Heat Transfer Su Yongkang
School of Mechanical Engineering
# 1
HEAT TRANSFER
CHAPTER 6
Introduction to convection
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 2
CH6 – INTRODUCTION
Where we’ve been ……
Basic Conduction Heat Transfer Finished
Fourier’s law:
Where we’re going,
Begin study of convective heat transfer.
Newton’s law of cooling:
dx
dtkq
tkq
)( TThq s
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
Convective transfer problem
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
CH6 INTRODUCTION
KEY POINTS THIS CHAPTER
What are the key variables when analyzing
convection heat transfer?
Review boundary layer concept and significance
General idea of relationship between velocity and
thermal profiles in a boundary layer.
Effect of laminar versus turbulent flow on heat
transfer potential
Boundary layer similarity
This chapter will be taught in two lectures,
the first includes text book sections § 6.1 to 6.4
the other includes text book sections § 6.5 to 6.10
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
Convection overview
Consider a flat plate of length L,in air flow with
velocity u? and temperature T?
Local heat flux is,where h is the
local heat transfer coefficient
Total heat transfer rate:
h average heat transfer coefficient
Determination of ‘h’ will rely on analytical as well
as empirical data
)( TThq s
ss A ssA s hdATTdAqq )(
)( TTAhq ss
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
Convection overview (Cont’d)
Same principal applies to any arbitrary shape,not
just a flat plate
Average convection heat transfer coefficient:
So,we need to know how h varies with x,the
distance from the leading edge……..
What do you think key parameters that might
influence h?
q?
sdA
ss TA,
o r,fo r u n i t w i d t h,
sA s
s
h d AAh 1 L hdxLh 01
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
Key parameters
Transfer potential,forced flow or free flow
Phase change,boiling and condensation
Flow conditions,laminar or turbulent flow
Geometries,shape,size,position and roughness.
Properties,density,viscosity,thermal
conductivity,specific heat,and so on.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
Example
Given
Experimental results for measured local heat
transfer coefficient h for flow over a flat plate
with a rough surface
where,a = coefficient
x = distance from leading edge
– Find expression for average heat transfer
coefficient,and the relation of average heat
transfer coefficient to the local coefficient
)( 3.0 axxh x
7.0
0
3.0
0
3.0
0,7
1
1
x
x
ah
dxx
x
adxxa
x
h
x
xx
x
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
The Convection Boundary Layers
Velocity Boundary Layer
For fluid flow over a flat plate,which disturbs
the fluid flow:
– As y,where u is velocity in
x-direction
– As y?0,(no-slip condition)
– The boundary layer thickness is defined as
the value at which:
– The boundary layer thickness? varies with x
Shear Stress
Local friction coefficient
0
y
s y
u
2
2uC
s
f
uu
0?u
uyu 99.0)(
Dynamic viscosity
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
The Convection Boundary Layers
Thermal Boundary Layer
A hot or cold plate alters the temperature
distribution in the air
– As y:
– As y?0:
– The thermal boundary layer thickness is
defined as the value at which,
– The thermal boundary layer thickness,?t
also varies (increases) with x
99.0)( TT yTT
s
a i rs
TTs
TyT )(
sTyT?)(
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
The Convection Boundary Layers
Thermal Boundary Layer (Cont’d)
Heat Flux
Heat flux analogous to shear stress in velocity
boundary layer
Heat flux proportional to the temperature
gradient at the surface,
AND
since u(y=0) =0,energy transfer to/from fluid
occurs by conduction only!
Since thermal boundary layer gets larger along x
direction,the temperature gradient changes with
x,and therefore
TTs
fluid thermal
conductivity
wall
temperature
gradient
____________________
0
y
fs y
Tkq
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
The Convection Boundary Layers
Thermal Boundary Layer (Cont’d)
Heat Flux (Cont’d)
Using Newton’s law of cooling:
We obtain
While? increases with increasing x,temperature
gradients in the boundary layer must decrease
with increasing x,
Accordingly,and h decrease with increasing x,
)( TThq s
TT
yTkh
s
yf 0/
sq?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
The Convection Boundary Layers
Laminar Versus Turbulent Flow
Characterization of laminar flow
Low amount of,mixing” of fluid within the
boundary layer (smooth flow)
Characterization of turbulent flow
High amount of mixing of fluid within the
boundary layer (irregular flow)
High amount of mixing means increased surface
friction as well as convection transfer rates
(heat and mass)
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 14
Convection Heat Transfer Variations
Along Flat Plate
Consider flat plate with:
and all laminar flow
Thermal boundary layer defined by
Compare temperature gradients at points 1 and 2
to evaluate the heat flux rate (and hence the heat
transfer coefficient)
C0 a n d C100TT x
2?
1?
C0T
C100sT1 2
99.0
TT
TT
s
s
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 15
Convection Heat Transfer Variations
Along Flat Plate (Cont’d)
Consider flat plate with,C0 a n d C100TT x
2?
1?
C100?C1? C100?C1?
at x1 at x2
To determine the location that transition begins,we
define the Reynolds number,
And the critical Reynolds number,
xu
x
Re
5
,105Re
c
cx
xu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 16
Convective Transfer Equations
Topic of the Day
Project Teams
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 17
Convective Transport Equations
Where we’ve been ……
Last section:
Overview of the topic of convective transport of heat.
Where we’re going:
Convection transfer detailed equations
Heat and mass transfer analogy
Eventually get to applications in external and
internal flow.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 18
Convective Transport Equations
KEY POINTS THIS SECTION
Detailed development of boundary layer
equations for velocity,temperature and species
concentrations
What approximations can be made in the
boundary layers?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 19
Recall the convection overview
Local heat flux is:
where h is the local heat transfer coefficient
)( TThq s
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 20
Develop convection transfer equations
Consider steady,2-D flow of a viscous,
incompressible fluid with constant properties.
Key point to remember:
At each point in the fluid,conservation of mass,
energy and momentum must be satisfied.
Appendix E contains detailed development of
the full boundary layer equations,for example:
Conservation of mass (continuity)
0 yvxu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 21
The magnitude of variables in the thermal
boundary layer
variables x
(main flow direction)
y u v t
magnitude 1 1 1
x
T
y
T
Thermal boundary layer
y
v
y
u,
x
v
y
u,
x
u
y
u a n d ;
vu
Velocity boundary layer
Boundary Layer Approximations
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 22
Continue convection transfer equations
Quick overview of fluid equations
Conservation of mass (continuity):
OR
x-momentum equation:
y-momentum equation:
0 0?
y
v
x
u
y
v
x
u
0 yvxu
f o r c e s "B o d y " 2
2
2
2
y
u
x
u
x
P
y
uv
x
uu
xuyu 0 0
2
2 1
y
u
x
P
y
uv
x
uu
,w h e re
f o r c e s "B o d y " 2
2
2
2
y
v
x
v
y
P
y
vv
x
vu
0 0 0 0 0
0yPSo:
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 23
Continue convection transfer equations
Energy equation:
q
y
v
x
u
x
v
y
u
y
T
x
T
k
y
T
v
x
T
uc p
222
2
2
2
2
2?
0 0 0 0
0
2
yuc
p
NOTE,is usually small unless u is high
(as in sonic flows) or? is high (such as flow of oils).
Result is 4 equations and 4 unknowns:
Unknowns are,u,v,P,and T
Since:
P(x) can be obtained from free stream flow.
dx
dP
x
P a n d o n l y,)x(fP t h e n 0
y
P?
xTyT W h y
xvyu W h y
2
2
2
yucy TyTvxTu
p
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 24
Review boundary layer concept.
General relationship between velocity and
thermal boundary layers.
Convective heat transfer is dependent on
the temperature gradient at the fluid/solid
interface
Boundary layer grows with distance from
the leading edge,and this decreases the
local heat transfer rates.
Turbulent boundary layers have much
greater potential for heat and mass transfer
due to velocity fluctuations.
Convection transfer equations (4).
Go back and review fluids course notes!
SUMMARY
School of Mechanical Engineering
# 1
HEAT TRANSFER
CHAPTER 6
Introduction to convection
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 2
CH6 – INTRODUCTION
Where we’ve been ……
Basic Conduction Heat Transfer Finished
Fourier’s law:
Where we’re going,
Begin study of convective heat transfer.
Newton’s law of cooling:
dx
dtkq
tkq
)( TThq s
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
Convective transfer problem
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
CH6 INTRODUCTION
KEY POINTS THIS CHAPTER
What are the key variables when analyzing
convection heat transfer?
Review boundary layer concept and significance
General idea of relationship between velocity and
thermal profiles in a boundary layer.
Effect of laminar versus turbulent flow on heat
transfer potential
Boundary layer similarity
This chapter will be taught in two lectures,
the first includes text book sections § 6.1 to 6.4
the other includes text book sections § 6.5 to 6.10
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
Convection overview
Consider a flat plate of length L,in air flow with
velocity u? and temperature T?
Local heat flux is,where h is the
local heat transfer coefficient
Total heat transfer rate:
h average heat transfer coefficient
Determination of ‘h’ will rely on analytical as well
as empirical data
)( TThq s
ss A ssA s hdATTdAqq )(
)( TTAhq ss
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
Convection overview (Cont’d)
Same principal applies to any arbitrary shape,not
just a flat plate
Average convection heat transfer coefficient:
So,we need to know how h varies with x,the
distance from the leading edge……..
What do you think key parameters that might
influence h?
q?
sdA
ss TA,
o r,fo r u n i t w i d t h,
sA s
s
h d AAh 1 L hdxLh 01
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
Key parameters
Transfer potential,forced flow or free flow
Phase change,boiling and condensation
Flow conditions,laminar or turbulent flow
Geometries,shape,size,position and roughness.
Properties,density,viscosity,thermal
conductivity,specific heat,and so on.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
Example
Given
Experimental results for measured local heat
transfer coefficient h for flow over a flat plate
with a rough surface
where,a = coefficient
x = distance from leading edge
– Find expression for average heat transfer
coefficient,and the relation of average heat
transfer coefficient to the local coefficient
)( 3.0 axxh x
7.0
0
3.0
0
3.0
0,7
1
1
x
x
ah
dxx
x
adxxa
x
h
x
xx
x
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
The Convection Boundary Layers
Velocity Boundary Layer
For fluid flow over a flat plate,which disturbs
the fluid flow:
– As y,where u is velocity in
x-direction
– As y?0,(no-slip condition)
– The boundary layer thickness is defined as
the value at which:
– The boundary layer thickness? varies with x
Shear Stress
Local friction coefficient
0
y
s y
u
2
2uC
s
f
uu
0?u
uyu 99.0)(
Dynamic viscosity
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
The Convection Boundary Layers
Thermal Boundary Layer
A hot or cold plate alters the temperature
distribution in the air
– As y:
– As y?0:
– The thermal boundary layer thickness is
defined as the value at which,
– The thermal boundary layer thickness,?t
also varies (increases) with x
99.0)( TT yTT
s
a i rs
TTs
TyT )(
sTyT?)(
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
The Convection Boundary Layers
Thermal Boundary Layer (Cont’d)
Heat Flux
Heat flux analogous to shear stress in velocity
boundary layer
Heat flux proportional to the temperature
gradient at the surface,
AND
since u(y=0) =0,energy transfer to/from fluid
occurs by conduction only!
Since thermal boundary layer gets larger along x
direction,the temperature gradient changes with
x,and therefore
TTs
fluid thermal
conductivity
wall
temperature
gradient
____________________
0
y
fs y
Tkq
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
The Convection Boundary Layers
Thermal Boundary Layer (Cont’d)
Heat Flux (Cont’d)
Using Newton’s law of cooling:
We obtain
While? increases with increasing x,temperature
gradients in the boundary layer must decrease
with increasing x,
Accordingly,and h decrease with increasing x,
)( TThq s
TT
yTkh
s
yf 0/
sq?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
The Convection Boundary Layers
Laminar Versus Turbulent Flow
Characterization of laminar flow
Low amount of,mixing” of fluid within the
boundary layer (smooth flow)
Characterization of turbulent flow
High amount of mixing of fluid within the
boundary layer (irregular flow)
High amount of mixing means increased surface
friction as well as convection transfer rates
(heat and mass)
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 14
Convection Heat Transfer Variations
Along Flat Plate
Consider flat plate with:
and all laminar flow
Thermal boundary layer defined by
Compare temperature gradients at points 1 and 2
to evaluate the heat flux rate (and hence the heat
transfer coefficient)
C0 a n d C100TT x
2?
1?
C0T
C100sT1 2
99.0
TT
TT
s
s
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 15
Convection Heat Transfer Variations
Along Flat Plate (Cont’d)
Consider flat plate with,C0 a n d C100TT x
2?
1?
C100?C1? C100?C1?
at x1 at x2
To determine the location that transition begins,we
define the Reynolds number,
And the critical Reynolds number,
xu
x
Re
5
,105Re
c
cx
xu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 16
Convective Transfer Equations
Topic of the Day
Project Teams
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 17
Convective Transport Equations
Where we’ve been ……
Last section:
Overview of the topic of convective transport of heat.
Where we’re going:
Convection transfer detailed equations
Heat and mass transfer analogy
Eventually get to applications in external and
internal flow.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 18
Convective Transport Equations
KEY POINTS THIS SECTION
Detailed development of boundary layer
equations for velocity,temperature and species
concentrations
What approximations can be made in the
boundary layers?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 19
Recall the convection overview
Local heat flux is:
where h is the local heat transfer coefficient
)( TThq s
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 20
Develop convection transfer equations
Consider steady,2-D flow of a viscous,
incompressible fluid with constant properties.
Key point to remember:
At each point in the fluid,conservation of mass,
energy and momentum must be satisfied.
Appendix E contains detailed development of
the full boundary layer equations,for example:
Conservation of mass (continuity)
0 yvxu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 21
The magnitude of variables in the thermal
boundary layer
variables x
(main flow direction)
y u v t
magnitude 1 1 1
x
T
y
T
Thermal boundary layer
y
v
y
u,
x
v
y
u,
x
u
y
u a n d ;
vu
Velocity boundary layer
Boundary Layer Approximations
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 22
Continue convection transfer equations
Quick overview of fluid equations
Conservation of mass (continuity):
OR
x-momentum equation:
y-momentum equation:
0 0?
y
v
x
u
y
v
x
u
0 yvxu
f o r c e s "B o d y " 2
2
2
2
y
u
x
u
x
P
y
uv
x
uu
xuyu 0 0
2
2 1
y
u
x
P
y
uv
x
uu
,w h e re
f o r c e s "B o d y " 2
2
2
2
y
v
x
v
y
P
y
vv
x
vu
0 0 0 0 0
0yPSo:
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 23
Continue convection transfer equations
Energy equation:
q
y
v
x
u
x
v
y
u
y
T
x
T
k
y
T
v
x
T
uc p
222
2
2
2
2
2?
0 0 0 0
0
2
yuc
p
NOTE,is usually small unless u is high
(as in sonic flows) or? is high (such as flow of oils).
Result is 4 equations and 4 unknowns:
Unknowns are,u,v,P,and T
Since:
P(x) can be obtained from free stream flow.
dx
dP
x
P a n d o n l y,)x(fP t h e n 0
y
P?
xTyT W h y
xvyu W h y
2
2
2
yucy TyTvxTu
p
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 24
Review boundary layer concept.
General relationship between velocity and
thermal boundary layers.
Convective heat transfer is dependent on
the temperature gradient at the fluid/solid
interface
Boundary layer grows with distance from
the leading edge,and this decreases the
local heat transfer rates.
Turbulent boundary layers have much
greater potential for heat and mass transfer
due to velocity fluctuations.
Convection transfer equations (4).
Go back and review fluids course notes!
SUMMARY
