Pressure vs rate of flow


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  • Water is flowing through a pipe having ID 154 mm & gauge pressure 4Kg/cm2. What is the flow rate ? We do not know about velocity.
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Pressure vs rate of flow

  1. 1. Non Sereeyothin 01/10/11 IB HL Physics Year 2 Pressure vs. Rate of FlowIntroduction:Pressure is the exertion of force upon a unit area by a surface of an object in a perpendiculardirection. [1]Where is the density of water, is the gravity (acceleration), and is the height of water.When fluid flows through pipe, there are two main forces acting on it. One is the frictional forcethat is made by the side of the pipe, and the other one is the viscous force in the fluid. Near thewall of the pipe, there is a thin layer of fluid that sticks to the pipe. In the middle of the pipe, thewater moves faster and consistently. The viscous force of fluid makes a shearing action whichwill result in a small layer of fluid that will keep on increasing until it reach the speed of the freeflowing in the center of the pipe. The energy is lost from these two forces.Ideal fluid has a steady flow, nonviscous flow, irrational flow, and incompressible flow. Steadyflow is when the particles flow after each other in a stream line and all particles has the samevelocity. Nonviscous flow is when there are no shearing forces in the fluid and will result inproducing heat as the fluid flows. Irrational flow is when there will be no turmoil in the form ofeddy currents or whirlpools. Incompressible flow is when the density of the fluid is constant.Fluid pressure is defined as the pressure at some point within a fluid. This occurs in two differentconditions, one is an open condition (open channel flow) and the other one is a closed condition(conduits). In an open condition, the pressure stays the same which follow the principle of fluidstatics. In a closed condition can be static when the fluid does not move and it can be dynamicwhen the fluid can move in a pipe. This follows the principle of fluid dynamics. The fluidpressure is a characteristic of the discoveries of Daniel Bernoulli. As the kinetic energy of thewater decreases, the pressure increases. When a cross sectional area of a pipe decreases, thekinetic energy of water increases leading to a decrease in pressure. This is called the BernoulliEffect.
  2. 2. The equation for the flow rate of water coming out of the hole is shown below: [2]Where v is the velocity of the water, Q is the flow rate of the water, and A is the cross sectionalarea of the hole. The equation for the pressure of the water inside the hole is shown below(derived from Bernoulli’s equation). [3]Where is the pressure inside the container, is the pressure outside the container, is thepotential energy outside the hole, is the pressure of the atmosphere, is the density of water, is thegravity at the surface of the water, is the height of water, and is the pressure of the water insidethe container.Design:Research Question:How does the water pressure inside a cylindrical container affect its water flow rate?Variables:The independent variable is the water pressure inside the cylindrical container which iscontrolled by the height of water in the container. The dependent variable is the water flow rate.The controlled variables in this experiment was the amount of time per trials (10 seconds) whichcan be measured using a stopwatch, the temperature of water using the same source of water, theheight where the experiment was done (constant acceleration or gravity), the cylindricalcontainer, and the density of water using the same source of water.
  3. 3. Materials and Procedure:Cut a small 5.7 millimeter diameter hole on the side 7 centimeters above the bottom of the watercontainer. Use modeling clay to close the hole. Fill the water into the container only up to the most toppart of the largest diameter. Make sure that the water does not leak out or push the clay out. Stick themeter stick into the middle of the container and measure the height of the water in the container. Start thestopwatch and pull the clay out at the same time. Use 1000 ml graduated cylinder to catch the waterthat’s leaking out of the hole. When the stopwatch reaches 10 seconds, close the hole on the watercontainer with the clay. Measure the height of the water in the container again and measure the height ofwater in the 1000 ml graduated cylinder. Empty out the water in the graduated cylinder. Repeat theseitalicize steps for the next five trials with different height of water in the container (ranging from 7 to 33centimeters). Water Container Hole in the water container 1000 ml Graduated CylinderFigure 1: Shows the experimental setup. Note that the diameter of the hole is 5.7 mm and the concavebump under the container is 3 cm high. The bottom of the container to the hole is 7 cm.Data Collection and Processing: Measured Height Attuned Average Height (cm) ±0.2 (cm) ±0.4 Initial Final Average 7.4 6.6 7.0 3.0 10.2 9.3 9.8 5.8 15.5 14.5 15.0 11.0 20.8 19.7 20.3 16.3 26.7 25.5 26.1 22.1 32.4 30.8 31.6 27.6Table 1: Shows the calculated heights of the water inside the container when doing the experiment. Notethat the calculated average height of water is found by subtracting the average height by 4 since there is asmall concave bump at the bottom of the container.
  4. 4. Change in Volume (cm^3) ±9 Trial 1 Trial 2 Trial 3 Average 201 203 198 201 274 287 276 279 385 399 387 390 474 471 464 470 545 541 528 538 598 593 583 591Table 2: Shows the average change in volume of the water that flows out of the hole. Calculated Average Flow Average Pressure Average Flow Rate Squared Rate Difference (Pa) ±40 (cm^3) ±9 (cm^6/s^2) ±100 290 20 400 560 28 800 1080 39 1520 1590 47 2200 2160 54 2900 2690 59 3500Table 3: Shows the calculated average pressure difference and the average flow rate of the water.Figure 2: Shows the quadratic relationship between the pressure and flow rate.
  5. 5. Figure 3: Shows the proportional relationship between the pressure and the flow rate squared.Figure 4: Shows the high-low fit for the pressure and flow rate squared graph according to figure 3. Therange of the two slopes is 0.1 with an uncertainty of . The range of the two y-intercepts is 150 withan uncertainty of
  6. 6. Sample Calculations:Finding the attuned average heightFinding the calculated average pressure difference:Finding the Uncertainty for the calculated average pressure difference:Finding the Uncertainty for the change in volume:
  7. 7. Finding the average flow rate of the water:Finding the uncertainty for the average flow rate of the water:Finding uncertainty of the average flow rate squared:Conclusion:The relationship of the equation between the pressure and the water flow rate squared accordingto figure 3 and figure 4 is shown below: [4]The equation 3 states that the relationship between the pressure and the water flow rate squaredshould be proportional to each other. The results clearly support this relationship. This showsthat the results in this experiment are highly confident because the line of best fit in figure 3 goesthrough all the data point in their uncertainties.
  8. 8. The slope of equation 4 is a constant and will remain constant even though the water flowrate squared changes. This is because the density of water ( ) and the area of the hole ( ) areconstant. The y-intercept should be zero because if the pressure difference is zero, then the waterflow rate squared should be zero too due to the fact that the water does not flow out the containerresulting in a directly proportional relationship.The limitation of this experiment only applies to water flowing out of a standard 5 gallon watercontainer with a small opening hole with a height of water ranging from 7 to 33 centimeters.Evaluation:A systematic error in this experiment is the kinetic energy of the water in the water container isassumed to be zero. As the water flows out of the container, the water in the container woulddecrease therefore the kinetic energy of the water cannot be zero. This could be improved byusing a high speed video camera to calculate the velocity of the water and the level of the water.A human error in this experiment is the time taken to close the hole. It is very hard to close thehole in an exact 10 seconds. There might be problems when closing the hole with modeling clay.This error could be improved by using a smaller hole and a larger time frame.Another error might be finding the pressure using the average height of the water. This is anaccurate way of measuring the pressure because if the height is measured in a more accurateway, then the height must be measured every one second instead of ten seconds. This error couldbe fixed by using a high speed video camera to measure the heights of water during each trialsand use this value to calculate the average height for ten seconds.