Table of Contents
Pr incip le
3
Object ive
3
Background
3
? Fourier’s la w of Heat Conduct ion
3
? The Pin F in
6
? G overning Equat ion and Boundary Condit ions
6
? Temperat ure Dist r ibut ion
8
Apparat us
9
1) Linear heat conduct ion equipment :
10
• Descript ion of t he Linear Heat Conduct ion Equipment
10
2) Radial heat conduct ion equipment :
11
• Descript ion of t he Radial Heat Conduct ion Equipment
12
• Usef ul inf ormat ion:
12
3) Ext ended Surf ace Heat Transf er
12
• Descript ion of t he Ext ended Surf ace Heat Transf er Equipment
12
• Usef ul inf ormat ion:
13
Procedure
13
Ref erences
13
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INME 4032
Uni versi ty o f Puer to Rico
Mayagüez Campus
Depar tmen t of Mechanical Engineering
INME 4032 - LABORATORY II
Spring 2004
I nst ructor: G uillermo Ara ya
Experimen t 1: Conduction Heat Transfer Analysi s
Principle
A heat source placed in a mat eria l causes t emperat ure changes due t o heat conduct ion.
The relat ionsh ip bet ween t emperat ure and t he dist ance f rom t he heat source must be
linear af t er some t ime in t he case of linear heat conduct ion and it must have a
logarit hm ic dist r ibut ion in t he case of radial heat conduct ion.
Objecti ve
The exper iment demonst rat es heat conduct ion in three dif f erent experiment al models . I t
allo ws us t o obt ain e xper iment ally t he coef f icient of t hermal conduct ivit y of some
unknown mat erials and in t his way, t o underst and t he f act ors and paramet ers t hat af f ect
t he rat es of heat t ransf er.
Background
Four ier’s la w of Heat Conduct ion
A general st at ement of t he Fourier’s law is : The conduct ion heat f lu x in a specif ied
direct ion equals t he negat ive of t he product of t he medium t hermal conduct ivit y and t he
t emperat ure derivat ive in t hat direct ion. I n Cart esian coordinat es, wit h t emperat ure
var ying in t he x direct ion on ly,
q = −k
dT
dx
I n cylindr ical or spher ical coo rdinat es, wit h t emperat ure va r ying in t he r d irect ion on ly,
q = −k
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dT
dr
INME 4032
Fig 1.1. A c ylindr ical shell showing an
element al cont rol vo lume f or applicat ion of t he
energy conser vat ion p rinc iple .
Figure 1.1 sho ws a c ylindrica l she ll of lengt h L, wit h inner rad ius r1 and out er radius r2.
The inner surf ace is ma int ained at t emperat ure T1 and t he out er surf ace is ma int ained at
t emperat ure T2. An e lement al cont ro l vo lume is locat ed bet ween radii r and r+? r . I f
⋅
t emperat ures are unchanging in t ime and Q v = 0 , t he energy conser vat ion princ iple
requires t hat t he heat f low across t he f ace at r equal t hat at t he f ace r+? r
⋅
⋅
Q r = Q r +∆r
That is,
⋅
Q r = Constant, independent of r
Using Fourier ’s law
⋅
dT 

Q r = Aq = 2πrL − k

dr 

Dividing b y 2π kL and assuming t hat the conduct ivit y k is independent of t emperat ure
gives
⋅
Q
dT
= −r
= Constant = C1
2πkL
dr
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INME 4032
which is a f irst -order ord inar y d if f erent ial equat ion f or T(r) and can be int egrat ed easily:
C
dT
=− 1
dr
r
T = −C1 ln r + C2
Two boundar y condit ions are required t o e va luat e t he t wo const ant s; t hese are
r = r1 :
T = T1
r = r2 :
T = T2
Subst it ut ing;
T1 = −C1 ln r1 + C2
T2 = −C1 ln r2 + C 2
which are t wo a lgebraic equat ions f or t he unknowns C1 and C2. Subt ract ing t he second
equat ion f rom t he f irst :
T1 − T2 = −C1 ln r1 + C1 ln r2 = C1 ln( r2 / r1 )
or
C1 =
T1 − T2
ln( r2 / r1 )
Using eit her of t he t wo equat ions t hen gives
C 2 = T1 +
T1 − T2
ln r1
ln( r2 / r1 )
Subst it ut ing back and rearranging g ives t he t emperat ure dist ribut ion as
T1 − T
ln( r / r1 )
=
T1 − T2 ln( r2 / r1 )
which is a logar it hmic var iat ion, in cont rast t o t he linear va riat ion f ound f or t he plane
⋅
wall. The heat f low is Q = 2πkLC1 or:
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INME 4032
⋅
Q=
2πkL(T1 − T2 )
ln( r 2 / r1 )
The Pin F in
Simple pin f ins, such as t hose used t o cool elect ronic component s, will be analyzed t o
develop t he essent ial concept s of f in t heory. The f irst law is used t o derivat e t he
govern ing dif f erent ial equat ion, which, when solved subject t o appropriat e boundary
condit ions, g ives t he t emperat ure dist ribut ion along t he f in.
G ove rning Equat ion and Boundary Cond it ions
Fig. 1.2 A pin f in sho wing t he coordinat e
syst em and an energ y ba lance on a f in
element .
Consider t he pin f in shown in F ig. 1 .2. The cross sect ional area is Ac =π R2 where R is t he
radius of t he pin, and t he perimet er P =2π R. Bot h Ac and R are unif orm, t hat is, t hey do
not var y a long t he f in in t he x direct ion . The energ y conser vat ion p rinc iple is applied t o
an element of t he f in locat ed bet ween x and x + ? x. Heat can ent er and lea ve t he
element by conduct ion along t he f in and can also be lost by convect ion f rom t he surf ace
of t he element t o t he ambient f luid at t emperat ure Te. The surf ace area of t he element is
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INME 4032
P ? x; t hus.
qAc x − qAc x +∆x − hc P ∆x( T − Te )
Dividing b y ? x and let t ing ? x à 0 g ives
−
d
( qAc )− hc P( T −Te ) = 0
dx
For t he pin f in, Ac is independent of x; using Fourier’s law q = – k dT/dx wit h k const ant
gives
d 2T
kAc 2 − hc P ( T − Te ) = 0
dx
which is a second order ordinar y d if f erent ial equat ion f or T=T( x). Not ice t hat modeling
t he conduct ion along t he f in as one-dimensional has caused t he convect ive heat loss
f rom t he sides of the f in it self , it is appropriat e t o t ake it s base t emperat ure as known;
t hat is,
T
x =0
= TB
At t he ot her end, t he f in losses heat by Newt on’s law of cooling:
− Ac k
dT
dx
= Ac hc (T
x =L
x= L
− Te )
where t he convect ive heat t ransf er coeff icient here is, in general, dif f erent f rom t he one
f or t he sides of t he f in because t he geomet ry is dif f erent . However , because t he area of
t he end, Ac , is small compa red t o t he side area PL, t he heat loss f rom t he end is
correspondingly s mall and usua lly can be ignored . Then Equat ion becomes
dT
dx
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≅0
x =L
7
INME 4032
Temperat ure Dist r ibut ion
We will use t he last equat ion f or t he second boundary condit ion as a compro mise
bet ween accuracy and simplic it y of the result . For mat hemat ical convenience, let
? = T – Te and ß2=hc P/ kAc ; so
d 2θ
− β 2θ = 0
2
dx
For ß a const ant , Equat ion has t he solut ion
θ = C1 e β x + C 2 e − β x
or
θ = B1 sinh βx + B2 cosh βx
The second f orm pro ves more con venient ; t hus, we ha ve
T − Te = B1 sinh βx + B 2 cosh β x
Using t he t wo boundary condit ions, we ha ve t wo algebra ic equat ions f or t he unknown
const ant s B1 and B2,
TB − Te = B1 sinh( 0) + B2 cosh(0) ; B2 = TB − Te
dT
dx
= β B1 cosh β L + βB 2 sinh β L = 0 ;
x= L
B1 = − B2 tanh βL
Subst it ut ing B1 and B2 and rearranging gives t he t emperat ure dist ribut ion as
T − Te coshβ( L − x )
=
,
TB − Te
coshβL
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1/ 2
where
h P
β =  c 
 kA c 
INME 4032
Apparatus
There are t hree d if f erent experiment al models t o st udy t he heat t ransf er by conduct ion:
1. Linear Heat Conduct ion
2. Radial Heat Conduct ion
3. Ext ended surf ace heat t ransf er
1. L inear Heat Conduct ion Equipment :
Heat transfer module
Linear Heat Conduction Equipment
Fig. 1.3 Schematic Diagram showing the Heat
Conduction Equipment
Descript ion of t he Linear Heat Conduct ion Equip ment :
The heat t ransf er module is cylindr ica l and mount ed wit h it s axis vert ical t o t he base
plat e. The heat ing sect ion houses a 25 mm d iamet er c ylindr ica l brass sect ion wit h a
nominally 60W (at 24 VDC) cart ridge heat er in t he t op end. The t hree f ixed
t hermocouples T1 , T2, T3 , posit ioned along t he heat ed sect ion, are at 15mm int erva ls.
The cooling sect ion is a lso manuf act ured f rom 25mm dia met er brass t o mat ch t he
heat ed t op sect ion and t his is coo led at it s bot t om end by wat er f lo wing t hrough galle ries
in t he mat erial. Three f ixed t hermocouples T6, T7, T8 are posit ioned along t he cooled
sect ion at 15mm int ervals. There are f our dif f erent specimens available t o be placed
bet ween t he heat ed and t he cooled sect ions. These specimens are t he f ollo wing :
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INME 4032
Brass Specimen:
30mm long, 25m m dia met er f it t ed wit h t wo t hermocouples T4, T5 at 15mm int er vals
along t he axis. W it h t his specimen c lamped bet ween t he heat ed and cooled sect ions, a
unif orm 25mm diamet er brass bar is f ormed wit h 8 unif ormly spaced (15mm int erva ls)
t hermocouples (T1 t o T8) .
St ainless St eel specimen
30mm long , 25mm diamet er. No t hermocouples f it t ed.
Aluminum Alloy specimen
30mm long , 25mm diamet er. No t hermocouples f it t ed.
Brass specimen W it h Reduced Diamet er
30mm long , 13mm diamet er. No t hermocouples f it t ed.
NO TE: If t he experiment al session requires t he use of t wo specimens, use f irst t he
specimen wit h t he heat er at no more t han 10V, t hen use t he second specimen wit h t he
heat er at dif ferent volt s according t he inst ruct ions of your inst ruct or. I f more t han 10V
are used f or t he f irst specimen, t he second one will not f it in t he shallo w shou lders of t he
heat ed and cooled sect ions because of the t hermal expans ion.
2. Radia l Heat Conduct ion Equipment :
Radial Heat Conduction Equipment
Fig. 1.4 Schematic Diagram showing the Radial
Conduction Equipment
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INME 4032
Descript ion of t he Radial Heat Conduct ion Equip ment :
The rad ial heat conduct ion equipment allows in vest igat e t he basic la ws of heat
conduct ion t hrough a cylindrica l solid. The heat t ransf er module comprises an insulat ed
solid disc of brass (110mm diamet er) wit h a solid copper core (14mm diamet er), and an
elect ric heat er at t he cent er. The brass disc is wat er cooled around it s circumf erence.
Six f ixed t hermocouples T1 , T2, …, T6 are locat ed at increas ing rad io f rom t he heat ed
cent er.
Usef ul inf ormat ion :
Heat ed disc
Mat erial: brass
Out side diamet er: 0.110m
Diamet er of heat ed copper core: 0.014m
Thickness of d isc: 0 .0032m
Radial posit ion of t hermocouples :
T1 = 0.007m
T4 = 0.030m
T2 = 0.010m
T5 = 0.040m
T3 = 0.020m
T6 = 0.050m
3. E xt ended Surf ace Heat Transf er
Descript ion of t he Ext end Surf ace Heat Transf er Equipment :
The ext ended surf ace heat t ransf er equipment allows in vest igat ion of one-dimensional
conduct ion f rom a f in. A small diamet er met al rod is heat ed at one end and t he
remain ing e xposed lengt h is a llowed t o cool b y nat ural con vect ion and radiat ion .
The equipment comp rises a 10mm d iamet er brass rod of appro ximat ely 350mm
ef f ect ive lengt h mount ed horizont ally. Eight t hermocouples are locat ed at 50mm
int erva ls along t he rod t o record t he surf ace temperat ure. The rod is coat ed wit h a mat t
black paint in order t o pro vide a const ant radiant emissivit y close t o 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
Usef ul inf ormat ion
Heat ed Rod Diamet er D = 0.01m
Heat ed Rod Eff ect ive Lengt h L = 0 .35m
Therma l Conduct ivit y of t he Heat ed Rod Mat erial k = 121 W/m K
Procedure
Allow t he syst em t o reach st abilit y, and t ake read ings and make ad just ment s as
inst ruct ed in t he individual procedures f or each e xper iment . Record t he t emperat ures,
vo lt age and current , wit h t hese dat a; calculat e t he power of t he heat source. Repeat t he
lect ures t hree t imes t o assure t hat t he syst em has reached st abilit y. You should
in vest igat e about how t o use your inf ormat ion in order t o ca lculat e t he t hermal
conduct ivit y coef f icient in all t he cases and you must graph your dat a t o st udy t he
behavior of t emperat ure against dist ance f rom t he heat source. A ll t his e xper iment al
st rat egy must be present ed in t he Task Discuss ion Meet ing.
Re ferences
P. A . Hilt on LTD. “ Expe riment al Operat ing and Maint enance Manual” . Heat Transf er
Service Unit . Nove mber, 2000.
A. Mills, “ Basic Heat and Mass Transf er” , Richard D. I rwin I NC, Los Angeles, USA,
1995.
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INME 4032