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lecture_6 - 16.512, Rocket Propulsion Prof. Manuel...

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16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 6: Heat Conduction: Thermal Stresses Effect of Solid or Liquid Particles in Nozzle Flow An issue in highly aluminized solid rocket motors. 2Al + 22 3 OA l O 2 3 m.p. 207 , b.p. 298 2 C D 0 C D In modern formulations, with 20% Al by mass, the mass fraction of the exhaust can be 35-40%. This material does not expand 23 Al O , so there must be a loss in exit velocity, hence in I sp . Assume mass flows (gas) (solids), non-converting g m i s m i . The momentum equation is gs md u u A d p 0 ++ ii = Call s ρ the (mass of solids)/(volume) (not the density of the solid, theory) gg g ss s u du u du dp 0 = ρρ Define a mass flux function s u m x uu mm == + + i g s x ud u d u d p0 1x ⎛⎞ ⇒+ + ⎜⎟ ⎝⎠ = g dp x udu =− The energy equation is similarly, () ( ) pg g g g s s s s 1 x c dT u du x c dT u du 0 −+ + + = 16.512, Rocket Propulsion Lecture 6 Prof. Manuel Martinez-Sanchez Page 1 of 10

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Substitute here from above: gg udu () pg g g s s s s s g dp x x cd T ud u T ud u 0 1x −− + + = ρ pg g s s s g s g dp x c dT c dT u u du ⎡⎤ =+ + ⎣⎦ with no particles (x=0), this gives γ γ− ⎛⎞ =→ = ⎜⎟ ⎝⎠ 1 gp 00 dp P T RT cdT PP T With particles, we need to know the history of the velocity slip sg uu and of the temperature slip . This is a difficult problem, requiring detailed modeling of the motion and heating/cooling of the particle. But we can look at the extreme cases easily. s TT g (a) Very Small Particles good contact. For sub-micro particles (not a bad representation of reality), we can say that uuu = ± , TTT = ± . Then pg s g dp x cc d T Note that the mean specific heat ( and c are per unit mass) is pg c s pg s p g v g pv vg s v c1 x cx c Rcc 1 x 1 x R and also c 1 x c xc =− + =−=− + g So that p g c dp dT = pp g g dp dp dT dT P1 x PRT = and defining an effective γ by the usual γ= p v c c , γ = 1 PT 16.512, Rocket Propulsion Lecture 6 Prof. Manuel Martinez-Sanchez Page 2 of 10
The equation of motion is now ( ) gs udu dp 0 += ρ+ρ Or g udu dp 0 1x ρ () g g g PP

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