Diffusion of a species through a sphere

In summary, the speaker is seeking help with understanding diffusion in two different scenarios at work. They are struggling with the second scenario, which involves diffusion in three dimensions. They are using Fick's First Law and wondering if using spherical coordinates would make the problem easier to solve. They are also glad for the opportunity to brush up on this topic.
  • #1
chatoma
1
0
Hi guys,

This is my first post here, this place looks like a great resource. Well, jumping straight in, I have a couple of questions on diffusion.

At work, I did a couple of experiments with Ion Exchange Resins. Not getting the results we wanted, my boss asked me to do an analysis of diffusion in two scenarios: through a simple "plate" (in 1 dimension), and through a sphere (in 3 dimensions).

Now, I am a Chem Eng, so this should be easy, but I have found myself struggling way more than I should with this topic; its a bit embarrassing at this point. For the first scenario, I used Ficks First Law, and I think that process was rather straightforward.

However, for the second scenario, I am completely stuck. So basically, I'm looking at diffusion in 3 dimensions; from the center of the sphere and to the outside. I don't think I need to complicate myself too much with technical details, so I'm assuming steady-state and I want to use Fick's First Law again. As I understand it, for 3 dimensions, Fick's Law is: J = -D∇C (where J=Diffusive flux, D=Diffusion Constant, C=Concentrations).

Now if I'm correct, this would work out as: J = -D(∂C/∂x + ∂C/∂y + ∂C/∂z). Would you guys agree, thus far, I am on the right track? If so, now the embarrassing part is that I'm not sure what to do next. That means, I'm not sure how to solve for J. Also, I was going through some reading material, and I saw that using spherical coordinates might make my life easier; is that correct?

Although its been two years, I can't believe I've forgotten so much about mass transfer...I'm actually kind of glad I'm getting these assignments so that I can brush up on this essential stuff. But anyways, any help is appreciated, and any clarifications/questions, I'll try to answer them in a timely fashion. Thanks!

chatoma
 
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  • #2
That means, I'm not sure how to solve for J.
The equation is solved for J.

If you have a spherical symmetry, spherical coordinates are better, indeed.
 

1. What is diffusion and how does it occur?

Diffusion is the movement of particles from an area of high concentration to an area of low concentration. It occurs due to the random motion of particles and can be influenced by factors such as temperature, pressure, and concentration gradient.

2. How does diffusion of a species through a sphere differ from diffusion in other shapes?

The diffusion of a species through a sphere differs from diffusion in other shapes because the surface area to volume ratio is different. In a sphere, the surface area is smaller compared to the volume, which can affect the rate of diffusion. In other shapes, such as a cube or a cylinder, the surface area is larger compared to the volume, which can lead to a faster diffusion rate.

3. What is the equation for calculating the diffusion rate of a species through a sphere?

The equation for calculating the diffusion rate of a species through a sphere is known as Fick's law and is given by:
J = -D * (dC/dr)
Where J is the diffusion flux, D is the diffusion coefficient, and (dC/dr) is the concentration gradient.

4. How does the size and concentration of the species affect diffusion through a sphere?

The size of the species can affect diffusion through a sphere as smaller particles may diffuse more quickly than larger particles due to their ability to pass through small pores. Additionally, the concentration of the species can also affect diffusion as a higher concentration gradient can lead to a faster diffusion rate.

5. What are some real-life examples of diffusion of a species through a sphere?

Diffusion of a species through a sphere can be seen in various natural and industrial processes. For example, the diffusion of oxygen through a red blood cell or the diffusion of carbon dioxide through a plant leaf. In industrial processes, diffusion through a sphere can be seen in the production of nanoparticles or the diffusion of gases through a catalyst in a chemical reaction.

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