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
Boundary Layer Similarity Parameters
The boundary layer equations (velocity,mass,
energy continuity) represent low speed,forced
convection flow.
Advectionterms on the left side and diffusion
terms on the right side of each equation,
such as:
Advection Diffusion
Non-dimensionalize the equations by setting:
2*s*
/ a n d
T
T-T
a n d
)( v e l o c i t y f r e e s t r e a m t h eis V w h e r e
s u r f a c e t h eofl e n g t h s t i cc h a r a c t e r i is L w h e r e
VpP
T
T
U
V
v
va n d
V
u
u
L
y
ya n d
L
x
x
s

2
2
yTyTvxTu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
Boundary Layer Similarity Parameters (Cont’d)
The boundary layer equations can be rewritten
in terms of the non-dimensional variables
Continuity
x-momentum
energy
With boundary conditions
0**** yvxu
2*
*2
*
*
*
**
*
**
y
u
VLx
P
y
uv
x
uu

2* *2******
y
T
VLy
Tv
x
Tu

1 ),(
] if 1[
)(
),(,F r ee s t r e am
0 )0,(; 0 )0,( ; 0 )0,(,W al l
***
*
***
***
******

yxT
UV
V
xU
yxu
yxT
yxvyxu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
Boundary Layer Similarity Parameters (Cont’d)
From the non-dimensionalized boundary layer
equations,dimensionless groups can be seen
Reynolds #
Prandtl #
Substituting gives the boundary layer equations:
VL?
LRe
0**** yvxu
2*
*2
*
*
*
**
*
**
Re
1
y
u
x
P
y
uv
x
uu
L?

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

Continuity:
x-momentum:
Energy:
rP
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
Back to the convection heat transfer problem…
Solutions to the boundary layer equations are of
the form:
Rewrite the convective heat transfer coefficient
Define the Nusselt number as:

*
*
***
*
*
*
*
***
P r,,Re,,
p l at ef l at f o r 0,w h er e,Re,,
dx
dP
yxfT
dx
dP
dx
dP
yxfu
L
L
L
TT
L
y
TT
TT
TT
k
TT
y
T
k
TT
q
h s
y
s
s
s
f
s
y
f
s
x
x

0
0
0
*
*
*

y
f
x y
T
L
kh

*
**
0
*
*
P r,,Re,
* dx
dPxf
y
T
L
y
Nu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
Nusselt number for a prescribed geometry
(For a prescribed geometry,is known)
Many convection problems are solved using
Nusselt number correlations incorporating
Reynolds and Prandtl numbers
The Nusselt number is to the thermal boundary
layer what the friction coefficient is to the velocity
boundary layer.

A v e r a g e Pr,Re
k
Lh
L o c a l Pr,Re,
k
hL
f
*
f
L
L
fNu
xfNu

*
*
dx
dP

*
*
*
0
*
*
2,Re,Re
2
2
dx
dPxf
y
u
V
C L
yL
s
f
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
Heat transfer coefficient,simple example
Given:
Air at 20oC flowing over heated flat plate at
100oC,Experimental measurements of
temperatures at various distances from the
surface are as shown
Find,convective heat transfer coefficient,h
tsme a su r e me n alE x p e r i me n t
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
Heat transfer coefficient,simple example
Solution:
Recall that h is computed by
From Table A-4 in Appendix,at a mean fluid
temperature
(average of free-stream and surface temperatures)
the air conductivity,k is? 0.028 W/m-K
Temperature gradient at the plate surface from
experimental data is -66.7 K/mm = -66,700
K/m
So,convective heat transfer coefficient is:
0

TT
y
Tk
h
s
y
f
x
Km
W
345.23
80
)6 6 7 0 0(0,0 2 8-
2

h
2m s
TTT
CT?602)10020(m
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
Example,Experimental results for heat transfer
over a flat plate with an extremely rough surface
were found to be correlated by an expression of the
form
where is the local value of the Nusselt number
at a position x measured from the leading edge of
the plate,Obtain an expression for the ratio of the
average heat transfer coefficient to the local
coefficient.
3/19.0 PrRe04.0 xxNu?
xNu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
General boundary layer equations
Nusselt number for heat transfer coefficient in the
thermal boundary layer
Many convection problems are solved using
Nusselt number correlations incorporating
Reynolds and Prandtl numbers.
SUMMARY
0 yvxu
2
2 1
y
u
x
P
y
uv
x
uu

0
y
P
2
2
2?

y
u
cy
T
y
Tv
x
Tu
p

A v er ag e Pr,Re
L o ca l Pr,Re,
*
L
f
L
f
f
k
Lh
Nu
xf
k
Lh
Nu

Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
Momentum and Heat Transfer Analogy
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
Momentum and Heat Transfer Analogy
Where we’ve been ……
Development of convective transport of heat
transfer equations.
Where we’re going:
Momentum and heat transfer (Reynolds) analogy.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 14
Momentum and Heat Transfer Analogy
KEY POINTS THIS SECTION
Physical significance of the dimensionless
parameters (Reynolds,Prandtl numbers)
Describe how the convective heat transfer
equations can be related (Heat Transfer Analogy)
Review of the general convection equations,
prepare for application to external and internal
flows.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 15
Review boundary layer equations for heat
transfer
For heat transfer (conservation of energy)
usually small,except for high speed
or highly viscous flow
Begin to see where the momentum and heat
transfer analogy is going …..
2
2
2
yucy TyTvxTu
p

2*
*2
*
*
*
**
*
**
Re
1
y
u
x
P
y
uv
x
uu
L?

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

2
2 1
y
u
x
P
y
uv
x
uu

Momentum
equation
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 16
If and,we obtain,
These two equations are of precisely the same form.
We know that if,.
The boundary conditions for these two equations are:
The boundary conditions are equivalent,
Therefore,the boundary layer velocity and temperature
profiles must be of the same functional form,
0/*?dxdp 1Pr?
Re 1 2* *2****** yTyTvxTu
L?

2*
*2
*
*
*
*
*
*
Re
1
y
u
y
uv
x
uu
L?

0/*?dxdp Vu
1 ),(
1;
)(
),(,F r e e s t r e a m
0 )0,(; 0 )0,(,W a l l
***
*
***
***
***

yxT
V
xU
yxu
yxT
yxu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 17
Momentum and Heat Transfer Analogy
(continued)
For the solution,the function f must be the same,
As we know,
We conclude that

Pr,Re,,T
Re,,
***
***
L
L
yxf
yxfu
L
LyL
s
f xfy
u
VC Re,Re
2
Re
2
2
*
0
*
*
2

Pr,Re,N u *
0
*
*
*
L
y
xfyT
NuC Lf?2Re
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 18
Momentum and Heat Transfer Analogy
(continued)
Define the Stanton number St,
The analogy takes the form
The restrictions,the validity of the boundary
layer approximations,and,
The modified Reynolds,or Chilton-Colburn,
analogy has the form
PrRe
Nu
Vc
hSt
p

StC f?2
Reynolds analogy
0/*?dxdp 1Pr?
2 / 3P r ( 0,6 P r 6 0 )2fc S t j
Colburn j factor
For laminar flow,it’s only appropriate when 0/*?dxdp
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 19
Boundary Layer Similarity Parameters
Recall the non-dimensional parameters
Reynolds # -
Ratio of the inertia to the viscous forces of a
fluid flow
Prandtl #
Ratio of the momentum to the thermal
diffusivity in a fluid flow.
For laminar boundary layers,
Where n is a positive exponent,
L
s
l VL
LV
LV
F
F Re
/
/
2
2

k
c pPr
n
th
Pr
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 20
Example,Forced air at ℃ and is used to
cool electronic elements on a circuit board,One such
element is a chip,4 mm by 4 mm,located 120 mm from the
leading edge of the board,Experiments have revealed that
flow over the board is disturbed by the elements and that
convection heat transfer is correlated by an expression of the
form
Estimate the surface temperature of the chip if it is
dissipating 30 mW.
3/185.0 PrRe04.0 xxNu?
25T smV /10?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 21
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 22
General boundary layer equations
Nusselt number for heat transfer coefficient in the
thermal boundary layer
Empirical evaluation of Nusselt number involves
correlations incorporating Re and Pr
Convective Transport Equations Summary
0 yvxu
2
2 1
y
u
x
P
y
uv
x
uu

0
y
P
2
2
2?

y
u
cy
T
y
Tv
x
Tu
p

A v er ag e Pr,Re
L o ca l Pr,Re,
*
L
f
L
f
f
k
Lh
Nu
xf
k
Lh
Nu

Local heat flux is,where h is the
local heat transfer coefficient )( TThq s
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 23
Convective Transport Equations Summary
(Cont’d)
VLVLLRe
th
Pr Pr npkμ c
Reynolds #
Prandtl #
Reynolds analogy
StC f?2
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 24
Nothing Is Impossible To A Willing Heart