AME 436	Assigned: 4/5/2008
Problem Set #5	Due: 4/12/2008 at 4:30 pm in OHE 430 (fax to me at 213-740-8071 if you’re off campus)

Problem #1

For each of the following equations a) – c) used frequently in this course, state whether or not each of the following restrictions i) – viii) apply.

i. Steady

ii. one-dimensional flow

iii. ideal gas

iv. constant specific heats

v. reversible

vi. adiabatic

vii. no work transfer

viii. negligible change in kinetic energy

a) Thrust equation: T = _a[(1+FAR)u_e-u₁] + (P_e-P_a)A_e

b) Enthalpy/velocity relation between states 1 and 2:

c) Temperature/pressure relation between states 1 and 2:

Problem #2

Using the Brequet range equation, estimate within a factor of 10 the range of a seagull. When estimating the heating value of seagull food, note that 1 diet calorie = 1000 thermodynamic calories. (This question always throws students for a loop. The point is not to get an exact answer, but to estimate each of the terms in the Brequet range equation, and see if the result is reasonable or not.)

Problem #3

Consider a very simple propulsion system operating at a flight Mach number of 5 that consists of 2 processes:

Process 1: Shock at entrance to duct

Process 2: Heat addition in a constant-area duct until thermal choking occurs

a) Compute all of the following properties of this system:

i. Static (not stagnation) temperature relative to T₁ after the shock

ii. Static (not stagnation) pressure relative to P₁ after the shock

iii. Static (not stagnation) temperature relative to T₁ at the exit

iv. Static (not stagnation) pressure relative to P₁ at the exit

v. Dimensionless heat addition {q_in divided by RT₁ = C_P(T_3t-T_2t)/RT₁ = [g/(g‑1)](T_3t-T_2t)/T₁}

vi. Specific thrust = Thrust /(_ac₁) (c₁ = sound speed at ambient conditions = (gRT)^1/2) (assume FAR << 1 in the thrust calculation)

vii. Overall efficiency

viii. Draw this cycle on a T - s diagram. Include appropriate Rayleigh and Fanno curves.

b) Repeat (a) if a nozzle is added after station 3 that expands the flow isentropically back to P = P₁.

c) Repeat (a) if there is no shock and no nozzle (but still constant area heat addition.)

d) Repeat (a) if there is a shock followed by constant temperature (not constant area) heat addition until the pressure is equal to ambient pressure. (Are we having fun yet?)

e) Answer the following questions about parts (a) through (d):

i. Why was no thrust generated in parts (a) and (c)?

ii. Why did parts (b) generate thrust whereas part (a) did not?

iii. Why did part (d) generate thrust whereas parts (a) and (c) did not?

iv. Why was the performance of part (b) so much better than that of part (d)?

Problem #4

Consider a flowing gas with g = 1.4, R = 287 J/kgK, M = 3, T_t = 840K and P_t = 36.73 atm.

a) If heat is added at constant area until thermal choking (M = 1), how much heat has been added to the gas (in J/kg)?

b) At this condition, what is the stagnation pressure?

c) How much heat could be added at constant pressure (in J/kg, starting at M = 3, T_t = 840K and P_t = 36.73 atm) before the stagnation pressure in part (b) is reached?

d) How much heat could be added at constant temperature (in J/kg, starting at M = 3, T_t = 840K and P_t = 36.73 atm) before the stagnation pressure in part (b) is reached?

Problem #5

Two hypersonic engine designs, A and B, are being considered for a high-speed transport aircraft operating at a flight Mach number of 5.

Engine A produces a flow at the exit with a stagnation pressure 80 times the ambient pressure and a stagnation temperature 12 times the ambient temperature.

Engine B produces a flow at the exit with a stagnation pressure 100 times the ambient pressure and a stagnation temperature 10 times the ambient temperature.

Because these two engines are made by rival companies with trade secrets, little is known about what happens inside the engines. It is not known for either engine it uses a compressor or not, whether combustion occurs at constant P, T, A or none of the above, if the diffuser is reversible or not, nor is known. All that is known is that for both engines (1) the same fuel is used, (2) reversible adiabatic expansion occurs in the exhaust nozzle to ambient pressure, (3) during the expansion the gas has constant specific heats with g = 1.4, and (4) the fuel to air ratio (FAR) is much less than 1.

a) Which engine, A or B, has the higher exhaust velocity?

b) Which engine, A or B, has the higher mass flow per unit throat area?