Chapter 9:
Natural Convection
Yoav Peles
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Objectives
When you finish studying this chapter, you should be able to:
• Understand the physical mechanism of natural convection,
• Derive the governing equations of natural convection, and obtain the
dimensionless Grashof number by nondimensionalizing them,
• Evaluate the Nusselt number for natural convection associated with
vertical, horizontal, and inclined plates as well as cylinders and
spheres,
• Examine natural convection from finned surfaces, and determine the
optimum fin spacing,
• Analyze natural convection inside enclosures such as double-pane
windows, and
• Consider combined natural and forced convection, and assess the
relative importance of each mode.
• Buoyancy forces are responsible for the fluid motion
in natural convection.
• Viscous forces appose the fluid motion.
• Buoyancy forces are expressed in terms of fluid
temperature differences through the volume expansion
coefficient
1 ⎛ ∂V ⎞
1 ⎛ ∂ρ ⎞
β= ⎜
(1 K ) (9-3)
⎟ = ⎜
⎟
V ⎝ ∂T ⎠ P ρ ⎝ ∂T ⎠ P
Viscous
Force
Buoyancy
Force
volume expansion coefficient (β)
• The volume expansion coefficient can be expressed
approximately by replacing differential quantities by
differences as
1 ∆ρ
1 ρ∞ − ρ
β ≈−
=− (
)
ρ ∆T
ρ T∞ − T
or
ρ∞ − ρ = ρβ (T − T∞ )
( at constant P )
( at constant P )
(9-4)
(9-5)
• For ideal gas (Pv = RT)
βideal gas
1
=
T
(1/K )
(9-6)
Equation of Motion and the Grashof Number
• Consider a vertical hot flat plate
immersed in a quiescent fluid body.
• Assumptions:
–
–
–
–
–
steady,
laminar,
two-dimensional,
Newtonian fluid, and
constant properties, except the density
difference ρ -ρ∞ (Boussinesq
approximation).
g
• Consider a differential volume element.
• Newton’s second law of motion
δ m ⋅ ax = Fx
(9-7)
g
其中 δ m = ρ ( dx ⋅ dy ⋅1)
• The acceleration in the x-direction is
obtained by taking the total
differential of u(x, y)
du ∂u dx ∂u dy
ax =
=
+
dt ∂x dt ∂y dt
∂u
∂u
ax =u
+v
∂x
∂y
(9-8)
in
m
Zoo
P∞
• The net surface force acting in the x-direction
Net pressure force
Net viscous force
Gravitational force
⎛ ∂τ ⎞
⎛ ∂P ⎞
Fx = ⎜ dy ⎟ ( dx ⋅1) − ⎜
dx ⎟ ( dy ⋅1) − ρ g ( dx ⋅ dy ⋅1)
⎝ ∂x ⎠
⎝ ∂y ⎠
⎛ ∂ 2u ∂P
⎞
= ⎜µ 2 −
− ρ g ⎟ ( dx ⋅ dy ⋅1) (9-9)
∂x
⎝ ∂y
⎠
• Substituting Eqs. 9–8 and 9–9 into Eq. 9–7 and
dividing by ρ·dx·dy·1 gives the conservation of
momentum in the x-direction
⎛ ∂u
∂u ⎞
∂ 2u ∂P
ρ ⎜u + v ⎟ = µ 2 −
− ρg
∂y ⎠
∂y
∂x
⎝ ∂x
(9-10)
• The x-momentum equation in the quiescent fluid
outside the boundary layer (setting u = 0)
∂P∞
(9-11)
= − ρ∞ g (靜力學)
∂x
• Noting that
– v<<u in the boundary layer and thus ∂v/ ∂x≈ ∂v/∂y ≈0, and
– there are no body forces (including gravity) in the ydirection,
the force balance in the y-direction is
∂P
∂P ∂P∞
=0
=
= − ρ∞ g
∂y
∂x
∂x
Substituting into Eq. 9–10
⎛ ∂u
∂u ⎞
∂ 2u
ρ ⎜ u + v ⎟ = µ 2 + ( ρ∞ − ρ ) g
∂y ⎠
∂y
⎝ ∂x
(9-12)
• Substituting Eq. 9-5 it into Eq. 9-12 and dividing both
sides by ρ gives
∂u
∂u
∂ 2u
u
+v
= ν 2 + g β (T − T∞ )
∂x
∂y
∂y
(9-13)
• The momentum equation involves the temperature, and
thus the momentum and energy equations must be solved
simultaneously.
• The set of three partial differential equations (the
continuity, momentum, and the energy equations) that
govern natural convection flow over vertical isothermal
plates can be reduced to a set of two ordinary nonlinear
differential equations by the introduction of a similarity
variable.
The Grashof Number
• The governing equations of natural convection
and the boundary conditions can be
nondimensionalized
T − T∞
x
y
u
v
*
*
*
*
; y =
; u = ; v = ; T =
x =
Lc
Lc
V
V
Ts − T∞
*
• Substituting into the momentum equation and
simplifying give
3
*
*
*
2 *
⎡
⎤
−
g
β
T
T
L
(
)
∂
∂
∂
u
u
T
u (9-14)
1
s
∞
c
*
*
+v
=⎢
u
⎥ 2 +
*
*
2
*2
∂x
∂y
ν
Re L Re L ∂y
⎣
⎦
GrL
• The dimensionless parameter in the brackets represents the
natural convection effects, and is called the Grashof
number GrL
GrL =
GrL=
g β (Ts − T∞ ) L3c
ν
2
(9-15)
Viscous
force
Buoyancy force
Viscous force
• The flow regime in natural convection is
governed by the Grashof number
GrL > 109 flow is turbulent
Buoyancy
force
Natural Convection over Surfaces
• Natural convection heat transfer on a surface depends on
–
–
–
–
geometry,
orientation,
variation of temperature on the surface, and
thermophysical properties of the fluid.
• The simple empirical correlations for the average
Nusselt number in natural convection are of the form
hLc
n
(9-16)
Nu ≡
= C ⋅ ( GrL ⋅ Pr ) = C ⋅ RaLn
k
• Where RaL is the Rayleigh number
g β (Ts − T∞ ) L3c
Pr
RaL = GrL ⋅ Pr =
2
ν
(9-17)
• The values of the constants C and n depend on the geometry of
the surface and the flow regime (which depend on the Rayleigh
number).
• All fluid properties are to be evaluated at the film temperature
Tf = (Ts+T∞).
• The Nusselt number relations for the constant surface
temperature and constant surface heat flux cases are nearly
identical.
• The relations for uniform heat flux is valid when the plate
midpoint temperature TL/2 is used for Ts in the evaluation of the
film temperature.
• Thus for uniform heat flux:
qs L
hL L qs
Nu =
= (
)=
k
k ∆T
k TL 2 − T∞
(
)
(9-27)
Empirical
correlations
for Nuavg
1010
例題
g
Ts = 1000C, T∞ = 300C, g = 9.8 m/s2, Pr = 0.7
Lc = 0.3 m, ν = 1.93 x 10-5 m2/s
β ≈ 1 / Tavg = 1 / [(373.15+303.15) / 2] = 0.00296 K-1,
k = 0.026 W/m-K
RaL = GrL ⋅ Pr =
g β (Ts − T∞ ) L3c
ν
2
Pr
= 1.03 x 108
GrL = 1.47 x 108
(laminar flow)
Nu = havgLc/k = 0.59 (RaL)1/4
Lc=0.3 m
= 80.3
Hence,
havg = 7 (W/m2-K)
Q = havg A (Ts – T∞) = havg (Lc w) (Ts – T∞)
空氣
Q/w = 147 (Watts)
Natural Convection from Finned Surfaces
• Natural convection flow through a channel formed by
two parallel plates is commonly encountered in
practice.
• Long Surface
– fully developed channel flow.
• Short surface or large spacing
– natural convection from two
independent plates in a quiescent
medium.
• The recommended relation for the average Nusselt number
for vertical isothermal parallel plates is
hS ⎡
576
2.873 ⎤
Nu =
=⎢
+
⎥
2
0.5
k ⎢ ( Ras S L ) ( Ras S L ) ⎥
⎣
⎦
−0.5
(9-31)
• Closely packed fins
– greater surface area
– smaller heat transfer coefficient.
• Widely spaced fins
– higher heat transfer coefficient
– smaller surface area.
Ras ≡
gβ (Ts − T∞ ) S 3
ν
2
Pr
;
RaL ≡
gβ (Ts − T∞ ) L3
ν2
L3
Pr = Ra s 3
S
固定 WL，自然熱傳量最大時 (Ts =constant) ， 可推導:
⎛S L⎞
= 2.714 ⎜
⎟
Ra
⎝ s⎠
3
Sopt
S opt = 2.714[
S = S opt
;
S4
0.25
S
Ras ( )
L
]
0.25
= 2.714
L
RaL0.25
(9-32)
S
= 2.714
S
[ Ras ( )]0.25
L
T∞
S
Ra s ( ) = 54.255
L
代回 (9-31)
Nu = h Sopt / k = 1.307
n =片數
(9-33)
熱傳模組之總散熱量 = Q = h (2nLH) (Ts –T∞)
(9-34)
Natural Convection Inside Enclosures
• In a vertical enclosure, the fluid adjacent to the hotter
surface rises and the fluid adjacent to the
cooler one falls, setting off a rotationary
motion within the enclosure that enhances
heat transfer through the enclosure.
• Heat transfer through a horizontal enclosure
– hotter plate is at the top ─ no convection
currents (Nu=1).
– hotter plate is at the bottom
• Ra<1708 no convection currents (Nu=1).
• 3x105>Ra>1708 Bénard Cells.
• Ra>3x105 turbulent flow.
Nusselt Number Correlations for Enclosures
• Simple power-law type relations in the form of
Nu = C ⋅ Ra
where C and n are constants, are sufficiently accurate,
but they are usually applicable to a narrow range of
Prandtl and Rayleigh numbers and aspect ratios.
n
L
• Numerous correlations are widely available for
–
–
–
–
–
horizontal rectangular enclosures,
inclined rectangular enclosures,
vertical rectangular enclosures,
concentric cylinders,
concentric spheres.
Combined Natural and Forced Convection
• Heat transfer coefficients in forced convection are typically much
higher than in natural convection.
• The error involved in ignoring natural convection may be
considerable at low velocities.
• Nusselt Number:
– Forced convection (flat plate, laminar flow):
Nuforced convection ∝ Re1 2
– Natural convection (vertical plate, laminar flow):
Nunatural convection ∝ Gr1 4
• Therefore, the parameter Gr/Re2 represents the importance of
natural convection relative to forced convection.
• Gr/Re2 < 0.1 : natural convection is negligible.
• Gr/Re2 > 10 : forced convection is negligible.
• 0.1 < Gr/Re2 <10 : forced and natural convection are
not negligible.
hot isothermal vertical
plate
• Natural convection may help or hurt forced convection
heat transfer depending on the relative directions
of buoyancy-induced and the forced convection
motions.
Nusselt Number for Combined Natural and
Forced Convection (Mixed Convection)
• A review of experimental data suggests a
Nusselt number correlation of the form
(
Nucombined = Nu
n
forced
± Nu
n
natural
)
1n
(9-66)
• Nuforced and Nunatural are determined from the
correlations for pure forced and pure natural
convection, respectively.