therefore solved numerically on the NE 803 computer by using a Runge-KuttaMerson procedure [ 161. 3. PHYSICAL
r
140
INTERPRETATION PARAMETERS
OF MODEL
The coefficient of backmixing between stages is the fraction of the feed which is recirculated
Residence time distribution and fractional conversion for a multistage reactor
between adjacent stages. It therefore has the same significance as in the tanks in series model with backmixing between adjacent tanks. The extent of mixing inside individual stages is expressed by a series of S ideally mixed vessels and a coefficient of backmixing b. It has been verified experimentally [ 161 that the parameter S depends only on the geometry of the system. For a mechanically agitaged multistage column with stage height to dia. ratio h/d = 1, the best fit to experimental curves was obtained for a value of S = 3. It was further shown [ 161 that the coefficient of backmixing b between vessels in a stage depends on the number of revolutions according to the well known expression for the circulation ratio in circulation models
[ 131. Our parameter S seems to be an analogy of van de Vusse’s parameter n, which expresses the diffusivity inside the circulation loop by the number of ideally mixed vessels in series. It also depends only on the geometry of the system and does not depend on the other variables. Our parameter b can be compared with van de Vusse’s circulation ratio r,. In order to substantiate these analogies we calculated several residence time distribution curves according to both models for equal variances. Van de Vusse shows that the variance of his model is
The variance of a series of vessels with backmixing was calculated by Miyauchi and Vermeulen [2] as
b = const. $. In view of the above relations we can consider the model bf a series of vessels with backmixing b,tween vessels which describes the mixing inside a stage to be analogous to the circulation model as shown on Fig. 2. Let us now compare
2_ 1+2b CT2--- S
2b(l+b)
{l-(&r}.
S2
(4)
Equating the variances we obtain the following relation for van de Vusse’s circulation ratio r,,
S(S-n,-22n&)+2n,(b+b2)
(
r, =
l-exp
(
-Sin
9 (3
Sn,(1+2b-S)-2n,(b+b2)(l-exp(-Slny)) the parameters of this model with one of the more recent circulation models of van de Vusse
vb
A value of S = 3, giving the best fit of experimental data[ 161, and a value of n, = 4 as recommended by van de Vusse, was substituted into Eq. (5). For several values of b the corresponding values of r, from this equation were then calculated. On Fig. 3 are shown the curves obtained by calculation from these parameters. The similarity of the general trend of these curves supports the foregoing discussion.
v(l ‘+bl
Fig. 2. Analogy with the circulation model.
’
4. EVALUATION
OF EXPERIMENTAL
DATA
In order to determine the best parameter corresponding to an experimental values residence time distribution curve it is necessary to compare this curve with a theoretical curve calculated from known parameters. As a criterion of agreement we may choose equal variances or 141
M. REHAKOVA
and Z. NOVOSAD
1c E(e)
Fig. 3. Comparison with model of van de Vusse. l-van de Vusse, n,,= 4, r, = 4.357; 2-our model, S = 3, b = 5; 3-van de Vusse, n, = 4, r, = 0.928; 4-our model, S = 3, b = 1.
equal coordinate values of characteristic points on the curve. Using equal variances is not recommended since for its determination great accuracy of the tail part of the residence time distribution curve is required. It is preferable to use the method of characteristic points. A suitable point frequently used is the maximum of the curve. In practical evaluation of curve parameters it is convenient to prepare charts for each number of stages of the measured equipment by numerical solution of Eq. (1). As an example on Fig. 4 is shown the relation between the ordinate of the maximum point E,,,,, and the abscissa 19,~~ for various parameter values b and r (S = 3) for a threestage column which is being used in research work for this is the least number of stages which includes both end stages and an intermediate stage as well. By reading off the coordinates of the maximum point of the experimental residence time distribution curve, the parameters r and b can be directly determined from this figure.
5. FRACTIONAL
CONVERSION ORDER
FOR
A FIRST
REACTION
In order to determine the relative importance of the parameters of the model we shall derive an expression for the fractional conversion of a first order homogeneous reaction as a function of these parameters. For this case Danckwerts [ 171 showed that X= LmE(t)(l-e-KL)
dt
= JbmE(t) dt -jre-ktE(t) = 1 -E(k)
dt (6)
where k is the reaction velocity constant and j?(k) is the Laplace transform of the residence time distribution curve. This can be obtained by solving the set of differential equations (1) by means of the Laplace transformation. This will be shown for a value of S = 3 and any number of stages N. The solution is presented in dimensional time t = BfNS.
142
Residence time distribution and fractional conversion for a multistage reactor
095
090
085
OHJ
Fig. 4. Determination of curve parameters r and b for a three-stage column and S = 3 from the coordinates of the maximum point ,?Z,- and O,,,,.
By a step. by-step elimination of concentration terms we obtain a relation between the Laplace transforms of the tracer concentration in any three subsequent stages (j- l), (~7 and (j+ 1) (see Fig. 1) C33(1+1)-
~q,+Ji;,(,-l,
=
0
where
(7)
where
K=b(l+b)(FG-(l+r)rG+2TGF -Fb(l+b)-TTb(l+II)) -G2T(P-r(l+r)) L=@(b(l+b)-GT) T= (l+fk+b) M=
(~+!J)~(FG--b(l+b)).
I=b(l+b)(3PG-r(l+r)-%(l+b))+G2F(r(l+r)-F) rb2(b(l+b)-GF) J=
(l+b)2(l+r) rb2
F=
(l-t-fk+b+r)
G = (l+fk+2b). Equation (7) is a difference equation which (3) The third boundary condition is the mass must be solved with the following boundary balance for the last stage in Laplace transforms conditions: Q&9 - PC33(~__1) = 0 (8~) (1) at t = 0 cfj = 0 for all i, j. @a) (2) The second boundary condition is the mass where balance for the first stage in Laplace transforms Q= TGF-b(l+b)(t+F) KC331_Lii33+M=
0
(8b) 143
P=
(l+b)2(l+r).
M. REHAK~vA
NOVOSAD
mdz.
Solving Eq. (7) with these boundary conditions we obtain an expression for the Laplace transform of the concentration from anyj-th stage
between stages complicates the mathematics somewhat but this complication is not too serious since the solution has to be performed numerical-
A4(uzJ ul+l (Q-ju1)-ua,ju2N-’ h=
(Q
u~~-Z’U~~--‘)
(Ku~-Lu~~)-
where u1,2are roots of characteristic U 1,2 =
z+A@--4J) 2
(Ku~--Lu~~)
Eq. (7)
*
Substituting j = N into Eq. (9) we obtain the Laplace transform of the tracer concentration from the last stage or the Laplace transform of the residence time distribution curve
E’(k)=
t& =
(Q-j&)
~ ’
(Q~$‘-f’a,~-‘)
ly in any case. To show the relative importance of the parameters of the model we have calculated for several cases the fractional conversion for a homogeneous reaction of first order according to Eq. (6) and (10). The results are presented graphically on Fig. 5 for values of r = 0, 1,5 and reaction numbers 4 = k(fN.9) = 0*9,1+$, 3.6. From the curves it is apparent that for,b > 10
M(u~~u N-‘(Q-Nul)-u,Nu2N-1 (Q-Nu,)) (Kul--~a,~)(Qu,~-Pu,~-')' (QulN-PulN-’ )(Ku~-L~~)-
The fractional conversion can then be calculated by substituting for g(k) from Eq. (10) into Eq. (6).
(9)
(10)
the stages can be considered as being ideally mixed. For lower values of b the non-ideal behaviour in a stage must be taken into &count and the effect of the value of this parameter increases with increasing r. It is further interesting to note that this kind of non-ideality improves reactor performance so that under certain conditions it may be a desirable effect. NOTATION
b
I
0
I
I
lo
20
I b
30
Fig. 5. Fractional conversion X for various reaction numbers
q and parameter values r vs. coefficient of backmixing b for a three-stage column and S = 3. 6. THE RELATIVE
IMPORTANCE
OF MODEL
PARAMETERS
The inclusion of non-ideal mixing within a stage into the multistage model with backmixing 144
coefficient of backmixing between vessels, B/v B flow rate of backmixing between vessels in a stage L3P c dimensionless concentration of tracer E Laplace transform of c of tracer in CU dimensionless concentration the i-th vessel of the j-th stage d stage dia. L time distribution E(@ impulse residence function Laplace transform of E(8) ordinate of the maximum point on the E(8) curve stage height L reaction rate constant for first order reaction t-l
,
Residence time distribution and fractional conversion for a multistage reactor
t f
time tmean residence time in one ideally mixed vessel t V feed flow rate L3t-’ X fractional conversion 6 delta function 6 dimensionless time t/SNf e max abscissa of the maximum point on the E(O) curve
number of revolutions per unit time t-l diffusion parameter of van de Vusse n, N number of stages 4 reaction number, k(tNS) r coefficient of backmixing between stages, R/v R flow rate of backmixing between stages II
LV-’
ru
s
circulation ratio of van de Vusse number of ideally mixed vessels in a stage
REFERENCES [l] NOVOSAD Z., Colln Czech. them. Commun. 1967 32 3842. [2] MIYAUCHI T. and VERMEULEN T., Ind. Engng Chem. Fundls 1963 2 304. [3] PROCHAiXA J. and LANDAU J., Colln Czech. them. Commun. 1963 28 1927. [4] STAINTHORP F. P. and SUDALL N., Trans. Instn them. Engrs. 1964 42T198. [5] RETALLICK W. B., Ind. Engng Chem. Fundls 1965 4 88. [6] BELL R. L. and BABB A. L., Chem. Engng Sci. 1965 24llOOl. [7] MI&A J., Colln Czech. them. Commun. 1967 32 1518. [8] GUTOFF E. B.,A. I. Ch. E. Jll960 6 347. 191 HOVORKA R. B., Ph. D. Thesis, Case Institute of Technology, Ohio 1961. [lo] MARR G. R. and JOHNSON E. F., Chem. Engng Progress; Symp. Ser. 196157 36,109. [ 111 SINCLAIR G. G.,A. I. Ch. E. Jll9617 709. [12] LEVENSPIEL O., Chemical Reaction Engineering, John Wiley 1962. [13] van de VUSSE J. G., Chem. Engng Sci. 1962 17 507. [14] WOLF D. and RESNICK W., Ind. Engng Chem Fundls 1963 2 287. [15] VONCKEN R. M., ROTTE J. W., and HOUTEN A. T., A. 1. Ch. E.-Inst. Chem. Engng Joint Meeting, London 13-17June 1965. [16] REHAKOVA M., Ph.D. Thesis, Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Prague 1966. [17] DANCKWERTS P. V., Chem. Engng Sci. 1953 2 1.
R&u&l On sug&re un modele mathematique pour exprimer la distribution du temps de rksidence dans un systtme a stages multiples avec melange posterieur entre les stages et un mtlangenon-ideal a chaque stage. On tient compte de ce comportement non-ideal en considerant chaque stage comme &ant compose dune s&e de recipients avec mtlange posterieur entre ceux-ci. Les courbes de distribution du temps de residence sont obtenues par une solution nurmbique dun ensemble d’equations differentielles d’tquilibre de masse. On montre comment l’ordonnee et l’abscisse du point maximum de la courbe de distribution du temps de residence peut &tre utilisb pour tvaluer les parametres des courbes experimentales. Une expression analutique est tiree du calcul de la conversion fractionnelle pour un reacteur homogene base sur le modele ci-dessus et l’importance relative des parametres d’un modele individuel est indiquee. Zusammenfassung-Ein mathematisches Model1 wird zur Erklarung der Verweilzeitverteilung in einem Vielstufensystem mit Rllckmischung zwischen den Stufen und nicht-idealer Mischung in jeder Stufe vorgeschlagen. Das nicht ideale Verhalten wird bertlcksichtigt, indem man jede Stufe als eine Reihe von Gefassen mit Riickmischung zwischen ihnen betrachtet. Die Kurven der Verweilzeitverteihmg werden durch numerische Liisung einer Reihe von Massengleichgewicht-DifIerentialgleichungen erhalten. Es wird aufgezeigt, wie die Ordinate und Abzisse am Maximalpunkt der Verweilzeitverteilungskurve dazu verwendet werden kiinnen, die Parameter der experimentellen Kurven auszuwerten. Ein analytischer Ausdruck ftir fraktionelle Umwandlungsrechnung wird tiir einen homogenen Reaktor auf der Grundlage des oben angeflihrten Modells abgeleitet, und die relative Bedeutung der individuellen Modell-Parameter wird eriirtert.
145
C.E.S.-D