Chapter 5.7, Problem 17P

(A)

Interpretation Introduction

Interpretation:

The flow rate at which the water/stream circulates through the process.

Concept Introduction:

Set an energy balance around CURRENT 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 $\dot{Q}$.

(B)

Interpretation Introduction

Interpretation:

The efficiency of the turbine in the current cycle.

Concept Introduction:

The generalized entropy balance 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, time 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 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 labeled n occurs is $T_n$, and the rate at which entropy is generated within the boundaries of the system is ${\dot{S}}_{gen}$.

The expression to obtain the quality of the steam $\left(q\right)$ is,

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

The expression to obtain the efficiency of the turbine in the CURRENT cycle $\left({\eta }_{CURRENT}\right)$ is,

${\eta }_{CURRENT}=\frac{{\dot{W}}_S}{-\frac{{\dot{W}}_{S,reversible}}{\dot{m}}}$

(C)

Interpretation Introduction

Interpretation:

The overall efficiency of the current cycle.

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 $\widehat{V}$.

The expression to obtain the overall efficiency of the CURRENT cycle $\left({\eta }_{H.E}\right)$ is,

${\eta }_{H.E}=\frac{-\left(\frac{{\dot{W}}_S}{\dot{m}}-{\dot{W}}_S\right)}{\dot{Q}/\dot{m}}$

(D)

Interpretation Introduction

Interpretation:

The power produced by the turbine in the UPGRADED cycle.

Concept Introduction:

The expression to obtain the power produced by the turbine in the UPGRADED cycle $\left(\dot{W}\right)$ is,

$\dot{W}=\frac{{\dot{W}}_{S,reversible}}{\dot{m}}\left({\eta }_{turbine}\right)$

Here, efficiency of the turbine is ${\eta }_{turbine}$.

(E)

Interpretation Introduction

Interpretation:

The overall efficiency of the UPGRADED cycle

Concept Introduction:

The expression to obtain the overall efficiency of the UPGRADED cycle $\left({\eta }_{H.E}\right)$ is,

${\eta }_{H.E}=\frac{-\left(\frac{{\dot{W}}_S}{\dot{m}}-\dot{W}\right)}{\dot{Q}/\dot{m}}$

(F)

Interpretation Introduction

Interpretation:

How long will the new turbine have to operatein order to pay for the $1.2 million cost of theupgrade?

Concept Introduction:

The expression to obtain the addition power produced $\left({\dot{W}}_{additional}\right)$ is,

${\dot{W}}_{additional}={\dot{W}}_{upgraded}-{\dot{W}}_{original}$

Here, upgraded power produced is ${\dot{W}}_{upgraded}$, and original power produced is ${\dot{W}}_{original}$.

The expression to obtain how long $\left(t\right)$ it will take to generate the revenue of $1.2 million is,

$t=\frac{\$1.2\text{million}}{\text{rate}\text{of}\text{revenue}}$