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11. NATURAL CONVECTION

No mechanical force to push the fluid pump, fan etc.

No predefined fluid flowrate and velocity cant

prescribeReynolds number Fluid moves as a result of density difference

Fluid velocity established as a result of the temperature

field Fluid can move downward and upward

Examples:

Cold air cools

the egg

Warm air heats

the can

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Further examples:

Atmospheric inversion -

no vertical exchange ofmass of air

T(z)z

(z)

T(z), (z)

high temperature

low temperature

Vertical exchange of

mass of air

T(z)(z)

T(z),(z)

high temperature

low temperature

z

Cold window

Space heater

radiator

Atmospheric circulation

Room circulation

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Heated vertical wall

Which force makes the fluid to raise?

The force called the buoyancy (vztlak)

Which role does the gravity play??

How the velocity is established

P.D.E. for momentum conservation

2

2

y

ug

x

p

1

y

uv

x

uu

+

=

+

2

2

y

ug

x

p

1

y

uv

x

uu

+

=

+

simply add gravity in xdirection

Fluid moves upwards

Heat flux

gravity

L

Individual terms can be expressed in units of force per mass [N/kg]or in units of acceleration [m/s2]

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Fluid moves upwards

gravity

2

2

y

ugx

p

1

y

uvx

uu

+

=

+

0yp =No movement iny - direction

xp Same in the boundary layerand outside it

We need temperature difference how to replace by T ?

( )

gg

x

p

1=

g

p

=

Pressure difference results fromthe weight of the fluid column

Volume expansion coefficient

pT

1

=

TT

1

=

or

=

T

1For ideal gas

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Fluid moves upwards

gravity

Momentum conservation equation

2

2

y

T

ay

T

vx

T

u

=

+

Energy conservation equation

Continuity equation

0y

v

x

u

=

+

( )2

2

y

uTTg

y

uv

x

uu

+=

+

buoyancy force

So called coupled problem cant solve velocity field unless

we know temperature field which is a function of velocity field.

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Heat transfer coefficient

Similarity parameter dimensionless number. Cant use

Reynolds number fluid velocity or flow rate not defined

a priori.

Grashofnumber for vertical wall

rceviscous.fo

buoyancy

LTTg

Gr 2

3

L

Functional relation for Heat Transfer coefficient Nusseltnumber

)(Ra.PrGrfNuLLL

==L

Ra Rayleigh number

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Laminar versus turbulent

Under certain conditions, laminar regime can change to turbulent.

93

w

kritx,kritx,10

a

xTTg.PrGrRa =

== Vertical wall:

Characteristic length is always dimension in the direction of the

fluid movement:

Vertical wall: Height of the wallVertical cylinder: Lengthof the cylinder if:

41

L

Gr

35

L

d

Horizontal cylinder: Diameter of the cylinder

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Horizontal plates

insulation

B. Lower surface of a cold

plate TwT

A. Upper surface

of a cold plate

TwT

L=A/P = surface area/surface perimeterCharacteristic dimension:

Nusselt number41

LL 0,27RaNu =A.

41LL 0,54RaNu =B.

7L

4 10Ra10

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Inclined plates

Use vertical plate equations for

the upper surface of a cold plate

and the lower surface of a hot plate

equations A.

Nusselt number 41LL 0,27RaNu =A.

( )a

LTTg

Ra

3w

L

=

Replacegbygcos angle from the vertical

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Cavities

Applications: plate solar collectors, double glazed windows,sandwich walls, etc.

Air trapped inside good insulator

Complications: air doesnt remain stationary it movesupwards and downwards

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Horizontal cavity

When the hotter plate at the top noconvection occurs pure conduction

transfer of heat

When the hotter plate at the bottom

tendency for the lighter air to rise to the top

( )1708

3

21

1708 natural convection occurs Bnard cells

ForRa > 3.105

Bnard cells break down turbulence occurs

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Horizontal cavity

41LL 0.195RaNu = for 10

4 < Ra < 4.105

For air:

31LL Ra0.06Nu 8= for 4.105 < Ra < 107

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Vertical cavityForRa < 1000, no natural convection

pure conduction heat transfer across

the cavity Nu = 1

ForRa > 1000, natural convection occurs -

along the hot surface air rises, along the cold surface air flows down

410.28

LLLHRa

Pr0.2Pr0.22Nu

+=

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Heat Transfer Rate

L

Nu

=

LTTNuS)TS(TQ 2121 ==&

It resembles the equation for

heat conduction

L

TTSQ 21effcond

=&

.Nueff =Effective conductivity

Conclusions: Heat transfer rate can be determined from heat

conduction using effective thermal conductivity eff

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Natural versus forced convection

Forced convection much higher heat transfer coefficients

Tendency to ignore natural convection

Error in ignoring natural convection negligible at high velocities

Error considerable at low velocities

2Re

GrParameter representing the importance ofnatural convection

0,1ReGr2 < Natural convection negligibleIf

10Re

Gr

2>If Forced convection negligible

If 10Re

Gr0,1

2

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Natural versus forced convection

Natural convection may help or hurt

forced convection depending on relative

directions of buoyancy - induced andforced convection motion

Assisting flow sign +

Opposing flow sign -)( nnatural

nforced

nNuNuNu =

Exponent n recommended 3