HYDROSTATICS LABORATORY

Transcription

1 HYDROSTATICS LABORATORY ME 13: Mechanical Engineering Laboratory II: Fluids Dr. J. M. Meyers Dr. D. G. Fletcher Dr. Y. Dubief

2 1. Introduction In this experiment, we will study the effects of hydrostatic pressure and buoyancy on a submerged object blocking a drain. The main objectives of this lab are to: Learn how to effectively collect data within a group environment Experimentally understand the concepts of hydrostatic pressure and buoyancy Compare theory with that of experiment Perform uncertainty analysis of reduced data and comment on the fidelity of your methodology. 1.1 Theory of experimental case When the ball is submerged and rested upon a drain of diameter,, there will be an atmospheric pressure force acting on the bottom face of the ball. An interesting phenomenon occurs as you increase the water depth. At some point the ball will remain fixed to the bottom of the tank! What is the buoyancy effect in this case? Figure 1: Hydrostatic test tube apparatus #1 (left) and # (right). Imagine the orifice diameter projected to a column of fluid above the ball with a height of. The liquid pressure force distribution (owing to ) to the left of this water column cancels out with the pressure force distribution to the right of the column. From Newton s first law we know that the sum of all of the forces on the ball equals zero: 0 With the ball submerged and blocking the drain, we can envision four main forces acting upon the ball:

3 1) Ball weight ( ) acting in the negative -direction ) Atmospheric pressure force ( ) acting below the ball 3) Force of the cylinder of water ( ) above the ball acting in the negative -direction 4) Buoyancy force ( ) acting in the positive -direction (but not due to the ball s complete volume!) From these forces, we create a free body diagram and force balance equation: + 0 At this point we will define R and r to be the radius of the ball and the radius of the orifice, respectively: The weight of the ball,, is the first force we ll consider is quite simple to derive. Force through orifice due to atmospheric pressure,, is determined from the atmospheric pressure acting on the orifice area:! " # Force due to water column acting on upper-surface of ball,, requires consideration of both water column height and atmospheric pressure: $#% ( +' ( ) ( +' ( )" # The force due to buoyancy of ball,, is not as straightforward. The most general definition, laid out by Archimedes states that a body immersed in a fluid experiences a vertical buoyancy force equal to the weight of the fluid it displaces. Here, the ball is not completely immersed in fluid, so what volume do we use?

4 Figure : Volume used for buoyancy force calculation. * * * + ' (,* -' (. 4 3 "1 " # Substituting individual force calculations into the force balance relation: ( +' ( )" # ' (,* * - + " # The theoretical minimum water depth, 3!, required to hold the ball in place is the water column height plus the ball diameter (and including and ): Expanding: " # +' ( ( 3! )" # ' (. 4 3 "1 " # + " # " # +' ( 3! " # ' ( " # ' ( 4 3 "1 ' ( " # + " # Rearranging and solving for water depth: 3! 4"' ( 1 3 3"' ( #

5 . Experimental Arrangement There are two hydrostatic experiment set-ups in the lab. Choose one or the other. The set-ups consist of a large plastic tube (about 3 to 4 ft. in height) mounted to a plastic base and sealed to water-tight criteria with two O-rings. At the center of the base is an open hole with a diameter of 9.53±0.05 mm. This will act as your orifice as described in the theory earlier in this handout. A second hole of similar diameter is also located on the base off of the center. Underneath there is a tube and ball valve connected to this hole. You will be testing three simple balls in your experiment: a table tennis ball, a squash ball, and a racquet ball. These balls were selected owing to their variation in diameters and weights. These balls are numbered and you will note which ball and number you are performing your experiments on. Buckets and a water hose are available to handle water delivery to the test apparatus. Be sure to write a concise, yet sufficiently detailed, explanation of how the different components of the experiment work as this information will be used to describe your experiment in your report. Figure 3: Hydrostatic test tube apparatus #1 (left) and # (right).

6 3. Measurements Measure the ball weights and diameters (each member does this once) Begin by deriving the theoretical height of water in tank where each ball lifts off Record all uncertainties of individual measurements Perform multiple measurements of each ball (at least three) Compare experimental measurements with theoretical results Discuss within your group the influence of ball characteristics, diameter of orifice, and measurement of water height on uncertainty on h from theory and experiment Set up data record sheet similar to the following (Do not throw this away. Retain your original data sheet and add it to your report!): Hydrostatics Lab Raw Data Sheet (Date) Group#: Section#: Group Members: Lab Monitor: Ball Table Tennis Racquet Squash Mass [g] ±σ m Mass [kg] ±σ m Diameter [mm] ±σ d Diameter [mm] ±σ d Orifice Diameter [mm] ±σ do Orifice Diameter [m] ±σ do Theoretical Liftoff Depth [m] ±σ hho Theoretical Liftoff Depth [dft] ±σ hho Hydrostatic Test Apparatus #: Test # AVG Water Depth [dft] ±σ h Table Tennis Ball Racquet Ball Squash Ball Figure 4: Example data record sheet

7 A manometer will be used to measure pressure along the Venturi and at other relevant locations during the experiment. At each rheostat setting, cycle through the manometer measurements AT LEAST 3 times. More data will not lead to extra effort in data reduction. For each cycle, one person will read the manometer level with a single transparent ruler (1, or 3). Measure the manometer level of each tube with a common base location for each tube. The fluid in the manometer will have a slight meniscus owing to capillary effects. Be sure to measure your manometer height at the same meniscus location for each reading. Make note of which ruler and stick with it and don t change as each ruler has a slightly different offset. Another person will record what the manometer reader measures to the raw data record. Third person can oversee tunnel (monitor Pitot-static probe orientation, for example). Students will rotate tasks once so that each has an opportunity to read the manometer. Compare the total pressure reading with that of the atmospheric pressure reading. What can you ascertain from this comparison? Retain original measured units on raw data sheet. Ultimately, all measured units will be put into common units of [kg], [m], and [s] in another column. Keep this original hand written data sheet (if used laptop recording of data is OK) as it will be added to final report. 4. Reduction of Data Lecture notes covering material for this laboratory have already been introduced. These notes along with literature and notes from ME 143 Fluids contain ample information regarding how to reduce your measured values of ball diameter and weight to a theoretical ball liftoff water depth. 5. Analysis and Discussion for Report You are also required to ascertain the uncertainty of liftoff water depth due to all error in measurements for all three balls. To do so you are required to develop a water depth uncertainty equation derived from all measured parameters for both methods. Recall the general expression for uncertainty as explained in your notes: # # +4 9: # 7<. 9 # +4 ; 9: # 7=. 9 # + +4 # 9: # 7?. 9 # 1 Using a measured parameter sensitivity analysis (described in Lecture 1), determine which measured values affect the precision of your measurements the most (and least). When presenting your results graphically, think about what is important to show and how can you most efficiently and effectively communicate you results in a single figure of representative data.