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\markboth{ Gas--solid reaction with porosity change }
{ Ivar Stakgold }
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Nonlinear Differential Equations, \newline
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 247--252\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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Gas--solid reaction with porosity change
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\thanks{ {\em Mathematics Subject Classifications:} 35K57
\hfil\break\indent
{\em Key words:} Reaction-diffusion, Parabolic equation, Porous medium, Gas-solid
reaction
\hfil\break\indent
\copyright 2000 Southwest Texas State University.
\hfil\break\indent Published October 25, 2000. } }
\date{}
\author{ Ivar Stakgold \\[12pt]
{\em Dedicated to Alan Lazer} \\ {\em on his 60th birthday }}
\maketitle
\begin{abstract}
For a gas diffusing through a porous solid and reacting with it isothermally and
irreversibly, the mathematical formulation consists of a nonlinear parabolic PDE
for the gas concentration coupled with an ODE for the solid concentration.
Under the assumption of constant porosity, a fairly complete analysis was
provided by Diaz and Stakgold, [3]. Here some of the results are extended to
the case when the porosity increases as the solid is consumed. In particular,
estimates are given for the time to full conversion of the solid when the
reaction rate is proportional to the product of the gas concentration and a
fractional power of the solid concentration.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\section{Introduction and Preliminary Results}
Consider a gas diffusing through a porous solid and reacting irreversibly and
isothermally with the solid matrix (or some specified component of it). The solid is
being consumed as the reaction proceeds causing an increase in the porosity, that is,
the fraction of the volume available to the gas. Although porosity changes have been
considered in geophysical problems (see [2], for instance), the methods and results do
not seem to apply here. In [3], we dealt with gas--solid reactions while neglecting
porosity changes; we established existence and uniqueness as well as various estimates
of physical interest. In the present article, we do take into account the porosity
changes but we also restrict ourselves to simpler reaction rates, boundary conditions,
and initial conditions than in [3]. We assume that the reaction can be regarded as
distributed throughout the domain $\Omega$ occupied by the porous solid with a rate
(per unit volume) proportional to $CS^m$, where $C$ and $S$ are the
nondimensional gas and solid concentrations, respectively. The positive exponent $m$
can be smaller than 1 in some realistic settings: in the Sohn--Szekely model (see
[4]) for instance, the porous solid consists of a matrix of very small spherical
grains between which the gas diffuses; if the reaction is confined to the grain
surface and is proportional to its area, then the equivalent distributed reaction rate
has $m = 2/3$. An interesting feature of the nonlipschitz case $m < 1$ is that the
solid is fully converted in a {\it finite} time $T$. One of our principal aims is to
provide an estimate for $T$.
Let us now turn to the formulation of the reaction--diffusion problem in an initially
homogeneous porous solid occupying the bounded domain $\Omega$ in $R^n$, when the
gas concentration on the boundary $\partial \Omega$ is maintained at a constant
positive value. We take the diffusivity as constant but allow the porosity to vary
with the solid concentration. After straightforward nondimensionalization and
rescaling of the time variable, the mass balances for the solid and gas concentrations
yield the system
\begin{eqnarray}
&S_t = - S^m C \quad x \in \Omega , \; t > 0 \, ,&\\
&(\epsilon C)_t - \Delta C = - \lambda S^m C ( = \lambda S_t)
\quad x \in \Omega, \; t > 0 \, ,&
\end{eqnarray}
where the positive constant $\lambda$ is Thiele's modulus. The porosity $\epsilon$
is related to $S$ through
\begin{equation}
\epsilon = \epsilon_0 + \epsilon_1 ( 1 - S ) \, ,
\end{equation}
with $\epsilon_0$ and $\epsilon_1$ positive constants. The non--dimensional
concentration $C$ and $S$ satisfy the initial and boundary conditions
\begin{equation}
S(x,0)= 1\, , \; C ( x,0) = C_0 (x) \, , \; C ( \partial \Omega , t ) = 1 \, ,
\end{equation}
where $0 \le C_0 (x) \le 1$. We seek solutions of (1)--(4) with $S \ge 0$, $C \ge
0$.
A few preliminary remarks are in order:
\begin{description}
\item{a)} $S ( \cdot , t )$ is decreasing so that $\epsilon $ increases with time.
\item{b)} If $S ( x_0 , t_0 ) = 0$, then $S ( x_0 , t ) \equiv 0$ for $t \ge
t_0$. (A similar result is not necessarily true for $C$ as gas is being supplied
through the boundary.)
\item{c)} $0 \le C ( x,t) \le 1$. This follows from the maximum principle:
expanding (2), we obtain the inequality
$$
C_t - \frac{1}{\epsilon} \; \Delta C + \frac{\epsilon_t}{\epsilon} \; C \le 0
$$
with $\epsilon_t \ge 0$ and $0 < \epsilon_0 \le \epsilon \le \epsilon_0 +
\epsilon_1$. Thus the coefficient of $C$ is nonnegative and the
coefficient of $\Delta C$ is a bounded positive function. Hence $C$
obeys a uniform parabolic inequality (see [5]) and its maximum occurs on
the parabolic boundary so that $C \le 1$.
\item{d)} On $\partial \Omega$, $C \equiv 1$ so that (1) becomes an
ordinary differential equation for $S ( \partial \Omega, t)$ with solution
\begin{equation}
S = B_m (t) = \left\{ \begin{array}{ll}
[ 1 - ( 1-m) \, t ]_+^{1/1-m} & m \ne 1\\
e^{-t} & m = 1 \end{array} \right.
\end{equation}
where $z_+$ is defined as max $(0,z)$. For $m \ge 1$, $B_m (t) $ is
positive for all $t$ and tends to 0 as $t \to \infty$; for $m < 1$,
$B_m (t)$ decreases until $t = \frac{1}{1-m}$ where it vanishes and remains
zero thereafter (note that $B_m (t)$ is continuously differentiable so that
it is a classical solution of the differential equation).
At interior points, (1) yields
\begin{equation}
S(x,t) = B_m \left( \int_0^t \, C ( x, \tau ) \; d \tau \right)
\end{equation}
which could be used to convert (2) into an integro--differential equation for $C$
alone, but we shall not avail ourselves of that formulation. Let us note that if
$m \ge 1$, $S ( x,t) > 0$ for all $x$ and $t$. For $m < 1$, full conversion
of the solid occurs when the argument of $B_m$ in (6) has reached the value
$\frac{1}{1-m}$ for all $x$; thus we can characterize the time $T$ to full
conversion from
\begin{equation}
\min_{x \in \bar \Omega} \; \int_0^T \; C ( x, \tau ) \; d \tau = \frac{1}{
1-m} \, .
\end{equation}
\item{e)} The steady state for (1)--(4) is $S \equiv 0$, $C \equiv 1$.
\item{f)} We shall not deal here with existence and uniqueness, although the
methods of [3] can be adapted for that purpose. We can also show that $S$
and $C$ approach their respective steady states uniformly on $\bar \Omega$
as $t \to \infty$.
\end{description}
\section{Upper Bound for the Time $T$ to Full \\ Conversion}
Following [6], we introduce the time--integrated difference between the gas
concentration and its steady--state value:
\begin{equation}
\eta ( x,t) = \int_0^t \; [ 1- C ( x, \tau ) ] \; d \tau \, ; \quad \eta_t = 1 -
C \ge 0 \, .
\end{equation}
In terms of $\eta$, (6) reduces to
\begin{equation}
S = B_m ( t - \eta ) \, .
\end{equation}
Integrating (2) with respect to time from 0 to $t$, we obtain
\begin{equation}
\epsilon \eta_t - \Delta \eta = \epsilon_0 ( 1-C_0 ) +
(\epsilon_1 + \lambda) ( 1-S) \, ; \quad
\eta ( x,0) = \eta ( \partial \Omega, t ) = 0 \, .
\end{equation}
Since $\epsilon$ and $S$ are functions of $\eta$ through (9) and (3), (10)
is a scalar PDE for $\eta$. In (10), unlike (2), $\epsilon$ is not
differentiated. From (10) it is clear that $\eta ( x,t)$ increases in time to
the steady state value
\begin{equation}
\eta_\infty (x) = ( \epsilon_1 +\lambda) w (x) + ( \epsilon _0 ) \; z (x) \, ,
\end{equation}
where $w$ and $z$ satisfy the Poisson equations
\begin{eqnarray}
&- \Delta w = 1 \, , \; x \in \Omega \, ; \quad w ( \partial \Omega )= 0&\\
&- \Delta z = 1 - C_0 (x) \, , \; x \in \Omega \, ; \quad z ( \partial
\Omega ) = 0 \, .&
\end{eqnarray}
We observe that $t - \eta ( x,t) = \int\limits_0^t \; C ( x, \tau ) \; d \tau$
is an increasing function of time tending to $\infty$ as $t \to \infty$. We
can therefore rewrite (7) as
\begin{equation}
\min_{x \in \bar \Omega} \; [ T - \eta ( x,T) ] = \frac{1}{1-m} \, ,
\end{equation}
and, hence,
\begin{equation}
T = \frac{1}{1-m} + \max_{x \in \bar \Omega} \, \eta ( x,T) \le \frac{1}{1-m} + (
\epsilon_0 ) \; \Vert z \Vert + ( \lambda + \epsilon_1) \; \Vert w \Vert \, ,
\end{equation}
where $\Vert \; \Vert$ stands for the sup norm.
\section{The Pseudo--Steady--State Problem and a Lower Bound for $T$}
The rescaling which occurs in passing from the original mass balances to the
nondimensional versions (1) and (2) usually yields $\epsilon \ll 1$. It is
therefore common practice among chemical engineers to set $\epsilon =0$ in
(1)--(4). This yields the {\it pseudo--steady--state (or P.S.S.) problem}
\begin{eqnarray}
& \hat S_t = - \hat S^m \hat C \quad x \in \Omega \, , \; t > 0&\\
& - \Delta \hat C = - \lambda \hat S^m \hat C ( = \lambda \hat S_t )
\quad x \in \Omega \, , \; t > 0 &
\end{eqnarray}
with initial and boundary conditions
\begin{equation}
\hat S ( x,0) = 1 \, , \quad \hat C ( \partial \Omega , t ) = 1 \, .
\end{equation}
No initial condition can be imposed on $\hat C$ since $\hat C ( x,0)$ is
determined as the solution of the elliptic equation
\begin{equation}
- \Delta \hat C ( x,0) = - \lambda \hat C ( x,0) \, ; \quad \hat C ( \partial \Omega
, 0 ) = 1 \, .
\end{equation}
The solution of (19) is unique and positive. As in (8) we introduce
\begin{equation}
\hat \eta ( x,t) = \int_0^t \; [ 1 - \hat C ( x, \tau ) ] \; d \tau \, , \quad
\hat \eta_t = 1 - \hat C \ge 0
\end{equation}
so that
\begin{equation}
\hat S = B_m ( t - \hat \eta ) \, .
\end{equation}
Integrating (17) from time 0 to time $t$, we obtain
\begin{equation}
- \Delta \hat \eta = \lambda ( 1 - \hat S ) \, , \quad \hat \eta ( \partial
\Omega , t ) = 0 \, .
\end{equation}
In view of (21), we see that (22) is a scalar elliptic equation for $\hat \eta$.
It is easy to show that, as $t \to \infty$, $\hat S ( x,t)$ decreases
monotonically to zero and $\hat C ( x,t)$ increases monotonically to 1. For
$m < 1$, the time $\hat T$ to full conversion in the PSS model is
characterized by
\[
\min_{x \in \bar \Omega} \; [ \hat T - \hat \eta ( x , \hat T) ] = \frac{1}{1-m}
\, .
\]
Since $\hat S \equiv 0$ for $t \ge \hat T$, we see from (22) that $\hat \eta
( x, \hat T) = \lambda w (x)$, where $w$ is defined from (12). Therefore
\begin{equation}
\hat T = \frac{1}{1-m} + \lambda \; \Vert w \Vert \, ,
\end{equation}
and we have an explicit expression for $\hat T$ in terms of the solution of the
simple Poisson problem (12) about which much is known (see, for instance, [1]).
If, for instance, $\Omega$ is a ball of radius $a$ in $R^n$, then
\[
w = \frac{a^2 - |x|^2}{2n} \quad \mbox{and} \quad \Vert w \Vert = \frac{a^2}{2n}
\, .
\]
Next, we show that $\hat T$ is a lower bound to $T$ if $C_0 (x)$ in (4)
is smaller than $\hat C ( x,0)$ as given by (19). The result will follow
easily if we can prove that $\hat \eta \le \eta$. We shall do so by showing
that $\hat \eta$ is a subsolution to the parabolic problem (10) rewritten as
$A \eta = 0$, where
\begin{equation}
A\eta = \epsilon [ \eta_t - 1] + \epsilon_0 C_0
- \Delta \eta - \lambda [ 1-S ]
\end{equation}
and $S, \epsilon$ are regarded as functions of $\eta$ through (9) and (3).
Since $\hat \eta$ satisfies (22) with $\hat S$ given by (21), the last two
terms on the right side of (24) cancel out for $A \hat \eta$. Hence $A \hat
\eta \le 0$ if $\hat \eta_t \le 1 - C_0$ or, equivalently, if $C_0 (x) \le
\hat C ( x,t)$. This latter condition holds if
\begin{equation}
C_0 (x) \le \hat C ( x,0)
\end{equation}
because $\hat C ( x,t)$ is monotonically increasing in time. With $\hat \eta$
satisfying the same initial and boundary conditions as $\eta$, it follows that
$\hat \eta$ is a subsolution of (24). Appealing to uniqueness and the theory of
monotone operators, we conclude that $\hat \eta \le \eta$. Hence $t - \hat
\eta ( x,t) \ge t - \eta ( x,t)$ and since both $t - \hat \eta$ and $t -
\eta$ are increasing in $t$, $t - \hat \eta$ will reach $\frac{1}{1-m}$
throughout $\bar \Omega$ before $t - \eta$. We therefore conclude that, for
$C_0 (x)$ satisfying (25),
\begin{equation}
\hat T \le T \, .
\end{equation}
Combining (26) and (15) we obtain the bounds
\begin{equation}
\hat T \le T \le \hat T + \epsilon_0 \; \Vert z \Vert + \epsilon_1 \; \Vert w
\Vert
\end{equation}
where $\hat T$ is given explicitly by (23) and $z$ is the solution of (13).
Note that $\Vert z \Vert \le \Vert w \Vert$ so that we can replace $\Vert z
\Vert$ by $\Vert w \Vert$ in the upper bound in (27).
\begin{thebibliography}{0}
\bibitem{1.} Bandle, C., {\it Isoperimetric Inequalities and Their
Applications}, Pitman, 1980.
\bibitem{2.} Chadam, J., Chen, X., Comparini, E., and Ricci, R., Travelling wave
solutions of a reaction--infiltration problem and a related free boundary
problem, {\it Euro. J. Appl. Math.} {\bf 5} (1994), pp. 255--265.
\bibitem{3.} Diaz, J. I., and Stakgold, I., Mathematical aspects of the combustion
of a solid by a distributed isothermal gas reaction, {\it SIAM J. Math. Anal.}
{\bf 26} (1995), pp. 305--328.
\bibitem{4.} Froment, G. F. and Bischoff, K. B., {\it Chemical Reactor Analysis
and Design}, Wiley, 1990.
\bibitem{5.} Protter, M. H. and Weinberger, H., {\it Maximum Principles in
Differential Equations}, Prentice--Hall, 1967.
\bibitem{6.} Stakgold, I. and McNabb, A., Conversion estimates for gas--solid
reactions, {\it Math. Modelling} {\bf 5} (1984), pp. 325--330.
\end{thebibliography}
\noindent{\sc Ivar Stakgold }\\
Department of Mathematics, University of Delaware \\
Newark, DE 19716 \\
e-mail: stakgold@math.udel.edu
\end{document}