NATURAL CONVECTION
• No mechanical force to push the fluid – pump, fan etc.
• No predefined fluid flowrate and velocity – can’t
prescribe Reynolds number
• Fluid moves as a result of density difference
• Fluid velocity established as a result of the temperature
field
• Fluid can move downward and upward
Examples:
Cold air cools
the egg
Warm air heats
the can
1
• Further examples:
high temperature
z
ρ(z)
Atmospheric circulation
low temperature
T(z)
low temperature
z
high temperature
T(z), ρ(z)
Atmospheric inversion no vertical exchange of
mass of air
Room
T(z)
ρ(z)
T(z), ρ(z)
Vertical exchange of
mass of air
circulation
Cold window
Space heater –
radiator
2
Heated vertical wall
•
•
•
•
Which force makes the fluid to raise?
The force called the buoyancy (vztlak)
Which role does the gravity play??
How the velocity is established
Fluid moves upwards
gravity
Heat flux
P.D.E. for momentum conservation
L
∂u
∂u
1 ∂p
∂ 2u
u +v =−
− g+ν 2
ρ ∂x
∂x
∂y
∂y
Individual terms can be expressed in units of force per mass [N/kg]
or in units of acceleration [m/s2]
∂u
∂u
1 ∂p
∂ 2u
u +v =−
− g+ν 2
∂x
∂y
ρ ∂x
∂y
simply add gravity in –x direction
3
1 ∂p
∂ 2u
∂u
∂u
u +v =−
− g+ν 2
∂x
∂y
ρ ∂x
∂y
Fluid moves upwards
No movement in y - direction ∂p ∂y = 0
gravity
Same ∂p ∂x in the boundary layer
and outside it
∂p
= − ρ∞ g
∂x
Pressure difference results from
the weight of the fluid column
1 ∂p
g
−
− g = ( ρ∞ − ρ )
ρ ∂x
ρ
Volume expansion coefficient
We need temperature difference –
how to replace Δρ by ΔT ?
1 ⎛ ∂ρ ⎞
1 ρ∞ − ρ
β = − ⎜ ⎟ or β = −
ρ ⎝ ∂T ⎠ p
ρ T∞ − T
For ideal gas
1
β=−
T
4
∞
Momentum conservation equation
∂ 2u
∂u
∂u
u + v = gβ (T − T∞ ) + ν 2
∂y
∂x
∂y
Fluid moves upwards
gravity
buoyancy force
Energy conservation equation
∂T
∂T
∂ 2T
u
+v
=a 2
∂y
∂x
∂y
Continuity equation
∂u ∂v
+ =0
∂x ∂y
So called coupled problem – can’t solve velocity field unless
we know temperature field which is a function of velocity field.
5
Heat transfer coefficient
Similarity parameter – dimensionless number. Can’t use
Reynolds number – fluid velocity or flow rate not defined
a priori.
Grashof number for vertical wall
Gr ≡
L
(
)
gβ T − T∞ L3
ν2
buoyancy
≈
viscous.force
Functional relation for Heat Transfer coefficient – Nusselt number
(
)
Nu = f Gr .Pr = f(Ra )
L
L
L
Ra
L
Rayleigh number
6
Laminar versus turbulent
Under certain conditions, laminar regime can change to turbulent.
Vertical wall:
Ra x,krit = Grx,krit .Pr =
(
)
gβ Tw − T∞ x 3
νa
= 10 9
Characteristic length is always dimension in the direction of the
fluid movement:
Vertical cylinder: Length
of the cylinder if:
Vertical wall: Height of the wall
d
35
≥ 14
L Gr
L
Horizontal cylinder: Diameter of the cylinder
7
Horizontal plates
insulation
A. Upper surface
of a cold plate
Tw<T∞
Nusselt number
gβ (Tw − T∞ )L3
Ra L =
νa
B. Lower surface of a cold
plate Tw<T∞
B. Upper surface of a hot plate
Tw>T∞
A.
B.
NuL = 0,27RaL1 4
NuL = 0,54Ra L1 4
NuL = 0,15RaL1 3
A. Lower surface
of a hot plate
Tw>T∞
Can you sketch graphically?
10 4 < Ra L < 107
107 ≤ Ra L ≤ 10 11
Characteristic dimension: L=A/P = surface area/surface perimeter
8
Inclined plates
Use vertical plate equations for
the upper surface of a cold plate
and the lower surface of a hot plate –
equations A.
Nusselt number
A.
NuL = 0,27RaL1 4
gβ (Tw − T∞ )L3
Ra L =
νa
Replace g by gcosθ
θ angle from the vertical
9
Cavities
Applications: plate solar collectors, double glazed windows,
sandwich walls, etc.
Air trapped inside – good insulator
Complications: air doesn’t remain stationary – it moves
upwards and downwards
10
Horizontal cavity
When the hotter plate at the top – no
convection occurs – pure conduction
transfer of heat
When the hotter plate at the bottom –
tendency for the lighter air to rise to the top
L – distance between hot and
cold plates
But if
gβ (T1 − T2 )L3
< 1708
Ra L =
aν
buoyancy force too
week compared to viscous force – still conduction heat transfer
λ(T1 − T2 )
= α (T1 − T2 )
L
α=λ/L and NuL=1.
For Ra > 1708 – natural convection occurs – Bénard cells
For Ra > 3.105 – Bénard cells break down – turbulence occurs
11
Horizontal cavity
For air:
Nu L = 0.195Ra L1 4
for 104 < Ra < 4.105
Nu L = 0.068 Ra L1 3
for 4.105 < Ra < 107
12
Vertical cavity
For Ra < 1000, no natural convection –
pure conduction heat transfer across
the cavity – Nu = 1
For Ra > 1000, natural convection occurs ⇒ along the hot surface air rises,
⇒ along the cold surface air flows down
⇒ Convection enhances heat transfer
⇒ As Ra increases, circulation region gets closer to walls,
a centre is created with almost no movement
0.28
⎞ ⎛H⎞
⎛ Pr
Nu L = 0.22⎜
Ra L ⎟ ⎜ ⎟
⎠ ⎝ L⎠
⎝ 0.2 + Pr
−0.3
H
⎛ ⎞
Nu L = 0.42Ra L1 4 Pr 0.012 ⎜ ⎟
⎝ L⎠
−1 4
H
⎛
⎞
2
<
< 10 ⎟
⎜
L
⎝
⎠
H
⎛
⎞
⎜ 10 < < 40 ⎟
L
⎝
⎠
13
Heat Transfer Rate
T1 − T2
&
Q = αS(T1 − T2 ) = λNuS
L
λNu
α=
L
It resembles equation for
heat conduction
T1 − T2
&
Qcond = λeff S
L
Effective conductivity λeff = λ.Nu
Conclusions: Heat transfer rate can be determined from heat
conduction using “effective” thermal conductivity λeff
14
Natural versus forced convection
Forced convection – much higher heat transfer coefficients
Tendency to ignore natural convection
Error in ignoring natural convection negligible at high velocities
Error considerable at low velocities
Parameter representing the importance of
natural convection
If
If
Gr
Re
2
Gr
Gr
Re 2
< 0,1
Natural convection negligible
> 10
Forced convection negligible
Re 2
Gr
< 10
If 0,1 <
2
Re
Both convections important
15
Natural versus forced convection
Natural convection may help or hurt
forced convection – depending on relative
directions of buoyancy - induced and
forced convection motion
⇒Assisting flow sign +
⇒ Opposing flow sign -
n
Nu n = ( Nu nforced ± Nunatural
)
Exponent n recommended 3
16