ME 150 – Heat and Mass Transfer
Chap. 15: Natural Convection
Natural Convection - No (externally) forced flow - Flow is driven by density differences in the gravity field Example: Space between two horizontal plates T1
e.g. Floor heating dT <0 dy
→
dρ >0 dy
y
→
dρ <0 dy
T1
ρ1
instabile Bewegung Instable flow der Flüssigkeit Τ,ρ T2 > T1
e.g. Ceiling heating dT >0 dy
y
T1
y
y
ρ2
T2
ρ1
T1
Stable fluid, stabile Flüssigkeit (keine Bewegung) no flow Τ,ρ T2 < T 1
Prof. Nico Hotz
T2
ρ2
1
ME 150 – Heat and Mass Transfer
T >T∞ ρ < ρ∞
T∞ , ρ ∞
Natural convection from a heated wire
Chap. 15: Natural Convection
Example: thermal T∞ , ρ ∞ buoyancy flow from T>T∞ a ρ< ρ ∞ hot wire (development of a plume) T >T∞ ρ < ρ∞
Outlet of a hot exhaust gas with buoyancy
T∞ , ρ ∞
T>T∞ ρ< ρ ∞
T∞ , ρ ∞
Mix of forced and natural convection: Natürliche Konvektion geheizten hot jetvon ofeinem exhaust gas Draht
Prof. Nico Hotz
Outlet of a hot exhaust
gas with buoyancy
2
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
Natural Convection on a Vertical Plate
Main flow direction: in y-direction Velocity bounary layer: v(0) and v(∞) = 0 vmax is within the boundary layer
Prof. Nico Hotz
3
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
Boundary Layer Equations (with y-momentum)
C
∂u ∂v + =0 ∂ x ∂ y
M
⎛ ∂ v ∂ v ⎞ ∂ 2v dP ⎟⎟ = − ρ ⋅ ⎜⎜ u ⋅ +v⋅ −ρ⋅g+µ⋅ ∂ y ⎠ dy ∂ x2 ⎝ ∂ x
E
∂T ∂T ∂ 2T u⋅ +v⋅ =α ⋅ ∂x ∂ y ∂ x2
Gravity is relevant
As for forced convection: no pressure gradient within the boundary layer: P( x, y) ≈ P( y) ≈ P∞ ( y)
Prof. Nico Hotz
dP∞ = −ρ∞ ⋅ g dy 4
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
Pressure term used in momentum equation: constant
⎛ ∂ v ∂ v ⎞ ∂ 2v ⎟⎟ = ( ρ ∞ − ρ ) ⋅ g + µ ⋅ ρ ⋅ ⎜⎜ u ⋅ +v⋅ ∂ y ⎠ ∂ x2 ⎝ ∂ x
Boussinesq Approximation: density differences can be neglected, except where they appear in terms multiplied by g Series expansion for density:
Thermal expansion coefficient:
ρ = ρ ∞ − ρ ∞ ⋅ β ⋅ (T − T∞ )
⎛ ∂ ρ ⎞ ⎟⎟ ⋅ (T − T∞ ) + .... . . ... ρ = ρ ∞ + ⎜⎜ ⎝ ∂ T ⎠ ∞
β =−
→
Prof. Nico Hotz
1 ⎛ ∂ ρ ⎞ ⎟ ⋅ ⎜ ρ ⎜⎝ ∂ T ⎟⎠ p
( ρ ∞ − ρ ) = ρ ∞ ⋅ β ⋅(T − T∞ )
5
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
Momentum equation: Density difference in terms of temperature difference ∂v ∂v ∂ 2v u⋅ +v⋅ = g ⋅ β ⋅(T − T∞ ) + ν ⋅ ∂x ∂ y ∂ x2 Buoyancy
ν=
Inertia
µ ρ
Friction
Entire equation system: C
∂u ∂v + =0 ∂ x ∂ y
∂v ∂v ∂ 2v + v⋅ = g ⋅ β ⋅ (T − T∞ ) + ν ⋅ M u⋅ ∂x ∂ y ∂ x2
Boundary conditions: x = 0:
T = TW
2
E
u⋅
u=v=0
∂T ∂T ∂ T +v⋅ =α ⋅ ∂x ∂ y ∂ x2
x →∞:
Prof. Nico Hotz
v =0 T = T∞ 6
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
Orders of magnitude for temperature boundary layer: x ≈ δt y≈H Orders of magnitude
T ≈ TW − T∞ u≈u
Used in governing equations:
u
C
δt
M
u⋅
E
u⋅
≈ v
δt
v≈v
v H +v⋅
v v ≈ g ⋅ β ⋅ (Tw − T∞ ) + ν ⋅ 2 H δt ΔT
Tw − T∞
δt
+v ⋅
Tw − T∞ T −T ≈α ⋅ w 2 ∞ H δt
Prof. Nico Hotz
7
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
This leads to: v ⋅ δt u = H
v2 v ≈ g ⋅ β ⋅ ΔT + ν ⋅ 2 H δt Buoyancy Inertia Friction
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
Prof. Nico Hotz
8
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
Introduction of dimensionless Rayleigh-Number Ra, characteristic for natural convection (forced convection: Re, Pr) g ⋅ β ⋅ ΔT ⋅ H 3 RaH = ν ⋅α
ΔT = Tw − T∞
Solution (without derivation): ⎛ g ⋅ β ⋅ ΔT ⋅ α 3 ⎞ ⎟⎟ u = ⎜⎜ ν ⋅H ⎝ ⎠
1
⎛ ν ⋅ α ⋅ H ⎞ ⎟⎟ δ t = ⎜⎜ ⎝ g ⋅ β ⋅ ΔT ⎠
1
= Ra H 4 ⋅
⎛ g ⋅ β ⋅ ΔT ⋅ α ⋅ H ⎞ v = ⎜ ⎟ ν ⎝ ⎠ 1
4
1
4
2
α H
1
= Ra H 2 ⋅
α H
−1
= Ra H 4 ⋅ H
Prof. Nico Hotz
9
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
Convective heat transfer coefficient through Fourier‘s Law: qʹ′ʹ′ = h ⋅ ΔT ≈ k ⋅
k
ΔT
δt 1
k ⋅ Ra H4 h ≈ ≈ δt H
Nusselt number for natural convection on a vertical plate (friction dominant): Nu =
1 h ⋅H ≈ Ra 4 k
Order of magnitude analysis: shows functional relations, but not exact values !
Prof. Nico Hotz
10
ME 150 – Heat and Mass Transfer
Chap. 15.1: Natural Convection – Vertical Plate
Limit 2: Buoyancy = Inertia (no friction) Solution (without derivation): u ≈
α
v≈
α
H
H
⋅ (Ra H ⋅ Pr )
⋅ (RaH ⋅ Pr )
1
1
4
2
δ t ≈ H ⋅ (Ra H ⋅ Pr )−
1
4
Nusselt number for natural convection on a vertical plate (inertia dominant) k ⋅ (RaH ⋅ Pr ) h ≈ ≈ δt H k
1
4
Nu ≈
Prof. Nico Hotz
h ⋅H 14 ≈ (Ra H ⋅ Pr ) k
11
ME 150 â&#x20AC;&#x201C; Heat and Mass Transfer
Prof. Nico Hotz
12