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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