AME 436	Assigned: 4/30/08
Problem Set #7	“Due” 5/2/08 but no late penalty until 5/6/08, 12:00 noon; no assignments accepted after that date – NO EXCEPTIONS! (Fax to me at 213-740-8071 if you’re off campus.)

NOTE: since your lowest homework grade will be eliminated anyway, if you’re satisfied with your other grades you can skip this one. (Not recommended, however.)

Problem #1

Draw each of the following hypersonic propulsion cycles on the attached T-s diagrams (starting in the lower left corner of the figure). Draw additional Rayleigh/Fanno lines as needed.

a) Conventional ramjet with poor diffuser (non-isentropic) and maximum (constant-area) heat addition

b) "Scramjet" with (isentropic) inlet diffuser, normal shock, and heat addition at increasing area

c) "Scramjet" with no inlet diffuser, normal shock, maximum (constant-area) heat addition, partial (isentropic) expansion (i.e. not all the way to P_e = P_a, supersonic afterburner (!) with decreasing area, and finally isentropic expansion to P_e = P_a.

a)

b)

c)

Problem #2

A hypersonic aircraft is in level flight at M₁ = 10 at 100,000 ft altitude and uses H₂ fuel. The ambient air temperature is 227K and the ambient pressure is 0.0108 atm. The maximum pressure inside the engine is limited to 100 atm by structural considerations. Using GASEQ, determine the performance of this pressure-limited stoichiometric-burning H₂-air engine in the following way:

a. Choose as reactants “hydrogen-air flame” and as products “H2/O2/N2 products.” The default mixture strength is stoichiometric, so you shouldn’t have to change that. Choose 227K and 0.0108 atm as the reactant conditions. Then select Problem Type “Adiabatic compression/expansion” and check the “frozen composition” box. Choose a product pressure P₂ = 100 atm and hit the “Calculate” button. Note the enthalpy (h₁) and sound speed (c₁) of the reactants at ambient conditions, and calculate the flight velocity u₁ = c₁M₁.

b. Now do the combustion. Hit the “R<<P” button to make the products of the first (compression) process become the reactants of the second (combustion) process. For the Problem Type choose “adiabatic T and composition at constant P.” This corresponds to constant h (that is, h₂ = h₃.) This is ok since conservation of energy requires h₂ + u₂²/2 = h₃ + u₃²/2, and for constant pressure processes the momentum balance yields u₂ = u₃ (see lecture 11, slide 28).

c. Now do the expansion back to ambient pressure. Select Problem Type “Adiabatic compression/expansion” and make sure the that “frozen composition” box is not checked. Hit the “R<<P” button to make the products of the second (combustion) process become the reactants of the third (expansion) process. Choose a product pressure of 0.0108 atm, hit “Calculate” one more time and you’re done. Note the final enthalpy h₄.

d. Compute the product velocity from h₁ + u₁²/2 = h₄ + u₄²/2. You have everything except u₄. Note that GASEQ gives you enthalpies in kJ/kg, not J/kg, so you need to multiply GASEQ’s values of h by 1000 to get the units right.

e. Compute the specific thrust = (u_e – u₁)/c₁.

f. Compute TSFC = (Heat input)/Thrust*c₁ = c₁*FAR*Q_R/((u_e – u₁)*c₁²) = (1/(Specific thrust)) FAR_stoichQ_R/c₁². Also calculate the Specific Impulse = (1/TSFC)(Q_R/c₁g_earth) = (Specific thrust * c₁)/(FAR*g_earth).

Problem #3

Compare the results of problem #2 to those using aircycles4hypersonics.xls for the same flight mach number, ambient temperature and pressure, fuel type (which means you’ll change the fuel heating value to match that of hydrogen), fuel mass fraction, and heat addition process (i.e. constant pressure, which means you’ll set “Const P?” = TRUE, “Const. A” = FALSE, and “Const. T” = FALSE.) You’ll need to adjust the property “Mach No. after diffuser” until the pressure after the diffuser (state 2) is 100 atm, then adjust Tau_lamda until f (fuel mass fraction) is equal to the stoichiometric fuel mass fraction for hydrogen.

Problem #4 (from last year’s final exam) (Continuation of a problem from HW #6).

The following 5 changes to a t_l-limited turbofan engine flying at subsonic conditions (M₁ < 1) are being considered:

1) Increase the fan air bypass ratio (a) by a factor of 2

2) Increase the flight Mach number (M₁) by a factor of 2

3) Increase the turbine inlet temperature limit (t_l) by a factor of 2 (yeah, right…)

4) Increase the compressor pressure ratio (π_c) by a factor of 2 (ditto…)

5) Increase the fuel heating value (Q_R) by a factor of 2

Briefly explain:

a) Which of these would decrease NO_x emissions the most?

Problem #5 (from last year’s final exam) (continuation of a problem from HW #6)

Ronney Oil & Gas Company claims to have developed a fuel, called PDR®, whose chemical formula is C₈H₁₈ (octane) and has all the same thermodynamic properties, transport properties, etc. as C₈H₁₈. The only difference between C₈H₁₈ and PDR® is that using PDR® leads to 10% lower activation energy (E) for all chemical reactions. If PDR® fuel were used instead of C₈H₁₈, how would each of the following be affected? In particular, state whether the property would increase, decrease or remain the same, and if there is a change, would it be by more than, less than, or equal to 10%. (Notice the operative words: LOWER ACTIVATION ENERGY.) No credit without explanation!

a) Amount of unburned hydrocarbon emissions in a premixed-charge engine

b) Amount of soot emission from a rich premixed flame

Problem #6

Consider a premixed-charge engine with the following characteristics: equivalence ratio 1.0, stoichiometric air to fuel mass ratio 14.7, compression ratio 8:1, bore 100 mm, stroke 100 mm, piston diameter above top ring 99.4 mm, distance from piston top to upper surface of top piston ring 9.52 mm, volumetric efficiency 0.8, temperature in cylinder at start of compression 333K, pressure in cylinder at start of compression = 1 atm, mixture temperature before entering cylinder 30˚C, maximum cylinder pressure 3 MPa, wall temperature 400K, brake specific fuel consumption 300 g/kW-hr. Calculate:

(a) the total mass of fuel in the cylinder

(b) the volume of the "crevice" between the top surface of the top piston ring

(c) the mass of the fuel in the crevice at the time of maximum cylinder pressure (assume the temperature at this time is the same as the wall temperature)

(d) the mass fraction of fuel in the crevice = (c)/(a)

(e) the emitted mass of hydrocarbons, assuming 1/3 of the fuel in the crevice is burned, 1/2 of that remaining is oxidized within the combustion chamber and 1/3 of that remaining is oxidized in the exhaust system.

(f) the volume fraction (in parts per million) of unburned hydrocarbons in the exhaust assuming the hydrocarbons can be modeled as CH₂

(g) the ratio of brake specific hydrocarbon emission to brake specific fuel consumption

(h) the brake specific hydrocarbon emissions in grams of fuel per kilowatt-hour

Problem #7

Consider a turbojet (not turbofan) engine using conventional octane fuel, operating at 0.25 atm ambient pressure, 225K ambient pressure, compressor pressure ratio of 30, turbine inlet temperature limit of 1700K, flight Mach number 0.8. Now consider the following modifications to this “baseline” cycle:

1. Compressor pressure ratio increased to 40

2. Turbine inlet temperature limit raised to 1800K

3. Fuel changed from hydrocarbon to hydrogen

4. Flight velocity doubled

5. A fan is added

(a) Use aircycles4propulsion.xls and assume ideal cycles, determine the temperature and pressure at state 4 (after the combustor) for the baseline cycle and each of these cycle modifications.

(b) For all 6 temperatures/pressure combinations, determine the equilibrium NO concentrations using GASEQ.

(c) For all 6 temperatures/pressure combinations, determine the time scale for NO formation using Heywood’s relationship

(d) The amount of NO_x formed is proportional to the equilibrium NO for the given temperature/pressure divided by the NO_x formation time scale (as per Heywood’s relation). Determine the amount of NO formation (relative to the baseline case) for all 5 cycle modifications. Which cycle modification produces the highest NO? The lowest NO?