Apuntes termofluidos de mecanismos de transferencia de calor

1. Chapter 16
MECHANISMS OF HEAT
TRANSFER
Mehmet Kanoglu
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Fundamentals of Thermal-Fluid Sciences, 3rd Edition
Yunus A. Cengel, Robert H. Turner, John M. Cimbala
McGraw-Hill, 2008

2. 2
Objectives
 Understand how thermodynamics and heat transfer are related to each other
 Distinguish thermal energy from other forms of energy, and heat transfer from
other forms of energy transfer
 Perform general energy balances as well as surface energy balances
 Understand the basic mechanisms of heat transfer, which are conduction,
convection, and radiation, and Fourier's law of heat conduction, Newton's law of
cooling, and the Stefan–Boltzmann law of radiation
 Identify the mechanisms of heat transfer that occur simultaneously in practice
 Develop an awareness of the cost associated with heat losses
 Solve various heat transfer problems encountered in practice

3. 3
INTRODUCTION
• Heat: The form of energy that can be transferred from one system to
another as a result of temperature difference.
• Thermodynamics concerned with the amount of heat transfer as a
system undergoes a process from one equilibrium state to another.
• Heat Transfer deals with the determination of the rates of such energy
transfers as well as variation of temperature.
• The transfer of energy as heat is always from the higher-temperature
medium to the lower-temperature one.
• Heat transfer stops when the two mediums reach the same temperature.
• Heat can be transferred in three different modes:
conduction, convection, radiation

4. 4
Applications

5. 5
Historical
Background
Kinetic theory: Treats molecules as tiny balls
that are in motion and thus possess kinetic
energy.
Heat: The energy associated with the random
motion of atoms and molecules.
Caloric theory: Heat is a fluidlike substance
called the caloric that is a massless, colorless,
odorless, and tasteless substance that can be
poured from one body into another
It was only in the middle of the nineteenth century
that we had a true physical understanding of the
nature of heat.
Careful experiments of the Englishman James P.
Joule published in 1843 convinced the skeptics
that heat was not a substance after all, and thus
put the caloric theory to rest.

6. 6
Heat conduction through
a large plane wall of
thickness x and area
A.
CONDUCTION
Conduction: The transfer of energy from the more energetic
particles of a substance to the adjacent less energetic ones
as a result of interactions between the particles.
In gases and liquids, conduction is due to the collisions and
diffusion of the molecules during their random motion.
In solids, it is due to the combination of vibrations of the
molecules in a lattice and the energy transport by free
electrons.
The rate of heat conduction through a plane layer is
proportional to the temperature difference across the layer
and the heat transfer area, but is inversely proportional to
the thickness of the layer.

7. 7
When x → 0 Fourier’s law of heat
conduction
Thermal conductivity, k: A measure of the ability of a
material to conduct heat.
Temperature gradient dT/dx: The slope of the temperature
curve on a T-x diagram.
Heat is conducted in the direction of decreasing
temperature, and the temperature gradient becomes
negative when temperature decreases with increasing x.
The negative sign in the equation ensures that heat
transfer in the positive x direction is a positive quantity.
The rate of heat conduction
through a solid is directly
proportional to its thermal
conductivity.
In heat conduction
analysis, A represents
the area normal to the
direction of heat transfer.

8. 8
Thermal
Conductivity
Thermal conductivity: The rate of
heat transfer through a unit
thickness of the material per unit
area per unit temperature
difference.
The thermal conductivity of a
material is a measure of the
ability of the material to conduct
heat.
A high value for thermal
conductivity indicates that the
material is a good heat
conductor, and a low value
indicates that the material is a
poor heat conductor or insulator.
A simple experimental setup to
determine the thermal conductivity of
a material.

9. 9
The range of
thermal
conductivity of
various materials at
room temperature.

10. 10
The mechanisms of heat conduction
in different phases of a substance.
The thermal conductivities of gases such as air
vary by a factor of 104 from those of pure metals
such as copper.
Pure crystals and metals have the highest thermal
conductivities, and gases and insulating materials
the lowest.

11. 11
The variation of the thermal
conductivity of various solids,
liquids, and gases with
temperature.

12. 12
Thermal Diffusivity
cp Specific heat, J/kg · °C: Heat capacity per unit mass
cp Heat capacity, J/m3 · °C: Heat capacity per unit
volume
 Thermal diffusivity, m2/s: Represents how fast heat
diffuses through a material
A material that has a high thermal conductivity or a
low heat capacity will obviously have a large thermal
diffusivity.
The larger the thermal diffusivity, the faster the
propagation of heat into the medium.
A small value of thermal diffusivity means that heat is
mostly absorbed by the material and a small amount
of heat is conducted further.

13. 13
CONVECTION
Convection: The mode of energy
transfer between a solid surface and
the adjacent liquid or gas that is in
motion, and it involves the combined
effects of conduction and fluid
motion.
The faster the fluid motion, the
greater the convection heat transfer.
In the absence of any bulk fluid
motion, heat transfer between a solid
surface and the adjacent fluid is by
pure conduction. Heat transfer from a hot surface to air
by convection.

14. 14
Forced convection: If the fluid
is forced to flow over the
surface by external means
such as a fan, pump, or the
wind.
Natural (or free) convection: If
the fluid motion is caused by
buoyancy forces that are
induced by density differences
due to the variation of
temperature in the fluid. The cooling of a boiled egg by forced and
natural convection.
Heat transfer processes that involve change of phase of a fluid are also
considered to be convection because of the fluid motion induced during the
process, such as the rise of the vapor bubbles during boiling or the fall of
the liquid droplets during condensation.

15. 15
Newton’s law of cooling
h convection heat transfer coefficient, W/m2 · °C
As the surface area through which convection heat transfer takes place
Ts the surface temperature
T the temperature of the fluid sufficiently far from the surface.
The convection heat transfer coefficient h is
not a property of the fluid.
It is an experimentally determined parameter
whose value depends on all the variables
influencing convection such as
- the surface geometry
- the nature of fluid motion
- the properties of the fluid
- the bulk fluid velocity

16. 16
RADIATION
• Radiation: The energy emitted by matter in the form of electromagnetic waves (or
photons) as a result of the changes in the electronic configurations of the atoms or
molecules.
• Unlike conduction and convection, the transfer of heat by radiation does not require
the presence of an intervening medium.
• In fact, heat transfer by radiation is fastest (at the speed of light) and it suffers no
attenuation in a vacuum. This is how the energy of the sun reaches the earth.
• In heat transfer studies we are interested in thermal radiation, which is the form of
radiation emitted by bodies because of their temperature.
• All bodies at a temperature above absolute zero emit thermal radiation.
• Radiation is a volumetric phenomenon, and all solids, liquids, and gases emit,
absorb, or transmit radiation to varying degrees.
• However, radiation is usually considered to be a surface phenomenon for solids.

17. 17
Stefan–Boltzmann law
 = 5.670  108 W/m2 · K4 Stefan–Boltzmann constant
Blackbody: The idealized surface that emits radiation at the maximum rate.
Blackbody radiation represents the maximum amount of
radiation that can be emitted from a surface at a specified
temperature.
Emissivity  : A measure of how closely a surface
approximates a blackbody for which  = 1 of the
surface. 0   1.
Radiation emitted
by real surfaces

18. 18
Absorptivity : The fraction of the radiation energy incident on a surface
that is absorbed by the surface. 0   1
A blackbody absorbs the entire radiation incident on it ( = 1).
Kirchhoff’s law: The emissivity and the absorptivity of a surface at a given
temperature and wavelength are equal.
The absorption of radiation
incident on an opaque surface of
absorptivity .

19. 19
Radiation heat transfer
between a surface and the
surfaces surrounding it.
Net radiation heat transfer: The
difference between the rates of
radiation emitted by the surface and
the radiation absorbed.
The determination of the net rate of
heat transfer by radiation between two
surfaces is a complicated matter since
it depends on
• the properties of the surfaces
• their orientation relative to each other
• the interaction of the medium
between the surfaces with radiation
Combined heat transfer coefficient hcombined Includes
the effects of both convection and radiation
When radiation and convection occur
simultaneously between a surface and a gas
Radiation is usually significant
relative to conduction or natural
convection, but negligible relative
to forced convection.

20. 20
SIMULTANEOUS HEAT
TRANSFER MECHANISMS
Although there are three mechanisms of
heat transfer, a medium may involve only
two of them simultaneously.
Heat transfer is only by conduction in opaque solids, but by
conduction and radiation in semitransparent solids.
A solid may involve conduction and radiation but not
convection. A solid may involve convection and/or radiation on
its surfaces exposed to a fluid or other surfaces.
Heat transfer is by conduction and possibly by radiation in a
still fluid (no bulk fluid motion) and by convection and radiation
in a flowing fluid.
In the absence of radiation, heat transfer through a fluid is
either by conduction or convection, depending on the presence
of any bulk fluid motion.
Convection = Conduction + Fluid motion
Heat transfer through a vacuum is by radiation.
Most gases between two solid surfaces do not
interfere with radiation.
Liquids are usually strong absorbers of radiation.

21. 21
SIMULTANEOUS HEAT
TRANSFER MECHANISMS
Combined heat transfer coefficient hcombined
Includes the effects of both convection and radiation

22. 22
THE FIRS LAW OF THERMODYNAMICS
The conservation of energy principle (or the energy balance) for any system
undergoing any process may be expressed as follows: The net change
(increase or decrease) in the total energy of the system during a process is
equal to the difference between the total energy entering and the total
energy leaving the system during that process. That is,

23. 23
THE FIRS LAW OF THERMODYNAMICS
In steady operation, the rate of
energy transfer to a system is equal
to the rate of energy transfer from
the system.

24. 24
THE FIRS LAW OF THERMODYNAMICS
In heat transfer analysis, we are usually interested only in the forms of energy
that can be transferred as a result of a temperature difference, that is, heat or
thermal energy. In such cases it is convenient to write a heat balance and to
treat the conversion of nuclear, chemical, mechanical, and electrical energies
into thermal energy as heat generation. The energy balance in that case can be
expressed as

25. 25
THE FIRS LAW OF THERMODYNAMICS
Surface Energy Balance
A surface contains no volume or mass and thus no
energy. Therefore, a surface can be viewed as a
fictitious system whose energy content remains
constant during a process (just like a steady-state
or steady-flow system). Then the energy balance
for a surface can be expressed as
Energy interactions at the outer wall
surface of a house.

26. 28
Summary
• Conduction
 Fourier’s law of heat conduction
 Thermal Conductivity
 Thermal Diffusivity
• Convection
 Newton’s law of cooling
• Radiation
 Stefan–Boltzmann law
• Simultaneous Heat Transfer
Mechanisms

27. 29
Modeling in engineering
Mathematical modeling of
physical problems.
Modeling is a powerful engineering
tool that provides great insight and
simplicity at the expense of some
accuracy.

28. 30
Heat transfer, Modeling in engineering
Heat transfer equipment such as heat exchangers, boilers, condensers, radiators, heaters, furnaces,
refrigerators, and solar collectors are designed primarily on the basis of heat transfer analysis.
The heat transfer problems encountered in practice can be considered in two groups:
(1) rating and (2) sizing problems.
The rating problems deal with the determination of the heat transfer rate for an existing system at a
specified temperature difference.
The sizing problems deal with the determination of the size of a system in order to transfer heat at a
specified rate for a specified temperature difference.
An engineering device or process can be studied either experimentally (testing and taking
measurements) or analytically (by analysis or calculations).
The experimental approach has the advantage that we deal with the actual physical system, and the
desired quantity is determined by measurement, within the limits of experimental error. However, this
approach is expensive, timeconsuming, and often impractical.
The analytical approach (including the numerical approach) has the advantage that it is fast and
inexpensive, but the results obtained are subject to the accuracy of the assumptions, approximations,
and idealizations made in the analysis.

29. 31
Heat transfer, Modeling in engineering
• Step 1: Problem Statement
• Step 2: Schematic
• Step 3: Assumptions and Approximations
• Step 4: Physical Laws
• Step 5: Properties
• Step 6: Calculations
• Step 7: Reasoning, Verification, and Discussion

30. HEAT CONDUCTION EQUATION
•Mehmet Kanoglu
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Fundamentals of Thermal-Fluid Sciences, 3rd Edition
Yunus A. Cengel, Robert H. Turner, John M. Cimbala
McGraw-Hill, 2008

31. 41
Objectives
• 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.
• Evaluate heat conduction in solids with temperature-dependent thermal
conductivity.

32. 42
Introduction
Unlike temperature, heat transfer
has direction as well as
magnitude, and thus it is a vector
quantity (see figure).
To avoid such questions, we can work with a
coordinate system and indicate direction with plus or
minus signs. The generally accepted convention is that
heat transfer in the positive direction of a coordinate
axis is positive and in the opposite direction it is
negative. Therefore, a positive quantity indicates heat
transfer in the positive direction and negative quantity
indicate heat transfer in the negative direction.

33. 43
Introduction
The specification of the temperature at a point in a medium first requires the
specification of the location of that point. This can be done by choosing a
suitable coordinate system such as the rectangular, cylindrical, or spherical
coordinates, depending on the geometry involved, and a convenient
reference point (the origin).
• The driving force for any form of heat
transfer is the temperature difference.
• The larger the temperature difference, the
larger the rate of heat transfer.
• Three prime coordinate systems:
• rectangular T(x, y, z, t)
• cylindrical T(r, , z, t)
• spherical T(r, , , t).

34. 44
Steady versus Transient Heat Transfer
Heat transfer problems are often classified as
being steady (also called steady-state) or
transient (also called unsteady).
The term steady implies no change with time
at any point within the medium, while
transient implies variation with time o time
dependence.
Therefore, the temperature or heat flux
remains unchanged with time during steady
heat transfer through a medium at any
location, although both quantities may vary
from one location to another.

35. 45
Multidimensional 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 one-dimensional depending on
the relative magnitudes of heat transfer rates in different directions and the level of
accuracy desired.
• One-dimensional if the temperature in the medium varies in one direction only and
thus heat is transferred in one direction, and the variation of temperature and thus
heat transfer in other directions are negligible or zero.
• Two-dimensional if the temperature in a medium, in some cases, varies mainly in
two primary directions, and the variation of temperature in the third direction (and
thus heat transfer in that direction) is negligible.

36. 46
Multidimensional Heat Transfer

37. 47
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:
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.

38. 48
• 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

39. 49
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.

40. 50
One Dimensional Heat Conduction Equation
Consider heat conduction through a large plane wall such as the wall of a
house, the glass of a single pane window, the metal plate at the bottom of a
pressing iron, a cast-iron steam pipe, a cylindrical nuclear fuel element, an
electrical resistance wire, the wall of a spherical container, or a spherical
metal ball that is being quenched or tempered.
Heat conduction in these and many other geometries can be approximated
as being one-dimensional since heat conduction through these geometries
is dominant in one direction and negligible in other directions.
Next we develop the onedimensional heat conduction equation in
rectangular, cylindrical, and spherical coordinates.

41. 51
(2-6)
Heat Conduction Equation
in a Large Plane Wall

42. 52
The simplification of the onedimensional heat conduction equation in
a plane wall for the case of constant conductivity for steady
conduction with no heat generation.

43. 53
Heat Conduction
equation in
Long Cylinder

44. 54
Heat Conduction
equation in
Long Cylinder
Two equivalent forms of the differential equation for the onedimensional
steady heat conduction in a cylinder with no heat generation.

45. 55
Heat Conduction
equation in Sphere

46. 56
Combined One-Dimensional Heat
Conduction Equation
An examination of the one-dimensional transient heat conduction equations for
the plane wall, cylinder, and sphere reveals that all three equations can be
expressed in a compact form as
n = 0 for a plane wall
n = 1 for a cylinder
n = 2 for a sphere
In the case of a plane wall, it is customary to replace the variable r by x.
This equation can be simplified for steady-state or no heat generation cases as
described before.

47. 57
General Heat Conduction Equation
In the last section we considered one-dimensional heat conduction
and assumed heat conduction in other directions to be negligible.
Most heat transfer problems encountered in practice can be
approximated as being one-dimensional, and we mostly deal with
such problems in this text.
However, this is not always the case, and sometimes we need to
consider heat transfer in other directions as well.
In such cases heat conduction is said to be multidimensional, and in
this section we develop the governing differential equation in such
systems in rectangular, cylindrical, and spherical coordinate systems.

48. 58
Rectangular coordinates

49. 59
Heat Conduction
equation in
Cartesian coordinates

50. 60
Heat Conduction
equation

51. 61
Heat Conduction equation in
cylindrical coordinates
Relations between the coordinates of a point in
rectangular and cylindrical coordinate systems:

52. 62
Heat Conduction equation in
spherical coordinates
Relations between the coordinates of a point in
rectangular and spherical coordinate systems:

53. 63
Boundary and initial conditions
The description of a heat transfer problem in a medium is not complete without a full
description of the thermal conditions at the bounding surfaces of the medium.
Boundary conditions: The mathematical expressions of the thermal conditions at the
boundaries.
The temperature at any point on the
wall at a specified time depends on
the condition of the geometry at the
beginning of the heat conduction
process.
Such a condition, which is usually
specified at time t = 0, is called the
initial condition, which is a
mathematical expression for the
temperature distribution of the
medium initially.

54. 64
Boundary and initial conditions
• Specified Temperature Boundary Condition
• Specified Heat Flux Boundary Condition
• Convection Boundary Condition
• Radiation Boundary Condition
• Interface Boundary Conditions
• Generalized Boundary Conditions

55. 65
Specified Temperature Boundary Condition
The temperature of an exposed surface can
usually be measured directly and easily.
Therefore, one of the easiest ways to specify the
thermal conditions on a surface is to specify the
temperature.
For one-dimensional heat transfer through a
plane wall of thickness L, for example, the
specified temperature boundary conditions can be
expressed as
where T1 and T2 are the specified temperatures at
surfaces at x = 0 and x = L, respectively.
The specified temperatures can be constant, which
is the case for steady heat conduction, or may vary
with time.

56. 66
Specified Heat Flux Boundary Condition
For a plate of thickness L subjected to heat flux of
50 W/m2 into the medium from both sides, for
example, the specified heat flux boundary
conditions can be expressed as
The heat flux in the positive x-direction anywhere
in the medium, including the boundaries, can be
expressed by

57. 67
Special Case: Insulated Boundary
A well-insulated surface can be modeled as
a surface with a specified heat flux of zero.
Then the boundary condition on a perfectly
insulated surface (at x = 0, for example) can
be expressed as
On an insulated surface, the first derivative
of temperature with respect to the space
variable (the temperature gradient) in the
direction normal to the insulated surface is
zero.

58. 68
Another Special Case: Thermal Symmetry
Some heat transfer problems possess thermal symmetry as a
result of the symmetry in imposed thermal conditions.
For example, the two surfaces of a large hot plate of thickness
L suspended vertically in air is subjected to the same thermal
conditions, and thus the temperature distribution in one half of
the plate is the same as that in the other half.
That is, the heat transfer problem in this plate possesses
thermal symmetry about the center plane at x = L/2.
Therefore, the center plane can be viewed as an insulated
surface, and the thermal condition at this plane of symmetry
can be expressed as
which resembles the insulation
or zero heat flux boundary
condition.

59. 69
Convection Boundary Condition
For one-dimensional heat transfer in the x-direction in a plate
of thickness L, the convection boundary conditions on both
surfaces:

60. 70
Radiation Boundary Condition
For one-dimensional heat transfer in the x-
direction in a plate of thickness L, the
radiation boundary conditions on both
surfaces can be expressed as
Radiation boundary condition on a surface:

61. 71
Interface Boundary Conditions
The boundary conditions at an interface are
based on the requirements that
(1) two bodies in contact must have the same
temperature at the area of contact and
(2) an interface (which is a surface) cannot store
any energy, and thus the heat flux on the two
sides of an interface must be the same.
The boundary conditions at the interface of two
bodies A and B in perfect contact at x = x0 can
be expressed as

62. 72
Generalized Boundary Conditions
In general, however, 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

63. 73
Generalized Boundary Conditions
1. Dirichlet boundary condition:
A known value is imposed on boundary for
temperature as shown in Fig.(A). Also known as
boundary condition of first kind.
1. 2. Neumann boundary condition:
Gives a condition for the first derivative of
temperature and is shown in Fig.(B). Also known
as boundary condition of second kind
3. Robin boundary condition (or) mixed condition (or)
third kind of boundary condition:
This concerns convection or radiation or both at the
surfaces in question. In Fig.(C), the key point is that both
T1 and T2 are unknown.

64. 74
Solution of steady one-dimensional heat
equation conduction
In this section we will solve a wide range of heat conduction
problems in rectangular, cylindrical, and spherical geometries.
We will limit our attention to problems that result in ordinary
differential equations such as the steady one-dimensional heat
conduction problems. We will also assume constant thermal
conductivity.
The solution procedure for solving heat conduction problems
can be summarized as
(1) formulate the problem by obtaining the applicable differential
equation in its simplest form and specifying the boundary
conditions,
(2) Obtain the general solution of the differential equation, and
(3) apply the boundary conditions and determine the arbitrary
constants in the general solution.

65. 75
Heat generation in a solid
Many practical heat transfer applications involve the
conversion of some form of energy into thermal energy
in the medium.
Such mediums are said to involve internal heat
generation, which manifests itself as a rise in
temperature throughout the medium.
Some examples of heat generation are
- resistance heating in wires,
- exothermic chemical reactions in a solid, and
- nuclear reactions in nuclear fuel rods
where electrical, chemical, and nuclear energies are
converted to heat, respectively.
Heat generation in an electrical wire of outer radius ro
and length L can be expressed as

66. 76
Heat generation in a solid
The quantities of major interest in a medium with heat
generation are the surface temperature Ts and the maximum
temperature Tmax that occurs in the medium in steady operation.

67. 77
Heat generation in a solid

68. 78
Variable thermal conductivity, k(T)
When the variation of thermal conductivity with temperature
in a specified temperature interval is large, it may be
necessary to account for this variation to minimize the error.
When the variation of thermal conductivity with temperature
k(T) is known, the average value of the thermal conductivity
in the temperature range between T1 and T2 can be
determined from

69. 79
 temperature coefficient
of thermal conductivity.
The average value of thermal conductivity in the
temperature range T1 to T2 in this case can be
determined from
The average thermal conductivity in this case is
equal to the thermal conductivity value at the
average temperature.
The variation in thermal conductivity of a material with
temperature in the temperature range of interest can
often be approximated as a linear function and
expressed as

70. 80
Summary
• Introduction
• Steady versus Transient Heat Transfer
• Multidimensional Heat Transfer
• Heat Generation
• One-Dimensional Heat Conduction Equation
• Heat Conduction Equation in a Large Plane Wall
• Heat Conduction Equation in a Long Cylinder
• Heat Conduction Equation in a Sphere
• Combined One-Dimensional Heat Conduction Equation
• General Heat Conduction Equation
• Rectangular Coordinates
• Cylindrical Coordinates
• Spherical Coordinates
• Boundary and Initial Conditions
• Solution of Steady One-Dimensional Heat Conduction Problems
• Heat Generation in a Solid
• Variable Thermal Conductivity k (T )

71. 81
Examples
Steady, 1-D
Heat Conduction in a Plane Wall
A Wall with Various Sets of Boundary Conditions
Heat Conduction in the Base Plate of an Iron
Fire Hazard Prevention of Oil Leakage on Hot Engine Surface
Heat Conduction in a Solar Heated Wall
Heat Loss Through a Steam Pipe
Heat Conduction Through a Spherical Shell
Variation of Temperature in a Resistance Heater
Heat Conduction in a Two-Layer Medium
Heat Conduction in a Plane Wall with Heat Generation