How Much Work Can Be Extracted from Copper Blocks Using a Carnot Process?

[8614smith]

Homework Statement

Four identical blocks of copper, of mass 20Kg each, are 'perfectly' isolated from the environment. One is kept at a temperature of 573K and the other three at 288K. The heat capacity of copper is 375 J/K/Kg. Assume a carnot process is being used to exploit the temperature differences between the se 4 blocks of copper. How much work can maximally be extracted?
(Hint: the temperature of the copper blocks change continuously from initial to final temperature. Remember that a carnot process is reversible, the total entropy change is zero)

Homework Equations

1. $$T_{f}=T_{1}^{\frac{m_{1}}{m_{1}+m_{2}}}*T_{2}^{\frac{m_{2}}{m_{1}+m_{2}}}$$

The Attempt at a Solution

"Treating it as 1 block of 573K at 20Kg and 1 block of 288K at 60Kg and using T1 = 573, T2 = 288 and m1 = 20, m2 = 60 i get;

$$T_{mix}=359.25K$$

$$T_{f}=573^{\frac{20}{80}}*288^{\frac{60}{80}}$$

$$T_{f}=342.04K$$

now using $$\Delta{E}=mc\Delta{T}$$

$$\Delta{E}=80*375*(359.25-342.04)=516300J$$

$$\Delta{S}=\frac{\Delta{E}}{T}-\frac{\Delta{W}}{T}$$

As there's no entropy change the delta S terms is zero which leaves,
$$\frac{\Delta{E}}{T}=\frac{\Delta{W}}{T}$$

the T's cancel and your left with extractable work = 516300J"


Does this seem correct? I am not sure about the 'T_mix' part and the last part where delta E is converted to delta W.

 

[Andrew Mason]

I am having difficulty following your reasoning. I think you have to apply the equation for efficiency to determine the work as a function of temperature. Initially, the extractable work is:

$$dW = \eta dQ_h = (1-\frac{T_c}{T_h}) cmdT$$

Write out the relationship between Tc and Th and do the integration.

AM