Maximizing Temperature in Identical Blocks with Constant Heat Capacity

[TheTank]

Homework Statement

Hope to get a little help here..
3 identical blocks with constant heatcapasity have temperatures T1, T1, T2, with T1>T2.
What is the highest possible temperature T we can achive in one of the blocks, when no heat/work exchange with the envoriment.

Answere $$T=2T_{1}+\frac{1}{2}T_{2}-\sqrt{2T_1 T_2 +\frac{1}{4}{T_2}^2}$$

Homework Equations

TdS=dU, dU=dQ=cdT

The Attempt at a Solution

I end up with a 3. grade equation for T. Very sceptical..

 

[TheTank]

Anyone?

 

[Andrew Mason]

Homework Statement

Hope to get a little help here..
3 identical blocks with constant heatcapasity have temperatures T1, T1, T2, with T1>T2.
What is the highest possible temperature T we can achive in one of the blocks, when no heat/work exchange with the envoriment.

Answere $$T=2T_{1}+\frac{1}{2}T_{2}-\sqrt{2T_1 T_2 +\frac{1}{4}{T_2}^2}$$

Homework Equations

TdS=dU, dU=dQ=cdT

The Attempt at a Solution

I end up with a 3. grade equation for T. Very sceptical..

I am not sure how one gets that answer.

Heat flow of $\Delta Q$ will flow from 2M to M where $\Delta Q = 2MC(T_f - T_1) = -MC(T_f - T_2)$.

So, it seems to me that the equilibrium temperature is:

$$T_f =(T_2 + 2T_1)/3$$


AM

 

[TheTank]

The odd thing here is that one of the blocks will get a higher end temperature. Maybe if we imagine two Carnot engines working between the cold and the hot blocks? So that we can say that change in entropy S is zero. I can't get the "correct" answere either.

Hope for more response. Thanks

 

[Andrew Mason]

The odd thing here is that one of the blocks will get a higher end temperature. Maybe if we imagine two Carnot engines working between the cold and the hot blocks? So that we can say that change in entropy S is zero. I can't get the "correct" answere either.

Ok. Let's call them A, B and C (A and B being at T1). In order to increase the temperature of A you have to move heat from B. To do that you have to run a heat pump between B and A. The work to run the heat pump comes from running a heat engine between B and C. If the heat pump and heat engines are as efficient as possible, you will maximize the amount of heat delivered to A.

But I don't think the given answer can be correct. The limit would be where T1 is arbitrarily close to T2 and it would be T1. That equation gives a higher limit.

AM

 

[TheTank]

Thanks :) appreciated!