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Thermodynamics: Room Air Conditioner

  1. May 5, 2008 #1
    [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?
  2. jcsd
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