Solving Thermodynamic Problem: Mass, m, Cylinder, h, r, Tmi, To, Tme

[vector3]

Mass, m, with initial temperature, Tmi, is flowing along the z axis. The mass enters a right circular cylinder at the plane x=0. The cylinder is centered and oriented on the z axis (which is to the right). The cylinder's length is h and radius is r. The outside temperature of the cylinder is maintained at To.

As the mass exits the cylinder, the temperature at the center of the mass is Tme.

Find the function, T(r,z), that represents the temperature as a function of the radius and distance along the cylinder.

Given the heat equation:

$$\frac{1 \partial }{r \partial{r}} \left(r \frac{\partial{T}} {\partial {r}} \right) + \frac{\partial{T^2}}{\partial {z^2}} = \frac{1}{\alpha} \frac{\partial {T}}{\partial {t}}$$

and material properties:
c = specific heat
k = thermal conductivity
$$\alpha =$$ a constant

Any comments on how to attack this problem?

 

[chaoseverlasting]

I think you would just need Stephens Law, in this case. $$\frac{dq}{dt}=\sigma A\epsilon t^4$$

where,
$$\sigma$$ is the stephens constant,
$$\epsilon$$ is the emissivity,
A is the area over which emission/absorption is happening
T is the temperature of the body.

 

[genneth]

I don't think has anything to do with Stephan-Boltzmann Law. The equation given is just one for diffusion of heat.

My understanding is that it's asking for the equilibrium temperature distribution, i.e. dT/dt = 0. Then you've just got Laplace's equation, in axisymmetric cylindrical coordinates. T can be separated into a radial R part and an axial Z part: $$T(r,z) = R(r)Z(z)$$. Then you're just finding eigenfunctions of the appropriate operators. In fact, your textbook/course may have already derived them for the general case, and you can just plug in the boundary conditions.

 

[vector2]

I'm sure I'll use the wrong terms in what follows:

Restating what genneth posted in different words:
The temperature field will assume a steady state condition for a constant flow of mass. In other words, the temperature will be a function of it's position only. The temperature field will assume a similar steady state shape for a different mass flow rate, given the same boundary conditions.

The given equation is the only one that I could find that is remotely related. It lacks a mass flow or velocity aspect. The temperature field that will set up depends on the boundary conditions and the rate at which the mass flows.

There is another real nice post in the forum addressing a related problem but the mass is not flowing and the boundary conditions are different. I'm in the process trying to understand the application of that solution to this current problem.

Just as a note, I "own" this problem. It is not in a textbook that I have or read. Wish it was... I'm sure flowing masses must be heated in cylinders in manufacturing or production processes somewhere...

As a clarification: vector2 = vector3 not sure how different user names got assigned...?