Heat Transfer Through a Cylinder

[ChemIncognito]

OK, this has been driving me crazy. It's basically straight out of heat transfer, but for some reason I can't get it. I don't care about the numbers here, I want to know if I'm making a mistake in my logic/setup

Description:
A hollow cylinder with inner radius r1, outer radius r2, and length L. No radiation present

I have gas flowing through it with a known temperature, T_hot, and a known convection coefficient, H_hot

I have gas circulating externally with T_cold, H_cold.

The cylinder has a thermal conductivity of k_cyl. Thus, my heat flowing through the cylinder, q, is

(T_hot - Tcold) / ( (1/(H_hot*2*pi*r1*L)) + (log(r2/r1) / (2*pi*k_cyl*L)) + (1/(H_cold*2*pi*r2*L))

I can find my surface temperatures, T_inner, and T_outer with:

T_inner = T_hot - q*(1/(H_hot*2*pi*r1*L))
T_outer = T_cold + q*(1/(H_cold*2*pi*r2*L))


Here's my frustration:
I can check that my surface temperatures are right by calculating the heat loss or gain at either surface (H*A*dT) and comparing that to the q from above. Shouldn't I also be able to calculate the rate of heat loss by using the temperature at r1 (T_inner) and T_cold? That is, does q also equal:

(T_inner-T_cold)/((1/(H_cold*2*pi*r_outer*l))+(log(r2/r1)/2*pi*k_cyl*L))

The rate of heat I get from the above equation is less than that calculated from the value obtained for q, and the double-checked values for convection into and out of the system.

If you need a diagram (or equations in graphical form), I'm happy to help. This has been driving me crazy because I have a system where I have a specified internal surface temperature, the rate of heat gained by internal convection, and I want to make sure that the two balance (heat in by convection = heat lost through cylinder wall and external convection).

FYI, this is a steady state problem...The cylinder wall is not accumulating energy.

Thanks for your help, chaps.

 

[prex]

Your last equation appears correct to me (supposing r_outer=r2), and by using it I get the same q as needed. So you should just recheck your numbers.

prex

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[IRstuff]

Your 1st equation is derived by substitution of Tin and Tout from two convection equations into the conduction equation. Your last equation is derived from substituting Tout from the outer convection equation into the conduction equation.

So all the algebra checks out.

TTFN

FAQ731-376