Chapter 4.8, Problem 17P

(A)

Interpretation Introduction

Interpretation:

The heat transfer, work done, change in internal energy, change in enthalpy, and change in entropy.

Concept Introduction:

Write the expression to determine the work done $\left(W_{EC}\right)$ through expansion or contraction process.

$\begin{array}{c}{W}_{EC}=-{\displaystyle {\int }_{T_1}^{T_2}PdV}\\ =-P{\displaystyle {\int }_{T_1}^{T_2}dV}\end{array}$

Here, pressure is $P$, initial temperature is $T_1$, final temperature is $T_2$, and change in volume is $dV$.

Write the expression to calculate the change in molar enthalpy $\left(d\underset{\_}{H}\right)$.

$\begin{array}{c}d\underset{\_}{H}={\displaystyle \int C_P^{\ast }}dT\\ =C_P^{\ast }\left(T_2-T_1\right)\end{array}$

Here, constant pressure heat capacity for an ideal gas is $C_P^{\ast }$ and change in temperature is $dT$.

Write the expression to determine the change in enthalpy $\left(\Delta H\right)$.

$\Delta H=n\left(d\underset{\_}{H}\right)$

Write the expression to determine the change in molar internal energy $\left(d\underset{\_}{U}\right)$.

$\begin{array}{c}d\underset{\_}{U}=-{\displaystyle \int C_V^{\ast }}dT\\ =-\left(1-C_P^{\ast }\right)\left(T_2-T_1\right)\end{array}$

Write the expression to determine the change in molar entropy $\left(\Delta \underset{\_}{S}\right)$.

$\Delta \underset{\_}{S}=C_P^{\ast }\ln \left(\frac{T_2}{T_1}\right)+R\ln \left(\frac{P_2}{P_1}\right)$

Here, initial pressure is $P_1$ and final pressure is $P_2$.

Write the expression to determine the change in entropy $\left(\Delta S\right)$.

$\Delta S=n\left(\Delta \underset{\_}{S}\right)$

Write the steady state energy balance equation for closed system.

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

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 done through expansion or contraction of the system is $W_{\text{EC}}$.

(B)

Interpretation Introduction

Interpretation:

The heat transfer, work done, change in internal energy, change in enthalpy, and change in entropy.

Concept Introduction:

Write the expression to determine work done $\left(W_{EC}\right)$ through expansion or contraction of the system.

$\begin{array}{c}{W}_{EC}=-{\displaystyle {\int }_{P_1}^{P_2}PdV}\\ =-P{\displaystyle {\int }_{P_1}^{P_2}dV}\end{array}$