Chapter 4.8, Problem 16P

(A)

Interpretation Introduction

Interpretation:

To calculate the heat added to the gas.

Concept Introduction:

The expression of steady state energy balance equation on the gas.

$\Delta \left\{M\left(\widehat{U}+\frac{v^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\dot{m}}_{j,in}\left({\widehat{H}}_j+\frac{v_j^2}{2}+g{h}_j\right)-{\dot{m}}_{k,out}\left({\widehat{H}}_k+\frac{v_k^2}{2}+g{h}_k\right)+\\ W_{EC}+W_S+Q\end{array}\right]$

Here, total mass of the system is M, specific internal energy of the system is $\widehat{U}$, , time taken is t 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 are ${\dot{m}}_{j,in}$ and ${\dot{m}}_{k,out}$, specific enthalpies of streams inlet and outlet are ${\widehat{H}}_j$ and ${\widehat{H}}_k$, heights at which streams enters and leave the system are $h_j$ and $h_k$, total expansion/contraction work added throughout the process is $W_{EC}$, total shaft work added throughout the process is $W_S$, and total heat added throughout the process is Q.

(B)

Interpretation Introduction

Interpretation:

To calculate the heat added to the container.

Concept Introduction:

Write the steady state equation for a closed system by neglecting the shaft work and the potential energy and kinetic energy. Rewrite the equation in terms of specific heat capacity.

$M_2{\widehat{U}}_2-M_1{\widehat{U}}_1=Q$

$\begin{array}{l}M\left({\widehat{U}}_2-{\widehat{U}}_1\right)=Q\\ Q=M{C}_V\left(T_2-T_1\right)\end{array}$

Here, mass of a container is M and heat capacity is $C_V$.

(C)

Interpretation Introduction

Interpretation:

To calculate the change in entropy of the gas

Concept Introduction:

Write the expression to determine the molar change in entropy of the gas.

$\begin{array}{c}{\underset{\_}{S}}_2-{\underset{\_}{S}}_1=C_V^{*}\ln \frac{T_2}{T_1}+R\ln \frac{{\underset{\_}{V}}_2}{{\underset{\_}{V}}_1}\\ =C_V^{*}\ln \frac{T_2}{T_1}\end{array}$

Here, initial and final molar entropy is ${\underset{\_}{S}}_1$ and ${\underset{\_}{S}}_2$, initial and final molar volume is ${\underset{\_}{V}}_1$ and ${\underset{\_}{V}}_2$ respectively.

Write the expression to calculate the change in entropy of the gas.

$\Delta S=N\left({\underset{\_}{S}}_2-{\underset{\_}{S}}_1\right)$

(D)

Interpretation Introduction

Interpretation:

To calculate the change in entropy of the universe.

Concept Introduction:

Write the formula to calculate the change in entropy of furnace.

$\Delta S_{res}=\frac{Q_{res}}{T_{res}}$

Here, heat and temperature of reservoir are $Q_{res}$ and $T_{res}$.

Write the expression to calculate the change in entropy.

$\begin{array}{c}d{S}_{container}=\frac{d{Q}_{rev}}{T}\\ d{S}_{container}=\frac{d\left(M\widehat{U}\right)}{T}\end{array}$

Specific internal energy is related to $T$ through the heat capacity and $M$ is constant.

$d{S}_{container}=M{C}_V\ln \frac{T_2}{T_1}$

Write the expression to calculate the total entropy changes.

$\Delta S=\Delta S_{gas}+\Delta S_{cont}+\Delta S_{fur}$

(E)

Interpretation Introduction

Interpretation:

Aspects of this process.