Question

(a) What is the best coefficient of performance for a heat pump that has a hot reservoir temperature of 50.0ºC and a cold reservoir temperature of −20.0ºC ? (b) How much heat transfer occurs into the warm environment if 3.60×10⁷ J of work ( 10.0kW⋅h ) is put into it? (c) If the cost of this work input is 10.0 cents/kW⋅h , how does its cost compare with the direct heat transfer achieved by burning natural gas at a cost of 85.0 cents per therm. (A therm is a common unit of energy for natural gas and equals 1.055×10⁸ J .)

Short answer

Coefficient of performance for heat pump is \(4.61\). Heat transferred to the warm environment is \(1.656 \times {10^8}\;{\rm{J}}\). The total cost of this pump is \(100\;{\rm{cents}}\) whereas total cost of burning natural gas is \(29\;{\rm{cents}}\). It is lesser than that of pump.

Step-by-step solution

Introduction

Coefficient of performance is a index which tells us how much heat is produced by the pump, per unit intake of energy.

 Given parameters & formula for efficiency of heat engine and performance coefficient

The temperature of a cold environment\({T_c} = - 20\;^\circ {\rm{C}} = {\rm{253}}\;{\rm{K}}\)

The temperature of the warm environment\({T_h} = 50\;^\circ {\rm{C}} = {\rm{323}}\;{\rm{K}}\)

The efficiency of the Carnot engine \[\eta = 1 - \frac{{{T_c}}}{{{T_h}}}\]

Heat pump’s performance coefficient\({\beta _{hp}} = \frac{1}{\eta }\)

Also, Heat pump’s performance coefficient\({\beta _{hp}} = \frac{W}{Q}\)

Here,

\(\eta \)- efficiency of the engine.

\({T_c}\)- the temperature of the cold environment

\({T_h}\)- temperature of the warm environment

\({\beta _{hp}}\)- heat pump’s performance coefficient

\(Q\)-heat transfer.

\(W\) -work done.

 Calculate efficiency and coefficient of performance of ideal heat pump

Maximum efficiency can be calculated by calculating Carnot efficiency as

\[\begin{array}{c}\eta = 1 - \frac{{{T_c}}}{{{T_h}}}\\ = 1 - \frac{{253\;{\rm{K}}}}{{323\;{\rm{K}}}}\\ = 0.217\end{array}\]

Heat pump’s performance coefficient can be calculated as

\[\begin{array}{c}{\beta _{hp}} = 1/\eta \\ = 1/0.217\\ = 4.61\end{array}\]

 Calculate heat transfer

Heat that transfers to warm environment is

\[\begin{array}{c}Q = W \times {\beta _{hp}}\\ = \left( {3.6 \times {{10}^7} \times 4.61 } \right)\;{\rm{J}}\\ = 1.656 \times {10^8}\;{\rm{J}}\end{array}\]

 Calculate total cost of pump and natural gas

For heat pump:

Total work\( = 10\;{\rm{kWh}}\)

For heat pump:

Total work = 10kWh

Cost = 10 cents per kWh

Total cost of pump =100 cents

For natural gas:

Cost of 85 cents per therms

\[\begin{array}{l}1 {\rm{therm}} = 1.055 \times {10^8}\;{\rm{J}}\\3.6 \times {10^7}\;{\rm{J}} = \frac{{3.6 \times {{10}^7}}}{{1.055 \times {{10}^8}}}\;therms\end{array}\]

Total cost of natural gas

\[\begin{array}{l} = 85 \times \frac{{3.6 \times {{10}^7}}}{{1.055 \times {{10}^8}}} {\rm{cents}}\\ = 29 {\rm{cents}}\end{array}\]

Therefore, the heat pump’s performance coefficient is\(4.61\). Heat that transfers to warm environment is \(1.656 \times {10^8}\;{\rm{J}}\). The total cost of this pump is \(100\)cents whereas total cost of burning natural gas is \(29\)cents which is very much less than by pump.