Chapter 5.7, Problem 15P

(A)

Interpretation Introduction

Interpretation:

The efficiency of the turbine.

Concept Introduction:

The expression of the efficiency of the turbine $\left(\eta \right)$ is,

$\eta =\frac{W_{S,actual}}{W_{S,reversible}}$

Here, actual shaft work is $W_{S,actual}$, and reversible shaft work is $W_{S,reversible}$.

Set up an entropy balance around the reversible turbine to obtain the work done by the turbine if it was reversible as,

$M_2{\widehat{S}}_2-M_1{\widehat{S}}_1=m_{in}{\widehat{S}}_{in}-m_{out}{\widehat{S}}_{out}+\frac{Q}{T}+S_{gen}$

Here, final mass of the actual stream is $M_2$, initial mass of the actual stream is $M_1$, final specific entropy of actual stream is ${\widehat{S}}_2$, initial specific entropy of actual stream is ${\widehat{S}}_1$, initial mass of the stream is $m_{in}$, final mass of the stream is $m_{out}$, final specific entropy of stream is ${\widehat{S}}_{out}$, initial specific entropy of stream is ${\widehat{S}}_{in}$, heat added or removed from the system is Q, temperature is T, and entropy is generated within the boundaries of the system is $S_{gen}$.

Set an energy balance around the reversible turbine as,

$\frac{d}{dt}\left\{M\left(\widehat{U}+\frac{V^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\dot{m}}_{in}\left({\widehat{H}}_{in}+\frac{V_{in}^2}{2}+g{h}_{in}\right)-{\dot{m}}_{out}\left({\widehat{H}}_{out}+\frac{V_{out}^2}{2}+g{h}_{out}\right)\\ +{\dot{W}}_{\text{S}}+W_{\text{EC}}+\dot{Q}\end{array}\right]$

Here, time is $t$, total mass is $M$, specific internal energy is $\widehat{U}$, velocity is $V$, acceleration due to gravity is $g$, height is $h$, initial mass flow rate is ${\dot{m}}_{in}$, initial specific enthalpy is ${\widehat{H}}_{in}$, initial velocity is $V_{in}$, initial height of the gas is $h_{in}$, final mass flow rate is ${\dot{m}}_{out}$, final height of the gas is $h_{out}$, rate at which shaft work is added to the system is ${\dot{W}}_{\text{S}}$, rate at which work is added to the system through expansion or contraction of the system is ${\dot{W}}_{\text{EC}}$, and rate at which heat is added to the system is $\dot{Q}$.

The expression to obtain the reversible quality of the steam $\left(q_{rev}\right)$ is,

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

(B)

Interpretation Introduction

Interpretation:

The work required by the pump.

Concept Introduction:

The expression of work required by the pump by modeling the liquid as constant specific volume is,

$\begin{array}{c}{\dot{W}}_S={\displaystyle {\int }_{P=P_{in}}^{P=P_{out}}\dot{V}dP}\\ =\dot{V}{\displaystyle {\int }_{P=P_{in}}^{P=P_{out}}dP}\\ =\dot{m}\widehat{V}\left(P_{out}-P_{in}\right)\\ \frac{{\dot{W}}_S}{\dot{m}}=\widehat{V}\left(P_{out}-P_{in}\right)\end{array}$

Here, outside pressure is $P_{out}$, inside pressure is $P_{in}$, volume rate is $\dot{V}$, change in pressure is $dP$, and specific volume is $\widehat{V}$.

(C)

Interpretation Introduction

Interpretation:

The overall efficiency of the heat engine.

Concept Introduction:

The expression of the overall efficiency of the heat engine $\left({\eta }_{H.E}\right)$ is,

${\eta }_{H.E}=\frac{-{\dot{W}}_{S,actual}/\dot{m}}{\dot{Q}/\dot{m}}$

(D)

Interpretation Introduction

Interpretation:

The efficiency of the Carnot heat engine.

Concept Introduction:

The expression of the efficiency of a Carnot heat engine $\left({\eta }_{Carnot}\right)$ is,

${\eta }_{Carnot}=1-\frac{T_C}{T_H}$

Here, temperature of low temperature heat reservoir is $T_C$, and temperature of high temperature heat reservoir is $T_H$.