A mixture of methane, ethane, and argon at \(25^{\circ} \mathrm{C}\) is burned
with excess air in a power-plant boiler. The hydrocarbons in the fuel are
completely consumed. The following variable definitions will be used
throughout this problem:
\(x_{\mathrm{M}}=\) mole fraction of methane in the fuel \(x_{\mathrm{A}}=\) mole
fraction of argon in the fuel \(P_{\mathrm{xs}}(\%)=\) percent excess air fed to
the furnace \(T_{\mathrm{u}}\left(^{\circ} \mathrm{C}\right)=\) temperature of
the entering air \(T_{\mathrm{s}}\left(^{\circ} \mathrm{C}\right)=\) stack gas
temperature
\(r=\) ratio of \(\mathrm{CO}_{2}\) to \(\mathrm{CO}\) in the stack gas
\(\left(\mathrm{mol} \mathrm{CO}_{2} / \mathrm{mol} \mathrm{CO}\right)\)
\(\dot{Q}(\mathrm{k} \mathrm{W})=\) rate of heat transfer from the furnace to
the boiler tubes
(a) Without doing any calculations, sketch the shapes of the plots you would
expect to obtain for plots
of \(\dot{Q}\) versus (i) \(x_{\mathrm{M}, \text { (ii) } x_{\mathrm{A}}, \text {
(iii) } P_{\mathrm{xs}}, \text { (iv) } T_{\mathrm{a}},(\mathrm{v})
T_{\mathrm{s}}, \text { and }(\mathrm{vi})} r,\) assuming in each case that the
other variables are held constant. Briefly state your reasoning for each plot.
(b) Take a basis of 1.00 mol/s of fuel gas, draw and label a flowchart, and
derive expressions for the molar flow rates of the stack gas components in
terms of \(x_{\mathrm{M}}, x_{\mathrm{A}}, P_{\mathrm{xs}},\) and \(r .\) Then
take as references the elements at \(25^{\circ} \mathrm{C}\), prepare and fill
in an inlet-outlet enthalpy table for the furnace, and derive expressions for
the specific molar enthalpies of the feed and stack gas components in terms of
\(T_{\mathrm{a}}\) and \(T_{\mathrm{s}}\)
(c) Calculate \(\dot{Q}(\mathrm{kW})\) for \(x_{\mathrm{M}}=0.85 \mathrm{mol}
\mathrm{CH}_{\mathcal{J}} / \mathrm{mol}, x_{\mathrm{A}}=0.05 \mathrm{mol}
\mathrm{Ar} / \mathrm{mol}, P_{\mathrm{xs}}=5 \%, r=10.0 \mathrm{mol}\)
\(\left.\mathrm{CO}_{2} / \mathrm{mol} \mathrm{CO}, T_{\mathrm{a}}=150^{\circ}
\mathrm{C}, \text { and } T_{\mathrm{s}}=700^{\circ} \mathrm{C} \text {
(Solution: } \dot{Q}=-655 \mathrm{kW} .\right)\)
(d) Prepare a spreadsheet that has columns for \(x_{\mathrm{M}},
x_{\mathrm{A}}, P_{\mathrm{xs}}, T_{\mathrm{a}}, r, T_{\mathrm{s}},\) and
\(\dot{Q},\) plus columns for any other variables you might need for the
calculation of \(\dot{Q}\) from given values of the preceding six variables
(e.g., component molar flow rates and specific enthalpies). Use the
spreadsheet to generate plots of
\(\dot{Q}\) versus each of the following variables over the specified ranges:
$$\begin{aligned}&x_{\mathrm{M}}=0.00-0.85 \mathrm{mol} \mathrm{CH}_{4} /
\mathrm{mol}\\\
&x_{\mathrm{A}}=0.01-0.05 \mathrm{mol} \mathrm{Ar} / \mathrm{mol}\\\
&P_{\mathrm{xs}}=0 \%-100 \%\\\
&T_{\mathrm{a}}=25^{\circ} \mathrm{C}-250^{\circ} \mathrm{C}\\\
&r=1-100 \mathrm{mol} \mathrm{CO}_{2} / \mathrm{mol} \mathrm{CO} \text { (make
the } r \text { axis logarithmic) }\\\
&T_{\mathrm{s}}=500^{\circ} \mathrm{C}-1000^{\circ} \mathrm{C}\end{aligned}$$
When generating each plot, use the variable values given in Part (c) as base
values. (For example, generate a plot of \(\dot{Q}\) versus \(x_{\mathrm{M}}\) for
\(x_{\mathrm{A}}=0.05, P_{\mathrm{xs}}=5 \%,\) and so on, with \(x_{\mathrm{M}}\)
varying from 0.00 to 0.85 on the horizontal axis.) If possible, include the
plots on the same spreadsheet as the data.
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