What Is the Coefficient of Performance of a Carnot Heat Pump?

[~angel~]

This question is killing me because I just can't seem to get it.

A heat pump is used to heat a house in winter; the inside radiators are at T_h and the outside heat exchanger is at T_c. If it is a perfect (e.g., Carnot cycle) heat pump, what is K_pump, its coefficient of performance?

Give your answer in terms of T_h and T_c.

According to the hints, you're meant to work out the efficiency of the pump in terms of Q_c and Q_h. I thought it was 1 + (Q_c/Q_h)...

The textbook states that Q_c/Q_h = - T_c/T_h, but in the hints, it states Q_h/Q_c = T_h/T_c.

I'm totally confused.

Please help.

 

[whozum]

I don't know how to solve the problem, but I wanted to point out that Q_h/Q_c = T_h/T_c and Q_c/Q_h = T_c/T_h are the same equation written differently. I don't know what the negative symbolizes, but if it was left out, then it must be somewhat arbitrary in meaning.

Hope this helps, good luck.

 

[quark]

It might be confusing with redundant data. The expression should be Qh/(Qh-Qc) or Th/(Th-Tc)

 

[Clausius2]

The textbook states that Q_c/Q_h = - T_c/T_h, but in the hints, it states Q_h/Q_c = T_h/T_c.

I'm totally confused.

Please help.

I would say this equation represents such an ideal behavior of a Carnot heat pump. Entropy variation of the system must be 0 for being a cyclic machine, and entropy variation of universe (system+surroundings must be also 0 for being a reversible machine). So that the variation of entropy of the surroundings must counterbalance each other (in each focus).

On the other hand you should be careful with the sign convention. Maybe the book refers to different sign convention in each sentence. I always take the absolute value of the heats and put externally the convenient sign.

Also be careful because the COP of a heat pump is defined as $$COP=Q_h/W$$.