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

ME150_Lect14-1_Natural Convection-1
ME150_Lect14-1_Natural Convection-1

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 H...