CHAPTER 1: BASIC CONCEPTS
1.1 Examples of Conduction Problems
(1) Design
(2) Nuclear Reactor Core
(3) Glaciology
(4) Re-entry Shield
(5) Cryosurgery
(6) Rocket Nozzle
(7) Casting
(8) Food Processing
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1.2 Focal Point in Conduction Heat Transfer
Determination of the temperature distribution in a region
1.3 Fourier's Law of Conduction
A(Tsi  Tso )
qx 
L
A(Tsi  Tso )
(1.1)
qx  k
L
k = thermal conductivity
Tso
Tsi
A
qx
dx
0
L
x
Fig.1.1
2
Apply (1.1) to an element dx
T ( x )  T ( x  dx )
T ( x  dx )  T ( x )
q x  kA
  kA
dx
dx
dT
q x   kA
dx
(1.2)
• Definition: heat flux qx
qx
qx 
A
(1.3)
dT
qx   k
dx
(1.4)
3
For 3-D:
T
T
T (1.5)
qx   k
, qy   k
, qz   k
x
y
z
1.4 Conservation of Energy: Differential
Formulation of the Heat Conduction
Equation in Rectangular Coordinates
y
Consider:
• Uniform velocity
• Volumetric energy generation q
x
z
Fig. 1.2
4
Conservation of energy for element:
Rate of energy added + Rate of energy generated
– Rate of energy removed
= Rate of energy change within element
E in  E g  E out  E
(1.6)
Express each term in eq. (1.6) in terms of the dependent
variable T.
Assumptions:
(1) Negligible radiation
(2) Negligible work done due to surface stress.
5
 ˆ hˆ 
V  h  dy dxdz
y 

q 

 qy  y dy dxdz


y


qx dydz
dy
q


 qx  x dx dydz
x


dx
Uhˆ dydz
qy dxdz
 ˆ hˆ 
U  h  dx dydz
x 

dy
dx
Vhˆ dxdz
(b)
(a)
Figure 1.3
E in  qx dydz  qy dxdz  qzdxdy  Uhˆ dydz
 Vhˆ dxdz  Whˆ dxdy
(a)
6
E g  qdxdydz
E out
(b)
qy 

qx


  qx 
dx dydz   qy 
dy dxdz
x
y




 ˆ hˆ 
qz 

  qz 
dz dxdy  U  h 
dx dydz
z
x 



 ˆ hˆ 
 ˆ hˆ 
 V  h 
dy dxdz  W  h 
dz dxdy
y 
z 


(c)
U, V and W are constant since material motion is
uniform.
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Energy change within element E
uˆ

E
dxdydz
t
(d)
Substituting (a) - (d) into eq. (1.6)
qx qy qz
hˆ
hˆ
hˆ



 U
 V
 W
x
y
z
x
y
z
uˆ
(e)
 q  
t
Enthalpy ĥ
ˆh  uˆ  P

(f)
8
(f) into (e)
ˆ
ˆ
ˆ
 hˆ
qx qy qz

h

h

h




 q     U
V
W

x
y
z

t

x

y

z


(g)
Enthalpy change for constant P
d hˆ  cp dT
(h)
Eq. (1.5) and (h) into (g)
  T    T    T 
   k
k
   k
  q
x  x  y  y  z  z 
(1.7)
 T
T
T
T 

 c p 
U
V
W
x
y
z 
 t
9
Stationary material, constant k
  2T  2T  2T  q
T



 2  2 

2
 x
 c
t

y

z
p


(1.8)
 = thermal diffusivity
k

c p
(1.9)
1.5 The Heat Conduction Equation in
Cylindrical and Spherical Coordinates
Assumptions:
(1) Constant k,
(2) Constant  and
(3) No dissipation
10
z
z
(r,z,)
•
y

 (r,,)
r
•
x
x
y
r
(a) cylindrical

(b) spherical
Fig. 1.4
Cylindrical Coordinates (r,,z)
 1   T  1  2T  2T  q

 2 
r
 2
2
z  c p
 r r  r  r 
(1.10)
T V T
T 
 T

 Vr

 Vz

r
r 
z 
 t
Assume:
(4) Stationary material
 1   T  1  2T  2T  q
T

 2 

r
 2
2
z  c p t
 r r  r  r 
(1.11)
Spherical Coordinates (r,,)
 1   2 T 
1
 
T 
1
 2T 
 sin 
  2 2
 2 r
 2

2
  r sin   
 r r  r  r sin   
V T
q T
T V T


 Vr


c p t
r
r  r sin  
(1.12)
For stationary material
 1   2 T 
1
 
T 
1
 2T 
 2 r
 2
 sin
 2 2

2
  r sin   
 r r  r  r sin  
q T


c p t
(1.13)
1.6 Boundary Conditions
Boundary conditions are mathematical equations
describing what takes place physically at a boundary
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An initial condition describes the temperature
distribution at time t = 0
1.6.1 Surface Convection: Newton's Law of
Cooling
Energy interchange between a surface and a fluid
moving over it
qs  Ts  T 
qs  hTs  T 
(1.14)
h = heat transfer coefficient
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1.6.2 Surface Radiation: Stefan-Boltzmann
Law
The Stefan-Boltzmann law for an ideal surface called
blackbody is
qb  Ts4
(1.15)
qb  blackbody radiation flux
Ts  surface temperature, absolute degrees
  Stefan-Boltzmann constant
8
  5.67  10 W m  K
2
4
(1.16)
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Emissivity :
qr

qb
(1.17)
Combining (1.15) and (1.17)
qr   Ts4
(1.18)
• Special case: small gray surface which is completely
enclosed by a much larger surface

qr   1A1 T14  T24
( )1 = small surface
( )2 = large surface
T = absolute temperature

(1.19)
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1.6.3 Examples of Boundary Conditions
Typical boundary conditions are shown in Fig. 1.5 and Fig. 1.6
y
insulation
qo
W
To
00
L
convection, h, T
x
Fig. 1.5
k1
k2
T1
T2
00
x
Fig. 1.6
To write boundary conditions we must:
• Select an origin
• Select coordinate axes
• Identify the physical conditions at the boundaries
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y
insulation
qo
W
To
00
L
convection, h, T
x
Fig. 1.5
(1) Specified temperature
Along (0,y) the temperature is To
T (0, y )  To
(1.20)
(2) Specified flux
The heat flux along (L,y) is given as qo
T ( L, y )
qo   k
x
(1.21)
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(3) Convection
Heat is exchanged at (x,0) by convection with a fluid
at temperature T. Conservation of energy, Newton’s
law and Fourier’s law:
T ( x ,0)
hT  T  x ,0   k
y
(1.22)
(4) Insulated Boundary
Boundary (x,W) is thermally insulated. Fourier’s
law:
T  x ,W 
0
y
(1.23)
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(5) Interface
+
For a perfect interface contact,
continuity of temperature
T1 0, y   T2 0, y 
k1
1 T
1
(1.24)
k2
T2 2
0
q i
Equality of flux gives
T1 0, y 
T2 0, y 
k1
 k2
x
x
Fig. 1.7
(1.25)
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(6) Interface with a heat source
Examples:
(1) An electrical heating element which is sandwiched
between two materials
(2) An interface with frictional heat
due to relative motion
Boundary conditions:
Continuity of temperature: For perfect contact,
eq. (1.31) applies.
21
Conservation of energy at the interface
T1 0, y 
T2 0, y 
 k1
 qi   k 2
x
x
(1.26)
(7) Radiation
Boundary (x,0) exchanges heat by
radiation and convection.
y
Conservation of energy gives
0
  qrad
  qcond

qconv

h, T
x
Tsur
22
Use Fourier, Newton, and Stefan-Boltzmann laws:
hT  T  x ,0

4
  Tsur

T  x ,0
 T  x ,0   k
y
(1.27)
4
1.7 Problem Solving Format
Conduction problems lend themselves to a systematic
solution procedure.
(1) Observations
• Read the problem statement very carefully.
• Note essential facts and features: geometry,
temperature, heat flow direction, symmetry, number
of independent variables, etc.
23
• Identify characteristics that require special
attention: variable properties, composite domain,
non-linearity, radiation, phase change, etc.
• Show a sketch
(2) Origin and Coordinates
• Select an origin and a coordinate system appropriate
to the geometry under consideration.
(3) Formulation
• Express problem mathematically
• Define all terms
24
• Assign units to all symbols
• Proceed according to the following steps:
(i) Assumptions
Modeling the problem. List all assumptions.
(ii) Governing Equations
Select an appropriate heat equation.
(iii) Boundary conditions
Decide on the number of boundary conditions
needed. Identify the physical nature of each
boundary condition and express it
mathematically.
25
(4) Solution
• Examine equation to be solved. Select a method of
solution.
• Apply boundary conditions.
(5) Checking
• Check each step of the analysis as you proceed
• Apply dimensional checks and examine limiting cases
(6) Computations
• Execute the necessary computations and calculations
to generate the desired numerical results.
26
(7) Comments
1.8 Units
The basic units in the SI system are
Length (L): meter (m)
Time (t): second (s)
Mass (m): kilogram (kg)
Temperature ( T): kelvin (K)
Temperature on the Celsius scale is related to the Kelvin
scale by
(1.28)
T(oC) = T(K) - 273.15
27
Units of other derived quantities:
• Force:
Measured in newtons (N). One newton is the force
needed to accelerate a mass of one kilogram one meter
per second per second.
N = kg • m /s 2
• Energy:
Measured in joules (J). One joule is the energy
associated with a force of one newton moving a
distance of one meter.
2 2
J = N. m = kg • m /s
28
• Power:
Measured in watts (W). One watt is the energy rate of
one joule per second.
W = J/s = N. m/s = kg • m2 /s 3
29