/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 A turbojet engine has the follow... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 10

A turbojet engine has the following design parameters (which is at takeoff): \(\begin{aligned} M_{0} &=0 \\ p_{0} &=101.33 \mathrm{kPa}, T_{0}=15.2^{\circ} \mathrm{C}, \gamma_{\mathrm{c}}=1.4 \\ c_{p \mathrm{c}} &=1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \\ \pi_{\mathrm{d}} &=0.95 \\ \pi_{\mathrm{c}} &=30, e_{\mathrm{c}}=0.90 \\ Q_{\mathrm{R}} &=42,600 \mathrm{~kJ} / \mathrm{kg}, \pi_{\mathrm{b}}=0.95, \eta_{\mathrm{b}}=0.98, T_{\mathrm{t} 4}=1700^{\circ} \mathrm{C} \\ \eta_{\mathrm{m}} &=0.98, e_{\mathrm{t}}=0.85, \gamma_{\mathrm{t}}=1.33, c_{p \mathrm{t}}=1156 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \\ \pi_{\mathrm{n}} &=0.90 \\ p_{9} &=p_{0} \end{aligned}\) This engine powers an aircraft that cruises at \(M_{0}=0.80\) at an altitude where \(T_{0}=-35^{\circ} \mathrm{C}, p_{0}=20 \mathrm{kPa}\). The turbine entry temperature at cruise is \(T_{\mathrm{t} 4}=1500^{\circ} \mathrm{C}\). Assume that the engine has the same component efficiencies at cruise and takeoff, and the nozzle is perfectly expanded at cruise, as well. Calculate (a) the exhaust velocity \(V_{9}\) (in \(\mathrm{m} / \mathrm{s}\) ) at the design point, i.e., at takeoff (b) the thermal efficiency \(\eta_{\text {th }}\) at the design point (c) thrust-specific fuel consumption at the design point (d) the compressor pressure ratio at cruise (e) the exhaust velocity \(V_{9}\) (in \(\mathrm{m} / \mathrm{s}\) ) at cruise (f) the thermal efficiency \(\eta_{\text {th }}\) at cruise (g) the propulsive efficiency \(\eta_{\mathrm{p}}\) at cruise (h) the thrust-specific fuel consumption at cruise

Short Answer

Expert verified
This problem requires multiple calculations. These include the computation of exhaust velocity, thermal efficiency and fuel consumption at both takeoff and cruise, as well as the computation of compressor-pressure ratio at cruise and propulsive efficiency at cruise. The numbers for these calculations can be derived by substituting the given design parameters into specific equations for each element.

Step by step solution

01

Compute exhaust velocity at takeoff

The exhaust velocity \(V_{9}\) at the design point can be calculated using the equation: \(V_{9}=\sqrt{2 \cdot c_{p \cdot t} \cdot e_{t} \cdot (T_{t4} - T_{0} / \pi_{t})}\) where \(c_{p_t}\) is the specific heat at constant pressure for the turbine, \(e_t\) is the turbine efficiency, \(T_{t4}\) is the turbine entry temperature and \(T_0\) is the atmospheric temperature. Substitute the given values into the equation to find the value of \(V_9\).
02

Calculate thermal efficiency at takeoff

Thermal efficiency can be found using the equation: \(\eta_{\text {th }} = (V_9)^2 / 2 \cdot Q_R \cdot f\), where \(Q_R\) is the heating value of fuel and \(f\) is the fuel-to-air ratio, which is calculated using the formula: \(f = c_{p . c} \cdot T_{t4} / (e_b \cdot Q_R) - 1 / (\pi_c)^{(纬_c - 1)}\). Plug in the known values into the equations to obtain \(\eta_{\text {th }}\).
03

Find fuel consumption at takeoff

Thrust-specific fuel consumption is given by the formula: \(\text{TSFC} = f / (V_9 / a)\), where \(a\) is the speed of sound in the free stream which can be calculated by the formula: \(a=\sqrt{\gamma_c \cdot R \cdot T_0}\). Insert the known values into the equations to find the value of \(\text{TSFC}\).
04

Determine compressor pressure ratio at cruise

The compressor pressure ratio at the cruise condition is given by the formula: \(\pi_c = (1 + f \cdot 畏_b \cdot Q_R / (c_{pc} \cdot T_0))^{纬_c / (畏_c \cdot (纬_c - 1))}\), where 纬_c is the heat capacity ratio during the compressor stage. Use the known values to compute \(\pi_c\) at cruise.
05

Calculate exhaust velocity at cruise

With the updated values for \(T_{t4}\) and \(T_0\) at cruise, repeat step 1 to compute \(V_9\) at cruise.
06

Compute thermal efficiency at cruise

With the new values for \(f\), and \(V_9\) at cruise, repeat step 2 to find \(\eta_{\text {th }}\) at the cruise point.
07

Find propulsive efficiency at cruise

Propulsive efficiency is calculated using the equation: \(\eta_p = 2 / (1 + V_9 / U)\), where \(U = M_0 \cdot a\) is the flight speed. Substituting the given values and the results calculated during previous steps can obtain the propulsive efficiency.
08

Determine fuel consumption at cruise

Substitute the new value for \(f\), and the flight speed \(U\) into the TSFC formula used in step 3 to compute the fuel consumption at cruise.

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Key Concepts

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

Exhaust Velocity Calculation
Exhaust velocity is a crucial parameter in analyzing the performance of a turbojet engine. It represents the speed at which the exhaust gases leave the engine and is a direct contributor to the thrust produced. To find the exhaust velocity, we apply the equation: \[ V_9 = \sqrt{2 \cdot c_{p \cdot t} \cdot e_{t} \cdot (T_{t4} - T_{0} / \pi_{t})} \] Here,
  • \( c_{p \mathrm{t}} \) is the specific heat at constant pressure for the turbine.
  • \( e_t \) is the efficiency of the turbine.
  • \( T_{t4} \) is the turbine entry temperature.
  • \( T_0 \) is the atmospheric temperature.
We use the provided values for these parameters to substitute into the equation and calculate the exhaust velocity. This outcome helps determine not just the thrust, but also the efficiency and fuel consumption of the engine.
Thermal Efficiency
Thermal efficiency of a turbojet engine measures how effectively the engine converts the heat from fuel into work. This efficiency plays an essential role in assessing engine performance. The formula used to calculate thermal efficiency at takeoff is: \[ \eta_{\text{th}} = \frac{(V_9)^2}{2 \cdot Q_R \cdot f} \] Where:
  • \( V_9 \) is the exhaust velocity.
  • \( Q_R \) is the heating value of the fuel.
  • \( f \) is the fuel-to-air ratio.
The fuel-to-air ratio \( f \) is calculated using: \[ f = \frac{c_{p . c} \cdot T_{t4}}{e_b \cdot Q_R} - \frac{1}{{\pi_c}^{(\gamma_c - 1)}} \] In this equation, \( c_{p . c} \) denotes the specific heat capacity of the compressor stage, and \( \pi_c \) is the pressure ratio across the compressor. By substituting in the known values, we can solve for \( \eta_{\text{th}} \), thus understanding the engine's conversion efficiency.
Thrust-Specific Fuel Consumption
Thrust-specific fuel consumption (TSFC) is vital in evaluating how much fuel a turbojet engine uses to maintain a particular thrust. It gives a clear picture of the engine's fuel efficiency regarding thrust production. The formula for finding TSFC is: \[ \text{TSFC} = \frac{f}{V_9 / a} \] Here,
  • \( f \) is the fuel-to-air ratio.
  • \( V_9 \) is the exhaust velocity.
  • \( a \) indicates the speed of sound in the surrounding air, calculated with:
\[ a = \sqrt{\gamma_c \cdot R \cdot T_0} \] Utilizing the given values, we substitute into these equations to compute the TSFC. Knowing the TSFC helps in understanding the engine's operational cost-effectiveness and aids in planning long-duration flights.
Compressor Pressure Ratio
The compressor pressure ratio is a significant factor for turbojet engine performance, particularly during cruise. This ratio describes how much the pressure increases as air passes through the compressor. The formula used to derive it at cruise is: \[ \pi_c = \left(1 + \frac{f \cdot \eta_b \cdot Q_R}{c_{pc} \cdot T_0}\right)^{\frac{\gamma_c}{\eta_c \cdot (\gamma_c - 1)}} \] Parameters in the equation include:
  • \( \gamma_c \), the heat capacity ratio during the compressor stage.
  • \( \eta_b \), the burner efficiency.
By plugging in the engine's specific operating values, the compressor pressure ratio is computed, indicating how efficiently the compressor works under cruise conditions. This ratio is pivotal in determining overall engine efficiency and output.

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Most popular questions from this chapter

A multistage compressor is connected to a multistage turbine on the same shaft. The shaft speed is \(N=8000\) \(\mathrm{rpm}\). The throttle parameter is \(T_{\mathrm{t} 4} / T_{\mathrm{t} 2}=6.0\). The compressor inlet flow has a \(p_{\mathrm{t} 2}=p_{\mathrm{ref}}\) and \(T_{\mathrm{t} 2}=T_{\mathrm{ref}}\). Compressor discharge temperature is \(T_{\mathrm{t} 3}=872 \mathrm{~K}\). The engine corrected mass flow rate is \(\dot{m}_{\mathrm{c} 2}=360 \mathrm{~kg} / \mathrm{s}\). Calculate (a) the corrected shaft speed \(N_{\mathrm{c} 2}(\mathrm{rpm})\) (b) the corrected shaft speed \(N_{\mathrm{c} 4}(\mathrm{rpm})\) (c) the compressor pressure ratio \(\pi_{\mathrm{c}}\) assuming \(e_{\mathrm{c}}=\) \(0.90\) (d) the compressor shaft power in MW (e) the fuel-to-air ratio assuming \(\pi_{\mathrm{b}}=0.94, \eta_{\mathrm{b}}=0.99\), and \(Q_{\mathrm{R}}=42,000 \mathrm{~kJ} / \mathrm{kg}\) (f) turbine expansion parameters \(T_{\mathrm{t} 5} / T_{\mathrm{t} 4}\) and \(p_{\mathrm{t} 5} / p_{\mathrm{t} 4}\) for \(\eta_{\mathrm{t}}=0.85\) and \(\eta_{\mathrm{m}}=0.99\) (g) gas generator pumping characteristics \(p_{\mathrm{t} 5} / p_{\mathrm{t} 2}\) and \(T_{\mathrm{t} 5} / T_{\mathrm{t} 2}\)

An afterburning turbojet engine's design-point parameters are 1\. \(M_{0}=0, p_{0}=101.33 \mathrm{kPa}, T_{0}=288.2 \mathrm{~K}, \gamma_{\mathrm{c}}=1.4\), \(c_{p \mathrm{c}}=1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) 2\. \(\pi_{\mathrm{d}}=0.95\) 3\. \(\pi_{\mathrm{c}}=18, e_{\mathrm{c}}=0.90\) 4\. \(m_{\mathrm{c} 2}=67 \mathrm{~kg} / \mathrm{s}\) 5\. \(N_{\mathrm{c} 2}=7120 \mathrm{rpm}\) 6\. \(M_{z 2}=0.5\) 7\. \(Q_{\mathrm{R}}=42,800 \mathrm{~kJ} / \mathrm{kg}, \pi_{\mathrm{b}}=0.98, \eta_{\mathrm{b}}=0.97, T_{\mathrm{t} 4}=\) \(1773 \mathrm{~K}\) 8\. \(\gamma_{\mathrm{t}}=1.33, c_{p \mathrm{t}}=1156 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) 9\. \(e_{\mathrm{t}}=0.80, \eta_{\mathrm{m}}=0.995\) 10\. \(Q_{\mathrm{R}, \mathrm{AB}}=42,800 \mathrm{~kJ} / \mathrm{kg}, \pi_{\mathrm{AB}}=0.95, \eta_{\mathrm{AB}}=0.98, T_{\mathrm{t} 7}\) \(=2250 \mathrm{~K}\) 11\. \(\gamma_{\mathrm{AB}}=1.3, c_{p \mathrm{c}}=1243 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) 12\. \(\pi_{\mathrm{n}}=0.90, p_{9} / p_{0}=1.0\) The off-design conditions correspond to supersonic flight at high altitude $$ \begin{aligned} M_{0} &=2.5, p_{0}=15 \mathrm{kPa}, T_{0}=223 \mathrm{~K}, \gamma_{\mathrm{c}}=1.4, \\ c_{p \mathrm{c}} &=1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \\ \pi_{\mathrm{d}} &=0.82 \\ e_{\mathrm{c}} &=0.90 \\ Q_{\mathrm{R}} &=42,800 \mathrm{~kJ} / \mathrm{kg}, \pi_{\mathrm{b}}=0.98, \eta_{\mathrm{b}}=0.97, T_{\mathrm{t} 4}=1850 \mathrm{~K} \\ \gamma_{\mathrm{t}} &=1.33 c_{p \mathrm{t}}=1156 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \\ e_{\mathrm{t}} &=0.80, \eta_{\mathrm{m}}=0.995 \\ Q_{\mathrm{R}, \mathrm{AB}} &=42,800 \mathrm{~kJ} / \mathrm{kg}, \pi_{\mathrm{AB}}=0.95, \eta_{\mathrm{AB}}=0.98, \\ T_{\mathrm{t} 7} &=2450 \mathrm{~K} \\ \gamma_{\mathrm{AB}} &=1.3, c_{p \mathrm{c}}=1243 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \\ \pi_{\mathrm{n}} &=0.88, p_{9} / p_{0}=1.0 \end{aligned} $$ Calculate (a) compressor pressure ratio at off-design (b) the corrected and physical mass flow rates at the compressor face at off- design in \(\mathrm{kg} / \mathrm{s}\) (c) the fuel-to-air ratio at off-design (d) the exhaust speed at off-design in \(\mathrm{m} / \mathrm{s}\) (e) thrust specific fuel consumption in \(\mathrm{mg} / \mathrm{s} / \mathrm{N}\) at onand off-design

The corrected mass flow rate at the engine face is \(\dot{m}_{\mathrm{c} 2}=180 \mathrm{~kg} / \mathrm{s}\). Calculate the axial Mach number \(M_{z 2}\) at the engine face for \(A_{2}=1 \mathrm{~m}^{2}\). Also calculate the capture area \(A_{0}\) for a flight Mach number of \(M_{0}=0.85\) and assume an inlet total pressure recovery \(\pi_{\mathrm{d}}=0.995 .\) Assume \(\gamma_{\mathrm{c}}=1.4 . R_{\mathrm{c}}=\) \(287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, p_{0}=30 \mathrm{kPa}\) and \(T_{0}=250 \mathrm{~K}\).

A separate-flow turbofan engine has a dual spool configuration, as shown. Fan and core nozzles are convergent and choked. The design parameters for this engine are: 1\. \(M_{0}=0, p_{0}=101.33 \mathrm{kPa}, T_{0}=15.2^{\circ} \mathrm{C}\) 2\. \(\pi_{\mathrm{d}}=0.98\) 3\. \(\pi_{\mathrm{f}}=1.8, e_{\mathrm{f}}=0.90\) 4\. \(\alpha=5.0\) 5\. \(\pi_{\mathrm{cH}}=14, e_{\mathrm{cH}}=0.90\) 6\. \(T_{\mathrm{t} 4}=1600^{\circ} \mathrm{C}, Q_{\mathrm{R}}=42,800 \mathrm{~kJ} / \mathrm{kg}, \eta_{\mathrm{b}}=\) \(0.99, \pi_{\mathrm{b}}=0.95\) 7\. \(e_{\mathrm{tH}}=0.85, \eta_{\mathrm{mH}}=0.995\) 8\. \(e_{\mathrm{tL}}=0.89, \eta_{\mathrm{mL}}=0.995\) 9\. \(\pi_{\mathrm{n}}=\pi_{\mathrm{nf}}=0.98, p_{8}=p_{18}=p_{0}\) 10\. \(\gamma_{\mathrm{c}}=1.4, c_{p \mathrm{c}}=1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) 11\. \(\gamma_{\mathrm{t}}=1.33, c_{p \mathrm{t}}=1146 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) 12\. \(M_{4}=M_{4.5}=M_{9}=M_{19}=1.0\) An off-design operating condition is described by (a) \(M_{0}=0.90, p_{0}=20 \mathrm{kPa}, T_{0}=-20^{\circ} \mathrm{C}\) (b) \(T_{\mathrm{t} 4}=1300^{\circ} \mathrm{C}\) Assuming all component efficiencies remain constant, calculate (a) fan pressure ratio \(\pi_{\mathrm{f}}\) (b) high-pressure compressor pressure ratio \(\pi_{\mathrm{cH}}\) (c) the bypass ratio \(\alpha\)

A turbojet engine has choked nozzles both in turbine as well as exhaust. Its design point is at the standard sea level static condition \(\left(\mathrm{p}_{\mathrm{SL}}=101 \mathrm{kN}\right.\) and \(\left.\mathrm{T}_{\mathrm{SL}}=288 \mathrm{~K}\right)\) and the design calculations yield a turbine expansion parameter of \(\tau_{\mathrm{t}}=0.70 .\) Assuming design parameters are: \(\mathrm{T}_{\mathrm{t} 4-\mathrm{D}}=1950 \mathrm{~K}\), \(\mathrm{f}=0.023, \eta_{\mathrm{m}}=0.993\) and \(\mathrm{e}_{\mathrm{c}}=0.9\), estimate: (a) compressor pressure ratio at design, \(\pi_{\mathrm{c}-\mathrm{D}}\) (b) compressor pressure ratio at off-design, \(\pi_{\mathrm{c}-\mathrm{OD}}\) Where the off-design condition is described by: \(\mathrm{M}_{0}=2.0, \mathrm{p}_{0}\) \(=25 \mathrm{kPa}, \mathrm{T}_{0}=223 \mathrm{~K}, \mathrm{~T}_{\mathrm{t} 4, \mathrm{OD}}=1650 \mathrm{~K}\). Assume the same component efficiencies and fuel-to-air ratio in on- and off- design and constant gas properties, i.e., assume \(\gamma_{\mathrm{c}}=\gamma_{\mathrm{t}}=1.4, \mathrm{c}_{\mathrm{pc}}=\mathrm{c}_{\mathrm{pt}}=1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).
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