Table of Contents
Principle
3
Objective
3
Background
3
▪ Fourier’s law of Heat Conduction
3
▪ The Pin Fin
6
▪ Governing Equation and Boundary Conditions
6
▪ Temperature Distribution
8
Apparatus
9
1) Linear heat conduction equipment:

10
Description of the Linear Heat Conduction Equipment
10
2) Radial heat conduction equipment:
11

Description of the Radial Heat Conduction Equipment
12

Useful information:
12
3) Extended Surface Heat Transfer
12

Description of the Extended Surface Heat Transfer Equipment
12

Useful information:
13
Procedure
13
References
13
Experiment1
2
INME 4032
University of Puerto Rico
Mayagüez Campus
Department of Mechanical Engineering
INME 4032 - LABORATORY II
Spring 2004
Instructor: Guillermo Araya
Experiment 1: Conduction Heat Transfer Analysis
Principle
A heat source placed in a material causes temperature changes due to heat conduction.
The relationship between temperature and the distance from the heat source must be
linear after some time in the case of linear heat conduction and it must have a
logarithmic distribution in the case of radial heat conduction.
Objective
The experiment demonstrates heat conduction in three different experimental models. It
allows us to obtain experimentally the coefficient of thermal conductivity of some
unknown materials and in this way, to understand the factors and parameters that affect
the rates of heat transfer.
Background
Fourier’s law of Heat Conduction
A general statement of the Fourier’s law is: The conduction heat flux in a specified
direction equals the negative of the product of the medium thermal conductivity and the
temperature derivative in that direction. In Cartesian coordinates, with temperature
varying in the x direction only,
q  k
dT
dx
In cylindrical or spherical coordinates, with temperature varying in the r direction only,
q  k
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3
dT
dr
INME 4032
Fig 1.1. A cylindrical shell showing an
elemental control volume for application of the
energy conservation principle.
Figure 1.1 shows a cylindrical shell of length L, with inner radius r1 and outer radius r2.
The inner surface is maintained at temperature T1 and the outer surface is maintained at
temperature T2. An elemental control volume is located between radii r and r+Δr. If

temperatures are unchanging in time and Q v  0 , the energy conservation principle
requires that the heat flow across the face at r equal that at the face r+Δr


Q r  Q r  r
That is,

Q r  Constant, independent of r
Using Fourier’s law

dT 

Q r  Aq  2rL  k

dr


Dividing by 2kL and assuming that the conductivity k is independent of temperature
gives

Q
dT
 r
 Constant  C1
2kL
dr
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INME 4032
which is a first-order ordinary differential equation for T(r) and can be integrated easily:
C
dT
 1
dr
r
T  C1 ln r  C2
Two boundary conditions are required to evaluate the two constants; these are
r  r1 :
T  T1
r  r2 :
T  T2
Substituting;
T1  C1 ln r1  C2
T2  C1 ln r2  C2
which are two algebraic equations for the unknowns C1 and C2. Subtracting the second
equation from the first:
T1  T2  C1 ln r1  C1 ln r2  C1 ln( r2 / r1 )
or
C1 
T1  T2
ln( r2 / r1 )
Using either of the two equations then gives
C 2  T1 
T1  T2
ln r1
ln( r2 / r1 )
Substituting back and rearranging gives the temperature distribution as
T1  T
ln( r / r1 )

T1  T2 ln( r2 / r1 )
which is a logarithmic variation, in contrast to the linear variation found for the plane

wall. The heat flow is Q  2kLC1 or:
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INME 4032

Q
2kL(T1  T2 )
ln( r 2 / r1 )
The Pin Fin
Simple pin fins, such as those used to cool electronic components, will be analyzed to
develop the essential concepts of fin theory. The first law is used to derivate the
governing differential equation, which, when solved subject to appropriate boundary
conditions, gives the temperature distribution along the fin.
Governing Equation and Boundary Conditions
Fig. 1.2 A pin fin showing the coordinate
system and an energy balance on a fin
element.
Consider the pin fin shown in Fig. 1.2. The cross sectional area is Ac=R2 where R is the
radius of the pin, and the perimeter P =2R. Both Ac and R are uniform, that is, they do
not vary along the fin in the x direction. The energy conservation principle is applied to
an element of the fin located between x and x + Δx. Heat can enter and leave the
element by conduction along the fin and can also be lost by convection from the surface
of the element to the ambient fluid at temperature Te. The surface area of the element is
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INME 4032
P Δx; thus.
qAc x  qAc xx  hc P x( T  Te )
Dividing by Δx and letting Δx  0 gives

d
( qAc )  hc P ( T  Te )  0
dx
For the pin fin, Ac is independent of x; using Fourier’s law q = –k dT/dx with k constant
gives
d 2T
kAc 2  hc P ( T  Te )  0
dx
which is a second order ordinary differential equation for T=T(x). Notice that modeling
the conduction along the fin as one-dimensional has caused the convective heat loss
from the sides of the fin itself, it is appropriate to take its base temperature as known;
that is,
T
x 0
 TB
At the other end, the fin losses heat by Newton’s law of cooling:
 Ac k
dT
dx
 Ac hc (T
xL
xL
 Te )
where the convective heat transfer coefficient here is, in general, different from the one
for the sides of the fin because the geometry is different. However, because the area of
the end, Ac, is small compared to the side area PL, the heat loss from the end is
correspondingly small and usually can be ignored. Then Equation becomes
dT
dx
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0
x L
7
INME 4032
Temperature Distribution
We will use the last equation for the second boundary condition as a compromise
between accuracy and simplicity of the result. For mathematical convenience, let
θ = T – Te and β2=hcP/ kAc; so
d 2
  2  0
2
dx
For β a constant, Equation has the solution
  C1e x  C 2 e  x
or
  B1 sinh x  B2 cosh x
The second form proves more convenient; thus, we have
T  Te  B1 sinh x  B2 cosh x
Using the two boundary conditions, we have two algebraic equations for the unknown
constants B1 and B2,
TB  Te  B1 sinh( 0)  B2 cosh( 0) ; B2  TB  Te
dT
dx
 B1 cosh L  B2 sinh L  0 ;
xL
B1  B2 tanh L
Substituting B1 and B2 and rearranging gives the temperature distribution as
T  Te cosh (L  x )

,
TB  Te
cosh L
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1/ 2
where
h P
   c 
 kA c 
INME 4032
Apparatus
There are three different experimental models to study the heat transfer by conduction:
1.
2.
3.
Linear Heat Conduction
Radial Heat Conduction
Extended surface heat transfer
1. Linear Heat Conduction Equipment:
Heat transfer module
Linear Heat Conduction Equipment
Fig. 1.3 Schematic Diagram showing the Heat
Conduction Equipment
Description of the Linear Heat Conduction Equipment:
The heat transfer module is cylindrical and mounted with its axis vertical to the base
plate. The heating section houses a 25mm diameter cylindrical brass section with a
nominally 60W (at 24 VDC) cartridge heater in the top end. The three fixed
thermocouples T1, T2, T3, positioned along the heated section, are at 15mm intervals.
The cooling section is also manufactured from 25mm diameter brass to match the
heated top section and this is cooled at its bottom end by water flowing through galleries
in the material. Three fixed thermocouples T6, T7, T8 are positioned along the cooled
section at 15mm intervals. There are four different specimens available to be placed
between the heated and the cooled sections. These specimens are the following:
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INME 4032
Brass Specimen:
30mm long, 25mm diameter fitted with two thermocouples T4, T5 at 15mm intervals
along the axis. With this specimen clamped between the heated and cooled sections, a
uniform 25mm diameter brass bar is formed with 8 uniformly spaced (15mm intervals)
thermocouples (T1 to T8).
Stainless Steel specimen
30mm long, 25mm diameter. No thermocouples fitted.
Aluminum Alloy specimen
30mm long, 25mm diameter. No thermocouples fitted.
Brass specimen With Reduced Diameter
30mm long, 13mm diameter. No thermocouples fitted.
NOTE: If the experimental session requires the use of two specimens, use first the
specimen with the heater at no more than 10V, then use the second specimen with the
heater at different volts according the instructions of your instructor. If more than 10V
are used for the first specimen, the second one will not fit in the shallow shoulders of the
heated and cooled sections because of the thermal expansion.
2. Radial Heat Conduction Equipment:
Radial Heat Conduction Equipment
Fig. 1.4 Schematic Diagram showing the Radial
Conduction Equipment
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INME 4032
Description of the Radial Heat Conduction Equipment:
The radial heat conduction equipment allows investigate the basic laws of heat
conduction through a cylindrical solid. The heat transfer module comprises an insulated
solid disc of brass (110mm diameter) with a solid copper core (14mm diameter), and an
electric heater at the center. The brass disc is water cooled around its circumference.
Six fixed thermocouples T1, T2,…, T6 are located at increasing radio from the heated
center.
Useful information:
Heated disc
Material: brass
Outside diameter: 0.110m
Diameter of heated copper core: 0.014m
Thickness of disc: 0.0032m
Radial position of thermocouples:
T1 = 0.007m
T4 = 0.030m
T2 = 0.010m
T5 = 0.040m
T3 = 0.020m
T6 = 0.050m
3. Extended Surface Heat Transfer
Description of the Extend Surface Heat Transfer Equipment:
The extended surface heat transfer equipment allows investigation of one-dimensional
conduction from a fin. A small diameter metal rod is heated at one end and the
remaining exposed length is allowed to cool by natural convection and radiation.
The equipment comprises a 10mm diameter brass rod of approximately 350mm
effective length mounted horizontally. Eight thermocouples are located at 50mm
intervals along the rod to record the surface temperature. The rod is coated with a matt
black paint in order to provide a constant radiant emissivity close to 1.
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INME 4032
T8
Extended Surface Heat Transfer Equipment
T7
T6
T5
T4
T3
T2
T1
Fig. 1.3 Schematic Diagram showing the
Extended Surface Heat Transfer Equipment
Useful information
Heated Rod Diameter D = 0.01m
Heated Rod Effective Length L = 0.35m
Thermal Conductivity of the Heated Rod Material k = 121 W/mK
Procedure
Allow the system to reach stability, and take readings and make adjustments as
instructed in the individual procedures for each experiment. Record the temperatures,
voltage and current, with these data; calculate the power of the heat source. Repeat the
lectures three times to assure that the system has reached stability. You should
investigate about how to use your information in order to calculate the thermal
conductivity coefficient in all the cases and you must graph your data to study the
behavior of temperature against distance from the heat source. All this experimental
strategy must be presented in the Task Discussion Meeting.
References
P. A. Hilton LTD. “Experimental Operating and Maintenance Manual”. Heat Transfer
Service Unit. November, 2000.
A. Mills, “Basic Heat and Mass Transfer”, Richard D. Irwin INC, Los Angeles, USA,
1995.
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INME 4032