Chapter 9, Problem 117P

To determine

The rate at which heat is added in the boiler, the power produced to operate the pumps, the net power produced by the cycle and the thermal efficiency of the cycle.

Explanation of Solution

Given:

Pressure of water at the condenser $\left(P_1\right)$ is $\text{50}\text{kPa}$.

Pressure of water at the boiler $\left(P_3\right)$ is $\text{6000}\text{kPa}$.

Temperature of water at the turbine inlet $\left(T_3\right)$ is $\text{450}°\text{C}$.

Mass flow rate of water $\left(\dot{m}\right)$ is $\text{20}\text{kg}/\text{s}$.

Temperature of water leaving the condenser $\left(T\right)$ is $\text{6}\text{.3}°\text{C}$.

Isentropic efficiency of the turbine $\left({\eta }_{\text{T}}\right)$ is $\text{0}\text{.94}$.

Calculation:

Draw the $T-s$ diagram of the cycle as in Figure (1).

The pressures are constant for the process 2 to 3 and process 4 to 1.

  $\begin{array}{l}{P}_2=P_3\\ P_4=P_1\end{array}$

The entropies are constant for the process 1 to 2 and process 3 to 4.

  $\begin{array}{l}{s}_1=s_2\\ s_3=s_4\end{array}$

Refer Table A-5, “Saturated water-Pressure table”, obtain the saturation temperature corresponding to the pressure of $\text{50}\text{kPa}$.

  $T_{\text{sat}@50\text{kPa}}=81.3°\text{C}$

Calculate the temperature at state 1$\left(T_1\right)$ .

  $\begin{array}{c}{T}_1=T_{\text{sat}@50\text{kPa}}-T\\ =81.3°\text{C}-6.3°\text{C}\\ =\text{75}°\text{C}\end{array}$

Refer Table A-4, “Saturated water-Temperature table”, obtain the enthalpy and specific volume at state 1 corresponding to the temperature of $\text{75}°\text{C}$.

  $\begin{array}{c}{h}_1=h_{f@75°\text{C}}\\ =314.03\text{kJ}/\text{kg}\\ v_1=v_{f@75°\text{C}}\\ =0.001026{\text{m}}^3/\text{kg}\end{array}$

Refer Table A-5, “Saturated water-Pressure table”, obtain the following properties corresponding to the pressure of $\text{50}\text{kPa}$.

  $\begin{array}{l}{h}_f=340.54\text{kJ}/\text{kg}\\ h_{fg}=2304.7\text{kJ}/\text{kg}\\ s_f=1.0912\text{kJ}/\text{kg}\cdot \text{K}\\ s_{fg}=6.5019\text{kJ}/\text{kg}\cdot \text{K}\end{array}$

Calculate the work done by the pump during process 1-2$\left(w_{\text{p,in}}\right)$.

  $\begin{array}{c}{w}_{\text{p,in}}=v_1\left(P_2-P_1\right)\\ =\left(0.001026{\text{m}}^3/\text{kg}\right)\left(6000\text{kPa}-50\text{kPa}\right)\\ =\left(0.001026{\text{m}}^3/\text{kg}\right)\left(6000\text{kPa}-50\text{kPa}\right)\left(\frac{1\text{kJ}}{\text{1}\text{kPa}\cdot {\text{m}}^3}\right)\\ =6.10\text{kJ}/\text{kg}\end{array}$

Calculate the enthalpy at state 2$\left(h_2\right)$.

  $\begin{array}{c}{h}_2=h_1+w_{\text{p,in}}\\ =314.03\text{kJ}/\text{kg}+6.10\text{kJ}/\text{kg}\\ =320.13\text{kJ}/\text{kg}\end{array}$

Refer Table A-6, “Superheated water”, obtain the enthalpy and entropy at state 3 corresponding to the pressure of $\text{6000}\text{kPa}$ and temperature of $450°\text{C}$.

  $\begin{array}{l}{h}_3=3302.9\text{kJ}/\text{kg}\\ s_3=6.7219\text{kJ}/\text{kg}\cdot \text{K}\end{array}$

Calculate the quality of water at state 4s $\left(x_{4s}\right)$.

  $\begin{array}{c}{x}_{4s}=\frac{s_4-s_f}{s_{fg}}\\ =\frac{s_3-s_f}{s_{fg}}\\ =\frac{6.7219\text{kJ}/\text{kg}\cdot \text{K}-1.0912\text{kJ}/\text{kg}\cdot \text{K}}{6.5019\text{kJ}/\text{kg}\cdot \text{K}}\\ =0.8660\end{array}$

Calculate the enthalpy at state 4s $\left(h_{4s}\right)$.

  $\begin{array}{c}{h}_{4s}=h_f+x_{4s}{h}_{fg}\\ =340.54\text{kJ}/\text{kg}+\left(0.8660\right)\left(2304.7\text{kJ}/\text{kg}\right)\\ =2336.4\text{kJ}/\text{kg}\end{array}$

Calculate the enthalpy at state 4$\left(h_4\right)$.

  ${\eta }_{\text{T}}=\frac{h_3-h_4}{h_3-h_{4s}}$

  $\begin{array}{c}{h}_4=h_3-\left({\eta }_{\text{T}}\right)\left(h_3-h_{4s}\right)\\ =3302.9\text{kJ}/\text{kg}-\left(0.94\right)\left(3302.9\text{kJ}/\text{kg}-2336.4\text{kJ}/\text{kg}\right)\\ =2394.4\text{kJ}/\text{kg}\end{array}$

Calculate the rate at which heat is added in the boiler $\left({\dot{Q}}_{\text{in}}\right)$.

  $\begin{array}{c}{\dot{Q}}_{\text{in}}=\dot{m}\left(h_3-h_2\right)\\ =\left(20\text{kg}/\text{s}\right)\left(3302.9\text{kJ}/\text{kg}-320.13\text{kJ}/\text{kg}\right)\\ =59,660\text{kW}\end{array}$

Thus, the rate at which heat is added in the boiler is $59,660\text{kW}$.

Calculate the work produced by the turbine $\left({\dot{W}}_{\text{T, out}}\right)$.

  $\begin{array}{c}{\dot{W}}_{\text{T, out}}=\dot{m}\left(h_3-h_4\right)\\ =\left(20\text{kg}/\text{s}\right)\left(3302.9\text{kJ}/\text{kg}-2394.4\text{kJ}/\text{kg}\right)\\ =18,170\text{kW}\end{array}$

Calculate the power produced to operate the pumps $\left({\dot{W}}_{\text{P, in}}\right)$.

  $\begin{array}{c}{\dot{W}}_{\text{P, in}}=\dot{m}\left(w_{\text{p,in}}\right)\\ =\left(20\text{kg}/\text{s}\right)\left(6.10\text{kJ}/\text{kg}\right)\\ =122\text{kW}\end{array}$

Thus, the power produced to operate the pumps is $122\text{kW}$.

Calculate the net power produced by the cycle $\left({\dot{W}}_{\text{net}}\right)$.

  $\begin{array}{c}{\dot{W}}_{\text{net}}={\dot{W}}_{\text{T, out}}-{\dot{W}}_{\text{P, in}}\\ =18,170\text{kW}-122\text{kW}\\ =18,050\text{kW}\end{array}$

Thus, the net power produced by the cycle is $18,050\text{kW}$.

Calculate the thermal efficiency of the cycle $\left({\eta }_{\text{th}}\right)$.

  $\begin{array}{c}{\eta }_{\text{th}}=\frac{{\dot{W}}_{\text{net}}}{{\dot{Q}}_{\text{in}}}\\ =\frac{18,050\text{kW}}{59,660\text{kW}}\\ =0.3025\\ =30.25\%\end{array}$

Thus, the thermal efficiency of the cycle is $30.25\%$.