Aero335_W24_HW7

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AEROSP 335 (Winter 2024) Jorns AEROSP 335 (W24) Homework 7 Due: Wednesday, March 27 at 11:59 PM Instructions: Show all steps for credit . All homework submissions must be neat and legible. Show all your work and circle or box the final answer. Points will be deducted if the final answer has missing or incorrect units. At- tach all code used to compute answers, but note a piece of code (e.g. MATLAB ) is insufficient by itself to show your work: you must still work through your analysis. 1 Turbofan Design (35 pts) You are designing an engine for a long-range commercial aircraft that will fly at an altitude of 11,000 meters ( T a = 216.8 K, ρ a = 0.2978 kg/m 3 , a a = 295 m/s) at Mach 0.80. Assume the maximum turbine temperature is 1400 K, the engine operates at a matched condition, that the pressure drop across the combustor is 5%, and that Q r = 46 MJ/kg. Assume a fan pressure ratio of 1.4. Using the efficiencies and specific heat ratios given in Tab. 1 , answer the following questions about the engine’s design: a) (25 pts) Modify the program you developed for Homework 5 to work with turbofans, following the algorithm laid out on slide 28 of lecture III.5 (Turbofans). Use your code to plot the specific thrust and efficiency as a function of compressor pressure ratios π pr = 1 , 2 , ..., 100 and for bypass ratios β = 1 , 2 , ..., 10. Put π pr on the x axis and plot a separate line for each β , so there should be ten lines on each of the graphs. Make sure to clearly label each line or indicate in a legend which line corresponds to which bypass ratio. b) (5 pts) What combination of CPR and bypass ratio give the best ef- ficiency? AEROSP 335 (Winter 2024) 1 Jorns

AEROSP 335 (Winter 2024) Jorns Table 1: Non-ideal efficiencies and specific heat ratios Stage Efficiency Mean Specific Heat Ratio ( γ ) Diffuser η d = 0.97 1.4 Compressor η c = 0.87 1.37 Fan η f = 0.93 1.4 Fan nozzle η nf = 0.98 1.4 Combustion η b = 1 1.35 Turbine η t = 0.9 1.33 Nozzle η n = 0.98 1.36 c) (5 pts) What are the specific thrust, efficiency, TSFC, and fuel frac- tion at this optimal point? 2 Diffuser design (35 pts) We wish to design a diffuser for a turbojet designed to fly at M = 0.75 at an altitude of 15 km ( T a = 217 K, ρ a = 0.194 kg/m 3 , a a = 295 m/s). a) (20 pts) Plot (on the same graph) the specific area of the diffuser inlet A 1 ˙ m a and the specific area of the fan inlet A 2 ˙ m a as a function of Mach number at the fan, M 2 ∈ [0 . 1 , 0 . 5]. Assume that the maximum pressure coefficient in the diffuser is 0.5 and the diffuser efficiency is 1. b) (5 pts) What diffuser inlet diameter and compressor inlet diameter are required for ˙ m a = 500 kg/s if we design for M 2 = 0 . 5? c) (10 pts) Say the diffuser size remains the same as our original design and we need to draw the same ˙ m a , but we want to fly at M = 0 . 9 instead. What is the new Mach number at the fan inlet? How might this change our design? Justify your answer. 3 Compressor design (30 pts) There is the core section of a turbojet in the FXB atrium (see also Fig. 1 ). a) (5 pts) Firstly, label on the photo the different sections of the core of the turbojet (compressor, turbine, etc.). AEROSP 335 (Winter 2024) 2 Jorns

AEROSP 335 (Winter 2024) Jorns Figure 1: Turbojet core b) (5 pts) What is the blade angle and radius of the rotors of the axial compressor? Do not measure each rotor individually—simply measure one or two and assume the rest are the same. c) (15 pts) What RPM is required to drive the compressor? The com- pressor has a total pressure ratio of 30. Assume c p = 1000 J/kg-K, R = 287 J/kg-K, the axial velocity is 50 m/s, the stagnation temperature at the inlet is 300 K, and the pressure ratio across each stage is constant. You may also take the blade speed, U , to be the speed at the tip of the blade, the compressor to be 100% efficient, and the stator blade angle to be the same as the rotor blade angle, β 2 = α 1 . d) (5 pts) What is the ratio of stagnation densities from the first and last stages? AEROSP 335 (Winter 2024) 3 Jorns

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