Second Fick's law with nonconstant diffusion coefficient

In summary, a modified version of second Fick's law in nondimensional spherical coordinates can be solved by converting it to a set of ODEs and solving them numerically using an automatic integration package. It is not possible to find an analytic solution and there is no known user-friendly tool for numerical solution of this type of PDE.
  • #1
mabiondi
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Hello everybody.
I should solve a modified version of second Fick's law in nondimensional spherical coordinates; t is the time and rho [0,1] the nondimensional radius. In this equation the diffusion coefficient is vraiable with t and rho.

The equation is the following:

dC/dt = (1/rho^2)*d/d(rho)(rho^2*D(rho,t)*dC/d(rho))

Initial conditions:
C(0,rho) = a*rho^4 + b*rho^2 -(a + b);

Boundary conditions:
dC(t,0)/d(rho) = 0
C(t,1) = 0

Is it possible to solve such an equation by Laplace transforms?
Alternatively, is there a user friendly tool for numerical solution of a PDE like this?

Thanks in advance, cheers.
Marco
 
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  • #2
I don't think an analytic solution exists. It would have to be solved numerically. I would just convert it to a set of ODE's using the method of lines, and then solve the equations numerically in FORTRAN using an automatic (stiff) integration package.

Chet
 
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Likes mabiondi
  • #3
Thanks a lot for the suggestion, I really appreciate it. I'll try to develop a routine using MOL.
Cheers,
Marco
 

1. What is Second Fick's law with nonconstant diffusion coefficient?

Second Fick's law with nonconstant diffusion coefficient is a mathematical equation that describes the rate of diffusion of a substance through a medium with a varying diffusion coefficient. It is an extension of the original Fick's law, which assumes a constant diffusion coefficient.

2. How is Second Fick's law with nonconstant diffusion coefficient different from the original Fick's law?

The main difference between Second Fick's law and the original Fick's law is the inclusion of a nonconstant diffusion coefficient. This accounts for situations where the diffusion coefficient may change over time or in different regions of the medium, such as in a heterogeneous medium.

3. What are some real-world applications of Second Fick's law with nonconstant diffusion coefficient?

Second Fick's law with nonconstant diffusion coefficient has many applications in various fields such as chemistry, biology, and environmental science. It can be used to study diffusion in porous materials, drug delivery systems, and the movement of pollutants in soil and groundwater, among others.

4. How is the diffusion coefficient determined in Second Fick's law with nonconstant diffusion coefficient?

The diffusion coefficient in Second Fick's law with nonconstant diffusion coefficient can be determined experimentally by measuring the concentration of the diffusing substance at different times and locations. It can also be calculated using theoretical models and equations that take into account the properties of the medium.

5. What are some limitations of Second Fick's law with nonconstant diffusion coefficient?

While Second Fick's law with nonconstant diffusion coefficient is a useful tool for describing diffusion in complex systems, it does have some limitations. It assumes that the medium is isotropic and that there are no external forces acting on the diffusing substance. It also does not account for the effects of convection, which can significantly impact diffusion in certain situations.

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