Finite difference conjugate heat xfer: internal flow

[pm272]

Hi.

I have been trying to solve this problem that has been keeping me up at night for a coupe weeks at least. If anyone can help me, I would be greatly appreciated.

Hot air enters a cylindrical duct. The duct has some R-value and radiation and convection is being accounted for on the outside surface of the duct. What is the bulk temperature of the air and the duct surface after some length, L?

I approached this problem as follows:
1) I sectioned out the duct into n dx lengths each with their own energy balances.
2) The first dx will see air entering at T_entering. This is T_i. The air is leaving at T_i+1. Air properties are dictated by this entering air temperature. Internal convection coefficient can be found as an empirical calculation - same with outside convection coeff.
3) Now I analyze the conduction problem of the duct itself with an external boundary condition accounting for convection and radiation. The internal boundary condition would be a temperature. I am using a Newton-raphson iterative method to solve the non-linear equation.

The problem I am running into is that I can find an external wall temperature based upon whatever internal bulk air temp I am using but I am unsure how to account for the fact that T_i+1 will change from T_i, and therefore continue through the entire length. I believe that I require another equation.

It seems similar to solving a 1-D conduction problem with radiation except that the internal temperature is changing along the length of the duct (as is wall temp).

This problem must be possible since I can essentially solve this using any CFD package.

Thank you

 

[Chestermiller]

Hi.

I have been trying to solve this problem that has been keeping me up at night for a coupe weeks at least. If anyone can help me, I would be greatly appreciated.

Hot air enters a cylindrical duct. The duct has some R-value and radiation and convection is being accounted for on the outside surface of the duct. What is the bulk temperature of the air and the duct surface after some length, L?

I approached this problem as follows:
1) I sectioned out the duct into n dx lengths each with their own energy balances.
2) The first dx will see air entering at T_entering. This is T_i. The air is leaving at T_i+1. Air properties are dictated by this entering air temperature. Internal convection coefficient can be found as an empirical calculation - same with outside convection coeff.
3) Now I analyze the conduction problem of the duct itself with an external boundary condition accounting for convection and radiation. The internal boundary condition would be a temperature. I am using a Newton-raphson iterative method to solve the non-linear equation.

The problem I am running into is that I can find an external wall temperature based upon whatever internal bulk air temp I am using but I am unsure how to account for the fact that T_i+1 will change from T_i, and therefore continue through the entire length. I believe that I require another equation.

It seems similar to solving a 1-D conduction problem with radiation except that the internal temperature is changing along the length of the duct (as is wall temp).

This problem must be possible since I can essentially solve this using any CFD package.

Thank you

Let's see your differential equations (not you finite difference equations).

 

[pm272]

I have attached a drawing with equations that I would use. I suppose it is actually not a finite difference equation... I am merely breaking the duct into dx lengths in an attempt to account for an ever changing wall temperature traversing down the duct.

Basically, it appears that I have 2 equations but 3 unknowns (unknowns are: Twall_int, Twall_ext, and air T_exit).

How can I solve this?

 

[Chestermiller]

OK. So you have. $$\dot{m}C\frac{dT}{dx}=-\pi D q$$
Neglecting wall-thickness effects, we have:

$$q=h_{inside}(T-T_{inside wall})=\frac{k}{t}(T_{inside wall}-T_{outside wall})=h_{outside}(T_{outside wall}-T_{\infty})+\sigma \epsilon(T_{outside wall}^4-T_{\infty}^4)$$

Is this basically it? How do you know that the radiation is significant relative to the outside convection?

 

[pm272]

Well, the air would be around 700°F so I suppose that for the case where the r-value of the insulation is low, I would assume radiation to be significant. My intention was to formulate a script that would essentially solve for both of these edge cases: The case where we want to radiate heat from the duct (no insulation) and the case where we want to keep the air hot (high r-value).

Thank you.

 

[pm272]

So, I get to the point you are at and then it appears that unless we know what the wall temp is, we can't know what the internal air leaving our duct section is. Why is it possible that I can find the answer to this using a CFD package but it appears unsolvable by hand at this point?

I am sure that I am missing something.

 

[Chestermiller]

So, I get to the point you are at and then it appears that unless we know what the wall temp is, we can't know what the internal air leaving our duct section is. Why is it possible that I can find the answer to this using a CFD package but it appears unsolvable by hand at this point?

I am sure that I am missing something.

The 2nd equation I wrote represents 2 equations in the two unknowns, the inside wall temperature and the outside wall temperature, in terms of the gas temperature and the surroundings temperature. So, what's the problem?