Post on 02-Feb-2021
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− −
=+