91Ó°ÊÓ

A turbojet engine has the following design-point parameters: $$ \begin{aligned} M_{0} &=0, p_{0}=101.33 \mathrm{kPa}, T_{0}=15.2^{\circ} \mathrm{C} \\ \pi_{\mathrm{d}} &=0.98 \\ \pi_{\mathrm{c}} &=25, e_{\mathrm{c}}=0.90 \\ M_{4} &=1.0 \\ Q_{\mathrm{R}} &=42,800 \mathrm{~kJ} / \mathrm{kg}, \pi_{\mathrm{b}}=0.95, \eta_{\mathrm{b}}=0.98, \tau_{\lambda}=6.0 \\ e_{\mathrm{t}} &=0.85, \eta_{\mathrm{m}}=0.98 \\ \pi_{\mathrm{n}} &=0.97, p_{9}=p_{0} \\ \dot{m}_{\mathrm{c} 2} &=80 \mathrm{~kg} / \mathrm{s}, M_{z 2}=0.50 \end{aligned} $$ Calculate (a) fuel-to-air ratio \(f\) (b) turbine total temperature ratio \(\tau_{\mathrm{t}}\) For the following off-design operation $$ \begin{aligned} M_{0} &=0.85, p_{0}=20 \mathrm{kPa}, T_{0}=-15^{\circ} \mathrm{C} \\ \tau_{\lambda} &=6.5 \\ e_{\mathrm{t}} &=0.80, \eta_{\mathrm{m}}=0.98, \pi_{\mathrm{n}}=0.97, p_{9}=p_{0} \end{aligned} $$ Assume a calorically perfect gas with \(\gamma=1.4\) and \(c_{p}=1004\) \(\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\) constant throughout the engine, and calculate (c) \(\pi_{\mathrm{c}-\mathrm{off} \text {-design }}\) (d) the ratio of corrected shaft speeds \(N_{\mathrm{c} 2, \mathrm{O}-\mathrm{D}} / N_{\mathrm{c} 2, \mathrm{D}}\) (e) the corrected mass flow rate at off-design (kg/s) (f) the axial Mach number at the engine face, \(M_{z 2}\), atoff-design (g) thrust-specific fuel consumption at design and offdesign in \(\mathrm{mg} / \mathrm{s} / \mathrm{N}\)

Step by step solution

01

Calculations for Fuel-to-air Ratio

Firstly, the fuel-to-air ratio (f) can be calculated using the energy balance equation of the combustor. The energy given to the air by the burning fuel is \(fQ_{R}\) and the energy obtained by the air is \(c_{p}T_{4}(\tau_{\lambda}-1)\). Equating energy given and obtained provides the fuel-to-air ratio.

02

Calculation for Turbine Total Temperature Ratio

For the turbine total temperature ratio \(\tau_{t}\), we make use of the relationship derived from the energy balance at the turbine. The energy that the turbine extracts from the air is \(\tau_{t}=\tau_{\lambda}/\tau_{r}\) where \(\tau_{r}=(1+f)\pi_{r}\) with \(\pi_{r}\) for the ram compression ratio derived from the isentropic flow relations.

03

Calculation for Corrected Shaft Speed Ratio

For corrected shaft speed ratio, we can make use of the equation \(\frac{N_{C2, O-D}}{N_{C2, D}}=\sqrt{\frac{T_{0, O-D}}{T_{0, D}}}\times \frac{p_{0, D}}{p_{0, O-D}}\) by substituting the given temperature and pressure at off-design operation.

04

Calculation for Corrected Mass Flow Rate

The corrected mass flow rate can be obtained using the ratio \(\frac{\dot{m}_{C2, O-D}}{\dot{m}_{C2, D}}=\frac{T_{0, O-D}}{T_{0, D}}\sqrt{\frac{p_{0, D}}{p_{0, O-D}}}\) where we substitute the given temperature and pressure at off-design operation.

05

Calculation for Axial Mach Number

The axial mach number at off-design, \(M_z2\) can be obtained by solving mass flow rate per unit area \(A2_{O-D}\) which is \(\dot{m}_{C2, O-D}=\dot{m}_{C2, D}A2_{O-D}M_z2 \sqrt{\frac{\gamma RT_{0, O-D}}{1+\frac{\gamma -1}{2}M_z2^2}}\) for \(M_z2\) where A refers to the choked flow area at station 2.

06

Calculation for Thrust-specific Fuel Consumption

Finally, Thrust-specific fuel consumption can be calculated using \(TSFC=\frac{\dot{m}_f}{F}\) where thrust, F, can be calculated by using the equation \(F=\dot{m}(1+f)V9-A9p9+(A2p2-A9p9)\) for choked flow conditions in outlet plane 9 of the engine.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fuel-to-Air Ratio

The fuel-to-air ratio is crucial in a turbojet engine. It represents the proportion of fuel mass to air mass. This ratio is significant because it determines the energy input to the engine. The equation utilized for calculation is derived from the energy balance within the combustor.

Here, the burning fuel imparts energy, calculated as \(fQ_{R}\), to the air. Conversely, the air gains energy calculated by \(c_{p}T_{4}(\tau_{\lambda}-1)\). By equating these two energy expressions, one finds the fuel-to-air ratio.

• \(f\) - Fuel-to-air ratio
• \(Q_{R}\) - Heating value of the fuel
• \(c_{p}\) - Specific heat at constant pressure
• \(T_{4}\) - Total temperature at combustor exit
• \(\tau_{\lambda}\) - Total temperature ratio through combustion

Understanding this balance enables efficient engine operation and performance enhancements.

Turbine Total Temperature Ratio

The turbine total temperature ratio, \(\tau_{t}\), is a key parameter to evaluate the efficiency of energy extraction in a turbine. It relates the temperature before and after the turbine.

In our turbojet analysis, this ratio is informed by the energy balance within the turbine. The total temperature ratio is defined as \(\tau_{t}=\frac{\tau_{\lambda}}{\tau_{r}}\), with \(\tau_{\lambda}\) being the total temperature ratio through combustion and \(\tau_{r}\) a derived ram compression ratio. This ram compression ratio can be found using isentropic flow relations and the fuel-to-air ratio \(f\).

• Efficient calculation ensures optimal turbine energy extraction.
• A proper \(\tau_{t}\) reflects balanced combustion and turbine processing.

In summary, knowing \(\tau_{t}\) helps in designing engines with improved efficiency and performance.

Corrected Shaft Speed Ratio

The corrected shaft speed ratio assesses how well the turbojet engine can adapt to off-design conditions. Understanding this ratio is vital for maintaining engine performance under varying atmospheric conditions.

This calculation uses the relation \(\frac{N_{C2, O-D}}{N_{C2, D}}=\sqrt{\frac{T_{0, O-D}}{T_{0, D}}}\times \frac{p_{0, D}}{p_{0, O-D}}\). Here, you'll need the temperature and pressure data both at design and off-design states.

• \(N_{C2, O-D}\) - Corrected shaft speed off-design
• \(N_{C2, D}\) - Corrected shaft speed at design
• Variables \(T\) and \(p\) - Temperature and pressure conditions

Keen insight into this evaluation promotes a better understanding of performance limitations and necessary adjustments.

Corrected Mass Flow Rate

Knowing the corrected mass flow rate at off-design conditions assists engineers in determining how the engine manages airflow variances. This is integral to sustaining performance and efficiency.

The relationship used is \(\frac{\dot{m}_{C2, O-D}}{\dot{m}_{C2, D}}=\frac{T_{0, O-D}}{T_{0, D}}\sqrt{\frac{p_{0, D}}{p_{0, O-D}}}\). This formula relies on temperatures and pressures during different operational states.

• \(\dot{m}_{C2, O-D}\) - Off-design mass flow rate
• \(\dot{m}_{C2, D}\) - Design mass flow rate
• Inputs include environmental temperature and pressure readings.

Adjustments based on these calculations can significantly enhance engine reliability.

Thrust-Specific Fuel Consumption

Thrust-specific fuel consumption (TSFC) is used to measure the fuel efficiency of a turbojet engine. It indicates how much fuel is needed to produce a specific amount of thrust.

Calculated using \(TSFC=\frac{\dot{m}_f}{F}\), where \(\dot{m}_f\) is the fuel mass flow rate and \(F\) is the engine thrust. TSFC helps compare how different engines or settings affect performance.

• A lower TSFC indicates a more efficient engine.
• This can inform decisions on enhancements to reduce fuel consumption.

By reviewing TSFC across design and off-design conditions, engineers can fine-tune the engine for optimal efficiency.