‫بسم هللا الرحمن الرحيم‬
2. Transient Heat Conduction
1. Introduction
It is meant by transient heat conduction, that the
temperature distribution inside the body is varies with
time. Accordingly, the temperature inside the body is
function of position and also of time, i.e.:
z
𝑇 = 𝑇(𝑥, 𝑦, 𝑧, 𝑡)
The general form of conductive heat transfer, in case of
constant physical properties as ρ, k…etc. may be written
as:
𝜕2 𝑇
𝜕2 𝑇
𝜕2 𝑇
+ 𝜕𝑦 2 + 𝜕𝑧 2 + 𝑘1 𝑞̇ = 𝛼1
𝜕𝑥 2
z
x
𝜕𝑇
(1)
𝜕𝑡
y
y
Where 𝑞̇ is the rate of heat generation inside the body
per unit volume, and α is the thermal diffusivity, which
has the following definition:
x
Figure (1)
𝛼=
𝑘
𝜌 𝑐𝑝
Where k,  and cp are thermal conductivity [W/m.K], density [kg/m2] and constant
pressure specific heat; respectively.
Examining equation (1), one can distinguish between three possible situations for the
case of no heat generated inside the body:
a) Temperature variation throughout the body can be neglected, i.e. the
temperature distribution is uniform everywhere, but is varied with time, then:
𝑇 = f(t)
Such case is known as lumped system.
b) Temperature is function of time and heat is transferred in single direction, such
case satisfied the condition of large plane wall, long cylinder, Sphere and semiinfinite solid. The transient conduction in large plane wall can be simplified
according to equation (1) to the following form as:
𝜕2 𝑇
𝜕𝑥 2
= 𝛼1
1
𝜕𝑇
𝜕𝑡
(2)
Equation (2) is transient one dimensional heat conduction equation. Temperature
in this case is function of both t and x:
𝑇 = 𝑓(𝑥, 𝑡)
c) Temperature is function of time and heat is transferred in more than one
direction. In such case, one speaks about Transient heat conduction in
multidimensional system. Temperature, in such case, can express as:
𝑇 = 𝑇(𝑥, 𝑦, 𝑡)
𝑜𝑟
𝑇 = 𝑇(𝑥, 𝑦, 𝑧, 𝑡)
2.2 Lumped System Analysis
As it is mentioned before, in lumped system,
the temperature throughout the whole of body is
uniform but it varies from time to time. For the
shown body, Fig.( ) and taking in account the
conservation law of energy and for an incremental
time dt, one can write:
Convection
Conduction
h
Too
Solid Body
heat transferred 𝒊𝒏𝒕𝒐 the body = the increase in
the energy of the body during dt
T(t)
Ti
And
ℎ 𝐴𝑠 (𝑇∞ − 𝑇) 𝑑𝑡 = 𝑚 𝑐𝑝 𝑑𝑇
Since T∞ has a constant value, the previous equation
can be written as
Lumped system
ℎ 𝐴𝑠
𝑑(𝑇∞ − 𝑇)
𝑑𝑡 = −
𝑚 𝑐𝑝
𝑇∞ − 𝑇
Carrying out the integration of the previous equation and after some manipulations, the
integral has the form:
𝑇(𝑡) − 𝑇∞
= 𝑒 −𝑏𝑡
𝑇𝑖 − 𝑇∞
Where
ℎ 𝐴
𝑏 = 𝑚 𝑐 𝑠 and its reciprocal 1/b is called time constant. The maximum of
𝑝
transferred heat Qmax can be determined according to the following relation:
𝑄𝑚𝑎𝑥 = 𝑚 𝑐𝑝 (𝑇∞ − 𝑇𝑖 )
Heat transferred up to certain time t is given by:
𝑄 = 𝑚 𝑐𝑝 (𝑇(𝑡) − 𝑇𝑖 )
2
The assumption of lumped system is valid in case of heat transfer by conduction
throughout the body is greater than that of transferred by convection at the surface of
this body. This condition is achieved in case of very conductive material as copper and
relatively small convection coefficient of heat transfer. The measure of the ratio of
convection heat transfer to conduction heat transfer is given by Biot number, which is
defined as:
ℎ 𝐿𝑐
𝐵𝑖 =
𝑘
Where Lc is characteristic length defined as:
𝐿𝑐 =
𝑉
𝐴𝑠
Where V is the volume of the body and As is the surface area. As it is clear from the
definition of Biot number, the lumped system assumption is achieved as the value of Bi
smaller. It is proper to consider the system as lumped system if:
𝐵𝑖 ≤ 0.1
Example 1
A thermocouple junction, which may be approximated as a sphere, is to be used for
temperature measurement in a gas stream. The convection coefficient between junction
surface and the gas is known to be h = 400 W/m2.K, and the junction thermophysical
properties are k = 20 W/m.K, cp = 400 J/kg.K, and ρ = 8500 kg/m3. Determine the
junction diameter needed for the thermocouple to have time constant of 1 s. If the
junction is at 25˚C and is placed in a gas stream that is at 200˚C, how long will it take for
the junction to reach 199˚C.
Given:
T∞ = 200°C, Ti = 25°C, h = 400 W/m2.K,
k = 20 W/m.K, cp = 400 J/kg.K, ρ = 8500 kg/m3,
T =199°C, τt = 1/b = 1 s.
Leads
Too
Required:
d, t@199C
Gas Stream
Ti
Solution:
Thermocouple
Junction
𝑇(𝑡) − 𝑇∞
= 𝑒 −𝑏𝑡
𝑇𝑖 − 𝑇∞
𝑏=
ℎ 𝐴𝑠
𝑚 𝑐𝑝
h
=
1
= 1 𝑠 −1
𝜏𝑡
3
1
𝜌𝜋𝑑3
𝜏𝑡 =
×
𝑐
ℎ𝜋𝑑 2
6 𝑝
𝑑=
6ℎ𝜏𝑡 6 × 400 × 1
=
= 7.06 × 10−4 𝑚
𝜌𝑐𝑝
8500 × 400
𝐿𝑐 =
𝑟𝑜
3
𝐵𝑖 =
ℎ𝐿𝑐 400 × 3.53 × 10−4
=
= 2.35 × 10−4 < 0.1
𝑘
3 × 20
𝜋𝑑3
𝜌 ( 6 ) 𝑐𝑝
𝑇𝑖 − 𝑇∞ 𝜌𝑑𝑐𝑝
𝑇𝑖 − 𝑇∞
𝑡=
× 𝑙𝑛
=
𝑙𝑛
2
ℎ(𝜋𝑑 )
𝑇 − 𝑇∞
6ℎ
𝑇 − 𝑇∞
𝜋𝑑3
𝜌 ( 6 ) 𝑐𝑝
𝑇𝑖 − 𝑇∞ 8500 × 7.06 × 10−4
25 − 200
𝑡=
×
𝑙𝑛
=
𝑙𝑛
2
ℎ(𝜋𝑑 )
𝑇 − 𝑇∞
6 × 400
199 − 200
𝑡 = 5.2 𝑠 = 5 𝜏𝑡
4
2.3 Transient Heat Conduction in Large Plane Walls, Long cylinders, and Spheres
with Spatial Effects
One-dimensional systems with an initial uniform temperature subjected to sudden
convection conditions.
Consider the plane wall of thickness 2L,
shown in figure. If the thickness is small
relative to the width and height of the wall,
it is reasonable to assume that conduction
occurs exclusively in the x direction. If the
wall is initially at a uniform temperature,
T(x, 0) = T i and is suddenly immersed in a
fluid of T∞≠Ti, the resulting temperatures
may be obtained by solving Equation (2):
𝜕2 𝑇
𝜕𝑥 2
= 𝛼1
𝜕𝑇
𝜕𝑡
(2)
Equation (2) must satisfy the
boundary and initial conditions:
following
𝑇(𝑥, 0) = 𝑇𝑖
5
Transient temperature profiles in plane
wall
𝜕𝑇
=0
|
𝜕𝑥 𝑥=0
and
−𝑘
𝜕𝑇
= ℎ[𝑇(𝐿, 𝑡) − 𝑇∞ ]
|
𝜕𝑥 𝑥=𝐿
Since the convection conditions for the surfaces at x/L = ± 1 are the same, the
temperature distribution at any instant must be symmetrical about the midplane (x/L =
0). As it is clear from the governing equation (2) and their initial and boundary
conditions, the solution of this equation is a function of x, L, t, , h, Ti and T∞. To reduce
the problem parameters and variables, seeking for generalization and simplification, it is
proper to introduce the following dimensionless quantities:
𝜃(𝑥, 𝑡) =
Dimensionless temperature:
Dimensionless distance from the center: 𝑋 =
𝑥
Dimensionless heat transfer coefficient: 𝐵𝑖 =
Dimensionless time:
𝜏=
𝛼𝑡
𝑇(𝑥,𝑡)−𝑇∞
𝑇𝑖 −𝑇∞
𝐿
ℎ𝐿
𝑘
(Biot number)
(Fourier number)
𝐿2
The nondimensionalization enables us to present the temperature in terms of three
parameters only: X, Bi, τ. Substituting the definitions of dimensionless parameters into
energy equation (2), the heat equation becomes:
𝜕2 𝜃
𝜕𝑋 2
𝜕𝜃
=
𝜕𝜃
𝜕𝜏
|
=0
|
= −𝐵𝑖 𝜃(1, 𝜏)
𝜕𝑋 𝑋=0
𝜕𝜃
𝜕𝑋 𝑋=1
An approximate solution of the foregoing dimensionless equation, takes the form:
6
For Plane wall (P): 𝜃(𝑥, 𝑡) =
𝑇(𝑥,𝑡)−𝑇∞
𝑇𝑖 −𝑇∞
= 𝐴1 𝑒 −𝜆1
2
𝜏
𝑥
cos(𝜆1 𝐿 ) 𝜏 > 0.2
In the same manner, the solution for cylinder and sphere is listed as the following,
taking in account that the characteristic length in this case is given ro instead of L in
case of plane wall:
For Cylinder (C):
For Sphere:
𝜃(𝑟, 𝑡) =
𝜃(𝑟, 𝑡) =
𝑇(𝑟,𝑡)−𝑇∞
𝑇𝑖 −𝑇∞
𝑇(𝑟,𝑡)−𝑇∞
𝑇𝑖 −𝑇∞
= 𝐴1 𝑒 −𝜆1
= 𝐴1
2
𝑒−𝜆1 𝜏
2
𝜏
𝑟
Jo (𝜆1 )
𝑟𝑜
𝑟
𝑜
𝑠𝑖𝑛(𝜆1 𝑟 )
𝑟
𝑜
𝜆1 𝑟
𝜏 > 0.2
𝜏 > 0.2
The values of A1, 𝜆1 and Jo are listed in table(1) and table(2), where Jo is the
zeroth-order Bessel function of first kind. From the foregoing temperature distribution
equations at different time and position, the distribution at center of plane wall, cylinder
and sphere can be derived (setting x/L=0 and r/ro =0) as:
𝑇𝑜 −𝑇∞
= 𝐴1 𝑒 −𝜆1
Center of plane wall (x=0):
𝜃𝑜 =
Center of cylinder (r=0):
𝜃𝑜 = 𝑇𝑜−𝑇 ∞ = 𝐴1 𝑒−𝜆1
Center of sphere (r=0):
𝜃𝑜 =
𝑇𝑖 −𝑇∞
𝑇 −𝑇
𝑖
2
2
𝜏
𝜏
∞
𝑇𝑜 −𝑇∞
𝑇𝑖 −𝑇∞
= 𝐴1 𝑒 −𝜆1
2
𝜏
The maximum quantity of heat which the body can gain (or lose if T i > T∞) is simply the
change of energy content of the body, accordingly:
𝑄𝑚𝑎𝑥 = 𝑚 𝑐𝑝 (𝑇∞ − 𝑇𝑖 )
Where m is the mass, V is the volume, 𝝆 is the density, and cp is the specific heat of the
body. The quantity of heat gained or lost up to certain limited time t is given by:
7
𝑡
𝑄 = ℎ 𝐴𝑠 ∫0 (𝑇(𝐿, 𝑡) − 𝑇∞ ). 𝑑𝑡
In accordance the fraction of heat transfer can be calculated from the previous
temperature distribution relations as:
𝑄
For Plane wall:
𝑄𝑚𝑎𝑥
𝑄
For Cylinder:
𝑄𝑚𝑎𝑥
𝑄
For Sphere:
𝑄𝑚𝑎𝑥
sin 𝜆1
= 1 − 𝜃𝑜
𝜆1
= 1 − 2 𝜃𝑜
J1 ( 𝜆1 )
𝜆1
sin 𝜆1 − 𝜆1 𝑐𝑜𝑠𝜆1
= 1 − 3 𝜃𝑜
𝜆1 3
2.4 Transient Heat Conduction in Semi-Infinite Solid
A semi-infinite solid is an idealized body that
has a single plane surface and extends to
infinity in all directions as shown in figure.
The earth, for example, can be considered
semi-infinite medium in determining the
variation of temperature near its surface.
Also, a thick wall can be modeled as a semiinfinite medium if the variation of
temperature in the region near one of the
surfaces is of interest and other surfaces
are far enough such that they have no
impact on the region of interest. The
analytical solution of differential equation of
temperature describing heat transfer in
semi-infinite solid has the following form:
𝑇(𝑥,𝑡)−𝑇𝑖
𝑇∞ −𝑇𝑖
= 𝑒𝑟𝑓𝑐 (
𝑥
2√
Semi-infinite body
ℎ𝑥
) − 𝑒𝑥𝑝 ( 𝑘 +
𝛼𝑡
ℎ2 𝛼 𝑡
𝑘2
) . [𝑒𝑟𝑓𝑐 (2
𝑥
√𝛼 𝑡
+
ℎ √𝛼 𝑡
𝑘
)]
Where erfc(ξ) is error function and its numerical value is listed in table (3).
For special case of h→∞, the surface temperature Ts becomes equal to the fluid
temperature T∞ and the foregoing equation reduced to the simpler form
𝑇(𝑥,𝑡)−𝑇𝑖
𝑇𝑠 −𝑇𝑖
= 𝑒𝑟𝑓𝑐 (
𝑥
2√𝛼 𝑡
8
)
2.5 Transient Heat Conduction in Multidimensional Systems
Using superposition approach called the product solution; one can obtain the
temperature distribution in case of two or
three dimensional transient conduction
problems; by implying the solutions of onedimensional transient conduction. The
solution of case of large plane wall, infinite
cylinder and semi-infinite solid is the base of
solving many two and three dimensional
transient convection. Consider a short
cylinder of height (a) and radius (ro) initially at
a uniform temperature Ti. At time t=0, the
cylinder is subjected to convection from all
surfaces to a medium at temperature T∞ with
heat transfer coefficient h. The temperature in Short cylinder of radius ( ro) and height ( a), is the
intersection of long cylinder and a plane wall
such case is function of x, r and time t
[T=T(r,x,t)] and thus this is a two-dimensional
transient conduction problem. The solution of
this two dimensional problem can be expressed as:
𝑇(𝑟, 𝑥, 𝑡) − 𝑇∞
𝑇(𝑥, 𝑡) − 𝑇∞
𝑇(𝑟, 𝑡) − 𝑇∞
(
)
=(
)
×(
)
𝑇𝑖 − 𝑇∞
𝑇𝑖 − 𝑇∞
𝑇𝑖 − 𝑇∞
𝑠ℎ𝑜𝑟𝑡 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟
𝑝𝑙𝑎𝑛𝑒 𝑤𝑎𝑙𝑙
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟
The solution for two dimensional short cylinder of height (a) and radius ro is equal to the
product of the nondimensionalized solutions for one dimensional plane wall of thickness
a and long cylinder of radius ro, which are the two geometries whose intersection is the
short cylinder, as shown in figure. It can be generalized that:
The solution of a multidimensional geometry is the product of the solutions of the
one dimensional geometries whose intersection is multidimensional body
For convenience, the one-dimensional solutions are denoted by:
𝑇(𝑥,𝑡)−𝑇∞
S(x,t)≡ (
𝑇𝑖 −𝑇∞
𝑇(𝑥,𝑡)−𝑇∞
P(x,t)≡ (
𝑇𝑖 −𝑇∞
𝑇(𝑥,𝑡)−𝑇∞
C(r,t)≡ (
𝑇𝑖 −𝑇∞
)𝑆𝑒𝑚𝑖−𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑜𝑙𝑖𝑑
)𝑃𝑙𝑎𝑛𝑒 𝑤𝑎𝑙𝑙
)𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟
The proper forms of the product solutions for some other geometry are given
in table (4).
9
The transient heat transfer for a two-dimensional geometry formed by the
intersection of two one-dimensional geometries 1 and 2 is
(
𝑄
𝑄𝑚𝑎𝑥
=(
)
𝑡𝑜𝑡𝑎𝑙,2𝐷
𝑄
𝑄𝑚𝑎𝑥
) +(
1
𝑄
𝑄𝑚𝑎𝑥
) [1 − (
2
𝑄
𝑄𝑚𝑎𝑥
) ]
1
And for three-dimensional body:
𝑄
)
𝑄𝑚𝑎𝑥 𝑡𝑜𝑡𝑎𝑙,3𝐷
(
𝑄
)
𝑄𝑚𝑎𝑥 1
=(
𝑄
𝑄
) [1 − (
) ]
𝑄𝑚𝑎𝑥 2
𝑄𝑚𝑎𝑥 1
+(
𝑄
𝑄
) [1 − (
) ] [1 −
𝑄𝑚𝑎𝑥 3
𝑄𝑚𝑎𝑥 1
+(
10
𝑄
) ]
𝑄𝑚𝑎𝑥 2
(
11
12
TABLE 4
Multidimensional solutions expressed as product of one-dimensional solutions
13