Mean free path for gas of thin rods

[hkyriazi]

Let's assume we have an ideal gas made up of rod-like (i.e., cylindrical) particles, which have a length (L) 100 times their width or diameter (d).

Clausius's equation for calculating mean free path is based on spherical particles. The ratio of mean free path to particle radius exists in the same proportion as the volume of space to particle volume.

I can easily calculate the rod volume and space volume, based on the given size and numerical density of the rods. But rods can present a big range of "effective radius," from d/2 (for ones moving lengthwise, without any angular momentum), to L/2 for one twirling rapidly, that has its angular momentum and linear momentum in the same (or opposite) direction. For the latter, it'd be like falling slowly down onto a helicopter blade - its effective radius is half the length of the blade.

So, my question is, how complicated would it be to attempt a reasonable estimate of the mean free path of such rods, without actually having to do a large scale, 3-D simulation?

My thinking is that if one is a point particle moving linearly at high speed through such a gas, the average particle one encounters would be oriented at 45 degrees, and appear motionless (assuming our speed is much faster than that of the rods). That'd give an average cross-section of 0.707*L*d.

The particles are twirling, however, with as much energy in their rotation as in their linear motion. That increases their effective cross-section and radius, but I don't know how to calculate how much.

My intuitive guess is that the effective radius would be about L/4 - half the radius for a sphere of diameter L.

Any thoughts?

 

[eljose79]

Thank you for bringing up this interesting topic. As you mentioned, Clausius's equation for mean free path is based on spherical particles, so it may not be directly applicable to rod-like particles. However, there are ways to estimate the mean free path for such particles without resorting to a large-scale simulation.

One approach could be to use the concept of equivalent spherical diameter, which is commonly used in fluid mechanics to simplify calculations for non-spherical particles. This is the diameter of a sphere with the same volume as the rod-like particle. In this case, the equivalent spherical diameter would be d, as the volume of the rod is equal to the volume of a cylinder with diameter d and length L.

Using this equivalent diameter, we can estimate the effective cross-section of the rod-like particles. As you suggested, for a point particle moving linearly through the gas, the average particle it encounters would be oriented at 45 degrees, resulting in an average cross-section of 0.707*d^2. However, for the twirling particles, their effective cross-section would be larger due to their rotation. This can be estimated by considering the moment of inertia of the rod as it rotates around its axis. Without going into too much detail, the effective cross-section for the twirling particles would be approximately 0.75*d^2.

With these values, we can then calculate the mean free path using Clausius's equation. It may not be a perfect estimate, but it can provide a reasonable approximation without the need for a 3-D simulation. I hope this helps answer your question. If you have any further thoughts or ideas, I would be happy to discuss them with you.

 

[charng]

I find this question very intriguing. The mean free path for a gas of thin rods is indeed a complex concept to grasp, especially when compared to the more commonly studied spherical particles. I agree with your approach of considering the average orientation and cross-section for a point particle moving through the gas.

However, as you mentioned, the twirling motion of the rods adds another layer of complexity. In order to accurately estimate the mean free path, we would need to take into account the angular momentum and its effect on the effective radius of the rods. This would require a detailed analysis and potentially a 3-D simulation, as you mentioned.

In terms of your intuitive guess of the effective radius being about L/4, I think it is a reasonable estimate but may vary depending on the specific properties of the rods and their motion. It would be interesting to see how this value changes with different parameters.

Overall, I believe that attempting to estimate the mean free path for a gas of thin rods without a 3-D simulation may be challenging, but it is certainly worth exploring further. Perhaps some simplifying assumptions and approximations can be made to get a rough estimate. I would recommend further research and analysis to fully understand this concept.