Chapter 6.6, Problem 14P

A)

Interpretation Introduction

Interpretation:

The work produced in the turbine has to be determined.

Concept introduction:

Use the given equation of state derive the value of residual molar internal energy, enthalpy, and entropy in terms of constant, temperature, and coefficient of thermal expansion then use the energy balance and entropy expression to obtain the work produced in the turbine and the actual temperature of the exiting stream.

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 efficiency of the turbine;

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

theexpression 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 $T_2$.