# Homework Help: Using heat pump to make a house heater

1. Oct 25, 2015

### Pouyan

A heat pump in winter heat energy from the bottom of a lake, where temperature is 4 ° C and delivers thermal energy in a home where the temperature is 22 C. What is the theoretical minimum power the heat pump must be supplied to you at home must be able to take out 4000 watts heating power ?!

Relevant equations
What I learned is this :

for heat pump:
When we will use heat pump to make a place heater:
(ΔQ/ΔW) ≤ T(hot) / T(hot) - T(cold)

And η = (Useful energy/Supply energy)

The attempt at a solution
I see in my solution that the answer is near 244 W
My attempt is :

η=T(hot) / T(hot) - T(cold) = 295 / 22-4 ≈ 16.4

η = P useful/ P Supply

P useful = 4000W and η= 16.4 then P supply = 243.9 W

But is this right ?! Can "η" be more than 1 ?! If the answer is positive is this because of heat pump which is inverse of heat engine and actually is a reversible process ?!

2. Oct 25, 2015

### Mister T

${T_{hot}}$ will always be greater than ${T_{hot}-T_{cold}}$.

Therefore the ratio $\displaystyle \frac{T_{hot}}{T_{hot}-T_{cold}}$ will always be greater than one.

3. Oct 25, 2015

### Mister T

The heat pump is the reverse of the heat engine, but any one specific heat pump cannot be reversed and turned into a heat engine unless it's a reversible engine.

Note that the efficiency of an engine is always less than one, even in the case of the reversible engine.