sec11-022410

sec11-022410 - ChE 120B Estimating h Boundary Layer...

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ChE 120B 11 - 1 Estimating h — Boundary Layer Equations Before, we just assumed a heat transfer coefficient, but can we estimate them from first principles? Look at steady laminar flow past a flat plate, again: Clearly, the fluid motion is coupled to the heat transfer processes. Momentum Equations x comp: 22 x xx x xy x vv v v p g x yx x y        y comp: yy y y y v v p g x x y Continuity: 0 y x v v Thermal: px y x TT T T Cv v kv x y (assuming negligible viscous dissipation) Boundary conditions , x vu 0 y v x  , x y  [email protected]  0 y 0 x p p x p p y 0 x x 0 0 y y 0 x
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ChE 120B 11 - 2 ,, upT  are conditions far from plate. To get anywhere — need some idea of which terms are important. Hence we non- dimensionalize: / xx Uv u / yy u  22 // Pp pp u u   xH H X vu L  p u u  Re / Nu L Pr p NC k v  00 / TT T T This choice of (logical) variables gives the dimensionless B.L. Eqn: Momentum Re Re 1 1 x x xy y y UU U U P XY X N X Y U U P Y N X Y           Continuity x U X + 0 y U Y Energy Re Pr 1 X YN NX Y  The real coupling between these equations is the temperature dependence of the viscosity. If we are not concerned about this, then the momentum equations are decoupled. Now we can say something about magnitudes to identify the important terms: 0(1) means order of 1. That is, the expected value of this quantity is about 1.   / 01, Uu u X  and we can estimate the important derivatives so 01 x U X     10 1 x u and from continuity
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ChE 120B 11 - 3  00 1 yy x UU U XY Y    01 y U Y but changes in U y occur only over the boundary layer so y H U yH U  2 22 2 1 1 x x U U X X  2 1 1 x x H H U U Y Y  We can now put these estimates into the momentum B.L. Eqn. we also know for the T.E. Eqn. 2 11 0 , 0 TT X Y      In general , , TH L   so 1 T  so ,, a n d xx YX      2 1 1 / H H H Re 1 x xy U dP d x N Y 1 HH H  2 2 Re 1 y U dP x yd y N y    The second equation is always 0 H , and the first is 0(1). Hence, the pressure gradient PP x y  . Hence, let 0 P y and hence P only changes in the x direction.
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ChE 120B 11 - 4 P X function only of X.
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sec11-022410 - ChE 120B Estimating h Boundary Layer...

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