# Homework Help: Thermodynamics: Room Air Conditioner

1. May 5, 2008

### WlfordBrimley

[SOLVED] Thermodynamics: Room Air Conditioner

1. The problem statement, all variables and given/known data
A room air conditioner acts as a Carnot cycle refrigerator between an outside temperature T_h and a room at a lower temperature T_l. The room gains heat from the outdoors at a rate A(T_h - T_l); this heat is removed by the air conditioner. The power supplied to the cooling unit is P. Show that the steady state temperature of the room is:
T_l = (T_h + P/(2A) ) - SQRT[(T_h + P/(2A))^2 - T_h ^2]

2. Relevant equations
Carnot coefficient of refrigerator performance: gamma_C := ( Q_l / W ) = T_l / ( T_h - T_l )
Q_l = (T_l / T_h) Q_h
W = Q_h - Q_l = Q_l * (T_h - T_l)/ T_l

3. The attempt at a solution
At equilibrium, the flow of heat in = flow of heat out, so
P*gamma_C = A (T_h - T_l)
After a lot of algebra (done with maple), I get:
T_l = (P/2A _ T_h) + sqrt(P^2 + 4PAT_h) / 2A
...which is not the correct answer. What did I do wrong?