lecture_7

# lecture_7 - 16.512 Rocket Propulsion Prof Manuel...

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16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 7: Convective Heat Transfer: Reynolds Analogy Heat Transfer in Rocket Nozzles General Heat transfer to walls can affect a rocket in at least two ways: (a) Reducing the performance. This tends to be a 1-3% effect on only, and is therefore secondary. sp I (b) Creating great difficulties in the design of hot-side structures that have to survive heat fluxes in the range. 78 10 10 w /m 2 The principal modes of heat transfer to nozzle and combustor walls are convection and radiation . Of these, convection dominates, and radiation tends to be important only for particle-laden flows from solid propellant rockets. Convective Heat Transfer We will review here the compressible 2D boundary layer equations in order to extract information on wall heat transfer. The governing equations are (in the B.L. approximation) Continuity () ( ) uv 0 xy ∂∂ + ρρ = (1) X-Momentum xy uu p x yxyyy ∂τ ⎛⎞ ++ == µ ∂∂∂∂ ⎝⎠ u (2) Y-Momentum P 0 y = (3) 16.512, Rocket Propulsion Lecture 7 Prof. Manuel Martinez-Sanchez Page 1 of 16

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Total enthalpy () tt xy hh T uv u k xy y y ∂∂ ⎛⎞ y += τ + ⎝⎠ ρρ (4) where 2 t u 2 =+ is the specific total enthalpy, and µ is the viscosity. For a laminar flow, ( ) T µ=µ is a fluid property. Rocket boundary layers are almost always turbulent, and µ is then the “turbulent viscosity”, where momentum transport is effected by the random motion of turbulent “eddies”. If these eddies have a velocity scale u' and a length scale l', we have, in order-of-magnitude. turb. u'l' µ ρ (5) where u' is some fraction of the local u, and l'tends to be of the order of the wall distance y. The important points about (5) are (a) turb. µµ ± , mostly because l' mean free path and ± (b) turb. µ is proportional to density (whereas µ is not, because the m.f.p. is inversely proportional to ). Similarly, the last term on the right in the energy balance, representing the convergence of heat flux, contains the “turbulent thermal conductivity” p Kc u ' l ' . Once again, we notice that K is here proportional to density. We also note that the “turbulent Prandtl number” tp r t c P k µ = 1 (from the orders of magnitude) It is of some interest to note the origin and composition of the viscous term in equation (4). If we collect the dot products t u τ G G i around a fluid element as shown (in B.L. approximation), 16.512, Rocket Propulsion Lecture 7 Prof. Manuel Martinez-Sanchez Page 2 of 16
we obtain the term ( xy u y τ ) as written in (4). This can be expanded as () 2 xy xy xy uu u yy y y y ∂τ ⎛⎞ ∂∂ τ= + τ = µ + µ ⎜⎟ ⎝⎠ u y (6) The 1 st term in (6) is just the velocity times the viscous net force per unit volume, so it is the part of the total viscous work that goes to accelerate the local flow. The second term in (6) is positive definite, and it is the rate of dissipation of energy into heat due to viscous effects. We will return later to this heating effect.

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## This note was uploaded on 11/07/2011 for the course AERO 16.512 taught by Professor Manuelmartinez-sanchez during the Fall '05 term at MIT.

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lecture_7 - 16.512 Rocket Propulsion Prof Manuel...

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