Transcript Convective Heat Transfer

HEAT
TRANSFER
CHAPTER 7
External flow
Heat Transfer
#1
Su Yongkang
School of Mechanical Engineering
External Flow: Flat Plate
Topic of the Day
Heat Transfer
#2
Su Yongkang
School of Mechanical Engineering
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.
Nu 
hL
kf
Where we’re going:
• Applications to external flow
– Flat plate
 Today
– Other shapes
 Next time
Then onto internal flow ……
Heat Transfer
#3
Su Yongkang
School of Mechanical Engineering
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
#4
Su Yongkang
School of Mechanical Engineering
How this impacts convective heat transfer
• Recall the boundary layer convection equations:
Ts  T
qs   k f
fluid thermal
conductivity
T
y
wall
temperature
gradient
y 0
• 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, , 
–
–
Heat Transfer
so
and
T
y

y 0
qs
Also 
#5
Su Yongkang
School of Mechanical Engineering
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
#6
Su Yongkang
School of Mechanical Engineering
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………….
T , U 
Twing surface
• What are the parameters involved?
– Velocity,
U
–wing length, L
– Prandtl number, Pr –viscosity,

– Nusselt number, Nu
• Which of these can we control easily?
• Looking for the relation:
Experience has shown the following relation
works well:
Nu  C RemL Prn
Heat Transfer
#7
Su Yongkang
School of Mechanical Engineering
Empirical method to obtain heat transfer
coefficient
• Experimental test setup
 Power input
T , U 
L
insulation
• Measure current (hence heat transfer) with
various fluids and test conditions for T , U 
• Fluid properties are typically evaluated at the
mean film temperature
T  T
Tf  s
2
Heat Transfer
#8
Su Yongkang
School of Mechanical Engineering
Analytical Solution – Laminar Flow
• Assume:
– Steady, incompressible, laminar flow
– Constant fluid properties
– For flat plate,
y
T ,U


Ts
• Boundary layer equations
Continuity
Momentum
Energy
u v
 0
x y
u
u
 2u
u  v  2
x
y
y
T
T
 2T
u
v
 2
x
y
y
• Blasius developed a similarity solution to the
hydrodynamic equations in 1908 based on the
stream function, (x,y)
Heat Transfer
#9
Su Yongkang
School of Mechanical Engineering
Analytical Solution – Laminar Flow (Cont’d)
u

y
and
v

x
• Define new dependent and independent
variables,
f ( ) 
u

x / u
  y u /x
• The momentum equation can be rewritten as
d3 f
d2 f
2 3f
0
2
d
d
• And the boundary conditions are
df
 f ( 0)  0
d   0
Heat Transfer
and
df
1
d   
# 10
Su Yongkang
School of Mechanical Engineering
Analytical Solution – Laminar Flow (Cont’d)
• Blasius solution summary:
u x
5
5x

but, since Re x  
  

u
Re x
x
• Conclusions from the Blasius solution:
1
  x and   
and  
u
• Solution for the thermal boundary layer:
 2T * Pr T *

f
0
2
2


– For Pr  0.6
T *

 0.332 Pr
1
3
η0
– Expressing the local convection coefficient
u  T *
as:
hx  k
x  η0
– Then the Local Nusselt number is:
Eq.
7.21
Heat Transfer
Nu x 
hx x
 0.332 Re1x/ 2 Pr1/ 3
k
For 0.6  Pr  50
# 11
Su Yongkang
School of Mechanical Engineering
Analytical Solution – Laminar Flow (Cont’d)
• The Average Nusselt number over the whole
plate found by integrating:
hx x
x 1 x

Nu x 

h
dx

x

k
k  x 0

Eq.
7.25
Nux  0.664Re1x/ 2 Pr1/ 3
• Ratio of velocity to thermal boundary layer
thickness:
For large Pr (oils):
y
For small Pr (liquid metals):
y


 th

x
x
Pr > 1000
Pr < 0.1
Fluid viscosity greater
than thermal diffusivity

Fluid viscosity less than
thermal diffusivity

Heat Transfer
th
# 12
Su Yongkang
School of Mechanical Engineering
Analytical Solution – Laminar Flow (Cont’d)
• Solution for friction factor
 s , x  0.332u
C f ,x 
C f ,x 
 s, x
u2 / 2
u
x
 0.664Re x1/ 2
 s,x
 s,x
u / 2
2

1 x
   s , x dx
x 0
C f , x  1.328Rex1/ 2
• Textbook contains Nusselt number correlations
for low Pr (liquid metals) and large Pr (oils)
Heat Transfer
# 13
Su Yongkang
School of Mechanical Engineering
Analytical Solution – Turbulent Flow
• For flat plate in turbulent flow (more common)
C f , x  0.0592Re x1/ 5
Re x  107
  0.37 x Re -1x 5
Nu x  StRe x Pr  0.0296 Re 4x 5 Pr1 3
0.6  Pr  60
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
1  xc

hx    hlam dx   hturb dx 
L0

xc
Heat Transfer
# 14
Su Yongkang
School of Mechanical Engineering
Analytical Solution – Mixed Boundary Layer
• Integrating
Nu L  (0.037Re 4L/ 5 - 871)Pr1/3
0.6  Pr  60
5 105  Re L  108
0.074 1742
Cf,L  1/5 Re L Re L
5 105  Re L  108
Re
Re
x, c
x, c
 5 105
 5 105
Equations 7.33 and 7.34
Heat Transfer
# 15
Su Yongkang
School of Mechanical Engineering


Analytical Solution – Special Cases
• The existence of unheated starting length.
• When the boundary condition is a uniform
surface heat flux.
For laminar flow,
Nux  0.453 Re1x 2 Pr1 3
Pr  0.6
For turbulent flow,
Nux  0.0308 Re4x 5 Pr1 3
0.6  Pr  60
qs
Ts ( x)  T 
hx
Heat Transfer
# 16
Su Yongkang
School of Mechanical Engineering
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
# 17
Su Yongkang
School of Mechanical Engineering
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
# 18
Su Yongkang
School of Mechanical Engineering
Example – Cooling of automobile crankcase
(Cont’d)
1. Determine air properties at an average film
temperature T  Ts  T  325 K
f
2
kg
N s



m3
m2
W
Pr 
k
mK
2. Calculate Reynolds #
3. Calculate average Nusselt number (mixed b.l.)
4. Average convection coefficient is
5. BOTTOM SURFACE HEAT LOSS:
Heat Transfer
# 19
Su Yongkang
School of Mechanical Engineering
Example – Cooling of automobile crankcase
(Cont’d)
•
How to determine the heat loss from the
other surfaces?
– Assumptions …………..
– Analysis procedure ………
Heat Transfer
# 20
Su Yongkang
School of Mechanical Engineering
Example: Cooling air over electronic chips
• Given:
Cooling air drawn over electronic devices
mounted on board.
T = 27 º C
Q = 40 mW each device
V = 10 m/s
“turbulator”
15 mm
CL
• 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?
Heat Transfer
# 21
Su Yongkang
School of Mechanical Engineering
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
5
5
6
corresponding to 10 , 510 , and 10 .
Heat Transfer
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Su Yongkang
School of Mechanical Engineering
Heat Transfer
# 23
Su Yongkang
School of Mechanical Engineering
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:
Nu L 
hL
 C Re mL Pr n
kf
• 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.
Heat Transfer
# 24
Su Yongkang
School of Mechanical Engineering
Have a good time!
Go back and review lecture notes!
Heat Transfer
# 25
Su Yongkang
School of Mechanical Engineering