Chapter 4 transient heat condution

1/21/2018 Heat Transfer 1
HEAT TRANSFER
(MEng 3121)
TRANSIENT HEAT CONDUCTION
(One and two dimensional)
Chapter 4
Debre Markos University
Mechanical Engineering
Department
Prepared and Presented by:
Tariku Negash
Sustainable Energy Engineering (MSc)
E-mail: thismuch2015@gmail.com
Lecturer at Mechanical Engineering Department
Institute of Technology, Debre Markos
University, Debre Markos, Ethiopia
Jan, 2018

1/21/2018Heat Transfer
2
Objectives
• Assess when the spatial variation of temperature is negligible,
and temperature varies nearly uniformly with time, making the
simplified lumped system analysis applicable.
• Obtain analytical solutions for transient one-dimensional
conduction problems in rectangular, cylindrical, and spherical
geometries using the method of separation of variables,
• Understand why a one-term solution is usually a reasonable
approximation.
• Solve the transient conduction problem in large mediums using
the similarity variable, and predict the variation of temperature
with time and distance from the exposed surface.
• Construct solutions for multi-dimensional transient conduction
problems using the product solution approach.

3
 We now recognize that many heat transfer problems are time
dependent. Such unsteady, or transient, problems typically arise when
the boundary conditions of a system are changed.
 For example, a hot metal billet that is removed from a furnace and
exposed to a cool airstream.
i. Energy is transferred by convection and radiation from its surface to
the surroundings.
ii. Energy transfer by conduction also occurs from the interior of the
metal to the surface, and the temperature at each point in the billet
decreases until a steady-state condition is reached.
iii. The final properties of the metal will depend significantly on the
time-temperature history that results from heat transfer.
 Thus, controlling the heat transfer is one key to fabricating new
materials with enhanced properties.
4.1 Introduction

4
Our objectives in this chapter is to develop procedures for determining
1. the time dependence of the temperature distribution within a solid
(conduction) during a transient process, and
2. heat transfer between the solid and its surroundings (convection or
radiation).
A system in which the temperature of a solid varies with time but remains
uniform throughout the solid at any time is called Lumped Systems
It depends on the assumption that may be made for the process.
If, for example, temperature gradients within the solid may be neglected, a
comparatively simple approach, termed the lumped capacitance method,
may be used to determine the variation of temperature with time.

5
4.2 Lumped System Analysis
 In heat transfer analysis, some bodies are observed to behave like a
“lump” A compact mass
 Interior temperature of some bodies remains essentially uniform at all
times during a heat transfer process.
 The temperature of such bodies can be taken to be a function of time
only, T(t).
 Heat transfer analysis that utilizes this idealization is known as lumped
system analysis. which provides great simplification in certain classes
of heat transfer problems without much sacrifice from accuracy.

6
Consideration
Fig. a a small hot copper ball coming out of an
oven. Measurements indicate that the temperature
of the copper ball changes with time, but it does
not change much with position at any given time.
Fig b. a large roast in an oven. If you cut in
half, you will see that the outer parts of the
roast are well done while the center part is
barely warm.
Thus, which one is the two consideration that lumped system analysis is not
applicable?

7
4.3 Formulation of Lumped System Analysis
Consider a body of arbitrary shape shown fig.
At time 𝑡 = 0, the body is placed into a medium at
temperature 𝑇∞, and heat transfer takes place between
the body and its environment, with a heat transfer
coefficient h.
For the sake of discussion, we will assume that 𝑇∞ > 𝑇𝑖, but the analysis is
equally valid for the opposite case.
We assume lumped system analysis to be applicable, so that the temperature
remains uniform within the body at all times and changes with time only, T
= T(t).
During a differential time interval dt, the temperature of the body rises by a
differential amount dT .

8
An energy balance of the solid for the time interval dt can be expressed
as
and
(1)
𝑠𝑖𝑛𝑐𝑒 𝑇∞is constant
Can be rearranged as (2)
Integrating with
T = Ti at t = 0
T = T(t) at t = t
Taking the exponential of both sides and rearranging, we obtain
Where,(3) is + ve quantity

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Where, for, 𝜽 ≡ 𝑻 − 𝑻∞ 𝒂𝒏𝒅 𝜽𝒊 ≡ 𝑻𝒊 − 𝑻∞
Or
(4)
The reciprocal of b has time unit (usually s), and is called the time constant.
 We can consider from the fig the temperature of
a lumped system approaches the environment
temperature as time gets larger.
Observations from the fig and the relation above
equation:
i. Equation (3) enables us to determine the
temperature T(t) of a body at time t, or alternatively,
the time t required for the temperature to reach a
specified value T(t).

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i. The temperature of a body approaches the
ambient temperature 𝑻∞ exponentially.
ii. The temperature of the body changes
rapidly at the beginning, but rather slowly
later on.
iii. A large value of b indicates that the body
will approach the environment temperature
in a short time
iv. The larger the value of the exponent b, the
higher the rate of decay in temperature.
 Note that b is proportional to the surface area, but inversely proportional
to the mass and the specific heat of the body. Which means that it takes
longer to heat or cool a larger mass, especially when it has a large
specific heat
Cont.

11
The rate of convection heat
transfer between the body
and its environment at time t
The total amount of heat transfer
between the body and the surrounding
medium over the time interval t = 0 to t
(5)
(6)
 Once the temperature T(t) at time t is available from Eq. (3) , the rate
of convection heat transfer between the body and its environment at
that time can be determined from Newton’s law of cooling as
(7)
The maximum heat transfer between
the body and its surroundings when
body reaches the surrounding Tem (𝑻∞).

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4.4 Validity of the Lumped Capacitance Method
The lumped system analysis certainly provides great convenience in heat
transfer analysis, and naturally we would like to know when it is appropriate
to use it.
The first step in establishing a criterion for the applicability of the
lumped system analysis is to define as
characteristic lengthBiot number Where,
It can also be expressed as
Or,
(8)
(9)

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 When a solid body is being heated by the hotter fluid surrounding it
(such as a potato being baked in an oven), heat is first convected to the
body and subsequently conducted within the body.
 The Biot number is the ratio of the internal resistance of a body to heat
conduction to its external resistance to heat convection.
 Therefore, a small Biot number represents small resistance to heat
conduction, and thus small temperature gradients within the body.
 Lumped system analysis assumes a uniform temperature distribution
throughout the body, which will be the case only when the thermal
resistance of the body to heat conduction (the conduction resistance) is
zero.
 Thus, lumped system analysis is exact when Bi = 0 and
approximate when Bi > 0.
 Of course, the smaller the Bi number, the more accurate the
lumped system

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Then the question we must answer is, How much accuracy are we
willing to sacrifice for the convenience of the lumped system analysis?
Therefore, the accepted that lumped system analysis is applicable if
 When this criterion is satisfied, the temperatures within the body
relative to the surroundings (i.e., 𝑻 − 𝑻∞) remain within 5 percent of
each other even for well-rounded geometries such as a spherical ball.
 Thus, when Bi < 0.1, the variation of temperature with location within
the body will be slight and can reasonably be approximated as being
uniform.

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Example 4.1: Predicting the Time of Death
A person is found dead at 5 PM in a room whose temperature is 20°C. The
temperature of the body is measured to be 25°C when found, and the heat
transfer coefficient is estimated to be h= 8 W/m2· °C. Modeling the body
as a 30-cm-diameter, 1.70-m-long cylinder, estimate the time of death of
that person
Assumptions
1 The body can be modeled as a 30-cm-diameter,
1.70-m- long cylinder.
2 The thermal properties of the body and the heat transfer
coefficient are constant.
3 The radiation effects are negligible.
4 The person was healthy(!) when he died with a body
temperature of 37°C.
 The average human body is 72% water by mass, and thus we can assume the
body to have the prperties of water at the average temperaturee of (37+25)/2
=31 ℃, 𝑘 = 0.617 W/m℃ and 𝜌 =
996𝑘𝑔
𝑚3 𝑎𝑛𝑑 𝐶 𝑃 = 4178 𝐽/𝑘𝑔℃

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4.5 Transient Heat Conduction In Large Plane Walls, Long
Cylinders, and Spheres With Spatial Effects
Consider the variation of temperature with time and position in one-
dimensional problems such as those associated with a large plane wall, a
long cylinder, and a sphere.
(c) A sphere

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 At time t = 0, each geometry is placed in a large medium that is at a
constant temperature 𝑻∞ and kept in that medium for t > 0.
 Heat transfer takes place between these bodies and their environments
by convection with a uniform and constant heat transfer coefficient h.
 By neglecting radiation or incorporate the radiation effect into the
convection heat transfer coefficient h
The variation of the temperature profile with time in the
plane wall is illustrated in Fig.
 When the wall is first exposed to the surrounding
medium at 𝑻∞ < 𝑻𝒊 at t=0, the entire wall is at its
initial temperature 𝑻𝒊.
 But the wall temperature at and near the surfaces
starts to drop as a result of heat transfer from the
wall to the surrounding medium.
 This creates a temperature gradient in the wall and initiates heat
conduction from the inner parts of the wall toward its outer surfaces.

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Similar discussions can be given for the long cylinder or sphere.
 Note that the temperature at the center of the wall
remains at 𝑻𝒊 at 𝒕 = 𝒕 𝟐 and that the temperature
profile within the wall remains symmetric at all
times about the center plane.
 The temperature profile gets flatter and flatter
as time passes as a result of heat transfer, and
eventually becomes uniform at 𝑻 = 𝑻∞.
 That is, the wall reaches thermal equilibrium
with its surroundings. At that point, the heat
transfer stops since there is no longer a
temperature difference.

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4.6 Formulation of One Dimensional Transient Temperature
Distribution. T(x, t)
There is clear motivation to present the solution in tabular or graphical
form. However, the solution involves the parameters 𝒙, 𝑳, 𝒕, 𝒌, 𝜶, 𝒉, 𝑻𝒊 𝒂𝒏𝒅
𝑻∞ , which are too many to make any graphical presentation of the
results practical.
In order to reduce the number of parameters, we non-dimensionalize the
problem by defining the following dimensionless quantities:
(10)
(11)
(12)
(13)

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Non-dimensionalization reduces the number of independent variables in
one-dimensional transient conduction problems from 8 to 3, offering
great convenience in the presentation of results.
Dimensionless
differential equation
Dimensionless BC’s
Dimensionless initial
condition

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The non-dimensionalization enables us to present the temperature in
terms of three parameters only: X, Bi, and 𝝉 . This makes it practical
to present the solution in graphical form.
The dimensionless quantities for cylinder or sphere
By replacing the space variable x by r and the half-thickness L by the outer
radius 𝒓 𝟎

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4.6.1 Exact Solution of One-Dimensional Transient Conduction
Problem

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Table 4.1
(14)
(15)
(16)
Summary of Analytical equation one dimensional transient
heat conduction

24
The analytical solutions of
transient conduction problems
typically involve infinite
series, and thus the evaluation
of an infinite number of terms
to determine the temperature
at a specified location and
time.

25
 The one-dimensional transient heat conduction problem just described
can be solved exactly for any of the three geometries, but the solution
involves infinite series, which are difficult to deal with.
 However, the terms in the solutions converge rapidly with increasing
time, and for 𝝉 > 𝟎. 𝟐, keeping the first term and neglecting all the
remaining terms in the series results in an error under 2 percent.
 We are usually interested in the solution for times with 0.2, and thus it is
very convenient to express the solution using this one term
approximation (one term solution), given as
4.6.2 Approximate Analytical and Graphical Solutions
A. Approximate Analytical Solution (𝝉 > 𝟎. 𝟐)

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 Where, 𝐴1 𝑎𝑛𝑑 𝜆1 ar e a function of Bi number and their values are listed in
Table 4–2 against the Bi number for all three geometries.
 The function 𝐽0 is the zeroth-order Bessel function of the first kind, whose
value can be determined from Table 4–3.
 Noting that for x=0 cos (0) =1, for r=0 𝐽0(0)=1 and the limit of (sin x)/x is
also 1, these relations simplify to the next ones at the center of a plane wall,
cylinder, or sphere:
(18)
(19)
(20)
(16)
(17)
(18)
Heat transfer from the center plane

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The dimensionless temperatures anywhere in a plane (𝜃 𝑤𝑎𝑙𝑙) wall,
cylinder, and sphere are related to the center temperature (𝜃0, 𝑤𝑎𝑙𝑙) by
(21-23)
There are three charts associated with each geometry:
1st chart is to determine the temperature 𝑇𝑜 at the center of the geometry at a
given time t.
2nd chart is to determine the temperature at other locations at the
same time in terms of 𝑇𝑜.
3rd chart is to determine the total amount of heat transfer up to the
time t.
Notice:- these plots are valid for 𝝉 = 𝟎. 𝟐

29
(a) Mid plane temperature
Transient temperature and heat transfer charts (Heisler and GrÖber
charts) for a plane wall of thickness 2L initially at a uniform
temperature Ti subjected to convection from both sides to an
environment at temperature T with a convection coefficient of h.
A. Plane Wall
B Approximate Graphical Solution (𝝉 > 𝟎. 𝟐)

30
(b) Temperature distribution

32
(a) Centerline temperature (from M. P. Heisler)
B. Cylinder

34
(a) Midpoint temperature (from M. P. Heisler)
C. Sphere

35
Note that the case 1/Bi = k/hL =0
corresponds to h → ∞ which
corresponds to the case of specified
surface temperature 𝐓∞. That is,
the case in which the surfaces of the
body are suddenly brought to the
temperature 𝑻∞ 𝒂𝒕 𝒕 = 𝟎 and
kept at 𝑻∞ all times can be handled
by setting h to infinity

36
(26)
The fraction of total heat transfer Q/Qmax up to a specified time t is determined
using the Gröber charts.
(25)
(24)
(27)
Where,
and
The physical significance of the Fourier number
(28)
The Fourier number is a measure
of heat conducted through a body
relative to heat stored.

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Example 4–2 Plane Wall
In a production facility, large brass plates of 4 cm thickness that are initially at a
uniform temperature of 20°C are heated by passing them through an oven that is
maintained at 500°C (Fig.). The plates remain in the oven for a period of 7 min.
Taking the combined convection and radiation heat transfer coefficient to be h=120
W/𝑚2°C, determine the surface temperature of the plates when they come out of the
oven.
Assumptions
1 Heat conduction in the plate is one-dimensional since the
plate is large relative to its thickness and there is thermal
symmetry about the center plane.
2 The thermal properties of the plate and the heat transfer
coefficient are constant.
3 The Fourier number is 𝝉 > 𝟎. 𝟐 so that the one-term
approximate solutions are applicable.
 The properties of brass at room temperature 𝑘 = 110 W/m℃ and
𝜌 =
8530𝑘𝑔
𝑚3 𝐶 𝑃 = 380 𝐽/𝑘𝑔℃ and 𝛼 =
𝑘
𝜌𝐶 𝑃
= 33.9 𝑋10−6 𝑚2/𝑠
However more accurate results are obtained by using properties at average temperature

38
An ordinary egg can be approximated as a 5-cm-diameter sphere as
shown o the fig. The egg is initially at a uniform temperature of 5°C and
is dropped into boiling water at 95°C. Taking the convection heat transfer
coefficient to be h=1200 W/𝑚2
· °C, determine how long it will take for
the center of the egg to reach 70°C.
Example 4–3 Sphere
Assumptions
1. The egg is spherical in shape with a radius of
𝑟0 = 2.5 cm.
2. Heat conduction in the egg is one-dimensional
because of thermal symmetry
about the midpoint.
3. The thermal properties of the egg and the heat transfer coefficient are constant.
4 The Fourier number is 𝝉 > 𝟎. 𝟐 so that the one-term approximate solutions are
applicable.
5. The water content of eggs is about 74 percent, and thus the thermal conductivity
and diffusivity of eggs can be approximated by those of water
𝑘 = 0.627 W/m℃ and 𝛼 =
𝑘
𝜌𝐶 𝑃
= 0.151𝑋10−6 𝑚2/𝑠

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Example 4–4 Cylinder
A long 20-cm-diameter cylindrical shaft made of stainless steel 304 comes out
of an oven at a uniform temperature of 600°C (Fig). The shaft is then allowed
to cool slowly in an environment chamber at 200°C with an average heat
transfer coefficient of h=80 W/m · °C. Determine
a) the temperature at the center of the shaft 45 min after the start of the
cooling process.
b) the heat transfer per unit length of the shaft during this time period.
Assumptions
1. Heat conduction in the shaft is one-dimensional since
it is long and it has thermal symmetry about the
centerline.
2.The thermal properties of the shaft and the heat transfer coefficient are constant.
3. The Fourier number is 𝝉 > 𝟎. 𝟐 so that the one-term approximate solutions are
applicable.
 The properties of stainless steel 304at room temperature 𝑘 = 14.9 W/m℃ and
𝜌 =
7900𝑘𝑔
𝑚3 𝐶 𝑃 = 477 𝐽/𝑘𝑔℃ and 𝛼 =
𝑘
𝜌𝐶 𝑃
= 3.95 𝑋10−6 𝑚2/𝑠
More accurate results are obtained by using properties at average temperature

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4.7 Transient Heat Conduction In Semi-infinite Solids
 A semi-infinite solid is an idealized body that has a
single exposed plane surface and extends to infinity
in all directions, as shown in Fig.
 For example, Earth can be considered to be a semi-
infinite medium in determining the variation of
temperature near its surface.
 Also, a thick wall can be modeled as a semi-infinite medium if all we
are interested in is the variation of temperature in the region near one of
the surfaces, and the other surface is too far to have any impact on the
region of interest during the time of observation.
For short periods of time, most bodies can be modeled as semi-infinite
solids since heat does not have sufficient time to penetrate deep into the
body.

41
 If a sudden change of conditions is imposed at this surface,
transient, one-dimensional conduction will occur within the solid.
 These early portions of the transient might correspond to very
small Fourier numbers (𝛕), and the approximate solutions of
(analytical and graphical) would not be valid.
 Although the exact solutions of the preceding sections could be
used to determine the temperature distributions, many terms might
be required to evaluate the infinite series expressions.
 Now let’s eliminate the need to evaluate the cumbersome (difficult)
infinite series exact solutions at small Fo/𝛕.
 It will be shown that a plane wall of thickness 2L can be accurately
approximated as a semi-infinite solid for 𝛕 = 𝛼𝑡/𝐿2
≤ 0.2

42
The heat equation for transient conduction in a semi-infinite solid
initial condition and
interior boundary condition
At t=0 three d/t
applications are applied
instantaneously,
i. application of a
constant surface
temperature 𝑇𝑠 ≠ 𝑇𝑖 ,
ii. application of a
constant surface heat
flux 𝑞 𝑜
′′, and
iii. exposure of the surface to a fluid characterized by 𝑇∞ ≠ 𝑇𝑖 and the
convection coefficient h.
(29)

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4.7.1 Analytical solution for case 1: Temperature distribution
It involving two independent variables (x and t), to an ordinary differential
equation expressed in terms of the single similarity variable.
Let a similar variable (𝜂)
first transform the pertinent differential operators, such that
Substituting a and b into Equation (29) , the heat equation becomes
(30)
a
b

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With x = 0 corresponding to 𝜂 = 0, the surface condition may be
expressed as (case 1 T(0,t) =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
Let’s see temperature as a unique function of 𝜂. (what does it mean T (η → ∞))
The specific form of the temperature dependence, T(𝜂), may be obtained by
separating variables in Equation (30) , such that
Integrating, it follows that
Or,
(31)
(32)

45
Integrating a second time, we obtain
Applying the boundary condition at η = 0, Equation (31) , it follows
that 𝑪 𝟐 = 𝑻 𝒔 and from the second boundary condition, Equation
(32) , we obtain
where u is a dummy variable.(33)
or, evaluating the definite integral,
Substitute the value of 𝑪 𝟏 𝒂𝒏𝒅 𝑪 𝟐 in to equation (33), then the
temperature distribution may be expressed as
(34)
(35)
where erf η: is the Gaussian error function, is a standard mathematical
function that is tabulated in Appendix B (Incropera book).

46
Note that erf (η) asymptotically approaches unity as becomes infinite.
Thus, at any nonzero time (t≠ 0), temperatures everywhere are predicted
to have changed from 𝑇𝑖 (become closer to 𝑇𝑠).
𝑇 − 𝑇𝑠
𝑇𝑖 − 𝑇𝑠
≈ 1
Which is
Error function
Complementary error
function
Note that : (erf 𝜂 𝑜𝑟 𝑒𝑟𝑓𝑤 𝑜𝑟 𝑒𝑟𝑓𝜉 )

47
4.7.2 Analytical solution for case 2: Heat flux
The surface heat flux may be obtained by applying Fourier's law at x = 0,
in which case
(36)
4.7.3 Analytical solution for case 3: Surface Convection
(37)
where erf𝐜 𝐰 is the complementary error function

48
Summary:
Analytical solutions for different boundary conditions on the surface
(38)
(39)
(40)

49
 Temperatures within the medium monotonically
approach 𝑇𝑠 with increasing t, while the magnitude of the
surface temperature gradient, and hence the surface heat
flux, decreases as
1
𝑡
..
 A thermal penetration depth 𝛿 𝑝 can be defined as the
depth to which significant temperature effects propagate
within a medium.
 For example, defining, 𝛿 𝑝 as the x location at which (T-
Ts)/(Ts-Ti) =0.90, the above equation result in 𝛿 𝑝
= 2.3 𝛼𝑡2.
 Hence, the penetration depth increases as 𝑡 and is
larger for materials with higher thermal diffusivity.
4.7.4 Reveals of three case eqautions
A. step change in the surface temperature Case 1,

50
B. For a fixed surface heat flux (case 2),
The above equn. reveals that T(0, t) = 𝑇𝑠(t)
increases monotonically as 𝑡 .
The surface temperature and temperatures within
the medium approach the fluid temperature𝑇∞with
increasing time.
As 𝑇𝑠 approaches T , there is, of course, a reduction
in the surface heat flux,
C. For surface convection (case 3),

51
For ℎ → ∞ , the surface instantaneously achieves the imposed fluid temperature
(𝑇𝑠 = 𝑇∞), and with the second term on the right-hand side of Equation 37
reducing to zero, the result is equivalent to Equation 37.
37

52
Error function is a standard mathematical
function, just like the sine and cosine
functions, whose value varies between 0
and 1.
(29)

53
Dimensionless temperature distribution for transient conduction in a semi-infinite
solid whose surface is maintained at a constant temperature Ts.

54
4.7.5 Contact between two semi-inﬁnite solids
 An interesting permutation of 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 as shown on the fig.
 If the contact resistance is negligible,
the requirement of thermal equilibrium
dictates that, at the instant of contact
(t = 0), both surfaces must assume the
same temperature Ts , for which
𝑻 𝑩,𝒊< Ts < 𝑻 𝑨,𝒊.
 Since Ts does not change with increasing time, it follows that the
transient thermal response and the surface heat flux of each of the solids
are determined by Equations (35) and (36) , respectively.
 If the two bodies are of the same material, the contact surface
temperature is the arithmetic average, Ts = (TA,I + TB,i)/2

55
The equilibrium surface temperature from the above fig, can be
determined from a surface energy balance, which requires that
x-coordinate of Figure requires a sign change for , it follows that
Substituting from Equation (36) for and recognizing that the
For Ts (42)

56
On a hot and sunny day, the concrete deck surrounding a swimming pool is
at a temperature of 𝑇𝑑 = 55℃. A swimmer walks across the dry deck to the
pool. The soles of the swimmer's dry feet are characterized by an 𝐿 𝑠𝑓
= 3𝑚𝑚 skin/fat layer of thermal conductivity 𝑘 𝑠𝑓 = 0.3 𝑊/𝑚𝑘. Consider
two types of concrete decking; (i) a dense stone mix and (ii) a lightweight
aggregate characterized by density, specific heat, and thermal
conductivity of
respectively. The density and specific heat of the skin/fat layer may be
approximated to be those of liquid water, and the skin/fat layer is at an
initial temperature of 𝑇𝑠𝑓,𝑖 = 37℃. What is the temperature of the
bottom of the swimmer's feet after an elapsed time of t = 1s?
Assumptions:
1. One-dimensional conduction in the x-direction.
2. Constant and uniform properties.
3. Negligible contact resistance.
Example 4–5

57
Example 4–6
A large cast iron container (k = 52 W/m · °C and 𝛼 = 1.7𝑥10−5
𝑚2
/𝑠) with 5-cm-
thick walls is initially at a uniform temperature of 0°C and is filled with ice at
0°C. Now the outer surfaces of the container are exposed to hot water at 60°C
with a very large heat transfer coefficient. Determine
a) how long it will be before the ice inside the container starts melting.
b) the rate of heat transfer to the ice through a 1.2-m-wide and 2-m-high section
of the wall when steady operating conditions are reached. Assume the ice
starts melting when its inner surface temperature rises to 0.1°C and heat
transfer coefficient on the inner surface of the container to be 250 W/𝑚2
·°C,
Assumptions
1.The temperature in the container walls is affected
by the thermal conditions at outer surfaces only
and the convection heat transfer coefficient outside
is given to be very large. Therefore, the wall can
be considered to be a semi-infinite medium with a
specified surface temperature.
2. The thermal properties of the wall are constant.

58
4.8 Transient Heat Conduction In Multidimensional Systems
• Using a superposition approach called the product solution, the transient temperature
charts and solutions can be used to construct solutions for the two-dimensional and three-
dimensional transient heat conduction problems encountered in geometries such as a
short cylinder, a long rectangular bar, a rectangular prism, or a semi-infinite
rectangular bar, provided that all surfaces of the solid are subjected to convection to the
same fluid at temperature T, with the same heat transfer coefficient h, and the body
involves no heat generation.
• The solution in such multidimensional geometries can be expressed as the product of the
solutions for the one-dimensional geometries whose intersection is the multidimensional
geometry.
The temperature in a short
cylinder exposed to
convection from all
surfaces varies in both the
radial and axial directions,
and thus heat is transferred
in both directions.

59
 A short cylinder of radius ro and height a is the intersection of a long
cylinder of radius ro and a plane wall of thickness a.
 The solution for the two-dimensional short cylinder of height a and
radius ro is equal to the product of the nondimensionalized solutions
for the one-dimensional plane wall of thickness a and the long
cylinder of radius ro.
(43)

60
A long solid bar of rectangular profile a
 b is the intersection of two plane walls
of thicknesses a and b.
The transient heat transfer for a two-dimensional geometry formed by
the intersection of two one-dimensional geometries 1 and 2 is
(44)
(15)
(46)
(47)
(48)

61
Transient heat transfer for a three-dimensional body formed by the
intersection of three one-dimensional bodies 1, 2, and 3 is
(49)
7.6 Multidimensional solutions expressed as products of one-dimensional
solutions for bodies that are initially at a uniform temperature Ti and exposed to
convection from all surfaces to a medium at T

63
A short brass cylinder (𝑘 = 110 W/m℃ , 𝜌 =
8530𝑘𝑔
𝑚3 𝐶 𝑃 = 0.389 𝑘𝐽/𝑘𝑔℃ and
𝛼 =
𝑘
𝜌𝐶 𝑃
= 3.39 𝑋10−5 𝑚2/𝑠) of diameter D = 8 cm and height H = 15 cm is
initially at a uniform temperature of 𝑇𝑖= 150°C. The cylinder is now placed in
atmospheric air at 20°C, where heat transfer takes place by convection with a
heat transfer coefficient of h = 40 W/𝑚2℃ . Calculate
(a) the center temperature of the cylinder,
(b) the center temperature of the top surface of the
cylinder, and
(c) the total heat transfer from the cylinder 15 min after
the start of the cooling.
Example 4–7
Assumptions
1 Heat conduction in the short cylinder is two-dimensional, and thus the
temperature varies in both the axial x- and the radial r- directions. 2 The thermal
properties of the cylinder are constant.
3 The heat transfer coefficient is constant and uniform over the entire surface.

64
 The finite difference solution of
transient problems requires
discretization in time in addition to
discretization in space.
 This is done by selecting a suitable
time step t and solving for the
unknown nodal temperatures
repeatedly for each t until the solution
at the desired time is obtained.
 In transient problems, the superscript i
is used as the index or counter of time
steps, with i = 0 corresponding to the
specified initial condition.
4.9 Numerical Method for Transient Heat Conduction
4.9.1 Finite Difference Solution: Energy Balance Method

65
the energy balance on a volume element during a time interval
∆𝑡 can be expressed as
where 𝑸 conduction (interior nodes), convection,
heat flux, and radiation for boundary nodes.
(49)

66
Explicit method: If temperatures at the previous time step i is used.
Implicit method: If temperatures at the new time step i + 1 is used.
4.9.2 Explicit method and Implicit method
(50)
(51)
It appears that the time derivative is expressed in forward difference form in the
explicit case and backward difference form in the implicit case.
The formulation of explicit and implicit methods
differs at the time step (previous or new) at
which the heat transfer and heat generation terms
are expressed.

67
4.9.2 One dimensional Transient Heat Conduction in a Plane Wall
Mesh Fourier
number
(52)
(53)
(54)
Where,

68
The temperature of an interior node at the new time step is simply the average of
the temperatures of its neighboring nodes at the previous time step.
4.9.4 Transient Heat Conduction and convection
in a Plane Wall
(54)
For explicit method
Calculate 𝑇 𝑚
𝑖+1 for no heat generation and  = 0.5 from heat conduction
𝑇 𝑚
𝑖+1
=
𝑇 𝑚+1
𝑖
+ 𝑇 𝑚−1
𝑖
2
b. For implicitly method equation 𝑇 𝑚
𝑖+1 =
𝑇 𝑚+1
𝑖+1
+ 𝑇 𝑚−1
𝑖+1
2
+ 𝑇 𝑚
𝑖
a. For explicitly method equation

69
4.9.3 Stability Criterion for Explicit Method: Limitation on t
 The explicit method is easy to use, but it suffers from an
undesirable feature that severely restricts its utility: the
explicit method is not unconditionally stable, and the
largest permissible value of the time step t is limited
by the stability criterion.
 If the time step t is not sufficiently small, the solutions
obtained by the explicit method may oscillate wildly and
diverge from the actual solution.
 To avoid such divergent oscillations in nodal
temperatures, the value of t must be maintained below
a certain upper limit established by the stability
criterion.
Example

70
The implicit method is unconditionally stable, and thus we can use any time step we
please with this method (of course, the smaller the time step, the better the accuracy
of the solution).
The disadvantage of the implicit method is that it results in a set of equations that
must be solved simultaneously for each time step.
Both the implicit and explicit methods are used in practice.
The violation of the stability criterion in the
explicit method may result in the violation of
the second law of thermodynamics and thus
divergence of solution.

71
4.9.5 Two-Dimensional Transient Heat Conduction
Where,
(55)
(56)

72
Explicit finite difference formulation by expressing the left side at time step i as
This equation can be solved explicitly for the new temperature 𝑻 𝒏𝒐𝒅𝒆
𝒊+𝟏
to give
(58)
(57)
For all interior nodes (m, n) where m 1, 2, 3, . . . , M - 1 and
n = 1, 2, 3,..., N - 1 in the medium.
In the case of no heat generation 𝝉 = 𝟏/𝟒 explicit
finite difference

73
 The stability criterion that requires the coefficient of 𝑻 𝒎
𝒊 and 𝑻 𝒎
𝒊+𝟏in the
expression to be greater than or equal to zero for all nodes is equally valid for
two or three-dimensional cases and severely limits the size of the time step ∆𝒕
That can be used with the explicit method.
4.9.4 Stability criterion
 In the case of transient two-dimensional heat transfer in rectangular
coordinates, the coefficient of 𝑻 𝒎
𝒊
and 𝑻 𝒎
𝒊+𝟏
in the expression is 1 − 4𝜏 and
thus the stability criterion for all interior nodes in this case is 1 − 4𝜏 > 0 or
(interior nodes, two-dimensional heat
transfer in rectangular coordinates)
where ∆𝒙 = ∆𝒚 = 𝒍
 When the material of the medium and thus its thermal diffusivity 𝜶 are known
and the value of the mesh size l is specified, the largest allowable value of the
time step ∆𝑡 can be determined from the relation above.

74
Consider two-dimensional transient heat transfer in an L-shaped solid bar
that is initially at a uniform temperature of 140°C and whose cross section
is given in the figure. The thermal conductivity and diffusivity of the body
are 𝑘 = 15 𝑊/·°C and 𝛼 = 3.2𝑥10−6 𝑚2/𝑠, respectively, and heat is
generated in the body at a rate of 𝑔 = 2𝑥107
𝑤/𝑚3
. The right surface of
the body is insulated, and the bottom surface is maintained at a uniform
temperature of 140°C at all times. At time t = 0, the entire top surface is
subjected to convection with ambient air at 𝑇∞ = 25°C with a heat transfer
coefficient of ℎ = 80 𝑊/𝑚2°C, and the left surface is subjected to
uniform heat flux at a rate of °C 𝑞𝑙 = 8000 W/𝑚2
. The nodal network of
the problem consists of 13 equally spaced nodes with ∆𝑥 = ∆𝑦 = 1.5 𝑐𝑚.
Using the explicit method, determine the temperature at the top corner
(node 3) of the body after 2, 5, and 30 min.
Example 4–8

75
Assumptions
1 Heat transfer through the body is given to be transient and two-
dimensional.
2 Thermal conductivity is constant.
3 Heat generation is uniform.

76
Reference
This heat transfer lecture power point adapted from
1. Yunus Cengel, Heat and Mass Transfer A Practical Approach,
3rd edition
2. Jack P. Holman, Heat Transfer, Tenth Edition.
3. Frank P. Incropera, Theodore l. Bergman, Adrienne S.
Lavine, and David P Dewitt, fundamental of Heat and Mass
Transfer, 7th edition
4. Lecture power point of heat transfer by Mehmet Kanoglu
University of Gaziantep
You can download the previous chapter one- three by using the following link
https://www.slideshare.net/TarikuNegash/heat-transfer-chapter-one-and-two-
81503043?qid=aa518005-0a35-4103-9310-b8b6608a72e4&v=&b=&from_search=3
https://www.slideshare.net/TarikuNegash/two-dimensional-steady-state-heat-
conduction?qid=aa518005-0a35-4103-9310-b8b6608a72e4&v=&b=&from_search=1

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