Grand Valley State University The Padnos School of Engineering VISCOSITY LAB EGR 365 – FLUID MECHANICS Brad Vander Veen May 13, 2003 Lab Partners Julie Watjer Thomas Freundl PURPOSE: The purpose of this lab is to experimentally determine the absolute viscosity, , of a fluid using a Stormer Viscometer and to determine whether the fluid being tested is newtonian or non-newtonian. THEORY: The viscosity of a fluid is a transport property associated with the transport of momentum through that fluid via intermolecular collisions and intermolecular forces. Fluid immediately adjacent to a solid surface will move at the surface velocity. This is known as the no-slip boundary condition and is a direct result of momentum transfer between the fluid molecules and the solid surface. Moving away from the solid surface the fluid velocities can change indicating fluid shear stresses. Fluid shear stresses acting over the surface of a solid body result in fluid forces on that body. Consider Newton’s Second Law for rotating bodies: applied resist I (1) -where is the torque, I is the mass moment of inertia, and is the angular acceleration Now consider the experimental setup in Figure 1 below: Applying Equation (1) to the setup: dw (W rs ) viscous _ side viscous _ bottom I (2) dt Since the angular velocity is constant: (W rs ) viscous _ side viscous _ bottom (3) FIRST FIND viscous_side: It is known that: viscous _ side Fviscous _ side R (4) By definition: velocity Fviscous _ side A (5) clearance -where is the viscosity of the fluid, and A is the lateral surface area of the rotating cylinder Substituting: R Fviscous _ side 2 R L (6) hs Simplifying: R2 Fviscous _ side 2 L (7) hs Substituting (7) into (4): R2 viscous _ side 2 L R (8) hs Simplifying: R3 viscous _ side 2 L (9) hs NOW FIND viscous_bottom: It is known that: viscous _ bottom R dFviscous _ bottom (10) By definition: velocity dFviscous _ bottom dA (11) clearance -where A is a differential area (ring) on the bottom of the cylinder, and is the viscosity of the fluid Substituting (11) into (10): velocity viscous _ bottom R dA (12) clearance Simplifying: R viscous _ bottom R 2 R dR (13) hb Simplifying: 2 viscous _ bottom R dR 3 (14) hb Integrating: 2 R 4 viscous _ bottom (15) hb 4 Simplifying: 2 R 4 viscous _ bottom (16) 2hb The governing equations are now set up for the Viscometer: For Steady State: R3 2 R 4 (W rs ) 2 L (17) hs 2hb For Full, Unsteady State: R3 2 R 4 dw (W rs ) 2 L I (18) hs 2hb dt NOTE: In this lab the viscous torque from the bottom will be neglected because it is small compared to the torque created by the viscous torque on the side. APPARATUS: ITEM Stormer Viscometer Meter stick Stopwatch Calipers Various Masses PROCEDURE: 1). Measure all pertinent dimensions of the Stormer Viscometer. 2). Set the Viscometer on the edge of a table or something similar at a height of at least 2 meters off the ground. 3). Mark a distance of 1.5 meters off the ground. 4). Hang various masses on the string of the Viscometer, allowing the mass to reach terminal velocity before the 1.5-meter mark. Time how long it takes for the mass to go from a height of 1.5 meters to 0 meters (ground). RESULTS: Below is a list of the measured properties of the Stormer Viscometer: rs =30.10 mm hs =2.52 mm hb = 12.0 mm R = 57.65 mm L = 74.2 mm m = .95 kg NOTE: Each measurement above has the last significant digit after the decimal place estimated In Table 2 below, the results of the time trials can be seen. TRIAL MASS(kg) TIME(seconds) 1 0.05 66.3 2 0.10 32.6 3 0.20 16.1 4 0.30 12.7 Table 2 – Time Trial Results The distance traveled by the mass was 1.5 meters. In Table 3 below, the calculated velocity of the mass can be seen as well as the angular velocity of the spool. TRIAL VELOCITY (m/s) ANGLULAR VELOCITY (rad/s) 1 0.0226 0.7516 2 0.0460 1.5286 3 0.0932 3.0953 4 0.1181 3.9239 Table 3 – Velocity Values ANALYSIS: Using results from Table 2, the experimental viscosity can be found. Consider the following equation: R3 viscous _ side 2 L (9) hs By substituting W rs in for , and solving for : (rs W h s ) (19) R 3 2L The result of inserting velocity values into this equation is seen below in Table 4. TRIAL VISCOSITY (Ns/m) 1 0.5536 2 0.5444 3 0.5377 4 0.6362 Table 4 – Viscosity Values CONCLUSION: The experimental viscosity of the glycerin used in this lab experiment was found by averaging the results from trails one, two, and three. The fourth trial was excluded because of the difficulty in measuring the velocity of the falling mass. It was hard to measure the velocity because of the errors in starting and stopping the stopwatch on time. Since in the fourth trail, the mass was falling the fastest, the error in starting and stopping was amplified. Therefore, the experimental viscosity is as follows: 1 2 3 3 .55 Ns/m2 Since the experimental viscosity of the fluid did not change as we doubled, and even quadrupled the falling mass, it can be assumed that glycerin is Newtonian Fluid. ERROR CALCULATION: The propagated error in this lab is as follows: U rs U t 2 U hs 2 U U U error 4 ( )2 ( ) ( ) 9 ( R ) 2 ( L ) 2 ( L ) 2 rs t hs R L L .01 2 1 .01 2 .01 2 .1 2 .1 2 error 4 ( ) ( )2 ( ) 9( ) ( ) ( ) 60 .20 32 2.52 57 .65 1500 .0 74 .2 propagated error = .03 Therefore, the viscosity of the glycerin with error calculation is .55 .03 Ns/m. The recognized value of the viscosity of glycerin at room temperature is 0.8 Ns/m The experimental value found in this lab was .55 Ns/m. Therefore the discrepancy between the two is 0.25 Ns/m. This turns out to be a % error of 31.25%.
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