Grand Valley State University The Padnos School of Engineering DRAG COEFFICIENT OF A SPHERE EGR 365 FLUID MECHANICS Brad Vander Veen July 8 2003 Lab Partners Julie Watjer PURPOSE 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 Ball Writing a force balance Fw FB 1 2 A CD V t 2 1 where Fw is the weight FB is the buoyant force 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 is the fluid density g is gravitational acceleration and V is the displaced volume The expression for the weight force is FW mg 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 terminal velocity l m ln 19 A CD where l is the path length m is the mass is the fluid density A is the crosssectional area and CD is the drag coefficient APPARATUS 4 ITEM Water Tank Ping Pong Balls BB s Tape Stopwatch Meterstick Rubber band PROCEDURE 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 region of 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 0 0734 0 0781 0 0844 0 0884 0 0888 0 1072 3 93 2 31 2 20 1 75 1 69 1 63 1 34 Table 2 Experiment Results of Procedure The terminal velocity of the ball can also be plotted against the weight Velocity vs Weight 2 5000 term inal velocity ft s 2 0000 1 5000 1 0000 0 5000 0 0000 0 0600 0 0700 0 0800 0 0900 0 1000 0 1100 w eight lb Figure 3 Terminal Velocity vs Weight NOTE 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 0 0734 0 0781 0 0844 0 0884 0 0888 0 1072 3 93 2 31 2 20 1 75 1 69 1 63 1 34 0 7634 1 2987 1 3636 1 7143 1 7751 1 8405 2 2388 0 0056 0 0134 0 0181 0 0243 0 0284 0 0287 0 0471 0 8352 0 6937 0 8497 0 7237 0 7877 0 7408 0 8224 7753 13190 13849 17410 18028 18692 22737 Table 4 Calculated Values The drag coefficient CD can now be plotted versus the Reynolds number CD vs Re 100 0000 Experimental CD CD 10 0000 1 0000 0 1000 0 5000 10000 15000 20000 25000 Re Figure 5 Plot of CD versus Reynolds Number The 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 78 DISCUSSION In Figure 6 below the experimental drag coefficient is plotted along side the nominal drag coefficient for a sphere CD 0 45 CD vs Re 100 0000 Experimental CD Theoretical CD CD 10 0000 1 0000 0 1000 0 5000 10000 15000 20000 25000 Re Figure 6 Plot of Experimental and Theoretical CD The percent discrepancy between the experimental CD and the nominal CD is error 78 45 78 There 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 2 error 2 2 error 2 U f u t u d u l 4 4 4 f t d l 2 2 2 001 4 5 4 001 4 0625 0236 2 12 1 470 36 2 propagated error 240 Below in Figure 7 is a plot of CD with the experimental propagated error CD vs Re 100 0000 Experimental CD Theoretical CD CD 10 0000 1 0000 0 1000 0 5000 10000 15000 Re Figure Propagated error of CD CONCLUSION 20000 25000 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 42