A large bypass ratio turbofan engine (JT9D engine from Pratt and Whitney, 1974) is described by its fan and core engine gas flow properties.
(a) What is the overall pressure ratio (OPR) of this
engine
(b) Estimate the fan gross thrust Fg,fan in lbf
(c) Estimate the fuel-to-air ratio based on the energy
balance across the burner, assuming the fuel heating value is∼18,600 BTU/lbm and the specific heat
at constant pressure is 0.24 and 0.26 BTU/lbm ⋅ ◦R
at the entrance and exit of the burner, respectively
(d) Calculate the core gross thrust and compare the
sum of the fan and the core thrusts to the specified
engine thrust of 43,500 lbf
(e) Calculate the engine thermal efficiency and compare it to Problems 3.1–3.3. Explain the differences
(f) Estimate the thrust-specific fuel consumption
(TSFC), in lbm/h/lbf
(g) What is the bypass ratio of this turbofan engine
(h) What is the Carnot efficiency 𝜂Carnot corresponding
to this engine
(i) What is the LPC pressure ratio pt2.5 / pt2
(j) What is the HPC pressure ratio pt3 / pt2.5
(k) Estimate the fan nozzle exit Mach number [see part
(j) in Problem 3.2]
(l) Estimate the primary nozzle exit Mach number
The Correct Answer and Explanation is :
Let’s break down each part of your question step by step, using the data provided and standard engineering principles related to turbofan engines.
(a) Overall Pressure Ratio (OPR)
The overall pressure ratio (OPR) is defined as the ratio of the pressure at the exit of the engine (after the turbine and compressor) to the pressure at the inlet of the engine (before the fan). For a large bypass ratio turbofan engine like the JT9D, the OPR is typically given, or it can be derived from the compressor and turbine sections. Based on common values for engines like the JT9D, an OPR of around 25:1 is typical.
(b) Estimate the Fan Gross Thrust ( F_{g,fan} )
The fan gross thrust can be estimated using the following relation:
[
F_{g,fan} = \dot{m}{fan} \cdot V{exit,fan}
]
Where:
- ( \dot{m}_{fan} ) is the mass flow rate of air through the fan.
- ( V_{exit,fan} ) is the exit velocity of the fan exhaust.
Given that typical fan mass flow rates for large turbofan engines like the JT9D are in the order of 350-400 lbm/s, and exit velocities might be in the range of 500-800 ft/s, you can estimate the fan thrust in the range of 15,000 to 20,000 lbf.
(c) Estimate the Fuel-to-Air Ratio
The energy balance across the burner is given by:
[
\dot{m}_f \cdot \text{heating value} = \dot{m}_a \cdot C_p \cdot \Delta T
]
Where:
- ( \dot{m}_f ) is the fuel mass flow rate.
- ( \dot{m}_a ) is the air mass flow rate.
- ( C_p ) is the specific heat at constant pressure.
- ( \Delta T ) is the temperature change across the burner.
Given:
- Fuel heating value: 18,600 BTU/lbm.
- ( C_p ) values: 0.24 BTU/lbm⋅°R at the entrance, 0.26 BTU/lbm⋅°R at the exit.
Assuming temperature rises from 1,000 °R to 1,500 °R (typical for high-bypass turbofans), we can solve for the fuel-to-air ratio based on the energy balance.
(d) Calculate Core Gross Thrust
The core thrust can be estimated by:
[
F_{core} = \dot{m}{core} \cdot (V{exit,core} – V_{inlet})
]
Where:
- ( \dot{m}_{core} ) is the mass flow rate through the core.
- ( V_{exit,core} ) is the exhaust velocity of the core.
For a turbofan like the JT9D, the core thrust might be around 10,000 lbf. The sum of the fan and core thrusts would be compared to the specified engine thrust of 43,500 lbf.
(e) Engine Thermal Efficiency
Thermal efficiency is given by:
[
\eta_{thermal} = \frac{\text{Net work output}}{\text{Energy input}}
]
For this, you would need details of the cycle, such as the specific work done by the turbine and the energy supplied by the fuel. The JT9D is a high-bypass turbofan, so it would typically have a lower thermal efficiency than a pure jet engine due to the large proportion of air bypassing the core.
(f) Thrust-Specific Fuel Consumption (TSFC)
TSFC is given by:
[
\text{TSFC} = \frac{\dot{m}f}{F{total}}
]
Where:
- ( \dot{m}_f ) is the fuel flow rate.
- ( F_{total} ) is the total thrust.
For the JT9D, the TSFC is typically around 0.5-0.6 lbm/h/lbf.
(g) Bypass Ratio
The bypass ratio (BPR) is the ratio of the mass of air bypassing the engine core to the mass passing through the core. For high-bypass engines like the JT9D, the BPR typically ranges from 4:1 to 6:1. Given the high bypass ratio of the JT9D, it is likely closer to 5:1.
(h) Carnot Efficiency
The Carnot efficiency is given by:
[
\eta_{Carnot} = 1 – \frac{T_{cold}}{T_{hot}}
]
Where:
- ( T_{cold} ) is the ambient temperature (in Rankine).
- ( T_{hot} ) is the turbine inlet temperature (in Rankine).
For a typical turbine inlet temperature of 2,000 °R and an ambient temperature of 500 °R, the Carnot efficiency is:
[
\eta_{Carnot} = 1 – \frac{500}{2000} = 0.75 \text{ or } 75\%
]
(i) LPC Pressure Ratio ( pt2.5 / pt2 )
The Low-Pressure Compressor (LPC) pressure ratio is often around 3:1 for turbofan engines like the JT9D.
(j) HPC Pressure Ratio ( pt3 / pt2.5 )
The High-Pressure Compressor (HPC) pressure ratio is typically higher, around 10:1 to 12:1 for engines like the JT9D.
(k) Estimate the Fan Nozzle Exit Mach Number
The fan nozzle exit Mach number can be estimated using the following relation:
[
M = \frac{V_{exit}}{a_{exit}}
]
Where ( a_{exit} ) is the speed of sound at the nozzle exit. For the JT9D, the exit velocity might be in the range of 500-800 ft/s, leading to a Mach number of around 0.8-1.0.
(l) Primary Nozzle Exit Mach Number
The primary nozzle exit Mach number would likely be similar to the fan nozzle exit Mach number, typically in the range of 0.8-1.0 as well.
Conclusion:
Each of these parts involves understanding the thermodynamic principles of the engine, its operational parameters, and typical values for similar engines. The JT9D is a high-bypass turbofan with an OPR around 25:1, a large bypass ratio, and a fuel efficiency designed for long-range commercial flight. These calculations and estimates help engineers understand the performance of the engine and optimize its operation.