lect - 12 natural convection, empirical equations for free and forced convection.pptx

8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx

1/18

Natural convection, Empirical

equations for free and forced

convection

Dr. Senthilmurugan S. Department of Chemical Engineering IIT Guwahati - Part 12


2/18

5/12/16 | Slide 2

Objectives

nderstand t!e p!"sical mec!anism of natural convection

#erive t!e $overnin$ equations of natural convection, and

obtain t!e dimensionless %ras!of number b"

nondimensionali&in$ t!em

Evaluate t!e Nusselt number for natural convection

associated 'it! vertical, !ori&ontal, and inclined plates as'ell as c"linders and sp!eres


3/18

5/12/16 | Slide (

)!"sical *ec!anism of Natural +onvection

*an" familiar !eat transfer applications

involve natural convection as t!e

primar" mec!anism of !eat transfer

E-amples.

Natural convection in $ases is usuall"

accompanied b" radiation of comparablema$nitude e-cept for lo'emissivit"

surfaces

0!e motion t!at results from t!e

continual replacement of t!e !eated air

in t!e vicinit" of t!e e$$ b" t!e cooler air

nearb" is called a natural convectioncurrent, and t!e !eat transfer t!at is

en!anced as a result of t!is current is

called natural convection !eat transfer

The cooling of a boiled egg in acooler environment by naturalconvection.


4/18

5/12/16 | Slide

)!"sical *ec!anism of Natural +onvection

uo"anc" force3 0!e up'ard force e-erted b"

a fluid on a bod" completel" or partiall"

immersed in it in a $ravitational field 0!e

ma$nitude of t!e buo"anc" force is equal to

t!e 'ei$!t of t!e fluid displaced b" t!e bod"

0!e net vertical force actin$ on a bod"

4rc!imedes principle3 4 bod" immersed in a

fluid 'ill e-perience a 'ei$!t loss7 in anamount equal to t!e 'ei$!t of t!e fluid it

displaces

0!e c!imne" effect7 t!at induces t!e up'ard

flo' of !ot combustion $ases t!rou$! a

c!imne" is due to t!e buo"anc" effect


5/18

5/12/16 | Slide 5

Effect of 8olume e-pansion

8olume e-pansion coefficient3 8ariation of t!e

densit" of a fluid 'it! temperature at constant

pressure

'!en infinite temperature is far from surface replacement differential quantities b" differences as

0!e lar$er t!e temperature difference bet'een t!efluid adjacent to a !ot 9or cold: surface and t!e fluid

a'a" from it, t!e larger t!e buo"anc" force and t!e

stronger t!e natural convection currents, and t!us

t!e higher t!e !eat transfer rate

0!e coefficient of volume e-pansion

is a measure of t!e c!an$e in

volume of a substance 'it!

temperature at constant pressure


6/18

5/12/16 | Slide 6

Natural +onvection

;n natural convection, no blo'ers are

used, and t!erefore t!e flo' rate cannot

be controlled e-ternall"

0!e flo' rate in t!is case is establis!ed

b" t!e d"namic balance of buoyancy

and friction. 4n interferometer produces a map of

interference frin$es, '!ic! can be

interpreted as lines of constant

temperature.

0!e smoot! and parallel lines in 9a:

indicate t!at t!e flo' is laminar, '!ereast!e eddies and irre$ularities in 9b:

indicate t!at t!e flo' is turbulent. 0!e lines are closest near t!e surface,

indicatin$ a higher temperature gradient.


7/18

5/12/16 | Slide =

Equation of *otion

0!e t!ic>ness of t!e boundar" la"er

increases in t!e flo' direction nli>e forced convection, t!e fluid

velocit" is zero at t!e outer ed$e of t!e

velocit" boundar" la"er as 'ell as at t!e

surface of t!e plate 4t t!e surface, t!e fluid temperature is

equal to t!e plate temperature, and

$raduall" decreases to t!e temperature

of t!e surroundin$ fluid at a distance

sufficientl" far from t!e surface

;n t!e case of cold surfaces, t!e s!apeof t!e velocit" and temperature profiles

remains t!e same but t!eir direction is

reversed

0"pical velocit" and temperature profiles for

natural convection flo' over a !ot vertical

plate at temperature T s inserted in a fluid at

temperature T ∞.


8/18

5/12/16 | Slide ?

Equation of *otion

+onsider a differential volume element of !ei$!t dx, len$t! dy, and unit dept! in t!e

z direction for anal"sis

0!e forces actin$ on t!is volume element are s!o'n in @i$

Ne'tons second la' of motion for t!is volume element can be e-pressed as

where δ m = ρ (dx·dy·1) is the mass of the fluid withinthe olume element.

The acceleration in the x!direction is obtained by ta"ing

the total differential of u(x, y), which is y, and diiding it

by dt. #e get

pressure forces

pressure forces

shearstresses

shearstresses

Assumption: the normal stresses acting on thetop and bottom surfaces are small and aredisregarded

the force ofgravity acting onthe entirevolume element


9/18

5/12/16 | Slide A

Equation of *otion

0!en t!e net surface force actin$ in t!e

x direction becomes

Substitutin$

by diiding ρ (dx·dy·1)

0!e x momentum equation in t!e

quiescent fluid outside t!e boundar"

la"er can be obtained from t!e relation

above as a special case b" settin$ u B

C ;t $ives

0!e x momentum equation,


10/18

5/12/16 | Slide 1C

Equation of *otion

0!is is t!e equation t!at $overns t!e fluid motion in t!e boundar" la"er due to t!e effect of

buo"anc" Note t!at t!e momentum equation involves t!e temperature, and t!us t!e

momentum and ener$" equations must be solved simultaneousl"

0!e above set of t!ree partial differential equations can be reduced to a set of t'o ordinar"

nonlinear differential equations b" t!e introduction of a similarit" variable ut t!e resultin$

equations must still be solved alon$ 'it! t!eir transforred boundar" conditions numericall"


11/18

5/12/16 | Slide 11

Solution for Equation of *otion

0!e $overnin$ equations of natural convection and t!e boundar" conditions can be

nondimensionali&ed b" dividin$ all dependent and independent variables b" suitable

constant quantities3

Substitutin$ t!em into t!e momentum equation and simplif"in$ $ive

%ras!of number3 Depresents t!e natural convection effects in momentum equation

%ras!of Number


12/18

5/12/16 | Slide 12

@ran& %ras!of


13/18

5/12/16 | Slide 1(

%ras!of Number

0!e %ras!of number provides t!e main

criterion in determinin$ '!et!er t!e fluid flo'

is laminar or turbulent in natural convection@or vertical plates, t!e critical %ras!of

number is observed to be about 1CA !en a surface is subjected to e-ternal flo',

t!e problem involves bot! natural and forcedconvection

0!e relative importance of eac! mode of

!eat transfer is determined b" t!e value of

t!e coefficient %r/De23Natural convection effects are ne$li$ible if

%r/De2 FF 1@ree convection dominates and t!e forced

convection effects are ne$li$ible if %r/De2 GG

1ot! effects are si$nificant and must be

considered if %r/De2 ≈ 1 9mi-ed convection:

#efination

0!e %ras!of number %r is a measure of

t!e relative ma$nitudes of t!e buoyancy

force and t!e opposin$ iscous force actin$

on t!e fluid


14/18

5/12/16 | Slide 1

0!e relative importance of convection !eat transferre$imes for flo' near a !ot sp!ere


15/18

5/12/16 | Slide 15

Natural convection over surfaces

Natural convection !eat transfer on a

surface depends on t!e $eometr" of t!e

surface as 'ell as its orientation, t!e

variation of temperature on t!e surface and

t!e t!ermop!"sical properties of t!e fluid

involved it! t!e e-ception of some simple cases,

!eat transfer relations in natural convection

are based on e-perimental studies 0!e constants $ and n depend on t!e

geometry of t!e surface and t!e flow

regime, '!ic! is c!aracteri&ed b" t!e ran$e

of t!e Da"lei$! number 0!e value of n is usuall" 1/ for laminar

flo' and 1/( for turbulent flo' 4ll fluid properties are to be evaluated at

t!e film temperature T f = 9T s % T ∞:/2

Da"lei$! number


16/18

5/12/16 | Slide 16

Empirical correlations for t!e avera$e Nusselt numberfor natural convection over surfaces


17/18

5/12/16 | Slide 1=


18/18

lect - 12 natural convection, empirical equations for free and forced convection.pptx

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