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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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Natural convection, Empirical
equations for free and forced
convection
Dr. Senthilmurugan S. Department of Chemical Engineering IIT Guwahati - Part 12
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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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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
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)!"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.
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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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)!"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
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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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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
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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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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.
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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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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 ∞.
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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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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
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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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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,
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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"
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8/17/2019 Lect - 12 Natural convection, Empirical equations for free and forced convection.pptx
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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$ $ive
%ras!of number3 Depresents t!e natural convection effects in momentum equation
%ras!of Number
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@ran& %ras!of
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%ras!of Number
0!e %ras!of number provides t!e main
criterion in determinin$ '!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
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0!e relative importance of convection !eat transferre$imes for flo' near a !ot sp!ere
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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$! number
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Empirical correlations for t!e avera$e Nusselt numberfor natural convection over surfaces
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