External Forced Convection Notes

1. 27-10-2022
1
Chapter 7: External Forced
Convection
Andallib Tariq
Department of Mechanical & Indl Engg
Objectives
When you finish studying this chapter, you should be
able to:
• Distinguish between internal and external flow,
• Develop an intuitive understanding of friction drag and
pressure drag, and evaluate the average drag and
convection coefficients in external flow,
• Evaluate the drag and heat transfer associated with
flow over a flat plate for both laminar and turbulent
flow,
• Calculate the drag force exerted on cylinders during
cross flow, and the average heat transfer coefficient,
and
• Determine the pressure drop and the average heat
transfer coefficient associated with flow across a tube
bank for both in-line and staggered configurations.
Drag and Heat Transfer in External
flow
• Fluid flow over solid bodies is responsible for numerous
physical phenomena such as
– drag force
• automobiles
• power lines
– lift force
• airplane wings
– cooling of metal or plastic sheets.
• Free-stream velocity ─ the velocity of the fluid relative to
an immersed solid body sufficiently far from the body.
• The fluid velocity ranges from zero at the surface (the no-
slip condition) to the free-stream value away from the
surface.
Friction and Pressure Drag
• The force a flowing fluid exerts on a body in the flow
direction is called drag.
• Drag is compose of:
– pressure drag,
– friction drag (skin friction drag).
• The drag force FD depends on the
– density  of the fluid,
– the upstream velocity V, and
– the size, shape, and orientation of the body.
• The dimensionless drag coefficient CD is defined as
2
1 2
D
D
F
C
V A

 (7-1)
• At low Reynolds numbers, most drag is due to friction
drag.
• The friction drag is also proportional to the surface area.
• The pressure drag is proportional to the frontal area and to
the difference between the pressures acting on the front
and back of the immersed body.
• The pressure drag is usually dominant for blunt bodies
and negligible for streamlined bodies.
• When a fluid separates from a body,
it forms a separated region between
the body and the fluid stream.
• The larger the separated region, the
larger the pressure drag.
Heat Transfer
• The phenomena that affect drag force also affect heat
transfer.
• The local drag and convection coefficients vary along
the surface as a result of the changes in the velocity
boundary layers in the flow direction.
• The average friction and convection coefficients for
the entire surface can be determined by
,
0
1
L
D D x
C C dx
L
  (7-7)
0
1
L
x
h h dx
L
  (7-8)
1 2
3 4
5 6

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Parallel Flow Over Flat Plates
• Consider the parallel flow of a fluid over a flat plate of
length L in the flow direction.
• The Reynolds number at a distance
x from the leading edge of a flat
plate is expressed as
• In engineering analysis, a generally accepted value for
the critical Reynolds number is
• The actual value of the engineering critical Reynolds
number may vary somewhat from 105 to 3X106.
Rex
Vx Vx

 
  (7-10)
5
Re 5 10
cr
cr
Vx


   (7-11)
Local Friction Coefficient
• The boundary layer thickness and the local friction
coefficient at location x over a flat plate
– Laminar:
– Turbulent:
, 1/2
5
, 1/2
4.91
Re
Re 5 10
0.664
Re
v x
x
x
f x
x
x
C


 

 





(7-12a,b)
, 1/5
5 7
, 1/5
0.38
Re
5 10 Re 10
0.059
Re
v x
x
x
f x
x
x
C


 

  





(7-13a,b)
Average Friction Coefficient
• The average friction coefficient
– Laminar:
– Turbulent:
• When laminar and turbulent flows are significant
5
1/ 2
1.33
Re 5 10
Re
f L
L
C    (7-14)
5 7
1/5
0.074
5 10 Re 10
Re
f L
L
C     (7-15)
, laminar , turbulent
0
1 cr
cr
x L
f f x f x
x
C C dx C dx
L
 
 
 
 
 
  (7-16)
5 7
1/5
0.074 1742
- 5 10 Re 10
Re Re
f L
L L
C     (7-17)
5
Re 5 10
cr  
Heat Transfer Coefficient
• The local Nusselt number at location x over a flat plate
– Laminar:
– Turbulent:
• hx is infinite at the leading edge
(x=0) and decreases by a factor
of x0.5 in the flow direction.
1/ 2 1/3
0.332Re Pr Pr 0.6
x
Nu   (7-19)
(7-20)
0.8 1/3
0.0296Re Pr
x x
Nu  5 7
0.6 Pr 60
5 10 Re 10
x
 
  
Average Nusset Number
• The average Nusselt number
– Laminar:
– Turbulent:
• When laminar and turbulent flows are significant
, laminar , turbulent
0
1 cr
cr
x L
x x
x
h h dx h dx
L
 
 
 
 
 
 
(7-23)
 
0.8 1 3
0.037Re 871 Pr
L
Nu   (7-24)
5
Re 5 10
cr  
0.5 1/3 5
0.664Re Pr Re 5 10
L
Nu    (7-21)
(7-22)
0.8 1/3
0.037Re Pr
L
Nu  5 7
0.6 Pr 60
5 10 Re 10
x
 
  
Uniform Heat Flux
• When a flat plate is subjected to uniform heat flux
instead of uniform temperature, the local Nusselt
number is given by
– Laminar:
– Turbulent:
• These relations give values that are 36 percent higher
for laminar flow and 4 percent higher for turbulent
flow relative to the isothermal plate case.
0.5 1/3
0.453Re Pr
x L
Nu  (7-31)
(7-32)
0.8 1/3
0.0308Re Pr
x x
Nu  5 7
0.6 Pr 60
5 10 Re 10
x
 
  
7 8
9 10
11 12

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Flow Across Cylinders and Spheres
• Flow across cylinders and spheres is frequently
encountered in many heat transfer systems
– shell-and-tube heat exchanger,
– Pin fin heat sinks for electronic cooling.
• The characteristic length for a circular cylinder or
sphere is taken to be the external diameter D.
• The critical Reynolds number for flow across a circular
cylinder or sphere is about
Recr=2X105.
• Cross-flow over a
cylinder exhibits complex
flow patterns depending on the Reynolds number.
• At very low upstream velocities (Re≤1), the fluid
completely wraps around the cylinder.
• At higher velocities the boundary layer detaches from
the surface, forming a separation region behind the
cylinder.
• Flow in the wake region is characterized by periodic
vortex formation and low pressures.
• The nature of the flow across a cylinder or sphere
strongly affects the total drag coefficient CD.
• At low Reynolds numbers (Re<10) ─ friction drag
dominate.
• At high Reynolds numbers (Re>5000) ─ pressure
drag dominate.
• At intermediate Reynolds numbers ─ both pressure
and friction drag are significant.
Average CD for circular cylinder and
sphere
• Re≤1 ─ creeping flow
• Re≈10 ─ separation starts
• Re≈90 ─ vortex shedding
starts.
• 103<Re<105
– in the boundary
layer flow
is laminar
– in the separated
region flow is
highly turbulent
• 105<Re<106 ─
turbulent flow
Effect of Surface Roughness
• Surface roughness, in general, increases the drag coefficient in
turbulent flow.
• This is especially the case for streamlined bodies.
• For blunt bodies such as a circular cylinder or sphere, however,
an increase in the surface roughness may actually decrease the
drag coefficient.
• This is done by tripping the
boundary layer into
turbulence at a lower Reynolds
number, causing the fluid to close
in behind the body, narrowing the
wake and reducing pressure drag considerably.
Heat Transfer Coefficient
• Flows across cylinders and spheres, in general, involve flow
separation, which is difficult to handle analytically.
• The local Nusselt number Nu around the periphery of a cylinder
subjected to cross flow varies considerably.
Small  ─ Nu decreases with increasing  as a
result of the thickening of the laminar boundary
layer.
80º< <90º ─ Nu reaches a minimum
– low Reynolds numbers ─ due to separation in laminar flow
– high Reynolds numbers ─ transition to turbulent flow.
 >90º laminar flow ─ Nu increases with increasing
 due to intense mixing in the separation zone.
90º< <140º turbulent flow ─ Nu decreases due to
the thickening of the boundary layer.
 ≈140º turbulent flow ─ Nu reaches a second minimum due to
flow separation point in turbulent flow.
Average Heat Transfer Coefficient
• For flow over a cylinder (Churchill and Bernstein):
ReꞏPr>0.2
• The fluid properties are evaluated at the film temperature
[Tf=0.5(T∞+Ts)].
• Flow over a sphere (Whitaker):
• The two correlations are accurate within ±30%.
 
4 5
5 8
1 2 1/3
1 4
2/3
0.62Re Pr Re
0.3 1
282,000
1 0.4 Pr
cyl
hD
Nu
k
 
 
   
 
 
 
   
  
 
(7-35)
1 4
1 2 2 3 0.4
2 0.4Re 0.06Re Pr
sph
s
hD
Nu
k



 
 
     
 
 
(7-36)
13 14
15 16
17 18

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• A more compact correlation
for flow across cylinders
where n=1/3 and the
experimentally
determined constants C and
m are given in Table 7-1.
• Eq. 7–35 is more accurate,
and thus should be preferred
in calculations whenever
possible.
Re Pr
m n
cyl
hD
Nu C
k
  (7-37)
Flow Across Tube Bank
• Cross-flow over tube banks is commonly encountered
in practice in heat transfer equipment such heat
exchangers.
• In such equipment, one fluid
moves through the tubes while
the other moves over the tubes
in a perpendicular direction.
• Flow through the tubes can be analyzed by considering
flow through a single tube, and multiplying the results
by the number of tubes.
• For flow over the tubes the tubes affect the flow pattern
and turbulence level downstream, and thus heat transfer
to or from them are altered.
• Typical arrangement
– in-line
– staggered
• The outer tube diameter D is the characteristic length.
• The arrangement of the tubes are characterized by the
– transverse pitch ST,
– longitudinal pitch SL , and the
– diagonal pitch SD between tube centers.
Staggered
In-line
• As the fluid enters the tube bank, the flow area
decreases from A1=STL to AT (ST-D)L between the
tubes, and thus flow velocity increases.
• In tube banks, the flow characteristics are dominated
by the maximum velocity Vmax.
• The Reynolds number is defined on the basis of
maximum velocity as
• For in-line arrangement, the maximum velocity
occurs at the minimum flow area between the tubes
max max
ReD
V D V D

 
  (7-39)
max
T
T
S
V V
S D


(7-40)
• In staggered arrangement,
– for SD>(ST+D)/2 :
– for SD<(ST+D)/2 :
• The nature of flow around a tube in the first row
resembles flow over a single tube.
• The nature of flow around a tube in the second and
subsequent rows is very different.
• The level of turbulence, and thus the heat transfer
coefficient, increases with row number.
• there is no significant change in turbulence level after
the first few rows, and thus the heat transfer
coefficient remains constant.
max
T
T
S
V V
S D


(7-40)
 
max
2
T
D
S
V V
S D


(7-41)
• Zukauskas has proposed correlations whose general
form is
• where the values of the constants C, m, and n depend
on Reynolds number.
• The average Nusselt number relations in Table 7–2 are
for tube banks with 16 or more rows.
• Those relations can also be used for tube banks with NL
provided that they
are modified as
• The correction factor F
values are given in
Table 7–3.
 
0.25
Re Pr Pr Pr
m n
D D s
hD
Nu C
k
  (7-42)
, L
D N D
Nu F Nu
  (7-43)
19 20
21 22
23 24

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Pressure drop
• the pressure drop over tube banks is expressed as:
• f is the friction factor and  is the correction factor.
• The correction factor () given in the insert is used to
account for the effects of deviation from square
arrangement (in-line) and from equilateral
arrangement (staggered).
2
max
2
L
V
P N f


  (7-48)
25