Chapter 4.8, Problem 15P

Interpretation Introduction

Interpretation:

The value of entropy and the process is possible or not.

Concept Introduction:

Theentropy balance equation for the turbine.

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

Here, total mass of container 2 is $M_2$, molar entropy of container 2 is ${\underset{\_}{S}}_2$, total mass of container 1 is $M_1$, molar entropy of container 1 is ${\underset{\_}{S}}_1$, initial mass is $m_{in}$, initial molar entropy is ${\underset{\_}{S}}_{in}$, final mass is $m_{out}$, final molar entropy is ${\underset{\_}{S}}_{out}$, temperature is $T$, heat is added or removed from the system is $Q$, and entropy is generated within the boundaries of the system is $S_{gen}$.

Thesteady state energy balance equation for the turbine.

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

Here, time taken 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 is $m_{in}$, initial specific enthalpy is ${\widehat{H}}_{in}$, initial velocity is $V_{in}$, initial height of the gas is $h_{in}$, final mass flow is $m_{out}$, final height of the gas is $h_{out}$, shaft work is added to the system is $W_{\text{S}}$, and work is added to the system through expansion or contraction of the system is $W_{\text{EC}}$.

The ideal gas equation.

$PV=nRT$

Here, number of mole is $n$, gas constant is $R$, temperature is $T_1$, and pressure is $P$.

The expression to determine the net molar entropy $\left({\underset{\_}{S}}_2-{\underset{\_}{S}}_1\right)$.

$\begin{array}{c}{\underset{\_}{S}}_2-{\underset{\_}{S}}_1=C_V^{\ast }\ln \left(\frac{T_2}{T_1}\right)+R\ln \left(\frac{R{T}_2/P_2}{R{T}_1/P_1}\right)\\ =C_V^{\ast }\ln \left(\frac{T_2}{T_1}\right)+R\ln \left(\frac{T_2{P}_1}{T_1{P}_2}\right)\end{array}$

Here, final temperature is $T_2$, final pressure is $P_2$,initial temperature is $T_1$, and initial pressure is $P_1$.