Chapter 5.7, Problem 6P

(A)

Interpretation Introduction

Interpretation:

Estimate the efficiency of the cycle and the quality of stream leaving the turbine when the turbine is reversible.

Concept Introduction:

The expression of efficiency of heat engine is,

$\begin{array}{c}{\eta }_{\text{H}\text{.E}}=\frac{\left|W_{\text{net}}\right|}{\left|Q_{\text{added}}\right|}\\ =\frac{\left|\frac{{\dot{W}}_{S,turbine}}{\dot{m}}+\frac{{\dot{W}}_{S,pump}}{\dot{m}}\right|}{\frac{{\dot{Q}}_H}{\dot{m}}}\end{array}$

Here, net work done on turbine and pump is $W_{net}$, mass flow rate is $\dot{m}$, heat added to the heat engine system is $Q_{added}$, rate of shaft work for turbine and pump is ${\dot{W}}_{S,turbine}$ and ${\dot{W}}_{S,pump}$, and rate of heat transfer to the heat engine is ${\dot{Q}}_H$.

The general expression for an entropy balance equation is,

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

Here, 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 is ${\dot{m}}_{j,in}$, ${\dot{m}}_{k,out}$, specific entropies of streams entering and leaving the system is ${\widehat{S}}_j,{\widehat{S}}_k$, 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 labelled n occurs is $T_n$, and the rate at which entropy is generated within the boundaries of the system is ${\dot{S}}_{gen}$ and time is t.

The expression of specific entropy of an outlet for a reversible process is,

${\widehat{S}}_{out,rev}=\left(1-q_{rev}\right){\widehat{S}}_L+q_{rev}{\widehat{S}}_V$

Here, mole fraction of the system for reversible process is $q_{rev}$.

The expression of the enthalpy for outlet steam in reversible case is,

${\widehat{H}}_{out,rev}=\left(1-q_{rev}\right){\widehat{H}}_L+q_{rev}{\widehat{H}}_V$

The energy balance equation for adiabatic steady state turbine is,

$\frac{{\dot{W}}_{S,turbine}}{\dot{m}}={\widehat{H}}_{out}-{\widehat{H}}_{in}$

Here, specific enthalpy at inlet and outlet is ${\widehat{H}}_{in}$ and ${\widehat{H}}_{out}$, mass flow rate is $\dot{m}$, and rate at which shaft work is added to the system is ${\dot{W}}_S$.

The energy balance equation around condenser is,

$\frac{d}{dt}\left\{M\left(\widehat{U}+\frac{v^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\displaystyle \sum _{j=1}^{j=J}{\dot{m}}_{j,in}\left({\widehat{H}}_j+\frac{v_j^2}{2}+g{h}_j\right)}-{\displaystyle \sum _{k=1}^{k=K}{\dot{m}}_{k,out}\left({\widehat{H}}_k+\frac{v_k^2}{2}+g{h}_k\right)}+\\ {\dot{W}}_S+{\dot{W}}_{EC}+\dot{Q}\end{array}\right]$

Here, total mass of the system is M, specific internal energy of the system is $\widehat{U}$, velocity of the system is v, height of the system is h, acceleration due to gravity is g, mass flow rate for inlet and outlet streams is ${\dot{m}}_{j,in}$ and ${\dot{m}}_{k,out}$, specific enthalpies of streams inlet and outlet is ${\widehat{H}}_j$ and ${\widehat{H}}_k$, heights at which streams enters and leave the system is $h_j$ and $h_k$, rate at which work is added to the system through expansion or contraction of the system is ${\dot{W}}_{EC}$, rate at which shaft work is added to the system is ${\dot{W}}_S$, and the rate at which heat is added to the system is $\dot{Q}$.

The expression to obtain the pump work is,

$\frac{{\dot{W}}_{S,pump}}{\dot{m}}=\widehat{V}\left(P_{out}-P_{in}\right)$

Here, inlet and outlet pressure at $40°\text{C}$ and $250°\text{C}$ is $P_{in}$ and $P_{out}$ respectively.

The equation of energy balance around whole engine is,

$0=\frac{{\dot{Q}}_C}{\dot{m}}+\frac{{\dot{Q}}_H}{\dot{m}}+\frac{{\dot{W}}_{S,pump}}{\dot{m}}+\frac{{\dot{W}}_{S,turbine}}{\dot{m}}$

Here, rate of heat exchange with low temperature reservoir is ${\dot{Q}}_H$.

(B)

Interpretation Introduction

Interpretation:

Estimate the efficiency of the cycle and the quality of stream leaving the turbine when the turbine efficiency is 75%.

Concept Introduction:

The expression of efficiency of turbine is,

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

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

The energy balance for an adiabatic steady state actual turbine is,

$\begin{array}{l}\frac{{\dot{W}}_{S,actualturbine}}{\dot{m}}={\widehat{H}}_{out}-{\widehat{H}}_{in}\\ {\widehat{H}}_{out}=\frac{{\dot{W}}_{S,actualturbine}}{\dot{m}}+{\widehat{H}}_{in}\end{array}$

Here, specific enthalpy at inlet and outlet is ${\widehat{H}}_{in}$ and ${\widehat{H}}_{out}$, mass flow rate is $\dot{m}$, and rate at which shaft work is added to the system for actual turbine is ${\dot{W}}_{S,actualturbine}$.

The expression to obtain the specific enthalpy at outlet state for VLE mixture at $40°\text{C}$ is,

${\widehat{H}}_{out}={\widehat{H}}_L+q\left({\widehat{H}}_V-{\widehat{H}}_L\right)$

(C)

Interpretation Introduction

Interpretation:

The flow rate of circulating water.

Concept Introduction:

Write the expression to obtain the mass flow rate.

$\dot{m}=\frac{\frac{{\dot{W}}_{net}}{\dot{m}}}{-\left(\frac{{\dot{W}}_{S,pump}}{\dot{m}}+\frac{{\dot{W}}_{S,actualturbine}}{\dot{m}}\right)}$

Here, net power is ${\dot{W}}_{net}$.