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sec10-022410

# sec10-022410 - ChE 120B Heat Transfer Review of General...

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Heat Transfer Review of General Transport Equations 1. Consider an observer moving through space at a velocity w , which may be different than the fluid velocity v . The observer is continually measuring some fluid property designated by S (scalar) such as ,, , x Tv etc. What is the time rate of change of S as measured by observer: lim 0 tt t SS dS t dt t   In general, S is a function of t, x, y, z and the spatial coordinates are a function of time  (, , , dS d Sx ty tz dt dt Using the chain rule, we can write with respect to fixed ( x, y, z ) ,, and yzt xyt xyz xzt dS dS dx dS dy dS dx dS dt dx dt dy dt dz dt dt        we see that x dx w dt y dy w dt z dz w dt so d S ssss S www S dt t x y z t  w in the specific case the w = v , this is called the material derivative: DS S S Dt t  v ChE 120B 10–1

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ChE 120B If the observer is fixed in space v = w = 0 and DS S Dt t With sufficient space, we can also show that () = + mm Vol t V t A DS SdV dV S v n dA Dt t      Reynold's Transport Theorem a particular example of Leibnig formula (page 732) Review of Transport Equations Material Derivative DS S vS S Dt t  Scalar Conservation of Mass 0 V D dV Dt and the Reynold's transport theorem states VV A SdV dV S dA Dt t vn Divergence Theorem  , hence V SdV S dV Dt t    v V D dV dV Dt t  v and arbitrary volumes implies t    0 v continuity equation equivalent to conversation of mass 10—2
ChE 120B Conservation of Linear Momentum     time rate of change of linear momentum the force on body dA body force surface (shear) force m V A dV n D dV Dt  gt v n  tT n T stress tensor a n d  mm VA D dV gdV dA Dt   VT n Showing using Reynold’s theorem, S V D dV dV Dt t    v vv v tt  v v v v DD dv

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sec10-022410 - ChE 120B Heat Transfer Review of General...

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