MCE 855: CONDUCTIVE HEAT TRANSFER
MCE 855
COURSE SYNOPSIS
CONDUCTIVE HEAT TRANSFER
PROFESSOR M. A. WAHEED
VISITING PROFESSOR
COVENANT UNIVERSITY, OTA
akindoye@gmail.com
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Course Outling
Course Outling Contd
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Derivation of Heat Conduction Equation for Heterogeneous, Isotropic
Materials in Cartesian Coordinates. Heat conduction equation for
homogeneous, isotropic materials in Cartesian, Cylindrical and
Spherical Coordinates. Summary of basic steady 1D heat
conduction solutions including concept of resistances.
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Illustration #2: 2D Steady State Heat Conduction with Constant Heat
Generation in a Long Rod of Rectangular Cross-section with
Boundaries at the ambient temperature (large heat transfer
coefficient)
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Heat transfer from a fin of uniform cross-section. Fin efficiency and
fin effectiveness. Fin with variable cross-section.
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Two-dimensional Steady State Heat Conduction: Illustration # 1: A
rod with rectangular cross-section with three sides having
temperature, To and other side at T = f(x). Solution by Method of
Separation of Variables. Isotherms and Heat Flux Lines.
Steady 2D Conduction in Cylindrical Coordinates: Examples of
various 2D conduction problems in cylindrical
coordinates. Illustration #1: T (r, z), Circular Cylinder of Finite Length
(Axi-symmetric Problem with top surface at T = f (r) and other
surfaces at T = Tc). Fourier-Bessel Series Solution.
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Illustration #2: Long Cylinder having Circumferential Surface
Temperature Variation: T (r, φ) Problem: Periodic boundary
conditions in φ-direction. Justification of orthogonality in φdirection. Solution by Separation of Variables method.
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Course Outling Contd
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Treatment of variable conductivity by Kirchhoff transformation.
Unsteady State Conduction: Applications. Definition of Lumped and
Distributed Systems. Biot Number and its Physical
Significance. Characteristic lengths for plane wall, long cylinder and
sphere. Lumped System Analysis: Derivation of the governing
equation. Solution. T vs. t as a function of hA/ρcV for the cases of
heating and cooling. Time Constant and its Physical
Significance. Distributed Systems Analysis: Plane Wall: CaseI:
Large Heat Transfer Coefficient. Case II: Moderate Heat Transfer
Coefficient.
Long Cylinder: Case I: Large Heat Transfer Coefficient. Case II:
Moderate Heat Transfer Coefficient. Introduction to Heisler
Charts. Multi-dimensional transient heat conduction: Nondimensional temperature expressed as a product of 1D transient
solution in each direction.
Course Outling Contd.
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Semi-Infinite Solid: Definition. 1D Transient Solution by Laplace
Transform and Similarity technique (Error function solution) when
temperature of the surface at x = 0 is suddenly changed to T∞ (<
Ti). Expression of heat flux at x = 0. Other surface boundary
conditions: (i) Surface Convection (ii) Constant surface heat
flux. Penetration depth.
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Time-dependent Boundary Conditions-Duhamel’s Superposition
Integral: Principle. Derivation of the integral. Solidification and
Melting: Introduction. 1D Solidification Analysis: Stefan (1891)
Problem. Melting of a Solid: 1D Analysis.
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Inverse heat conduction: Determination of unknown boundary
conditions; Experimental determination of thermal conductivity and
heat capacity.
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Microscale heat transfer: hyperbolic heat conduction, speed of
propagation of thermal waves, time lag, solution for a thin slab.
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Grading Policy:
ding Policy
Resources for our learning
• Textbook: Fundamental of Heat and Mass Transfer: F.
P. Incropera and D. P. Dewitt
Homework
10%
• Heat Transfer: Yunus A. Cengel
Mid-Semester Exam
15%
• Heat Transfer: J.P. Holman
Class Attendance/Quiz
• Read assigned sections before coming to class.
Final Semester Exam
• Class participation welcome and essential.
• Other Instructors, i. e., Classmates, Organized Learning
Groups
• Homework: Submission, grading, and return policies will
be announced in the class.
Total
(Date?)
5%
70%
(Date?)
100%
Please turn in homework on time! May discuss, but do not
copy solutions from any source!
10% penalty for late homework.
No credit after solutions have been posted, except in serious
situations.
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IMPORTANT INFORMATION
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Homework:
You are encouraged to work along with your colleagues, but each of you must
provide your own individual solution set. Plagiarism will result in a zero for that set.
Exam:
There will be a midterm and comprehensive final exams
Student responsibilities:
You have certain responsibilities and rights as an adult and a student. Please refer to
the student manual for a description of what these entail.
Use of Laptop/Cell Phone in Classroom:
No Laptop or Cell Phone (please turn off your cell phones) is allowed during the class
periods. You may use a PC Tablet strictly for note-taking.
Academic Integrity:
Academic integrity is the cornerstone of the university and will be strongly enforced in
this course. Any student found in violation of the academic integrity policy will be
given an “F” for the course and will be referred to the Student Disciplinary Committee.
For additional information about FUNAAB’s Academic Integrity policy/procedures
please contact the office of the College officer.
Your Performance in the Course
• You will find that this is a very interesting course about
how many useful devices work: cars, air planes, heaters,
power plants, air conditioners, refrigerators etc.
• If you don’t do well on early homework, quizzes and
exams?
• Ensure you understand the material (Ask questions in
class, Form a Study Group- use Blackboard and/or
Mixable to stay in touch with your study group, See
one of the Teaching Assistants or one of the
Instructors in their Office Hours)
• Ensure you are doing homework on your own (A little
help that allows you to complete homework well may
not be available in quizzes and exams).
• See me early and discuss your situation (in
confidence).
Heat Transfer as a Course
CONDUCTIVE HEAT TRANSFER
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INTRODUCTORY CHAPTER – HEAT
TRANSFER
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Has a “reputation” for being one of the most challenging,
fundamental, conceptual courses in ME. It is the “heart” of thermal
engineering
Why??
– Physically diverse: thermodynamics, material science, diffusion
theory, fluid mechanics, radiation theory
– Higher-level math: vector calculus, ODEs, PDEs, numerical
methods
– Physically elusive: heat is invisible; developing intuition takes time
– Appropriate assumptions: required to simplify and solve most
problems
However, Heat Transfer is interesting, fun, and readily applicable to
the real world
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THERMODYNAMICS AND HEAT TRANSFER
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Heat: The form of energy that can be transferred from one system
to another as a result of temperature difference.
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Thermodynamics is concerned with system in equilibrium; may be
used to predict the amount of heat transfer as a system undergoes
a process from one equilibrium state to another.
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Heat Transfer deals with the determination of the rates of such
energy transfers as well as variation of temperature.
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The transfer of energy as heat is always from the highertemperature medium to the lower-temperature one.
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Heat transfer stops when the two mediums reach the same
temperature.
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Heat can be transferred in three different modes:
All modes of heat transfer require the existence of a temperature
difference.
• Net radiation heat transfer occurs when there exists a
temperature difference between two or more surfaces
emitting radiation energy
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Application Areas of Heat Transfer
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• Conduction heat transfer is due to a temperature
gradient in a stationary medium or media
• Radiation heat transfer occurs due to emission of
energy in the form of electromagnetic waves by all
bodies above absolute zero temperature
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Application Areas of Heat Transfer
indeed a relevant subject in many industrial and environmental problem
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• Different types of heat transfer processes are called
different modes of heat transfer
• Convection heat transfer occurs between a surface and
a moving fluid at different temperatures
conduction, convection, radiation
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Definition
• Heat transfer is energy transfer due to a temperature
difference in a medium or between two or more media
Power generation and distribution: In energy production and
conversion, i.e. in the generation of electrical power whether through
nuclear fission or fusion, the combustion of fossil fuels,
magnetohydrodynamic processes, or the use of geothermal energy
sources, there are numerous heat transfer problem that must be
solved, i.e. in boilers, condensers, cooling towers, feed heaters,
transformer cooling, transmission cable cooling, etc.
Renewable energy system: Development of solar energy
conversion systems for space heating as well as for electric power
production, flat plate collectors, thermal energy storage, photovoltaic
cell (PVC) module cooling, etc.
Transportation: In propulsion systems such as internal
combustion, gas turbine, and rocket engines, engine cooling,
automobile radiators, etc.
Comfort heating, ventilation, and air conditioning: Design of
convectional space and water heating systems in the design of
incinerator and cryogenic storage equipment, in the design of
refrigeration and air conditioning systems, and in many
manufacturing processes, mobile food storage,
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Aviation and space exploration: Gas turbine blade cooling,
vehicle heat shields, rocket engine/nozzles cooling, space suits,
space power generation, etc.
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Chemical, petrochemical and process industry: heat
exchangers, reactors, reboilers, etc.
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Electrical machines and electronic equipment: Cooling of
motors, generators, computers and microelectronic devices, etc.
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Manufacturing and material processing: Metal processing, heat
treatment, composite material processing, crystal growth,
micromachining, laser machining, etc.
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Fire and combustion: combustion plant
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Domestic applications: ovens, stoves, toaster, etc.
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Health care and biomedical applications: blood warmers, organ
and tissue storage, hypothermia, etc.
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Weather and environmental changes: Also relevant to air and
water pollution and strongly influences local and global climate,
climate control, etc
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Application Areas of Heat Transfer
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Heat Transfer Applications
Human body
• • Heat transfer is commonly encountered in engineering systems
and other aspects of life, and one does not need to go very far to
see some application areas of heat transfer.
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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. 21
HEAT AND OTHER FORMS OF ENERGY
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Energy can exist in numerous forms such as:
 thermal,
 mechanical,
 kinetic,
ENGINEERING HEAT TRANSFER
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.
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Internal energy: May be viewed as the sum of the kinetic and
potential energies of the molecules.
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Sensible heat: The kinetic energy of the molecules.
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Latent heat: The internal energy associated with the phase of a
system.
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Chemical (bond) energy: The internal energy associated with
the atomic bonds in a molecule.
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Nuclear energy: The internal energy associated with the bonds
within the nucleus of the atom itself.
 potential,
 electrical,
 magnetic,
 chemical,
 nuclear.
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Their sum constitutes the total energy E (or e on a unit
mass basis) of a system.
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The sum of all microscopic forms of energy is called the
internal energy of a system.
What is thermal energy?
What is the difference between thermal
energy and heat?
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Internal Energy and Enthalpy
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Specific Heats of Gases, Liquids, and Solids
In the analysis of systems
that involve fluid flow, we
frequently encounter the
combination of properties u
and Pv.
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Specific heat: The energy required to
raise the temperature of a unit mass of a
substance by one degree.
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Two kinds of specific heats:
 specific heat at constant volume cv
The combination is defined
as enthalpy (h = u + Pv).
 specific heat at constant pressure cp
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The specific heats of a substance, in
general, depend on two independent
properties such as temperature and
pressure.
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At low pressures all real gases approach
ideal gas behavior, and therefore their
specific heats depend on temperature
only.
The term Pv represents the
flow energy of the fluid (also
called the flow work).
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Energy Transfer
Energy can be transferred to or from a given
mass by two mechanisms:
when
is constant:
heat transfer and work.
Heat transfer rate: The amount of heat
transferred per unit time.
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Heat flux: The rate of heat transfer per unit
area normal to the direction of heat transfer.
Incompressible substance: A
substance whose specific volume (or
density) does not change with
temperature or pressure.
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The constant-volume and constantpressure specific heats are identical
for incompressible substances.
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The specific heats of incompressible
substances depend on temperature
only.
Power: The work
done per unit time.
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Energy Balance for
Steady-Flow Systems
Surface Energy Balance
A surface contains no volume or mass,
and thus no energy. Thereore, a surface
can be viewed as a fictitious system
whose energy content remains constant
during a process.
A large number of engineering devices such as
water heaters and car radiators involve mass flow
in and out of a system, and are modeled as
control volumes.
Most control volumes are analyzed under steady
operating conditions.
The term steady means no change with time at a
specified location.
Mass flow rate: The amount of mass flowing
through a cross section of a flow device per unit
time.
Volume flow rate: The volume of a fluid flowing
through a pipe or duct per unit time.
This relation is valid for both steady and
transient conditions, and the surface
energy balance does not involve heat
generation since a surface does not
have a volume.
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MECHANISMS OF HEAT TRANSFER
MECHANISMS
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Heat as the form of energy that can be transferred from one
system to another as a result of temperature difference.
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A thermodynamic analysis is concerned with the amount of heat
transfer as a system undergoes a process from one equilibrium
state to another.
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The science that deals with the determination of the rates of such
energy transfers is the heat transfer.
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The transfer of energy as heat is always from the highertemperature medium to the lower-temperature one, and heat
transfer stops when the two mediums reach the same temperature.
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MECHANISMS OF HEAT TRANSFER BY 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.
Heat can be transferred in three basic modes:
 conduction
 convection
 radiation
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All modes of heat transfer require the existence of a temperature
difference.
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Heat conduction
through a large plane
wall of thickness x
and area A.
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When x → 0
Thermal
Conductivity
Fourier’s law of
heat conduction
Thermal conductivity, k: A measure of the ability of
a material to conduct heat.
Thermal conductivity:
The rate of heat transfer
through a unit thickness
of the material per unit
area per unit
temperature difference.
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.
In heat conduction
analysis, A represents
the area normal to the
direction of heat
transfer.
The thermal conductivity
of a material is a
measure of the ability of
the material to conduct
heat.
The rate of heat conduction
through a solid is directly
proportional to its thermal
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conductivity.
A high value for thermal
conductivity indicates
A simple experimental setup
that the material is a
to determine the thermal
good heat conductor,
conductivity of a material.
and a low value indicates
that the material is a
poor heat conductor or
insulator.
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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.
The range of
thermal
conductivity of
various
materials at
room
temperature.
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The mechanisms of heat
conduction in different
phases of a substance.
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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 variation of
the thermal
conductivity of
various solids,
liquids, and gases
with temperature.
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.
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CONVECTION
Forced convection: If
the fluid is forced to flow
over the surface by
external means such as
a fan, pump, or the wind.
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.
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 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.
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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.
Heat transfer from a hot surface to air
by convection.
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Newton’s law of cooling
h
As
Ts
T
RADIATION
convection heat transfer coefficient, W/m2 · °C
the surface area through which convection heat transfer takes place
the surface temperature
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
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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.
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Unlike conduction and convection, the transfer of heat by radiation does
not require the presence of an intervening medium.
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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.
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In heat transfer studies we are interested in thermal radiation, which is
the form of radiation emitted by bodies because of their temperature.
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All bodies at a temperature above absolute zero emit thermal radiation.
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Radiation is a volumetric phenomenon, and all solids, liquids, and
gases emit, absorb, or transmit radiation to varying degrees.
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However, radiation is usually considered to be a surface phenomenon
for solids.
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Stefan–Boltzmann law
 = 5.670 
108
W/m2 ·
K4
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Absorptivity  : The fraction of the radiation energy incident on a
surface that is absorbed by the surface. 0   1
Stefan–Boltzmann constant
A blackbody absorbs the entire radiation incident on it ( = 1).
Blackbody: The idealized surface that emits radiation at the maximum rate.
Kirchhoff’s law: The emissivity and the absorptivity of a surface at
a given temperature and wavelength are equal.
Radiation emitted
by real surfaces
Emissivity  : A measure of how closely
a surface approximates a blackbody for
which  = 1 of the surface. 0   1.
Blackbody radiation represents the maximum
amount of radiation that can be emitted from
a surface at a specified temperature.
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The absorption of radiation incident on
an opaque surface of absorptivity .
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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
Radiation is usually
significant relative to
conduction or natural
convection, but
negligible relative to
forced convection.
When a surface is completely enclosed by a
much larger (or black) surface at temperature
Tsurr separated by a gas (such as air) that
does not intervene with radiation, the net rate
of radiation heat transfer between these
two surfaces is given by
When radiation and convection occur
simultaneously between a surface and a gas:
Combined heat transfer coefficient hcombined
Includes the effects of both convection and radiation
Radiation heat transfer between a
surface and the surfaces surrounding it.
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SIMULTANEOUS HEAT
TRANSFER MECHANISMS
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
Although there are three mechanisms of
Heat transfer through a vacuum is by radiation.
heat transfer, a medium may involve
only two of them simultaneously.
Most gases between two solid surfaces
do not interfere with radiation.
Liquids are usually strong absorbers of
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radiation.
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