Chapter 4.8, Problem 9P

(A)

Interpretation Introduction

Interpretation:

The work produced per kg of entering steam and the quality.

Concept Introduction:

Write the energy balance equation for an adiabatic steady state turbine.

$\begin{array}{c}\frac{{\dot{W}}_S}{\dot{m}}={\widehat{H}}_{out}-{\widehat{H}}_{in}\\ ={\widehat{H}}_{out,rev}-{\widehat{H}}_{in}\end{array}$

Here, rate of shaft work is ${\dot{W}}_S$, final reversible specific enthalpy is ${\widehat{H}}_{out,rev}$, mass flow rate is $\dot{m}$, final specific enthalpy is ${\widehat{H}}_{out}$, and initial specific enthalpy is ${\widehat{H}}_{in}$.

Write the entropy balance equation.

$\frac{d\left(M\widehat{S}\right)}{dt}={\displaystyle \sum _{j=1}^{j=J}{\dot{m}}_{j,in}{\widehat{S}}_j}-{\displaystyle \sum _{k=1}^{k=K}{\dot{m}}_{k,out}{\widehat{S}}_k}+{\displaystyle \sum _{n=1}^{n=N}\frac{{\dot{Q}}_n}{T_n}}+{\dot{S}}_{gen}$

Here, time taken is t, mass of the system is M, specific entropy of the system is $\widehat{S}$, mass flow rates of individual streams entering and leaving the system are ${\dot{m}}_{j,in}$ and ${\dot{m}}_{k,out}$, specific entropies of streams entering and leaving the system are ${\widehat{S}}_j$ and ${\widehat{S}}_k$, actual rate at which heat is added to or removed from the system at one particular location is ${\dot{Q}}_n$, the temperature of the system at the boundary where the heat transfer labeled n occurs is $T_n$, and the rate at which entropy is generated within the boundaries of the system is ${\dot{S}}_{gen}$.

Write the reversible quality $\left(q_{rev}\right)$ using entropy relation.

${\widehat{S}}_{out,rev}=\left(1-q_{rev}\right){\widehat{S}}^L+q_{rev}{\widehat{S}}^V$

Here, entropy of liquid phase is ${\widehat{S}}^L$ and entropy of vapor phase is ${\widehat{S}}^V$.

Write the final reversible specific enthalpy $\left({\widehat{H}}_{out,rev}\right)$.

${\widehat{H}}_{out,rev}=\left(1-q_{rev}\right){\widehat{H}}^L+q_{rev}{\widehat{H}}^V$

Here, enthalpy of liquid phase is ${\widehat{H}}^L$ and enthalpy of vapor phase is ${\widehat{H}}^V$.

(B)

Interpretation Introduction

Interpretation:

The final temperature of the leaving fluid

Concept Introduction:

Write the energy balance equation for an adiabatic steady state turbine.

$\begin{array}{c}\frac{{\dot{W}}_S}{\dot{m}}={\widehat{H}}_{out}-{\widehat{H}}_{in}\\ ={\widehat{H}}_{out,rev}-{\widehat{H}}_{in}\end{array}$

Here, rate of shaft work is ${\dot{W}}_S$, final reversible specific enthalpy is ${\widehat{H}}_{out,rev}$, mass flow rate is $\dot{m}$, final specific enthalpy is ${\widehat{H}}_{out}$, and initial specific enthalpy is ${\widehat{H}}_{in}$.

Write the entropy balance equation.

$\frac{d\left(M\widehat{S}\right)}{dt}={\displaystyle \sum _{j=1}^{j=J}{\dot{m}}_{j,in}{\widehat{S}}_j}-{\displaystyle \sum _{k=1}^{k=K}{\dot{m}}_{k,out}{\widehat{S}}_k}+{\displaystyle \sum _{n=1}^{n=N}\frac{{\dot{Q}}_n}{T_n}}+{\dot{S}}_{gen}$

Here, time taken is t, mass of the system is M, specific entropy of the system is $\widehat{S}$, mass flow rates of individual streams entering and leaving the system are ${\dot{m}}_{j,in}$ and ${\dot{m}}_{k,out}$, specific entropies of streams entering and leaving the system are ${\widehat{S}}_j$ and ${\widehat{S}}_k$, actual rate at which heat is added to or removed from the system at one particular location is ${\dot{Q}}_n$, the temperature of the system at the boundary where the heat transfer labeled n occurs is $T_n$, and the rate at which entropy is generated within the boundaries of the system is ${\dot{S}}_{gen}$.

Write the reversible quality $\left(q_{rev}\right)$ using entropy relation.

${\widehat{S}}_{out,rev}=\left(1-q_{rev}\right){\widehat{S}}^L+q_{rev}{\widehat{S}}^V$

Here, entropy of liquid phase is ${\widehat{S}}^L$ and entropy of vapor phase is ${\widehat{S}}^V$.

Write the final reversible specific enthalpy $\left({\widehat{H}}_{out,rev}\right)$.

${\widehat{H}}_{out,rev}=\left(1-q_{rev}\right){\widehat{H}}^L+q_{rev}{\widehat{H}}^V$

Here, enthalpy of liquid phase is ${\widehat{H}}^L$ and enthalpy of vapor phase is ${\widehat{H}}^V$.