DRAG COefficient of a sphereEGR 365 – FLUID MECHANICSPURPOSE:THEORY:APPARATUS:ITEMPROCEDURE:RESULTS:ANALYSIS:DISCUSSION:CONCLUSION:Grand Valley State UniversityThe Padnos School of EngineeringDRAG COEFFICIENT OF A SPHEREEGR 365 – FLUID MECHANICSBrad Vander Veen July 8, 2003Lab PartnersJulie WatjerPURPOSE:The purpose of this lab is to measure the drag coefficient of a sphere as a function of Reynolds number.THEORY:Consider the free body diagram of the submerged ping-pong ball below in Figure 1:Figure 1 – FBD of Ping Pong BallWriting a force balance: FwFB12 A CD Vt2 (1)-where Fw is the weight, FB is the buoyant force, ρ is the fluid density, A is the cross-sectional area, CD is the drag coefficient, and Vt is the terminal velocity of the ball in the fluid.The expression for the buoyant force is: FBgV (2)-where ρ is the fluid density, g is gravitational acceleration, and V is the displaced volume.The expression for the weight force is: FWmg (3)-where m is the mass of the ball, and is gravitational acceleration.The following equation gives the distance it will take the ball to get to 90% of its terminalvelocity:lm ln 19( ) A CD (4)-where l is the path length, m is the mass, ρ is the fluid density, A is the cross-sectional area, and CD is the drag coefficient.APPARATUS:ITEM Water Tank Ping Pong Balls BB’sTapeStopwatchMeterstickRubber bandPROCEDURE:1). Choose some weighted ping-pong balls with various masses (BB’s) added.2). Using Equation (4) and the mass of the heaviest ball, calculate the acceleration regionof the ball. Then place a rubber band at that distance down from the surface of the water on the outside of the water tank.3). Place another rubber band at a distance down from the first rubber band.4). Drop a ball into the tank. Start timing when the ball reaches the first rubber band, and stop timing when the ball reached the second rubber band.5). Run this time trial for each ball, and record the weight and time of each trail.RESULTS:In Table 2 below, the results of the procedure can be seen:weight [lb] time [s] 0.0656 3.930.0734 2.310.0781 2.200.0844 1.750.0884 1.690.0888 1.630.1072 1.34Table 2 – Experiment Results of ProcedureThe terminal velocity of the ball can also be plotted against the weight:Velocity vs. Weight0.00000.50001.00001.50002.00002.50000.0600 0.0700 0.0800 0.0900 0.1000 0.1100w eight [lb]term inal velocity [ft/s]Figure 3 – Terminal Velocity vs. WeightNOTE: The curve is concave down due to the drag force being proportional to the velocity squared!ANALYSIS:Using Equation (1), (2), and (3), the following values can be calculated:weight [lb] time [s] velocity [ft/s] force_down [lb] CD Re 0.0656 3.93 0.7634 0.0056 0.8352 77530.0734 2.31 1.2987 0.0134 0.6937 131900.0781 2.20 1.3636 0.0181 0.8497 138490.0844 1.75 1.7143 0.0243 0.7237 174100.0884 1.69 1.7751 0.0284 0.7877 180280.0888 1.63 1.8405 0.0287 0.7408 186920.1072 1.34 2.2388 0.0471 0.8224 22737Table 4 – Calculated ValuesThe drag coefficient (CD) can now be plotted versus the Reynolds number:CD vs Re0.10001.000010.0000100.00000 5000 10000 15000 20000 25000ReCDExperimental CDFigure 5 – Plot of CD versus Reynolds NumberThe drag coefficient for this range of Reynolds number can be found by taking an average of the values above. This yields a drag coefficient of CD = 0.78DISCUSSION:In Figure 6 below, the experimental drag coefficient is plotted along side the nominal drag coefficient for a sphere. (CD = 0.45)CD vs Re0.10001.000010.0000100.00000 5000 10000 15000 20000 25000ReCDExperimental CDTheoretical CDFigure 6 – Plot of Experimental and Theoretical CDThe percent discrepancy between the experimental CD and the nominal CD is:%error.78 .45.78There was also propagated error in this lab experiment. The error was primarily in the timing of the ping-pong ball, but there was also some other measurement errors. Below is the calculation of propagated error.errorUff24utt2 4udd2 4ull2error.001.023624.52.122 4.0011.4702 4.0625362propagated error = .240Below in Figure 7 is a plot of CD with the experimental propagated error:CD vs Re0.10001.000010.0000100.00000 5000 10000 15000 20000 25000ReCDExperimental CDTheoretical CDFigure – Propagated error of CDCONCLUSION:In this lab, a sphere was dropped into a fluid and was allowed to reach terminal velocity. Using this terminal velocity and a force balance, an experimental drag coefficient was found. This drag coefficient was found to 0.78. This drag coefficient disagreed with the published drag coefficient by a percent error of