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NATURAL/FREE CONVECTION
Associate Professor
IIT Delhi
E-mail: [email protected]
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convection
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The upward force exerted by a fluid on a body completely or partially
immersed in it is called the buoyancy force. The magnitude of the
uoyancy orce s equa o e we g o e u sp ace y e o y
F = fluidgVbody
Fnet = W Fbouyancy
= bodygVbody -fluidgVbody= - V
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s e uoyancy orce a eeps e s ps a oa
in water (W = Fbuoyancy ) for floating objects
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The coefficient of volume expansion
is a measure of the change in volume of a
substance with temperature at constant
pressure
PP T
1
T
1
=
=
TTT
=
At constant P
For an ideal gas P = RT
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T
1gasideal =
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ndNewtons 2nd law gives:
x,bodyx,surfacex FFam +=
Mass )1dydx(m =
cce era on
y
u
x
u
udt
dy
y
u
dt
dx
x
u
dt
du
a x
+
=
+
==
Forces
)1.dy.dx(g)1dy(dxP
)1dx(dyFx
=
)1dydx(gx
P
y
u
xy
2
2
=
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=y
u
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( )2
2
L2
L
2
3cs
y
u
Re
1
Re
TLTTg
y
uv
x
uu
+
=
+
3cs
LL)TT(g
Gr
=
The Grashof number Gr is a measure
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force and the opposing viscous force
acting on the fluid.
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For a vertical plate
Gr < 109 Laminar
> 109 Turbulent
Forced convection dominates
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nLnLc CRaPr)Gr(Ck
hLNu ===
( )Pr
LTTgPrGrRa 2
3
csLL
==
Values of n and C depend on geometry
of the surface and flow regime
The value of n is usually for laminar flow and 1/3 for turbulent flow. The
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.
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surface:Samerelationasconstant.
)TT(kkNu 2/L
s
==
Verticalcylinders:
L35DLGr
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Inclined Hot Plate
Inclinedhotplatethatmakes
anangle fromtheverticalina
cooler
environment.
ThenetforceF= g( )
buoyancyandgravity)acting
onaunit
volume
of
the
fluid
intheboundarylayerisalways
intheverticaldirection
component Fy initiates upward motion in addition to the parallel motion
along the plate, and thus the boundary layer breaks up and forms
plumes, as shown in the figure
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Inthecaseofacoldplateinawarmer
environment??
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Use vertical plate equations for the
upper surface of a cold plate and the
lower surface of a hot plate L.
Inclined Plate
Replace g by g cos for Ra < 109
and < 60
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Horizontal SurfaceFor a hot surface in a cooler
,
upward, forcing the heated fluidto rise.
If the hot surface is facing
upward, the heated fluid rises
freely, inducing strong naturalconvection currents and thus
effective heat transfer.
u e o sur ace s ac ngdownward, the plate will block
the heated fluid that tends to rise
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,
heat transfer
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Identifytheconditions:Ts T
Ts < T Ts < T
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Ts > T
Ts > T
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The local Nusselt number is highest at the
bottom, and lowest at the top of thecylinder when the boundary layer flow
What will happen for a cold cylinder?
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Natural Convectioninside Enclosure
Vertical enclosure
Horizontal enclosure
( )Pr
LTTgPrGrRa
2
3c21
LL
==
Initially, the heat transfer is by pure conduction and Nu = 1.
When Ra > 1708, the buoyant force overcomes the fluid resistance and initiates
natural convection currents which are observed to be in the form of hexa onal
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cells called Bnard cells.
For Ra > 3 x 105
, the cells break down and the fluid motion becomes turbulent
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http://www.youtube.com/watch?v=xb_pHQzEFJg
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c
21
s21s LkNuA)TT(hAQ
==&
c
21scond
L
TTkAQ
=&
The fluid in an enclosure behaves like a fluid
whose thermal conductivity is kNu as a result of
convection currents.
Therefore, the quantity kNu is called
the effective thermal conductivity of the
enclosure.
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keff= kNu
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Hot plate at the top : Nu = ?
Hot plate at the bottom: Significantconvective current occurs when Ra > 1708
For horizontal enclosures that contain air,
Jakob recommends
can also be used for other gases
LRa195.0Nu=3/1
LRa068.0Nu=L 10x4Ra10
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Small aspect ratio 2L/H1 +
ny pran num er
4/128.0L )
L
H()Ra
Pr2.0
Pr(22.0Nu
+
=
10L/H2
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Concentric Cylinders and
Spheres
Raithby and Hollands (1975):
c=
o-
i.
for 0.70 Pr 6000 and 102 FcylRaL 107.
For F Ra 100, natural convection currents
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are negligible and thus keff = k.
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Combined Natural Convection
and Radiation
low compared to those for forced convection. Therefore, radiationis usually disregarded in forced convection problems, but it must
be considered in natural convection roblems that involve a as.
This is especially the case for surfaces with high emissivities. Forexample, about half of the heat transfer through the air space of a
double pane window is by radiation
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Combined Free and Forced
Convection NaturalconvectionisnegligiblewhenGr/Re2 10,
and
neitherisne li iblewhen0.1
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Nu combined = (Nunforced Nunnatural)1/n
The value of the exponent n varies
natural convection correlations
,
geometry involved. It is observed that n
= 3 correlates experimental data for
vertical surfaces well. Larger values of nare better suited for horizontal surfaces
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