Measuring Viscosity with a Stormer Viscometer by Dan Schwarz School of Engineering Grand Valley State University EGR 365 Fluid Mechanics Section 01 Instructor Dr S Fleischmann May 29 2007 Outline I II Purpose Statement a The viscosity of glycerin was experimentally determined using a Stormer Viscometer Background a Viscosity is a constant value that relates the shear stress applied at a fluid boundary to the transfer of velocity through the fluid b In this experiment shear stress will be applied to glycerin by a submerged cylinder spinning within a cylindrical container Glycerin at the boundary of the spinning cylinder will move at the same speed as the outer cylinder wall Adjacent layers of fluid will move at gradually slower velocities in accordance with the viscosity of the Glycerin c The experimental device shown in Figure 1 provides a means of measuring the viscosity of glycerin when the cylinder rotates at a constant speed due to the falling weight W rs 0 0298 5m R 0 0573 5m W L 0 142 6m Hs 0 0026 m hb 0 006 m Figure 1 Stormer Viscometer Dimensions in meters d The torque applied to the inner cylinder by the weight is given by Equation 1 T rsW 1 e The viscosity of the glycerin resists the applied torque along the side surface of the rotating cylinder This torque is given by Equation 2 See Appendix A for details R 3 2 L Tside 2 hs f The viscosity of the glycerin also resists the applied torque along the bottom surface of the rotating cylinder This torque is given by Equation 3 See appendix B for details R 4 Tbottom 3 2hb g The ratio of the resistive torque at the side and the resistive torque at the bottom is approximately 23 Consequently it was determined that the resistive torque at the bottom of the cylinder is negligible See Appendix C for details h At constant velocity the applied torque balances the resistive torque from the side of the cylinder This torque balance can be solved for to obtain Equation 4 See Appendix D for details rhW 3s s 4 R 2 L i For experimental purposes it is necessary to determine the distance the weight must fall before it reaches terminal velocity The angular acceleration of the system is given by Equation 5 See Appendix E and Appendix F for details rsW d R 3 2 L 5 dt 0 5m ro2 ri 2 0 5m ro2 ri 2 hs j For simplicity Equation 5 can be written in the form given by Equation 6 See Appendix G for details d C C 6 dt k Equation 6 can be integrated to obtain the angular velocity of the cylinder as a function of time The results are given by Equation 7 d C C tC e 7 dt C C l The path length required for the falling weight to reach terminal velocity is given by Equation 8 See Appendix H for details r C ln 10 Lter min al s 8 C 2 m Experimental Method i Equation 8 was used to determine the path length required for each of the test weights to reach terminal velocity ii The Stormer viscometer was filled with glycerin and placed on the edge of a high surface iii A meter stick was fastened beneath the viscometer to measure the height of the weight as it falls to the floor iv The smallest weight was connected to the fish line III v The spool of fish line was wound up until the bottom of the weight was at the top of the meter stick vi The weight was released and the clock was started as soon as the weight traveled the required path length for terminal velocity vii The weight was allowed to drop 0 5 meters before the timer was stopped viii Steps iv through vii were repeated until all of the weights were tested ix The linear velocity of the falling weight was calculated by dividing the fall distance by the time duration x The angular velocity of the cylinder was calculated from the linear velocity of the falling weight xi The viscosity of the glycerin was obtained using Equation 4 Results Discussion a The maximum free fall distance required for the weight to reach terminal velocity was less than a millimeter as shown in Table 1 Consequently the terminal path was considered negligible when collecting data from the experimental procedure Table 1 Terminal path lengths required for each test weight Mass kg Terminal Path m 0 100 0 00007 0 070 0 00011 0 050 0 00021 0 040 0 00028 b Table 2 shows the data taken during the experimental procedure The data shows that the terminal velocity decreases as the mass of the test weights decrease Table 2 Experimental data Mass kg Time s Fall Distance m 0 100 11 90 0 500 0 070 17 80 0 500 0 050 28 87 0 500 0 040 36 99 0 500 c Table 3 compares the published viscosity value of glycerin with the experimental values The discrepancy decreases significantly as the mass of the test weights decrease Thus light weights produce more accurate viscosity measurements Table 3 Comparison of experimental and published viscosity values Published Viscosity Experimental Mass kg Discrepancy Error Ns m Viscosity Ns m 0 100 1 500 0 320 369 2 0 070 1 500 0 335 348 2 0 050 1 500 0 388 287 2 0 040 1 500 0 398 277 2 d The error propagation for this procedure was calculated by Equation 9 Table 3 shows that the error propagation was approximately 2 This indicates that the measuring devices were appropriate for obtaining accurate calculations of the viscosity of glycerin ur 4 s rs u IV V 2 2 2 2 2 u t u hs u R u L u Lter min al 9 t h R L Lter min al s 2 9 Conclusions a The extremely high discrepancy between the experimental and published values indicates that there may have been a calculation error b Another possible cause of the high discrepancy is that the glycerin has absorbed water This would decrease the viscosity of the glycerin since water has a very low viscosity c The low percentage of error propagation suggests that the measuring devices were sufficiently accurate to produce good result Thus experimental error was most likely not a contributing factor to the high discrepancies d Due to the high discrepancy this method of calculating viscosity using a Stormer Viscometer has not been proven valid Appendices a Appendix A Resisting Torque From the Side of Viscometer i Start with general torque equation Tside RFv ii Substitute the product of shear stress and area for the viscous force Tside R A iii Substitute the definition of fluid shear stress Tside R du A dy iv Substitute the linear velocity and gap size for the differential terms Tside R R A hs v Substitute the surface area of the outer cylinder wall R 3 2 L Tside hs b Appendix B Resisting Torque From the Bottom of Viscometer i Start with …