NATURAL CONVECTION

• Laminar Free Convection on a Vertical Surface

• Empirical Correlations: External Free Convection Flows

• Free Convection within Parallel Plate Channels

• Empirical Correlations: Enclosures

Free-convection boundary layers

Laminar Free Convection on a Vertical PlateBoussinesq approximationBasic idea : In many problems, δρ/ρ∞

Ex) water at 15°C and 1 atmthermal expansion coefficient : 4 1

1 1.5 10 KTρβ

ρ− −

∞ ∞

∂= − = ×

∂

isothermal compressibility : 51 5 10pρκ

ρ−

∞ ∞

∂= = ×

∂bar-1

When bar,10 K, 1T T p p∞ ∞− = − =

whilst the contribution of p – p∞ to δρ/ρ∞ is only 5×10-5.

( ) ( )1 1T T p pT p

ρ ρ ρ ρρ ρ ρ

∞∞ ∞

∞ ∞ ∞∞ ∞

− ∂ ∂= − + − +

∂ ∂

( ) ( )( , )T p T T p pT pρ ρρ ρ∞ ∞ ∞

∞ ∞

∂ ∂= + − + − +

∂ ∂

the contribution of T – T∞ to δρ/ρ∞ is 1.5×10-3,

When the dynamic pressure is negligible,

( ) ( )1 1T T p pT p

ρ ρ ρ ρρ ρ ρ

∞∞ ∞

∞ ∞ ∞∞ ∞

− ∂ ∂= − + − +

∂ ∂

( )T Tβ ∞≈ − −

Thus, ( )1 T Tρ ρ β∞ ∞⎡ ⎤= − −⎣ ⎦dp gx pρ∞= − +

dp gdx

ρ∞= −

2

2

u u dp uu v gx y dx y

ρ ρ μ ρ∂ ∂ ∂+ = − + −∂ ∂ ∂

( )2

2 1u u uu v g T T gx y y

ρ ρ ρ μ ρ β∞ ∞ ∞∂ ∂ ∂

⎡ ⎤+ = + − − −⎣ ⎦∂ ∂ ∂

( ) ( )1 1u uT T u T T vx y

ρ β ρ β∞ ∞ ∞ ∞∂ ∂

⎡ ⎤ ⎡ ⎤− − + − −⎣ ⎦ ⎣ ⎦∂ ∂

( )2

2

u g T Ty

μ ρ β∞ ∞∂

= + −∂

( )2

2

u u uu v g T Tx y y

ν β ∞∂ ∂ ∂

+ = + −∂ ∂ ∂

When ( ) 1,T Tβ ∞−

Similarity Solution to 2-D Boundary Layer Equations with Boussinesq Approximation

boundary conditions

0u vx y∂ ∂

+ =∂ ∂

( )2

2

u u uu v g T T vx y y

β ∞∂ ∂ ∂

+ = − +∂ ∂ ∂

2

2

T T Tu vx y y

α∂ ∂ ∂+ =∂ ∂ ∂

at y = 0, 0, su v T T= =at x = 00,u T T∞= =as y→∞0,u T T∞→ →

, ,s

T Tu vy x T Tψ ψ θ ∞

∞

∂ ∂ −= = − =∂ ∂ −

Scaling : ( )2

2

u u uu v g T T vx y y

β ∞∂ ∂ ∂

+ = − +∂ ∂ ∂

( )2

2~ ~u uu g T T vx y

β ∞∂ ∂

−∂ ∂

( )0 00 2~ ~u uu g T T vx

βδ∞

→ −

( )2 0

0 2~ , ~s

vu vxug T T

δβ δ∞−

( )2

2~s

v vxg T T

δβ δ∞

→−

similarity variable :

Grashof number :

( )2

4 ~s

v xg T T

δβ ∞−

or( )

1/ 42

s

v xg T T

δβ ∞

⎛ ⎞= ⎜ ⎟

−⎝ ⎠

( ) 1/ 42~sg T Ty y

v xβ

ηδ

∞⎛ ⎞−= ⎜ ⎟⎝ ⎠

( ) 1/ 43 1/ 42 Gr

sx

g T T xy yx v x

β ∞⎛ ⎞−= ≡⎜ ⎟⎝ ⎠

( ) 32Gr

sx

g T T xv

β ∞−=

Let ( )1/ 41/ 4

2

Gr4 4

sx g T Ty yx v x

βη ∞

⎛ ⎞−⎛ ⎞= = ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

0~u uyψ ψ δ∂= →∂ 0 2

~ vxuδ

⎛ ⎞⎜ ⎟⎝ ⎠

( ) 1/ 42 2~ ~ ~

sg T Tvx vx vxv x

βδ

δ δ∞⎛ ⎞−

⎜ ⎟⎝ ⎠

( ) 1/ 43 1/ 42 Gr

sx

g T T xv v

vβ ∞⎛ ⎞−= =⎜ ⎟

⎝ ⎠

Let( )1/ 4

( )4 Gr / 4x

fv

ψ η≡

or1/ 4Gr4 ( )

4xv fψ η⎛ ⎞= ⎜ ⎟

⎝ ⎠

similarity equations:

boundary conditions:

1/ 4Gr44

x dfu vy d yψ η

η∂ ∂⎛ ⎞= = ⎜ ⎟∂ ∂⎝ ⎠

1/ 4 1/ 41/ 2Gr1 24 Gr

4 4x x

xGr vv f f

x x⎛ ⎞ ⎛ ⎞ ′ ′= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

similarly for other terms2 2

2 2, , , , , , , u u uvx y y x y y

θ θ θθ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

23 2 0, 3Pr 0f ff f fθ θ θ′′′ ′′ ′ ′′ ′+ − + = + =

(0) 0, (0) 0, ( ) 0, (0) 1, ( ) 0f f f θ θ′ ′= = ∞ = = ∞ =

Nusselt number1/ 4Gr,

4x

s

T T yT T x

θ η∞∞

− ⎛ ⎞= = ⎜ ⎟− ⎝ ⎠

( )2

0 0s

sy

T v x dq k ky g T T d yη

θ ηβ η∞= =

∂ ∂′′ = − = −∂ − ∂

( )

1/ 42 1 1(0) Gr4 xs

v xkg T T x

θβ ∞

⎛ ⎞′= − ⎜ ⎟− ⎝ ⎠

( )sh T T∞= −1/ 4 1/ 4Gr Gr(0) (Pr)

4 4x xk kh g

x xθ⎛ ⎞ ⎛ ⎞′= − = −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

1/ 4(0)Nu Gr2x x

hxk

θ ′= = −

1/ 4 1/ 4Gr Gr(0) (Pr)4 4

x xk kh gx x

θ⎛ ⎞ ⎛ ⎞′= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1/ 4GrNu (Pr)2x

x g=

( )1/ 2

1/ 41/ 2

0.75Pr(Pr)0.609 1.221Pr 1.238Pr

g =+ +

Average Nusselt number1/ 4

0 0

Gr1 1 (Pr)4

L Lxkh hdx g dx

L L x⎛ ⎞= = ⎜ ⎟⎝ ⎠∫ ∫

( ) 1/ 42 1/ 40

(Pr)4

Lsg T Tk dxgL v x

β ∞⎛ ⎞−= ⎜ ⎟⎝ ⎠

∫( ) 1/ 43

2

4 (Pr)3 4

sg T T Lk gL v

β ∞⎛ ⎞−= ⎜ ⎟⎝ ⎠

1/ 44 Gr(Pr)3 4

Lk gL

⎛ ⎞= ⎜ ⎟⎝ ⎠

1/ 44 Gr 4Nu (Pr) Nu3 4 3

LL L

hL gk

⎛ ⎞= = =⎜ ⎟⎝ ⎠

limiting cases:

Rayleigh number :

( )( )

1/ 4 1/ 4

1/ 4

0.600 Gr Pr Pr Pr 0Nu

0.503 Gr Pr Prx

x

x

⎧ →⎪= ⎨→∞⎪⎩

asas

Ra Gr Prx x=

( ) ( )32

3ssg T T x T

v vv g T xβα

βα

∞ ∞−=−

=

Critical Rayleigh number

( ) 3 9, ,Ra Gr Pr 10

sx c x c

g T T xβνα

∞−= = ≈

laminar:

turbulent:

isothermal plates

Nusselt number

Vertical PlateEmpirical Correlations: External Flows

Nu RanL LhL Ck

= =

10.59, 2

C n= =

10.10, 3

C n= =

• Churchill and Chu (1975)

for all RaL

( )

2

1/ 6

8 / 279 /16

0.387RaNu 0.8251 0.429 / Pr

LL

⎧ ⎫⎪ ⎪= +⎨ ⎬

⎡ ⎤⎪ ⎪+⎣ ⎦⎩ ⎭

( )

( )

1/ 4

4 / 99 /16

9

0.670RaNu 0.68 1 0.429 / Pr

Ra 10

LL

L

= +⎡ ⎤+⎣ ⎦

≤

Inclined and Horizontal PlatesInclined plates

Use correlations for the vertical plates by replacing g by gcosθ for 0 60θ≤ ≤

Horizontal plates

upper surface of heated plate or lower surface of cooled plate

(As: plate surface area, P: perimeter)sALP

≡

1/ 4Nu 0.54Ra L L=

( )4 710 Ra 10L< <1/ 3Nu 0.15Ra L L=

( )7 1110 Ra 10L<

lower surface of heated plate or upper surface of cooled plate

( )1/ 4 5 10Nu 0.27Ra 10 Ra 10L L L= <

Long Horizontal Cylinder

• Morgan (1975) C, n Table 9.1

• Churchil and Chu (1975)

Nu Ra ,nD DhD Ck

= =

( )

2

1/ 612

8 / 279 /16

0.387RaNu 0.60 , Ra 101 0.559 / Pr

DD D

⎧ ⎫⎪ ⎪= +

Spheres• Churchil (1983)

( )

1/ 4

4 / 99 /16

0.589RaNu 21 0.469 / Pr

DD = +

⎡ ⎤+⎣ ⎦

11Pr 0.7, Ra 10D≥

Example 9.3

air flow

long duct

Find: Heat loss from duct per meter of length

Assumption: Radiative effects are negligible.

45 CsT = °0.75 mw =

0.3 mH =

15 CT∞ = °quiescent air

heat loss from two side walls, top wall and bottom wallside wall: vertical platetop wall: upper surface of heated platebottom wall: lower surface of heated plate

air flow

long duct

45 CsT = °0.75 mw =

0.3 mH =

15 CT∞ = °quiescent air

2 s t bq qq q= + ′ + ′′ ′

( ) ( ) ( )2 s ss t bsH T T w Th h hT w T T∞ ∞ ∞= − + − + −

side wall: vertical plate

Churchill and Chu (1975)

for all RaL( )

2

1/ 6

8 / 279/16

0.387RaNu 0.8251 0.429 / Pr

LL

⎧ ⎫⎪ ⎪= +⎨ ⎬

⎡ ⎤⎪ ⎪+⎣ ⎦⎩ ⎭

( )( )

1/ 49

4/ 99 /16

0.670RaNu 0.68 Ra 101 0.429 / Pr

LL L= + ≤

⎡ ⎤+⎣ ⎦air: 6 2 6 2

1

16.2 10 m /s , 22.9 10 m / s0.0265 W/m K , 0.0033 K , Pr 0.71k

ν α

β

− −

−

= × = ×

= ⋅ = =

( ) 3 7Ra 7.07 10sLg T T Hβ

να∞−= = ×

2Nu 4.23 W/m KLskH

h = = ⋅

top wall: upper surface of heated plate

1/ 4 4 7Nu 0.54Ra (10 Ra 10 ),L L L= < <

1/ 3 7 11Nu 0.15Ra (10 Ra 10 )L L L= < <

(As: plate surface area, P : perimeter)sALP

≡

( )0.375m

2 2 2wL wL wLw L L

= ≈ = =+

( ) ( )3 8/ 2Ra 1.38 10sLg T T wβ

να∞−= = ×

( ) 1/ 3/ 2 0.15RaLth w

k=

( ) 1/ 3 2/ / 2 0.15Ra 5.47 W/m Kt Lkh w⎡ ⎤= × = ⋅⎣ ⎦

bottom wall: lower surface of heated plate

1/ 4 5 10Nu 0.27Ra (10 Ra 10 )L L L= < <

( ) ( )3 8/ 2Ra 1.38 10sLg T T wβ

να∞−= = ×

( ) 1/ 4 2/ / 2 0.27Ra 2.07 W/m Kb Lkh w⎡ ⎤= × = ⋅⎣ ⎦

( )( )2 2s t b s t b sq q qq h H h w h w T T∞′ ′ ′= + + = ⋅ + ⋅ + ⋅ −′246 W / m=

Example 9.4

total heat loss per unit length of pipe

Find: Heat loss from the pipe per unit length [W/m]q′

quiescent air23 CT∞ = °

0.1m

0.85ε = 165 CsT = °

sur 23 CT = °convq′ radq′

conv radq qq = +′ ′′ ( ) ( )4 4surs sD T T Th D Tπ επ σ∞= − + −

long horizontal cylinder

Churchil and Chu (1975)2

1/ 612

8/ 279/16

0.387RaNu 0.60 Ra 101 (0.559 / Pr)

DD D

⎧ ⎫⎪ ⎪= +

Vertical Channels• Elenbaas (1942)

symmetrically heated isothermal plates

fully developed limit (S/L→0)

Free Convection in Parallel Plate Channels

( )

3 / 4

1 35Nu Ra 1 exp24 Ra /S S S

SL S L

⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= − −⎨ ⎬⎢ ⎥⎜ ⎟⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

( )(fd)

Ra /Nu

24S

s

S L=

/Nu ,Ss

q A ST T k∞

⎛ ⎞= ⎜ ⎟−⎝ ⎠

( ) 3Ra sS

g T T Sβαν

∞−=

S: channel widthL: channel length

Inclined Channels• Azevedo and Sparrow (1985)

0 45θ≤ ≤

( ) 1/ 4Nu 0.645 Ra /S S S L⎡ ⎤= ⎣ ⎦

Rectangular CavitiesEmpirical Correlations: Enclosures

( )1 2q h T T′′ = −

Horizontal cavity heated from below

longitudinal roll cells

conduction regime( ) 31 2Ra 1708L

g T T Lβαν−

= <

Nu 1L =

41708 < Ra 5 10L ≤ ×

• Globe and Dropkin (1959)1/ 3 0.074Nu 0.069Ra PrL L

hLk

= =

5 93 10 Ra 7 10L× < < ×

Vertical rectangular cavityconduction regime

• Catton (1978)for aspect ratio

0.29

3 5

3

PrNu 0.18 Ra0.2 Pr

1 2

10 Pr 10Ra Pr100.2 Pr

L L

L

HL

−

⎛ ⎞= ⎜ ⎟+⎝ ⎠⎡ ⎤<

for large aspect ratio• MacGregor and Emery (1969)

0.31/ 4 0.012 4

4 7

10 40

Nu 0.42Ra Pr 1 Pr 2 1010 Ra 10

L L

L

HLH

L

−

⎡ ⎤<

Concentric Cylinders

keff: effective thermal conductivity

for

for

( ) ( )eff2

ln / i oo i

kq T TD Dπ′ = −

( )1/ 4

1/ 4*eff Pr0.386 Ra0.861 Pr c

kk

⎛ ⎞= ⎜ ⎟+⎝ ⎠

2 * 710 Ra 10c≤ ≤

[ ]( )

4

*53 3 / 5 3 / 5

ln( / )Ra Rao ic L

i o

D D

L D D− −=

+

effk k=

*Ra 100c

Concentric Spheres• Raithby and Hollands (1975)

for

( )eff i o i oD Dq k T T

Lπ ⎛ ⎞= −⎜ ⎟⎝ ⎠

2 * 410 Ra 10c≤ ≤

( )1/ 4

1/ 4*eff Pr0.74 Ra0.861 Pr c

kk

⎛ ⎞= ⎜ ⎟+⎝ ⎠

( ) ( )*

4 57 / 5 7 / 5

RaRa/

Ls

o i i o

LD D D D− −

=+