Schematic t u h lam c lam x 05 h turb c turb x 02

SCHEMATIC:
T
∞
, u
∞
h
lam
=C
lam
x -0.5
h
turb
=C
turb
x -0.2
Turbulent Laminar x
x
c
T
∞
, u
∞
h
lam
=C
lam
x -0.5
h
turb
=C
turb
x -0.2
Turbulent Laminar x
x
c
ASSUMPTIONS: (1) Steady-state conditions, (2) Constant properties.
PROPERTIES: Table A.4, air (T = 350 K): k = 0.030 W/m ⋅ K, ν = 20.92 × 10
-6
m
2
/s, Pr = 0.700.
ANALYSIS:
(a) Using air properties evaluated at 350 K with x
c
= 0.5 m,
5
c
x,c
u x
Re 5 10
ν
∞
= = ×
5 5 6
c
2
u 5 10 x 5 10 20.92 10 m / s 0.5m 20.9m s
−
∞
=× =× × × = ν
<
(b) From Eq. 6.13, the average coefficient in the laminar region, 0 ≤ x ≤ x
c
, is
( ) ( )
lam lam
x x
-0.5 0.5
C C
lam lam lam lam
0 o
1 1 1
-0.5
h x = h x dx = x dx = x 2C 2
x x x
= x h (x) =
∫ ∫
(1)
<
(c) The average coefficient in the turbulent region, x
c
≤ x ≤ L, is
( ) ( ) ( )
c
c
c
c
x x
0.5 0.8
x x
turb lam turb lam turb
0 x
0 x
1 x x
h x h x dx h x dx C C
x 0.5 0.8
= + = +
 
 
 
 
 
 
 
 
∫ ∫
Continued…

PROBLEM 6.15 (Cont.)
( )
( )
0.5 0.8 0.8
turb lam c turb c
1
h x 2C x 1.25C x x
x
= + −
 
 
 
(2)
<
(d) The local and average coefficients, Eqs. (1) and (2) are plotted below as a function of x for the
range 0 ≤ x ≤ L.
0 0.5 1
Distance from leading edge, x (m)
0
50
100
150
Convection coefficient (W/m^2.K)
Local - laminar, x <= xc
Local - turbulent, x => xc
Average - laminar, x <= xc
Average - turbulent, x => xc

PROBLEM 6.16
KNOWN: Air speed and temperature in a wind tunnel.
FIND: (a) Minimum plate length to achieve a Reynolds number of 10
7
, (b) Distance from
leading edge at which transition would occur.
SCHEMATIC:
Air,
u
∞
= 30 m/s
T
∞
= 23°C
ASSUMPTIONS: (1) Isothermal conditions, T
s
= T
∞
.
PROPERTIES: Table A-4 , Air (23 ° C = 296K): ν = 15.53 × 10
-6
m
2
/s.
ANALYSIS: (a) The Reynolds number is
x
u x u x
Re .
ρ
µ ν
∞ ∞
=
To achieve a Reynolds number of 1 × 10
7
, the minimum plate length is then
(
)
7 6 2
x
min
1 10 15.53 10 m / s
Re
L
u 30 m/s
ν
−
∞
× ×
= =
min
L 5.18 m. =
<
(b) For a transition Reynolds number of 5 × 10
5
( )
5 -6 2
x,c
c
5 10 15.53 10 m / s
Re
x
u 30 m/s
ν
∞
× ×
= =
c
x 0.259 m. =
<
COMMENTS: Note that
x,c
c
L
Re
x
L Re
=
This expression may be used to quickly establish the location of transition from knowledge of
Re .
x,c L
and Re

PROBLEM 6.17
KNOWN: Velocity and temperature of water flowing over a flat plate. Length of plate. Variation of
local convection coefficient with x for laminar and turbulent flow.
FIND: Minimum and maximum average convection coefficient for roughness applied over the range
x
r
≤ x ≤ L .
Temperature at which extreme values of average convection coefficient occur and
corresponding values of x
r
.
SCHEMATIC :
T = 300 K or 350 K
u
∞
L = 0.6 m
x
x
r
Roughness
ASSUMPTIONS : (1) Steady-state conditions, (2) Constant properties, (3) Transition occurs at a
critical Reynolds number of 5 × 10
5
for the smooth plate, (4) Incompressible flow.
PROPERTIES: Table A.6 , Liquid water ( T = 300 K): ρ = v
f
-1
= 997 kg/m
3
, µ = 855 × 10
-6
N ⋅ s/m
2
;
Liquid water ( T = 350 K): ρ = v
f
-1
= 974 kg/m
3
, µ = 3655 × 10
-6
N ⋅ s/m
2
.
ANALYSIS: The smooth plate transition location, x
c
, was found in Example 6.4 to be 0.43 m and
0.19 m, for T = 300 K and 350 K, respectively. For roughness applied over the range 0 ≤ x
r
≤ x
c
,
transition occurs at x
r
. From Eq. 6.14,
( )
0.5 0.8 lam turb
lam turb
0 0
0
0.5 0.8 0.8
lam turb
1 1 1
0.5 0.8
1
0.5 0.8
r
r
r
r
x L
L x L
x
x
r r
C C
h hdx h dx h dx x x
L L L
C C
x L x
L
 
 
= = + = +
 
 
 
 
 
 
= + −
 
 
∫ ∫ ∫
(1)
Roughness applied over the range x
r
> x
c
has no effect on the transition since the transition occurs at x
c
for the smooth plate. Thus, from Example 6.4,
2
1620 W/m K h = ⋅
or
2
3710 W/m K ⋅
for T = 300 K or
350 K, respectively for x
r
> x
c
.