# Homework Help: Refrigerators and heat pumps

1. Feb 8, 2006

### Nusc

Show that the relationship betwee efficiency (n) of a Carnot engine and the coefficient of performance (w) of the same engine when operated as a refrigerator is given by nw= Tc/Th

Is a Carnot engine whose efficiency is very high particularly suited as a refrigerator? Why?

Well we know that for Carnot engines the efficiency is n = [Th-Tc]/Th
and that the coefficieny of performance for a refrigerator is Qh/w = Qh/(Qh-Qc)

nw = (1 - Tc/Th)(Qh-Qc)
= Qh - TcQh/Th - Qc +TcQc/Th
=/= Tc/Th

What is wrong?

2. Feb 9, 2006

### Andrew Mason

The coefficient of performance of a refrigerator is given by $\omega = Q_C/W$ and the efficiency of a heat engine is: $\eta = W/Q_H$

So $$\eta\omega = \frac{Q_C}{Q_H}$$

In a Carnot cycle, $\Delta S = 0$ so:
$$\frac{Q_H}{T_H} - \frac{Q_C}{T_C} = 0$$
and:
$$\frac{T_C}{T_H} = \frac{Q_C}{Q_H}$$

AM

3. Feb 11, 2006

### Nusc

Is a Carnot engine whose efficiency is very high particularly suited as a refrigerator? Why?

A high efficiency would imply that Qc >> Qh based on the result Tc/Th for part a).

But what specific justification is neccessary to explain that for Carnot engines?

4. Feb 12, 2006

### Andrew Mason

A Carnot engine and a Carnot refrigerator both employ the reversible (Carnot) cycle. The refrigerator just reverses the direction of the heat flow by making an infinitessimal change in the starting conditions.

In the question, the temperatures are fixed. High efficiency does not mean that Qc >> Qh. This can never be the case. Whatever heat is removed from the cold reservoir is transferred to the hot reservoir (along with the added work which is converted to heat as well).

The reversible cycle represents the limit of the efficiency of the ratio of work to heat transfer between two temperatures. In the case of heat moving from hot to cold, the Carnot cycle provides the maximum amount of useable work that can be extracted. In moving heat from cold to hot, the Carnot cycle provides the minimum amount of work that must be added.

Theoretically, you can make a Carnot engine into a Carnot refrigerator by storing all the work output of the engine cycle (say by lifting a weight) and then using that work to reverse the cycle and move the heat from the cold reservoir back to the hot reservoir.

AM

5. Feb 14, 2006

### Nusc

Is a Carnot engine whose efficiency is very high particularly suited as a refrigerator? Why?

So how's this, if a Carno engine has alot of efficiency then we can see from n = 1 - T2/T1 that T2<<T1 so it makes a good refrigerator.

6. Feb 14, 2006

### Andrew Mason

A Carnot refrigerator is not very efficient if T2<<T1. But it is more effiient than any other refrigerator between those two temperatures.

Between two given temperatures, the Carnot engine is more efficient than any other and the Carnot refrigerator is more efficient than any other refrigerator.

AM

7. Feb 14, 2006

### Nusc

Was what I wrote sufficient in answering the question, that's all I ask for. Because you just confused me.

Last edited: Feb 15, 2006
8. Feb 15, 2006

### Andrew Mason

A Carnot engine is highly efficient only if Th>>Tc. With this temperature difference, a refrigerator (Carnot or otherwise) is not going to be very efficient (ie. in the reverse cycle). I think that is the answer they are looking for. But you can add that the Carnot will be more efficient than any other.

AM