(23 24) Free Convection

8/13/2019 (23 24) Free Convection

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NATURAL/FREE CONVECTION

Associate Professor

IIT Delhi

E-mail: [email protected]


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convection

P.Talukdar/Mech-IITD 2


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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


s e uoyancy orce a eeps e s ps a oa

in water (W = Fbuoyancy ) for floating objects


5/31

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


T

1gasideal =


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7/31

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

=


=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


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



11/31

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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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


Inthecaseofacoldplateinawarmer

environment??


15/31

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

P.Talukdar/Mech-IITD


16/31

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


,

heat transfer


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Identifytheconditions:Ts T

Ts < T Ts < T


Ts > T

Ts > T


18/31

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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20/31

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


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.


keff= kNu


25/31

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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27/31

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


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



30/31

Combined Free and Forced

Convection NaturalconvectionisnegligiblewhenGr/Re2 10,

and

neitherisne li iblewhen0.1


31/31

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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