Question

An idealized diesel engine operates in a cycle known as the air-standard diesel cycle shown in the figure below. Fuel is sprayed into the cylinder at the point of maximum compression, B. Combustion occurs during the expansion B-C, which is modeled as an isobaric process. Show that the efficiency of an engine operating in this idealized diesel cycle is given by the following expression. (Submit a file with a maximum size of 1 MB.) 1/10-TA VTC e-1- Adiabatic processes V Vi= Choose File no file selected

Answer 1

Heat is provided only during the process from B to C, because here volume increases and the process is non-adiabatic.

Therefore, heat provided = .

Q provided nCp(Tc - TB) nR -(Tc – TB) 7-1

Now, work done in the process from A to B = [since A to B is an adiabatic process]

W AB nR(TB -TA) 1-7

Work done in the process from B to C = $W_{BC}=nR(T_{C}-T_{B})$ [since B to C is an isobaric process]

Work done in the process from C to D = $W_{CD}=\frac{nR(T_{D}-T_{C})}{1-\gamma}$ [since C to D is an adiabatic process]

Work done in the process from D to A = 0 [since D to A is an isochoric process].

Therefore, total work done = $W_{total}=W_{AB}+W_{BC}+W_{CD}+W_{DA}$

$=\frac{nR(T_{B}-T_{A})}{1-\gamma}+nR(T_{C}-T_{B})+\frac{nR(T_{D}-T_{C})}{1-\gamma}$

$=nR\left (\frac{(T_{B}-T_{A}+T_{D}-T_{C})}{1-\gamma}+(T_{C}-T_{B}) \right )$

Now, efficiency = Total work done / heat provided

$\Rightarrow \eta=\frac{nR\left (\frac{(T_{B}-T_{A}+T_{D}-T_{C})}{1-\gamma}+(T_{C}-T_{B}) \right )}{\frac{nR\gamma}{\gamma - 1}(T_{C}-T_{B})}$

  $=\frac{\left (\frac{(T_{B}-T_{A}+T_{D}-T_{C})}{1-\gamma}+(T_{C}-T_{B}) \right )}{\frac{\gamma}{\gamma - 1}(T_{C}-T_{B})}$

$=\frac{\left ((T_{B}-T_{A}+T_{D}-T_{C})+(1-\gamma)(T_{C}-T_{B}) \right )}{\gamma(T_{B}-T_{C})}$

$=\frac{\left ((-T_{A}+T_{D})-\gamma(T_{C}-T_{B}) \right )}{\gamma(T_{B}-T_{C})}$

$=\frac{\left (-T_{A}+T_{D} \right )}{\gamma(T_{B}-T_{C})}+\frac{-\gamma(T_{C}-T_{B})}{\gamma(T_{B}-T_{C})}$

  $=1-\frac{\left (T_{D}-T_{A} \right )}{\gamma(T_{C}-T_{B})}$. [Hence proved]