Mathcad Functions for Conduction heat transfer calculations

Mathcad Functions for
Conduction heat
transfer calculations:
by
Dr. M. Thirumaleshwar
formerly:
Professor, Dept. of Mechanical Engineering,
St. Joseph Engg. College, Vamanjoor,
Mangalore,
India

Preface:
This file contains notes on Mathcad Functions
for Conduction heat transfer calculations.
Some problems are also included.
These notes were prepared while teaching Heat
Transfer course to the M.Tech. students in
Mechanical Engineering Dept. of St. Joseph
Engineering College, Vamanjoor, Mangalore,
India.
It is hoped that these notes will be useful to
teachers, students, researchers and
professionals working in this field.
For students, it should be particularly useful to
study, review the subject and solve the home-
work problems.
• M. Thirumaleshwar
• August 2016

References:
1. M. Thirumaleshwar: Fundamentals of Heat & Mass Transfer,
Pearson Edu., 2006. See: https://books.google.co.in/books?id=b2238B-
AsqcC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false
2. Cengel Y. A. Heat and Mass Transfer: 3rd Ed. McGraw Hill Co.,
3. Incropera , Dewitt, Bergman, Lavine: Fundamentals of Heat
and Mass Transfer, 6th
Ed., Wiley Intl.
4. Cengel, Y. A. and Ghajar, A. J., Heat and Mass Transfer -
Fundamentals and Applications, 5th Ed., McGraw-Hill, New
York, NY, 2014.
5. M. Thirumaleshwar: Software Solutions to Problems on Heat
Transfer – Conduction – Part-I- 2013
http://bookboon.com/en/software-solutions-to-problems-on-heat-
transfer-ebook
6. M. Thirumaleshwar: Software Solutions to Problems on Heat
Transfer – Conduction – Part-II- 2013
http://bookboon.com/en/software-solutions-problems-on-heat-transfer-
cii-ebook
7. Mathcad Functions for fluid properties – useful in convection heat
transfer:
http://www.slideshare.net/tmuliya/mathcad-functions-for-fluid-properties-for-
convection-heat-transfer-calculations

Mathcad Functions for Conduction heat
transfer calculations:
Contents:
Summary of formulae with and without internal heat generation.
Mathcad Functions for: Plane slab, solid cylinder, cylindrical
shell, and spherical shell with and without internal heat generation.
Cases of constant thermal conductivity and conductivity varying
linearly with temp are also considered.

When there is no internal heat generation:

When there is internal heat generation:

Work out an example:
Solution:

For temp. distribution, we have given the eqn above.
To sketch the temperature distribution in the shell:

Heat is carried away at the boundaries by a fluid at a temp Tf, with heat transfer coeff. h:

Example: Heat is generated uniformly in a stainless steel plate having k= 20 W/(m.K). The
thickness of the plate is 1 cm and heat generation rate is 500 MW/m3. The temperatures
on either side of the plate are maintained at 100 C, calculate:
(i) the temperature on the centre line
(ii) temperature at one-quarter of the thickness from the surface
(iii) draw the temperature profile
At the centre line: x = 0:

Example[M.U.]: Heat is generated uniformly in a stainless steel plate having k= 20
W/(m.K). The thickness of the plate is 1 cm and heat generation rate is 500 MW/m3. If
the two sides of the plate are maintained at 200 and 100 C respectively, calculate:
(i) the temperature at the centre of the plate
(ii) the position and value of maximum temperature
(iii) heat transfer at the left and right faces
(iv) sketch the temp. profile in the slab
Plot: T vs x:
To find max. temp:

Heat transfer from left and right faces: Apply Fourier’s Law:
Tmax occurs at the insulated surface, i.e. at x = 0.

Example: A plane wall of thickness 0.1 m and k = 25 W/(m.K), having uniform volumetric
heat generation of 0.3 MW/m3 is insulated on one side and is exposed to a fluid at 92 C.
The convective heat transfer coeff. between the wall and the fluid is 500 W/(m2.K).
Determine:
(i) the max. temperature in the wall
(ii) temp. at the surface exposed to the fluid
(iii) Draw the temperature profile.
We get:

Example: Heat is generated uniformly in a stainless steel plate having k= 20 W/(m.K). The
thickness of the plate is 1 cm and heat generation rate is 500 MW/m3. The two sides of the
plate are maintained at 200 and 100 C respectively. If the thermal conductivity of the
material varies as:
k(T) = ko (1 + β T), (W/(m.C) where ko = 14.695 W/(m.C) and β = 10.208 x 10-4 (C-1), and
T is in deg. C.
(i) calculate the temperature on the centre line
(ii) find location and value of max. temp. in the plate
(iii) find heat transfer rate to the left and right sides, and
(iv) draw the temp. profile

In the above graph also, one can note the max. temp and its location.

Example (M.U.): (a). A 3.2 mm diameter stainless steel wire, 30 cm long has a voltage of 10
V impressed on it. The outer surface temperature of the wire is maintained at 93 C.
Calculate the centre temperature of the wire. Take the resistivity of the wire as 70 micro-
Ohm.cm and the thermal conductivity as 22.5 W/(m.K)
(b) The heated wire in the above example is submerged in a fluid maintained at 93 C. The
convection heat transfer coefficient is 5.7 kW/(m2
.K). Calculate the centre temperature of
the wire.

Convection boundary condition:
Analysis with variable thermal conductivity:

Example: Consider a hollow cylinder, 1 m long, inner radius of 0.1 m, outer radius of 0.15
m, thermal cond = 0.5 W/m.C, and heat gen. rate qg = 5000 W/m^3. Outer surface temp =
50 C. Find the innerr surface temp. if the inner surface is insulated.

Example (P.U.): A nuclear fuel element is in the form of a hollow cylinder insulated at the
inner surface. Its inner and outer radii are 5 cm and 10 cm respectively. The outer surface
gives heat to a fluid at 50 C where the unit surface conductance is 100 W/(m2.K). k of the
material is 50 W/(m.K). Find the rate of heat generation so that max. temp. in the system
will not exceed 200 C.

With convection boundary condition:
Analysis with variable thermal conductivity:

Example 5.13 (M.U.): A hollow conductor with ri = 0.6 cm, ro = 0.8 cm is made up of metal
of k = 20 W/(m.K) and electrical resistance per metre of 0.03 ohms. Find the maximum
allowable current if the temperature is not to exceed 50 C anywhere in the conductor. The
cooling fluid inside is at 38 C. (Conductor is insulated on the outside).
Solution:

Since heat is transferred from both the inside and outside surfaces, max. temperature, Tm must
occur somewhere in the shell. Let it occur at a radius rm . Obviously, rm lies in between ri and ro.
So, the cylindrical shell may be thought of as being made up of two shells; the inner shell,
between r = ri and r = rm , insulated on its 'outer periphery' and, an outer shell, between r = rm and
r = ro, insulated at its 'inner periphery'.

Example: A hollow cylinder 6 cm ID, 9 cm OD, has a heat generation rate of
5 x 106 W/m3. Inner surface is maintained at 450 C and outer surface at 350 C. k of the
material is 3 W/(m.K).
(i) determine the location and value of max. temperature
(ii) what is the temp. at mid-thickness of the shell?
(iii) determine the fraction of heat generated going to the inner surface, and
(iv) sketch the temp. profile

Example: A solid sphere of radius, R = 10 mm and k = 18 W/(m.C) has an uniform heat
generation rate of 2 x 106 W/m3. Heat is conducted away at its outer surface to ambient air
at 20 C by convection, with a heat transfer coeff. of 2000 W/(m2.C).
(i) Determine the steady state temp. at the centre and outer surface of the sphere.
(ii) Draw the temp. profile along the radius.
Solution:

Mathcad Functions for Conduction heat transfer calculations

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Mathcad Functions for Conduction heat transfer calculations