12. HEAT TRANSFER OF IMPINGING JETS
12.1 J ET CHARACTERISTICS
12.1.1 Air discharge through nozzle
A jet is a stream of air blown out from a nozzle in Figure 12.1. The behaviour of the discharging air flow depends on the shape of the nozzle. In a well-rounded orifice, the flow area is the same as the minimum cross area of the orifice, whereas in a sharply edged orifice the flow contracts. The velocity of air in the vena contracta for choked flow when p1/p>1.9 is the sonic velocity
1
2 1
M
T C R
uvc v u (12.1)
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The velocity in the vena contracta for incompressible flow is
p C p
uvc v 2 1
(12.2) The air flow can be assumed to be incompressible at least when p1/p< 1.2, which is usually true when fans are used. The corresponding equation for mass flow through the nozzle is
4 1
2 1 2
1
1
C D u A
u
mvc vc vc vc vc a
(12.3)
for a choked flow and
4 D2
C u
mvc vc a (12.4)
for a low velocity flow. In the equations above Ca is the coefficient considering the contraction of a jet at vena contracta. Coefficient Cv considers the difference between true and theoretical velocity at vena contracta. Both coefficients are always below 1. Values for the coefficients depend on the shape of the nozzle. Some values are shown in Figure 12.2, and more can be found from the literature [86][87]. According to [88] the velocity coefficient Cv of sharp-edged orifice is even higher than 0.98.
Figure 12.1. Details of jet flow.
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Figure 12.2. Types and coefficients of velocity and contraction for different orifices.
12.1.2 Single impinging jet
Air discharging from a circular nozzle forms an axisymmetric jet. Figure 12.1 shows the time-averaged details of a developing turbulent jet. After some nozzle diameters from the nozzle exit, the jet starts driving the surrounding air with it and spreading, with decreasing velocity and increasing mass flow. The momentum remains the same. The velocity profile of a fully developed (x/D >10) axisymmetric jet is according to White [89] as follows:
)
in which δ = 0.0848x is radius of jet at distance x from the nozzle exit and
x
In Eq. (12.7) ueff is the effective jet velocity, which governs the development of the jet. For pressure ratios p1/p< 1.9 expansion is completed at the vena contracta, and pvc = p. For overcritical pressures p1/p>1.9 the remaining overpressure in the vena contracta pvc = 0.528p1 creates a series of expansion waves, which affect the jet’s momentum and behaviour. For incompressible jets
The mass flow of a jet after distance x from the orifice is
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J x Jx x rdr
r x u x
m( ) 2 , 0.404 ( ) 0.404
0
(12.9)
In Eq. (12.9) it is assumed that the density of a jet is the same as in the surrounding air. This is true when the temperature of discharging and surrounding air are the same. In other cases the density assumption is more valid when x/D increases.
12.1.3 Measurement of jet momentum
Figure 12.3 is a schematic view of an air jet blowing against the surface of a weighing disk, which gives the weight m on the screen of the weighing machine. The jet causes the force F2 = mg to the disk. The momentum equation of the impinging jet in Figure 12.3 is:
F2
A p u m A
pvc vc vc vc vc (12.10)
Combining that with Eq. (12.7) gives J = F2 = mg. Thus, an electric weighing machine can be used to measure the jet momentum.
During the measurements it was discovered that the distance affected the force so that it dropped clearly at a distance of less than 20 mm, because the surface-orientated air velocity (wall jet in Figure 12.1) lowered the static pressure on the surface of the weighing disk and produced a lifting force. Due to that the surface of the weighing disk surface was equipped with flow blocks to prevent the development of wall jet.
Figure 12.3. Schematic view of jet hitting the surface.
Table 12.1 shows some examples of measured and calculated momentums. The main inaccuracy in measured momentums arises from inaccurate measuring technique. The main inaccuracy in calculated momentums arises from the inaccuracy of nozzle diameters (± 0.05 mm) and discharge coefficient. The type names for orifices and values of discharge coefficients CD at the vena contracta for calculations were taken from [87] and it was assumed that Cv = 0.98 at the vena contracta in all cases.
p
vc, u
vc,
mvc, A
vcp
∞, T
∞u
2=0 F
2manifold
disk
p
1, T
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Table 12.1. Examples of measured and calculated momentums.
Nozzle
Orifice type Discharge coefficient
With incompressible jets the typical difference between measured and calculated momentums was less than 10%, whereas with choked compressible air jets with a sonic velocity in the vena contracta, the measured momentums were from 4 to 35 % higher than the calculated ones. The difference tended to increase with pressure difference, which indicates that the term (pvc-p)Avc in Eq. (12.7) does not fully account for the effect of the remaining overpressure at the vena contracta on the momentum. In addition to the unknown effect of expansion waves, the difference between measured and calculated momentums may be caused by the temperature drop due to substantial pressure decrease in discharging air, which was not taken into account. In any case, the remaining overpressure in the vena contracta clearly affected the results. Thus, the sonic velocity condition which defines the maximum discharging velocity of a choked jet does not restrict the increasing of the jet momentum due to increasing overpressure. What happens to the jet after the vena contracta because of the remaining overpressure is beyond the scope of this thesis and is not easy to solve. For instance, Chapman and Walker [90] bypass this detail by saying that “adjustment of the flow to be imposed must take place externally by multidimensional means.” Oosthuizen and Carscallen [91] include some figures showing multidimensional expansion after the jet exit.