Page 1

122 Advanced Heat Transfer (ME 211) Younes Shabany

Natural Convection Heat Transfer

123 Advanced Heat Transfer (ME 211) Younes Shabany

Net Force on a Body Completely in a Fluid ❑ The net force applied to a body completely submerged

in a fluid is

bodyfluidbody

bodyfluidbodybody

buoyancynet

gV

gVgV

FWF

)( ρρ

ρρ

−=

−=

−=

Fluid Fbuoyancy

W

Body

q The body can be a bulk of hot fluid (same fluid but at a higher temperature).

bodyfluidfluidhotnet gVF )( ρρ −=

q But ρhot fluid < ρfluid. Therefore Fnet is negative and it means that the hot fluid will move up.

124 Advanced Heat Transfer (ME 211) Younes Shabany

Volumetric Thermal Expansion Coefficient ❑ Rate of change of a unit volume of material per unit

temperature change at constant pressure is called volumetric thermal expansion coefficient.

PTv

v ⎟⎠⎞

⎜⎝⎛∂∂

=1

β

where v is the specific volume of the material. q Since v=1/ρ and therefore ∂v=-∂ρ/ρ2

PT ⎟⎠⎞

⎜⎝⎛∂∂

−=ρ

ρβ

1

q For an ideal gas

PR

Tv

PRTvRTPv

P=⎟

⎠⎞

⎜⎝⎛∂∂

⇒=⇒=

TRTR

PvR

PRvT

vv P

111=⇒===⎟

⎠⎞

⎜⎝⎛∂∂

= ββ

125 Advanced Heat Transfer (ME 211) Younes Shabany

Net Buoyancy Force and Temperature ❑ If V is the volume of a bulk of hot fluid, the net upward

force applied to this bulk of fluid is

where Δρ=ρhot fluid - ρfluid is the density difference between the hot and cold fluids.

q A correlation between this force and temperature difference can be obtained by using an approximate expression for β.

TT

Δ−=Δ⇒ΔΔ

−≈ ρβρρ

ρβ 1

gVFnet ρΔ=

TgVFnet Δ−= ρβq This shows that the larger the temperature difference

between hot and cold fluids the larger the net upward force and therefore the stronger the natural convection.

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126 Advanced Heat Transfer (ME 211) Younes Shabany

Natural Convection Boundary Layer Equations

❑ Consider natural convection boundary layer along a vertical plate with temperature Ts > T∞.

❑ Since the body force in the x direction is X=-ρg, boundary layer equations are

gyu

xp

yuv

xuu −

∂∂

+∂∂

−=∂∂

+∂∂

2

21ν

ρ

x,u

y,v

0=∂∂

+∂∂

yv

xu

2

2

yT

yTv

xTu

∂∂

=∂∂

+∂∂

α

❑ If there is no body force in y direction, ∂p/∂y=0. Therefore

gxp

∞−=∂∂

ρ

127 Advanced Heat Transfer (ME 211) Younes Shabany

Natural Convection Boundary Layer Equations

❑ Therefore, boundary layer equations are

)()(1∞

∞∞∞ −=−−

=−

=−=−∂∂

− TTggTTggggxp

βρ

ρβρρρ

ρρ

ρ

2

2

)(yuTTg

yuv

xuu

∂∂

+−=∂∂

+∂∂

∞ νβ

0=∂∂

+∂∂

yv

xu

2

2

yT

yTv

xTu

∂∂

=∂∂

+∂∂

α

128 Advanced Heat Transfer (ME 211) Younes Shabany

Normalized Boundary Layer Equations ❑ Let’s introduce normalized variables

∞

∞

−−

=====TTTTT

uvv

uuu

Lyy

Lxx

s*****

00

where L is the length of the plate and u0 is a characteristic velocity.

q The x-component momentum equation can be normalized as follows.

2

2

)(yuTTg

yuv

xuu

∂∂

+−=∂∂

+∂∂

∞ νβ

2

2

200

00

0 ***)(

***

***

yu

LuTTTg

yu

Luuv

xu

Luuu s ∂

∂+−=

∂∂

+∂∂

∞ νβ

2

2

020 *

**)(***

***

yu

LuT

uLTTg

yuv

xuu s

∂∂

+−

=∂∂

+∂∂ ∞ νβ

2

2

022

0

2

2

3

***)(

***

***

yu

LuT

LuLTTg

yuv

xuu s

∂∂

+−

=∂∂

+∂∂ ∞ νν

νβ

2

2

2L

**

Re1*

ReGr

***

***

yuT

yuv

xuu

LL ∂∂

+=∂∂

+∂∂

129 Advanced Heat Transfer (ME 211) Younes Shabany

Grashof and Rayleigh Numbers ❑ Grashof number is defined as

2

3)(Grν

δβδ

∞−=

TTg s

where δ is a characteristic length of the geometry, and Ts and T∞ are surface and free stream temperatures.

q It can be shown that Grashof number is a measure of ratio of buoyancy force to viscous force acting on the fluid.

q Rayleigh number is the product of Grashof and Prandtl numbers.

PrGrRa ⋅= δδ

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130 Advanced Heat Transfer (ME 211) Younes Shabany

Similarity Solutions of Boundary Layer Equations ❑ Boundary layer equations can be transformed into two

ordinary differential equations by introducing similarity variable 4/1

x

4Gr

⎟⎠⎞

⎜⎝⎛≡xy

η

and representing the velocity components in terms of a stream function defined as

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛≡

4/1x

4Gr4)(),( νηψ fyx

❑ The two ordinary differential equations for f and T* are

0*'Pr3'*'0*)'(2''3''' 2

=+

=+−+

fTTTffff

and the boundary conditions are

0*0'1*0'0

→→⇒∞→

===⇒=

TfTff

η

η

131 Advanced Heat Transfer (ME 211) Younes Shabany

Similarity Solutions of Boundary Layer Equations

❑ Numerical solution of these equation are shown below.

132 Advanced Heat Transfer (ME 211) Younes Shabany

Similarity Solutions of Boundary Layer Equations

❑ The local Nusselt number may be expressed as

kxTTq

khx sx )]/([Nu

''

x∞−

==

where

0

4/1x

0

'' *4Gr)(

=∞

=

⎟⎠⎞

⎜⎝⎛−−=

∂∂

−=ηηd

dTTTxk

yTkq sy

s

Therefore

(Pr)Gr*4GrNu 4/1

x0

4/1x

x gddT

khx

=⎟⎠

⎞⎜⎝

⎛−===ηη

4/1

2/1

2

Pr)2Pr21(5Pr2

43(Pr) ⎥

⎦

⎤⎢⎣

⎡

++=g

❑ Average Nusselt number over the entire length of the plate, L, is

L

4/1L

L Nu34(Pr)

4Gr

34Nu =⎟

⎠⎞

⎜⎝⎛== g

kLh

133 Advanced Heat Transfer (ME 211) Younes Shabany

Integral Solution of Laminar Natural Convection Boundary Layer

00

00

0

2

)(

)(

=

∞

∞

=

∂

∂−=−

−+∂

∂−=

∫

∫∫

y

Y

Y

y

Y

yTdyTTu

dxd

dyTTgyudyu

dxd

α

βν

22

1and1 ⎟⎠

⎞⎜⎝

⎛ −=−

−⎟⎠

⎞⎜⎝

⎛ −=∞

∞

δδδy

TTTTyy

Uu

s

❑ The integral form of the momentum and energy equations for natural convection boundary layers are

❑ Let’s assume the velocity and temperature profiles are given by

U is a reference velocity such as maximum velocity in the boundary layer.

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134 Advanced Heat Transfer (ME 211) Younes Shabany

Integral Solution of Laminar Natural Convection Boundary Layer

❑ Substituting into the integral equations gives

❑ If we assume U=C1xm and δ=C2xn, it can be shown that m = 1/2 and n = 1/4 and

❑ Boundary layer thickness and Nusselt number are

δα

δ

δβδ

νδ

2)(301

)(31)(

1051 2

=

−+−= ∞

Udxd

TTgUUdxd

s

2/14/1

3

4/14/1

2

2/1

3

2/12/1

1 PrGrPr2120

16154andGrPr

2120

354 −

−−

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛=x

Cx

C xxν

4/14/1

4/14/1

2 RaPr952.0

Pr508.0NuandPr

Pr952.093.3 xxxGrx⎟⎠

⎞⎜⎝

⎛+

=⎟⎠

⎞⎜⎝

⎛ += −δ

135 Advanced Heat Transfer (ME 211) Younes Shabany

❑ The integral form of the momentum and energy equations for turbulent natural convection boundary layers are

❑ Let’s assume the velocity and temperature profiles are given by

❑ Assuming U=C1xm and δ=C2xn, it can be shown that

Integral Solution of Turbulent Natural Convection Boundary Layer

p

sY

YsY

cqdyTTu

dxd

dyTTgdyudxd

ρ

βρτ

"

0

00

2

)(

)(

=−

−+−=

∫

∫∫

∞

∞

7/147/1

1and1 ⎟⎠

⎞⎜⎝

⎛−=−

−⎟⎠

⎞⎜⎝

⎛ −⎟⎠

⎞⎜⎝

⎛=∞

∞

δδδy

TTTTyy

Uu

s

5/25/23/2

15/1

Ra)Pr494.01(

Pr0295.0Nu xx +=

136 Advanced Heat Transfer (ME 211) Younes Shabany

Empirical Correlations for Vertical Plates ❑  For a vertical plate with length L the characteristic

length δ=L.

❑  The natural convection flow is laminar if RaL<109 and is turbulent if RaL>109.

❑  Average Nusselt number correlations for an isothermal vertical plate are

1393/1

944/1

10Ra10Ra1.0Nu

10Ra10Ra59.0Nu

<<=

<<=

LLL

LLL

L

where

ναβ 3

L)(PrGrRaNu LTTg

kLh s

LL∞−

=== and

❑  The following correlation may be used for the entire range of RaL

2

27/816/9

6/1

L]Pr)/492.0(1[

Ra387.0825.0Nu⎭⎬⎫

⎩⎨⎧

++= L

137 Advanced Heat Transfer (ME 211) Younes Shabany

Nusselt Number Correlations above Hot Horizontal Plates

❑ For a horizontal plate the characteristic length is δ=As/P where As and P are surface area and perimeter of the plate.

❑ Average Nusselt number correlations are

1173/1

744/1

10Ra10Ra15.0Nu

10Ra10Ra54.0Nu

≤≤=

≤≤=

δδδ

δδδ

Page 5

138 Advanced Heat Transfer (ME 211) Younes Shabany

Nusselt Number Correlations under Hot Horizontal Plates

❑ The characteristic length is δ=As/P where As and P are surface area and perimeter of the plate.

❑ Average Nusselt number correlations are

1154/1 10Ra10Ra27.0Nu ≤≤= δδδ

139 Advanced Heat Transfer (ME 211) Younes Shabany

Nusselt Number Correlations for Cylinders

❑ Vertical Cylinder

The characteristic length is δ=L.

If D≥35L/GrL1/4, use vertical plate correlations.

If cylinder is thin, i.e. D<35L/GrL1/4,

L

D

q Horizontal Cylinder

The characteristic length is δ=D.

Average Nusselt number, if RaD < 1012, is given by D

( )

2

27/816/9

6/1

Pr)/559.0(1Ra387.06.0Nu

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++= D

D

DLRa

kLhNu L

L Pr)6364(35Pr)315272(4

Pr)2120(5Pr7

34

4/1

++

+⎥⎦

⎤⎢⎣

⎡

+==

140 Advanced Heat Transfer (ME 211) Younes Shabany

Natural Convection in Enclosures

L

W

H

T1

T2

g

L

W

H

T1

T2

g )(" 21 TThq conv −=

ανβ 3

21 )(Ra HTTgH

−=

kLh

LTTkTTh

qqNucond

convL =

−

−==

/)()(

""

21

21

141 Advanced Heat Transfer (ME 211) Younes Shabany

Natural Convection in Enclosures (Conduction Limit)

LTTkqqRa

condconv

H

/)(""1

21 −==

≤

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142 Advanced Heat Transfer (ME 211) Younes Shabany

Natural Convection in Enclosures (Tall Enclosure Limit)

4/1RaHLH> LTTkqq condconv /)("" 21 −==

143 Advanced Heat Transfer (ME 211) Younes Shabany

Natural Convection in Enclosures (Shallow Enclosure Limit)

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

H / L = 1.0H / L = 0.2H / L = 0.1H / L = 0.05H / L = 0.0191H / L = 0.0103

102 103 104 105 106 107

104

103

102

10

1

LNu

)/(Ra LHH

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

H / L = 1.0H / L = 0.2H / L = 0.1H / L = 0.05H / L = 0.0191H / L = 0.0103

102 103 104 105 106 107

104

103

102

10

1

LNu

)/(Ra LHH

4/1Ra−< HLH

144 Advanced Heat Transfer (ME 211) Younes Shabany

Natural Convection in Enclosures (Boundary Layer Regime)

4/14/1 RaRa HH LH<<−

4/11 RaHLHLC

kLhNu ==

0

0.1

0.2

0.3

0.4

0.5

1 10 100 1000

1C

7/17/4 Ra)/( HLH

0

0.1

0.2

0.3

0.4

0.5

1 10 100 1000

1C

7/17/4 Ra)/( HLH