Steady- state-heat-transfer-in-multiple-dimensions

Lectures on Heat Transfer --
Steady State Heat Conduction in
Multiple Dimensions
by
Dr. M. ThirumaleshwarDr. M. Thirumaleshwar
formerly:
Professor, Dept. of Mechanical Engineering,
St. Joseph Engg. College, Vamanjoor,
Mangalore
India

Preface
• This file contains slides on Steady State
Heat Conduction in Multiple Dimensions.
• 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.
Aug. 2016 2MT/SJEC/M.Tech.

• It is hoped that these Slides will be useful
to teachers, students, researchers and
professionals working in this field.
• For students, it should be particularly
useful to study, quickly review the subject,useful to study, quickly review the subject,
and to prepare for the examinations.
•
Aug. 2016 3MT/SJEC/M.Tech.

References
• 1. M. Thirumaleshwar: Fundamentals of Heat &
Mass Transfer, Pearson Edu., 2006
• https://books.google.co.in/books?id=b2238B-
AsqcC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false
• 2. Cengel Y. A. Heat Transfer: A Practical
Approach, 2nd Ed. McGraw Hill Co., 2003
Aug. 2016 MT/SJEC/M.Tech. 4
Approach, 2nd Ed. McGraw Hill Co., 2003
• 3. Cengel, Y. A. and Ghajar, A. J., Heat and
Mass Transfer - Fundamentals and Applications,
5th Ed., McGraw-Hill, New York, NY, 2014.

References… contd.
• 4. Incropera , Dewitt, Bergman, Lavine:
Fundamentals of Heat and Mass Transfer, 6th
Ed., Wiley Intl.
• 5. M. Thirumaleshwar: Software Solutions to• 5. M. Thirumaleshwar: Software Solutions to
Problems on Heat Transfer – CONDUCTION-
Part-II, Bookboon, 2013
• http://bookboon.com/en/software-solutions-problems-on-heat-
transfer-cii-ebook

Steady State Heat Conduction in
Multiple Dimensions…
Outline..
• 2-D conduction - Various methods of
solution – Analytical - Graphical -
solution – Analytical - Graphical -
Analogical – Numerical – Shape factors for
2-D conduction - Problems

Two-dimensional conduction..
• Practical examples of multi-dimensional heat transfer
are: heat treatment of engineering components of
irregular shapes, heat transfer in I.C.Engine blocks,
chimneys, air conditioning ducts etc.
• To solve multi-dimensional heat transfer problems,
basically, there are four methods:
basically, there are four methods:
• Analytical methods - solutions are quite
cumbersome
• Graphical methods - for two dimensional
problems with isothermal and adiabatic
boundaries. This is an approximate method.

Two-dimensional conduction ..
• Analogical methods - Special conducting paper
(or, conducting solution in a bath) is used to make a
model of the geometry being investigated and the
isothermal (equi-potential) lines are traced using a
probe.
probe.
• Numerical methods - Numerical methods have
taken over other methods because of availability of high
speed computers and the ability to analyze complex
shapes and deal with complicated boundary conditions.

Analytical methods:
• For 2D conduction, we have to solve:
Analytical solution becomes complicated and difficult
since now we have to deal with a partial differential
equation. We illustrate this with a problem:
Method used is Separation of variables, and it is briefly
illustrated below for a specific problem (Ref: Incropera
et al.)

• Consider the system of Fig. 4.2: three sides of a thin rect. plate
maintained at a const. temp. T1 and the fourth side is at T2.
Introduce the transformation:
Substituting eqn. (4.2) in (4.1) we get:

• B.C’s are:
In (4.5) LHS depends only on x and RHS depends only on y.
So, both sides are equal to the same constant. We write:

Now, applying the B.C’s and after much manipulation,
we get the final solution:

• Consider one more problem (Ref: Schaum’s Series, Pitts
& Leighton):
B.C’s are:

Proceeding as earlier,
Now, applying the B.C’s and simplifying, final expression for temp.
distribution is:

Graphical methods:
(Ref: Schaum’s Series, Pitts & Leighton):
• Consider a heated pipe with thick insulation,
with inside temp. Ti and outside temp. T0 as
shown. Constructing perpendiculars to
isothermal lines result in heat flow lanes.
Because of symmetry consider only one
quadrant as shown, and in this quadrant
quadrant as shown, and in this quadrant
there are 4 heat flow lanes.
• Method is to determine the heat flow in a
single lane and then find the total.

Graphical methods:
(Ref: Schaum’s Series, Pitts & Leighton):

• Applying Fourier’s Law to element a-b-c-d of a typical lane, heat
transfer per unit depth is:
Now, if the isotherms are uniformly spaced, and if there are M such
curvilinear squares in a flow lane, then temp. difference across one square is:
Then, for N flow lanes:

• And, the conductive shape factor per unit depth is:
Freehand plotting:
As shown above, a graphical plot of equally spaced
isotherms and adiabatics is enough to determine the
isotherms and adiabatics is enough to determine the
Shape factor.
The graphical net can usually be obtained by
freehand plotting. Typical one-eighth section is
shown, and its shape factor is one-eighth of overall
shape factor.

Shape factors for 2-D
conduction:
• This is a simple method to analyse a particular type of 2-
D conduction problems where steady state heat
transfer occurs between two surfaces at fixed
temperatures, T1 and T2, with an intervening solid
medium in between.
• If Q is the rate of heat transfer between two temperature
potentials T1 and T2, with the thermal conductivity of
intervening material being k, with no heat generation in
the medium, we write:
• Q = k S (T1 – T2)……….(4.76)
where S is known as ‘Shape factor’ and has dimension
of length.

conduction:
• From eqn.(4.76), immediately it follows that thermal
resistance of the medium is given by:
Rth = 1/(k S) ………..(4.77)
• Then, since we can write: S = 1/(R.k), we get:

conduction:
• In calculation of heat transfer in a furnace, separate
shape factors are used to calculate the heat flow through
the walls, edges and corners.
• When all the interior dimensions are greater than one–
fifth of the wall thickness, we get:
fifth of the wall thickness, we get:
where, A = inside area of the wall, L = wall thickness,
and D = length of edge

Ref: Cengel

Steady- state-heat-transfer-in-multiple-dimensions

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