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In steady state diffusion, concentration of mass does not change over time. However, concentration can change with position, and this gradient drives the transfer of mass. We can consider a few different situations of steady-state diffusion: in a slab, in a composite slab, and in the presence of a chemical reaction.
Steady State Diffusion in a Slab
Concentration Profile:
where
Steady State Diffusion in a Composite Slab
With a composite slab, convection occurs on either side of the slab while diffusion occurs within the slab. This means mass transfer is occuring between two phases at the surface. The concentrations at the interface are not equal. For example, the surface concentration is different in liquid vs in the solid.
Mass flow can be calculated by concentration differences and mass transfer resistances:
where
Steady State Diffusion in a Slab with a Chemical Reaction
Chemical reactions can occur during diffusion. This produces a decay with time. Consequently, the surface flux increases while the concentration gradient gets higher.
Concentration Profile:
For thick material (semi-infinite), the limit as L approaches \( \infty \) is taken:
where
In unsteady state diffusion, also called transient diffusion, concentration changes over time. There are three situations of unsteady state diffusion that will be examined here. First, characteristic length and the mass transfer Biot number will need to be defined.
The Characteristic Length (\( L \)) of an object is its path of least resistance. In other words, it is the shortest distance heat or mass must move through the object.
| Geometry | Characteristic Length (\( L \)) |
|---|---|
| Infinite Slab | \( \frac{1}{2} \) thickness |
| Long Cylinder | Radius |
| Sphere | Radius |
Table 2: Characteristic length for slab, cylindrical, and spherical geometries.
The mass transfer Biot number (\( Bi_m \)) is used to determine if the internal resistance within an object is negligible. If it is negligible, this means there is very little concentration variation within the solid. It is calculated as a fraction of internal and external resistance:
where
If \( Bi_m < 0.1 \), the internal resistance can be ignored, and no spatial variation in concentration can be assumed. In other words, concentration changes only as a function of time. If \( Bi_m > 0.1 \), the internal resistance cannot be ignored due to spatial variation in concentration within the solid.
If concentration does not change with position and only changes with time, a lumped parameter approximation can be used. When internal resistance is very low compared to the external resistance, the total resistance can be assumed to be external resistance and the concentration will not vary inside the object.
To determine if there is no spatial variation in concentration, use the mass transfer Biot Number!
For mass transfer, the lumped parameter approach is commonly used to describe drying. For example, moisture can evaporate from the surface of a wet solid and be removed by air flowing over it. If water diffuses quickly inside the solid, the moisture concentration inside the solid can be treated as spatially uniform.
A mass balance on the water content in the solid gives:
where
When the solid surface has free water, the surface concentration is constant. In this case, moisture changes linearly with time:
When the solid surface does not have free water, the surface concentration is no longer constant. In this case, the drying rate decreases over time, and the moisture content follows an exponential decay.
If the internal resistance is significant, concentration variation inside cannot be ignored. There are multiple cases for unsteady diffusion when concentration changes with both position and time. For a simple slab geometry where concentration changes with position, a series solution can be used.
Use the mass transfer Biot Number to determine if internal resistance is significant!
For a slab with concentration varying along its thickness, the governing equation is:
This applies when there is no bulk flow and no chemical reaction.
Series Solution:
You can solve the series solution two different ways: using more than one term of the series solution or using a Heisler Chart.
Heisler Charts are used to describe the relationship between the concentration, position, and time variables. They plot the 1st term of the series solution and are used for the long-time approximation.
The Fourier Number is used to determine long times:
where
In order to use the Heisler Chart your system must satisfy the following:
If \( F_o < 0.2 \), more than one term of the series solution may be needed.
Learn More!
To learn how to use the Heisler Chart, visit the Reference Library & Resources Page!
The second method to analyze systems with spatial and time variation of concentration \( (Bi_m > 0.1) \) is for short time and/or thick material. This can be solved with the semi-infinite region solution.
A semi-infinite region is an idealized geometry that extends to infinity in two directions and has a single identifiable surface in the other direction. This approximation is useful for practical situations of mass transfer in thin materials over shorter time or thick materials over longer time.
The material can be approximated as semi-infinite when:
where
Semi-infinite Region Solution:
where
For Reference!
A chart with the error function (erf) is provided on the formula sheet found on the Reference Library & Resources Page!
| Situation | Biot Number | Thickness / Time Condition | Solution Equation |
|---|---|---|---|
| Lumped | \( Bi_m < 0.1 \) | N/A | \( w-w_i = -\frac{h_mA}{m_s}(c_s-c_{\infty})t \) |
| Series | \( Bi_m > 0.1 \) | Finite geometry; \( F_o > 0.2 \) for Heisler Chart | \( \frac{c-c_s}{c_i-c_s} =\sum_{n=0}^{\infty} \frac{4(-1)^n}{(2n+1)\pi} \cos\left(\frac{(2n+1)\pi x}{2L}\right)\exp\left[-\left(\frac{(2n+1)\pi}{2}\right)^2 \frac{D_{AB}t}{L^2}\right] \) |
| Semi-Infinite | \( Bi_m > 0.1 \) | \( L \geq 4\sqrt{D_{AB}t} \) | \( \frac{c-c_i}{c_s-c_i} = 1 - erf\left[\frac{x}{2\sqrt{D_{AB}t}}\right] \) |