Ht 02 Conduction

7/23/2019 Ht 02 Conduction

1/37

HEAT TRANSFERCONDUCTION HEAT TRANSFER

gjv


2/37

Recap

Heat fows rom hot to cold regions (Third

Law)

Ease o movement o electrons in metals thereason or greater distribution o energy

compared to other substances and explainsthe relationship between thermal andelectrical conductivities

onducting medium necessary or conduction


3/37

!asic law o conduction

Temperature gradient leads to energy

transer

Heat transer per unit area (heat fux)proportional to normal temperaturegradient

"roportionality constant

dx

dT

dA

dQ


4/37

#ourier$s Law o heat conduction

#or steady state one dimensional heat fow%the rate o heat fow in thexdirection normal

to the surace area% is directly proportional tothe temperature gradient% the area o fow andinversely proportional to the distance&

dx

dTk

dA

dQ=


5/37

#ourier$s Law

'ey eatures o the law

ot an expression that may be derived romrst principle

* generali+ation based on experimentalevidence

,enes the important material property othermal conductivity


6/37

-ne dimensional steady fow

x

.

x

/

x

0

T.

T/

T

dx

dT

1sothermal

surace

Temperature

prole


7/37

Assumptions

Uniform temperatures over the surface perpendicular tox

which is the direction of heat conduction ( isothermal

surface)

Steady flow ( Temperature does not vary with time)

Rate of heat flow constant

Consider an element of thickness dxwith surfaces at

temperature of T and T+dT


8/37

#rom #ourier$sLaw2

Thermal conductivity o

material

x

TTkAQ

kAdtQdx

dx

dTkAQ

T

T

21

1

2

1

=

=

=


9/37

Thermal Conductivity

k% though not a unction o temperaturegradient% is a unction o temperature androm experimental data2

( )

( )A

qdxdTTkkkdT

grearranginand

dx

dTkAqcombining

Tkkk

=+=

=+=

!

!

1

1


10/37


( ) ( )

( )

=

=

++=

2

1

2

1

2

1

21

21!21

sin

21

x

xa

x

x

T

T

A

dx

qTTk

Toffunctionlinearaiskceand

A

dxQ

TTkkTTkdT


11/37


#or a nonlinear kthe mean value is $iven %y"

Thermal conductivities o metalsare usually very high

on3metallic solids and li4uids

=2

112

1 T

Tm kdT

TTk


12/37

The Solid State

5odern view o solids highlights ree electrons

and atomic lattice structure

Thermal energy determined by Latticevibrations which are additive2

k = ke+kl but we 6now that

ke= 1/e electrical resistivity

For pure metals with low e ke>>kl


13/37

The Solid State

Hence contribution o kl to kis negligible

#or alloys where eis large contribution o klto 6 is no longer negligible

#or non3metals% kis determined primarily by6l and depends on the re4uency o

interaction between atoms o the lattice

hrystalline% well ordered substances such asdiamond 7 4uart+ have high kvalues

compared with amorphous substances (glass)


14/37

The fluid state

Larger intermolecular spacing and greater random

motion lead to lower thermal energy transport

Thermal conductivity o gases and li4uids moresimilar than solids

'inetic theory o gases gives a good account o

their thermal conductivities

Thermal conductivity o gases is directlyproportional to the number o particles per unitvolume% mean molecular speed and mean reepath (*verage distance travelled by a molecule

beore a collision)


15/37

#luids

k 8 n c 9

Thermal conductivities o gasesincrease with increasing temperatureand decreasing molecular weight

since c increases accordingly

Thermal conductivities o gases are

independent o pressure since n 7 are directly and indirectly proportionalto gas pressure respectively


16/37

Thermal conductivity

onductivity o alloys less than the puremetals

:ases have very low conductivities and or

ideal gases k is proportional to meanmolecular velocity% mean ree path and molarheat capacity

mKWtyconductiviThermalk

diametercollisionEffective

weightolecular

KeTem!eraturT

Tk

gasesmonoatomic"or

&'

'

'

()2*

"

2&1

2

==

==

=


17/37

+i,uid metals

"hysical mechanisms o the

thermal conductivity o li4uidmetals are still not wellunderstood

Li4uid metals are commonlyused in high heat fux applicationsuch as in nuclear power plants&

Li4uid metals thermal


18/37

Temperature dependence of conductivity

Thermal insulation comprise low conductivity materials

which when com%ined achieve even lower system thermal

conductivity

#i%er' powder and flake type insulation have solidmaterial finely dispersed in air spaces

The nature and volumetric fraction of the solid to void

ratio characteri-es the thermal conductivity of theinsulation


19/37

.nsulation

Cellular insulation / hollow spaces or small voids are

sealed from each other and formed %y fusion or %ondin$of solid material in a ri$id matri0

#oam systems (plastic or $lass)

Reflective insulation / thin sheets of hi$h reflectivity foil

spaced to reflect radiant ener$y

vacuation of air from voids reduce effective thermalconductivity


20/37

Thermal conductivity of materials oC

5etals

;ilver?m'opper @AB >?m'

*luminium /=/ >?m'

1ron C@ >?m'

Lead @B >?m'

hrome3nic6el steel (.ADr% ADi) .&@ >?m'

on3metallic solids

,iamond /@== >?m'5arble /&=A3/&F?m'

>ater =&BB

Lube oil% ;*E B= =&.


22/37

0ample

3ne face of a copper plate * cm thick is maintained at

4* oC and the other face is kept at 1* oC* 5ow much

heat is transferred throu$h the plateG

6u @C=&= >?m' I /B=&=o

Estimate the heat loss per m/ through a bric6wall =&B m thic6 when the inner surace is at

?m' I @B=&=o


23/37

Thermo physical properties

1mportant ant properties or heattranser calculations2

'inematic viscosity% J (m/?s),ensity% K (6g?m@)

Heat capacity% cp% cv(?6g')

Thermal diusivity% 8 (m/?s)


24/37

Thermal diffusivity

The ability to conduct thermal energy

relative to the ability to store it2

5aterials with large 8 respond4uic6ly to changes in their thermalenvironment

*ccuracy o engineering calculationsdepend on the accuracy odetermining the thermophysical

properties

#

c

k

=


25/37

0ample

Mse tables to calculate 8 or theollowing2

"ure *luminium I @==' 7 C== '

;ilicon arbide I .=== '

"aran I @== '


26/37

Steadystate conduction

Heat fow into 7 rom tan6Tan6wall

rerigerant

*ir

!oiling H/-

*ir

insulation

1nsulation

x x

TT

onsider a fat walled insulated tan6 containing a rerigerantat 3.=&= o with outside air at /A&= o


27/37

Steady flow conduction

#or xdistance rom the hot side2

Thermalresistance

( ) ( )

$

T

%

Tk

xx

TTk

A

Q

TTkxxA

Q

dTkdxA

Q

kdTdxAQ

T

T

x

x

=

=

=

=

=

=

12

21

2112

2

1

2

1


28/37

Compound resistance in series

onsider a fat wall with three layers% *%! 7

Let thic6nesses be !*% !!7 !and average

thermal conductivities be 6*% 6!7 6or the

layers respectively& Temperature

drop

T

x

NT* NT! NT

O*O! O

!*

!!!

0

*s

C d i i i


29/37


++=

++=

===

=

++=

&

&

%

%

A

A

s

sc

cc

s%

%%

sA

AA

sc

ccc

s%

%%%

sA

AAA

s

&%A

k

%

k

%

k

%

A

Q

T

Ak%Q

Ak%Q

Ak%QT

Ak

%QT

Ak

%QT

Ak

%QT

then%

Tk

A

Qce

TTTT

6

sin

C d i t i i


30/37


$

T

A

Q

$$$T

k%

k%

k%

TAQ

s

&%A

&

&

%

%

A

As

=

++=++=


31/37

0ample

*n exterior wall o a house consists o a m3. o3.Q and a.&B cm layer o gypsum plaster Pk 0"#> m3.

o3.Q& >hat thic6ness o loosely pac6ed roc63wool Pk=00$% > m3. o3.Q insulation should beadded to reduce the heat loss (or gain)through the wall by A=&= DG


32/37

Radial Systems

ylindrical shape (Thic6 walled tube)

r.

r/

r

dr

T1T&

T&

T

1 TdT

T+dT

Assumptions"

.nternal 6 e0ternal

temperatures areconstant

Area e0posed to heatflow proportional to

the radius


33/37

Thic6 walled tube

=

=

=

=

==

12

21

1

2

21

1

2

212

ln

2

ln

2

2

2

2

1

2

1

rr

TTlkr

r

r

TTlk

r

r

TTlkq

dTlkrdrq

dr

dT

rlkdr

dT

kAq

mr

T

T

r

rr

r


34/37

Thick walled tu%e

12

21

12

21

1

2

12

2

ln

rr

TTlkrq

rr

TTkAq

rr

rr

r

ar

mr

m

=

=

=


35/37

0ample

* tube o =&= mm -, is insulated with a B=&=mm layer o silica oam P6=&=BB >?moQ anda ?moQ&alculate the heat loss per unit length o pipe

given that the temperature at the outersurace o the pipe is .B=&= o while theouter surace o the cor6 is 6ept at @=&= o &


36/37

Conduction throu$h a spherical shell

T1

r.

r/

rrRdr

dr

T&

Sery important or applications such as heat transer in fuidi+ed beds %rotary 6ilns 7 spray dryers where conduction ta6es place through a

stationary fuid to a spherical particle or droplet o radius r

( )

21

21

2

2

114

4

4

rr

TT

kq

dTkr

drq

dr

dTrk

dr

dTkAq

=

=

==

; h i l h ll


37/37

;pherical shells

7hen T'(T) is spread over lar$e distances so that r)8 9And T' is the temperature of the surface of the drop then:

( )

( ) h

TTr

qrwhere

*uk

hd

or

kTTr

qr

=

==

=

21

2

21

2

4

2

14

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Ht 02 Conduction