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The Free Air Jet Experiment is designed to give insight into the fundamentals of a free jet at various locations inside and outside the core region. The core region is a location in the flow field where the flow has a velocity that is approximately the same as the velocity coming from inside the jet. This experiment will provide data to describe the location of the core region. Besides defining the core region this experiment will provide information on the velocity changes outside the core region, mass flow rates at various locations, the momentum flow at various locations, showing that the exiting jet width varies with distance, and how the speed varies along the center streamline as a function of distance from the exit.
In order to best interpret the results obtained in this lab there are several assumptions that must be made. In this situation the flow is in a steady state, the air from the jet and the stationary air in the surroundings is assumed to be constant, the flow is incompressible, and the produced flow is axis symmetric.
The local fluid speed can be determined from equation (1)
V = (2*(po - p)/ρ)1/2 (1)
where the variable V is the magnitude of the velocity, po is the stagnation pressure, p is the static pressure of the fluid, and ρ is the density of the fluid. From equation (2) the mass flow rate can be determined
md = ïƒ²ïƒ²A (ρV)dA = ïƒ²02πïƒ²0R (ρVr)drdθ (2)
where md is the mass flow rate, A is the surface area that is being integrated over, ρ is the density, r is the radius, and R is the maximum radius. The momentum flow can also be determined via equation (3)
Pd = ïƒ²ïƒ²A (ρV)VdA = ïƒ²02πïƒ²0R (ρV2r)drdθ (3)
where Pd is the momentum flow rate. The local sound speed, c, was found from equation (4)
c = (kRT)1/2 (4)
where k and R are constants defined by the physical properties of air and T is the temperature of the medium. In this experiment k = 1.4, R = 287, and T = 298. Knowing c, the mach speed can be calculated via equation (5)
Ma = V/c (5)
where Ma is the mach speed.
An apparatus was constructed in such a way that a tube that emits air is placed horizontally and blows into a Pitot tube that can be moved horizontally or in a radial outward direction. The volumetric flow rate is a constant for this experiment. From here the first set of data to be recorded is the centerline speed of the jet at various horizontal distances away from the center of the tube. This is first to be done by recording the pressure close to the pipe's exit and then taking pressure measurements increasing the distance from the Pitot tube to the pipe's exit by small intervals. This will provide a relationship of mass flow rate and momentum flow to the distance from the air exiting the pipe. Change the volumetric flow rate and repeat the preceding procedure.
To determine how the mass flow rate and momentum flow rate will vary radially from the center streamline, another experiment is to be conducted. In this case a measurement is to be taken at the center streamline at some fixed horizontal displacement with a constant volumetric flow rate. From here the Pitot tube is to be moved radially outward in small increments such that several data points can be obtained at that horizontal displacement. At a few other horizontal displacements the same procedure is to be followed.
Results and Discussion
From Figure 1 it can be seen that up to about 0.03 m from the exit, the centerline speed doesn't change much. This defines the core region starting from the exit of the tube to 0.03 m away from the tube's exit. Outside the core region the speed of the air decreases as the distance from the exit is increased. When measuring the pressure from the Pitot tube the pressure had a precision of +/- 0.005 inches of H2O. This margin of error created an uncertainty in the centerline speed of about +/- 1 m/s. Knowing that the uncertainty of the speed is about 1 m/s, this uncertainty will propagate into the length of the core region. The core region can then be determined to have a length of 0.03 m +/- 0.01 m. Centerline speeds were recorded for a volumetric flow rate at 70 L/min and at 50 L/min. As expected, as the volumetric flow rate increases the centerline speed also increases.
Velocities of varying radial distances from the centerline were measured and compared with each other at different horizontal distances from the tube in Figure 2. At a radial distance of 1 cm, the velocity doesn't change much with respect to the velocity measured at the centerline for all horizontal positions. This defines the average radial component of the core region as 1 cm. This radial component decreases as a function of the distance from the pipe's exit. The farther the Pitot tube is moved outward from the core region the slower the velocity becomes. It can also be noticed that at the closest horizontal displacement the velocity drops off quicker as a function of radial displacement as apposed to the larger horizontal displacements. This is caused by the energy dissipating out to the sides as the horizontal displacement increases. The energy dissipation is caused by eddies or more commonly swirling in air. An eddy is the terminology used to describe the circular motion a fluid takes as it displaces from the source. This plays an even bigger role in mass and momentum flow rates.
Looking at Figure 3 it can be seen that the mass flow rate increases as the horizontal displacement increases. This increase is caused by eddies. What happens here is the source puts out a finite amount of mass at some constant rate. Eddies then form and this swirling motion of the fluid reaches out into the stagnant fluid and pulls more mass in to the system. Now more mass is being brought into the system causing the mass flow rate to increase. As the horizontal displacement increases the mass flow rate begins to level off, as seen in Figure 3, and will eventually begin to decrease. Here more mass is still being brought into the system but now the velocity has decreased significantly and this decrease is now causing the mass flow rate to decrease.
Similarly to the mass flow rate the momentum flow rate is effected by eddies. In this case the momentum flow rate has reached a peak where the mass flow rate is still increasing and is decreasing where the mass flow rate begins to reach a maximum, as seen in Figures 3 and 4. The momentum flow equation and mass flow rate equation only differ by one term. In the mass flow rate equation there is a V component and in the momentum flow equation there is a V2 component. Having this extra component is what causes the momentum flow to peak before the mass flow rate. The velocity is decreasing and the mass is increasing as a function of horizontal displacement, but the momentum flow depends more heavily on the velocity component.
The mach speed was then calculated from the maximum velocity obtained. In this situation the mach speed was found to be 0.087 with a local sound speed of 346 m/s. If the mach speed is greater than or equal to 0.3 than this implies that the flow is compressible. By having a mach speed that is smaller than 0.3 implies that the flow is incompressible.
Conclusion and Recommendations
By conducting this experiment a fairly accurate core region was able to be defined. The core region was defined as having a horizontal displacement of 0.03 m +/- 0.01m and an average radius of 0.01 m. The mass flow rate and momentum flow were both found to be heavily dependent on mass and velocity. Both the mass flow rate and the momentum flow were affected by eddies, which is the swirling motion of air, that pulled stagnant mass into the system causing the mass to increase as the flow got further away from the core region. The velocity of the air decreased as the displacement from the pipe exit increased. Momentum flow was affected by the velocity more so than the mass flow rate because of the V2 component in the momentum equation. This flow was deemed incompressible due to the mach speed being smaller than 0.3.
For better results in the future, supplying the jet with an independent compressor would eliminate any variance in volumetric flow rates caused by other users of the compressor. This would then generate a higher precision when measuring pressures.
Figure 1. This graph shows the relationship between the centerline speed and the distance from the exit.
Figure 2. This graph shows the relationship between the normalized velocity and the radial distance from the tube's exit.
Figure 3. This graph shows the relationship between the calculated per measured mass flow rate and horizontal position.
Figure 4. This graph shows the relationship between the rate of momentum flow and horizontal position.