Chapter 5, Problem 5.13P

(a)

Interpretation Introduction

Interpretation:

A Carnot engine operates between two finite heat reservoirs of total heat capacity $C_H{}^t$ and $C_C{}^t$ . Determine the expression relating their temperatures $T_H$ and $T_C$ at any time.

Concept Introduction:

The heat change of any system at any time is defined as below:

For hot reservoir,

  $d{Q}_H=C_H{}^{t}d{T}_H$

And

For cold reservoir,

  $d{Q}_C=C_C{}^{t}d{T}_C$

(b)

Interpretation Introduction

Interpretation:

Determine the expression for the work obtained as the function of $T_H,C_H{}^t,C_C{}^t$, and initial temperatures $T_{C0}$, $T_{H0}$ .

Concept Introduction:

The heat change of any system at any time is defined as below:

For hot reservoir,

  $d{Q}_H=C_H{}^{t}d{T}_H$

And

For cold reservoir,

  $d{Q}_C=C_C{}^{t}d{T}_C$

(c)

Interpretation Introduction

Interpretation:

What is the maximum work obtainable or work at infinite time when reservoir attains the same temperature.

Concept Introduction:

The maximum work in any Carnot cycle obtainable when the difference between two temperatures or hot and cold reservoir is zero and both reservoirs attains same temperature.

Hence for maximum work

  $T_H=T_C$