lect - 3 basic equations of one-dimensional, two-dimensional and three-dimensional conduction.pptx

8/17/2019 Lect - 3 Basic equations of one-dimensional, two-dimensional and three-dimensional conduction.pptx

1/17

Heat ConductionDr. Senthilmurugan S. Department of Chemical Engineering IIT Guwahati - Part 3

Basic equations of one-dimensional, two-dimensional and three-dimensional

conduction


2/17

5/12/16 | Slide 2

O!ecti"es

#eri"e asic equations for conduction heat transfer

One dimensional

$wo-dimensional

$hree-dimensional conduction

Boundary and initial conditions to solve asicequations for conduction heat transfer


3/17

5/12/16 | Slide %

One #imensional Heat Conduction &quation

'et is consider consider the (eneral

case where the tem)erature ma* e

chan(in( with time and heat sources

ma* e )resent within the od*+

or the element of thicness d., the

followin( ener(* alance ma* e made &ner(* conducted in left face 0 heat

(enerated within element chan(e in

internal ener(* 0 ener(* conducted out

ri(ht face

nstead* State and Constant Heat $ransfer 3rea

&lemental "olume for one-dimensional

heat conduction anal*sis

&ner(* in left face

&ner(* (enerated within element

= energy generated per unit volume, W / m3


4/17

5/12/16 | Slide 4

One #imensional Heat Conduction &quation

Chan(e in internal ener(*

&ner(* out ri(ht face

Cominin( the relations ao"e (i"es

c s)ecific heat of material, /( °C

densit*, (/m%

nstead* State and Constant Heat $ransfer 3rea

one-dimensional heat-conduction equation

7hen constant

$a*lor series e.)ansion 8first two term9


5/17

5/12/16 | Slide 5

$hree - #imensional Heat Conduction &quation

$he ener(* alance *ields

or constant thermal conducti"it*,


6/17

5/12/16 | Slide 6

$hree - #imensional Heat Conduction &quationor constant thermal conducti"it*,

0

7hen constant


7/17

5/12/16 | Slide :

$hree - #imensional Heat Conduction &quationC*lindrical Coordinates

.


8/17

5/12/16 | Slide ;

$hree - #imensional Heat Conduction &quationS)herical Coordinates

.


9/17

5/12/16 | Slide <

Heat Conduction &quation

Stead*-state one-dimensional heat flow

8no heat (eneration9


with heat sources

$wo-dimensional stead*-state

conduction without heat sources


in c*lindrical coordinates 8no heat

(eneration9

=ulti)le cases

q constant


10/17

5/12/16 | Slide 1>

Boundar* conditions

$o determine tem)erature distriution

alon( coordinates, one has to sol"e

a))ro)riate heat conduction equation+

Stead* state $hree dimensional Heat

Conduction &quation is sol"ed alon( with

oundar* conditions

$ransient condition $hree dimensional

Heat Conduction &quation is sol"ed alon(

?nitial condition of the s*stem and

oundar* condition+

Since heat equation is second order in

s)atial coordinates, two oundar*conditions must e e.)ressed for each

coordinates needed to e descried for

(i"en s*stem+

One initial condition is must @ first order

with res)ect to time de)ended

?n (eneral three ind of oundar*

conditions commonl* encountered in heat

transfer a))lications

Dirichlet condition $he condition are

s)ecified at the surface+

$8>,t9$s 8constant surfacetem)erature9

?nterface Boundar* Conditions

eumann condition Constant surface

heat flu.

inite heat flu.

3diaatic or insulated surface

$hermal s*mmetr*

Third !ind of condition" &.istence of

con"ection, radiation or comined mode

of heat transfer at surface

Sol"e asic equations for conduction heat transfer


11/17

5/12/16 | Slide 11


12/17

5/12/16 | Slide 12

Aariation in $hermal Conducti"it*

7hen the "ariation of thermal conducti"it*

with tem)erature 8$9 is nown, the

a"era(e "alue of the thermal conducti"it*

in the tem)erature ran(e etween $1 and

$2 can e determined from

$his relation is ased on the requirement

that the rate of heat transfer throu(h a

medium with constant a"era(e thermalconducti"it* a"( equals the rate of heat

transfer throu(h the same medium with

"ariale conducti"it* 8$9+

7hen thermal conducti"it* is constant

then a"(

7hen 8$9 is linear function of

tem)erature

is called the temperaturecoefcient o thermalconductivity

$hen the a"era(e thermal conducti"it*

in this case is equal to the thermalconducti"it* "alue at the a"era(e

tem)erature+

=


13/17

5/12/16 | Slide 1%

$he "ariation of tem)erature in a )lane wall durin(stead* one-dimensional heat conduction for the casesof constant and "ariale thermal conducti"it*

7e ha"e mentioned earlier that in a

)lane wall the tem)erature "aries

linearl* durin( stead* one-dimensional

heat conduction when the thermal

conducti"it* is constant+

But this is no lon(er the case when thethermal conducti"it* chan(es with

tem)erature, e"en linearl*


14/17

5/12/16 | Slide 14

Stead* State Heat Conduction

?f the s*stem is in a stead* state, i+e+, if

the tem)erature does not chan(e with

time, then the )rolem is a sim)le one,

and we need onl* inte(rate ouriers law

of heat conduction equation and

sustitute the a))ro)riate "alues tosol"e for the desired quantit*+

7hen is constant

One #imension Constant 3rea


15/17

5/12/16 | Slide 15

Stead* State Heat Conduction

?f the s*stem is in a stead* state, i+e+, if

the tem)erature does not chan(e with

time, then the )rolem is a sim)le one,

and we need onl* inte(rate ouriers law

of heat conduction equation and

sustitute the a))ro)riate "alues tosol"e for the desired quantit*+

7hen f8$9 o810β$9

One #imension Constant 3rea


16/17

5/12/16 | Slide 16

?m)ortance on includin( as function of tem)erature

DSoure htt)//www+en(ineerin(toolo.+com/thermal-conducti"it*-dE42 5> 1>> 15> 2>> 25>>

5>

1>>

15>

2>>

25>

%>>

f8.9 >+2%. 0 1.F2 - >+>6+41

$em)erature C

$hermal Conducti"it* w/m2 C

3luminium $hermal conducti"it*D 3luminiumBar

$12>> °C

$2 5> °C

Ga"(22:+5 w/m °C 8 2)oint9

3ssum)tion is constant

3ssum)tion "aries with tem)erature

first order )ol*nomial

> 1+>>115%. 1m

%4125 w/m2 1> I de"iation from 2nd order

= %%5>> w/m2 , f8$9 1st order )ol*nomial

= %:


17/17

lect - 3 basic equations of one-dimensional, two-dimensional and three-dimensional conduction.pptx

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