Heat Transfer Su Yongkang
School of Mechanical Engineering
# 1
HEAT TRANSFER
CHAPTER 9
Free Convection
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 2
Natural Convection
Where we’ve been ……
Up to now,have considered forced convection,
that is an external driving force causes the flow.
Where we’re going:
Consider the case where fluid movement is by
buoyancy effects caused by temperature
differential
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
When natural convection is important
Weather events such as a thunderstorm
Glider planes
Radiator heaters
Hot air balloon
Heat transfer with pipes and electrical lines
Heat flow through and on outside of a double
pane window
Just sitting there
Oceanic and atmospheric motions
Coffee cup example ….
Small velocity
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
Natural Convection
KEY POINTS THIS LECTURE
New terms
– Volumetric thermal expansion coefficient
– Grashof number
– Rayleigh number
Buoyancy is the driving force
– Stable versus unstable conditions
Nusselt number relationship for laminar free
convection on vertical surface
Boundary layer impacts,laminar? turbulent
Text book sections,§ 9.1 – 9.5
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
Buoyancy is the driving force
Buoyancy is due to combination of
– Differences in fluid density
– Body force proportional to density
– Body forces gravity,also Coriolis force in
atmosphere and oceans
Convection flow is driven by buoyancy in
unstable conditions
Fluid motion may be
(no constraining surface) or
along a surface
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
Buoyancy is the driving force (Cont’d)
Free boundary layer flows
Heated wire or hot pipe
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
A heated vertical plate
We focus on free convection flows bounded by
a surface,
The classic example is
TTs
u(x,y)
y
g
sT
T
x
v
u
Extensive,
quiescent fluid
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
Governing Equations
The difference between the two flows (forced
flow and free flow) is that,in free convection,a
major role is played by buoyancy forces.
Consider the x-momentum equation.
As we know,,hence the x-pressure
gradient in the boundary layer must equal that in
the quiescent region outside the boundary layer.
gX
Very important
2
2
g 1 y uxPyuvxuu
0/ yp
g-xP
2
2
g y uyuvxuu

Buoyancy force
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
Governing Equations (Cont’d)
Define?,the volumetric thermal expansion
coefficient.
In general,
T
T h u s
RT
PRT
P
T
P
1
,
,g as i d ealan F o r
1

For liquids and non-ideal
gases,see appendix A
TTT?

11
)( TT
Density gradient is due to the temperature gradient
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
Governing Equations (cont’d)
Now,we can see buoyancy effects replace
pressure gradient in the momentum equation
The buoyancy effects are confined to the
momentum equation,so the mass and energy
equations are the same.
2
2
)( y uvTTgyuvxuu
0 yvxu
2
2
2

y
u
cy
T
y
Tv
x
Tu
p

Strongly coupled and must be solved simultaneously
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
Dimensionless Similarity Parameter
The x-momentum and energy equations are
v el o c i t yr ef er en c ear b i t r ar y an is u
an d l en g t h,s t i cc h ar ac t er i a is L w h er e
T
TT
0
*
00?

s
T
T
u
v
va n d
u
u
u
L
y
ya n d
L
x
x
2*
*2
*
2
0
*
*
*
*
*
*
Re
1 T )(
y
u
u
LTTg
y
uv
x
uu
L
s

PrRe 1 2
*
*2
*
*
*
*
*
*
y
T
y
Tv
x
Tu
L?

Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
Dimensionless Similarity Parameter (cont’d)
Define new dimensionless parameter,
Grashof number in natural convection is
analogous to the Reynolds number in forced
convection.
Grashof number indicates the ratio of the
buoyancy force to the viscous force.
– Higher Gr number means increased natural
convection flow
1Re 2
L
LGr 1
Re 2L
LGr
Eq
9.12
2
32
0
2
0
)()(
LTTgLu
u
LTTgGr ss
L

forced natural
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
Example #1,Consider a object having a characteristic
length of 0.25m and a situation for which the temperature
difference is 25℃,Using thermophysical properties
evaluated at 350K,calculate the Grashof number for air,
hydrogen,water,and ethylene glycol,Assume a pressure of
1 atm,
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 14
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 15
TTs
u(x,y)
y
g
sT
T
x
v
u
Laminar Free Convection on Vertical Surface
As y,u = 0,T = T?
As y? 0,u = 0,T = Ts
With little or no external
driving flow,Re? 0 and
forced convection effects can
be safely neglects
–
P r ),( LL GrfNu?
1
Re 2

L
LGr
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 16
Analytical similarity solution for the local
Nusselt number in laminar free convection
( P r )4
4/1
gGrkhxNu Lx

4/1Pr238.1Pr 1,2 2 10,6 0 9
Pr 75.0( P r )
:by f o u n d is ( P r )f u n c t i o n t h ew h e r e

g
g
( P r )434
4/1
gGrk LhNu LL
Average
Nusselt # =
Eq
9.21
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 17
Effects of Turbulence
Just like in forced convection flow,hydrodynamic
instabilities may result in the flow.
For example,illustrated for a heated vertical surface:
Define the
Rayleigh number
for relative
magnitude of
buoyancy and
viscous forces
TTs
For vertical surface,transition to turbulence at
910

3
,,
)(
Pr
xTTg
GrRa
s
cxcx

Heat Transfer Su Yongkang
School of Mechanical Engineering
# 18
Effects of Turbulence (cont’d)
Transition to turbulent flow greatly effects
heat transfer rate.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 19
Example #2,Consider a large vertical plate with a uniform
surface temperature of 130℃ suspended in quiescent air
at 25℃ and atmospheric pressure.
a) Estimate the boundary layer thickness at a location
0.25m measured from the lower edge,
b) What is the maximum velocity in the boundary layer
at this location and at what position in the boundary
layer does the maximum occur?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 20
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 21
Example #3,A number of thin plates are to be cooled by
vertically suspending them in a water bath at a temperature
of 20 ℃,If the plates are initially at 54 ℃ and 0.15m long,
what minimum spacing would prevent interference between
their free convection boundary layers?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 22
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 23
Have a good time!