1) The chapter discusses heat conduction and the governing equation for one-dimensional, steady-state heat conduction through a plane wall.
2) It derives the transient, one-dimensional heat conduction equations for plane walls, long cylinders, and spheres. These equations can be simplified for steady-state and cases without heat generation.
3) The chapter also covers boundary and initial conditions like specified temperature, heat flux, convection, radiation, and interfaces. Governing equations are developed for multidimensional and transient heat conduction problems.
This chapter contains:-.
Analytical Methods of two dimensional steady state heat conduction
Finite difference Method application on two dimensional steady state heat conduction.
Finite difference method on irregular shape of a system
Lectures on Heat Transfer - Introduction - Applications - Fundamentals - Gove...tmuliya
This file contains Introduction to Heat Transfer and Fundamental laws governing heat transfer.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
This chapter contains:-.
Analytical Methods of two dimensional steady state heat conduction
Finite difference Method application on two dimensional steady state heat conduction.
Finite difference method on irregular shape of a system
Lectures on Heat Transfer - Introduction - Applications - Fundamentals - Gove...tmuliya
This file contains Introduction to Heat Transfer and Fundamental laws governing heat transfer.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Numerical methods in Transient-heat-conductiontmuliya
This file contains slides on Numerical methods in Transient heat conduction.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Finite difference eqns. by energy balance – Explicit and Implicit methods – 1-D transient conduction in a plane wall – stability criterion – Problems - 2-D transient heat conduction – Finite diff. eqns. for interior nodes – Explicit and Implicit methods - stability criterion – difference eqns for different boundary conditions – Accuracy considerations – discretization error and round–off error - Problems
this is my presentation about 2nd law of thermodynamic. this is part of engineering thermodynamic in mechanical engineering. here discussed about heat transfer, heat engines, thermal efficiency of heat pumps and refrigerator and its equation for perfect work done with best figure and table wise discription, entropy and change in entropy, isentropic process for turbines and compressor and many more.
This file contains slides on Transient Heat conduction: Part-I
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010. Contents: Lumped system analysis – criteria for lumped system analysis – Biot and Fourier Numbers – Response time of a thermocouple - One-dimensional transient conduction in large plane walls, long cylinders and spheres when Bi > 0.1 – one-term approximation - Heisler and Grober charts- Problems
heat conduction and its mechanisms ,thermal conductivity,Fourier law,variation of thermal conductivity with temperature in metals and solids,steady and unsteady states,biot and Fourier numbers and their significance, Lumped heat analysis
Numerical methods in Transient-heat-conductiontmuliya
This file contains slides on Numerical methods in Transient heat conduction.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Finite difference eqns. by energy balance – Explicit and Implicit methods – 1-D transient conduction in a plane wall – stability criterion – Problems - 2-D transient heat conduction – Finite diff. eqns. for interior nodes – Explicit and Implicit methods - stability criterion – difference eqns for different boundary conditions – Accuracy considerations – discretization error and round–off error - Problems
this is my presentation about 2nd law of thermodynamic. this is part of engineering thermodynamic in mechanical engineering. here discussed about heat transfer, heat engines, thermal efficiency of heat pumps and refrigerator and its equation for perfect work done with best figure and table wise discription, entropy and change in entropy, isentropic process for turbines and compressor and many more.
This file contains slides on Transient Heat conduction: Part-I
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010. Contents: Lumped system analysis – criteria for lumped system analysis – Biot and Fourier Numbers – Response time of a thermocouple - One-dimensional transient conduction in large plane walls, long cylinders and spheres when Bi > 0.1 – one-term approximation - Heisler and Grober charts- Problems
heat conduction and its mechanisms ,thermal conductivity,Fourier law,variation of thermal conductivity with temperature in metals and solids,steady and unsteady states,biot and Fourier numbers and their significance, Lumped heat analysis
This PowerPoint is one small part of the Geology Topics unit from www.sciencepowerpoint.com. This unit consists of a five part 6000+ slide PowerPoint roadmap, 14 page bundled homework package, modified homework, detailed answer keys, 12 pages of unit notes for students who may require assistance, follow along worksheets, and many review games. The homework and lesson notes chronologically follow the PowerPoint slideshow. The answer keys and unit notes are great for support professionals. The activities and discussion questions in the slideshow are meaningful. The PowerPoint includes built-in instructions, visuals, and review questions. Also included are critical class notes (color coded red), project ideas, video links, and review games. This unit also includes four PowerPoint review games (110+ slides each with Answers), 38+ video links, lab handouts, activity sheets, rubrics, materials list, templates, guides, 6 PowerPoint review Game, and much more. Also included is a 190 slide first day of school PowerPoint presentation.
Areas of Focus within The Geology Topics Unit: -Plate Tectonics, Evidence for Plate Tectonics, Pangea, Energy Waves, Layers of the Earth, Heat Transfer, Types of Crust, Plate Boundaries, Hot Spots, Volcanoes, Positives and Negatives of Volcanoes, Types of Volcanoes, Parts of a Volcano, Magma, Types of Lava, Viscosity, Earthquakes, Faults, Folds, Seismograph, Richter Scale, Seismograph, Tsunami's, Rocks, Minerals, Crystals, Uses of Minerals, Types of Crystals, Physical Properties of Minerals, Rock Cycle, Common Igneous Rocks, Common Sedimentary Rocks, Common Metamorphic Rocks.
This unit aligns with the Next Generation Science Standards and with Common Core Standards for ELA and Literacy for Science and Technical Subjects. See preview for more information
If you have any questions please feel free to contact me. Thanks again and best wishes. Sincerely, Ryan Murphy M.Ed www.sciencepowerpoint@gmail.com
This file contains slides on Transient Heat conduction: Part-II
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in the year 2010.
Contents: Semi-infinite solids with different BC’s - Problems - Product solution for multi-dimension systems -
Summary of Basic relations for transient conduction
Mathcad Functions for Condensation heat transfertmuliya
This file contains slides on Mathcad Functions for Condensation heat transfer.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Functions for properties of sat. water and steam- Film condensation of steam on vertical plate and inclined plate – on vertical cylinder and horizontal cylinder – on horizontal cylinders in vertical tier - on a sphere – inside horizontal tubes – on copper surfaces – Mathcad Functions for properties of sat. Ammonia – Film condensation of Ammonia on vertical plate, horizontal cylinder and tube banks, and inside horizontal tubes
Heat transfer from extended surfaces (or fins)tmuliya
This file contains slides on Heat Transfer from Extended Surfaces (FINS). The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Governing differential eqn – different boundary conditions – temp. distribution and heat transfer rate for: infinitely long fin, fin with insulated end, fin losing heat from its end, and fin with specified temperatures at its ends – performance of fins - ‘fin efficiency’ and ‘fin effectiveness’ – fins of non-uniform cross-section- thermal resistance and total surface efficiency of fins – estimation of error in temperature measurement - Problems
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
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Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
2. Objectives
When you finish studying this chapter, you should be able to:
• 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, and
• Evaluate heat conduction in solids with temperature-dependent
thermal conductivity.
3. Introduction
• Although heat transfer and temperature are
closely related, they are of a different nature.
• Temperature has only magnitude
it is a scalar quantity.
• Heat transfer has direction as well as magnitude
it is a vector quantity.
• We work with a coordinate system and indicate
direction with plus or minus signs.
4. Introduction ─ Continue
• 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, φ, z, t)),
– spherical (T(r, φ, θ, t)).
5. Classification of conduction heat transfer problems:
• steady versus transient heat transfer,
• multidimensional heat transfer,
• heat generation.
Introduction ─ Continue
6. Steady versus Transient Heat Transfer
• Steady implies no change with time at any point
within the medium
• Transient implies variation with time or time
dependence
7. 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.
8. • 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.
(W)cond
dT
Q kA
dx
= −& (2-1)
9. General Relation for Fourier’s Law of
Heat Conduction
• 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
(W)n
dT
Q kA
dn
= −& (2-2)
10. General Relation for Fourier’s Law of
Heat Conduction-Continue
• In rectangular coordinates, the heat conduction
vector can be expressed in terms of its components as
• which can be determined from Fourier’s law as
n x y zQ Q i Q j Q k= + +
r rr r
& & & &
x x
y y
z z
T
Q kA
x
T
Q kA
y
T
Q kA
z
∂
= − ∂
∂
= −
∂
∂
= −
∂
&
&
&
(2-3)
(2-4)
11. Heat Generation
• Examples:
– electrical energy being converted to heat at a rate of I2
R,
– 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.
• The total rate of heat generation in a medium of
volume V can be determined from
(W)gen gen
V
E e dV= ∫& & (2-5)
12. One-Dimensional Heat Conduction
Equation - Plane Wall
xQ&
Rate of heat
conduction
at x
Rate of heat
conduction
at x+∆x
Rate of heat
generation inside
the element
Rate of change of
the energy content
of the element
- + =
,gen elementE+ &
x xQ +∆− & elementE
t
∆
=
∆
(2-6)
13. • The change in the energy content and the rate of heat
generation can be expressed as
• Substituting into Eq. 2–6, we get
( ) ( )
,
element t t t t t t t t t
gen element gen element gen
E E E mc T T cA x T T
E e V e A x
ρ+∆ +∆ +∆∆ = − = − = ∆ −
= = ∆
& & &
,
element
x x x gen element
E
Q Q E
t
+∆
∆
− + =
∆
& & & (2-6)
(2-7)
(2-8)
x x xQ Q +∆−& & (2-9)
gene A x+ ∆& t t tT T
cA x
t
ρ +∆ −
= ∆
∆
1
gen
T T
kA e c
A x x t
ρ
∂ ∂ ∂
+ = ÷
∂ ∂ ∂
& (2-11)
• Dividing by A∆x, taking the limit as ∆x 0 and ∆t 0,
and from Fourier’s law:
14. The area A is constant for a plane wall the one dimensional
transient heat conduction equation in a plane wall is
gen
T T
k e c
x x t
ρ
∂ ∂ ∂
+ = ÷
∂ ∂ ∂
&Variable conductivity:
Constant conductivity:
2
2
1
;
geneT T k
x k t c
α
α ρ
∂ ∂
+ = =
∂ ∂
&
1) Steady-state:
2) Transient, no heat generation:
3) Steady-state, no heat generation:
2
2
0
gened T
dx k
+ =
&
2
2
1T T
x tα
∂ ∂
=
∂ ∂
2
2
0
d T
dx
=
The one-dimensional conduction equation may be reduces
to the following forms under special conditions
(2-13)
(2-14)
(2-15)
(2-16)
(2-17)
15. One-Dimensional Heat Conduction
Equation - Long Cylinder
rQ&
Rate of heat
conduction
at r
Rate of heat
conduction
at r+∆r
Rate of heat
generation inside
the element
Rate of change of
the energy content
of the element
- + =
,gen elementE+ & elementE
t
∆
=
∆r rQ +∆− &
(2-18)
16. • The change in the energy content and the rate of heat
generation can be expressed as
• Substituting into Eq. 2–18, we get
( ) ( )
,
element t t t t t t t t t
gen element gen element gen
E E E mc T T cA r T T
E e V e A r
ρ+∆ +∆ +∆∆ = − = − = ∆ −
= = ∆
& & &
,
element
r r r gen element
E
Q Q E
t
+∆
∆
− + =
∆
& & & (2-18)
(2-19)
(2-20)
r r rQ Q +∆−& & (2-21)
gene A r+ ∆& t t tT T
cA r
t
ρ +∆ −
= ∆
∆
1
gen
T T
kA e c
A r r t
ρ
∂ ∂ ∂
+ = ÷
∂ ∂ ∂
& (2-23)
• Dividing by A∆r, taking the limit as ∆r 0 and ∆t 0,
and from Fourier’s law:
17. Noting that the area varies with the independent variable r
according to A=2πrL, the one dimensional transient heat
conduction equation in a plane wall becomes
1
gen
T T
rk e c
r r r t
ρ
∂ ∂ ∂
+ = ÷
∂ ∂ ∂
&
1
0
gened dT
r
r dr dr k
+ = ÷
&
The one-dimensional conduction equation may be reduces
to the following forms under special conditions
1 1geneT T
r
r r r k tα
∂ ∂ ∂
+ = ÷
∂ ∂ ∂
&
1 1T T
r
r r r tα
∂ ∂ ∂
= ÷
∂ ∂ ∂
0
d dT
r
dr dr
= ÷
Variable conductivity:
Constant conductivity:
1) Steady-state:
2) Transient, no heat generation:
3) Steady-state, no heat generation:
(2-25)
(2-26)
(2-27)
(2-28)
(2-29)
18. One-Dimensional Heat Conduction
Equation - Sphere
2
2
1
gen
T T
r k e c
r r r t
ρ
∂ ∂ ∂
+ = ÷
∂ ∂ ∂
&
2
2
1 1geneT T
r
r r r k tα
∂ ∂ ∂
+ = ÷
∂ ∂ ∂
&
Variable conductivity:
Constant conductivity:
(2-30)
(2-31)
19. General Heat Conduction Equation
x y zQ Q Q+ +& & &
Rate of heat
conduction
at x, y, and z
Rate of heat
conduction
at x+∆x, y+∆y,
and z+∆z
Rate of heat
generation
inside the
element
Rate of change
of the energy
content of the
element
- + =
x x y y z zQ Q Q+∆ +∆ +∆− − −& & &
,gen elementE+ elementE
t
∆
=
∆
(2-36)
20. Repeating the mathematical approach used for the one-
dimensional heat conduction the three-dimensional heat
conduction equation is determined to be
2 2 2
2 2 2
1geneT T T T
x y z k tα
∂ ∂ ∂ ∂
+ + + =
∂ ∂ ∂ ∂
&
2 2 2
2 2 2
0
geneT T T
x y z k
∂ ∂ ∂
+ + + =
∂ ∂ ∂
&
2 2 2
2 2 2
1T T T T
x y z tα
∂ ∂ ∂ ∂
+ + =
∂ ∂ ∂ ∂
2 2 2
2 2 2
0
T T T
x y z
∂ ∂ ∂
+ + =
∂ ∂ ∂
Two-dimensional
Three-dimensional
1) Steady-state:
2) Transient, no heat generation:
3) Steady-state, no heat generation:
Constant conductivity: (2-39)
(2-40)
(2-41)
(2-42)
21. Cylindrical Coordinates
2
1 1
gen
T T T T T
rk k k e c
r r r r z z t
ρ
φ φ
∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + = ÷ ÷ ÷
∂ ∂ ∂ ∂ ∂ ∂ ∂
&
(2-43)
22. Spherical Coordinates
2
2 2 2 2
1 1 1
sin
sin sin
gen
T T T T
kr k k e c
r r r r r t
θ ρ
θ φ φ θ θ θ
∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + = ÷ ÷ ÷
∂ ∂ ∂ ∂ ∂ ∂ ∂
&
(2-44)
24. Specified Temperature Boundary
Condition
For one-dimensional heat transfer
through a plane wall of thickness
L, for example, the specified
temperature boundary conditions
can be expressed as
T(0, t) = T1
T(L, t) = T2
The specified temperatures can be constant, which is the
case for steady heat conduction, or may vary with time.
(2-46)
25. Specified Heat Flux Boundary
Condition
dT
q k
dx
= − =&
Heat flux in the
positive x-
direction
The sign of the specified heat flux is determined by
inspection: positive if the heat flux is in the positive
direction of the coordinate axis, and negative if it is in
the opposite direction.
The heat flux in the positive x-
direction anywhere in the medium,
including the boundaries, can be
expressed by Fourier’s law of heat
conduction as
(2-47)
26. Two Special Cases
Insulated boundary Thermal symmetry
(0, ) (0, )
0 or 0
T t T t
k
x x
∂ ∂
= =
∂ ∂
( ),
2 0
LT t
x
∂
=
∂
(2-49) (2-50)
27. Convection Boundary Condition
[ ]1 1
(0, )
(0, )
T t
k h T T t
x
∞
∂
− = −
∂
[ ]2 2
( , )
( , )
T L t
k h T L t T
x
∞
∂
− = −
∂
Heat conduction
at the surface in a
selected direction
Heat convection
at the surface in
the same direction
=
and
(2-51a)
(2-51b)
28. Radiation Boundary Condition
Heat conduction
at the surface in a
selected direction
Radiation exchange
at the surface in
the same direction
=
4 4
1 ,1
(0, )
(0, )surr
T t
k T T t
x
ε σ
∂
− = − ∂
4 4
2 ,2
( , )
( , ) surr
T L t
k T L t T
x
ε σ
∂
− = − ∂
and
(2-52a)
(2-52b)
29. Interface Boundary Conditions
0 0( , ) ( , )A B
A B
T x t T x t
k k
x x
∂ ∂
− = −
∂ ∂
At the interface the requirements are:
(1) two bodies in contact must have the same
temperature at the area of contact,
(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.
TA(x0, t) = TB(x0, t)
and
(2-53)
(2-54)
30. Generalized Boundary Conditions
In general 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
Heat transfer
to the surface
in all modes
Heat transfer
from the surface
In all modes
=
Heat Generation in Solids
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.
31. The heat transfer rate by convection can also be
expressed from Newton’s law of cooling as
( ) (W)s sQ hA T T∞= −&
gen
s
s
e V
T T
hA
∞= +
&
Rate of
heat transfer
from the solid
Rate of
energy generation
within the solid
=
For uniform heat generation within the medium
(W)genQ e V=& &
-
Heat Generation in Solids -The Surface
Temperature
(2-64)
(2-65)
(2-66)
(2-63)
32. Heat Generation in Solids -The Surface
Temperature
For a large plane wall of thickness 2L (As=2Awall and
V=2LAwall)
,
gen
s plane wall
e L
T T
h
∞= +
&
For a long solid cylinder of radius r0 (As=2πr0L and
V=πr0
2
L) 0
,
2
gen
s cylinder
e r
T T
h
∞= +
&
For a solid sphere of radius r0 (As=4πr0
2
and V=4
/3πr0
3
)
0
,
3
gen
s sphere
e r
T T
h
∞= +
&
(2-68)
(2-69)
(2-67)
33. Heat Generation in Solids -The maximum
Temperature in a Cylinder (the Centerline)
The heat generated within an inner
cylinder must be equal to the heat
conducted through its outer surface.
r gen r
dT
kA e V
dr
− = &
Substituting these expressions into the above equation
and separating the variables, we get
( ) ( )2
2
2
gen
gen
edT
k rL e r L dT rdr
dr k
π π− = → = −
&
&
Integrating from r =0 where T(0) =T0 to r=ro
2
0
max, 0
4
gen
cylinder s
e r
T T T
k
∆ = − =
&
(2-71)
(2-70)
34. Variable Thermal Conductivity, k(T)
• The thermal conductivity of a
material, in general, varies with
temperature.
• An average value for the
thermal conductivity is
commonly used when the
variation is mild.
• This is also common practice
for other temperature-
dependent properties such as
the density and specific heat.
35. Variable Thermal Conductivity for
One-Dimensional Cases
2
1
2 1
( )
T
T
ave
k T dT
k
T T
=
−
∫
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
The variation in thermal conductivity of a material
with can often be approximated as a linear function
and expressed as
0( ) (1 )k T k Tβ= +
β the temperature coefficient of thermal conductivity.
(2-75)
(2-79)
36. Variable Thermal Conductivity
• For a plane wall the
temperature varies linearly
during steady one-
dimensional heat conduction
when the thermal conductivity
is constant.
• This is no longer the case
when the thermal conductivity
changes with temperature
(even linearly).