Chapter 9, Problem 9.1P

Interpretation Introduction

Interpretation:

A coefficient of performance ∅ for a heat pump and ∅for a Carnot heat pump should be defined.

Concept Introduction:

  $Thermalefficiency=Workout/Workin$

  $COP=Whatwewant/Whatwepay$

Answer to Problem 9.1P

Coefficient of performance of Carnot heat pump is greater than unity.

Given Information:

Measure of performance is defined as what you get compared to what you pay for. So, for an engine, the thermal efficiency is η = |W|/ |Q_H| while for a refrigerator, the coefficient of performance is ω = |Q_H|/ |W|.

Explanation:

  $\begin{array}{l}Thermalefficiency=Workout/Workin\hfill \\ =Benefit/Cost\hfill \\ Carnotheatengine\hfill \\ Q_H=W+Q_C\hfill \\ W=Q_H-Q_C\hfill \\ Efficiency\left(e\right)=\frac{W}{Q_H}\hfill \\ =\frac{Q_H-Q_C}{Q_H}=1-\frac{Q_C}{Q_H}=1-\frac{T_c}{T_H}\hfill \end{array}$

Efficiency of refrigerator and heat pump is given by coefficient of performance (COP).

  $\begin{array}{l}COP=Whatwewant/Whatwepay\hfill \\ COP=\frac{Q_C\left(Wewantheattoescape\right)}{W}\hfill \\ Q_H=W+Q_C\hfill \\ So,COP=\frac{Q_H-W}{W}\times \text{ (}\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}) after multiply\&divideby1/Q_H\hfill \\ =\frac{1-\raisebox{1ex}{$W$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}{\raisebox{1ex}{$W$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}\hfill \\ COP=\frac{\text{1- }e}{e} sincee=W/Q_H\hfill \end{array}$

Heat pump

  $\begin{array}{l}COP=Whatwewant/Whatwepay\hfill \\ =\frac{Q_H}{W}\left(wewantheattoenter\right)\hfill \\ COP=\frac{Q_H}{W} Hence,COP>1\hfill \end{array}$

Carnot pump is a Carnot engine that is driven backward by another engine. It removes heat from cold reservoir and exhaust heat into hot reservoir.

Here, coefficient of performance is equal to reciprocal of Carnot efficiency.

This figure is reverse of Carnot heat engine

  $\begin{array}{l}q{`}_4=-q_1\hfill \\ q{`}_2=-q_3\hfill \\ e`=\frac{q{`}_4}{W`}=\frac{-q_1}{-q{`}_2+q{`}_4}\hfill \\ =\frac{q_1}{q_1-q_4}\hfill \\ e`=\frac{1}{1-\raisebox{1ex}{$q_3$}\!\left/ \!\raisebox{-1ex}{$q_1$}\right.} Carnotheatengine\hfill \\ \begin{array}{l}=\frac{1}{1-\raisebox{1ex}{$Q_c$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}\\ e\text{'}=\frac{1}{e}\end{array}\hfill \end{array}$

COP of Carnot heat pump is greater than unity.

Carnot cycle is a speculative thermodynamic process which offers a ceiling limit on the efficacy that any conventional thermodynamic system can acquire while converting energy to work or on the other hand, the efficacy of a refrigerating system in generating a difference in temperature by applying work to the system.

Explanation of Solution

Given Information:

Measure of performance is defined as what you get compared to what you pay for. So, for an engine, the thermal efficiency is η = |W|/ |Q_H| while for a refrigerator, the coefficient of performance is ω = |Q_H|/ |W|.

  $\begin{array}{l}Thermalefficiency=Workout/Workin\hfill \\ =Benefit/Cost\hfill \\ Carnotheatengine\hfill \\ Q_H=W+Q_C\hfill \\ W=Q_H-Q_C\hfill \\ Efficiency\left(e\right)=\frac{W}{Q_H}\hfill \\ =\frac{Q_H-Q_C}{Q_H}=1-\frac{Q_C}{Q_H}=1-\frac{T_c}{T_H}\hfill \end{array}$

Efficiency of refrigerator and heat pump is given by coefficient of performance (COP).

  $\begin{array}{l}COP=Whatwewant/Whatwepay\hfill \\ COP=\frac{Q_C\left(Wewantheattoescape\right)}{W}\hfill \\ Q_H=W+Q_C\hfill \\ So,COP=\frac{Q_H-W}{W}\times \text{ (}\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}) after multiply\&divideby1/Q_H\hfill \\ =\frac{1-\raisebox{1ex}{$W$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}{\raisebox{1ex}{$W$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}\hfill \\ COP=\frac{\text{1- }e}{e} sincee=W/Q_H\hfill \end{array}$

Heat pump

  $\begin{array}{l}COP=Whatwewant/Whatwepay\hfill \\ =\frac{Q_H}{W}\left(wewantheattoenter\right)\hfill \\ COP=\frac{Q_H}{W} Hence,COP>1\hfill \end{array}$

Carnot pump is a Carnot engine that is driven backward by another engine. It removes heat from cold reservoir and exhaust heat into hot reservoir.

Here, coefficient of performance is equal to reciprocal of Carnot efficiency.

This figure is reverse of Carnot heat engine

  $\begin{array}{l}q{`}_4=-q_1\hfill \\ q{`}_2=-q_3\hfill \\ e`=\frac{q{`}_4}{W`}=\frac{-q_1}{-q{`}_2+q{`}_4}\hfill \\ =\frac{q_1}{q_1-q_4}\hfill \\ e`=\frac{1}{1-\raisebox{1ex}{$q_3$}\!\left/ \!\raisebox{-1ex}{$q_1$}\right.} Carnotheatengine\hfill \\ \begin{array}{l}=\frac{1}{1-\raisebox{1ex}{$Q_c$}\!\left/ \!\raisebox{-1ex}{$Q_H$}\right.}\\ e\text{'}=\frac{1}{e}\end{array}\hfill \end{array}$

COP of Carnot heat pump is greater than unity.

Conclusion

Carnot cycle is a speculative thermodynamic process which offers a ceiling limit on the efficacy that any conventional thermodynamic system can acquire while converting energy to work or on the other hand, the efficacy of a refrigerating system in generating a difference in temperature by applying work to the system.