Chapter 15.7, Problem 87P

To determine

The entropy generated during the combustion of n-Octane.

The exergy destructed during the combustion of n-Octane.

Answer to Problem 87P

The entropy generated during the combustion of n-Octane is $103.4\text{kJ}/\text{K}\cdot \text{kg of }{\text{C}}_8{\text{H}}_{18}$.

The exergy destructed during the combustion of n-Octane is $30,805\text{kJ}/\text{K}\cdot \text{kg of }{\text{C}}_8{\text{H}}_{18}$.

Explanation of Solution

Write the complete balanced reaction for combustion of octane $\left({\text{C}}_3{\text{H}}_8\right)$ with 70% excess theoretical amount of air.

$\begin{array}{l}{\text{C}}_8{\text{H}}_{18}+1.7a_{\text{th}}\left({\text{O}}_2+3.76{\text{N}}_2\right)\\ \to 8{\text{CO}}_2+9{\text{H}}_2\text{O}+\text{0}\text{.7}{a}_{\text{th}}{\text{O}}_2+\left(1.7\right)3.76a_{\text{th}}{\text{N}}_2\end{array}$ (I)

Write the energy balance equation for the combustion of octane $\left({\text{C}}_3{\text{H}}_8\right)$.

$-Q_{\text{out}}={\displaystyle \sum N_P{\left({\overline{h}}_f^o+\overline{h}-{\overline{h}}^o\right)}_P}-{\displaystyle \sum N_R{\left({\overline{h}}_f^o+\overline{h}-{\overline{h}}^o\right)}_R}$ (II)

Here, the amount of heat transfer for the combustion of octane gas is $Q_{\text{out}}$, number of moles of product is $N_P$, number of moles of reactants is $N_R$, enthalpy of vaporization is ${\overline{h}}_f^o$, enthalpy of formation at a temperature is $\overline{h}$, and the enthalpy of formation at the standard reference state is ${\overline{h}}^o$.

Write the formula to calculate the entropy of an $i$ component $\left(S_i\right)$.

$S_i=N_i{\overline{s}}_i\left(T,P_i\right)$

$S_i=N_i\left[{\overline{s}}_i^o\left(T,P_o\right)-R_u\ln \left(y{i}_{}{P}_m\right)\right]$ (III)

Here, number of moles of i component is $N_i$, entropy of the i component is ${\overline{s}}_i\left(T,P_i\right)$, entropy of vaporization is ${\overline{s}}_i^o\left(T,P_o\right)$, universal gas constant is $R_u$, mole fraction of i component is $y_i$, and the total pressure is $P_m$.

Write the formula to calculate the entropy generation $\left(S_{\text{gen}}\right)$ for the combustion process.

$S_{\text{gen}}=S_P-S_R+\frac{Q_{\text{out}}}{T_{\text{surr}}}$ (IV)

Here, entropy of the products is $S_P$, entropy of the reactants is $S_R$, and the temperature of the surroundings is $T_{\text{surr}}$.

Write the expression to calculate the entropy generated per unit mass $\left({\overline{S}}_{\text{gen}}\right)$.

${\overline{S}}_{\text{gen}}=\frac{S_{\text{gen}}}{M_{\text{fuel}}}$ (V)

Here, molar mass of the fuel is $M_{\text{fuel}}$.

Write the expression to calculate the exergy destructed per unit mass $\left({\overline{X}}_{\text{dest}}\right)$.

${\overline{X}}_{\text{dest}}=\frac{T_o{S}_{\text{gen}}}{M_{\text{fuel}}}$ (VI)

Here, temperature of surroundings is $T_o$.

Conclusion:

Balance for the oxygen from combustion equation (I).

$\begin{array}{l}1.7a_{\text{th}}=8+4.5+0.7a_{\text{th}}\\ a_{\text{th}}=12.5\end{array}$

Rewrite the combustion reaction of propane with 95% theoretical amount of air.

${\text{C}}_8{\text{H}}_{18}\left(l\right)+1.7\left(12.5\right)\left[{\text{O}}_2+3.76{\text{N}}_2\right]\to \left\{\begin{array}{l}8{\text{CO}}_2+9{\text{H}}_2\text{O}+\left(1.7\times 12.5\right){\text{O}}_2\\ +\left(1.7\times 12.5\times 3.76\right){\text{N}}_2\end{array}\right\}$

${\text{C}}_8{\text{H}}_{18}\left(l\right)+21.25\left[{\text{O}}_2+3.76{\text{N}}_2\right]\to 8{\text{CO}}_2+9{\text{H}}_2\text{O}+8.75{\text{O}}_2+79.9{\text{N}}_2$ (VII)

Rewrite the Equation (II) for the heat transfer from combustion.

$\begin{array}{c}-Q_{\text{out}}={{\displaystyle \sum N_P\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}}_P+{{\displaystyle \sum N_R\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}}_R\\ =\left\{\begin{array}{l}{N}_{{\text{CO}}_2}{\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}_{{\text{CO}}_2}+N_{{\text{H}}_2\text{O}}{\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}_{{\text{H}}_2\text{O}}\\ +N_{{\text{O}}_2}{\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}_{{\text{O}}_2}+N_{{\text{N}}_2}{\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}_{{\text{N}}_2}\\ -N_{{\text{C}}_8{\text{H}}_{\text{18}}}{\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}_{{\text{C}}_8{\text{H}}_{\text{18}}}-N_{{\text{O}}_2}{\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}_{{\text{O}}_2}\\ -N_{{\text{N}}_2}{\left({\overline{h}}_f^{\circ }+\overline{h}-{\overline{h}}^{\circ }\right)}_{{\text{N}}_2}\end{array}\right\}\end{array}$ (VIII)

Here, number of moles of product is $N_P$, number of moles of reactants is $N_R$, number of moles of ${\text{CO}}_{\text{2}}$ is $N_{{\text{CO}}_{\text{2}}}$, number of moles of ${\text{H}}_{\text{2}}\text{O}$ is $N_{{\text{H}}_2\text{O}}$, number of moles of ${\text{O}}_{\text{2}}$ is $N_{{\text{O}}_{\text{2}}}$, number of moles of ${\text{N}}_{\text{2}}$ is $N_{{\text{N}}_{\text{2}}}$, number of moles of ${\text{C}}_8{\text{H}}_{18}$ is $N_{{\text{C}}_8{\text{H}}_{18}}$, enthalpy at reference state is ${\overline{h}}_f^{\circ }$, sensible enthalpy at specified state is $\overline{h}$, and sensible enthalpy at reference state is ${\overline{h}}^{\circ }$.

From the Table A-18 through Table A-26, obtain the enthalpies of vaporization and the enthalpies of formation for different substance as follows,

Substance	${\overline{h}}_f^{°}$ $\text{kJ}/\text{kmol}$	${\overline{h}}_{298\text{K}}^{°}$ $\text{kJ}/\text{kmol}$	${\overline{h}}_{600\text{K}}^{°}$ $\text{kJ}/\text{kmol}$	${\overline{h}}_{1340\text{K}}$ $\text{kJ}/\text{kmol}$
${\text{C}}_8{\text{H}}_{18}\left(l\right)$	$-249,950$	0	0	0
${\text{O}}_2$	0	$8682$	$17,929$	$43,475$
${\text{N}}_2$	0	$8669$	$17,563$	$41,539$
${\text{H}}_2\text{O}\left(g\right)$	$-241,820$	$9904$	0	$50,612$
${\text{CO}}_2$	$-393,520$	$9364$	0	$61,813$

Substitute $8\text{kmol}$ for $N_{{\text{CO}}_2}$, $-393,520\text{kJ}/\text{kmol}$ for ${\overline{h}}_{f,{\text{CO}}_2}^{°}$, $9,364\text{kJ}/\text{kmol}$ for ${\overline{h}}_{298\text{K},{\text{CO}}_2}^{°}$, $61,183\text{kJ}/\text{kmol}$ for ${\overline{h}}_{1340\text{K},{\text{CO}}_2}$, $9\text{kmol}$ for $N_{{\text{H}}_2\text{O}}$, $-241,820\text{kJ}/\text{kmol}$ for ${\overline{h}}_{f,{\text{H}}_2\text{O}}^{°}$, $50,612\text{kJ}/\text{kmol}$ for ${\overline{h}}_{1340\text{K},{\text{H}}_2\text{O}}$, $9,904\text{kJ}/\text{kmol}$ for ${\overline{h}}_{298\text{K},{\text{H}}_2\text{O}}^{°}$, $8.75\text{kmol}$ for $N_{{\text{O}}_2}$, 0 for ${\overline{h}}_{f,{\text{O}}_2}^{°}$, $8,682\text{kJ}/\text{kmol}$ for ${\overline{h}}_{298\text{K},{\text{O}}_2}^{°}$, $43,475\text{kJ}/\text{kmol}$ for ${\overline{h}}_{1340\text{K},{\text{O}}_2}$, $\text{79}\text{.9}\text{kmol}$ for $N_{{\text{N}}_2}$, 0 for ${\overline{h}}_{f,N_2}^{°}$, $8,669\text{kJ}/\text{kmol}$ for ${\overline{h}}_{298\text{K},{\text{N}}_2}^{°}$, $41,539\text{kJ}/\text{kmol}$ for ${\overline{h}}_{1340\text{K},{\text{N}}_2}$, $1\text{kmol}$ for $N_{{\text{C}}_8{\text{H}}_{18}}$, $-208,450\text{kJ}/\text{kmol}$ for ${\overline{h}}_{f,{\text{C}}_8{\text{H}}_{18}}^{°}$, 0 for ${\overline{h}}_{298\text{K},{\text{C}}_8{\text{H}}_{18}}^{°}$, and 0 for ${\overline{h}}_{600\text{K},{\text{C}}_8{\text{H}}_{18}}$, $21.25\text{kmol}$ for $N_{{\text{O}}_2}$, $17,929\text{kJ}/\text{kmol}$ for ${\overline{h}}_{600\text{K},{\text{O}}_2}$, and $17,563\text{kJ}/\text{kmol}$ for ${\overline{h}}_{600\text{K},{\text{N}}_2}$ in Equation (VIII).

$\begin{array}{c}-Q_{\text{out}}=\left\{\begin{array}{c}\left(8\text{kmol}\right)\left(\begin{array}{l}-393,520\text{kJ}/\text{kmol}+61,183\text{kJ}/\text{kmol}-\\ 9,364\text{kJ}/\text{kmol}\end{array}\right)\\ +\left(9\text{kmol}\right)\left(\begin{array}{l}-241,820\text{kJ}/\text{kmol}+50,162\text{kJ}/\text{kmol}-\\ 9,904\text{kJ}/\text{kmol}\end{array}\right)\\ +\left(8.75\text{kmol}\right)\left(0+43,475\text{kJ}/\text{kmol}-8,682\text{kJ}/\text{kmol}\right)\\ +\left(79.9\text{kmol}\right)\left(0+41,539\text{kJ}/\text{kmol}-8,669\text{kJ}/\text{kmol}\right)\\ -\left(1\text{kmol}\right)\left(-208,450\text{kJ}/\text{kmol}+0-0\right)\\ -\left(21.25\text{kmol}\right)\left(0+17,929\text{kJ}/\text{kmol}-8,682\text{kJ}/\text{kmol}\right)\\ -\left(79.9\text{kmol}\right)\left(0+17,563\text{kJ}/\text{kmol}-8,669\text{kJ}/\text{kmol}\right)\end{array}\right\}\\ =-2,265,004\text{kJ}/\text{kmol}\text{of}{\text{C}}_{\text{8}}{\text{H}}_{\text{18}}\end{array}$

From the Table A-18 through Table A-26, obtain the entropy values for different substance and tabulate using the Equation (III).

Represent the entropy calculations for the reactants as in Table below.

Substance	$N_i$	$y_i$	${\overline{s}}_i$, $\left(T,1\text{atm}\right)$	$R_u\ln \left(y_i{P}_m\right)$	$N_i{\overline{s}}_i$
${\text{C}}_8{\text{H}}_{18}\left(l\right)$	1		466.73	14.79	451.94
${\text{O}}_2$	21.25	0.21	226.35	1.81	4771.48
${\text{N}}_2$	79.9	0.79	212.07	12.83	15,919.28
Entropy of the reactant, $S_R$					$S_R=21,142.70\text{kJ}/\text{K}$

Represent the entropy calculations for the products as in Table below.

Substance	$N_i$	$y_i$	${\overline{s}}_i$, $\left(T,1\text{atm}\right)$	$R_u\ln \left(y_i{P}_m\right)$	$N_i{\overline{s}}_i$
${\text{CO}}_2$	8	0.0757	292.11	–6.673	2390.26
${\text{H}}_2\text{O}\left(g\right)$	9	0.0852	250.45	–5.690	2305.26
${\text{O}}_2$	8.75	0.0828	257.97	–5.928	2309.11
${\text{N}}_2$	79.9	0.7563	241.77	–12.46	18,321.87
Entropy of the product, $S_P$					$S_P=25,326.50\text{kJ}/\text{K}$

Substitute $25,326.50\text{kJ}/\text{K}$ for $S_P$, $21,142.70\text{kJ}/\text{K}$ for $S_R$, $2,265,004\text{kJ}/\text{kmol}\text{of}{\text{C}}_{\text{8}}{\text{H}}_{\text{18}}$ for $Q_{\text{out}}$, and $\text{298}\text{K}$ for $T_{\text{surr}}$ in Equation (IV).

$\begin{array}{c}{S}_{\text{gen}}=25,326.50\text{kJ}/\text{K}-21,142.70\text{kJ}/\text{K}+\frac{2,265,004\text{kJ}/\text{kmol}\text{of}{\text{C}}_{\text{8}}{\text{H}}_{\text{18}}}{\text{298}\text{K}}\\ =11,784.5\text{kJ}/\text{K}\text{per kmol of}{\text{C}}_{\text{8}}{\text{H}}_{\text{18}}\end{array}$

From the table A-2 of “Molar mass, gas constants, and critical-point properties”, select the molar masses of liquid octane as,

$M_{\text{fuel}}=114\text{kg}/\text{kmol}$

Substitute $11,784.5\text{kJ}/\text{K}\cdot \text{kmol}$ for $S_{\text{gen}}$, and $114\text{kg}/\text{kmol}$ for $M_{\text{fuel}}$ in Equation (V).

$\begin{array}{c}{\overline{S}}_{\text{gen}}=\frac{11,784.5\text{kJ}/\text{K}\cdot \text{kmol}}{114\text{kg}/\text{kmol}}\\ =103.4\text{kJ}/\text{K}\cdot \text{kg}{\text{C}}_{\text{8}}{\text{H}}_{\text{18}}\end{array}$

Thus, the entropy generated during the combustion of n-Octane is $103.4\text{kJ}/\text{K}\cdot \text{kg of }{\text{C}}_8{\text{H}}_{18}$.

Substitute $\text{298}\text{K}$ for $T_0$, $11,784.5\text{kJ}/\text{K}\cdot \text{kmol}$ for $S_{\text{gen}}$, and $114\text{kg}/\text{kmol}$ for $M_{\text{fuel}}$ in Equation (VI).

$\begin{array}{c}{\overline{X}}_{\text{dest}}=\frac{\left(\text{298}\text{K}\right)\left(11,784.5\text{kJ}/\text{K}\cdot \text{kmol}\right)}{114\text{kg}/\text{kmol}}\\ =30,805\text{kJ}/\text{K}\cdot \text{kg}{\text{C}}_{\text{8}}{\text{H}}_{\text{18}}\end{array}$

Thus, the exergy destructed during the combustion of n-Octane is $30,805\text{kJ}/\text{K}\cdot \text{kg of }{\text{C}}_8{\text{H}}_{18}$.