Drag Coefficient of a Sphere EGR 365 Lab 6b Performed by Lee C Groeneweg Professor Dr Fleischmann Outline I Purpose To measure drag coefficient of a sphere as a function of Reynolds number II Background A Drag is Primarily form drag Fv A CD V2 B Force balance on a sphere Figure 1 C Force balance equation Fw Fb A CD Vt2 III Procedure A Equation to determine drag coefficient CD mg gVol AVt2 B Table of values used data IV Results A Experimental B Theoretical C Propagated error assessment V Conclusions Purpose To measure the drag coefficient of a sphere as a function of Reynolds number Background When an object falls in a viscous fluid it experiences a gravity force Fw a buoyant force Fb and a viscous drag force Fv The gravity force is constant and acts upwards and the buoyant force is also constant and acts upwards The viscous force acts against the direction of motion and is an increasing function of the speed of the object The drag force is primarily form drag and can be written as Fv A CD V2 1 Figure 1 shows a free body diagram that is used in determining the force balance on the sphere Figure 1 FBD for force balance on sphere Considering a sphere that sinks in a viscous fluid Fw Fb if it is released from rest it will accelerate until the viscous plus buoyant force balance the gravity force At this point a force balance yields Fw Fb A CD Vt2 2 If the weight of the sphere in the fluid is known if the density of the fluid is known and the sphere dimensions are know then the drag coefficient CD can be determined from the force balance In this experiment the drag coefficient of the sphere was determined at terminal velocity Staring with a force balance on the sphere it is possible to show that the velocity as a function of time with the initial condition being at rest is V Fw Fb A CD 1 e 2t m A CD Fw Fb 1 e 2t m A CD Fw Fb 3 The derivation for this equation is attached as Appendix A1 Procedure Before any data could be collected the path length to reach terminal velocity had to be determined Using equation 4 assuming CD 0 45 the path length could be estimated L0 9Vt m ln 19 A CD 4 Solving equation 2 for CD yields the following equation used to experimentally determine the drag coefficient CD CD mg gVol AVt2 5 Table 1 contains the data values used to determine the CD for three differently weighted spheres Mass 31 5 g 28 5 g 29 7 g Time avg 2 56 s 3 26 s 2 81 s Length 1 0414 m 1 0414 m 1 0414 m Diameter 37 35 mm 37 35 mm 37 35 mm Area 0 0011m2 0 0011m2 0 0011m2 Volume 2 75e 3 m3 2 75e 3 m3 2 75e 3 m3 density 1000 kg m3 1000 kg m3 1000 kg m3 Table 1 Data to calculate CD using equation 5 A Results Table 2 contains the values for the drag coefficients using equation 5 and the Reynolds number Re VtD CD Experimental Theoretical CD1 0 4522 0 55 CD2 0 210 0 55 CD3 0 311 0 55 Table 2 Experimental and Theoretical Values for CD For this lab since there were many measurable data values propagated error was introduced and had to be calculated The propagated error was found to be 0 88 The calculation for the propagated error is found in Appendix A4 Conclusions As shown in Table 2 there is a discrepancy in the 2nd and 3rd values for the drag coefficient It is worth noting that the initial entrance length was determined using a CD equal to 0 45 which is agreement for the first value of CD There may have been some error in the data collection that may have propagated through the results but with an error propagation calculated to be 0 889 it may not have that large of an effect The theoretical values may be off slightly since they were read from a log log graph in the textbook with a limited amount of readability and accuracy