1
TRANSIENT CONDUCTION ANALYSIS
Spatial Effects
Class N° 7
Facultad de minas
2
2
OBJECTIVES
WHAT WE KNOW
• The term transient, or unsteady, refers to a situation for which conditions
change with time.
• Transient problems typically arise when the boundary conditions of a system
are changed.
• When the Biot <0.1 the lumped model is considering.
OBJECTIVES
• To develop procedures for determining the time and spatial dependence of
the temperature distribution within a solid during a transient process, as well
as for determining heat transfer between the solid and its surroundings.
• To Identify, depending on assumptions that may be made for a specific process,
the best-fitting procedure to be applied.
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3
SPATIAL EFFECTS
In their most general form, transient conduction problems are described by the
heat equation. With no internal generation and the assumption of constant
thermal conductivity, the heat equation then reduces to
𝜕 2 𝑇 1 𝜕𝑇
=
𝜕𝑥 2 𝛼 𝜕𝑡
To solve for the temperature distribution 𝑇(𝑥, 𝑡), it is necessary to specify an
initial condition and two boundary conditions.
𝑇 𝑥, 0 = 𝑇𝑖
𝜕𝑇
=0
𝜕𝑥 𝑥=0
𝑘
𝜕𝑇
= ℎ 𝑇 𝐿, 𝑡 − 𝑇∞
𝜕𝑥 𝑥=𝐿
In addition to depending on 𝑥 and 𝑡, temperatures in the wall also depend on a
number of physical parameters. In particular
𝑇 = 𝑇(𝑥, 𝑡, 𝑇𝑖 , 𝑇∞ , 𝐿, 𝑘, 𝛼, ℎ) = 0
And the boundary conditions are
4
SPATIAL EFFECTS
It is convenient nondimensionalizing the governing equations by arranging the
relevant variables into suitable groups. Consider the dependent variable 𝑇 . A
dimensionless form of the dependent variable may be defined as:
𝜃
𝑇 − 𝑇∞
𝜃 ≡ =
𝜃𝑖 𝑇𝑖 − 𝑇∞
∗
Accordingly, 𝜃 ∗ must lie in the range 0 ≤ 𝜃 ∗ ≤ 1. A
dimensionless spatial coordinate may be defined,
where 𝐿 is the half-thickness of the plane wall, and a
dimensionless time may be defined as:
𝑥∗ ≡
𝑥
𝐿
𝑡∗ ≡
𝛼𝑡
≡ 𝐹𝑜
𝐿2
Figure 1. Plane wall with an initial uniform
temperature
subjected
to
sudden
convection conditions.
5
SPATIAL EFFECTS
Introducing the dimensionless forms, the heat equation becomes:
𝜕 2 𝜃 ∗ 𝜕𝜃 ∗
=
𝜕𝑥 ∗2 𝜕𝐹𝑜
And the initial and boundary conditions become
𝜃∗ 𝑥 ∗, 0 = 1
𝜕𝜃 ∗
=0
𝜕𝑥 ∗ 𝑥 ∗ =0
𝜕𝜃 ∗
∗ (1, 𝑡 ∗ )
=
−𝐵𝑖𝜃
𝜕𝑥 ∗ 𝑥 ∗=1
Where the Biot number is 𝐵𝑖 ≡ ℎ𝐿 𝑘 . In dimensionless form the functional
dependence may now be expressed as
𝜃 ∗ = 𝑓(𝑥 ∗ , 𝐹𝑜, 𝐵𝑖)
For a prescribed geometry, the transient temperature distribution is a universal
function of 𝑥 ∗ , 𝐹𝑜, and 𝐵𝑖.
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SPATIAL EFFECTS (The Plane Wall With Convection)
EXACT SOLUTION
Consider the plane wall of thickness 2𝐿. We assume that:
• Conduction occurs in the 𝑥 −direction.
• The wall is initially at a uniform temperature, 𝑇 𝑥, 0 = 𝑇𝑖 .
• The wall is suddenly immersed in a fluid of 𝑇∞ ≠ 𝑇𝑖 .
• The convection conditions for the surfaces at 𝑥 ∗ = ±1 are the same.
• The temperature distribution at any instant must be symmetrical about
the midplane (𝑥 ∗ = 0).
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SPATIAL EFFECTS (The Plane Wall With Convection)
An exact solution to this problem is of the form
∞
𝜃∗ =
𝐶𝑛 exp(− 𝜁𝑛2 𝐹𝑜) cos(𝜁𝑛 𝑥 ∗ )
0 ≤ 𝐹𝑜 ≤ ∞
𝑛=1
Where 𝐹𝑜 = 𝛼𝑡 𝐿2 , the coefficient 𝐶𝑛 is
4 sin 𝜁𝑛
𝐶𝑛 =
2𝜁𝑛 + sin(2𝜁𝑛 )
And the discrete values of 𝜁𝑛 (eigenvalues) are positive
roots of the transcendental equation
𝜁𝑛 tan 𝜁𝑛 = 𝐵𝑖
Figure 2. Plane wall with an initial uniform temperature
subjected to sudden convection conditions.
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SPATIAL EFFECTS (The Plane
Wall With Convection)
Table 1. The first four roots of the Transcendental
Equation, 𝜉𝑛 tan 𝜉𝑛 = 𝐵𝑖 , for Transient Conduction in a
plane wall
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SPATIAL EFFECTS (The Plane Wall With Convection)
APPROXIMATE SOLUTION
For values of 𝐹𝑜 > 0.2, the infinite series solution, can be approximated by the first term
of the series, 𝑛 = 1. The dimensionless form of the temperature distribution then
becomes
𝜃 ∗ = 𝐶1 exp (−𝜁12 𝐹𝑜) cos(𝜁1 𝑥 ∗ )
𝜃 ∗ = 𝜃𝑜∗ cos(𝜁1 𝑥 ∗ )
Where 𝜃𝑜∗ ≡ 𝑇𝑜 − 𝑇∞
temperature
(𝑇𝑖 − 𝑇∞ )
represents
the
midplane
(𝑥 ∗ = 0)
𝜃𝑜∗ = 𝐶1 exp(−𝜁12 𝐹𝑜)
The time dependence of the temperature at any location within the wall is function of
the midplane temperature.
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SPATIAL EFFECTS (The Plane Wall With Convection)
APPROXIMATE SOLUTION
Table 2. The coefficients 𝐶1 and 𝜁1 are
evaluated from Equations, and are given in
Table a range of Biot numbers.
4 sin 𝜁1
𝐶1 =
2𝜁1 + sin(2𝜁1 )
𝜁1 tan 𝜁1 = 𝐵𝑖
a𝐵𝑖 = ℎ𝐿
𝑘 for the plane wall and ℎ𝑟𝑜 𝑘 for the
infinite cylinder and sphere.
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SPATIAL EFFECTS (Total Energy Transfer)
To know the total energy that has left (or entered) the wall up to any time 𝑡, we may
apply the conservation of energy requirement. That is
𝐸𝑖𝑛 − 𝐸𝑜𝑢𝑡 = ∆𝐸𝑠𝑡
The energy transferred from the wall 𝑄 to 𝐸𝑜𝑢𝑡 and setting 𝐸𝑖𝑛 = 0 and ∆𝐸𝑠𝑡 =
𝐸 𝑡 − 𝐸 0 , it follows that
𝑄=−
𝑄 = − 𝐸 𝑡 − 𝐸(0)
𝜌𝑐 𝑇 𝑥, 𝑡 − 𝑇𝑖 𝑑𝑉
Where the integration is performed over the volume of the wall. It is convenient to
nondimensionalize this result by introducing the quantity
𝑄𝑜 = 𝜌𝑐𝑉(𝑇𝑖 − 𝑇∞ )
𝑄𝑜 is the initial energy of the wall relative to the fluid temperature. It is also the
maximum amount of energy transfer if the process were continued to time 𝑡 → ∞.
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SPATIAL EFFECTS (Total Energy Transfer)
The ratio of the total energy transferred from the wall over the time interval 𝑡 to the
maximum possible transfer is
𝑄
=
𝑄𝑜
− 𝑇 𝑥, 𝑡 − 𝑇𝑖 𝑑𝑉 1
=
𝑇𝑖 − 𝑇∞
𝑉
𝑉
1 − 𝜃 ∗ 𝑑𝑉
Using the approximate form of the temperature distribution for the plane wall, we can
integrate to obtain:
𝑄
sin 𝜁1 ∗
=1−
𝜃
𝑄𝑜
𝜁1 𝑜
The foregoing results may also be applied to:
• A plane wall that is insulated on one side and experiences convection transport on
the other side.
• Determine the transient response of a plane wall to a sudden change in surface
temperature (ℎ → ∞, 𝐵𝑖 → ∞).
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SPATIAL EFFECTS (Radial Systems With Convection)
Consider an infinite cylinder (if 𝐿 𝑟𝑜 ≥ 10 one-dimensional conduction in the radial
direction, and it is reasonable) or sphere of radius 𝑟𝑜 , which is at an initial uniform
temperature and experiences a change in convective conditions.
EXACTS SOLUTIONS
Infinite cylinder: in dimensionless form the temperature is
∞
𝜃∗ =
𝐶𝑛 exp(− 𝜁𝑛2 𝐹𝑜)𝐽𝑜 𝜁𝑛 𝑟 ∗
0 ≤ 𝐹𝑜 ≤ ∞
𝑛=1
Where 𝐹𝑜 = 𝛼𝑡 𝑟𝑜2 ,
𝐶𝑛 =
2
𝐽1 (𝜁𝑛 )
𝜁𝑛 𝐽𝑜2 𝜁𝑛 + 𝐽12 (𝜁𝑛 )
Figure 3.
convection.
Infinite
And the discrete values of 𝜁𝑛 are positive roots of the transcendental equation
𝜁𝑛
𝐽1 (𝜁𝑛 )
= 𝐵𝑖
𝐽𝑜 (𝜁𝑛 )
Where 𝐵𝑖 = ℎ𝑟𝑜 𝑘 .
Cylinder
whit
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SPATIAL EFFECTS (Radial Systems With Convection)
Sphere: Similarly, for the sphere
𝜃∗ =
1
∞
2
∗
𝑛=1 𝐶𝑛 exp(− 𝜁𝑛 𝐹𝑜) 𝜁 𝑟 ∗ sin(𝜁𝑛 𝑟 )
𝑛
0 ≤ 𝐹𝑜 ≤ ∞
4 sin 𝜁𝑛 − 𝜁𝑛 cos(𝜁𝑛 )
𝐶𝑛 =
2𝜁𝑛 − sin(2𝜁𝑛 )
And the discrete values of 𝜁𝑛 are positive roots of the transcendental equation
1 − 𝜁𝑛 cot 𝜁𝑛 = 𝐵𝑖
Where
𝐵𝑖 = ℎ𝑟𝑜 𝑘 .
Where
𝐹𝑜 = 𝛼𝑡 𝑟𝑜2 ,
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SPATIAL EFFECTS (Approximate Solutions)
For the infinite cylinder and sphere, the foregoing series solutions can again be
approximated by a single term, 𝑛 = 1, for 𝐹𝑜 > 0.2.
Infinite cylinder: The one-term approximation is:
𝜃 ∗ = 𝐶1 exp(− 𝜁12 𝐹𝑜)𝐽𝑜 (𝜁1 𝑟 ∗ )
𝜃 ∗ = 𝜃𝑜∗ 𝐽𝑜 (𝜁1 𝑟 ∗ )
Where 𝜃𝑜∗ represents the center temperature
𝜃𝑜∗ = 𝐶1 exp(−𝜁12 𝐹𝑜)
Sphere: the one-term approximation is
𝜃 ∗ = 𝐶1 exp(− 𝜁12 𝐹𝑜)
1
sin(𝜁1 𝑟 ∗ )
∗
𝜁1 𝑟
𝜃𝑜∗ = 𝐶1 exp(−𝜁12 𝐹𝑜)
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SPATIAL EFFECTS (Radial Systems With Convection)
Infinite cylinder
𝜃 ∗ = 𝐶1 exp(− 𝜁12 𝐹𝑜)𝐽𝑜 (𝜁1 𝑟 ∗ )
Sphere
𝜃 ∗ = 𝐶1 exp(− 𝜁12 𝐹𝑜)
1
sin(𝜁1 𝑟 ∗ )
∗
𝜁1 𝑟
Where 𝐵𝑖 = ℎ𝑟𝑜 𝑘 .
Where 𝐹𝑜 = 𝛼𝑡 𝑟𝑜2 ,
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SPATIAL EFFECTS (Total Energy Transfer)
An energy balance may be performed to determine the total energy transfer from the
infinite cylinder or sphere over the time interval ∆𝑡 = 𝑡. The results are as follows.
Infinite cylinder
𝑄
2𝜃𝑜∗
=1−
𝐽 (𝜁 )
𝑄𝑜
𝜁1 1 1
Sphere
𝑄
3𝜃𝑜∗
= 1 − 3 sin 𝜁1 − 𝜁1 cos(𝜁1 )
𝑄𝑜
𝜁1
ADDITIONAL CONSIDERATIONS
The foregoing results may be used to predict the transient response of long cylinders
and spheres subjected to a sudden change in surface temperature.
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SPATIAL EFFECTS (Hessler Graphical Representation)
The results obtained for one-dimensional conduction for 𝐹𝑜 > 0.2, can be conveniently
be represented in graphical forms.
The Plane Wall
Figure 4. Midplane temperature as a function of time for a plane wall of thickness 2𝐿.
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SPATIAL EFFECTS (Hessler Graphical Representation)
The Plane Wall
Figure 5. Temperature distribution in a
plane wall of thickness 2𝐿.
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SPATIAL EFFECTS (Hessler Graphical Representation)
The Plane Wall
Figure 6. Internal energy change as a function of time for a plane wall of thickness 2𝐿.
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SPATIAL EFFECTS (Graphical Representation)
The Infinite
Cylinder
The results obtained for
one-dimensional
conduction for 𝐹𝑜 > 0.2 ,
can be conveniently be
represented in graphical
forms.
Figure 7. Centerline temperature as a function of time for an infinite cylinder of radius 𝑟𝑜
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SPATIAL EFFECTS (Graphical Representation)
The results obtained for one-dimensional conduction for 𝐹𝑜 > 0.2, can be conveniently
be represented in graphical forms.
The Infinite
Cylinder
Figure 8. Temperature distribution in an
infinite cylinder of radius 𝑟𝑜 .
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SPATIAL EFFECTS (Graphical Representation)
The Infinite
Cylinder
Figure 9. Internal energy change as a function of time for an infinite cylinder of radius 𝑟𝑜 .
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SPATIAL EFFECTS (Graphical Representation)
The results obtained for one-dimensional conduction for 𝐹𝑜 > 0.2, can be conveniently
be represented in graphical forms.
The Sphere
Figure
10.
Centerline
temperature as a function of
time for an sphere of radius 𝑟𝑜
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SPATIAL EFFECTS (Graphical Representation)
The results obtained for one-dimensional conduction for 𝐹𝑜 > 0.2, can be conveniently
be represented in graphical forms.
The Sphere
Temperature distribution in an sphere of
radius 𝒓𝒐 .
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SPATIAL EFFECTS (Graphical Representation)
The Sphere
Internal energy change as a function of time for an sphere of radius 𝒓𝒐 .
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SPATIAL EFFECTS (The Semi-Infinite Solid)
The semi-infinite solid extends to infinity in all but one direction and is characterized by a
single identifiable surface. The heat equation for transient conduction is:
𝜕 2 𝑇 1 𝜕𝑇
=
𝜕𝑥 2 𝛼 𝜕𝑡
The initial conditions is prescribed by
𝑇 𝑥, 0 = 𝑇𝑖
the boundary condition is of the form
𝑇 𝑥 → ∞, 𝑡 = 𝑇𝑖
Transient temperature distributions in a
semi-infinite solid for three surface
conditions:
• constant surface temperature.
• constant surface heat flux.
• and surface convection.
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SPATIAL EFFECTS (The Semi-Infinite Solid)
Application Of A Constant Surface Temperature 𝑻𝒔 ≠ 𝑻𝒊
There is a similarity variable, 𝜂 ≡ 𝑥 (4𝛼𝑡)1 2 , which permits the transformation of the
differential equation. The heat equation then becomes
𝑑2𝑇
𝑑𝑇
=
−2𝜂
𝑑𝜂2
𝑑𝜂
With 𝑥 = 0 corresponding to 𝜂 = 0, the
surface condition may be expressed as
𝑇 𝜂 = 0 = 𝑇𝑠
And with 𝑥 → ∞, as well as 𝑡 = 0, corresponding to 𝜂 → ∞, both the initial condition
and the interior boundary condition correspond to the single requirement that
𝑇 𝜂 → ∞ = 𝑇𝑖
𝑇(𝑥, 𝑡) − 𝑇𝑠
𝑥
= erf
𝑇𝑖 − 𝑇𝑠
2 𝛼𝑡
𝑇 0, 𝑡 = 𝑇𝑠
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SPATIAL EFFECTS (The Semi-Infinite Solid)
𝑞𝑠" 𝑡
=
𝑘(𝑇𝑠 − 𝑇𝑖 )
𝜋𝛼𝑡
Constant Surface Heat Flux: 𝒒"𝒔 = 𝒒"𝑶
2𝑞𝑂" (𝛼𝑡 𝜋)
𝑇 𝑥, 𝑡 − 𝑇𝑖 =
𝑘
𝜕𝑇
Surface Convection: −𝑘 𝜕𝑥
𝑥=0
𝑇 𝑥,𝑡 −𝑇𝑖
𝑥
=
erfc
𝑇∞ −𝑇𝑖
2 𝛼𝑡
1 2
−𝑥 2
𝑞𝑂" 𝑥
𝑥
exp
−
erfc
4𝛼𝑡
𝑘
2 𝛼𝑡
= ℎ 𝑇∞ − 𝑇(0, 𝑡)
− exp
ℎ𝑥
ℎ 2 𝛼𝑡
+
𝑘
𝑘2
𝑥
erfc 2 𝛼𝑡 +
ℎ 𝛼𝑡
𝑘
The complementary error function, erfc (), is defined as erfc 𝑤 ≡ 1 − erf 𝑤 .
30
SPATIAL EFFECTS (The Semi-Infinite Solid)
31
SPATIAL EFFECTS (The Semi-Infinite Solid)
Temperature histories in a semi-infinite solid with surface convection.
32
SPATIAL EFFECTS (The Semi-Infinite Solid)
A permutation of the case 1 occurs when two semi-infinite solids, initially at uniform
temperatures 𝑇𝐴,𝑖 and 𝑇𝐵,𝑖 , are placed in contact at their free surfaces. there is an
equilibrium surface temperature that may be determined from a surface energy balance:
"
"
𝑞𝑠,𝐴
= 𝑞𝑠,𝐵
𝑞𝑠" 𝑡 =
"
"
Substituting for 𝑞𝑠,𝐴
and 𝑞𝑠,𝐵
, it follows that
1/2
𝑇𝑠 =
1/2
(𝑘𝜌𝑐)𝐴 𝑇𝐴,𝑖 + (𝑘𝜌𝑐)𝐵 𝑇𝐵,𝑖
1/2
1/2
(𝑘𝜌𝑐)𝐴 +(𝑘𝜌𝑐)𝐵
𝑚 ≡ (𝑘𝜌𝑐)1/2 is a weighting factor that determines
whether 𝑇𝑠 will more closely approach 𝑇𝐴,𝑖 or 𝑇𝐵,𝑖 .
𝑘(𝑇𝑠 − 𝑇𝑖 )
𝜋𝛼𝑡
33
3
REFERENCES
Bergman, T. et al. Fundamentals of Heat and Mass Transfer. John Wiley & Sons, 2011.
Video 5.1: Termoaço Tratamentos Térmicos LTDA. Têmpera Tratamento térmico nos aços [on line].
Available on: < http://www.youtube.com/watch?v=nLo7VQVtqWE> . Last view: September 10th 2012.
Bergman, T. et al. Fundamentals of Heat and Mass Transfer. John Wiley & Sons, 2011.