Fluids Lab Report_ The Force of a Jet.docx

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Lab 2: The Force of a Jet I pledge my honor that I have abided by the Stevens Honor System.

Introduction: The objective of this experiment was to understand the fluid forces on solid objects that deflect the momentum of an impinging jet. This was accomplished by theoretically and experimentally calculating the force of a circular fluid jet as it acted on several differently shaped deflecting surfaces. The forces were calculated by comparing physical data and theoretical models derived from Linear Momentum Equations, Newton’s 2nd Law, and Reynolds Transport Theorem. The experiment utilized a Controllable Flow Apparatus to shoot a stream of water with varying pressure at a flat plate at 30, 45, 60, and 90 degrees, a cone at 45 and 60 degrees, and a bucket at 160 and 90 degrees (Shown below in Figure 1). The controllable flow apparatus has a flow control valve, pressure gage, and a digital flow meter which aids in conducting and performing the needed calculations. Water flow pressures of 0, 30, 40, and 50 psi were used to find the volumetric flow rate and force both theoretically and experimentally. The experiment took place in a water bucket with a cover to prevent any water from spilling out during testing as well as to provide a controlled environment. A force balance was connected to the test base to calculate the force of the jet. These recorded values, although not 100% accurate to theoretical models, are a close representation and provide physical evidence to theories learned in Fluid Mechanics. The hypothesis that this lab intends to prove is that force increases as larger vertical angles are introduced to an impinging jet of water, hence, it is expected to see the 160° pelton bucket to have the largest momentum transfer out of all the test specimens. Theory: In order to predict the behaviors produced by the fluid within these experiments - Newton’s 2nd Law, the Linear Momentum equations, and Reynolds transport theorem must be used to derive equations that are capable of determining the expected forces experienced by the various plates. Therefore the following is a comprehensive derivation that expresses the theoretical forces for each plate used in the laboratory. Figure 1: Plate, Bucket, and Cone Flow Diagrams

The significant equation behind all of these relationships is the linear momentum equation: This equation can be manipulated to determine other equations that will help represent the correlations between multiple variables of each solid surface at various angles. To simplify this equation, the time derivative on the right side can be eliminated because the control volume does not change with time. Another way to simplify the equation is to remove the F(B) term because no body forces were present in this lab. The weight of the water in the stream was negligible, and the weight of the deflecting surface was ignored because the scale was reset to zero after the plate was placed on the stand. Now that the equation can be simplified, it is much easier to derive equations to demonstrate relationships between variables. As seen above in figure 1, the first schematic shows a flat plate inclined at some value 𝜃 and has an inlet volume flow of 𝑸 . When the inlet flow splits into two streams, each stream has the same velocity but unequal volume flows, 𝛂𝑄 and (1- 𝛂 ) 𝑄 where 𝛂 ∈ [0,1], due to the fact that for frictionless flow, the fluid can exert no tangential force Ft on the plate. A relationship between 𝜃 and 𝛂 can be derived as follows: → There are no body forces. 𝐹 ?? = 0 → There are no surface forces acting in the x-direction because the flow is frictionless 𝐹 𝑆? = 0 → , The volume does not change over time → ? 1 = 𝑉???θ → ? 2 = ? 3 = 𝑉 → Divide both sides by ρ𝑄 − 𝑉???θ − 𝑉(1 − α) + 𝑉α = 0 → Divide both sides by 𝑉 − ???θ − (1 − α) + α = 0 ⇒ − ???θ − 1 + 2α = 0

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⇒ α = (1 + ???θ)/2 (2) The linear momentum equation can also be used to derive an equation to formulate the magnitude of the vertical force Fy to hold the flat plate in place as a function of incoming volume flow rate 𝑄 , fluid density ρ, get area A, and inclination angle . θ

After the data was obtained, the applied force was plotted on a graph as a function of the square of the mass flow rate. A line of best fit was found from the data points and the slope of the line was found. Next, in order to determine the uncertainty in the force measurements (σ y ), the principle of least squares was applied, and the following formula was used: In this formula, N represents the number of data points, and B represents the slope of the line of best fit made from data points. Next, the uncertainty in the slope (σ B ) was calculated using the following formula: Materials and Methods:

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In this lab, water is supplied from a tap where it flows through a throttling valve, a digital flow meter, and a quick acting “ball” valve. The water then discharges through a nozzle that has a diameter of 2.8mm. The schematic of the set up for the experiment is shown in Figure 2 below. The flow rate of the water can be adjusted to specified values needed for the experiment by adjusting the pressure through the pressure control valve. Figure 2: A Schematic of the Experimental Set Up The jet sprayed water onto the test specimens, a flat plate, two different buckets, and two cones, which are shown in Figure 3 below. The specimens sat on the vertical rod, which is connected to a force balance. The force balance measured the vertical component of the force acting on the specimens. Figure 3: Deflecting Surface Specimens (Flat Plate, 2 Cones, and 2 Buckets)

To conduct this experiment, the first specimen, the flat plate, was set on the support rod at a 90 ° angle. The force balance displayed the weight of the specimen, so the scale was reset to zero so that it only measured the force of the water acting on the plate. Then, the faucet was turned on and the gage pressure control valve was adjusted to 20 psig. After waiting 15-30 seconds for the system to stabilize, the volume flow rate from the flow meter and the weight from the force balance were recorded. The gage pressure was adjusted to 30, 40, and 50 psig and the same measurements were recorded. These steps were then repeated for the flat plate at 60 ° , 45 ° , 30 ° , and for the two buckets and two cones which is shown in Figure 3 above. The lab setup was adjusted so that the jet stream was spraying the specimens at the center and observing the deflection as shown back in Figure 1. Materials: ● Protractor ● Metal Plate (inclination adjustable) ● Metal 160° Pelton Bucket ● Metal 90° Pelton Bucket ● Metal 45° Cone ● Metal 60° Cone ● Controllable Flow Apparatus ● Force Balance ● Flow Meter ● Bucket with lid Results: The graphs below show the relationship between the force of the jet and the square of the mass flow in each configuration tested during the experiment. This allows for visual analysis of the relationship between the experimental and theoretical values to occur. The red line represents the relationship for a theoretical situation, which was calculated. Figure 4: Bucket

Figure 5: Cone Figure 6: Flat Plate

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Discussion: The purpose of this experiment was to analyze the relationship between the force on the angle and shape of different surfaces. The first recognized relationship in this experiment was increased force with an increase in pressure. This increased pressure also caused increased volume flow rate and mass flow rate. Another trend throughout the experiment was a decrease in force with a decrease in angle between the surface and the jet stream. To compare the experimental results to general accepted or theoretical values, the following methodology was used: The Mass flow rate in gallons per minute and weight of the water were calculated and the units were then converted to kilograms per second. The theoretical force calculations were calculated using the individual geometries. The equations are derived and provided above in the theory section. The experimental results were compared to the theoretical results using percent error calculations (Refer to Appendix D). Although the results of the experiment were generally accurate, there were some errors while conducting the experiment that could explain the discrepancies. While conducting the experiment, each specimen was supposed to be perfectly centered underneath the jet stream to produce the most accurate angle and results. Due to human error, not every specimen was perfectly centered which would partially explain some of the inaccuracies. The jet stream was assumed to hit the center of the cone, but even though the cone was centered on the force balance, the stream would not always hit the center of the cone, causing a slight change in angle of the stream. Furthermore, from a theoretical perspective the tangential force and spray may be considered uniform, however in reality the steam splashes in every direction. This real life randomness is another way in which the theoretical models differ from reality. This measurement error also effects the results due to the splashing affecting the direct reading of the experiment as it contacts the plate in an unpredictable or quantifiable manner.

In the experiment, it was to be assumed that the jet stream was a uniform flow, which was not always the case. The jet stream fluctuated slightly due to an uneven pressure valve, due to either a faucet being turned on, a toilet flushing, among other reasons. These small factors caused both the volume flow rate and force to fluctuate frequently. The only way to get a reading for both of these variables was to pick the most consistent number in the set of constantly changing numbers. It was also slightly difficult to obtain the exact pressure for each trial of the experiment using the knob on the valve. The only way to address this was to be as precise as possible when adjusting the gage pressure. The tests that had the best agreement between theory and reality were the bucket at 90 degrees and the flat plate at 60 degrees. The cone tests and the flat plate at 30 degrees yielded the worst results. Conclusion: The deflecting plate that resulted in the highest applied force was the 160 degree Bucket, The applied force, and therefore the slope that was calculated for each specimen, increased as the angle of deflection increased. Since 160 was the largest deflection tested in this experiment, it saw the highest force applied. This occurs because the force is what allows the jet to change the direction of its momentum. The larger the change in momentum in the vertical direction, the larger the vertical force that is needed. Additionally, the jet impinging on the flat plate requires slightly more force at the same angles when compared to the cone or the bucket. This is due to the fact that the jet contacting the flat plate splits and some of the water travels up the plate. This fraction of the jet has a larger angle of deflection than the rest of the water, which required more force to allow for the larger change in momentum. For both the cone and the bucket, all water leaving the control volume experiences the same angle of deflection. The tests that achieved the smallest percent errors and best agreement between theory and reality were the bucket at 90 degrees and the flat plate at 60 degrees. The cone tests and the flat plate at 30 degrees yielded the worst results. The large areas of error could have come from measurement errors due to water hitting the bottom of the tank and thus affecting the reading of the scale. Errors such as these could be prevented in the future by reducing the distance from the nozzle to the objects, this would reduce the expansion of the water as it leaves the nozzle and is applied to the objects. To reduce the distance between the two simply raise the bucket system or lower the nozzle system. Appendix A: Figures Figure 1:Plate, Bucket, and Cone Flow Diagrams

Figure 2: A Schematic of the Experimental Set Up Figure 3: Deflecting Surface Specimens (Flat Plate, 2 Cones, and 2 Buckets)

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Figure 4: Bucket Figure 5: Cone

Figure 6: Flat Plate

Appendix B: Tables Raw Data Collected: Bucket 160° Pressure (psig) Volume Flow Rate (gal/min) Force (kg) 20 1.316 .1750 30 1.582 .2780 40 1.780 .3520 50 1.970 .4380 Bucket 90° Pressure (psig) Volume Flow Rate (gal/min) Force (kg) 20 1.334 .1330 30 1.506 .1830 40 1.745 .2250 50 1.990 .2912 Cone 45° Pressure (psig) Volume Flow Rate (gal/min) Force (kg) 20 1.296 .1080 30 1.550 .1515 40 1.799 .2025 50 1.981 .2390 Cone 60°

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Pressure (psig) Volume Flow Rate (gal/min) Force (kg) 20 1.280 .0925 30 1.555 .1245 40 1.757 .1545 50 2.000 .1945 Flat Plate 90° Pressure (psig) Volume Flow Rate (gal/min) Force (kg) 20 1.211 .124 30 1.553 .200 40 1.722 .255 50 1.916 .323 Flat Plate 60° Pressure (psig) Volume Flow Rate (gal/min) Force (kg) 20 1.246 .1125 30 1.515 .1590 40 1.793 .2190 50 1.960 .2650 Flat Plate 45° Pressure (psig) Volume Flow Rate (gal/min) Force (kg)

20 1.302 .1065 30 1.521 .1325 40 1.814 .1765 50 1.987 .2095 Flat Plate 30° Pressure (psig) Volume Flow Rate (gal/min) Force (kg) 20 1.307 .0915 30 1.537 .1145 40 1.751 .1410 50 1.963 .1660 Converted Experimental Data: Bucket 160° Pressure (psig) Mass Flow Rate (𝑘𝑔/?) 2 Force (N) Theoretical Force (N) 20 0.006873736464 1.71675 2.167158068 30 0.009933311556 2.72718 3.131783768 40 0.0125753796 3.45312 3.964777454 50 0.0154032921 4.29678 4.856364355 Bucket 90° Pressure (psig) Mass Flow Rate (𝑘𝑔/?) 2 Force (N) Theoretical Force (N)

20 0.007063057764 1.30473 1.14804077 30 0.009001834884 1.79523 1.463172721 40 0.01208570423 2.20725 1.964429804 50 0.0157176369 2.856672 2.554769983 Cone 45° Pressure (psig) Mass Flow Rate (𝑘𝑔/?) 2 Force (N) Theoretical Force (N) 20 0.006666395904 1.05948 1.849764084 30 0.0095355225 1.486215 2.645877517 40 0.01284527557 1.986525 3.564254169 50 0.01557578881 2.34459 4.321905739 Cone 60° Pressure (psig) Mass Flow Rate (𝑘𝑔/?) 2 Force (N) Theoretical Force (N) 20 0.0065028096 0.907425 1.585465698 30 0.009597141225 1.221345 2.339902157 40 0.01225249748 1.515645 2.987310973 50 0.015876 1.908045 3.870765864 Flat Plate 90° Pressure (psig) Mass Flow Rate (𝑘𝑔/?) 2 Force (N) Theoretical Force (N) 20 0.005820621849 1.21644 0.9460932377 30 0.009572469921 1.962 1.555924658 40 0.0117692122 2.50155 1.912986681

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50 0.01457042126 3.16863 2.368299709 Flat Plate 60° Pressure (psig) Mass Flow Rate (𝑘𝑔/?) 2 Force (N) Theoretical Force (N) 20 0.006161936004 1.103625 1.040797449 30 0.009109748025 1.55979 1.528408191 40 0.01275973568 2.14839 2.130435026 50 0.0152473104 2.59965 2.540026901 Flat Plate 45° Pressure (psig) Mass Flow Rate (𝑘𝑔/?) 2 Force (N) Theoretical Force (N) 20 0.006728264676 1.044765 1.151590794 30 0.009182047329 1.299825 1.560182923 40 0.01306037552 1.731465 2.203617614 50 0.01567028276 2.055195 2.635538466 Flat Plate 30° Pressure (psig) Mass Flow Rate (𝑘𝑔/?) 2 Force (N) Theoretical Force (N) 20 0.006780040281 0.897615 1.173307099 30 0.009376242561 1.123245 1.607839495 40 0.01216895797 1.38321 2.073440998 50 0.01529402156 1.62846 2.592953854 Experimental Slope Theoretical Slope Uncertainty

Bucket y=239.45x-0.1088 y = 242.19x 13.533 Cone y=126.31x+0.1508 y = 260.85x 47.66 Flat Plate y=138.9x+0.2098 y = 166.42x 18.853 Appendix C: Equations Newton’s 2nd Law for Fluids (linear momentum equation) ρdV + ρV • dA (1) 𝐹 𝑆 + 𝐹 ? = ? ?? ?𝑉 ∫ 𝑉 ?𝑆 ∫ 𝑉 Unequal Volume Flow for an Inclined Plane: (2) α = (???θ + 1) 2 Force of Jet on Inclined Plane at an Angle : θ (3) 𝐹 ? = ṁ 2 ⍴? ?𝑖? 2 θ Force of a Jet on a Bucket with a Deflection Angle : θ [1 + cos(180 - θ)] (4) 𝐹 ? = ṁ 2 ⍴? Force of a Jet on a Cone with a Deflection Angle : θ [1 cos(θ)] (5) 𝐹 ? = ṁ 2 ⍴? − Uncertainty of Measurements: (6) σ ? = 1 𝑁−1 Σ(? 𝑖 − ?? 𝑖 ) 2 Uncertainty of the Slope: (7) σ ? = σ ? Σ? 𝑖 2 Appendix D: Sample Calculations Flat Plate calculations 90 degrees-20psi

= (1.211 gal / min)(1 min / 60s)(3.79 L/gal)(0.001 m^3/L)(998 kg/m^3) =0.0765kg/sec Theoretical force = ((0.0765kg/sec)^2)*(Sin(90))/((998 kg/m^3)*( 0.000006157521601 m^2)) = 0.9460932377 N Measured Force on the flat plate: F=m*a m=0.124 kg g= weight= 9.81 m.sec F= 1.216 N Percent Error =(100)((1.216-0.946093)/(0.946093)) = 28.53% Volume Flow Rate (gal/min) Force (kg) 1.280 .0925 Cone piece 60 degrees- 20 psi = (1.280 gal / min)(1 min / 60s)(3.79 L/gal)(0.001 m^3/L)(998 kg/m^3) =.080692 kg/ sec Theoretical force

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= ((.080692 kg/sec)^2)*(1-cos 60)/ ((998 kg/m^3)*( 0.000006157521601 m^2)) =1.585465698 N Measured Force on the flat plate: F=m*a m=0.0925 kg g= weight= 9.81 m.sec F= .90743 N Percent Error =(100)((0.90743-1.5855)/(1.5855)) = -42.76%

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