Chapter 22, Problem 22.30P

Suppose you build a two-engine device with the exhaust energy output from one heat engine supplying the input energy for a second heat engine. We say that the two engines arc running in series. Let e₁ and e₂ represent the efficiencies of the two engines. (a) The overall efficiency of the two-engine device is defined as the total work output divided by the energy put into the first engine by heat. Show that the overall efficiency e is given by

$e=e_1+e_2-e_1{e}_2$

What If? For parts (b) through (e) that follow, assume the two engines are Carnot engines. Engine 1 operates between temperatures T_h and T_i. The gas in engine 2 varies in temperature between T_i and T_c. In terms of the temperatures, (b) what is the efficiency of the combination engine? (c) Does an improvement in net efficiency result from the use of two engines instead of one? (d) What value of the intermediate temperature T_i results in equal work being done by each of the two engines in series? (e) What value of T_i results in each of the two engines in series having the same efficiency?

To determine

The overall efficiency of the two engine is $e=e_1+e_2-e_1{e}_2$.

Answer to Problem 22.30P

The overall efficiency of the two engine is $e=e_1+e_2-e_1{e}_2$.

Explanation of Solution

The efficiency of the two engine is $e_1$ and $e_2$. The lower and higher temperature of the engine $1$ is $T_{\text{i}}$ and $T_{\text{h}}$ respectively. The lower and the higher temperature of the engine $2$ is $T_{\text{c}}$ and $T_{\text{i}}$ respectively.

Write the expression of the efficiency of the engine $1$.

    $\begin{array}{c}{e}_1=\frac{W_{\text{eng}\text{1}}}{Q_{1\text{h}}}\\ W_{\text{eng}\text{1}}=e_1{Q}_{1\text{h}}\end{array}$

Here, $W_{\text{eng}\text{1}}$ is the work done by the engine $1$ and $Q_{\text{1h}}$ is the heat supplied to the engine $1$.

Write the expression of the efficiency of the engine $2$.

    $\begin{array}{c}{e}_2=\frac{W_{\text{eng}2}}{Q_{\text{2h}}}\\ W_{\text{eng}2}=e_2{Q}_{\text{2h}}\end{array}$

Here, $W_{\text{eng}2}$ is the work done by the engine $2$ and $Q_{\text{2h}}$ is the heat supplied to the engine $2$.

The exhaust heat of engine $1$ is supplied to the engine $2$.

Equate the exhaust heat of engine $1$ to the engine $2$

    $\begin{array}{c}{Q}_{\text{2h}}=Q_{1\text{c}}\\ =Q_{1\text{h}}-W_{\text{eng}\text{1}}\end{array}$

Here, $Q_{1\text{c}}$ is the exhaust heat of engine $1$.

Substitute $e_1{Q}_{1\text{h}}$ for $W_{\text{eng}\text{1}}$ in above equation.

    $Q_{\text{2h}}=Q_{1\text{h}}-e_1{Q}_{1\text{h}}$

Write the expression of the efficiency of the two engine device.

    $e=\frac{W_{\text{eng}\text{1}}+W_{\text{eng}\text{2}}}{Q_{\text{1h}}}$

Substitute $e_1{Q}_{1\text{h}}$ for $W_{\text{eng}\text{1}}$ and $e_2{Q}_{\text{2h}}$ for $e_2{Q}_{\text{2h}}$ in above equation.

    $e=\frac{e_1{Q}_{1\text{h}}+e_2{Q}_{\text{2h}}}{Q_{\text{1h}}}$

Conclusion:

Substitute $Q_{1\text{h}}-e_1{Q}_{1\text{h}}$ for $Q_{\text{2h}}$ in above equation to find $e$.

    $\begin{array}{c}e=\frac{e_1{Q}_{1\text{h}}+e_2\left(Q_{1\text{h}}-e_1{Q}_{1\text{h}}\right)}{Q_{\text{1h}}}\\ =\frac{e_1{Q}_{1\text{h}}+e_2{Q}_{1\text{h}}-e_1{e}_2{Q}_{1\text{h}}}{Q_{1\text{h}}}\\ =e_1+e_2-e_1{e}_2\end{array}$

Therefore, the overall efficiency of the two engine is $e=e_1+e_2-e_1{e}_2$.

To determine

The efficiency of the combination engine.

Answer to Problem 22.30P

The efficiency of the combination engine is $1-\frac{T_{\text{c}}}{T_{\text{h}}}$.

Explanation of Solution

Write the expression for the overall efficiency of the two engine

    $e=e_1+e_2-e_1{e}_2$                                                                      (I)

Write the expression of the Carnot efficiency of the engine $1$.

    $e_1=1-\frac{T_{\text{i}}}{T_{\text{h}}}$

Write the expression of the Carnot efficiency of the engine $2$.

    $e_2=1-\frac{T_{\text{c}}}{T_{\text{i}}}$

Conclusion:

Substitute $1-\frac{T_{\text{i}}}{T_{\text{h}}}$ for $e_1$ and $1-\frac{T_{\text{c}}}{T_{\text{i}}}$ for $e_2$ in equation (I).

    $\begin{array}{c}e=\left(1-\frac{T_{\text{i}}}{T_{\text{h}}}\right)+\left(1-\frac{T_{\text{c}}}{T_{\text{i}}}\right)-\left(1-\frac{T_{\text{i}}}{T_{\text{h}}}\right)\left(1-\frac{T_{\text{c}}}{T_{\text{i}}}\right)\\ =2-\frac{T_{\text{i}}}{T_{\text{h}}}-\frac{T_{\text{c}}}{T_{\text{i}}}-1+\frac{T_{\text{c}}}{T_{\text{i}}}+\frac{T_{\text{i}}}{T_{\text{h}}}-\frac{T_{\text{i}}}{T_{\text{h}}}\times \frac{T_{\text{c}}}{T_{\text{i}}}\\ =1-\frac{T_{\text{c}}}{T_{\text{h}}}\end{array}$

Therefore, the efficiency of the combination engine is $1-\frac{T_{\text{c}}}{T_{\text{h}}}$.

To determine

Whether the efficiency is improved by using two engine instead of one.

Answer to Problem 22.30P

The efficiency is remains same even after combining the two engine.

Explanation of Solution

The Carnot engine has the maximum efficiency and no engine can have the efficiency more than the Carnot engine’s efficiency.

If the two Carnot engine is combined together than the efficiency is remain same because the combined engine is also a Carnot engine. So there is no requirement to use two engines simultaneously instead of one engine to improve the net efficiency. The work output increases but on the other hand the heat supplied also increase.

Conclusion:

Therefore, the efficiency is remains same even after combining the two engine.

To determine

The value of the intermediate temperature $T_{\text{i}}$ due to which the both the engine produce same work when connects in series.

Answer to Problem 22.30P

The value of the intermediate temperature $T_{\text{i}}$ due to which the both the engine produce same work when connects in series is $\frac{1}{2}\left(T_{\text{h}}+T_{\text{c}}\right)$.

Explanation of Solution

The work done by both the engine is same.

Equating both work done

    $W_{\text{eng}\text{2}}=W_{\text{eng}\text{1}}$

Write the expression of the efficiency of the two engine device.

    $e=\frac{W_{\text{eng}\text{1}}+W_{\text{eng}2}}{Q_{\text{1h}}}$

Substitute $W_{\text{eng}\text{1}}$ for $W_{\text{eng}2}$ in above equation.

    $\begin{array}{c}e=\frac{W_{\text{eng}\text{1}}+W_{\text{eng}1}}{Q_{\text{1h}}}\\ =\frac{2W_{\text{eng}\text{1}}}{Q_{\text{1h}}}\end{array}$

Substitute $e_1$ for $\frac{W_{\text{eng}\text{1}}}{Q_{\text{1h}}}$ in above equation.

    $e=2e_1$

Conclusion:

Substitute $1-\frac{T_{\text{c}}}{T_{\text{h}}}$ for $e$ and $1-\frac{T_{\text{i}}}{T_{\text{h}}}$ for $e_1$ in above equation.

    $\begin{array}{c}1-\frac{T_{\text{c}}}{T_{\text{h}}}=2\left(1-\frac{T_{\text{i}}}{T_{\text{h}}}\right)\\ 2T_{\text{i}}=T_{\text{h}}+T_{\text{c}}\\ T_{\text{i}}=\frac{1}{2}\left(T_{\text{h}}+T_{\text{c}}\right)\end{array}$

Therefore, the value of the intermediate temperature $T_{\text{i}}$ due to which the both the engine produce same work when connects in series is $\frac{1}{2}\left(T_{\text{h}}+T_{\text{c}}\right)$.

To determine

The value of the intermediate temperature $T_{\text{i}}$ due to which the efficiency of both the engine is same when connects in series.

Answer to Problem 22.30P

The value of the intermediate temperature $T_{\text{i}}$ due to which the efficiency of both the engine is same when connects in series is $\sqrt{T_{\text{h}}\cdot T_{\text{c}}}$.

Explanation of Solution

Write the expression for the efficiency of both the engine is same.

    $e_1=e_2$

Conclusion:

Substitute $1-\frac{T_{\text{i}}}{T_{\text{h}}}$ for $e_1$ and $1-\frac{T_{\text{c}}}{T_{\text{i}}}$ for $e_2$ in above equation.

    $\begin{array}{c}1-\frac{T_{\text{i}}}{T_{\text{h}}}=1-\frac{T_{\text{c}}}{T_{\text{i}}}\\ T_{\text{i}}=\sqrt{T_{\text{h}}\cdot T_{\text{c}}}\end{array}$

Therefore, the value of the intermediate temperature $T_{\text{i}}$ due to which the efficiency of both the engine is same when connects in series is $\sqrt{T_{\text{h}}\cdot T_{\text{c}}}$.