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SMJ 4463: HEAT TRANSFERSMJ 4463: HEAT TRANSFER
INSTRUCTOR: PM DR MAZLAN ABDUL WAHIDhttp://www.fkm.utm.my/~mazlan
TEXT: Introduction to Heat Transferby Incropera, DeWitt, Bergman, Lavine5th Edition, John Wiley and Sons
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Chapter 9Chapter 9
Natural ConvectionNatural Convection
Assoc Prof. Dr Mazlan Abdul WahidFaculty of Mechanical Engineering
Universiti Teknologi Malaysiawww.fkm.utm.my/~mazlan
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ObjectivesWhen 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.
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• 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 1 1 K
P P
V
V T T
ρβρ
∂ ∂ = = ∂ ∂ (9-3)
ViscousForce
BuoyancyForce
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volume expansion coefficient β• The volume expansion coefficient can be
expressed approximately by replacing differential quantities by differences as
or
• For ideal gas
( )1 1 at constant P
T T T
ρ ρρβρ ρ
∞
∞
−∆≈ − = −∆ −
(9-4)
( ) ( ) at constant T T Pρ ρ ρβ∞ ∞− = − (9-5)
( )ideal gas
1 1/K
Tβ = (9-6)
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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
differenceρ-ρ∞
(Boussinesq approximation).
g
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g
• Newton’s second law of motion
( )1x xm a F
m dx dy
δδ ρ
⋅ == ⋅ ⋅
(9-7)
(9-8)
x
du u dx u dya
dt x dt y dt
∂ ∂= = +∂ ∂
x
u ua
x y
∂ ∂= +∂ ∂
• Consider a differential volume element.
• The acceleration in the x-direction is obtained by taking the total differential of u(x, y)
u v
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• The net surface force acting in the x-direction
• 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
( ) ( ) ( )
( )
Net pressure forceNet viscous forceGravitational force
2
2
1 1 1
1
x
PF dy dx dx dy g dx dy
y x
u Pg dx dy
y x
τ ρ
µ ρ
∂ ∂ = ⋅ − ⋅ − ⋅ ⋅ ∂ ∂
∂ ∂= − − ⋅ ⋅ ∂ ∂
6447448 64474486447448
(9-9)
2
2
u u u Pu v g
x y y xρ µ ρ ∂ ∂ ∂ ∂+ = − − ∂ ∂ ∂ ∂
(9-10)
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• The x-momentum equation in the quiescent fluid outside the boundary layer (setting u=0)
• 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 y-
direction,
the force balance in the y-direction is
Substituting into Eq. 9–10
Pg
xρ∞
∞∂ = −∂
(9-11)
0P
y
∂ =∂
PPg
x xρ∞
∞∂∂ = = −
∂ ∂
( )2
2
u u uu v g
x y yρ µ ρ ρ∞ ∂ ∂ ∂+ = + − ∂ ∂ ∂
(9-12)
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• Substituting Eq. 9-5 it into Eq. 9-12 and dividing both sides by ρ gives
• 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.
( )2
2
u u uu v g T T
x y yν β ∞
∂ ∂ ∂+ = + −∂ ∂ ∂
(9-13)
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The Grashof Number• The governing equations of natural convection
and the boundary conditions can be nondimensionalized
• Substituting into the momentum equation and simplifying give
* * * * * ; ; ; ; c c s
T Tx y u vx y u v T
L L V V T T∞
∞
−= = = = =−
( )2
3* * * 2 ** *
* * 2 2 *
1
Re Re
L
s c
L L
Gr
g T T Lu u T uu v
x y y
βν
∞ −∂ ∂ ∂+ = + ∂ ∂ ∂ 144424443
(9-14)
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• The dimensionless parameter in the brackets represents the natural convection effects, and is called the Grashof number GrL
• The flow regime in natural convection is
governed by the Grashof number
GrL>109 flow is turbulent
( ) 3
2s c
L
g T T LGr
βν
∞−= (9-15)
GrL=Buoyancy forceViscous force
Buoyancy force
Viscous force
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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
• Where RaL is the Rayleigh number
( )Prn nc
L L
hLNu C Gr C Ra
k= = ⋅ ⋅ = ⋅ (9-16)
( ) 3
2Pr Prs c
L L
g T T LRa Gr
βν
∞−= ⋅ = (9-17)
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• 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:
( )2
s
L
q LhLNu
k k T T∞
= =−
&(9-27)
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Empirical correlations
for Nuavg
(9-26)
(9-30)
(9-31)
(9-34)
(9-35)
(9-26)
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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.
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• The recommended relation for the average Nusselt number for vertical isothermal parallel plates is
• Closely packed fins – greater surface area– smaller heat transfer coefficient.
• Widely spaced fins– higher heat transfer coefficient– smaller surface area.
• Optimum fin spacing for a vertical heat sink
( ) ( )
0.5
2 0.5
576 2.873
s s
hSNu
k Ra S L Ra S L
−
= = +
(9-31)
0.253
0.252.714 2.714opt
s L
S L LS
Ra Ra
= =
(9-32)
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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.
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Nusselt Number Correlations for Enclosures
• Simple power-law type relations in the form of
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.
• Numerous correlations are widely available for– horizontal rectangular enclosures,
– inclined rectangular enclosures,
– vertical rectangular enclosures,
– concentric cylinders,
– concentric spheres.
nLNu C Ra= ⋅
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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):
– Natural convection (vertical plate, laminar flow):
• Therefore, the parameter Gr/Re2 represents the importance of natural convection relative to forced convection.
1 2forced convection ReNu ∝
1 4natural convectionNu Gr∝
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• 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.
• Natural convection may help or hurt forced convection heat transfer
depending on the
relative directions
of buoyancy-induced
and the forced
convection motions.
hot isothermal vertical plate
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Nusselt Number for Combined Natural and Forced Convection
• A review of experimental data suggests a Nusselt number correlation of the form
• Nuforced and Nunatural are determined from the correlations for pure forced and pure naturalconvection, respectively.
( )1
combined forced natural
nn nNu Nu Nu= ± (9-66)
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CChapter hapter 99
NATURAL CONVECTIONNATURAL CONVECTION
Dr. Mazlan Abdul WahidFaculty of Mechanical Engineering
Universiti Teknologi Malaysiawww.fkm.utm.my/~mazlan
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The free (natural) convection is originated a thermal instability, i.e., when a body force acts on a fluid in which there are density gradients.
The net effect is a buoyancy force, which induces free convection currents.
The density gradient is mainly due to a temperature gradient and the body force is due to the gravity.
In free convection, the convection rate are smaller compared those in the forced convection.
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In many systems involving multimode heat transfer effects, free convection provides the largest resistance to heat transfer and thus plays an important role in the design or performance of the system.
When it is desirable to minimize heat transfer rates or to minimize operating cost, free convection is often preferred to forced convection.
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