Accuracy of the pseudo steady state approximation for moving boundary diffusion problems

Y=Yo
d2
43
R2 z Yl + **e
Y=Yo+
where t, is a characteristic time. If these are substituted into equations (4), (5) and (6) and coefficients of R2/tCD equated, the following sets of equations are obtained, co = A, + B,
solid-fluid chemical reactions, the solid often consists of an unreacted core surrounded by an envelope of reacted material [l-3]. The interface between reacted and unreacted solid moves inward from the particle surface until all of the solid is reacted. The fluid must thus diffuse through the reacted solid region to reach the interface and react. The rigorous mathematical treatment would consist of solving a non-steady state diffusion problem with a moving boundary. Similar equations occur in melting and freezing problems in heat conduction, and it is known that usually only problems involving plane geometry have analytical solutions [4]. Since most reacting systems consist of approximately spherical particles, these solutions are not directly applicable. For this reason, various approximation schemes are necessary for the calculations. One of the most common of these involves the assumption that the rate of movement of the interface is very much slower than the rate of diffusion. In this case, the interface can be taken to be stationary at any time and a steady-state diffusion problem solved to 6nd the concentration profile. The mass flux as found from this expression is then equated to the rate of disappearance of the unreacted solid in order to determine the location of the interface as a function of time. YAGI and KUNII [3], NARSMBAN [5] and LEVBNSPIEL [2] have recently given examples of the use of this method.
Department BISCHOFF
of Chemical Engineering, The University of Texas
(Received 1 October 1962; in revisedform 9 May 1963)
Abstract-The problem of heterogeneous solid-fluid chemical reactions with a moving boundary is A better approximation to the commonly treated by means of the pseudo steady state approximation. exact solution has been derived in order to estimate the accuracy of the pseudo steady state method. It is found that for solid-gas systems the pseudo steady state approximation is valid but for solidliquid systems it may be in error.
i a --r2_=_r2ar
acboundary conditions, c(R t) = co, c(r,, r) = 0, (2)
ac (I,, 0 ps dr, __% =.Ddt
An approximate solution can be obtained by the method used by PEKERIS and SLICHTER for ice formation on a cylindrical pipe [7]. The steady state solution of equation (1) is of the form c=A+T
BR A, B constant
711
IL B. BISCHOFF
This leads one to try the following more general solution,
c = A(t) + -
Equations (8) can be solved, and an expression for Y. found,
dy,+ 1 2y,
dYo
- 1 --y1=-2-YO YO- 1
coDt,
1
3R P~YO
(11)
-R2 !+ dv = _ D
D dt
Using the condition that at t = 0, y. = 1, Y, = 0 gives a solution
Yl
T+~D
where y = r,/R. Equations (4), (5), (6), are three simultaneous ordinary differential equations for y(t), A(t) and B(t). Approximate solutions sufficient for our purposes can be obtained by using
Chemical Engineering Science, 1963, Vol. 18, pp. 711-713. Pergamon Press Ltd., Oxford.
Printed in Great Britain.
Accuracy of the pseudo steady state approximation for moving boundary diffusion problems K. B.
IN HETEROGENOUS
A problem similar to the one just discussed was studied by IQRKALDY [6] who considered the case of diffusion in a system with infinite boundaries. He showed that the pseudo steady state approximation is not valid for the infinite boundary problem. It was also stated that for finite boundaries of characteristic length 1, an approximate steady state is reached after “diffusion time,” 12/D. The purpose of this paper is to approximately solve the full problem in order to get an estimate of the error involved in the pseudo steady state method. We shall assume constant density, no gas-phase resistance, and very rapid chemical reaction rate, although more complex cases could be treated if desired. The diffusion equation becomes for this case [4],
B(t)R + r2A(t) + 3rRB(t) + . .. r 60
(3)
COD
R2p,
t = &(l - 3yg + 2yZ)
(IO)
where the dots denote a time derivative. It can be shown by direct substitution that equation (3) satisfies equation (1). Using equation (3) in the boundary conditions, equation (2), c,=a+B+$(A+38) O=A+y+6D D R’ (yV + 3yB) R” (2yA + 3B) (4) (5) (6)