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6.5 Natural Convection in Enclosures

1

Chapter 6: Natural Convection

Advanced Heat and Mass Transfer by Amir a!hri" #u$en %han!" and &ohn '. Ho$ell

Enclosures are finite spaces bounded by walls and filled

with fluid. Natural convection in enclosures, also known

as internal convection, takes place in rooms and

buildings, furnaces, cooling towers, as well as electroniccooling systems. Internal natural convection is different

from the cases of external convection, where a heated or

cooled wall is in contact with the quiescent fluid and the

boundary layer can be developed without any restriction.Internal convection usually cannot be treated using

simple boundary layer theory because the entire fluid in

the enclosure engages to the convection.

6.5 Natural Convection in

Enclosures


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i!ure 6.()Different configuration of natural convection in enclosures

TH TC

L

H

(a)

L

HTH TC

(b)

TH

TC

HL

L

HTH TC

L H

D

TH TC

Di

Do

Di

Do

(a) shallow enclosure (b) tall enclosure (c) inclined enclosure

(d) enclosure with vertical partitions (e) concentric annulus

(f) box enclosure (g) truncated annular enclosure


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Twodimensional natural convection in a rectangular

enclosure with two differentially heated sides and

insulated top and bottom surfaces !"ig. #.$%& will be

considered. 'ssumed to be Newtonian and incompressible.

Initially at a uniform temperature of (ero.

't time (ero the two sides are instantaneously heated

and cooled to and , respectively. The transient behavior of the system during the

establishment of the natural convection is the sub)ect of

analysis

#.*.$ +cale 'nalysis

/ 2T / 2T


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i!ure 6.(*Two-dimensional natural convection in rectangular enclosure.

x, u

y,

0

0L

H

T=-T/2T=+T/2

g

T

Thermal boundar laer

!low circulation direction

T=T


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It is assumed that the fluid is singlecomponent and that

there is no internal heat generation in the fluid.Therefore, the governing equation for this internal

convection problem can be obtained by simplifying eqs.

!#.%&, !#.$& and !#.$-&

!#./0*&

!#./0#&

!#./01&

!#./0%&

"u v

x y

+ = 2 2

2 2

#u u u p u uu v

t x y x x y

+ + = + +

2 2

"2 2

#

$# ( )%

v v v p v v

u v g T T t x y y x y

+ + = + +

2 2

2 2

T T T T T u v

t x y x y

+ + = +


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Immediately after imposing of the temperature difference,

the fluid is still motionless, hence the energy equation!#./0%& reflects the balance between the thermal inertia

and the conduction in the fluid. The scales of the two

terms enclosed in the parentheses on the righthand side

of eq. !#./0%& are and , respectively. +ince ,

one can conclude that . Thebalance of scales for eq. !#./0%& then becomes

Thus, the scale of the thermal boundary layer thickness

becomes

!#./02&

2/ tT 2/T H

t H = 2 2 2 2/ /T y T x =

2

t

T T

t

:

#/ 2& ( )T t


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To estimate the scale of the velocity, one can combine eqs.

!#./0#& and !#./01& by eliminating the pressure to obtain

!#./$0&where the lefthand side represents the inertia terms, and

the righthand side represents the viscosity and

buoyancy terms. The scales of these three effects are

shown below

!#./$$&

v v v u u uu v u v

x t x y y t x y

+ + + +

2 2 2 2

2 2 2 2

v v u u T g

x x y y x y x

= + + +

'

nertia iscosit *uoanc

T T T

v v g T

t


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To examine the relative strength of each effect, one can

divide the above expression by the scale of viscosityeffect to obtain

where eq. !#./02& was used to simplify the inertia term. "or

the fluid with 3r4$, the momentum balance at requires a

balance between the viscosity and buoyancy terms

2

nertia iscosit *uoanc

# #

+r

Tg T

v

2

# & Tg Tv


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+ubstituting eq. !#./02& into the above expression and

rearranging the resultant expression, the scale of vertical

velocity at the initiation of the natural convection is

obtained as following

!#./$/&

's time increases, the effect of the inertia term in eq.!1./0%& weakens, hence the effect of advection becomes

stronger. This trend continues until a final time, tf, when

the energy balance requires balance between the

advection and conduction terms, i.e.,

&g T t

v

2

,

& &T f f

T T TvH t


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Thus, the scale of tfbecomes

!#./$&

The thermal boundary layer thickness at time tfis

!#./$-&

't time tf, natural convection in the rectangular enclosure

reaches steadystate and the thickness of the thermal

boundary layer no longer increases with time.

#/ 2

&f Htg T

#/ 2 #/ -

, & ( ) & aT f f H t H


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The wall )et thickness increases with time until t5 tf,when

the maximum wall )et thickness, v,f, is reached !see "ig.#.$2&. 6utside the thermal boundary layer, the buoyancy

force is absent and the thickness of the wall )et can be

determined by balancing the inertia and viscosity terms

in eq. !#./$0&

which can be rearranged to obtain

!#./$*&

"or t4 tf, steadystate has been reached, and the wall )etthickness is related to the thermal boundary layer

thickness by .

'&v v

v v

t

#/ 2 #/ 2& ( ) & +r v T

t

#/ 2

, ,& +rv f T f


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i!ure 6.(+Two-laer structure near the heated wall.

0 x

v

T

0 xt ,t f

t ,v f


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+imilarly, the condition to have distinct vertical wall )ets or

momentum boundary layers is , or equivalently

!#./$#&

7hen the vertical wall )et encounters the hori(ontal wall, it

will turn to the hori(ontal direction and become ahori(ontal )et. This hori(ontal )et will contribute to the

convective heat transfer from the heated wall to the

cooled wall

8onsidering eqs. !#./$/& and !#./$-&, the above scale of

convective heat transfer becomes

,v fL


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7hen a warm )et is formed at the top and a cold )et is

formed at the bottom, there will be a temperature

gradient along the vertical direction. The heat conduction

due to this temperature gradient is

The condition under which that the hori(ontal wall )ets can

maintain their temperature identity is that the heat

conduction along the vertical direction is negligible

compared to the energy carried by the hori(ontal )ets

&con"T

!LH

#/ aHT

!L ! T H

<


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or equivalently

!#./$1&

The characteristics of various heat transfer regimes are

summari(ed in Table #./.

#/ aH

H

L

>

egimes 0 1onduction 0 Tall stems 0 *oundar laer 0 hallow sstems

1ondition ofoccurrence

!low pattern 1loc3wise circulation Distinct boundar laer on topand bottom walls

*oundar laer on all fourwalls. 1ore remains stagnant

Two hori4ontal wall 5ets flowin opposite directions.

6ffect of flow on heattransfer

nsignificant nsignificant ignificant ignificant

7eat transfermechanism

1onduction in hori4ontaldirection

1onduction in hori4ontaldirection

*oundar laer convection 1onduction in verticaldirection

7eat transfer

Table 6., 8haracteristics of natural convection in a rectangular enclosure heated from the side

#/ / 8aHH La #H

i!ure 6.,0olls and hexagonal cells in natural convection in

enclosure heated from below (Eosthui4en and 9alor, #@@@).


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7hen the ;ayleigh number )ust exceeds the above critical;ayleigh number, the flow pattern is twodimensional

counter rotating rolls A referred to as =Bnard cells Csee"ig. #./!a&D. 's the ;ayleigh number further increasesto one or two orders of magnitude higher than the abovecritical ;ayleigh number, the twodimensional cellsbreakup to three dimensional cells whose top view ishexagons Csee "ig. #./!b&D. The function of the twodimensional rolls and threedimensional hexagonal cellsis to promote heat transfer from the heated bottom wallto the cooled top wall. ?lobe and ropkin suggested thefollowing empirical correlation

!#.//*&

where all thermophysical properties are evaluated atEquation !#.//*& is valid for . Inaddition, H


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1nclined 'ectan!ular Enclosure7hen the rectangular enclosure heated from the side istilted relative to the direction of gravity, additionalunstable stratification and thermal instability will affectthe fluid flow and heat transfer. The variation of Nusselt

number as function of tilt angle Fis qualitatively shown in"ig. #./-.

" @"G #?"G

i!ure 6.,26ffect of inclination angle on natural convection in enclosure

c


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The isotherms and the streamlines for ;a5$0*are shown in"ig. #./*. 't F5$*F, which, according to Table #., is

less than the critical inclination angle, the isotherms startto exhibit some behaviors of thermally unstableconditions. This is the correlation for natural convectionof air in a squared enclosure ! & in the region

!#.//#&

where is for pure conduction. 7hile eq. !#.//#& isgood for air in a squared enclosure, the followingcorrelation can be applied to other situations

!#.//1&

/ #H L= " @"< < o

9u ( ) 9u (" ) 2

sin9u (@" ) 9u (" )

H H

H H'

= =

o

o o

9u (" )Ho

# 9u (@" ) # sin " @"

9u ( )

9u (@" )(sin ) @"

H

H

H c

LL H

H L

H

+ <


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spect ratio,H/L # ' < #2 8#2

1ritical tilt angle, Fc

#AAG #2=G #2"G ##'G ##"G

Table ).0 8ritical inclination angle for different aspect ratio !'rnold et al., $21#&

sothermals treamlines

i!ure 6.,59atural convection in inclined s:uared enclosures

(Hhong #t a$,#@?').


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E3ample 6.5

' rectangular cavity is formed by two parallel plates, eachwith a dimension of 0.* m by 0.* m, which are separated

by a distance of * cm. The temperatures of the two

plates are 1 G8 and $1 G8, respectively. "ind the heat

transfer rate from hot plate to cold plate for the

inclination angles of 0G, -*G, 20G, and $%0G.

i!ure ).,)9atural convection in inclined s:uared enclosure.

T(

Tc

L

)i*


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The heat transfer coefficient is

Therefore, the heat transfer rate for is

7hen the inclination angle is , eq. !#.//1& yields

Thus, the Nusselt number is and the correspondingheat transfer coefficient is

o@" =

o

A / = =

oo 29u (@" ) "."2


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The heat transfer rate for is therefore

7hen the inclination angle is , the problem becomesnatural convection in an enclosure heated from thebelow. The ;ayleigh number is

The Nusselt number in this case can be obtained from eq.!#.//*&

oA =

' ' A

< isince Do4 Di. Equation will be valid only if theboundary layer thickness is less than the gap between the twocylinders, i.e. only if oJ DoA Di. Knder lower ;ayleigh numbers, on

the other hand, we have

!#./-0&

and the heat transfer mechanism between two cylinders will approachpure conduction.

( ) / 2i o

T T+

#/ - #/-& , &o io o D i i D

D Ra D Ra

#/aoo D o i

D D D >


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Instead of using eq. !#./-0& to check the validity of eq.!#./%&, another method is to calculate the heat transfer

rate via eq. !#./%& and pure conduction model, and thelarger of the two heat transfer rate should be used.

"or natural convection in the annulus between twoconcentric spheres, the trends for the evolution of theflow pattern and isotherms are similar to the concentriccylinder except the circulation between concentricspheres has the shape of a doughnut. The empiricalcorrelation for the heat transfer rate is

!#./-$&where the definition of ;ayleigh number is same as for eq.

!#./2&. Equation !#./-$& is valid for 0.1 J 3r J -000and ;a J $0-.

#/

= / A A/

+ra2.'2A ( )

$# ( / ) % ".?


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6.5 Natural Convection in Enclosures Chapter 6: Natural Convection


i!ure 6.,*9atural convection in a hori4ontal annulus (+r ;".=, K/Di;".?, a ;.=L#"B Date, #@?

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