Heat Transfer Su Yongkang
School of Mechanical Engineering
# 1
HEAT TRANSFER
CHAPTER 7
External flow
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 2
External Flow,Flat Plate
Topic of the Day
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
External Flow,Flat Plate
Where we’ve been ……
General overview of the convection transfer
equations.
Developed the key non-dimensional parameters
used to characterize the boundary layer flow and
convective heat and mass transfer.
Where we’re going:
Applications to external flow
– Flat plate? Today
– Other shapes? Next time
Then onto internal flow ……
fk
LhNu?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
Differences between external and internal flow
External flow:
Boundary layer develops freely,without
constraints
Internal flow:
Boundary layer is constrained and eventually
merges
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
How this impacts convective heat transfer
Recall the boundary layer convection equations:
As you go further from the leading edge,the
boundary layer continues to grow,Assuming
the surface and freestream T do not change:
with increasing distance ‘x’:
– Boundary layer thickness,?,?
– so
– and
fluid thermal
conductivity
wall
temperature
gradient
TTs
Also
0
y
fs y
Tkq
0
yy
T
sq?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
Methods to evaluate convection heat transfer
Empirical (experimental) analysis
– Use experimental measurements in a
controlled lab setting to correlate heat and/or
mass transfer in terms of the appropriate
non-dimensional parameters
Theoretical or Analytical approach
– Solving of the boundary layer equations for
a particular geometry.
– Example,
Solve for T*
Use evaluate the local Nusselt number,Nux
Compute local convection coefficient,hx
Use these (integrate) to determine the
average convection coefficient over the
entire surface
– Exact solutions possible for simple cases,
– Approximate solutions also possible using
an integral method
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
Empirical method to obtain heat transfer
coefficient
How to set up an experimental test?
Let’s say you want to know the heat transfer rate
of an airplane wing (with fuel inside) flying at
steady conditions………….
What are the parameters involved?
– Velocity,–wing length,
– Prandtl number,–viscosity,
– Nusselt number,
Which of these can we control easily?
Looking for the relation:
Experience has shown the following relation
works well:
UT,
surface wingT
U L
Pr?
Nu
nmLCNu PrRe?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
UT,
i n p u tP o w e r?
insulation
L
Empirical method to obtain heat transfer
coefficient
Experimental test setup
Measure current (hence heat transfer) with
various fluids and test conditions for
Fluid properties are typically evaluated at the
mean film temperature
UT,
2?
TTT s
f
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
Analytical Solution – Laminar Flow
Assume:
– Steady,incompressible,laminar flow
– Constant fluid properties
– For flat plate,
Boundary layer equations
Blasius developed a similarity solution to the
hydrodynamic equations in 1908 based on the
stream function,?(x,y)
0 yvxu
2
2
y
u
y
uv
x
uu
22y TyTvxTu
Continuity
Momentum
Energy
UT,
sT
y
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
Analytical Solution – Laminar Flow (Cont’d)
Define new dependent and independent
variables,
The momentum equation can be rewritten as
And the boundary conditions are
yu?
xv?
and
uxuf /)(
xuy /
02 2
2
3
3
d fdfd fd
0)0(
0
fddf
1?
d
dfand
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
Analytical Solution – Laminar Flow (Cont’d)
Blasius solution summary:
Conclusions from the Blasius solution:
Solution for the thermal boundary layer:
– For Pr? 0.6
– Expressing the local convection coefficient
as:
– Then the Local Nusselt number is:
x
xu
x
u Re
x5 Re s i n c e b u t,
5
x
u1 a n d a n d x
0 2Pr *2*2 TfT
31
0η
*
Pr 0,3 3 2
T
0η
*
T
x
ukh
x
For 0.6? Pr? 50Eq,7.21
3/12/1 PrRe3 32.0 xxx
k
xhNu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
Analytical Solution – Laminar Flow (Cont’d)
The Average Nusselt number over the whole
plate found by integrating:
Ratio of velocity to thermal boundary layer
thickness:
x1
0
x
xxx dxhk
x
k
xhuN
y
x
th?
For large Pr (oils):
Pr > 1000
y
x
th?
For small Pr (liquid metals):
Pr < 0.1
Fluid viscosity greater
than thermal diffusivity
Fluid viscosity less than
thermal diffusivity
Eq,
7.25 3/12/1 PrRe664.0
xxNu?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
Analytical Solution – Laminar Flow (Cont’d)
Solution for friction factor
Textbook contains Nusselt number correlations
for low Pr (liquid metals) and large Pr (oils)
2/1,Re328.1 xxfC
2/1
2
,
,Re664.02/
xxsxf uC
2/2
,
,
uC xsxf
x
uu
xs
332.0
,
x xsxs dxx 0,,1
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 14
Analytical Solution – Turbulent Flow
For flat plate in turbulent flow (more common)
60 P r 0,6 Pr Re 0,0 2 9 6PrS t R e
Re 0,3 7
3154
xx
5-1
x
xNu
x?
Important point:
– Typically a
turbulent boundary
layer is preceded by
a laminar boundary
layer first upstream
–? need to consider
case with mixed
boundary layer
conditions!
L
xc t u r b
xc
l a mx dxhdxhLh
1
0
75/1,10ReRe0592.0 xxxfC
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 15
Analytical Solution – Mixed Boundary Layer
Integrating
5
cx,
85
L
1 / 5
L
Lf,
5
cx,
85
1 / 35/4
105Re 10 Re105
Re
1 7 4 2
-
Re
0,0 7 4
C
105Re 10 Re105
60 P r 0,6
8 7 1 ) P r-Re0 3 7.0(
L
L
L
LNu
Equations 7.33 and 7.34
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 16
Analytical Solution – Special Cases
The existence of unheated starting length.
When the boundary condition is a uniform
surface heat flux.
For laminar flow,
For turbulent flow,
60P r 0,6 Pr Re 0,0 3 0 8 3154xxNu
0,6Pr Pr Re 0,4 5 3 3121xxNu
x
s
s h
qTxT
)(
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 17
Methodology for a Convection Calculation
Become immediately cognizant of the flow
geometry.
Specify the appropriate reference temperature
and evaluate the fluid properties.
Calculate the Reynolds number
Decide whether a local or surface average
coefficient is required.
Select the appropriate correlation.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 18
Example – Cooling of automobile crankcase
Given:
– Automobile crankcase with approximate
dimensions of 0.6 m long,0.2 m wide and
0.1 m deep.
– Surface temperature of 350 K
– Ambient temperature of 300 K
– Vehicle velocity of 30 m/s
Find:
– Heat loss from bottom surface exposed to air
stream
What other information or assumptions
needed?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 19
Example – Cooling of automobile crankcase
(Cont’d)
1,Determine air properties at an average film
temperature
2,Calculate Reynolds #
3,Calculate average Nusselt number (mixed b.l.)
4,Average convection coefficient is
5,BOTTOM SURFACE HEAT LOSS:
K 325 2TTT sf
Km
W Pr
m
sN
m
kg
23
k
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 20
Example – Cooling of automobile crankcase
(Cont’d)
How to determine the heat loss from the
other surfaces?
– Assumptions …………..
– Analysis procedure ………
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 21
Example,Cooling air over electronic chips
Given:
Cooling air drawn over electronic devices
mounted on board.
Devices are 4 x 4 mm in size,spacing = 0.25 mm
Find the surface temperature of the fourth device,
assumed uniform surface T.
Assumptions?
Solution Method?
T? = 27 o C
V = 10 m/s Q = 40 mW each device
,turbulator”
15 mm CL
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 22
Example,Consider atmospheric air at 25℃ and a
velocity of 25 m/s flowing over both surfaces of a 1-m
long flat plate that is maintained at 125 ℃,Determine
the rate of heat transfer per unit width from the plate
for values of the critical Reynolds number
corresponding to,,and,510 5105? 610
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 23
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 24
External Flow,Flat Plate
KEY POINTS THIS SECTION
What key characteristic of external flow
compared to internal flow?
Heat transfer rate generally decreases with
increasing distance from leading edge.
Turbulent convective heat transfer generally
higher than laminar due to mixing effect within
boundary layer.
Experimental tests indicate that heat transfer
coefficient will generally vary like:
Concept of transition Re number.
Difference in boundary layer growth for high
and low Pr number fluids.
General correlation for Nusselt number for flow
over flat plate in laminar,turbulent and mixed
flows.
nmL
f
L Ck LhNu Pr Re
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 25
Have a good time!
Go back and review lecture notes!
School of Mechanical Engineering
# 1
HEAT TRANSFER
CHAPTER 7
External flow
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 2
External Flow,Flat Plate
Topic of the Day
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
External Flow,Flat Plate
Where we’ve been ……
General overview of the convection transfer
equations.
Developed the key non-dimensional parameters
used to characterize the boundary layer flow and
convective heat and mass transfer.
Where we’re going:
Applications to external flow
– Flat plate? Today
– Other shapes? Next time
Then onto internal flow ……
fk
LhNu?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
Differences between external and internal flow
External flow:
Boundary layer develops freely,without
constraints
Internal flow:
Boundary layer is constrained and eventually
merges
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
How this impacts convective heat transfer
Recall the boundary layer convection equations:
As you go further from the leading edge,the
boundary layer continues to grow,Assuming
the surface and freestream T do not change:
with increasing distance ‘x’:
– Boundary layer thickness,?,?
– so
– and
fluid thermal
conductivity
wall
temperature
gradient
TTs
Also
0
y
fs y
Tkq
0
yy
T
sq?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
Methods to evaluate convection heat transfer
Empirical (experimental) analysis
– Use experimental measurements in a
controlled lab setting to correlate heat and/or
mass transfer in terms of the appropriate
non-dimensional parameters
Theoretical or Analytical approach
– Solving of the boundary layer equations for
a particular geometry.
– Example,
Solve for T*
Use evaluate the local Nusselt number,Nux
Compute local convection coefficient,hx
Use these (integrate) to determine the
average convection coefficient over the
entire surface
– Exact solutions possible for simple cases,
– Approximate solutions also possible using
an integral method
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
Empirical method to obtain heat transfer
coefficient
How to set up an experimental test?
Let’s say you want to know the heat transfer rate
of an airplane wing (with fuel inside) flying at
steady conditions………….
What are the parameters involved?
– Velocity,–wing length,
– Prandtl number,–viscosity,
– Nusselt number,
Which of these can we control easily?
Looking for the relation:
Experience has shown the following relation
works well:
UT,
surface wingT
U L
Pr?
Nu
nmLCNu PrRe?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
UT,
i n p u tP o w e r?
insulation
L
Empirical method to obtain heat transfer
coefficient
Experimental test setup
Measure current (hence heat transfer) with
various fluids and test conditions for
Fluid properties are typically evaluated at the
mean film temperature
UT,
2?
TTT s
f
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
Analytical Solution – Laminar Flow
Assume:
– Steady,incompressible,laminar flow
– Constant fluid properties
– For flat plate,
Boundary layer equations
Blasius developed a similarity solution to the
hydrodynamic equations in 1908 based on the
stream function,?(x,y)
0 yvxu
2
2
y
u
y
uv
x
uu
22y TyTvxTu
Continuity
Momentum
Energy
UT,
sT
y
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
Analytical Solution – Laminar Flow (Cont’d)
Define new dependent and independent
variables,
The momentum equation can be rewritten as
And the boundary conditions are
yu?
xv?
and
uxuf /)(
xuy /
02 2
2
3
3
d fdfd fd
0)0(
0
fddf
1?
d
dfand
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
Analytical Solution – Laminar Flow (Cont’d)
Blasius solution summary:
Conclusions from the Blasius solution:
Solution for the thermal boundary layer:
– For Pr? 0.6
– Expressing the local convection coefficient
as:
– Then the Local Nusselt number is:
x
xu
x
u Re
x5 Re s i n c e b u t,
5
x
u1 a n d a n d x
0 2Pr *2*2 TfT
31
0η
*
Pr 0,3 3 2
T
0η
*
T
x
ukh
x
For 0.6? Pr? 50Eq,7.21
3/12/1 PrRe3 32.0 xxx
k
xhNu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
Analytical Solution – Laminar Flow (Cont’d)
The Average Nusselt number over the whole
plate found by integrating:
Ratio of velocity to thermal boundary layer
thickness:
x1
0
x
xxx dxhk
x
k
xhuN
y
x
th?
For large Pr (oils):
Pr > 1000
y
x
th?
For small Pr (liquid metals):
Pr < 0.1
Fluid viscosity greater
than thermal diffusivity
Fluid viscosity less than
thermal diffusivity
Eq,
7.25 3/12/1 PrRe664.0
xxNu?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
Analytical Solution – Laminar Flow (Cont’d)
Solution for friction factor
Textbook contains Nusselt number correlations
for low Pr (liquid metals) and large Pr (oils)
2/1,Re328.1 xxfC
2/1
2
,
,Re664.02/
xxsxf uC
2/2
,
,
uC xsxf
x
uu
xs
332.0
,
x xsxs dxx 0,,1
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 14
Analytical Solution – Turbulent Flow
For flat plate in turbulent flow (more common)
60 P r 0,6 Pr Re 0,0 2 9 6PrS t R e
Re 0,3 7
3154
xx
5-1
x
xNu
x?
Important point:
– Typically a
turbulent boundary
layer is preceded by
a laminar boundary
layer first upstream
–? need to consider
case with mixed
boundary layer
conditions!
L
xc t u r b
xc
l a mx dxhdxhLh
1
0
75/1,10ReRe0592.0 xxxfC
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 15
Analytical Solution – Mixed Boundary Layer
Integrating
5
cx,
85
L
1 / 5
L
Lf,
5
cx,
85
1 / 35/4
105Re 10 Re105
Re
1 7 4 2
-
Re
0,0 7 4
C
105Re 10 Re105
60 P r 0,6
8 7 1 ) P r-Re0 3 7.0(
L
L
L
LNu
Equations 7.33 and 7.34
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 16
Analytical Solution – Special Cases
The existence of unheated starting length.
When the boundary condition is a uniform
surface heat flux.
For laminar flow,
For turbulent flow,
60P r 0,6 Pr Re 0,0 3 0 8 3154xxNu
0,6Pr Pr Re 0,4 5 3 3121xxNu
x
s
s h
qTxT
)(
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 17
Methodology for a Convection Calculation
Become immediately cognizant of the flow
geometry.
Specify the appropriate reference temperature
and evaluate the fluid properties.
Calculate the Reynolds number
Decide whether a local or surface average
coefficient is required.
Select the appropriate correlation.
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 18
Example – Cooling of automobile crankcase
Given:
– Automobile crankcase with approximate
dimensions of 0.6 m long,0.2 m wide and
0.1 m deep.
– Surface temperature of 350 K
– Ambient temperature of 300 K
– Vehicle velocity of 30 m/s
Find:
– Heat loss from bottom surface exposed to air
stream
What other information or assumptions
needed?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 19
Example – Cooling of automobile crankcase
(Cont’d)
1,Determine air properties at an average film
temperature
2,Calculate Reynolds #
3,Calculate average Nusselt number (mixed b.l.)
4,Average convection coefficient is
5,BOTTOM SURFACE HEAT LOSS:
K 325 2TTT sf
Km
W Pr
m
sN
m
kg
23
k
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 20
Example – Cooling of automobile crankcase
(Cont’d)
How to determine the heat loss from the
other surfaces?
– Assumptions …………..
– Analysis procedure ………
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 21
Example,Cooling air over electronic chips
Given:
Cooling air drawn over electronic devices
mounted on board.
Devices are 4 x 4 mm in size,spacing = 0.25 mm
Find the surface temperature of the fourth device,
assumed uniform surface T.
Assumptions?
Solution Method?
T? = 27 o C
V = 10 m/s Q = 40 mW each device
,turbulator”
15 mm CL
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 22
Example,Consider atmospheric air at 25℃ and a
velocity of 25 m/s flowing over both surfaces of a 1-m
long flat plate that is maintained at 125 ℃,Determine
the rate of heat transfer per unit width from the plate
for values of the critical Reynolds number
corresponding to,,and,510 5105? 610
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 23
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 24
External Flow,Flat Plate
KEY POINTS THIS SECTION
What key characteristic of external flow
compared to internal flow?
Heat transfer rate generally decreases with
increasing distance from leading edge.
Turbulent convective heat transfer generally
higher than laminar due to mixing effect within
boundary layer.
Experimental tests indicate that heat transfer
coefficient will generally vary like:
Concept of transition Re number.
Difference in boundary layer growth for high
and low Pr number fluids.
General correlation for Nusselt number for flow
over flat plate in laminar,turbulent and mixed
flows.
nmL
f
L Ck LhNu Pr Re
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 25
Have a good time!
Go back and review lecture notes!
