Thermodynamics resistance proof

In summary, thermal resistance is additive in series according to the equation Rtotal = R1 + R2. This applies to steady state conditions and is based on an energy balance at the point between the two slabs in thermal contact.
  • #1
skirmidirmi
3
0

Homework Statement


Prove that thermal resistance is additive in series

Homework Equations


H=A(TH-Tc)/R
where R=L/k

The Attempt at a Solution



For two slabs in thermal contact where TC is the outside cold temperature and TH is the outside hot temperature

A(TH-Tc)/R=A(TH-T)/R1+A(T-Tc)/R2)

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

R=(R1+R2)(TH-TC)/(R2(TH-T)+R1(T-Tc)

How can I get rid of the T's with no subscript and just be left with R1+R2 on the right side?
 
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  • #2
Try considering an energy balance at the point between the two slabs.
 
  • #3
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.
 
  • #4
Yes, the equation applies to steady state conditions only.
 
  • #5
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)
 
  • #6
Thank you very much! That is a lot more straightforward than how I was trying to prove it.
 

Related to Thermodynamics resistance proof

1. What is thermodynamics resistance?

Thermodynamics resistance refers to the ability of a system to resist changes in its state or temperature due to external influences. It is a fundamental concept in thermodynamics that helps us understand how energy is transferred and transformed within a system.

2. How is thermodynamics resistance measured?

Thermodynamics resistance is typically measured using the concept of entropy, which is a measure of disorder or randomness within a system. The higher the entropy, the more resistant a system is to changes in its state.

3. What role does thermodynamics resistance play in energy efficiency?

Thermodynamics resistance plays a crucial role in energy efficiency as it helps us understand how much energy is lost or wasted in a system due to various resistances. By minimizing these resistances, we can improve the efficiency of energy transfer and utilization within a system.

4. How can we reduce thermodynamics resistance in a system?

There are several ways to reduce thermodynamics resistance in a system, including insulating the system to minimize heat loss, using lubricants to reduce friction, and designing efficient pathways for energy transfer. Additionally, optimizing the design and materials of a system can also help reduce resistance.

5. What are some practical applications of thermodynamics resistance?

Thermodynamics resistance has various practical applications, such as in designing more efficient engines and power plants, improving the performance of heat exchangers, and developing better insulation materials. It is also crucial in understanding and mitigating energy losses in various industrial and environmental processes.

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