Chapter 2:
Heat Conduction Equation
Yoav Peles
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Objectives
When you finish studying this chapter, you should be able to:
• Understand multidimensionality and time dependence of heat transfer,
and the conditions under which a heat transfer problem can be
approximated as being one-dimensional,
• Obtain the differential equation of heat conduction in various
coordinate systems, and simplify it for steady one-dimensional case,
• Identify the thermal conditions on surfaces, and express them
mathematically as boundary and initial conditions,
• Solve one-dimensional heat conduction problems and obtain the
temperature distributions within a medium and the heat flux,
• Analyze one-dimensional heat conduction in solids that involve heat
generation, and
• Evaluate heat conduction in solids with temperature-dependent
thermal conductivity.
2.1. Introduction
• Although heat transfer and temperature are
closely related, they are of a different nature.
• Temperature has only magnitude
it is a scalar quantity.
• Heat transfer has direction as well as magnitude
it is a vector quantity.
• We work with a coordinate system and indicate
direction with plus or minus signs.
• The driving force for any form of heat transfer is the
temperature difference.
• Three prime coordinate systems:
– rectangular (T(x, y, z, t))
– cylindrical (T(r, φ, z, t))
– spherical (T(r, φ, θ, t)).
Classification of conduction heat transfer problems:
• steady versus transient heat transfer,
• multidimensional heat transfer,
• heat generation.
Steady versus Transient Heat Transfer
• Steady implies no change with time at any point
within the medium
• Transient implies variation with time or time
dependence
Multi-dimensional Heat Transfer
• Heat transfer problems are also classified as
being:
– one-dimensional,
– two dimensional,
– three-dimensional.
• In the most general case, heat transfer through a
medium is three-dimensional. However, some
problems can be classified as two- or onedimensional depending on the relative
magnitudes of heat transfer rates in different
directions and the level of accuracy desired.
• The rate of heat conduction through a medium in
a specified direction (say, in the x-direction) is
expressed by Fourier’s law of heat conduction
for one-dimensional heat conduction as:
dT
Qcond = − kA
dx
(W) (2-1)
• Heat is conducted in the direction
of decreasing temperature, and
thus the temperature gradient is negative
when heat is conducted in the positive x-direction.
General Relation for Fourier’s Law of Heat
Conduction
• The heat flux vector at a point P on the surface of the
figure must be perpendicular to the surface, and it must
point in the direction of decreasing temperature
• If n is the normal of the
isothermal surface at point P,
the rate of heat conduction at
that point can be expressed by
Fourier’s law as
dT
Qn = − kA
dn
(W) (2-2)
• In rectangular coordinates, the heat conduction
vector can be expressed in terms of its components as
G
G
G
G
Q n = Q x i + Q y j + Q z k
(2-3)
• which can be determined from Fourier’s law as
⎧
∂T
⎪Qx = − kAx ∂x
⎪
∂T
⎪
⎨Qy = −kAy
∂y
⎪
⎪
∂T
⎪Qz = −kAz
∂z
⎩
(2-4)
Heat Generation
• Examples:
– electrical energy being converted to heat at a rate
of I2R,
– fuel elements of nuclear reactors,
– exothermic chemical reactions.
• Heat generation is a volumetric phenomenon.
• The rate of heat generation units : W/m3 or Btu/h · ft3.
• The rate of heat generation in a medium may vary
with time as well as position within the medium.
• The total rate of heat generation in a medium of
volume V can be determined from
E gen = ∫ egen dV
V
(W)
(2-5)
六個面均可簡化考慮為onedimensional problem !
2.2 One-Dimensional Heat Conduction Equation
- (i) Plane Wall
Rate of heat
conduction
at x
-
Rate of heat
conduction
at x+∆x
+
Q x −Q x +∆x + E gen,element
(2-6)
Rate of heat
generation inside
the element
∆Eelement
=
∆t
=
Rate of change of
the energy content
of the element
∆Eelement
Qx − Qx+∆x + Egen,element =
∆t
(2-6)
• The change in the energy content and the rate of heat
generation can be expressed as
⎧∆
⎪ Eelement = Et +∆t − Et = mc (Tt +∆t − Tt ) = ρcA∆x (Tt +∆t − Tt ) (2-7)
⎨
(2-8)
⎪⎩Egen,element = egenVelement = egen A∆x
• Substituting into Eq. 2–6, we get
Tt +∆t − Tt
+
e
A
∆
x
Qx − Qx+∆x gen
= ρcA∆x
∆t
(2-9)
• Dividing by A∆x, taking the limit as ∆xÆ 0 and ∆tÆ 0,
and from Fourier’s law:
1 ∂ ⎛ ∂T
⎜ kA
A ∂x ⎝ ∂x
∂T
⎞
ρ
+
e
=
c
⎟ gen
∂t
⎠
(2-11)
The area A is constant for a plane wall Æ the one dimensional
transient heat conduction equation in a plane wall is
Variable conductivity:
∂ ⎛ ∂T
⎜k
∂x ⎝ ∂x
∂T
⎞
ρ
+
e
=
c
⎟ gen
∂t
⎠
Constant conductivity:
∂ 2T egen 1 ∂T
+
=
2
k
∂x
α ∂t
; α=
(2-13)
k
ρc
(2-14)
The one-dimensional conduction equation may be reduces
to the following forms under special conditions
1) Steady-state:
2) Transient, no heat generation:
3) Steady-state, no heat generation:
d 2T egen
+
=0
2
dx
k
(2-15)
∂ 2T 1 ∂T
=
2
∂x
α ∂t
(2-16)
d 2T
=0
2
dx
(2-17)
One-Dimensional Heat Conduction Equation
– (ii) Long Cylinder
Rate of heat
conduction
at r
-
Rate of heat
conduction
at r+∆r
+
Rate of heat
generation inside
the element
∆
E
element
=
Qr −Qr +∆r + Egen,element
∆t
(2-18)
=
Rate of change of
the energy content
of the element
∆Eelement
Qr − Qr +∆r + Egen,element =
∆t
(2-18)
• The change in the energy content and the rate of heat
generation can be expressed as
⎧∆
⎪ Eelement = Et +∆t − Et = mc (Tt +∆t − Tt ) = ρcA∆r (Tt +∆t − Tt ) (2-19)
⎨
(2-20)
⎪⎩Egen,element = egenVelement = egen A∆r
• Substituting into Eq. 2–18, we get
Tt +∆t − Tt
+
e
A
∆
r
Qr − Qr +∆r gen
= ρcA∆r
∆t
(2-21)
• Dividing by A∆r, taking the limit as ∆rÆ 0 and ∆tÆ 0,
and from Fourier’s law:
1 ∂ ⎛ ∂T
⎜ kA
A ∂r ⎝ ∂r
∂T
⎞
ρ
+
e
=
c
⎟ gen
∂t
⎠
(2-23)
Noting that the area varies with the independent variable r
according to A=2πrL, the one dimensional transient heat
conduction equation in a plane wall becomes
Variable conductivity:
Constant conductivity:
1 ∂ ⎛ ∂T
⎜ rk
r ∂r ⎝ ∂r
∂T
⎞
ρ
+
e
=
c
⎟ gen
∂t
⎠
1 ∂ ⎛ ∂T ⎞ egen 1 ∂T
=
⎜r
⎟+
α ∂t
r ∂r ⎝ ∂r ⎠ k
(2-25)
(2-26)
The one-dimensional conduction equation may be reduces
to the following forms under special conditions
1) Steady-state:
2) Transient, no heat generation:
1 d ⎛ dT
⎜r
r dr ⎝ dr
1 ∂ ⎛ ∂T
⎜r
r ∂r ⎝ ∂r
3) Steady-state, no heat generation:
⎞ egen
= 0 (2-27)
⎟+
⎠ k
⎞ 1 ∂T
⎟=
⎠ α ∂t
d ⎛ dT
⎜r
dr ⎝ dr
⎞
⎟=0
⎠
(2-28)
(2-29)
One-Dimensional Heat Conduction Equation
- (iii) Sphere
Variable conductivity:
Constant conductivity:
1 ∂ ⎛ 2 ∂T
⎜r k
2
∂r
r ∂r ⎝
∂T
⎞
ρ
+
e
=
c
⎟ gen
∂t
⎠
(2-30)
1 ∂ ⎛ 2 ∂T
⎜r
2
r ∂r ⎝ ∂r
⎞ egen 1 ∂T
=
⎟+
α ∂t
⎠ k
(2-31)
One-dimensional steady-state
heat conduction equation:
d 2T
=0
dx 2
2.3 General Heat Conduction Equation
Rate of heat
conduction
at x, y, and z
Rate of heat
Rate of heat
- conduction + generation
at x+∆x, y+∆y,
inside the
and z+∆z
element
Q x + Q y + Q z
=
Rate of change
of the energy
content of the
element
∆Eelement
(2-36)
−Qx +∆x − Qy +∆y − Qz +∆z + Egen ,element =
∆t
Repeating the mathematical approach used for the onedimensional heat conduction the three-dimensional heat
conduction equation is determined to be
Two-dimensional
Constant conductivity:
∂ 2T ∂ 2T ∂ 2T egen 1 ∂T
+ 2 + 2 +
=
2
∂x
∂y
∂z
α ∂t
k
(2-39)
Three-dimensional
∂ 2T ∂ 2T ∂ 2T egen
+ 2 + 2 +
= 0 (2-40)
2
∂x
∂y
∂z
k
1) Steady-state:
2) Transient, no heat
∂ 2T ∂ 2T ∂ 2T 1 ∂T
generation: ∂x 2 + ∂y 2 + ∂z 2 = α ∂t
(2-41)
∂ 2T ∂ 2T ∂ 2T
generation: 2 + 2 + 2 = 0
∂x
∂y
∂z
(2-42)
3) Steady-state, no heat
Cylindrical Coordinates
1 ∂ ⎛ ∂T
⎜ rk
r ∂r ⎝ ∂r
⎞ 1 ∂ ⎛ ∂T
k
⎟+ 2
⎜
⎠ r ∂φ ⎝ ∂φ
⎞ ∂ ⎛ ∂T
⎟ + ∂z ⎜ k ∂z
⎝
⎠
∂T
⎞ ⎟ + egen = ρ c
∂t
⎠
(2-43)
Spherical Coordinates
∂ ⎛ ∂T ⎞
∂ ⎛
∂T ⎞
∂T
1 ∂ ⎛ 2 ∂T ⎞
1
1
+
+
+
=
kr
k
k
sin
e
c
θ
ρ
⎜
⎟
⎜
⎟
⎜
⎟ gen
∂r ⎠ r 2 sin 2 θ ∂φ ⎝ ∂φ ⎠ r 2 sin θ ∂θ ⎝
∂θ ⎠
∂t
r 2 ∂r ⎝
(2-44)
2.4 Boundary and Initial Conditions
• (i) Specified Temperature Boundary Condition
• (ii) Specified Heat Flux Boundary Condition
• (iii) Convection Boundary Condition
• (iv) Radiation Boundary Condition
• (v) Interface Boundary Conditions
• (vi) Generalized Boundary Conditions
(i) Specified Temperature Boundary Condition
For one-dimensional heat
transfer through a plane wall of
thickness L, for example, the
specified temperature boundary
conditions can be expressed as
T(0, t) = T1
T(L, t) = T2
(2-46)
The specified temperatures can be constant, which is the
case for steady heat conduction, or may vary with time.
(ii) Specified Heat Flux Boundary Condition
The heat flux in the positive xdirection anywhere in the medium,
including the boundaries, can be
expressed by Fourier’s law of heat
conduction as
q = − k
dT
=
dx
Heat flux in
the positive
x-direction
(2-47)
The sign of the specified heat flux is determined by
inspection: positive if the heat flux is in the positive
direction of the coordinate axis, and negative if it is in
the opposite direction.
Two Special Cases – Insulated Boundary
Insulated boundary
∂T (0, t )
k
=0
∂x
or
∂T (0, t )
=0
∂x
(2-49)
Thermal symmetry
∂T ( L / 2, t )
= 0 (2-50)
∂x
(iii) Convection Boundary Condition
Heat conduction
at the surface in a
selected direction
=
Heat convection
at the surface in
the same direction
∂T (0, t )
−k
= h1 [T∞1 − T (0, t ) ]
∂x
(2-51a)
∂T ( L, t )
−k
= h2 [T ( L, t ) − T∞ 2 ]
∂x
(2-51b)
and
(iv) Radiation Boundary Condition
Heat conduction at the
surface in a selected
direction
=
Radiation exchange at
the surface in
the same direction
∂T (0, t )
4
4
⎡
−k
= ε1σ ⎣Tsurr ,1 − T (0, t ) ⎤⎦
∂x
(2-52a)
and
∂T ( L, t )
4
⎤
−k
= ε 2σ ⎡⎣T ( L, t ) 4 − Tsurr
,2 ⎦
∂x
(2-52b)
(v) Interface Boundary Conditions
At the interface the requirements are:
(1) the same temperature at the area of contact,
(2) the heat flux on the two sides of an
interface must be the same.
TA(x0, t) = TB(x0, t)
(2-53)
and
∂TA ( x0 , t )
∂TB ( x0 , t )
−k A
= −kB
(2-54)
∂x
∂x
(vi) Generalized Boundary Conditions
In general a surface may involve convection, radiation,
and specified heat flux simultaneously. The boundary
condition in such cases is again obtained from a surface
energy balance, expressed as
Heat transfer
to the surface
in all modes
=
Heat transfer
from the surface
in all modes
2.5 Solution of Steady One-Dimensional
Heat Conduction Problems
(a)
OK !
不成立 !
相依
2.6 Heat Generation (熱生成) in Solids
核能、電阻加熱
與化學反應熱於
過程中之轉換能
量，在分析時均
被當作生成熱。
The quantities of major interest in a medium
with heat generation are
(i) the surface temperature Ts
(ii) the maximum temperature Tmax
that occurs in the medium in steady operation.
Heat Generation in Solids
- (i) The Surface Temperature
Rate of
heat transfer
from the solid
=
Rate of
energy generation
within the solid
(2-63)
For uniform heat generation within the medium
Q = egenV (W)
(2-64)
The heat transfer rate by convection can also be
expressed from Newton’s law of cooling as
-
Q = hAs (Ts − T∞ )
Ts = T∞ +
egenV
hAs
(W)
(2-65)
(2-66)
(a) For a large plane wall of thickness 2L (As=2Awall
and V=2LAwall)
Ts , plane wall = T∞ +
egen L
h
(2-67)
(b) For a long solid cylinder of radius r0 (As=2πr0L
and V=πr02L)
egen r0
(2-68)
Ts ,cylinder = T∞ +
2h
(c) For a solid sphere of radius r0 (As=4πr02 and V=4/3πr03)
Ts , sphere = T∞ +
egen r0
3h
(2-69)
Heat Generation in Solids – (ii)The maximum
Temperature in a Cylinder (the Centerline)
The heat generated within an inner
cylinder must be equal to the heat
conducted through its outer surface.
dT
−kAr
= egenVr
dr
(2-70)
Substituting these expressions into the above equation
and separating the variables, we get
egen
dT
2
−k ( 2π rL )
= egen π r L → dT = −
rdr
2k
dr
(
)
Integrating from r =0 where T(0) =T0 to r=ro
∆Tmax,cylinder = T0 − Ts =
egen r02
4k
(2-71)
Variable Thermal Conductivity, k(T) - 略
• The thermal conductivity of a
material, in general, varies with
temperature.
• An average value for the
thermal conductivity is
commonly used when the
variation is mild.
• This is also common practice
for other temperaturedependent properties such as
the density and specific heat.
Variable Thermal Conductivity for OneDimensional Cases
When the variation of thermal conductivity with temperature
is known, the average value of the thermal conductivity in the
temperature range between T1 and T2 can be determined from
kave
∫
=
T2
T1
k (T )dT
(2-75)
T2 − T1
The variation in thermal conductivity of a material
with can often be approximated as a linear function
and expressed as
(2-79)
k (T ) = k0 (1 + β T )
β the temperature coefficient of thermal conductivity.
Variable Thermal Conductivity
• For a plane wall the
temperature varies linearly
during steady onedimensional heat conduction
when the thermal conductivity
is constant.
• This is no longer the case
when the thermal conductivity
changes with temperature
(even linearly).