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
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2.051 Introduction to Heat Transfer
Fall 2015
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