Chapter 7.7, Problem 21P

(A)

Interpretation Introduction

Interpretation:

The work produced in the turbine per mole

Concept Introduction:

The molar entropy equation for a reversible, steady-state, and adiabatic turbine.

${\underset{\_}{S}}_2-{\underset{\_}{S}}_1=0$

Here, final molar entropy of the fluid leaving the turbine is ${\underset{\_}{S}}_2$ and initial molar entropy of the fluid entering the turbine is ${\underset{\_}{S}}_1$.

The Lee Kesler correlation residuals entropy.

$\begin{array}{l}\left({\underset{\_}{S}}_2-{\underset{\_}{S}}_2^{ig}\right)+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-\left({\underset{\_}{S}}_1-{\underset{\_}{S}}_1^{ig}\right)=0\\ \left({\underset{\_}{S}}^{R,mix}\right)+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-\left({\underset{\_}{S}}_1-{\underset{\_}{S}}_1^{ig}\right)=0\end{array}$

Here, final molar entropy for an ideal gas state is ${\underset{\_}{S}}_2^{ig}$, initial molar enthalpy for an ideal gas state is ${\underset{\_}{S}}_1^{ig}$, and molar residual entropy of mixture is ${\underset{\_}{S}}^{R,mix}$.

The ideal gas entropy change.

${\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}=C_P^{\ast }\ln \left(\frac{T_2}{T_1}\right)-R\ln \left(\frac{P_2}{P_1}\right)$

Here, constant pressure heat capacity on a molar basis for ideal gas is $C_P^{*}$, final temperature is $T_2$, initial temperature is $T_1$, gas constant is R, final pressure is $P_2$, and initial pressure is $P_1$.

The expression for the initial residual entropy obtained from the van der Waals equation.

$\begin{array}{l}{\underset{\_}{S}}_1^R=R\ln \left(Z_1\right)+R\ln \left(\frac{{\underset{\_}{V}}_1-b}{{\underset{\_}{V}}_1}\right)\\ {\underset{\_}{S}}_1-{\underset{\_}{S}}_1^{ig}=R\ln \left(Z_1\right)+R\ln \left(\frac{{\underset{\_}{V}}_1-b}{{\underset{\_}{V}}_1}\right)\end{array}$

Here, initial molar residual entropy is ${\underset{\_}{S}}^R{}_1$, initial compressibility factor is $Z_1$, initial molar volume is ${\underset{\_}{V}}_1$, and parameter in Van der Waals, Soave, or Peng-Robinson EOS is b.

The initial molar volume $\left({\underset{\_}{V}}_1\right)$.

$P_1=\frac{R{T}_1}{{\underset{\_}{V}}_1-b}-\frac{a}{{\underset{\_}{V}}_1^2}$

Here, Van der Waals, Soave, or Peng-Robinson EOS parameter is a.

The final molar volume in liquid phase $\left({\underset{\_}{V}}_2^L\right)$.

$P_2=\frac{R{T}_2}{{\underset{\_}{V}}_2^L-b}-\frac{a}{{\left({\underset{\_}{V}}_2^L\right)}^2}$

The final compressibility factor in liquid phase $\left(Z_2^L\right)$.

$Z_2^L=\frac{P_2{\underset{\_}{V}}_2}{R{T}_2}$

The expression for the final residual entropy in liquid phase $\left({\underset{\_}{S}}_2^{R,L}\right)$ obtained from the van der Waals equation.

${\underset{\_}{S}}_2^{R,L}=R\ln \left(Z_2^L\right)+R\ln \left(\frac{{\underset{\_}{V}}_2^L-b}{{\underset{\_}{V}}_2^L}\right)$

The energy balance for the reversible turbine.

$\frac{{\dot{W}}_{S,rev}}{\dot{\eta }}={\underset{\_}{H}}_2-{\underset{\_}{H}}_1$

Here, rate of shaft work for reversible turbine is ${\dot{W}}_{S,rev}$, net efficiency is $\dot{\eta }$, and final and initial molar enthalpies are ${\underset{\_}{H}}_2\text{and}{\underset{\_}{H}}_1$ respectively.

The Lee Kesler correlation residuals enthalpy.

$\frac{{\dot{W}}_{S,rev}}{\dot{\eta }}=\left({\underset{\_}{H}}_2-{\underset{\_}{H}}_2^{ig}\right)+\left({\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}\right)-\left({\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}\right)$

Here, final molar enthalpy for an ideal gas state is ${\underset{\_}{H}}_2^{ig}$, and initial molar enthalpy for an ideal gas state is ${\underset{\_}{H}}_1^{ig}$.

The ideal gas change enthalpy.

${\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}={\displaystyle {\int }_{T_1}^{T_2}{C}_P^{\ast }}dT$

Here, change in temperature is $dT$, and Here, constant pressure heat capacity on a molar basis for ideal gas is $C_P^{*}$.

(B)

Interpretation Introduction

Interpretation:

The work required in the pump per mole

Concept Introduction:

The work required in the pump, per mole $\left({\dot{W}}_{S,pump}/\dot{\eta }\right)$.

$\frac{{\dot{W}}_{S,pump}}{\dot{\eta }}={\underset{\_}{V}}_2^L\left(P_1-P_2\right)$

(C)

Interpretation Introduction

Interpretation:

The efficiency of the cycle

Concept Introduction:

The parameter ${\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}$.

${\underset{\_}{H}}_2-{\underset{\_}{H}}_2^{ig}=R{T}_2\left(Z_2^L-1\right)-\frac{a}{{\underset{\_}{V}}_2^L}$

The energy balance for steady state system.

$\begin{array}{l}0=\dot{\eta }{\underset{\_}{H}}_{in}-\dot{\eta }{\underset{\_}{H}}_{out}+\dot{Q}+{\dot{W}}_{S,pump}\\ \frac{\dot{Q}}{\dot{\eta }}={\underset{\_}{H}}_{out}-{\underset{\_}{H}}_{in}-\frac{{\dot{W}}_{S,pump}}{\dot{\eta }}\\ =\left({\underset{\_}{H}}_2-{\underset{\_}{H}}_2^{ig}\right)+\left({\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}\right)-\left({\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}\right)-\frac{{\dot{W}}_{S,pump}}{\dot{\eta }}\end{array}$

Here, rate of heat added or removed from the system is $\dot{Q}$, initial molar enthalpy is ${\underset{\_}{H}}_{in}$, and final molar enthalpy is ${\underset{\_}{H}}_{out}$.

Write the efficiency of the cycle $\left(\eta \right)$.

$\eta =\frac{{\dot{W}}_{S,rev}-\frac{{\dot{W}}_{S,pump}}{\dot{\eta }}}{\frac{\dot{Q}}{\dot{\eta }}}$