P.C. Chau (UCSD, 1999)
Chapter 4 Study Guide
The topic is transient diffusion. However, the focus in not on solving partial differential
equations. The key is to learn now to interpret a given solution and to analyze limiting cases. This
is where most people have much trouble with. This is a fairly level play field whether you have
taken a course on PDE or not.
What follows are some of the key points. Further details are in our notes on combination of
variables and separation of variables. Read them before you tackle the text. When you read the text,
keep in mind that the writing is really on showing us how we may approach and solve different
problems.
• The four limiting cases are the "short" and "long" contact times and when Bi –> 0 or Bi –> ∞.
For more general circumstances, we may use the Gurney-Lurie charts, which can also handle
the case of Bi –> ∞. The challenge and the difficulty (or you say real pain) is to learn how to
assess the physics to see which assumption is applicable.
• For "short contact" time at an interface, we replace one of the boundary conditions by one at
infinity in such a way that we can apply the combination of variables to solve the problem.
From the solution, we can calculate the transient and time averaged flux at the interface.
Text relevant development: Example 4.2.2, middle of p. 131 to p. 135, (4.2.159) of Example
4.2.4, Example 4.2.5, Example 4.2.6.
• For diffusion problems of "long" duration, we make use of the large time asymptotes of an
analytical solution. Roughly speaking, we take one of those solutions from say, Crank, and
retain only the leading term in the Fourier series.
Text relevant development: pp. 121-122 of Example 4.2.1, (4.2.158) of Example 4.2.4,
• When we make an assumption, we always need to verify that it is valid. We either do a test
based on the data of a problem, or we check the validity of the assumption after we have
obtained the solution. Some of the analyses in pp. 135-137 do just that.
• To use the Gurney-Lurie charts, there are two items that we need to understand: (1) the initial
and boundary conditions that form the basis of the charts, and (2) the choice of a proper
dimensionless concentration Y which depends very much on the problem at hand. For planar
geometry, we also need to be careful with the choice of the length scale.
Text relevant development: Example 4.2.3, Example 4.3.1-3.
• The key to Sections 4.3 and 4.4 is the two limiting cases of Bi –> 0 and Bi –> ∞. This is a
brief summary of the underlying equations.
C
L
Solid
In general, we need to solve the problem of diffusion
within a solid:
Fluid
s
C
∂Cs
∂2 Cs
= D sAB 2
∂t
∂z
f
C∞
z=0
z=L
Note: the linear concentration within
the fluid boundary layer is simply a
schematic depiction of the mass
transfer coefficient model.
where the superscript "s" is used to denote quantities within
the solid, including the diffusion coefficient. The
concentration CA is shorthanded to C.
The symmetry boundary condition applies at the
center line (z = 0). At the solid-fluid interface (z = L), we
have the phase equilibrium condition
α CsL = CfL
1
P.C. Chau (UCSD, 1999)
and the flux boundary condition
– D sAB
∂Cs
∂z
L
= k c (CfL – Cf∞)
where we use a mass transfer coefficient model to describe the external mass transport. As written,
the unit of kc must be [cm/s in cgs units]. Since the solid diffusion problem solves for Cs, we
substitute for the fluid concentration with the phase equilibrium condition, and the boundary
condition becomes
– D sAB
∂Cs
∂z
L
1 Cf ) , where we can further define k = αk
= αk c (CsL – α
c
∞
If we define a dimensionless length x = z/L, we can put the condition in terms of the Biot number:
∂Cs
– B1
i ∂x
x=1
1 Cf ,
= CsL – α
∞
where Bi =
αk c L
D sAB
The key is to note that the Biot number compares the rate of external convective mass
transport to the rate of internal diffusion. When Bi becomes very large, the term on the LHS drops
out, and there is no external fluid phase concentration difference. When Bi becomes very small, the
internal concentration gradient becomes negligible.1 These two limiting cases are summarized in
the schematic diagram below.
CL
CL
Bi —> 0
Bi —> ∞
s
s
C
C
f
C∞
f
C∞
The two extreme cases of external mas transport. Left panel: When Bi
approaches zero, the mass transport resistance resides in the fluid phase. The
mass transport resistance in the solid is negligible, and the internal
concentration is uniform. In other words, we do not need to solve the diffusion
problem within the solid. Right panel: When Bi approaches infinity, the external
mass transfer resistance is negligible. The concentration in the fluid at the
surface is the same as in the bulk fluid.
• When is the Biot small or large enough to apply the limiting approximations? The analysis in
Section 4.4 is meant to establish that when Bi is less than approximately 0.2, the solution
may approach that of Bi = 0. From the Gurney-Lurie charts, the solutions of cases when Bi is
larger than 2 approach that of Bi = ∞. Thus in general, we may use the negligible internal
resistance model when Bi ≤ O(0.1), and we may use the negligible internal resistance model
when Bi ≥ O(10).
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This is how we may argue that the internal diffusion resistance is negligible: If there is a
finite mass transport rate, and if the quantities of the boundary condition remain "well behaved,"
then when Bi becomes very small, so must the gradient ∂Cs/∂x at x = 1. Note that the largest
gradient is at the surface. Recall that the concentration gradient is zero by symmetry at the center.
In addition, you may also note that as written, the concentration remains dimensional in the
boundary condition. Proper choice of a dimensional concentration Y should turn this condition into
the form used by the Gurney-Lurie charts. [More in our notes on separation of variables.]
2