Lecture by Jit Rana in Heat & Mass Transfer

Lecture 11
Natural Convection
1

Objectives
• Understand the physical mechanism of natural convection
• Evaluate the Grashof number
• 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
• Consider combined natural and forced convection, and assess
the relative importance of each mode.
2

Natural Convection
• Examples:
– Heat transfer from electric baseboard heaters
– Heat transfer from refrigeration coils
– Heat transfer from our body
• Natural convection in gases usually accompanied by radiation of comparable
magnitude
3

Natural Convection
• Buoyancy forces are responsible for the fluid motion in natural convection.
• Viscous forces oppose the fluid motion.
• In gravitational field, the upward force
exerted by a fluid on a body completely
or partially immersed in it 
buoyancy force
Net force:
4

Natural Convection
• In heat transfer  express the net buoyancy force in terms of temperature difference
• Buoyancy forces are expressed in terms
of fluid temperature differences
through the:
volume expansion coefficient
5
(1/K)
1
1
P
P
T
T
V
V























Volume expansion coefficient 
• The volume expansion coefficient can be expressed approximately
by replacing differential quantities by differences as
• For ideal gas
 
1 1
at constant P
T T T
 


 




 
 
(11-4)
   
at constant
T T P
  
 
   (11-5)
 
ideal gas
1
1/K
T
  (11-6)
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 – ∞ (Boussinesq
approximation).
g
7

g
Zoom in
• Newton’s second law of motion
 
1
x x
m a F
m dx dy

 
 
  
(11-7)
x
du u dx u dy
a
dt x dt y dt
 
  
 
x
u u
a
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
8

• The net surface force acting in the x-direction
• Substituting Eqs. 11–8 and 11–9 into Eq. 11–7 and
dividing by ·dx ·dy ·1 gives the conservation of
momentum in the x-direction
     
 
Net pressure force
Net viscous force
Gravitational force
2
2
1 1 1
1
x
P
F dy dx dx dy g dx dy
y x
u P
g dx dy
y x


 
 
 
 
      
   
 
 
 
 
 
    
 
 
 
            
    
(11-9)
2
2
u u u P
u v g
x y y x
  
 
   
   
 
   
 
(11-10)
9

• 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
P
g
x






(11-
11)
0
P
y



P
P
g
x x





 
 
 
2
2
u u u
u v g
x y y
   

 
  
   
 
  
 
(11-12)
10
P(x) = P(x) = P

• Substituting Eq. 11-5 it into Eq. 11-12 and dividing
both sides by 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 u
u v g T T
x y y
  
  
   
  
(11-13)

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 T
x y u v
x 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 L
u u T u
u v
x y y



 

  
  
 
  
 
      

(11-14)
12

• 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
2
s c
L
g T T L
Gr




 (11-15)
GrL=
Buoyancy force
Viscous force
Buoyancy
force
Viscous
force
13

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
 
Pr
n n
c
L L
hL
Nu C Gr C Ra
k
      (11-16)
  3
2
Pr Pr
s c
L L
g T T L
Ra Gr




   (11-17)
14

• The values of the constants C and n depend on the geometry of
the surface and the flow regime (which depend on the Ra).
• All fluid properties are to be evaluated at the film temperature
Tf=(Ts+T∞).
• Nu relations for constant Ts are applicable for the case of
constant qs, but 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 L
hL
Nu
k k T T
 


(11-27)
15

Empirical correlations for Nuavg
16

Empirical correlations for Nuavg
17

Review of Last Monday
• Driving force of natural convection?
• Volume expansion coefficient?
• Temperature and velocity profiles?
• Grashof number? Rayleigh number?
• Nusselt number relations?
19

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.
21

• 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
hS
Nu
k Ra S L Ra S L

 
  
 
 
 
(11-31)
0.25
3
0.25
2.714 2.714
opt
s L
S L L
S
Ra Ra
 
 
 
 
(11-32)
22

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. 23

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.
n
L
Nu C Ra
 
24

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):
• The parameter Gr/Re2
represents the importance of natural
convection relative to forced convection.
1 2
forced convection Re
Nu 
1 4
natural convection
Nu Gr

25

• 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.
26

• Natural convection may help or hurt forced convection
heat transfer
depending on the
relative directions
of buoyancy-induced
and the forced
convection motions.
27

Nusselt Number for Combined Natural
and Forced Convection
• A review of experimental data suggests a
Nusselt number correlation of the form
n ~ 3 – 4
• Nuforced and Nunatural are determined from the
correlations for pure forced and pure natural
convection, respectively.
 
1
combined forced natural
n
n n
Nu Nu Nu
  (11-66)
28

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