A heat pump heats the air in a rigid, insulated cuboid room of size 25m x 10m x 4m. The heat pump consumes 15 kW of power. The initial temperature and pressure in this room are 12°C and 1 bar, respectively. With an average coefficient of performance of COPHP= 3.0 over the range of air temperature in this room. Requried:
How long will it take to raise the temperature in the room to 27 °C?

Time required = 287.2 secs  

Explanation:

Volume  of room = 25 * 10 * 4 = 1000 m^3

power consumed by pump = 15 kW

T1 ( initial temperature ) = 12°C

P1 ( Initial pressure ) = 1 bar

COPhp = 3

Calculate time taken to raise room Temp to 27°C

average heat supplied ( ∅ ) = COPhp * power consumed by pump

                                             = 3 * 15 = 45 kW

Time required can be calculated using the relation below

∅t = P*V*Cv ( T2 - T1 )   [ p = 1.2 kg/m^3 , Cv = 0.718 KJ/kg ( air properties ) ]

45 * 10^3 ( t )  = 1.2*1000* 718 ( 27 - 12 )

∴ solving for t

t = 287.2 secs  ≈ 4.79 mins

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