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The main objective of this lab is to observe the nature of a free air jet and the relations among the variables pertaining to it. Free air jets are commonly found in products such as compressed air bottles used for cleaning keyboards or leaf blowers used for cleaning the yard. Examples of other types of free jets can be found in industry with giant smoke stacks and in nature with geysers. To accomplish the main objective, measure the core flow and the core length in the axial direction, calculate the velocity using the pressure difference, determine the mass flow rate and momentum flow rate, and calculate the Mach number. The main assumption is that the air used in the jet is incompressible, and this is later proven by calculating the Mach number. Two other important assumptions include that radial flow is constant and the room are is quiescent. After acquiring the needed values, compare the velocity to the axial tube distance and observe how the core region and the entrainment region contribute to the mass flow rate; Figure (1) shows a crude image of these regions.
To find the change in pressure (the change between the stagnation and the static pressure), use the following equation:
where is the measured water column height, is the change in pressure, is the density of water at 998 kg/m3, and is the gravitational acceleration. To find the fluid velocity, use the following equation:
where is the fluid velocity and is density of air at 1.2 kg/m3. The velocity is only in the axial direction that remains the same throughout the lab. It can be observed that the velocity of the fluid decreases as the axial distance increases due to the increased amount of shear from the room air which converts kinetic energy into dissipated heat; thus conservation of energy is assumed. The mass flow rate and the momentum flow rate are determined using the trapezoid rule in combination with the definitions of each variable due to multiple data points.
In these equations, is mass flow rate, is momentum flow rate, is the incremental radial distance, N is the number of trials and is the radial distance to the tube centerline. The subscripts 'o', 't', 'f' stand for initial, trial number, and final respectively.
To calculate the Mach number the following equation should be used:
where Ma is the Mach number, is 1.4, R is 287 J/(kg*K), d is 7.8 mm, Vx is the axial velocity, Q is the air flow rate and T is 298 K; if Ma < .3, then the fluid is considered incompressible.
The given apparatus includes a 7.8 mm diameter copper tube that supplies air, a rotameter that measures the air flow rate, a valve that controls the air flow, a Pitot probe that distinguishes between stagnation and static pressures, and water based U-Tube manometer used to measure air pressure. The other measuring tool that is needed is a small ruler.
For the first part of the experiment, use the slider to adjust the axial distance from the tube exit to the Pitot probe so that it is 1 cm and the radial distance from the centerline is 0 inches. Adjust the valve so that the air flow rate is 50 L/min. Use the rotameter by guiding the center of the "bead" to the 50 marking. One person makes sure that the bead is on the 50 marking; if the bead moves, that person needs to compensate by adjusting the valve. Another person needs to look at the manometer and be able to know when the water columns almost cease moving. At this point, the difference in vertical distance of the water column should be measured in inches. Repeat this step for nine other distances from the tube exit of 1 cm increments. Turn off the air flow, set the slider back to 1 cm from the tube exit, and repeat the previous process for a flow rate of 70 L/min. Convert all water column measurements from inches to meters by multiplying by 0.0254 and calculateusing Equation (1). Use Equation (2) to calculate the velocities in m/s.
For the second part, adjust the slider so that the probe is 2 cm from the tube exit. One person needs to turn on the air flow and set it to 70 L/min. Another person needs to set the radial distance from the centerline to 0 inches. Once the manometer and the rotameter are stable, measure the water column as done in the previous part of the lab. Increase the radial distance from the centerline by 0.05 inches, but keep the distance from the tube at 2 cm. To do this, turn the caliper-like spindle two full revolutions. Repeat this process with 0.05 inch increments until it is .45 inches away from the centerline. Turn off the air flow and repeat this process for 4, 6, and 8 cm distances away from the tube exit. Convert all the measured values to meters and use Equation (3) and Equation (4) to determine the mass flow rate and the momentum flow rate. Afterwards, use Equation (5) and Equation (6) to calculate the Mach number for 50 L/min and 70 L/min air flow.
Results and Discussion:
This lab contained a decent level of uncertainty providing overall good results. The three instruments that provided a good degree of precision were the ruler used to measure the axial distance from the tube exit, the manometer used to measure to water height, and the caliper spindle used to measure radial distance from the centerline. The only instrument that had a poor degree of precision was the rotameter because its markings were of 5 L/min intervals and the bead was relatively large. The rotameter was not only had inaccurate measurements, but seemed to have trouble responding to mild changes in air flow. The rotameter is the source of the greatest level of uncertainty in this lab, thus causing it to be the greatest source where errors may occur. A difference in 5 L/min at 50 L/min can cause the error to be 10%.
The relation between the distance from the tube exit and fluid velocity was as expected. Figure (2) shows a slightly negative slope turn into a moderate negative slope because as the distance from the tube increases, more kinetic energy is dissipated as heat; thus a decline in velocity. The dip in the slope indicates a significant growth in the entrainment region at about 3 cm away from the tube exit. The entrainment region becomes more dominating because it is larger farther away from the tube exit. It's not a surprise that the velocities of the 70 L/min are greater than the velocities of the 50 L/min because velocity is directly proportional to volume flow.
The relation between radial distance and the velocities ratio was as expected. The centerline, where the radial distance is zero, contains the maximum velocity. However, the velocity ratio diminishes faster at a smaller distance away from the tube than at a larger distance even though the maximum velocity at the smaller distance is greater than the maximum velocity at a larger distance. This is obvious in Figure (3) because at x = 2 cm, the plot hits the x-axis at 0.300 inches while x = 8 cm has not hit the x-axis anywhere in the plot. This makes sense because as the far axial distance can compensate for a large radial distance so that the core region does not miss Pitot probe completely. The plot also shows a clear arrangement of the order of the velocity ratios at r = 0.300 inches.
The ratio of the total mass flow rate to the tube mass flow rate is at first surprising that it is greater than 1 and increases as the distance from the tube increases as Figure (4) displays. However, after some further thought, the total mass flow rate is the sum of the individual mass flow rates at different axial positions. Since the trapezoid rule uses ten values, of which the first one is always zero, these combined values will create a greater value than the calculated tube mass flow rate, which is theoretically the constant radial mass flow rate out of the 7.8 mm tube. The trend that the ratio increases as the distance from the tube increases is understandable since there is a greater amount of air flowing in a larger range of the core region; however, the slope appears to be decreasing and will eventually reach zero because the core region is finite.
The relation between the total momentum rate and the distance from the tube exit appears to be a quadratic with a maximum at where the combination of the product of the air mass flow and velocity is the greatest. This relation is shown in Figure (5), but because there are only 4 data points, it is not clear whether or not this expectation is true. Farther away from the tube exit indicates a large total mass flow rate with a small velocity while closer toward the tube exit indicates a small total mass flow rate with a large velocity. Thus, somewhere around x = 6 cm lies the maximum for total momentum flow rate.
Using Equation (5) and Equation (6), the calculated Mach number for 50 L/min is about .05 and the Mach number for 70 L/min is about .07. Both of these values are well below 0.3 and thus confirm the assumption that the air used is incompressible. Some sources of error include general fluctuations in air flow through the pipe and any air blowing in the room. Fluctuations of air flow in the pipe cause the velocity of the fluid to change and shift the water levels inside the manometer. Any air blowing in the room will effect what the Pitot receives as non-static pressure, thus adding it pressure exerted by the fluid. Some human errors include misjudgment of when the manometer stops changing, inattentiveness toward the rotameter, and inaccurate adjustments of the air flow. These lead to low to moderate errors and are usually compensated with better data that came before and after these errors were made.
Conclusions and Recommendations
The free air jet flow was observed, and its variables were determined and related to each other. There were almost no surprises in the figures or data, and the results were good. One of the main trends was that the fluid velocity is greatest near the tube and decreases as the distance between it and the tube exit increases. This is because kinetic energy is lost by dissipating into heat because of the shear of the surrounding air. The total mass flow rate increases along the axial direction when there is enough radial distance away from the centerline. The total momentum flow rate appears to be a quadratic function of the distance from the tube exit; it has a maximum where the product of its velocity and mass flow rate is the greatest.
Some improvements that can be made for this lab include the use of a more precise rotameter for a greater degree of precision, a smaller pipe diameter for a more constant radial flow, and the use of a gas and a special setup that could better show the effect of the core flow and the entrainment region. Finally, the air flow rate should be higher to create a larger core flow region.
Figure 1: Free Jet Diagram.
Figure 2: Velocity vs. Distance From Tube Exit plot.
Figure 3: Velocity Ratio vs. Radial Distance plot.
Figure 4: Mass Flow Rate Ratio vs. Distance From Tube Exit plot.
Figure 5: Total Momentum Rate vs. Distance From Tube Exit plot.