MEEN 461 HEAT TRANSFER
Summary of Conduction
Fourier’s Law: q 
Newton’s Law: q 
q
T
 k
; k in [W/(mK)]
A
x
q
 h Tsurface  T fluid ; h in [W/(m2K)]
A
4
Stefan-Boltzmann Law: q   T ; emission of an ideal surface (blackbody);  = 5.67  10 8 W/(m2K4); T in
[K]
q   AT 4 ; emission of a real surface;
0.0    1.0
q12   A1 (T14  T24 ) ; radiation exchange --- surface 1 in a large enclosure 2
Steady state, 1-D conduction without internal heat generation:
resistance
Plane wall, T(x): q 
q
T1  T2  T

L /( kA) Rth
Convection: q 
Thermal resistance and thermal contact
Ts  T  T

1/(hA) Rth
Thermal
contact
Tinterface
Rtc
Cylindrical wall, T(r): q 
Ti  To
T

 ln (ro / ri )  Rth


 2 kL 
Spherical wall, T(r): q 
Ti  To
 1/ ro  1/ ri

 4 k
Analogy between conduction/convection and current flow: Ohm’s Law
Thermal resistances in series:
q  qA  qB  qC  ... 
Toverall
Toverall

Rth
( Rth, A  Rth, B  Rth,C  ...)
𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑡ℎ,𝐴 + 𝑅𝑡ℎ,𝐵 + 𝑅𝑡ℎ,𝐶 + ⋯
Thermal resistances in parallel:
 T T

 1

T
1
1
q  (q1  q2  q3  ...)  


 ...   T 


 ... 
R

R

 th ,1 Rth ,2 Rth ,3

 th ,1 Rth ,2 Rth ,3

1
𝑅𝑡𝑜𝑡𝑎𝑙
=
1
𝑅𝑡ℎ,𝐴
+
1
𝑅𝑡ℎ,𝐵
+
1
𝑅𝑡ℎ,𝐶
+⋯
Steady state, 1-D conduction with volumetric internal heat generation: q in [W/m3]
Plane wall, T(x):
T ( x) 
qL2  x 2 
1    Ts
2k  L2 
Cylinder, T(r):
T (r ) 
qro2  r 2 
1    Ts
4k  ro2 
Fins:
resistance:
Fin effectiveness:
 fin 
q fin
(qbase ) without fin

q fin
hAc (Tb  T )
 fin  1.0




T
Rth
Fin efficiency:
 fin 
q fin
(q fin ) k 

q fin
 fin  1.0
hAs (Tb  T )
Finite-difference numerical method: Steady one-dimensional or multi-dimensional conduction
Derivation of nodal equations --- based on conservation of energy,
qin  qout  qV  0
T
T

x
x
Transient Conduction:
Lumped capacitance system, T(t):
h V
Bi  
k  As
 hA
T (t )  T
 exp   s
Ti  T
  cV
Time
 t  constant

t   exp   

 t 

  0.1

Implicit and explicit finite-difference numerical methods: Transient conduction
Derivation of nodal equations --- based on conservation of energy,
qin  qout  qV  dE / dt
T
T

t
t
Solve a set of N equations simultaneously or explicitly for temperatures at N nodes after each t using the
computer
MEEN 461 --- Heat Transfer
Summary of Convection


Newton’s Law of Cooling --- Convective heat transfer coefficient in [W/(m2K)]
Local heat transfer coefficient, h, and average heat transfer coefficient, h .
h
h

1
As

As
qx
Tsurface  T fluid

qconvection
qconvection

Asurface Tsurface  T fluid
Tsurface  T fluid
h dAs
Local heat transfer on a surface exposed to a fluid
At y  0, u  0,


h
h(Ts  T f )  k
T
y
;
h
k (T / y ) y  0
y 0
Ts  T f

External flow forced convection --- Concepts of hydrodynamic and thermal boundary layers
Internal flow forced convection --- Concepts of developing flow (entrance region or length), and
hydrodynamically and thermally fully developed flows
Laminar flow and turbulent flow in forced convection --- Critical Reynolds number for transition from a
laminar boundary layer to a turbulent boundary layer, 5  105, and critical Reynolds number for determining
if an internal flow is laminar or turbulent, 2,300.
Developing flow (entrance region) and thermally fully developed flow in a channel

Thermally fully developed flow

q  h Tw  Tm
h
qw
 constant
Tw  Tm
for UHF and UWT
Laminar flow (ReDh < 2,300) --- use tables, NuDh = constant for UHF or UWT
Turbulent flow (ReDh > 2,300) --- use NuDh (ReDh, Pr) correlations, for a channel of any cross section, and UHF
and UWT.
qw  h ( PL)(Tw  Tm )
for UHF(Uniform heat flux)
qw  h ( PL) Tlm
for UWT(Uniform Wall temperature), where
Tlm  (Tmo  Tmi ) / ln[(Tw  Tmi )/(Tw  Tmo )]
 Px 
Ts  Tm ( x)
 exp  
h
 mc 
Ts  Tm,i
p


for constant surface temperature

Forced convection
Nu Dh  constant, Nu L  Nu L ( Re L , Pr ) , NuD  NuD ( ReD , Pr) , or NuDh  NuDh (ReDh , Pr)
External flow --- flat plate, cylinder, sphere, a bank of tubes, impinging jet, and …
Internal flow --- channels of various cross sections --- round, rectangular, parallel-plate, annular, and …

Natural convection
Nu L  Nu L (GrL , Pr ) , Nu L  Nu L ( RaL , Pr ) , Nu L  Nu L ( RaL ) , or …
External flow --- vertical, horizontal, and inclined plates, long horizontal cylinder, sphere, and …
Internal flow --- vertical channel, vertical, horizontal, and inclined rectangular enclosures, concentric
cylinders, and …
Summary of Radiation
1. Blackbody radiation --Blackbody, Planck’s distribution, Eb ( ) = function of  and T; maxT  2,898 μm  K; max  1/ T
Spectral (or monochromatic) emissive power, Eb (W/m2/m); emissive power, Eb   T 4 (W/m2)
2. Real surfaces --Spectral emissive power, E    Eb (W/m2/m), and emissive power E   Eb (W/m2), where  is
emissivity (dimensionless; 0 <   1.0). For a blackbody,  = 1.0
3. Definitions ---
 = J – G
Irradiation G (W/m2), radiosity J (W/m2), and net radiaiton flux qnet
Diffuse radiation; specular radiation; and non-participating medium
4. More Definitions --Absorptivity (), Reflectivity (), and Transmissivity () are fractions of the irradiation absorbed, reflected,
and transmitted. For a blackbody,  = 1.0, and  =  = 0. Kirchhoff’s law:     
      1.0
Similarly,        1.0. For any opaque surface,     1.0. and     1.0.
E (T )
 (T ) 

Eb (T )


0
  ( , T )  Eb ( , T ) d 
T
4
Use of Table 12.2; F(0 ) 

T
0
Eb
d (T )  f (T )
T 5
Intensity (I) is the radiation emitted by a surface per unit projected area of the surface and per unit solid
angle in a specific direction.
I1 
dq12
dq12

(dA1 cos 1 ) d1 2 (dA1 cos 1 ) [(dA2 cos 2 ) / r 2 ]
in W/(m2sr)
For a blackbody, I b  Eb / . For diffuse radiation, I  E / . Similarly, Ib  Eb / and I  E / .
 (dA cos 1 ) ( dA2 cos 2 ) 
 ( dA cos 1 ) ( dA2 cos 2 ) 
dq12  I b1  1
 Eb1  1
2

 , to be used to define view factor
r
 r2



View factor
F12 is the fraction of the radiation that leaves a surface 1 arrives at another surface 2.
Reciprocity relation (or reciprocity rule):
A1F12  A2 F21
Summation rule
For the interior surfaces of an enclosure,
n
 Fi  j  Fi 1  Fi 2  Fi 3  Fi 4  ...  1.0
j 1
Superposition rule
Symmetry rule
Blackbody Radiation Exchange
For a blackbody,     1.0,     0, and J  Eb
Assumptions: (1) steady state, (2) isothermal surfaces, and (3) non-participating medium separating surfaces
Between two black surfaces:
qnet ,12 
Eb1  Eb 2  (T14  T24 )

Rspace
1/ ( A1 F12 )
where “space” resistance, Rspace 
1
A1 F12
Gray body Radiation Exchange
“Gray” = radiation properties are not dependent on wavelength, 
Additional assumptions: (4) gray or black surfaces, (5) opaque surfaces ( = 0), and (6) diffuse radiation
Between two surfaces:
qnet ,12 
J1  J 2
J1  J 2

Rspace 1/ ( A1F12 )
where “space” resistance, Rspace 
1
A1 F12
On each of the surfaces (for instance, surface 1) exchanging radiation:
qnet ,1 
Eb1  J1
Eb1  J1

Rsurface
(1  1 ) / (1 A1 )
where “surface” resistance, Rsurface 
1  1
1 A1
For three or more surfaces exchanging radiation, first solve a set of equations simultaneously for the values of
J.
Special cases: A relatively small surface 1 in a large enclosure 2 (or the surroundings); A2
qnet ,12 
A1 or A2  
Eb1  Eb 2
 (T14  T24 )

 1 A1 (T14  T24 )
 1  1   1   1   2   1  1   1 

 0


 
 1 A1   A1 F12    2 A2   1 A1   A1 