net force on a body completely in a fluid · u ul t u g t l v x u u s ∂ ∂ + − = ∂ + ∞ ν...
TRANSCRIPT
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.
Page 2
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
Page 3
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.
Page 4
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 −==
≤
Page 6
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