David Mack Han Kuk No Nick Sovocool University of Rocester ME Department A Verification of Open Channel Flow Relations and Flow Straightening Open channel flow equations were evaluated and tested in a half round gutter 20 ft in length. The Manning equation for uniform flow in a sloped channel and a hydraulic jump relation were verified to a moderate degree of success, but our final results were not as conclusive as we had hoped. In addition, the forced development of laminar flow was attempted. A Manning roughness coefficient n for our galvanized steel was found to be 0.013, which compares favorably to our expected value of 0.012 for smooth steel. A hydraulic jump was achieved with a weir and the flow was straightened with flow straighteners. Wednesday, April 28th 2010 Introduction Open channel flow has been extensively studied in the past, and relations like the Manning equation and hydraulic jump length equation have been determined. The Manning equation relates flow velocity, hydraulic radius, energy grade line slope, and a Manning roughness coefficient of the channel material for steady, uniform flow. Steady uniform flow is obtained when the water level throughout the channel and the velocity distribution at any given cross sectional area in the channel remain constant with position and time. Uniform flow develops when there is a balance between potential energy lost by the fluid as it flows down the channel and the energy dissipated by viscous effects associated with shear stress throughout the fluid relating to the Manning coefficient (Young 2007). We will verify the Manning equation in this experiment as well as determine a Manning roughness coefficient for our material, galvanized steel. Current research involving uniform open channel flow involves the study of wetted perimeter effects on minimal environmental flows (Gippel & Stewardson 1998). The Manning equation provides a model that was used by Gippel to help determine the discharge rate of flow based on the wetted perimeter of the natural channel, and from that, determining a relation for the wetted perimeter as a function of discharge (Gippel & Stewardson 1998 56). Other research that has utilized the Manning equation involves municipal stormwater draining. The Manning equation allows for an estimation of water velocity and flow as it exits a river or lake, and it is necessary to have proper drainage if such events occur (Debo & Reese 2003).
When uniform flow is disturbed, different flow formations can be observed. A hydraulic jump is a dramatic increase in height in a fluid flow over a short distance that commonly occurs in open channels when water slows down abruptly by an opposing force such as a weir, an upward sloping channel angle, or a decrease in channel width. A hydraulic jump will only occur when the flow of the water is at or above the critical flow. Critical flow is defined as the minimum flow rate required for a hydraulic jump, which happens when the Froude Number, defined in Eq. 10, is equal to 1. If the fluid velocity is above the critical velocity, the flow is said to be supercritical flow while fluid velocities below the critical velocities are called subcritical flow. The Froude number, Fr is equal to the ratio of inertial and gravitational forces (Chaudhry 2008). As can be seen in Fig. 1, different Froude numbers will create different types of hydraulic jumps characterized by the strength of the rollers that appear on the surface of the jump, the energy loss in the prejump stage, and the initial velocity. The hydraulic jump relation we will be studying relates the length of a hydraulic jump for a given initial or final height and the Froude number at that height. All of these relationships are used by engineers for the construction of many types of systems such as irrigation canals, dams and power generation systems, aesthetic features of fountains, and others. While these relations provide analytical data, we tested their application to real world experimentation. Specifically, we analyzed the Manning equation for uniform flow for a given water depth and velocity as well as the hydraulic jump equation comparing the depths to the length of the jump using the Froude number relation. In order to take measurements and obtain the data, an experimental apparatus was constructed. One assumption made in the Manning equation is that the flow must be steady and uniform. Our first goal was to force laminar flow using various materials and flow straightening techniques. If we were able to achieve laminar flow, our data would be more accurate and taking measurements would be much easier. Formulation/Procedure This experiment called for a steady flow of water down an even channel at a variable angle. The construction of our apparatus began with a large tank to hold the water. Our tank was cylindrically shaped with an open top. It was approximately 2 meters in height and 1.2 meters in diameter having an exit nozzle 0.79 m above the ground. The channel, a 20 ft long half-round galvanized steel gutter, was supported and held in place from rolling by a wooden frame constructed from a base, side guards, and bottom supports with dimensions shown in Fig. 2. Measures were taken so that the frame did not compress the channel and deform its geometry by cutting away the segments of the side guards that protruded inwards. The beginning of the frame was raised off the ground and supported by four concrete blocks. The end of the frame had two concrete blocks and a mini scissor lift lab jack to vary the angle. To keep the center of the channel from bending, a second scissor jack atop two concrete blocks was used to support it. Fig. 3 depicts the channel in the frame on its supports. The channel began at the exit nozzle and emptied into a discharge basin that contained a sump pump. The water was pumped back into the tank at the same rate it was discharged from the gutter, maintaining a constant water depth in the tank. To get the flow out of the tank to match the flow back into it, we first measured the pumping ability of the pump in liters per second, and created a relation of the flow rate out of the nozzle of the tank for a given depth of water in the tank. We calculated the 2
flow rate of the pump to be roughly 0.72 liters per second. With this information, we found the depth above the nozzle to fill the tank to get equal flow rates to be approximately 1.1 m. This analysis can be seen in Appendix A. To straighten the flow after exiting the nozzle, we used a series of scouring pad configurations laid along the beginning of the channel. The first series consisted of three pads overlapping each other at their midpoint, the first held up to catch the free stream from the exit valve of the tank. A second pad was arranged in a similar fashion with a weight at the end to keep it pressed to the bottom of the channel to force the water to flow through it, rather than the water force the pad to the surface, nullifying its functionality as seen in Fig. 4. We created hydraulic jumps using an aluminum block in various orientations (resulting in different cross sections) as a weir. The force from the block caused the flow to quickly slow down, generating the jump as in Fig. 5. In the experiment, it was necessary to acquire angle, depth, velocity, and length measurements of the flow in the channel. To obtain these data, we used various measurement techniques were applied. To measure the angle in the channel, a digital readout was given by the accelerometer in an Apple iphone 3G properly calibrated on a level surface. The STMicroelectronics LIS302DL motion sensor 3-axis accelerometer in the device can be used to determine an angle through a ratio of the current passing through the weighted silicon springs in the two-axis plane whereby the springs are stretched proportionally by the angle and the force of gravity (STMicroelectronics 2008). Our readout was accurate to 0.1 degrees. Length measurements (for the length of hydraulic jumps and various measurements of the apparatus) were taken using a simple tape measure. In order to measure the depth of the water in the channel, a non-conventional method was used by first finding an equation of the geometry of the gutter. In order to obtain this equation we used computer imaging software. First, the cross section of the gutter was traced onto a piece of paper and scanned into Adobe Photoshop CS4. Overlain with a grid, X and Y pixel values along the cross section at regular horizontal intervals were recorded and exported into Microsoft Excel. Because the gutter was not even on both sides, Y values for corresponding X values on either side of the centerline were averaged, and a fifth order polynomial regression line was fitted to the halfcross-section plot of the gutter and converted into metric units from pixels using the ratio from the scanner. This polynomial is shown in Eq. 1. Substituting this polynomial into the arc-length equation, given below in equation 2, along with regular x-distance intervals we found the distances from the centerline that when marked on a flat piece of tape and transposed onto the channel gave back the same regular x-distance intervals. With the water flowing in the channel, the surface of the water matched up with the markings, which was translated back to a depth using the fifth order polynomial. These markings were traced at five regularly spaced intervals along the gutter to make multiple measurements for each angle. f x = 0.0007x 5 0.008x 4 + 0.306x 3 S = +0.0487x 2 + 0.0104x (1) b 1 + [f (x)] 2 dx (2) a The velocity at each of these five points was measured in two ways: by using a Pitot-tube, and through the relationship between the flow rate and cross section area of the flow. First we used the Pitot-tube to find the velocities at both the bottom of the channel and near the top of the surface of the water. The difference in pressure in the u-tube 3
manometer is then converted into velocity. We then calculated the velocity using the relationship mentioned above. The flow rate, which must be constant throughout the channel in order to maintain conservation of matter, was calculated by timing how long it took to fill the discharge basin with a set volume of water. This flow rate was then divided by the cross sectional area at each point in the channel (obtained by the integral of the fifth order polynomial subtracted from the height of the channel times the width of the water) to obtain velocity. We verified our overall velocity measurements with a sanity check. By placing small pieces of foam in the center on the surface of the flow and timing how long it took to travel a set distance, the average surface velocity was calculated. Results/Discussion Forcing Laminar Flow The first goal in our research was to force laminar flow as an ideal outcome, however it is not required for the equations we will be verifying. Due to the lack of equipment and limiting geometry of the channel to actually place a measuring device (such as a flow meter) in the channel without disrupting the flow, the majority of our observations were made by visual inspection, supplemented by imaging devices. After creating our flow, we used two flow relations including the Reynolds number to obtain numerical estimates of the smoothness of our flow. As the stream of water from the exit nozzle crashed onto the channel, it was somewhat steady, but extremely non-uniform. To reduce the initial splash, we raised the beginning of the channel as close to the exit nozzle as possible as seen in Fig. 6. From here, we experimented with a variety of materials and setups to straighten the flow. We found that scouring pads performed best overall to achieve this goal. As the water passed underneath and through the pads, air bubbles were trapped behind them and random motion in the flow was effectively redirected forward. The turbulence of this flow is depicted in Fig. 7. By placing the pads along a section of the bottom of the channel, the turbulence of the stream was greatly reduced. We found that it was difficult to keep the pads from shifting around in this configuration without permanently affixing them to the channel however. We did not have this option available to us, so we needed to find another solution. Although it is not a specifically scientific approach, a widely used technique among enthusiasts to create a laminar jet stream is to force the flow through an array of straws. We filled a section of the channel with straws and tested the flow. While this method worked with some success, it did not appear to perform any better than the scouring pads, and preventing the straws from separating and floating downstream proved to be difficult as well without some sort of permanent attachment. Along with straws, we tested other non-traditional materials, such as a variety of meshes and grates. These did not function at all however, as the water clung to the netting and climbed up it, creating small splashes and waves. Ultimately scouring pads proved to be the most effective material to straighten the flow. The arrangement that worked best was a combination of two methods. The first section consisted of the pads overlain on one another tied at the front end to hold it up and catch the stream of water from the nozzle, and the second section was only one scouring pad held up in the same way in the front, and held to the bottom of the channel by a weight (See Fig. 4). Upon inspection, this method appeared to create the smoothest flow as shown in Fig. 8. 4
In order to see how laminar or turbulent our flow was we used a laminar flow approximation obtained by Chaudhry defined in his open channel flow book. Chaudhry approximates how laminar or turbulent a fluid flow is through his dimensionless parameter R s given in Eq. 3. Where R s is a dimensionless number, V is velocity, g is the force of gravity, k is a characteristic length parameter for the size of channel-surface roughness, R is the hydraulic radius, S f is the slope at uniform flow, υ is the kinematic viscosity, and V* is the shear velocity. Eq. 6 was determined by solving Eq. 3 through Eq. 5 for the dimensionless number R s. Chaudhry states that fluid flow can be approximated to be laminar for R s values lower than 4, transitional for R s values between 4 and 100, and turbulent for R s values greater than 100. V = g k RS f (3) V = grs f (4) R s = kv ν (5) R s = gr S f 3 2 νv 2 (6) After finalizing our configuration using the material properties of water and velocity and geometry values for flow in our channel at an angle of 0.5 o, we found R s to be 206, which is considered fully turbulent (R s >100) (Chaudhry 2008). We further verified our fluid flow to be turbulent by using the Reynolds number (See Eq. 7), which is based hydraulic radius as the characteristic length, which is the cross sectional area made by the fluid over the wetted perimeter at that area. We calculated the Reynolds number to be 5,500 which is turbulent since the transition from laminar to turbulent flow in free surface flows occurs for Reynolds numbers of about 600 (Chaudhry 2008). This agrees with our previous turbulent flow estimation. Because our fluid was water (which has a very small viscosity), and the characteristic length generally used to compute the Re number is large (the length of the channel), it is unlikely from the beginning to have laminar flow. Open channel flow involves a free surface that can easily deform from an undisturbed flat configuration to form waves (Young 2007). Because the velocity at the edges of the free surface in contact with the gutter is less than the velocity in the center, ripples where shed off throughout the entire water surface along the channel as seen in Fig. 8. Therefore by these results, we cannot assume that our flow is laminar, and must consider the following equations simply under the steady and uniform assumptions. R e = ρvd μ Evaluation of the Manning Equation (7) In order to continue our analysis on open-channel flow we needed find the angle at which our channel had to be sloped in order for the flow of water to be uniform throughout the channel. Note that for our analysis we denoted a declining slope of the channel from the exit nozzle to the discharge basin to be a positive angle, regarding parallel to the ground to be zero degrees. The Manning equation (Chaudhry 2008) is as follows: Q = C 0 A 5/3 S 1/2 n P 2/3 0 (8) where Q is the flow rate; n is the Manning roughness coefficient; A is the cross sectional area of flow; P is the wetted perimeter; and S o is the slope. S o is determined as the tangent of the slope angle. The parameters A and P are dependent on the depth of the water in the channel, n is 5
dependent on the surface roughness, C o the unit conversion factor is equal to 1.49 for Customary English units and 1 for SI units, and Q is dependent of the fluid velocity and cross sectional area. To find the slope we needed to obtain uniform flow we had to test multiple angles. In order to evaluate the Manning equation for our experimental setup, we first found the angle at which the depth and velocity of the flow would remain uniform to satisfy the equation. Given our Q value of 7.2x10-4 m 3 /s, and estimating P = 0.07-0.15 m, A = 0.0015-0.0021 m 2, and n = 0.012, we found that our angle should be between approximately 0.2 o - 0.6 o. We calibrated the channel to the estimated angle, and recorded the depths along the channel at marked regular intervals. If the depths increased along the channel, the angle had to be increased to quicken the flow, and if depth decreased, the angle was lessened. By using this trial and error procedure we estimated the angle to be 0.5 degrees. Both water depth and velocity at the measurement points were plotted for multiple angles in Fig. 9 and Fig. 10. We observed that for gutter angles less than this, the water velocity decreased and depth increased, and for gutter angles greater, velocity increased and depth decreased. In addition, when plotting the water velocity versus channel angle, we found that the sets of data all crossed near the 0.5 degree mark as seen in Fig. 11. We can observe in the graphs that the measurements taken at 0.5 degrees are all approximately horizontal. This means that at this channel angle the depth of the water and the velocity remain constant throughout the length of the channel. However, it should be noted that in both of these plots, the last point measured in the 0.5 degree lines deviate from the supposed straight line. This might have happened because of a discrepancy in our gutter geometry at this point, which would affect the depth measurement at this point and therefore affect the velocity measurement and/or angle reading at this point. Once we found the appropriate angle, we had to determine the corresponding velocity of the flow to check the output of the Manning equation. We found that in our experiment, we were able to measure the velocity in several ways: using a Pitottube, timing a floating object between two points, and solving for it using the known volume flow rate and cross sectional area. Because these methods measured velocity at different locations on the cross section of the channel (surface, below the surface, and at the bottom), the velocity profile of the channel caused the results to vary. Fig. 12 gives a visualization of the velocity profile for a round channel. Measurements taken in the center of the surface of the channel had the greatest magnitude, while measurements taken closer to the walls of the gutter had smaller magnitudes. Between the most extreme values of these methods, we calculated a difference in measurements larger than 45%. Yet while our methods are spaced apart from each other, they remain consistent, as shown in Fig. 13, which shows the velocity as a function of angle, taken using several different methods. Therefore, while our accuracy may have been off for some of our measurements, they follow the same trend and our precision was sufficient. In order to find which of these methods was the most accurate, we compared them to the velocity we obtained by plugging our other measurements into the Manning equation. The slope variable was measured from the Apple iphone 3g accelerometer, the depth variable was measured from the channel markings, and the n value was obtained from the given value in the reference material for smooth steel (Young 2007). (Later on, we will attempt to verify this n value using our measured velocities.) Plotting the results, we found that the velocity obtained by the Manning equation was closest to the velocity obtained through flow rate 6
relationships, their average values separated by 6%. This is illustrated in Fig. 14. Given that our methods of data acquisition had a significant amount of built-in error, this is a positive and promising result. Flow rate measurement was a good method to choose because the velocity that was calculated from the flow rate was the average velocity, while all the other techniques measure something that is either above or below this average. Frictional forces acting on the water from the surface of the gutter cause the flow to be slower approaching the gutter and faster at the center of the surface as the frictional forces are escaped. Looking again at Fig. 14, we saw that the values given by the Manning equation fell between the data of the velocity from flow rate calculations and the average of the two Pitot-tube measurements. While this was promising in that the results of the Manning equation landed within expected values, our measurements were not sufficiently accurate enough to confidently confirm or deny the accuracy of the equation. However, since our flow was uniform but non-laminar, we were able to conclude that the Manning equation does in fact predict velocity for this type of flow within a reasonable margin of error. But while we had a difficult task addressing the widely practiced Manning equation due to imperfect data acquisition, we were able to use it to assess the Manning roughness coefficient n of our material. In order to perform out initial calculations, we used a value of 0.012 found in the resources for a material classified as smooth steel (Chaudhry 2008). Our wetted channel surface was constructed from galvanized steel however, which while similar, might have had a subtle enough difference in material properties to effect the coefficient. After plugging in the velocity calculated from the flow rate into the equation we solved for the n coefficient to find that n=0.013. The data may not be perfect, but it does suggest that our material is slightly rougher than smooth steel and therefore has a slightly higher Manning roughness coefficient. Smooth steel is manufactured in various ways such as rolling, drawing, or pulling, but the final outcome is a thin, flat, smooth piece of metal. To turn this steel into galvanized steel, it is coated in a thin layer of zinc that reacts with oxygen and carbon dioxide to form zinc carbonate (Hot-dip galvanizing 2010). The chemical reaction of this process is much more uneven than the machining of the steel since the zinc oxidizes on the surface without being controlled. Therefore the surface of galvanized steel is rougher than the smooth steel, and would have a higher Manning roughness coefficient. Hydraulic Jumps We analyzed the relation in Eq. 9 obtained by Hager in 1991 (Chaudhry 2008), which relates the length of a hydraulic jump for a given height and the Froude number at that height. L = 220 tan F r1 1 y 1 22 (9) For our experiment we obtained hydraulic jumps at a gutter slope of 0.5 degrees, which is where steady uniform flow was experimentally found to happen. We measured the initial height y 1, final height y 2, and length of the jump L for different hydraulic jumps obtained by varying the weir geometry. The length in Eq. 9 describes the hydraulic jump length which was measured from the start of the hydraulic jump up to the location where the jump levels off. In our experiment we obtained hydraulic jumps that continuously increased up to the location of our weir and never leveled off, so our length measurements were taken up to this location. This abnormality was not mentioned in the references however, and may be a product of real-world circumstances not 7
represented in the theoretical models. For this reason, our approach may differ from other measurement styles. Using the measured y 2 heights and our average flow rate of 0.72 L/s we were able to estimate the relevant velocities at the y 2 points and their equivalent Froude numbers. The Froude number Relation is defined below were D is the hydraulic depth defined as D=A/T and T is the width of the channel (Chaudhry 2008). The Froude number we calculated for our first jumps ranged from 1.6 to 2.5, and the difference in roller size can be compared in Fig. 15. Fr = V gd (10) Using both the velocity and Froude number measurements we calculated theoretical hydraulic jump lengths which we compared with the hydraulic jump lengths we measured in Fig 16. There was an apparent difference between the lengths of the experimental and theoretical jumps. As the Froude number increased, the lengths of the measured jumps became more widely spaced than the lengths of the theoretical jumps. There may have been several factors contributing to this discrepancy. A possible reason was that the empirical relations we used were for a circular cross section channel rather than our non-circular gutter. Finding a more precise empirical relation for hydraulic jumps for our geometry would be overly tedious and outside the scope of this report, so we simply approximated our cross section to be semicircular for these relations. Another main contributor is that our measuring technique (as mentioned above) probably did impact our results, however proper methods are not described in detail. In addition, our jumps usually never fully developed and straightened out, especially for smaller weir cross sections, so we were limited by what we could actually measure. The shape of the gutter was plotted overlaying a semicircular curve in Fig. 17, and in performing an rms error calculation of the difference between each point, we determined the error to be 9.3%. While the rms is not very large, this kind of error would assist in creating the systematic difference in the slopes of the two lines in Fig. 16. A second potential reason for this difference was that our hydraulic jumps never reach a perfect equilibrium height. Instead, the jump first increased quickly, then tapered down gradually (never stabilizing) until it reached the weir, where it then fell off. Still despite the errors in our apparatus, it is possible that the empirical relation in Eq. 9 may simply not be suitable for our setup due to the roughness of the surface, smaller dimensions, or measuring technique. Error Throughout our experiment there were a lot of compounded errors in our measurements. A large factor leading to irregular or simply incorrect data was that the gutter was not perfectly uniform. Not only did it not have a constant radius, but both ends were more spread out than the center of the channel. This means that when the cross section was traced, it was most likely wider than the sections we were examining, throwing off all of the depth measurements. In addition, while we were able to measure the gutter angle using a digital readout, it was at its minimum resolution, barely being able to measure angle differences to one tenth of a degree the range we were looking at. Another low resolution issue was encountered when taking Pitot-tube readings. Again, we were stuck trying to see differences between millimeters while the water in the tubes was fluctuating slightly. A way to correct this would be to obtain 8
an electronic measuring device, such as a flowmeter. Such a device would allow for the velocity and flow rate within the channel to be accurately determined, however there may not be enough room in our design for large measuring equipment. Another point of error occurred when we transposed the height markings onto the gutter. Sometimes our sharpie lines were too thick or misaligned, making accurate data acquisition difficult. Further Research As some suggestions for further research, we highly recommend avoiding a wooden frame if at all possible. As wood is not consistent and is often warped, especially for longer pieces, it is nearly impossible to maintain a constant slope across the 20 ft of gutter. Instead, a machined frame out of steel or some other material that is sturdy and will hold the gutter at a uniform slope would be a much better choice. Also, it is critical that the shape of the gutter remain constant and known at all points along the channel. In our experiment we used a simple half-round gutter purchased from a local gutter supply store, however future experiments would likely benefit from purchasing a custom, uniform channel, with a known cross-section. A rectangular channel made out of some kind of glass or plastic is suggested because not only will it be able to maintain a constant shape at all points in the channel allowing for uniform slope throughout, but also it will be much easier to see and measure the depth of the water at all points and length of hydraulic jumps. Due to the broad scope of research in the open channel flow field, specifically in the subject of hydraulic jumps, further research can be done by testing hydraulic jump lengths under varying initial velocities and channel slopes instead of different weir dimensions. The channel should also be sufficiently larger than the gutter we used for testing flow rates around 0.72 liters per second since most of the hydraulic jumps we obtained did not fully settle, and the water in many cases would overflow. If a bigger channel is used, the length measurement could be performed more accurately since the point where the water level stops increasing would be seen more clearly. If there is a large budget for the experiment, we would recommend the use of a hydraulic flow demonstrator, such as the Armfield S16 Hydraulic Flow Demonstrator depicted in Fig. 18 (Armfield 2008). Appendix A: Flow rates In order to prep experimental setup for data acquisition, it first needed to be properly calibrated. To maintain a steady flow throughout our system, it was critical that the flow rate at which the sump pump in the discharge basin pumps water to the tank was equal to the flow rate out of the main tank. We could theoretically find the velocity out of the tank nozzle using the Bernoulli principle in Eq. A-1, and experimentally we were able find the actual flow rate and discharge coefficient. To find the theoretical velocity out of the tank we used Bernoulli s equation, with our first point being the surface of the water in the tank and our second point being the exit of the nozzle in the tank, we were able to derive Eq. A-2 to find the velocity of the water out of the nozzle. To find the volumetric flow rate we used the velocity from the derived equation and the measured area of the nozzle (2.19 cm^2). We then plotted the theoretical flow rate (Q ideal ) over a range of water depths above the nozzle in the tank and plotted these points in Fig. A-1. 9
Flow Rate (L/s) ρgz 1 + 1 2 ρv 1 2 + P atm = ρgz 2 + 1 2 ρv 2 2 + P atm V 2 = 2gz 1 (A-1) (A-2) In testing for the flow rate out of the tank, we first marked a spot in the discharge basin at an initial height. We then proceed to fill this basin to the marked water level and add an additional 50 L of water and then mark the basin again at this new water level. This is done multiple times to obtain multiple 50 L level markings. After that, we drained the tank into the basin and measured the time it took to fill the basin to each 50 L line, recording the water depth above the nozzle of the tank along with the time at each point. Using the time taken to fill the given amount of water from one point to the other we were able to plot the flow rate (Q actual ) at each depth above the nozzle in Fig. A-1. Using these actual flow rate values we divided them by the ideal flow rate values to obtain a range of discharge coefficients, which we then average to obtain a discharge coefficient of 0.73. This is admittedly less than the expected value, because the nozzle in the tank had previously been designed to discharge water having laminar flow with a discharge coefficient of 1. However, it is likely that the reason it drops so low is that there is an added valve attached to control the flow of water. The flow rate times the discharge coefficient is also plotted in Fig. A-1 and an equation approximating this relation is placed in this plot. In order to find the flow rate of the pump we used the same method we used to find the actual flow rates of the water coming out of the nozzle. We obtained a pump flow rate of approximately 0.72 L/s. Using this flow rate and the equation for the flow rate out of the nozzle for a given water depth in the tank, we found that in order maintain a steady flow of water around the system, we needed to fill the tank to about 1.10 m above the exit nozzle. 1.2 Flow Rate vs. Depth for Cylindrical Tank 1.1 1 0.9 0.8 0.7 0.6 y = -0.101x 2 + 0.5606x + 0.2516 Qactual Qideal*Cd Qideal Poly. (Qideal*Cd) 0.5 0.5 0.7 0.9 1.1 1.3 1.5 Depth of Water above Nozzle (m) Figure A-1: flow rate for given depth of water above the nozzle for different flow rate calculations. Blue line shows the theoretical flow rate, the red line show the actual flow rate obtained experimentally, and the green line shows the theoretical flow rate accounting for a discharge coefficient. 10
Appendix B: Figures Figure 1: Froude number relations to hydraulic jumps (Chaudhry 2008) Figure 2: gutter support frame and dimensions. 11
Depth (cm) Figure 3: experimental setup (Tank, frames, cinder blocks, etc.) Depth of Water vs Channel Distance 1.70 0 0.5 1 1.5 2 2.5 Distance Along Channel (m) 0.4 0.5 0.6 1.3 Figure 4: flow straightening configuration. 12
Figure 5: WEIR. Figure 6: Exit Nozzle 13
Figure 7: Turbulence Figure 8: Smooth Flow Figure 9: From this plot it may be noted that, for every angle greater than 0.4 o, the depth of the water within the channel decreases as a function of distance. This is evidence that the slope that provides uniform flow is between 0.4 o and 0.5 o. 14
Velocity (m/s) Velocity (m/s) 0.70 Velocity vs Channel Distance 0.65 0.60 0.55 0.50 0.45 0.40 0.4 0.5 0.6 1.3 1.6 0.35 0 0.5 1 1.5 2 2.5 Distance (m) Figure 10: This plot shows the relationship between the slope and the velocity as a function of distance along the channel. As in Fig. 2.1, it may be noted that uniform flow occurs at a slope between 0.4 o and 0.5 o. 0.60 Average Water Velocity vs. Angle 0.55 0.50 0.45 0.40 0.35 0.45 m 0.93 m 1.532 m 2.12 m 0.30 0.20 0.40 0.60 0.80 1.00 1.20 1.40 Angle (degrees) Figure 11: average water velocity across the area using our Q value of 0.72 L/s 15
Velocity (m/s) Figure 12: Velocity Distribution 1.15 1.05 0.95 0.85 0.75 0.65 0.55 0.45 0.35 Velocity vs. Angle using Different Methods 0 0.5 1 1.5 2 Angle (degrees) Pitot a Pitot b Q = 0.72 L/s Floating Figure 13: The velocity vs. angle (using different measuring methods) is plotted. It is obvious that no two measurement techniques produced the same result. 16
Velocity (m/s) Velocity Found Using Different Methods at 0.5 Degrees vs. Distance 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0 0.5 1 1.5 2 2.5 Distance (m) Q=0.72 L/s Pitot Floating Manning EQ Figure 14: The velocity vs. distance along the channel is plotted at an angle of 0.5 o. The different lines correspond to different measuring techniques, and it may be observed that the values calculated from the manning equation (given n, R h, and S o ) are very nearly similar to those found using V=Q/A. Figure 15: Hydraulic Jumps 17
Height (cm) Hyraulic Jump Length (cm) 160.00 140.00 Experimental vs Theorical Hydraulic Jump Length Comparisson 120.00 100.00 80.00 60.00 40.00 Experimental Length Theoretical Length 20.00 0.00 3 3.5 4 4.5 5 5.5 6 Hydraulic Jump Height (cm) Figure 16: The Hydraulic Jump Length vs. the Hydraulic Jump Height for both the measured length and the theoretical length computed using Hager s relation. Height vs Width of Gutter and Semi- Circular Cross- Sections 8 7 6 5 4 3 2 1 0-10 -5 0 5 10 Circle Gutter Width (cm) Figure 17: Gutter Cross-Section 18
Figure 18: Armfield hydraulic flow demonstrator (Armfield 2008) 19
References: Chaudhry, M. Hanif. Open-channel Flow. 2nd ed. New York, NY: Springer, 2008. Print. Debo, Thomas N., and Andrew J. Reese. "Urban Hydrology." Municipal Stormwater Management. Boca Raton, Fla.: Lewis, 2003. 258-60. Print. Gippel, Christopher J., and Michael J. Stewardson. "Use of Wetted Perimeter in Defining Minimum Environmental Flows" Regulated Rivers: Research & Management 14 (1998): 53-67. Print. "Hot-dip Galvanizing." Wikipedia, the Free Encyclopedia. Web. 28 Apr. 2010. <http://en.wikipedia.org/wiki/hot-dip_galvanizing>. "Hydraulic Flow Demonstrator - S16." S SERIES: Applied Hydraulics and Hydrology 1 (2008): 1-4. Discover with Armfield. Armfield, 2008. Web. 25 Apr. 2010. "MEMS Motion Sensor." STMicroelectronics, 2008. Web. 27 Apr. 2010. <http://www.st.com/stonline/products/literature/ds/12726.pdf>. Young, Donald F. "Open-Channel Flow." A Brief Introduction to Fluid Mechanics. 4th ed. Hoboken, NJ: Wiley, 2007. 376-94. Print. 20