ME150_Lect14-1_Natural Convection
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Prof. Nico Hotz ME 150 – Heat and Mass Transfer 1 Natural Convection - No (externally) forced flow - Flow is driven by density differences in the gravity field Example: Space between two horizontal plates 0 0 > → < dy d dy dT ρ e.g. Floor heating e.g. Ceiling heating 0 0 < → > dy d dy dT ρ Chap. 15: Natural Convection y T 1 T 2 > T 1 der Flüssigkeit T 1 T 2 ρ 2 ρ 1 y Τ,ρ Instable flow y T 1 T 2 < T 1 T 1 T 2 ρ 2 ρ 1 y Τ,ρ Stable fluid, no flow
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dT dT ρ 1 y Chap. 15: Natural Convection Stable fluid, no flow T 2 ρ 2 instabile Bewegung der Flüssigkeit Instable flow T 2 ρ 2 Prof. Nico Hotz ρ ρ stabile Flüssigkeit (keine Bewegung) 1 T 1 T 1 T 1 T 1 T 2 > T 1 T 2 < T 1 y y Τ,ρ Τ,ρ gas with buoyancy T ∞ , ρ ∞ T>T ∞ ρ< ρ ∞ T ∞ , ρ ∞ T>T ∞ ρ< ρ ∞ Outlet of a hot exhaust Chap. 15: Natural Convection Prof. Nico Hotz 2 T ∞ , ρ ∞ T ∞ , ρ ∞ T >T ∞ ρ < ρ ∞ T >T ∞ ρ < ρ ∞
Transcript of ME150_Lect14-1_Natural Convection
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
1
Natural Convection - No (externally) forced flow - Flow is driven by density differences in the gravity field
Example: Space between two horizontal plates
00 >→<dyd
dydT ρ
e.g. Floor heating
e.g. Ceiling heating
00 <→>dyd
dydT ρ
Chap. 15: Natural Convection
y
T1
T2 > T1
instabileBewegung
der Flüssigkeit
T1
T2ρ2
ρ1
y
Τ,ρ
Instable flow
y
T1
T2 < T1
stabile Flüssigkeit(keine Bewegung)
T1
T2 ρ2
ρ1
y
Τ,ρ
Stable fluid, no flow
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
2
T ∞ , ρ ∞
T >T ∞ ρ < ρ ∞
Natural convection from a heated wire
Outlet of a hot exhaust gas with buoyancy
T ∞ , ρ ∞ T>T ∞ ρ< ρ ∞
T ∞ , ρ ∞
T >T ∞ ρ < ρ ∞
Natürliche Konvektion von einem geheizten Draht
Outlet of a hot exhaust gas with buoyancy
T ∞ , ρ ∞ T>T ∞ ρ< ρ ∞
Example: thermal buoyancy flow from a hot wire (development of a plume)
Mix of forced and natural convection: hot jet of exhaust gas
Chap. 15: Natural Convection
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
3
Natural Convection on a Vertical Plate
Velocity bounary layer: v(0) and v(∞) = 0
vmax is within the boundary layer
Main flow direction: in y-direction
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
4
0=+yv
xu
∂∂
∂∂
2
2
xvg
dydP
yvv
xvu
∂
∂µρ
∂∂
∂∂
ρ ⋅+⋅−−=⎟⎟⎠
⎞⎜⎜⎝
⎛⋅+⋅⋅
2
2
xT
yTv
xTu
∂∂
α∂∂
∂∂
⋅=⋅+⋅
Boundary Layer Equations (with y-momentum)
C
M
E Gravity is relevant
)()(),( yPyPyxP ∞≈≈
As for forced convection: no pressure gradient within the boundary layer:
gdydP
⋅−= ∞∞ ρ
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
5
2
2)(
xvg
yvv
xvu
∂
∂µρρ
∂∂
∂∂
ρ ⋅+⋅−=⎟⎟⎠
⎞⎜⎜⎝
⎛⋅+⋅⋅ ∞
Pressure term used in momentum equation:
Boussinesq Approximation: density differences can be neglected, except where they appear in terms multiplied by g
constant
..........)( +−⋅⎟⎟⎠
⎞⎜⎜⎝
⎛+= ∞
∞∞ TT
T∂ρ∂
ρρ
pT ⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−=∂ρ∂
ρβ
1
)()()( ∞∞∞∞∞∞ −⋅⋅=−→−⋅⋅−= TTTT βρρρβρρρ
Series expansion for density:
Thermal expansion coefficient:
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
6
ρµ
∂∂
β∂∂
∂∂
=⋅+−⋅⋅=⋅+⋅ ∞ νν
FrictionBuoyancy
Inertia
xvTTg
yvv
xvu 2
2
)(
Momentum equation: Density difference in terms of temperature difference
0=+yv
xu
∂∂
∂∂
2
2
)(xvTTg
yvv
xvu
∂∂
β∂∂
∂∂
⋅+−⋅⋅=⋅+⋅ ∞ ν
2
2
xT
yTv
xTu
∂∂
α∂∂
∂∂
⋅=⋅+⋅
Entire equation system:
∞=
=∞→
=
===
TTvx
TTvux
W
0:
0:0
Boundary conditions: M
C
E
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
7
vvuu
TTT
Hyx
W
t
≈≈
−≈
≈≈
∞
δ
Orders of magnitude for temperature boundary layer:
Hvu
t≈
δ
2)(tT
wt
vTTgHvvvu
δβ
δ⋅+−⋅⋅≈⋅+⋅ ∞ ν
Δ
2t
ww
t
w TTHTTvTTu
δα
δ∞∞∞ −
⋅≈−
⋅+−
⋅
Orders of magnitude
Used in governing equations:
C
M
E
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
8
Friction
tBuoyancyInertia
vTgHv
2
2
δβ ⋅+Δ⋅⋅≈ ν
This leads to:
Hvu tδ⋅=
2 Limits: - Inertia negligible → gases with high viscosity - Friction negligible (Buoyancy is always relevant)
Limit 1: Buoyancy = Friction (no inertia)
Unknown: u, v, δt to be determined from governing equations
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
9
∞−=Δ⋅
⋅Δ⋅⋅= TTTHTgRa wH α
βν
3
HRa
HTgu H
ααβ⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛
⋅
⋅Δ⋅⋅= 4
141
3
ν
HRaHTgv H
ααβ⋅=⎟
⎠
⎞⎜⎝
⎛ ⋅⋅Δ⋅⋅= 2
121
ν
HRaTgH
Ht ⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛
Δ⋅⋅
⋅⋅=
− 414
1
βα
δν
Introduction of dimensionless Rayleigh-Number Ra, characte-ristic for natural convection (forced convection: Re, Pr)
Solution (without derivation):
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
10
t
TkThqδΔ⋅≈Δ⋅=ʹ′ʹ′
HRakkh H
t
41
⋅≈≈
δ
41
RakHhNu ≈⋅
=
Convective heat transfer coefficient through Fourier‘s Law:
Nusselt number for natural convection on a vertical plate (friction dominant):
Order of magnitude analysis: shows functional relations, but not exact values !
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
11
Limit 2: Buoyancy = Inertia (no friction)
( ) 41Pr⋅⋅≈ HRaH
u α
( ) 21Pr⋅⋅≈ HRaH
v α
( ) 41
Pr −⋅⋅≈ Ht RaHδ
( ) 41Pr⋅≈⋅
≈ HRakHhNu
Solution (without derivation):
( )H
Rakkh H
t
41
Pr⋅⋅≈≈
δ
Nusselt number for natural convection on a vertical plate (inertia dominant)
Chap. 15.1: Natural Convection – Vertical Plate
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
12
