Question 7.13: In an adiabatic combustion chamber, a fuel mass flow rate of......
In an adiabatic combustion chamber, a fuel mass flow rate of 15 g/s of specific exergy 52.1 MJ/kg is consumed. A flow of hot gases, whose exergy is 120 kW, is produced. This flow is passed through an expander, where an electrical po-wer of 20 kW is produced, leaving a flow of gases still hot with an exergy content of 50 kW, which is used in subsequent processes for DHW production. Determine:
(a) The unit exergy cost and unit exergoeconomic cost (€c/kWh) of the gases at the combustion chamber outlet.
(b) The unit exergy cost and unit exergoeconomic cost (€c/kWh) of the electricity produced.
(a) Undertaking a balance of exergy costs in the combustion chamber we have
k_{g,1}^{*}\dot{B}_{g,1}=k_{F}^{*}\dot{B}_{F}We assume that k_{F}^{*}=1, so that from the previous equation we get
k_{g,1}^{*}=\frac{\dot{B}_{F}}{\dot{B}_{g,1}}=6.51To calculate the unit exergoeconomic cost, we carry out a balance of economic costs. Previously, to determine the capital cost rate of the combustion chamber, we calculate the annuity factor, which is
a_{c c}=\frac{i(1+i)^{n}}{(1+i)^{n}-1}=0.0735so that the capital cost rate of the combustion chamber is
Z_{c c}={\frac{a I_{c c}}{H}}=25.2{\frac{€}{h}}The economic costs balance in the combustion chamber gives
c_{g,1}{\dot{B}}_{g,1}=c_{F}{\dot{{B}}}_{F}+{\dot{Z}}_{c c}\ \ \rightarrow\ c_{g,1}=40.5\ {\frac{c€}{\mathrm{KWh}}}To express this cost per energy unit, one would have to know the relation between the energy and exergy of those gases.
(b) Undertaking an exergy costs balance in the expander, we have
Since the fuel of the turbine is {\dot{B}}_{g,1}-{\dot{B}}_{g,2}, according to P4 proposition of the Exergy Cost Theory, we have that k_{g,1}^{*}=k_{g,2}^{*} and, therefore, returning to the exergy costs balance equation, we get
k_{e}^{*}E=k_{g,1}^{*}\left(\dot{B}_{g,1}-\dot{B}_{g,2}\right)\rightarrow k_{e}^{*}=k_{g,1}^{*}\frac{\dot{B}_{g,1}-\dot{B}_{g,2}}{E}=22.78For the calculation of the exergoeconomic cost, we determine in the first place the capital cost rate of the expansor \dot Z_{e x p}, which is
a_{e x p}=\frac{i(1+i)^{n}}{(1+i)^{n}-1}=0.0578 \\ Z_{e x p}={\frac{a I_{e x p}}{H}}=9.9{\frac{{€}}{h}}Undertaking the exergoeconomic costs balance in the expander, we have
c_{e}E=c_{g,1}\left(\dot{B}_{g,1}-\dot{B}_{g,2}\right)+\dot{Z}_{e x p}\rightarrow\ c_{e}=1.91\ \frac{€}{\mathrm{kWh}}