Pearson 1959
Transcript of Pearson 1959
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Chemical Engineering S me. 1952, Vol. 10, pp. 281 to 284. Pergsmon Press Ltd. London
Printed in Great Britain
A note on the ( Danckwerts boundary conditions for continuous flow
reactors
J. R. A.
PEARSON
Imperial Chemical Industries Ltd., Akers Research Laboratories, The Frythe, Welwyn, Herts.
Abstract-A mathematical justification is given for the use of the Danckwerts boundary
conditions for continuous flow reactors. It is shown that the apparent indeterminacy, which
DANCE~ERTSesolves intuitively, is caused by the use of a discontinuous coefficient of diffusion.
By treating this as the limit of a cont~uous fusion and imposing eont~uity of the reactant
concentration as the physically relevant boundary condition, the Danekwerts solution is obtained
in the limit.
RBsum&--Lauteur donne une justification mathematiques des conditions aux limites utilisees
par DANCKWERTSans le cas dun reaeteur a Bcoulementcontinu.
I1 montre que lindttermination
apparente resolue ~t~tivement par D~~cxwmzrs est inherente it un coefficient de fusion
discontinu.
I1 con&d&e ce coefficient comme la limite dune fonction continue et il impose une
continuite a la concentration du reactant comme &ant la condition limite physiquement correcte :
la solution de DANCKWERTSest alors obtenue a la limite,
Zusammenfassnng-Die G~~be~n~ngen nach DANCKWERTSfur den kon~uierlich durch-
str6mten Reaktor werden mathematisch gerechtfertigt. Die anscheinende Unbestimmtheit, die
DANCKWERTSntuitiv aufltist, ist durch die Verwendung eines diskontinuierlichen Diffusions-
koeffizienten verursacht. Bebandelt man diesen als Grenzfall einer kontinuierlichen Funktion
und setzt die Stetigkeit der Konzentration des Reaktanten als die physikalisch entscheidende
Grenzbedingung fest, so erhalt man die DANCKWERTS-L~SIJNCm Grenzfall.
1. INTRODUCTION
IN an
oft-quoted paper (Chem. Etgng. Sci.
I953
2
I ,
ANCKWERTS has considered the steady state
ffow of r&&ant through a packed tubular reaction
vessel in terms of a second order ordinary differen-
tial equation (equation 30 in Eoc.
cit. .
This
equation for the steady state concentration, c,
of reactant in a first order reaction supposes that
c is a function of one space variable only, y, the
distance down the tube. The streaming velocity,
21, the rate constant,
k
and the (constant) coeffi-
cient of ~sion, 23, enter as parameters into
the diffusion equation, which may thus be written
= 0-t.
(1)
The boundary condition at the entry to the tube,
y = 0, where the diffusion coefficient discon-
tinuously changes from zero, is obtained by a
consideration of mass balance, and is
UC* = w: -
D .
dc
qj
Y =
0,
(2)
where c* is the concentration in the entering
stream.
A similar relation is obtained for c at the exit
from the tube, y = L, tihere D again changes
discontinuously, but is replaced by DAMXWERTS,
on intuitive grounds, by the stronger condition
de
- = 0
dY
y = L.
Conditions (2) and (3) lead to a unique solution
for c
;
however this solution, as presented in
t D~~c~vvnnrs notation is retained for ease of comparison.
231
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DANCKWERTS paper, appears to rely for its
uniqueness on the acceptance of an intuitive
boundary condition. If only because this boun-
dary condition has not been universally accepted,
it seems desirable to investigate a little more
closely the formal. mathematical implications of
the idealization represented by equation
1)
and
the boundary conditions (2) and (3).
First of all, we observe that in the general
solution to l), using boundary condition (2) at
y = 0 and a similar condition at y =
L, a discon-
tinuity in c, at either y = 0 or y = L, or both, is
necessarily consequent upon the imposed discon-
tinuities in
D. Within the reactor, i.e. where
0 < y