AME514-F06-PS4

AME514-F06-PS4 - AME 514 - Special topics in combustion -...

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AME 514 - Special topics in combustion - Fall 2006 Assignment #4 Due: Monday 11/27/06, 4:30 pm in my office (OHE 430J) Part 1: paper review Since they’re weren’t many references in this set of lectures, I’ve decided to skip Part I. Part 2 will count twice as much as did for the other homework sets. You’ll notice Part 2 is somewhat more time consuming than usual, though probably not twice as long as the others. (I think all the words in this problem set that make it look like a long problem set make it easier to do, not more difficult, since I’ve give you step by step instructions. Of course, your mileage may vary). Part 2. The usual type of homework questions Problem #1 a. Show that for heat addition at constant temperature (which really simplifies things) using the first law (h 1 + u 1 2 /2 + q = h 2 + u 2 2 /2), the entropy of an ideal gas (s 2 -s 1 = C P ln(T 2 /T 1 ) – Rln(P 2 /P 1 )), the enthalpy of an ideal gas (h 2 – h 1 = C P (T 2 – T 1 )), the definition of Mach number (M = u/c = u/( γ RT) 1/2 ) and the consequence of second law (ds = dq/T): P 2 /P 1 = exp [( γ /2)(M 1 2 - M 2 2 )] (yes you already have this result from the lecture notes but I’d like for you to show it…) b. By additionally using mass conservation, show that A 2 /A 1 = (M 1 /M 2 ) exp [( γ /2)(M 2 2 - M 1 2 )] (ditto comment in part a) c. Now consider a propulsion system based on this. First the air will be decelerated isentropically (not to M = 0) then heat will be added at constant temperature. For a flight Mach number of 15, an ambient atmosphere at 100,000 feet (227K and 0.0107 atm, with γ = 1.4), to what Mach number could the air be decelerated if the maximum allowable gas temperature is 3000K? What would the corresponding pressure be? d. From this condition, if heat is added at constant temperature until the ambient pressure was reached (not a good way to operate, but this represents a sort of maximum heat addition), what would the exit Mach (M e ) number be? What would the area ratio be? e. What would the specific thrust be? (Note for this case specific thrust = Thrust/ ˙ m a c 1
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= ˙ m a (u e – u 1 )/ ˙ m a a 1 = (M e c e – M 1 c 1 )/c 1 = M e (T e /T 1 ) 1/2 – M 1 , which is all stuff you already have) f. What would the Thrust Specific Fuel Consumption be? (Note that TSFC = (Heat input)/Thrust*c 1 = [ ˙ m a (C P (T 3t – T 2t )c 1 ]/[Thrust*c 1 2 ] = [( ˙ m a c 1 )/Thrust] [( γ /( γ -1))R(T 3t –T 2t )/( γ RT 1 )] = [1/(Specific thrust)] [1/( γ -1)] [(T 3t –T 2t )/T 1 ] and you have everything needed to calculate T 3t and T 2t ) g. Can any fuel generate enough heat to accomplish this? Look at stoichiometric hydrogen-air and see if the heat release per unit mass = f stoich Q R is equal to or greater than the heat input needed = C
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This note was uploaded on 07/22/2008 for the course AME 514 taught by Professor Ronney during the Fall '06 term at USC.

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AME514-F06-PS4 - AME 514 - Special topics in combustion -...

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