Temperature Variations in a fin 2

[theonlydave]

Hi,

I have a problem regarding the temperature distribution in a fin. The problem is different from the usual fin problem, instead of heat flowing from a body to the fin attached on it, there is heat generation at the fin tip and the heat transfer is actually to the environment through convection and also conduction to the body. The body is in fact a rotating cylinder and fins are attached to the cylinder surface. Convective heat coefficient is found to be 1705 W/m^2K. Fin tip temperature within the heat generation is 1200°C. Do I ned to derive the temperature profile from scratch? What are the boundary conditions I can use? I tried to setup the conditions similar to the derivations of the traditional fin problem but made great mistakes. The fin base temperature was found to be 5000°C , more than the heat generation region. Can anyone please help?

 

[sailoday28]

Prex-Please correct me if I'm wrong.
The convection coefficient for this problem seem squite high to me. The fin equations are based on no temp gradient in the direction normal to the convection surface.
OR a low film coefficient compared to high conductivity metal fins.

If so, then this fin analysis should be done with other techniques--that is---not the standard fin solutions.

 

[theonlydave]

Hi Prex and sailoday28,

I calculated the Biot number with (Cross Area)/(Perimeter) as the characteristic length. The Biot number is 0.07 << 1. Also, the fin is comparatively thin. So I can assume a one dimensional heat transfer.

Prex, I impose a temperature of 1200°C at the fin tip and 800°C at the fin base. I dont think i need to heat up the fin base as heat flows from a higher temperature to a lower one, that is it flows from the tip to the base. I don't understand your reason for heating the cylinder. What seems interesting is the result (see earlier post) that a higher heat flux at the tip will result in a lower temperature at the base, a lower heat flux will result in a higher temperature at the base. If I have a lower flux at the tip (x=0), I have erroneous result. Say if at x=0, I have flux q= 3W/mm^2. Then I have a temperature greater than 1200°C at the fin base. So, the reason I have for this is that now, heat flows from base to tip and the resulting heat flux at the tip is lower at 3W/mm^2. This I believe is also the reason why my earlier assumptions in earlier post gave me a result that is physically wrong. Anyone would like to comment?

 

[prex]

sailoday28,
I agree that the film coefficient is very high (I would say unrealistic, but at almost 400 m/s peripheral speed who knows).
Your second statement is less clear to me: of course there is a gradient along the fin longitudinal axis, but the fin equations assume a constant temperature in any fin cross section, so there is no temperature gradient in all directions normal to the longitudinal axis of the fin.
The fin equation holds for any combination of parameters, provided that assumption is valid, including at boundaries (constant temperature across the end surfaces).
Of course theonlydave's conditions represent quite a limit case: it is difficult to believe that such a high coefficient is effective on fin surface, and that this doesn't influence in some way the thermal behavior of cylinder wall.
It is also true BTW that finned surfaces are normally useful only with low convective coefficients and highly conductive metals, but here the heat generation at the fin tip reverses the point of view.

prex

http://www.xcalcs.com

Online tools for structural design

 

[sailoday28]

Prex- You state"The fin equation holds for any combination of parameters, provided that assumption is valid, including at boundaries (constant temperature across the end surfaces)."

This is approximately true if the metal conductivity is high and the film coef low. Otherwise temperature gradients will occur on the transverse axis making the basic fin problem 2D instead of 1D.

 

[hacksaw]

what is the emissivity of your fin, what is the stream velocity across the fin, how closely spaced are the (adjacent) fins, is the rotor enclosed, and what is the temperature and emissivity of the enclosure?


seems we have an incomplete problem with fill in the blank assumptions, small wonder the responses are all over the map...