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Midterm Formula Sheet...

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HEAT TRANSFER I

MECH 352/2 FALL 14

Mid-term Formula Sheet Fundamental equations for Heat Conduction; Convection; Radiation: ∂T 4  conv = hAs (Ts − T∞ ) ; Q Q cond = − kA Q rad = εσAs (Ts4 − Tsurr ; ) ∂x Heat Conduction Equation: a. Rectangular coordinates: ∂ 2T ∂2 T ∂ 2 T e gen 1 ∂T + = + + α ∂t k ∂ x2 ∂y 2 ∂z 2 b. Cylindrical coordinates:

1 ∂  ∂T  1 ∂  ∂T  ∂  ∂T  egen 1 ∂T = r  + r +  + r ∂r  ∂r  r 2 ∂ ϕ  ∂ϕ  ∂z  ∂z  k α ∂t 1D steady state conduction in plane, cylindrical and spherical walls: ∆T ∆T ∆T  Q Qcond = 2πkl Q cond = 4πr1r2 k cond = − kA ln( r2 r1 ) ∆x r2 − r1 For fins: T (x ) − T∞ = e − mx where m 2 = hP kAc Q fin = hPkAc (Tb − T∞ ) Very long fin: Tb − T∞ T ( x) − T∞ cosh(m(L − x )) Insulated fin tip: = Q fin = hPkAc (Tb − T∞ ) tanh(mL ) Tb − T∞ cosh(mL ) Q fin Q fin Fin efficiency: η fin = Fin effectiveness: ε fin = Q fin,max Q no_ fin T (t ) − T∞ = e −bt where b = hA ρVc p Lumped System Analysis: Ti − T∞

 Q  Q    =   Qmax  total,3 D  Qmax

  Q    Q   Q   +   1 −  +  1  Qmax 2   Qmax 1   Q max

Biot number: Bi = hx/k 

 

   Q  1 −  3   Qmax

   Q  1 −  1   Qmax

    2 ...