External Flow:
The Flat Plate in Parallel Flow
Chapter 7
Section 7.1 through 7.3
Appendix F
Physical Features
Physical Features
• As with all external flows, the boundary layers develop freely without constraint.
• Boundary layer conditions may be entirely laminar, laminar and turbulent,
or entirely turbulent.
• To determine the conditions, compute
Re L 
 u L u  L



and compare with the critical Reynolds number for transition to turbulence, Re x , c .
Re L  Re x , c  laminar flow throughout
Re L  Re x , c  transition to turbulent flow at xc / L  Re x , c / Re L
Physical Features (cont.)
• Value of Re x , c depends on free stream turbulence and surface roughness.
Nominally,
Re x , c  5  105.
• If boundary layer is tripped at the leading edge
Re x , c  0
and the flow is turbulent throughout.
• Surface thermal conditions are commonly idealized as being of uniform
temperature Ts or uniform heat flux qs . Is it possible for a surface to be
concurrently characterized by uniform temperature and uniform heat flux?
• Thermal boundary layer development may be delayed by an unheated
starting length.
Equivalent surface and free stream temperatures for x   and uniform Ts
(or qs ) for x   .
Similarity Solution
Similarity Solution for Laminar,
Constant-Property Flow over an Isothermal Plate
• Based on premise that the dimensionless x-velocity component, u / u ,
and temperature, T *  T  Ts  / T  Ts  , can be represented exclusively in
terms of a dimensionless similarity parameter
  y  u / x 
1/ 2
• Similarity permits transformation of the partial differential equations associated
with the transfer of x-momentum and thermal energy to ordinary differential
equations of the form
d3 f
d2 f
2 3f
0
d
d 2
where  u / u   df / d , and
d 2T * Pr dT *

f
0
2
d
2 d
Similarity Solution (cont.)
• Subject to prescribed boundary conditions, numerical solutions to the momentum
and energy equations yield the following results for important local boundary layer
parameters:
- with u / u  0.99 at   5.0,

5.0
 u / vx 
1/ 2
u
- with  s  
y

5x
Re1/x 2
  u
y 0
and d 2 f / d 2
C f ,x 
 0
d2 f
u / vx
d 2
 0.332,
 s, x
1/ 2

0.664Re
x
 u2 / 2
- with hx  qs / Ts  T   k T * / y
and dT * / d
and
 0
Nu x 
 0
 k  u / vx 
1/ 2
y 0
 0.332 Pr1/ 3 for Pr  0.6,
hx x
 0.332 Re1/x 2 Pr1/ 3
k

 Pr1/ 3
t
dT * / d
 0
Similarity Solution (cont.)
• How would you characterize relative laminar velocity and thermal boundary layer
growth for a gas? An oil? A liquid metal?
• How do the local shear stress and convection coefficient vary with distance from
the leading edge?
• Average Boundary Layer Parameters:
1
x
 1.328 Re x1/ 2
 s , x  0x  s dx
Cf , x
hx 
1 x
 hx dx
x 0
Nu x  0.664 Re1/x 2 Pr1/ 3
• The effect of variable properties may be considered by evaluating all properties
at the film temperature.
Tf 
Ts  T
2
Turbulent Flow
Turbulent Flow
• Local Parameters:
Empirical
Correlations
C f , x  0.0592 Rex 1/ 5
Nu x  0.0296 Re4x / 5 Pr1/ 3
How do variations of the local shear stress and convection coefficient with
distance from the leading edge for turbulent flow differ from those for laminar flow?
• Average Parameters:
1 xc
L
hL 
0 h1am dx  xc hturb dx
L


Substituting expressions for the local coefficients and assuming Re x ,c  5  105 ,
0.074 1742
C f , L  1/ 5 
Re L
Re L


Nu L  0.037 Re4L / 5  871 Pr1/ 3
For Re x , c  0 or L
xc  Re L
Re x ,c  ,
C f , L  0.074 Re L1/ 5
Nu L  0.037 Re 4L / 5 Pr1/ 3
Special Cases
Special Cases: Unheated Starting Length (USL)
and/or Uniform Heat Flux
For both uniform surface temperature (UST) and uniform surface heat flux (USF),
the effect of the USL on the local Nusselt number may be represented as follows:
Laminar
Nu x 
Nu x   0
b
1   / x a 


Nu x   0  C Re mx Pr1/ 3
Turbulent
UST
USF
UST
USF
a
3/4
3/4
9/10
9/10
b
1/3
1/3
1/9
1/9
C
0.332
0.453
0.0296
0.0308
m
1/2
1/2
4/5
4/5
Sketch the variation of hx versus  x    for two conditions:   0 and   0.
What effect does an USL have on the local convection coefficient?
Special Cases (cont.)
• UST:
qs  hx Ts  T  q  hL As Ts  T 
L 
 2 p 1 /  2 p  2   2 p /  2 p 1
Nu L  Nu L
1   / L 
 0  L    

p  1 for laminar flow throughout
p = 4 for turbulent flow throughout
hL  numerical integration for laminar/turbulent flow
hL 
• USF:
q
Ts  T  s
hx
1 xc
  h1am dx  xL hturb dx 
c

L
q  qs As
• Treatment of Non-Constant Property Effects:
Evaluate properties at the film temperature.
T T
Tf  s 
2
Problem: Orientation of Heated Surface
Problem 7.21: Preferred orientation (corresponding to lower heat loss) and the
corresponding heat rate for a surface with adjoining smooth and
roughened sections.
SCHEMATIC:
ASSUMPTIONS: (1) Surface B is sufficiently rough to trip the boundary layer when in the upstream position
(Configuration 2); (2) Rex,c  5  105 for flow over A in Configuration 1.
Orientation of Heated Surface (cont.)
-6
2
-3
PROPERTIES: Table A-4, Air (Tf = 333K, 1 atm):  = 19.2  10 m /s, k = 28.7  10
W/mK, Pr = 0.7.
ANALYSIS: With
u L
ReL   

20 m/s 1m
19.2 10-6m 2 / s
 1.04 106.
transition will occur just before the rough surface (xc = 0.48m) for Configuration 1. Hence,


4/5


Nu L,1  0.037 1.04  106
 871 0.71/3  1366


4
/
5
Since Nu L,2  0.037 1.04 106
 0.7 1/ 3  2139  Nu L,1 , it follows that the lowest heat


transfer is associated with Configuration 1.
For Configuration 1:
Hence

h L,1L
k
 Nu L,1  1366.

h L,1  1366 28.7 103 W/m  K /1m  39.2 W/m 2  K
q1  h L,1A  Ts  T   39.2 W/m2  K  0.5m 1m 100  20  K  1568 W
<
Comment: For a very short plate, a lower heat loss may be associated with Configuration 2. In
fact, parametric calculations reveal that for L< 30 mm, this configuration provides the preferred
orientation.
Problem: Conveyor Belt
Problem 7.25: Convection cooling of steel plates on a conveyor by
air in parallel flow.
KNOWN: Plate dimensions and initial temperature. Velocity and temperature of air in parallel flow
over plates.
FIND: Initial rate of heat transfer from plate. Rate of change of plate temperature.
Problem: Conveyor Belt (cont.)
SCHEMATIC:
 = 6 mm
q
Air
L=1m
Too = 20oC
uoo = 10 m/s
q
L=1m
Ti = 300oC
ASSUMPTIONS: (1) Negligible radiation, (2) Negligible effect of conveyor velocity on boundary
layer development, (3) Plates are isothermal, (4) Negligible heat transfer from edges of plate, (5)
5
Re x,c  5  10 .
PROPERTIES: Table A-1, AISI 1010 steel (573K): kp = 49.2 W/mK, c = 549 J/kgK,  = 7832
3
-6 2
kg/m . Table A-4, Air (p = 1 atm, Tf = 433K):  = 30.4  10 m /s, k = 0.0361 W/mK, Pr = 0.688.
ANALYSIS: The initial rate of heat transfer from a plate is
q  2 h As  Ti  T   2 h L2  Ti  T 
6 2
5
With ReL  u  L /   10 m / s  1m / 30.4  10 m / s  3.29  10 , flow is laminar over the entire surface.
Hence,


1/ 2
1/
2
1/
3
5
Nu L  0.664 ReL Pr
 0.664 3.29 10
 0.6881/ 3  336
h   k / L  Nu L   0.0361W / m  K /1m  336  12.1W / m2  K
q  2  12.1W / m2  K 1m 
2
300  20  C  6780 W
Problem: Conveyor Belt (cont.)
Performing an energy balance at an instant of time for a control surface about the plate, Eout  Est ,
  L2c
dT
 h 2L2  Ti  T 
dt i


2 12.1W / m 2  K  300  20  C
dT

 0.26C / s
3
dt i
7832 kg / m  0.006m  549 J / kg  K
COMMENTS: (1) With Bi  h  / 2  / k p  7.4  104 , use of the lumped capacitance method is
appropriate.
(2) Despite the large plate temperature and the small convection coefficient, if adjoining plates are in
close proximity, radiation exchange with the surroundings will be small and the assumption of negligible
radiation is justifiable.