Chapter 7, Problem 75P

To determine

The plot for power produced and the cycle efficiency against the source temperature.

Explanation of Solution

Given:

The high-temperature reservoir of heat engine $\left(T_H\right)$ varies from $300°\text{C}$ to $1000°\text{C}$.

The low-temperature reservoir of heat engine $\left(T_L\right)$ are $0°\text{C}$, $25°\text{C}$, and $50°\text{C}$.

The amount of the heat supplied to the engine $\left(Q_H\right)$ is $65,000\text{kJ}/\text{min}$.

Calculation:

When, $T_H=477°\text{C}$, and $T_L=25°\text{C}$.

Calculate the Carnot efficiency of the heat engine.

  ${\eta }_{\text{th,max}}=1-\frac{T_L}{T_H}$

  $\begin{array}{c}{\eta }_{\text{th,max}}=1-\frac{\left(25°\text{C}\right)}{\left(477°\text{C}\right)}\\ =1-\frac{\left(25°\text{C}+\text{273}\text{K}\right)}{\left(477°\text{C}+\text{273}\text{K}\right)}\\ =1-\left(0.3973\right)\\ =0.603\end{array}$

Calculate the net- work output of the heat engine $\left({\dot{W}}_{\text{net,out}}\right)$ using the thermal efficiency of a heat engine.

  $\begin{array}{l}{\eta }_{\text{th,max}}=\frac{{\dot{W}}_{\text{net,out}}}{{\dot{Q}}_H}\\ {\dot{W}}_{\text{net,out}}={\eta }_{\text{th,max}}\times {\dot{Q}}_H\end{array}$

  $\begin{array}{c}{\dot{W}}_{\text{net,out}}=\left(0.603\right)\left(65,000\text{kJ}/\text{min}\right)\\ =39,195\text{kJ}/\text{min}\\ =39,195\text{kJ}/\text{min}\times \left(\frac{0.016667\text{kW}}{1\text{kJ}/\text{min}}\right)\\ =653.25\text{kW}\end{array}$

Similarly, use excel spread sheet to calculate the ${\eta }_{\text{th,max}}$ and ${\dot{W}}_{\text{net,out}}$ for various source temperatures $\left(T_H\right)$ ranging from $300°\text{C}$ to $1000°\text{C}$ with the sink temperatures $\left(T_L\right)$ of $0°\text{C}$, $25°\text{C}$, and $50°\text{C}$. The calculated values are tabulated and shown below.

Source temperature $\left(T_H\right)$ in $°\text{C}$	Sink temperature  $\left(T_L\right)$ in $\left(°\text{C}\right)$	Thermal efficiency $\left({\eta }_{\text{th, max}}\right)$	Power produced $\left(W_{\text{net,out}}\right)$
300	0	0.524	34031.41
400	0	0.594	38632.99
500	0	0.647	42043.98
600	0	0.687	44673.54
700	0	0.719	46762.59
800	0	0.746	48462.26
900	0	0.767	49872.12
1000	0	0.786	51060.49
Table 1

Source temperature $\left(T_H\right)$ in $°\text{C}$	Sink temperature  $\left(T_L\right)$ in $\left(°\text{C}\right)$	Thermal efficiency $\left({\eta }_{\text{th, max}}\right)$	Power produced $\left(W_{\text{net,out}}\right)$
300	25	0.480	31195.46
400	25	0.557	36218.42
500	25	0.614	39941.79
600	25	0.659	42812.14
700	25	0.694	45092.50
800	25	0.722	46947.81
900	25	0.746	48486.79
1000	25	0.766	49783.97
Table 2

Source temperature $\left(T_H\right)$ in $°\text{C}$	Sink temperature  $\left(T_L\right)$ in $\left(°\text{C}\right)$	Thermal efficiency $\left({\eta }_{\text{th, max}}\right)$	Power produced $\left(W_{\text{net,out}}\right)$
300	50	0.436	28359.51
400	50	0.520	33803.86
500	50	0.582	37839.59
600	50	0.630	40950.74
700	50	0.668	43422.40
800	50	0.699	45433.36
900	50	0.725	47101.45
1000	50	0.746	48507.46
Table 3

Refer the Tables 1, 2, and 3. Plot the graph of source temperature $\left(T_H\right)$ versus power produced $\left(W_{\text{net,out}}\right)$. The plot is shown in Figure 1.

Refer the Tables 1, 2, and 3. Plot the graph of source temperature $\left(T_H\right)$ versus thermal efficiency $\left({\eta }_{\text{th}}\right)$. The plot is shown in Figure 2.