A building is maintained at a temperature T by means of an ideal heat pump, whic

Question

        A building is maintained at a temperature T by means of an ideal heat pump, which uses water from a lake at temperature T0 as a source of heat.  The heat pump consumes power W and the building loses heat to its surroundings at a rate ?(T-T0), where ? is a positve constant.  Show that the temperature T is given by:<?xml:namespace prefix = v />                                

                                               T = T0 + (W/2?)*{1+ [1+sqrt(1+(4?T(initial)/W))]}             <?xml:namespace prefix = v ns = "urn:schemas-microsoft-com:vml" />                   <?xml:namespace prefix = o />

   

A building is maintained at a temperature T by means of an ideal heat pump, which uses water from a lake at temperature T0 as a source of heat. The heat pump consumes power W and the building loses heat to its surroundings at a rate ?(T-T0), where ? is a positve constant. Show that the temperature T is given by: T = T0 + (W/2?)*{1+ [1+sqrt(1+(4?T(initial)/W))]}

Explanation / Answer

COPhp = QH/ (QH - QL)

= alpha*(T-To) /W

ideal heta pump so,

COPhp = 1/ (1 - TL/TH)

= 1/( 1 - To/T)

equating both equation

alpha*(T-To)^2 = T*W

alpha *(T^2 + To^2 - 2*T*To) = T*W

rearranging and solving for T,

T = T0 + (W/2?)*{1+ [1+sqrt(1+(4?T(initial)/W))]}

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