Chapter 3.9, Problem 14P

(A)

Interpretation Introduction

Interpretation:

Determine the work and heat transfer for the process.

Concept Introduction:

An energy balance equation piston cylinder arrangement.

$\Delta \left\{M\left(\widehat{U}+\frac{V^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\displaystyle \sum _{j=1}^{j=J}\left\{m_{j,in}\left({\widehat{H}}_j+\frac{V_j^2}{2}+g{h}_j\right)\right\}}\\ -{\displaystyle \sum _{k=1}^{k=K}\left\{m_{k,out}\left({\widehat{H}}_k+\frac{V_k^2}{2}+g{h}_k\right)\right\}}+W_{EC}+W_S+Q\end{array}\right]$

Here, mass of the system is $M$, specific internal energy is $\widehat{U}$, velocity is $V$, acceleration due to gravity is $g$, height is $h$, individual quantities of mass added to and removed from the process is $m_{j,in}\text{and}{m}_{k,out}$ respectively, initial specific enthalpy is ${\widehat{H}}_j$, final specific enthalpy is ${\widehat{H}}_k$, initial velocity is $V_j$, final velocity is $V_k$, initial height is $h_j$, final height is $h_k$, shaft work addition is $W_{\text{S}}$, work addition through expansion or contraction of the system is $W_{\text{EC}}$, and heat addition is $Q$.

Here, the height, velocity, shaft work is zero.

The expression to calculate the amount of work done $\left(W_{EC}\right)$:

$\begin{array}{c}{W}_{EC}=-PdV\\ =-P\left(V_{final}-V_{initial}\right)\\ =-P\left[\left({\widehat{V}}_{final}\times M\right)-\left({\widehat{V}}_{initial}\times M\right)\right]\end{array}$

Here, initial and final specific volume is ${\widehat{V}}_{initial}\text{and}{\widehat{V}}_{final}$, initial and final volume is $V_{initial}\text{and}{V}_{final}$, pressure is P.

(B)

Interpretation Introduction

Interpretation:

Determine the work, heat transfer for the process, and the percentage of part A far away from the Part B.

Concept Introduction:

The ideal gas equation in terms of initial volume:

$\begin{array}{c}{V}_{initial}=\frac{NR{T}_{initial}}{P}\\ =\frac{MR{T}_{initial}}{\left(MW\right)P}\end{array}$

Here, molecular weight of water is MW, number of moles is N, gas constant is R, and initial temperature is $T_{initial}$.

The ideal gas equation in terms of final volume:

$\begin{array}{c}{V}_{final}=\frac{NR{T}_{final}}{P}\\ =\frac{MR{T}_{final}}{\left(MW\right)P}\end{array}$

Here, molecular weight of water is MW, number of moles is N, gas constant is R, and initial temperature is $T_{initial}$.

The expression to calculate the work done through expansion or contraction:

$W_{EC}=-P\left(V_{final}-V_{initial}\right)$

The expression to calculate the specific change in internal energy:

$d\widehat{U}=C_V^{*}dt$

Integrate the above equation with initial and final condition.

$\begin{array}{l}{\displaystyle \underset{{\widehat{U}}_{initial}}{\overset{{\widehat{U}}_{final}}{\int }}d\widehat{U}}={\displaystyle \underset{initial}{\overset{final}{\int }}{C}_V^{*}dt}\\ {\widehat{U}}_{final}-{\widehat{U}}_{initial}={\displaystyle \underset{initial}{\overset{final}{\int }}\left(C_P^{*}-R\right)dt}\end{array}$

Here, constant volume heat capacity on a molar basis for ideal gas is $C_V^{*}$ , constant pressure heat capacity on a molar basis for ideal gas is $C_P^{*}$.

(C)

Interpretation Introduction

Interpretation:

When the steams expands and work done is zero in the part A, the percentage of part A far away from the Part B.