Chapter 7.7, Problem 9P

(A)

Interpretation Introduction

Interpretation:

The rate at which work is produced

Concept Introduction:

Write the entropy balance equation around a reversible turbine.

${\underset{\_}{S}}_{out}-{\underset{\_}{S}}_{in}=0$

Here, final molar entropy is ${\underset{\_}{S}}_{out}$ and initial molar entropy is ${\underset{\_}{S}}_{in}$.

Write the expression for difference in entropy in terms of residual properties.

$\begin{array}{l}0={\underset{\_}{S}}_{out}-{\underset{\_}{S}}_{in}\\ 0={\underset{\_}{S}}_{out}^R+\left({\underset{\_}{S}}_{out}^{ig}-{\underset{\_}{S}}_{in}^{ig}\right)-{\underset{\_}{S}}_{in}^R\end{array}$

Here, final residual molar entropy is ${\underset{\_}{S}}_{out}^R$, initial residual molar entropy is ${\underset{\_}{S}}_{in}^R$, change in final molar entropy for an ideal gas is ${\underset{\_}{S}}_{out}^{ig}$, and change in initial molar entropy for an ideal gas is ${\underset{\_}{S}}_{in}^{ig}$.

Write the molar entropy residual function.

$\frac{\left(\underset{\_}{S}-{\underset{\_}{S}}^{ig}\right)}{R}=\frac{{\left(\underset{\_}{S}-{\underset{\_}{S}}^{ig}\right)}^0}{R}+\omega \frac{{\left(\underset{\_}{S}-{\underset{\_}{S}}^{ig}\right)}^1}{R}$

Here, gas constant is $R$, molar entropy is $\underset{\_}{S}$, and change in molar entropy for an ideal gas is ${\underset{\_}{S}}^{ig}$.

Write the expression to calculate the reduced temperature.

$T_r=\frac{T}{T_c}$

Here, critical temperature is $T_c$ and system temperature is T.

Calculate the reduced pressure $\left(P_r\right)$.

$P_r=\frac{P}{P_c}$

Here, pressure is P.

Write the expression to calculate the compressibility factor $\left(Z\right)$.

$Z=Z^0+\omega Z^1$

Here, compressibility of compound with $\omega =0$ is $Z^0$ and difference between $Z^0\text{and}Z$ is $Z^1$.

Write the equation to calculate the ideal gas change in entropy.

$\begin{array}{l}d\underset{\_}{S}=\frac{C_V^{\ast }}{T}dT+\frac{R}{\underset{\_}{V}}d\underset{\_}{V}\\ {\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}=\frac{C_V^{\ast }}{T}dT+\frac{R}{\underset{\_}{V}}d\underset{\_}{V}\end{array}$

Here, constant volume heat capacity for ideal gas is $C_V^{\ast }$, temperature is $T$, gas constant is $R$, molar volume is $\underset{\_}{V}$, change in molar entropy is $d\underset{\_}{S}$, change in final molar entropy for an ideal gas is ${\underset{\_}{S}}_2^{ig}$, and change in initial molar entropy for an ideal gas is ${\underset{\_}{S}}_1^{ig}$

Write the formula to calculate the constant volume heat capacity for ideal gas $\left(C_V^{\ast }\right)$.

$C_V^{\ast }=\left(A-R\right)+BT+C{T}^2+D{T}^3+E{T}^4$

Here, ideal gas coefficients are $A,B,C,D\text{and}E$ respectively.

Write the energy balance for the turbine.

$\frac{{\dot{W}}_S}{\dot{n}}={\underset{\_}{H}}_2-{\underset{\_}{H}}_1$

Here, mass flow rate is $\dot{n}$, final molar enthalpy is ${\underset{\_}{H}}_2$, initial molar enthalpy is ${\underset{\_}{H}}_1$, and rate of shaft work is added to the system is ${\dot{W}}_{\text{S}}$.

Write the difference in entropy in terms of residual properties.

$\frac{{\dot{W}}_S}{\dot{n}}=\left({\underset{\_}{H}}_2-{\underset{\_}{H}}_2^{ig}\right)+\left({\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}\right)-\left({\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}\right)$

Here, molar enthalpy for an ideal gas state at state 2 and 1 is ${\underset{\_}{H}}_2^{ig},\text{and}{\underset{\_}{H}}_1^{ig}$ respectively.

Write the residual molar enthalpy for the entering toluene:

$\begin{array}{l}\frac{{\underset{\_}{H}}_1^R}{R{T}_c}=\frac{{\left(\underset{\_}{H}-{\underset{\_}{H}}^{ig}\right)}^0}{R{T}_c}+\omega \frac{{\left(\underset{\_}{H}-{\underset{\_}{H}}^{ig}\right)}^1}{R{T}_c}\\ \frac{\left({\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}\right)}{R}=\frac{{\left(\underset{\_}{H}-{\underset{\_}{H}}^{ig}\right)}^0}{R{T}_c}+\omega \frac{{\left(\underset{\_}{H}-{\underset{\_}{H}}^{ig}\right)}^1}{R{T}_c}\end{array}$

Here, initial residual molar enthalpy is ${\underset{\_}{H}}_1^R$, molar enthalpy is $\underset{\_}{H}$, and change in molar enthalpy for an ideal gas is ${\underset{\_}{H}}^{ig}$.

(B)

Interpretation Introduction

Interpretation:

The rate at which work is produced

Concept Introduction:

Write the relationship between the parameter, m and Soave’s EOS.

$m=0.480+1.574\omega -0.176{\omega }^2$

Here, the acentric factor is $\omega$.

Write the expression to calculate the $\alpha$ expressed as a function of the reduced temperature.

$\alpha ={\left[1+m\left(1-T_r^{0.5}\right)\right]}^2$

Here, reduced temperature is $T_r$.

Write the expression to calculate the reduced temperature.

$T_r=\frac{T}{T_c}$

Here, critical temperature is $T_c$ and system temperature is T.

Write the expression to calculate the value of a at the critical point.

$a=0.42747R^2\frac{T_c^2}{P_c}\times \alpha$

Write the expression to calculate the value of b using the Soave equation.

$b=0.08664R\left(\frac{T_c}{P_c}\right)$

Write the Peng-Robinson equation.

$P=\frac{RT}{\underset{\_}{V}-b}-\frac{a}{\underset{\_}{V}\left(\underset{\_}{V}+b\right)+b\left(\underset{\_}{V}-b\right)}$

Here, molar volume is $\underset{\_}{V}$, parameters of Robinson equation are a, b, gas constant is R, temperature and pressure is T and P respectively.

Write the dimensionless group parameter that includes the Peng-Robinson parameter.

$A=\frac{aP}{R^2{T}^2}$

Here, dimensionless group parameter is $A$.

Write the dimensionless group parameter that includes the Peng-Robinson parameter.

$B=\frac{bP}{RT}$

Here, dimensionless group parameter is $B$.

Write the residual molar enthalpy for the inlet.

${\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}=\left(Z-1\right)-\left\{\left(\frac{A}{B\sqrt{8}}\right)\left(1+\frac{\kappa \sqrt{T_r}}{\sqrt{{\alpha }_c}}\right)\ln \left[\frac{Z+\left(1+\sqrt{2}\right)B}{Z+\left(1-\sqrt{2}\right)B}\right]\right\}$

Write the difference in entropy in terms of residual properties.

$\frac{{\dot{W}}_S}{\dot{n}}=\left({\underset{\_}{H}}_2-{\underset{\_}{H}}_2^{ig}\right)+\left({\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}\right)-\left({\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}\right)$

Here, molar enthalpy for an ideal gas state at state 2 and 1 is ${\underset{\_}{H}}_2^{ig},\text{and}{\underset{\_}{H}}_1^{ig}$ respectively.