MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.051 Introduction to Heat Transfer Equation Sheet (Fall 2015) STEADY HEAT TRANSFER: Mode of Heat Transfer Equation   q  kT Conduction Fourier’s Law Newton’s law of cooling Convection Radiation heat transfer from a small grey object (1) to a large isothermal environment (2)   5.67 108 W/(m 2 K 4 ) where   emissivity of object (1) 1 Radiation THERMAL RESISTANCE: Mode of Heat Transfer Slab Conduction Resistance [K/W] Rcond  L / (kA) Cylindrical (radial direction) Rcond  ln(rout / rin ) 2 kL Spherical (radial direction) Rcond   1 rin  rout1   4 k  Convection Rconv  1/ (hA) Radiation Rrad  1/ (hr A) where hr  41 Tavg 3 when T1  T2 Steady state heat equation with energy generation: Planar coordinate:   T    T    T   k k  k   q gen  0 x  x  y  y  z  z  Cylindrical coordinate: 1   T  kr r r  r  1   T  2 k  r       T  k  z  z    q gen  0  FIN EQUATIONS Fin Equation with convection alone. d 2  m 2 (x)  0 2 dx Tip condition (x=L) Convection heat transfer: h (L)  k General solution: Temperature distribution cosh m  L  x     cosh mL  Adiabatic: h mk h   sinh m  L  x  sinh mL  M   sinh mL mk   cosh mL  b   T  T ; b   (0)  Tb  T ; hPkAc         sinh mL   cosh mL  1   sinh mL  M  L  cosh  mL  b       sinh mL M m 2  hP / kAc ;   sinh mL   cosh mL mk hPkAc tanh mL   1 sinh mL hPkAc    cosh  mL   L    b   1 emx h mk h   M tanh mL sinh mx  sinh m  L  x  sinh mL   cosh mL 1   L h mk h mk cosh mL Infinite fin  L   :  (L)  0 Equation Sheet Fin Resistance R fin cosh m  L  x  d 0 dx xL  (L)   L Fin heat transfer rate  / b d dx x  L Prescribed temperature:  (x)  Ae mx  Be mx hPkAc M  b hPkAc Page 1 of 2 HYPERBOLIC OPERATORS: Function: f(x) Definition Derivative: sinh(x) e e cosh(x) x  e x  / 2 cosh(x) x  e x  / 2 sinh(x) e x  e x e x  e x tanh(x) d  f (x) dx 1 tanh 2 (x) UNSTEADY HEAT TRANSFER = Heat diffusion equation: where a  k /  rc Biot number: Bi  Fourier number: and + ̇ is the heat generation rate per unit volume [W/m3] DTinternal Rinternal hLc where, possibly, Lc  V / As   DTexternal Rexternal k Fo  at L2 Lumped Parameter Model (for Bi<<1): General lumped equation: Solution: where t rVc hA Semi-Infinite Solid: Boundary Condition Temperature distribution Constant Surface Temperature: T  x  0,t   Ts Constant Surface Heat Flux: q  x  0   q0 Surf ace Convection: − =ℎ − ( = 0, ) ( , )− − ( , )− ( , )− − = erfc = q x0  √4 ⁄ 4 = erfc ⁄( ) − ∙ erfc √4 √4 − exp Equation Sheet Heat Flux at Surface ℎ + ℎ erfc √4 + ℎ√ Page 2 of 2 k  Ts  Ti  a t

Equation Sheet

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