Chapter 9

Natural Convection

Forced convection--with external forcing conditionNatural (or free) convection--driven by buoyancy force, which is induced by body force with the presence of density gradient

9.1 Physical Considerations

9.2 The Governing Equationsx-mom. eq.:

With from the y-mom. eq., the x-pressure gradient in the b.l. must equal to that in the quiescent region outside the b.l.,

i.e.,

So,

Introducing the volumetric thermal expansion coefficient

(9.1)

(9.2)

(9.3)

u ∂u∂x

+ v ∂u∂y

= − 1ρ∂p∂x

− g+ ν ∂2u

∂y2

∂p / ∂y =0

∂p∂x

= −ρ∞g

u ∂u∂x

+ v ∂u∂y

=gρ (ρ∞ −ρ) + ν

∂ 2u∂y2

1 1pT T T

ρ ρ∂ρβ ρ ∂ ρ∞

∞

−⎛ ⎞= − ≈ −⎜ ⎟ −⎝ ⎠

2

2( )u u uu v g T Tx y y

∂ ∂ ∂β ν∂ ∂ ∂∞

→ + = − +

(9.4)

(9.5)

211 1(Note: For ideal gases, )

p

pT RT T∂ρβ ρ ∂ ρ

⎛ ⎞= − = =⎜ ⎟⎝ ⎠ (9.9)

The set of governing equations for laminar free convection associated with a vertical heated plate are:

continuity eq.:

x-mom. eq.:

energy eq.:

Note the dissipation is neglected in (9.8) and Eqs. (9.6)-(9.8) are strongly coupled and must be solved simultaneously.

∂u∂x

+∂v∂y

= 0

u ∂u∂x

+ v ∂u∂y

= gβ(T − T∞ ) + ν∂ 2u∂y2

u ∂T∂x

+ v ∂T∂y

= α ∂2T∂y2

(9.6)

(9.7)

(9.8)

9.3 Similarity ConsiderationsDefining

Eqs. (9.7) and (9.8) reduce to

(9.10) →

where

* and * , is the characteristic lengthyxx y LL L≡ ≡

T* ≡ T −T∞Ts − T∞

(9.10)

(9.11)

(9.12)

(9.10a)

00 0

* and * , is an arbitrary reference velocityu vu v uu u≡ ≡

2*

1/2 2

* * *1* *( )* * *L

u u uu v TGrx y y

∂ ∂ ∂∂ ∂ ∂

+ = +3

2

( )sL

g T T LGr βν

∞−≡

2

2

* * 1 ** ** * *L

T T Tu vx y Re Pr y

∂ ∂ ∂∂ ∂ ∂

+ =

2*

2 20

( )* * *1* ** * *

s

L

g T T Lu u uu v T Rex y u yβ∂ ∂ ∂

∂ ∂ ∂∞−+ = +

1→ u0=[gβ(Ts-T∞)L]1/2

•GrL plays the same role in free convection that ReL plays in forced convection.

If there is a non-zero free stream velocity, u∞, we may use u0= u∞.Then

Generally,→both free & forced convection to be considered

→forced convection

→free convection

2( / ) 1L LGr Re ≈( , , )L L LNu f Re Gr Pr→ =

2( / ) 1L LGr Re >( , )L LNu f Gr Pr→ =

23

22 2

( ) ( )s LsL

g T T L Grg T T L u Lu Re

β βνν

∞ ∞ ∞

∞

− − ⎛ ⎞= =⎜ ⎟⎝ ⎠

(9.10b)2

*2 2

* * *1Eq. 9.10 becomes * ** * *

L

LL

Gru u uu v T Rex y Re y∂ ∂ ∂∂ ∂ ∂

→ + = +

Alternative derivation of Gr under purely natural convectionEqs. (9.10) can be also written as

If u0 is set to make u0L/ν≣1, or u0=ν/L

2*

2 200

( )* * ** ** * *

sg T T Lu u uu v T u Lx y u yβ∂ ∂ ∂ν

∂ ∂ ∂∞−+ = +

(9.10’)

(9.11’)

(9.12)

3 2*

2 2

( )* * ** ** * *

sg T T Lu u uu v Tx y y

β∂ ∂ ∂∂ ∂ ν ∂

∞−→ + = +

3

2

( )where sLg T T LGr β

ν∞−≡

2

2

* * 1 ** ** * *

T T Tu vx y Pr y

∂ ∂ ∂∂ ∂ ∂

+ =

9.4 Laminar Free Convection on a Vertical SurfaceIntroducing the similarity parameter

Eqs. (9.6 to 9.8) can be reduced to

The numerical results are shown in Fig. 9.4.

where g(Pr) is determined numerically determined as (9.20).And

1/ 4 1/ 4

and ( , ) ( ) 44 4x xGr Gry x y fxη ψ η ν

⎡ ⎤⎛ ⎞ ⎛ ⎞≡ ≡ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

f ' ' ' +3 ff "−2( f ' )2 +T * = 0*" *n3 0T Pr f T+ =

1/ 4 1/ 4*

0

( )4 4x x

xGr Grhx dTNu g Prk d ηη =

⎛ ⎞ ⎛ ⎞= = − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1/ 44 ( )3 4

xL

GrhLNu g Prk⎛ ⎞= = ⎜ ⎟⎝ ⎠

(9.17)

(9.18)

(9.19)

(9.21)

9.5 The Effects of TurbulenceFor vertical plates the transition occurs at

EX 9.1

9.6 Empirical Correlations: External Free Convection FlowsGenerally,

, n=1/4 for laminar, n=1/3 for turbulent flow

Table 9.2 (p. 583) summarizes the empirical correlations for different immersed geometries.

EX 9.2

39

, ,( ) 10sx c x c

g T T xRa Gr Pr βνα

∞−= = ≈

NuL = h Lk = CRaLn

(9.23)

F Kreith & MS Bohn, Principles of Heat Transfer, 2001

Some other flow conditions in 9.6

Flow Pattern

Reference: A. Bar-Cohen and W.M. Rohsenow, Thermally optimum spacing of vertical, natural convection cooled, parallel plates, ASME J. Heat Transfer, 106 (1984) 116-123.

9.7 Natural Heat Transfer Between Parallel Plates

1/ 2

1 22

3

,( / ) /

where ( ) /

S

S S

S s

C CNuRa S L Ra S L

Ra g T T Sβ αν

−

∞

⎡ ⎤= +⎢ ⎥⎢ ⎥⎣ ⎦

≡ − (9.45) (or qs”=c)

Cengel, Heat Transfer

Vertical Parallel Plates:

Eq. (9.45) is suitable for different thermal conditions of the plates, isothermal or isoflux plates, symmetric or with one plate adiabatic. The different values of C1 and C2 for each condition are given in Table 9.3.

Eq. (9.45) is commonly used for vertical plate heat sinks, although this can be inaccurate for short fins (H/S200,

1/ 40.645( / )S SNu Ra S L= (9.47)

Cengel, Heat Transfer