It may be submerged fluid discharged in the same ambient medium, or free surface a liquid discharged into ambient gas.. Typically, the jet is turbulent at the nozzle exit and can be chara
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The impinging jet may be circular(round) orplanar(rectangularorslot), based on its cross section It may be submerged (fluid discharged in the same ambient medium),
or free surface (a liquid discharged into ambient gas) The flows in each of these
cases may be unconfined or partially confined Moreover, in the case of multiple jets,
interaction effects arise
6.7.2 Submerged Jets
A schematic of a single submerged circularorplanarjet is seen in Fig 6.29 Typically, the jet is turbulent at the nozzle exit and can be characterized by a nearly uniform axial velocity profile With increasing distance from the nozzle exit, the potential core region within which the uniform velocity profile persists shrinks as the jet interacts with the ambient Farther downstream, in the free jet region, the velocity profile
is nonuniform across the entire jet cross section The centerline velocity decreases with distance from the nozzle exit in this region The effect of the impingement surface is not felt in this region The impingement surface influences the flow in
Figure 6.29 Transport regimes in a submerged circular unconfined jet impinging on a surface
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the transverse direction (in the wall jet region), the flow starts to decelerate due to
entrainment of the ambient fluid
Average Nusselt Number for Single Jets Martin (1977) provides an exten-sive review of heat transferdata forimpinging gas jets Forsingle nozzles, the average Nusselt number is of the form (Martin, 1977; Incropera and DeWitt, 1996)
Nu= f
Re, Pr, D r
h , D H
h
(6.188a) or
Nu= f
Re, Pr, D x
h , D H
h
(6.188b) where
Nu= ¯h
and
Re=V e D h
whereV eis the uniform exit velocity at the jet nozzle,D h = D fora round nozzle,
andD h = 2W for a slot nozzle Figure 6.29 defines the geometric parameters.
For a single round nozzle, Martin (1977) recommends
Nu
Pr0.42 = G
D
r ,
H D
where
f1(Re) = 2Re1/2 (1 + 0.005Re0.55 )1/2 (6.191a)
and
G = D r 1− 1.1(D/r)
1+ 0.1[(H/D) − 6](D/r) (6.191b)
ReplacingD/r by 2A1/2
r yields
G = 2A1/2
r
1− 2.20A1/2
r
1+ 0.20[(H/D) − 6]A1/2
r
(6.192)
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The ranges of validity of eqs (6.191) are
2000≤ Re ≤ 400,000 2≤ H
D ≤ 12
2.5 ≤ D r ≤ 7.5 and 0.004 ≤ A t ≤ 0.04
For r < 2.5D, the average heat transfer data are provided by Martin (1977) in
graphical form
For a single-slot nozzle, the recommended correlation is
Nu
Pr0.42 = 3.06Re m
wherex is the distance from the stagnation point,
m = 0.695 − x
2W
+
H
2W
1.33
+ 3.06
−1
(6.194)
and the corresponding ranges of validity are
3000≤ Re ≤ 90,000 4≤ W H ≤ 20 4≤ W x ≤ 50 Forx/W < 4, these results can be used as a first approximation and are within
40% of measurements More recent measurements by Womac et al (1993) suggest that with the Prandtl number effect included in Martin (1977), these correlations are also applicable as a reasonable approximation for liquid jets
Average Nusselt Number for an Array of Jets Martin (1977) also provides correlations for arrays of in-line and staggered nozzles, as well as for an array of slot jets These configurations are illustrated in Fig 6.30, and the following correlations are also provided by Incropera and DeWitt (1996)
For an array of round nozzles,
Nu
Pr0.42 = K
A r , H D
G
A r , H D
where
K
A r , H D
=
1 +
H/D
0.6/A1/2 r
6
−0.05
(6.196b)
whereG is the same as forthe single nozzle, eq (6.191b).
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S
D
D
S
S
S W
Figure 6.30 Commonly utilized jet array configurations (From Incropera and DeWitt, 1996.)
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The ranges of validity of eqs (6.196) are
2000≤ Re ≤ 100,000 2≤ H
D ≤ 12 0.004 ≤ A r ≤ 0.04
For an array of slot nozzles,
Nu
Pr0.42 = 2
3A3/4 r,0
2Re
A r /A r,0 + A r,0 /A r
2/3
(6.197) where
A r,0=
60+ 4
H
2W − 2
2−1/2
(6.198)
and
1500≤ Re ≤ 40,000 2≤ W H ≤ 80 0.008 ≤ A r ≤ 2.5A t,0
Free Surface Jets The flow regimes associated with a free surface jet are illus-trated in Fig 6.31 As the jet emerges from the nozzle, it tends to achieve a more uniform profile farther downstream, due to the elimination of wall friction There is
a corresponding reduction in jet centerline velocity (or midplane velocity for the slot jet) As forthe submerged jet, a stagnation zone occurs This zone is associated with the concurrent deceleration of the jet in a direction normal to surface and acceleration parallel to it and is also characterized by a strong favorable pressure gradient parallel
to the surface
Within the stagnation zone, hydrodynamic and thermal boundary layers are of uniform thickness Beyond this region, the boundary layers begin to grow in the wall jet region, eventually reaching the free surface The viscous effects extend throughout the film thicknesst(r), and the surface velocity V s starts to decrease with increasing radius The velocity profiles are similar to each other in a region that ends atr = r c, where the transition to turbulence begins
The flow development associated with the planarjet is less complicated Following the bifurcation at the stagnation line, the two oppositely directed films are of a fixed film thickness and the free surface velocity isV s = V i Boundary layers grow outside the stagnation zone These reach the film thickness, before or after transition to turbulence, depending on the initial conditions
Womac et al (1993) considered free surface impinging jets of water and FC-77
on a square nearly isothermal heater of side 12.7 mm Nozzle diametersD nranged
from 0.978 to 6.55 mm, and nozzle-to-surface spacings varied from 3.5 to 10 They correlated their average heat transfer coefficient data using an area-weighted average
of standard correlations for the impingement and wall jet regions The correlation for the impingement region is of the form
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Figure 6.31 Transport regions in a circular, unconfined, free surface jet impinging on a surface
Nu
Pr0.4 = C1· Rem
wherem = 0.5 and the Reynolds numberis defined in terms of the impingement
velocityV i diameterD i = D n (V n /V i )1/2 The wall jet correlation is of the form
Nu
Pr0.4 = C2· Ren
where the Reynolds number is defined in terms of the impingement velocityv i, and the average lengthL∗of the wall jet region for a square heater is
L∗= 0.5(
√
2L h − D i ) + 0.5(L h − D i )
Combining the correlations of eqs (6.199) and (6.200) in an area-weighted fashion gives (Incropera, 1999)
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Nu
Pr0.4 = C1· Rem
D i
L h
D i A r + C2· Ren
L∗L h
L∗(1 − A r ) (6.202) where
A r= πD i2
4l2
h
The data were found to be best correlated in the range 1000< Re D n < 51,000 for
C2 = 0.516, C2 = 0.491, and n = 0.532, where the fluid properties are evaluated at
the mean of the surface and ambient fluid temperature
• Isothermal flat plate in uniform laminar flow (Sections 6.4.3 through 6.4.5): Near
Pr= 1,
Nux = 0.332Re1/2
ForP r 1,
Nux = 0.565Re1/2
ForP r 1,
Nux = 0.339Re1/2
• Isothermal flat plate in uniform laminar flow with appreciable viscous dissipation (Section 6.4.6):
Nux = 0.332Re1/2
The local heat flux is given by
q= h x (T o − TAW )
where
TAW = T∞+ r c U2
2c p
and forgases
r c = b(Pr Pr)1/2
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T∗= T∞+ (TW − T∞) + 0.22(TAW− T∞)
The surface-averaged heat transfer coefficient in each of the foregoing cases for
an isothermal flat plate is determined from its local value atx = L(h L ) as
• Flat plate in uniform laminar flow with an unheated starting length (x0) and heated to a uniform temperature beyond Pr near 1 (Section 6.4.7):
Nu= 0.332Pr1/2· Re1x /2
[1− (x0 /x)3/4]1/3 (6.63)
• Wedge at uniform temperature with an included angle of βπ in a uniform laminar flow in the range 0.7 < Pr < 10 (Section 6.4.4):
Nux
Re1x /2 = 0.56A
where
β = m + 12m and A = (β + 0.2)0.11Pr0.333+0.067β−0.026β2
• Cylinder at uniform surface temperature in a laminar cross flow (Section 6.4.16):
NuD ≡ ¯hD
k = 0.30 +
0.62Re1/2
D · Pr1/3
[1+ (0.40/Pr)2/3]1/4
1+
ReD 282,000
5/84/5
(6.155)
• General two-dimensional object at uniform surface temperature in a uniform laminarflow (Section 6.4.9):
St= ρc q
p U(T0− T∞)=
k
ρc p Uδ =
c1(U∗) c2
x∗
0
(U∗) c3dx∗1/2
1
ReL
1/2 (6.72)
wherec1throughc3are as given in Table 6.1
• Laminar flow over a sphere at a uniform surface temperature (Section 6.4.16):
NuD= 2 +0.4Re1/2
D + 6Re2/3
D
Pr0.4
µ
µs
1/4
(6.156)
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• General axisymmetric object at uniform surface temperature in a uniform lami-narflow (Section 6.4.10):
St= c1(r∗
0) K (U∗) c2
x∗
3
0
(U∗) c3(r∗
0)2K dx∗ 3
1/2
1
ReL
1/2
(6.76)
wherec1throughc3are as given in Table 6.1
• Isothermal flat plate with a turbulent boundary layer from the leading edge for Prand PrT near1 (Section 6.4.14):
Nux = 0.0296Re0.80
• Isothermal flat plate with turbulent boundary layer transition from laminar to turbulent forPrand PrT near1 (Section 6.4.14):
NuL = 0.664Re0.5
T + 0.36Re0L .8− Re0.8
T
(6.110)
• Isothermal flat plate with turbulent boundary PrT near1 (Section 6.4.14):
Nux = 0.029Re0.8
where
(0.029/Re x )1/2 {5Pr + 5 ln[(1 + 5Pr)/6] − 5} + 1 (6.131)
• Flat plate with a turbulent boundary layer from the leading edge and with an unheated starting length followed by uniform surface temperature for Pr and PrT near1 (Section 6.4.14):
St· Pr0.8 = 0.287Re −0.2
x
1−x0 x
9/101/9
(6.140)
• Uniform flux plate with a turbulent boundary layer from the leading edge for Pr and PrT near1 (Section 6.4.14):
St· Pr0.4 = 0.03Re −0.2
• Isothermal rough flat plate with a turbulent boundary layer from the leading for Prand PrT near1 (Section 6.4.15):
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Stk 0.8Re −0.2
k · Pr−0.4
• Cross flow across a bank of cylinders at uniform surface temperature (Section 6.5.1):
NuD = C · Re m
D,max· Pr0.36
Pr
Prs
1/4
(6.157) for
N L≥ 20 0.7 ≤ Pr < 500 1< Re D,max < 2 × 106 whereC and m are given in Table 6.2.
• Plate stack (Section 6.5.2): The optimum numberof plates in a given cross-sectional area,L × H ,
nopt 0.26(H/L)Pr1/4· Re
1/2 L
1+ 0.26(t/L)Pr1/4· Re1/2
L
(6.161)
forPr≥ 0.7 and n 1.
• Offset strips (Section 6.5.2): In the laminar range (Re ≤ Re∗),
f = 8.12Re −0.74
d h
−0.15
j = 0.53Re −0.50
d h
−0.15
In the turbulent range (Re≤ Re∗+ 1000),
f = 1.12Re −0.36
d h
−0.65 t
d h
0.17
(6.168)
j = 0.21Re −0.40
d h
−0.24
t
d h
0.02
(6.169) The transition Reynolds number Re∗is obtained from the set of equations
Re∗ = Re∗
b d h
Re∗b = 257
s
1.23
t
0.58
(6.171)