Chapter 6.6, Problem 17P

A)

Interpretation Introduction

Interpretation:

Outlet pressure and the rate of work produced has to be determined.

Concept introduction:

The expression for the reversible process for a turbine is given as follows:

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

Here, turbine inlet and outlet molar entropy is ${\underset{\_}{S}}_1$ and ${\underset{\_}{S}}_2$, inlet and outlet turbine residual molar entropy is ${\underset{\_}{S}}_1^R$ and ${\underset{\_}{S}}_2^R$, turbine inlet and outlet molar entropy for an ideal gas state is ${\underset{\_}{S}}_1^{ig}$ and ${\underset{\_}{S}}_2^{ig}$ respectively.

The expression for the change in entropy as given as follows:

${\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}=C_P^{\ast }\ln \left(\frac{T_{2,rev}}{T_1}\right)+R\ln \left(\frac{P_1}{P_2}\right)$

Here, constant pressure heat capacity on a mass basis is $C_P^{\ast }$, gas constant is R, inlet temperature and pressure is $T_1$ and $P_1$, outlet pressure of turbine is $P_2$, and outlet temperature for a reversible process is $T_{2,rev}$.

The expression for the work produced in the turbine is given as follows:

$\frac{{\dot{W}}_{S,act}}{\dot{n}}=C_P^{\ast }\left(T_2-T_1\right)$

Here, actual temperature of the exiting stream is $T_2$.

B)

Interpretation Introduction

Interpretation:

The actual temperature of the exiting stream has to be determined.

Concept introduction:

Write the expression for the work produced in the turbine is given as follows:

$\frac{{\dot{W}}_{S,act}}{\dot{n}}=C_P^{\ast }\left(T_2-T_1\right)$

Here, actual temperature of the exiting stream is $T_2$.

The expression for the efficiency of the turbine;

$\eta =\frac{{\dot{W}}_{S,act}/\dot{n}}{{\dot{W}}_{S,rev}/\dot{n}}$

Here, actual rate of work produced is ${\dot{W}}_{S,act}/\dot{n}$, and efficiency is $\eta$.

Write the entropy balance for the reversible turbine.

$0=C_P^{\ast }\ln \left(T_{2,rev}/T_1\right)+R\ln \left(P_1/P_2\right)$

C)

Interpretation Introduction

Interpretation:

The actual temperature of the exiting stream has to be determined.

Concept introduction:

Write the energy balance to calculate the rate of work produced $\left({\dot{W}}_{S,rev}/\dot{n}\right)$.

$\begin{array}{c}{\dot{W}}_{S,rev}/\dot{n}={\underset{\_}{H}}_2-{\underset{\_}{H}}_1\\ =C_P^{\ast }\left(T_2-T_1\right)\end{array}$

Write the entropy balance for the reversible turbine.

$\begin{array}{l}{\underset{\_}{S}}_2-{\underset{\_}{S}}_1=0\\ {\underset{\_}{S}}_2^R+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-{\underset{\_}{S}}_1^R=0\\ \frac{-a{P}_2^3}{3}+C_P^{\ast }\ln \left(T_{2,rev}/T_1\right)+R\ln \left(P_1/P_2\right)-\frac{-a{P}_1^3}{3}=0\end{array}$        (4)

Here, molar entropy for an ideal gas state at state 2 and 1 is ${\underset{\_}{S}}_2^{ig},\text{and}{\underset{\_}{S}}_1^{ig}$ respectively, residual molar entropy at state 2 and 1 is ${\underset{\_}{S}}_2^R,\text{and}{\underset{\_}{S}}_1^R$ respectively, constant is $a$, pressure at state 2 or outlet pressure is $P_2$, constant pressure heat capacity for ideal gas is $C_P^{\ast }$ and inlet pressure is $P_1$.

D)

Interpretation Introduction

Interpretation:

The actual temperature of the exiting stream has to be determined.

Concept introduction:

Write the expression for the work produced in the turbine is given as follows:

$\frac{{\dot{W}}_{S,rev}}{\dot{n}}=C_P^{\ast }\left(T_{2_{rev}}-T_1\right)$

Here, actual temperature of the exiting stream is $T_2$.

The expression for the efficiency of the turbine;

$\eta =\frac{{\dot{W}}_{S,act}/\dot{n}}{{\dot{W}}_{S,rev}/\dot{n}}$

Here, actual rate of work produced is ${\dot{W}}_{S,act}/\dot{n}$, and efficiency is $\eta$.

Write the entropy balance for the reversible turbine.

$\begin{array}{l}{\underset{\_}{S}}_2-{\underset{\_}{S}}_1=0\\ {\underset{\_}{S}}_2^R+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-{\underset{\_}{S}}_1^R=0\\ \frac{-a{P}_2^3}{3}+C_P^{\ast }\ln \left(T_{2,rev}/T_1\right)+R\ln \left(P_1/P_2\right)-\frac{-a{P}_1^3}{3}=0\end{array}$        (4)

Here, molar entropy for an ideal gas state at state 2 and 1 is ${\underset{\_}{S}}_2^{ig},\text{and}{\underset{\_}{S}}_1^{ig}$ respectively, residual molar entropy at state 2 and 1 is ${\underset{\_}{S}}_2^R,\text{and}{\underset{\_}{S}}_1^R$ respectively, constant is $a$, pressure at state 2 or outlet pressure is $P_2$, constant pressure heat capacity for ideal gas is $C_P^{\ast }$ and inlet pressure is $P_1$.