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A unified treatment of the inlet boundary condition for dispersive flow models
Shorter Communications
250 Chemical Engiw&g
Scimcg
1976, VoL 31, pp. 250-252.
Pergamon Pnsa.
Printed in Great Britain
A unified treatment of the inlet boundary condition for dispersiveflow models (Received 4 June 1975;acceptedinrevisedfor The justification of boundary conditions for a tubular reactor requires some particular care in the case where the reactor model includes a dispersion term to account for axial mixing:
in which: 4=exp[-l[h($-u)]dy}
$ (Llg)-u+o.
=$exp
Following the pioneer work of Danckwertz[l], the boundary condition at the inlet position was examined by Pearson[2], Bischoff[3], Wehner and Wilhelm[rl], and Kershenbaum and Perkins [S].The consensus is that a proper condition at y = 0 is: dC -%=I@,-C);
17September 1975)
y=o+.
(2)
However, as detailed in Table 1, the underlying assumptions that lead to this result are quite varied and appear in some aspects to be inconsistent. In their recent review of the subject Karantb and Hughes [6] conclude that “a completely satisfactory solution . . . has not emerged as yet”. It is the purpose of this communication to provide a unified treatment that places the prior (sometimes divergent) studies into a single context. With the understanding that the dispersion coefficient is a function of position, but can never be identically zero over any finite interval, eqn (1) can also be written as (3)
[I
;dy
(5)
I
In order to focus attention on the inlet position it is convenient to identify as separate adjacent regions: (i) an inlet section from effectively infinite distance at y = -m to the reactor entrance at y = O-, (ii) a transition section within the reactor near the entrance position from y = 0’ to y = (l/A), (ii) the remainder of the reactor length, its main section from y = (l/A) to the exit at y = L-, and (iv) the effluent section that extends from the exit position at y = L + to the infinite distance at y = m. This notation and procedure is to this point essentially that of Pearson[2], but departs from his in what follows by not demanding a linear dependence of the dispersion coefficient on position within the transition region or a zero value of dispersion at any position. The model to be developed differs also from that used by Kershenbaum and Perkins[S] in that it does not necessarily assume an infinite dispersion at any point. Instead, it only calls for variation of dispersion throughout the transition region, such that continuity of D willbe preserved even in the limit as A --)m and the transition region shrinks to zero thickness. It is not necessary in what follows to fix any particular functional form for this B(y), but it may be heuristically useful to think of it as, for example, of the form:
or, in integral form:
Table 1. Characteristics of various models
Bischcff
Wehner and Wilhelm (1965)
Defined
Kershenbaum and Perkins (1974)
_
Yes
llor cornp1ctc1y matched
Ii0
Yea
NO
251
Shorter Communications where Di is the dispersion coefficient in the inlet section, and D applies in the main reactor section. Equation (6) will reduce to 9 = Di at y = 0 and D = D at y = (l/A), as sketched in Fig. 1.The function is continuous and has continuous tirst derivative over the entire interval of interest as well as at its end points. Related functions with continuous derivatives of all orders are also available[7] but are superfluous for the purposes of this discussion. In this connection it may be noted (as in Table 1) that the particular dispersion coefficient profiles used by Pearson and by Kershenbaum and Perkins are incompletely matched at the ends of the transition section.
continuity is a necessary consequence of continuity in the dispersion coe5cient, and furthermore that the diierent continuity results suggested by Pearson and by Kirshenbaum and Perkins respectively, result from limiting cases of the more general treatment. In summary, it is perhaps worth emphasizing that in spite of superficial appearances, the boundary condition eqn (2) does not necessarily imply discontinuity in either concentration or flux. The steady state mass conservation at each location assures that: = uC(O+)- D f!-
Yldy
(10)
o+
as well as: uC(O-)-Di
Fig. 1. A sketch of eqn (6). To demonstrate that the foregoing arguments are sufficient to establish continuity of the concentration at the inlet, define AC as the difference between the values of C(y) at the respective ends of the transition region. From eqn (4) this becomes:
(11)
Equation (10) reduces to eqn (2) for any 4 provided that (dC/dy)(-m) =0 in the inlet region; it should be noted that Co= C(-m) and C, # C(O-), even though the transition may in a practical case occur over a very short interval. It may further be deduced from eqn (10)that continuity of concentration at the inlet, C(O-)= C(O’), only permits continuity of the gradient at the inlet if D(0’) = D,. That this is in fact not the case has already been noted from eqn (6) which yielded D(0’) = D in the limit as y =(l/A)+O. The gradient is therefore discontinuous at the reactor entrance. These various points are summarized schematically as Fig. 2, consistent with the formulation given here as well
AC=I.“*+[j-&dy]dy+;. Since AC +O for A +m so long as D+ 0 for any usual r(C), it may be concluded that the concentration variation is continuous at the inlet position as the transition region shrinks to zero size. To reconcile this result with the earlier literature, it may be noted that Kershenbaum and Perkins’ finding is consistent, but it is not necessary to restrict the conclusion to systems showing infinite dispersion coefficients at the inlet. With regard to the discontinuity in concentration that is part of Pearson’s model, it is in the first instance a consequence of singular behavior arising from the particular choice of D, = 0. The same conclusion arises, however, if one starts from the presumption that Di # 0, but with the linear transition section profile: 9 = Di t (D - Di)Ay.
I
I
I
I l/A
0
I
L
Y
Fig. 2(a). Concentration profile for (l/L) < A < m.The concentration and its derivative are both continuous at y = 0.
(8)
This last point is most easily demonstrated from the solution to eqn (1) in the transition region where eqn (8) applies. For first order kinetics, for example, the solution may be written in terms of fractional order Bessel Functions: C = y’B’Z[a1~(2Ky”‘2) t bZ_,(2Ky”n)]
-0D
Fig. 2(b). Concentration profile with A + m. The concentration is continuous at y = 0, but its derivative is not.
(9) as those presented earlier by Wehner and Wilhelm, and Bischoff. The approach taken in this communication can readily be extended to the transient multiple reaction system, and the limiting behavior of a plug-flow tubular reactor can also be demonstrated.
where B=A(DU-4) k K=A(D-Di)
4
Acknowledgement-This research was supported by a grant from the National Science Foundation.
Y’=YfA(D-Di) CHA Y. CHOI D. D. PERLMUTTER
k = first order rate constant.
This function is discontinuous at y = -D,/A(D - D,) whether or not D, = 0. In essence the discontinuity is built into the linear assumption, and the location of the discontinuity approaches y = 0 as A +m. Pearson’s is the particular result for D, = 0. The formulation suggested here is also consistent with the models of Bischoff, and Wehner and Wilhelm,which are both built on the assumption of concentration continuity. The model developed in this communication demonstrates that concentration
Department of Chemical and Biochemical University of Pennsylvania Philadelphia PA 19174, U.S.A.
NOTATION
a, b
arbitrary constants
Engineering
252
Shorter Communications
A reciprocal of width of the inlet transition region C concentration D dispersion coefficient in a reactor 0, dispersion coefficient in the inlet section 9 dispersion coefficient in the transition region In modified Bessel function of the first kind of fractional order B k constant of integration or first order rate constant L reactor length r reaction rate u linear flow velocity
REFERENCES Ul Danckwertz P. V., Chem. Engng Sci. 19532 1. [2] Pearson J. R. A., Chem. En&-Sci. 195910281. 131Bischoff K. B.. Chem. Enpna Sci. 1%1 16 131. i4j Wehner J. F. and WilhelmR. ii., Chem. Engng Sei. 1%56 89. [5] Kershenbaum L. and Perkins J. D., Chem. Engng Sci. 197429 [6] F&tth
N. G. and Hughes R., Catalysis Rev. 1974 9(2)
[7j Milin S. G., hfathemnticul Physics, An Advanced Course. North Holland, Amsterdam 1970.
250 Chemical Engiw&g
Scimcg
1976, VoL 31, pp. 250-252.
Pergamon Pnsa.
Printed in Great Britain
A unified treatment of the inlet boundary condition for dispersiveflow models (Received 4 June 1975;acceptedinrevisedfor The justification of boundary conditions for a tubular reactor requires some particular care in the case where the reactor model includes a dispersion term to account for axial mixing:
in which: 4=exp[-l[h($-u)]dy}
$ (Llg)-u+o.
=$exp
Following the pioneer work of Danckwertz[l], the boundary condition at the inlet position was examined by Pearson[2], Bischoff[3], Wehner and Wilhelm[rl], and Kershenbaum and Perkins [S].The consensus is that a proper condition at y = 0 is: dC -%=I@,-C);
17September 1975)
y=o+.
(2)
However, as detailed in Table 1, the underlying assumptions that lead to this result are quite varied and appear in some aspects to be inconsistent. In their recent review of the subject Karantb and Hughes [6] conclude that “a completely satisfactory solution . . . has not emerged as yet”. It is the purpose of this communication to provide a unified treatment that places the prior (sometimes divergent) studies into a single context. With the understanding that the dispersion coefficient is a function of position, but can never be identically zero over any finite interval, eqn (1) can also be written as (3)
[I
;dy
(5)
I
In order to focus attention on the inlet position it is convenient to identify as separate adjacent regions: (i) an inlet section from effectively infinite distance at y = -m to the reactor entrance at y = O-, (ii) a transition section within the reactor near the entrance position from y = 0’ to y = (l/A), (ii) the remainder of the reactor length, its main section from y = (l/A) to the exit at y = L-, and (iv) the effluent section that extends from the exit position at y = L + to the infinite distance at y = m. This notation and procedure is to this point essentially that of Pearson[2], but departs from his in what follows by not demanding a linear dependence of the dispersion coefficient on position within the transition region or a zero value of dispersion at any position. The model to be developed differs also from that used by Kershenbaum and Perkins[S] in that it does not necessarily assume an infinite dispersion at any point. Instead, it only calls for variation of dispersion throughout the transition region, such that continuity of D willbe preserved even in the limit as A --)m and the transition region shrinks to zero thickness. It is not necessary in what follows to fix any particular functional form for this B(y), but it may be heuristically useful to think of it as, for example, of the form:
or, in integral form:
Table 1. Characteristics of various models
Bischcff
Wehner and Wilhelm (1965)
Defined
Kershenbaum and Perkins (1974)
_
Yes
llor cornp1ctc1y matched
Ii0
Yea
NO
251
Shorter Communications where Di is the dispersion coefficient in the inlet section, and D applies in the main reactor section. Equation (6) will reduce to 9 = Di at y = 0 and D = D at y = (l/A), as sketched in Fig. 1.The function is continuous and has continuous tirst derivative over the entire interval of interest as well as at its end points. Related functions with continuous derivatives of all orders are also available[7] but are superfluous for the purposes of this discussion. In this connection it may be noted (as in Table 1) that the particular dispersion coefficient profiles used by Pearson and by Kershenbaum and Perkins are incompletely matched at the ends of the transition section.
continuity is a necessary consequence of continuity in the dispersion coe5cient, and furthermore that the diierent continuity results suggested by Pearson and by Kirshenbaum and Perkins respectively, result from limiting cases of the more general treatment. In summary, it is perhaps worth emphasizing that in spite of superficial appearances, the boundary condition eqn (2) does not necessarily imply discontinuity in either concentration or flux. The steady state mass conservation at each location assures that: = uC(O+)- D f!-
Yldy
(10)
o+
as well as: uC(O-)-Di
Fig. 1. A sketch of eqn (6). To demonstrate that the foregoing arguments are sufficient to establish continuity of the concentration at the inlet, define AC as the difference between the values of C(y) at the respective ends of the transition region. From eqn (4) this becomes:
(11)
Equation (10) reduces to eqn (2) for any 4 provided that (dC/dy)(-m) =0 in the inlet region; it should be noted that Co= C(-m) and C, # C(O-), even though the transition may in a practical case occur over a very short interval. It may further be deduced from eqn (10)that continuity of concentration at the inlet, C(O-)= C(O’), only permits continuity of the gradient at the inlet if D(0’) = D,. That this is in fact not the case has already been noted from eqn (6) which yielded D(0’) = D in the limit as y =(l/A)+O. The gradient is therefore discontinuous at the reactor entrance. These various points are summarized schematically as Fig. 2, consistent with the formulation given here as well
AC=I.“*+[j-&dy]dy+;. Since AC +O for A +m so long as D+ 0 for any usual r(C), it may be concluded that the concentration variation is continuous at the inlet position as the transition region shrinks to zero size. To reconcile this result with the earlier literature, it may be noted that Kershenbaum and Perkins’ finding is consistent, but it is not necessary to restrict the conclusion to systems showing infinite dispersion coefficients at the inlet. With regard to the discontinuity in concentration that is part of Pearson’s model, it is in the first instance a consequence of singular behavior arising from the particular choice of D, = 0. The same conclusion arises, however, if one starts from the presumption that Di # 0, but with the linear transition section profile: 9 = Di t (D - Di)Ay.
I
I
I
I l/A
0
I
L
Y
Fig. 2(a). Concentration profile for (l/L) < A < m.The concentration and its derivative are both continuous at y = 0.
(8)
This last point is most easily demonstrated from the solution to eqn (1) in the transition region where eqn (8) applies. For first order kinetics, for example, the solution may be written in terms of fractional order Bessel Functions: C = y’B’Z[a1~(2Ky”‘2) t bZ_,(2Ky”n)]
-0D
Fig. 2(b). Concentration profile with A + m. The concentration is continuous at y = 0, but its derivative is not.
(9) as those presented earlier by Wehner and Wilhelm, and Bischoff. The approach taken in this communication can readily be extended to the transient multiple reaction system, and the limiting behavior of a plug-flow tubular reactor can also be demonstrated.
where B=A(DU-4) k K=A(D-Di)
4
Acknowledgement-This research was supported by a grant from the National Science Foundation.
Y’=YfA(D-Di) CHA Y. CHOI D. D. PERLMUTTER
k = first order rate constant.
This function is discontinuous at y = -D,/A(D - D,) whether or not D, = 0. In essence the discontinuity is built into the linear assumption, and the location of the discontinuity approaches y = 0 as A +m. Pearson’s is the particular result for D, = 0. The formulation suggested here is also consistent with the models of Bischoff, and Wehner and Wilhelm,which are both built on the assumption of concentration continuity. The model developed in this communication demonstrates that concentration
Department of Chemical and Biochemical University of Pennsylvania Philadelphia PA 19174, U.S.A.
NOTATION
a, b
arbitrary constants
Engineering
252
Shorter Communications
A reciprocal of width of the inlet transition region C concentration D dispersion coefficient in a reactor 0, dispersion coefficient in the inlet section 9 dispersion coefficient in the transition region In modified Bessel function of the first kind of fractional order B k constant of integration or first order rate constant L reactor length r reaction rate u linear flow velocity
REFERENCES Ul Danckwertz P. V., Chem. Engng Sci. 19532 1. [2] Pearson J. R. A., Chem. En&-Sci. 195910281. 131Bischoff K. B.. Chem. Enpna Sci. 1%1 16 131. i4j Wehner J. F. and WilhelmR. ii., Chem. Engng Sei. 1%56 89. [5] Kershenbaum L. and Perkins J. D., Chem. Engng Sci. 197429 [6] F&tth
N. G. and Hughes R., Catalysis Rev. 1974 9(2)
[7j Milin S. G., hfathemnticul Physics, An Advanced Course. North Holland, Amsterdam 1970.