Free Convection 1

Free Convection: Chapter 9

Free Convection 2

General Considerations

•  Free convection refers to fluid motion induced by buoyancy forces.

•  Buoyancy forces may arise in a fluid for which there are density gradients and a body force that is proportional to density.

•  In heat transfer, density gradients are due to temperature gradients and the body force is gravitational.

•  Stable and Unstable Temperature Gradients

General Considerations (cont.)

Free Boundary Flows Ø  Occur in an extensive (in principle, infinite), quiescent (motionless at locations far from the source of buoyancy) fluid.

Ø  Plumes and Buoyant Jets:

•  Free Convection Boundary Layers Ø  Boundary layer flow on a hot or cold surface induced by buoyancy forces.

( )sT T∞≠

General Considerations (cont.)

Combustion in Microgravity

•  Effective combustion occurs due to effective transfer of fresh oxidizer (air) by combined effect of buoyancy-driven convection molecular diffusion.

•  In space (microgravity), only molecular diffusion provides the reaction zone with fresh oxidizer (air)

On Earth In space

http://www.spaceflight.esa.int/impress/text/education/Microgravity/Why%20Do_Microgravity_Research.html

Free Convection 5

• In the previous discussions, a free stream velocity set up the conditions for convective heat transfer. • Due to friction with the surface, the flow must be maintained by a fan or pump– thus it is called forced convection. • An alternate situation occurs when a flow moves naturally due to buoyancy forces • This so called “free ” or “natural” convection and it is illustrated in the figure.

Free Convection

Free Convection 6

Free Convection [2]

•  Buoyancy is the result of difference in density between materials. •  In this case, the difference in density is due to the difference in temperature. • In the figure, the air next to the plate is heated, its density decreases, and the resulting buoyancy forces the air to rise. • However, note that only the flow inside the thermal boundary layer moves– the velocity is zero both at the wall and far away from it.

Free Convection 7

Ø  Grashof Number:

GrL =

gβ Ts −T∞( )L3

ν 2 ∼ Buoyancy ForceViscous Force

L → characteristic length of surface

thermal expansion coefficient (a thermodynamic property of the fluid)β →

1pT

ρβρ

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠

Liquids: Tables A.5, A.6β → ( )Perfect Gas: =1/ KTβ

Non-dimensional Coefficients

• The Grashof plays the same role in free convection that the Reynolds number plays in forced convection. • Practically, the Grashof number is the ratio of forcing (buoyancy) forces to restraining (viscous) forces.

Perfect Gas: =p/RTρ

Free Convection 8

Ø  Rayleigh number:

Non-dimensional Coefficients

Rax = Grx Pr =

gβ Ts −T∞( )x3

να

Vertical Plates

Vertical Plates •  Free Convection Boundary Layer Development on a Hot Plate:

Ø  Ascending flow with the maximum velocity occurring in the boundary layer and zero velocity at both the surface and outer edge.

Ø  How do conditions differ from those associated with forced convection? Ø  How do conditions differ for a cold plate ( )?sT T∞<

x-component velocity temperature

Vertical Plates

( )?sT T∞<

•  Boundary layer approximation: x momentum

u ∂u∂x

+ v ∂u∂ y

= − 1ρdp∞dx

− g+ µρ∂2u∂ y2

(9.1)

•  The pressure far from the wall is hydrostatic:

dp∞dx

= −ρ∞g (9.2)•  Substitute Eq. (9.2) into (9.1):

u ∂u∂x

+ v ∂u∂ y

= g Δρ /ρ( )+ µρ∂2u∂ y2

(9.3) Δρ = ρ∞ − ρ

•  The volumetric expansion coefficient:

β = − 1

ρ∂ρ∂T

⎛⎝⎜

⎞⎠⎟ p

(9.4)β = − 1

ρρ∞ − ρT∞ −T

Vertical Plates: Laminar Boundary Layer

Vertical Plates

Vertical Plates: Laminar Boundary Layer

( )?sT T∞<

•  Boussinesq Approximation

•  The x-momentum equation becomes:

•  All together: The laminar free convection boundary layer equations u ∂u∂x

+ v ∂u∂ y

= ρβ T∞ −T( )+ µρ∂2u∂ y2

(9.5)

β = − 1

ρ∂ρ∂T

⎛⎝⎜

⎞⎠⎟ p

≈ − 1ρρ∞ − ρT∞ −T

ρ∞ − ρ = ρβ T∞ −T( )

∂u∂x

+ ∂v∂ y

=0 (9.6)

u ∂u∂x

+ v ∂u∂ y

= ρβ T∞ −T( )+ µρ∂2u∂ y2

(9.7)

u∂T∂x

+ v ∂T∂ y

=α ∂2T∂ y2

(9.8)

Vertical Plates (cont.)

•  Form of the x-Momentum Equation for Laminar Flow

Net Momentum Fluxes ( Inertia Forces)

Buoyancy Force Viscous Force

Ø  Temperature dependence requires that solution for u (x,y) be obtained concurrently with solution of the boundary layer energy equation for T (x,y).

–  The solutions are said to be coupled.

u ∂u∂x

+ v ∂u∂ y

= ρβ T∞ −T( )+ µρ∂2u∂ y2

(9.5)

u∂T∂x

+ v ∂T∂ y

=α ∂2T∂ y2

(9.8)

Vertical Plates (cont.)

Ø  Based on existence of a similarity variable, , through which the x-momentum equation may be transformed from a partial differential equation with two-independent variables ( x and y) to an ordinary differential equation expressed exclusively in terms of .

,η

η

Ø  Transformed momentum and energy equations:

( ) ( )1/ 2

2 xs

T Tdf xf Gr u Td T T

ηη ν

− ∗ ∞

∞

−′ ≡ = ≡−

Similarity Solution

η

η

η ≡ y

xGrx4

⎛

⎝⎜⎞

⎠⎟

1/4

(9.13) and ψ (x , y)≡ f (η) 4ν Grx4

⎛

⎝⎜⎞

⎠⎟

1/4⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

(9.14)

f ''' +3 ff ' −2 f '( )2 +T * =0 (9.17)T *'' +3Pr fT *' =0 (9.18)

u= ∂ψ∂ y

= 2νxGrx

1/2 f '(η); T * ≡T −T∞Ts −T∞

Ø  Solve these coupled ODEs numerically

η =0: f = f ' =0; T * =1.η→∞ : f '→0; T * →0.

Vertical Plates (cont.)

( ) and :f Tη ∗′

Ø  Velocity boundary layer thickness dimensionless x-component velocity dimensionless temperature

Similarity Solution: Numerical Integration

Pr >0.6: δ =5x Grx

4⎛

⎝⎜⎞

⎠⎟

−1/4

=7.07 x

Grx( )1/4∼ x1/4

Vertical Plates (cont.)

•  The local Nusselt number

Nux =

hxk

= −Grx4

⎛

⎝⎜⎞

⎠⎟

1/4dT *

dηη=0

=Grx4

⎛

⎝⎜⎞

⎠⎟

1/4

g(Pr) (9.19)

Nusselt number

g(Pr)= 0.75Pr1/2

0.609+1.221Pr1/2+1.238Pr( )1/4(9.20)

•  The average Nusselt number

NuL =hLk

= 43GrL4

⎛

⎝⎜⎞

⎠⎟

1/4

g(Pr) (9.21)

NuL =43NuL

Free Convection 16

Transition to Turbulence

Transition in a free convection boundary layer depends on the relative magnitude of the buoyancy and viscous forces in the fluid. It is customary to correlate its occurence in terms of the Rayleigh number.

Ø  Rayleigh Number:

( ) 39

, , Pr 10sx c x c

g T T xRa Gr

βνα

∞−= = ≈

Free Convection 17

Empirical Correlations: The Vertical Plate

Ø  Laminar Flow (More accurate for laminar flow) ( )910 :LRa <

( )

1/ 4

4 /99 /16

0.6700.681 0.492/ Pr

LL

RaNu = +⎡ ⎤+⎣ ⎦

Ø  All Conditions (Churchill-Chu)

( )

2

1/ 6

4 /99 /16

0.3870.8251 0.492 / Pr

LL

RaNu⎧ ⎫⎪ ⎪= +⎨ ⎬

⎡ ⎤⎪ ⎪+⎣ ⎦ ⎭⎩

Free Convection 18

Ø  A condition for which forced and free convection effects are comparable.

Ø  Exists if

( )2/ Re 1L LGr ≈

- Free convection → GrL / ReL

2( )≫1 - Forced convection → GrL / ReL

2( )≪1

Ø  Heat Transfer Correlations for Mixed Convection:

n n nFC NCNu Nu Nu≈ ±

assisting and transverse flows- opposing flows+→→3n ≈

Mixed Convection