Thermodynamics resistance proof

[skirmidirmi]

Homework Statement

Prove that thermal resistance is additive in series

Homework Equations

H=A(T_H-T_c)/R
where R=L/k

The Attempt at a Solution

For two slabs in thermal contact where T_C is the outside cold temperature and T_H is the outside hot temperature

A(T_H-T_c)/R=A(T_H-T)/R₁+A(T-T_c)/R₂)

The A's cancel out, and after a bit of math, I've gotten the equation down to

R=(R₁+R₂)(T_H-T_C)/(R₂(T_H-T)+R₁(T-T_c)

How can I get rid of the T's with no subscript and just be left with R1+R2 on the right side?

 

[Mapes]

Try considering an energy balance at the point between the two slabs.

 

[skirmidirmi]

Are you suggesting that these slabs are in dynamic thermal equilibrium? If so, are you suggesting that I simply equate the two heat currents? I will not have "R" involved in my expression then, only R1 and R2...
I truly want to understand this. Thank you.

 

[Mapes]

Yes, the equation applies to steady state conditions only.

 

[LowlyPion]

I'd approach it more simply.

Heat Flow = ΔT/R

So R*Heat Flow = ΔT

The heat flow of R1 is then R1*H = ΔT = (Ti - T1)

And through R2 is R2*H = (T1 - T2)

Heat flow for the system then is

R1*H + R2*H = (Ti - T1) + (T1 - T2) = Ti - T2

If Rtotal*H for the system is Ti - T2, then Rtotal is (R1 + R2)

 

[skirmidirmi]

Thank you very much! That is a lot more straightforward than how I was trying to prove it.