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Viscosity of Fluids Lab: Ball Drop Method Objectives Solidify the concept of viscosity through experimentation Test viscosities of different samples by measuring the velocity of a sphere falling through a fluid Introduction Viscosity is a fluid property that measures the resistance of a fluid to flow and can simply be thought of as the “thickness” of a fluid. Fluids that have a high viscosity, such as honey or molasses, have a high resistance to flow while fluids with a low viscosity, such as a gas, flow easily. The resistance to deformation within a fluid can be expressed as both absolute (or dynamic) viscosity, µ [Ns/m2], and kinematic viscosity, υ [m2/s]. Absolute viscosity is determined by the ratio of the shear stress to the shear rate of the fluid. The shear stress is dependent on the fluid’s resistance force to flow over the area of the plate while the shear rate is the equivalent to the fluid’s gradient. 𝐹 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝜏 µ= = = 𝐴 𝑠ℎ𝑒𝑎𝑟 𝑟𝑎𝑡𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝛿µ 𝛿𝑦 These relationships shown in the equation above can be seen pictorially in Figure 1. Figure 1: Friction between fluid and boundaries causes shear stress at a specific gradient. While absolute viscosity is able to quantifiably compare various liquids and gases on the same scale, it does not account for an important characteristic of fluids – the density (ρ). Kinematic viscosity (υ) is highly dependent on density and is measured by the time required for a specific volume of fluid to flow through a capillary or restriction. µ 𝜐= 𝜌 Applications of Viscosity Viscosity is an important concept that is taken into consideration in a variety of fields ranging from cooking to oil rigging. Understanding the applications of viscosity can help in both flow characterization and quality control. Quality Control Since raw materials must be consistent from batch to batch, flow behavior can be used as an indirect measure of product consistency and quality. As mentioned earlier, similar viscosities is indicative of similar flows. Viscosity has a direct effect on the ability to be processed. When designing pumping and piping systems, it should be known that a high viscosity liquid requires more power to pump than a low viscosity one. The Viscosity Index of a liquid measures how variations in temperature directly affect the viscosity of a fluid. Liquids whose viscosity is greatly dependent on temperature have a high viscosity index. This is an important characteristic of a good lubricant. Flow Characterization Rheology is the study of the flow of matter, primarily in the liquid state. The viscosity of a fluid helps predict whether the flow will be laminar or turbulent and it can be categorized accordingly. Viscosity helps explain the behavior of fluids; thus, once the behaviors are understood, they can be manipulated according to specific needs. Measuring Fluid Viscosity from Drag on an Immersed Body The drag force on an immersed body is in the direction of the flow; thus it works to retard the motion of a body through a fluid. The diagram below is a schematic of a sphere of radius a falling freely in a fluid. W gV b The weight of the sphere is , the buoyancy force is FB gV , and D represents the drag force acting on the sphere. Here is the density of the fluid, b is the density of the sphere, and V is the volume of the sphere. In the schematic, the sphere is assumed to have reached its terminal velocity Ut. When it is released into the fluid, it accelerates to the terminal velocity. Once this velocity is reached, it no longer accelerates and all the forces on the sphere are in equilibrium. D FB a Ut . W The drag force on immersed bodies with simple shapes can be correlated to the speed with which the body moves through the fluid. This is achieved by specifying the drag coefficient CD defined by CD drag D 1 inertial force 2 U 2 S , where D is the drag, is the density of the fluid, U is the speed of the fluid approaching the body, and S is the projected frontal area, i.e., the maximum area perpendicular to the flow direction. The subscript indicates “freestream” quantities, i.e. quantities that are measured in the undisturbed fluid far upstream of the body. In general, the overall drag force is composed of a component purely from friction and another component, called profile drag that results from the finite size and shape of the body. A number of experiments have been performed to determine CD for several geometries. These experiments show that the variation of CD depends primarily on a parameter called the Reynolds number Re, defined by Re inertial force U L viscous force , where L is some characteristic length (diameter in the case of the sphere) and the other quantities are as defined earlier. A flow with a relatively large value for Re is dominated by inertial forces, thus appears nearly inviscid. In the case of a very low-Re flow, called creeping flow or Stokes’ flow, the inertial forces can be neglected and Newton’s second law of motion reduces to Stokes’ equation for a sphere, valid for Re < 1, D 6 Ua . If the velocity (speed) V in this equation is the terminal velocity Ut of the sphere of radius a, it provides a means for computing the absolute viscosity by writing the equation for the balance of forces on the sphere, D FB W . Or substituting with Stokes’ equation, we have finally, W FB W FB 6 U t a 3 U t d , where d is the sphere diameter. In the following experiment, use this relation to compute and compare the viscosities of a few common liquids. Ball Drop Experiment This experiment uses one of the oldest and easiest ways to measure viscosity: we will simply see how fast a sphere falls through a fluid. The measurement involves determining the velocity of the falling sphere. This is accomplished by dropping each sphere through a measured distance of fluid and measuring how long it takes to traverse the distance. Thus, you know distance and time, so you also know velocity, which is distance divided by time. Additionally you will have to measure the mass and diameter of the sphere. The formula for determining absolute viscosity () is : VT 1 d 2 S F g 18 Where d = diameter of sphere S = density of sphere = m/V = (mass of sphere/volume of sphere) F = density of fluid = 1367g/m3 g = acceleration of gravity = 9.81 m/s2 VT = Terminal Velocity = D/t = (distance sphere falls)/(time of it takes to fall) Materials Thermometer Graduated Cylinders Airsoft BB balls Syringes Stopwatch Test Liquids (e.g. liquid soap, corn syrup, vegetable oil, motor oil, etc) Procedure 1. Measure the diameter and weight of a BB ball and compute the volume and density in Table 1.. Table 1: Properties of a BB ball. Value Units Diameter (d) Mass (m) Volume (V) Density (S) 2. Calculate the density of the liquid samples. Table 2: Properties of liquid samples. Cylinder # Liquid Product Weight of empty cylinder Weight of cylinder + liquid Density (F) 1 2 3 4 5 1. Drop a ball into the center of the cylinder and record time between timing marks. Repeat three trials for each fluid sample and record data in Table 3. a. Alternative timing method: Record video of ball drop and import in Logger Pro for video analysis to determine time. 2. Calculate the velocity for each drop time in Table 3. Table 3: Time of ball drop in each liquid sample. Cylinder # Liquid Product Trial 1 Ball Drop Time (sec) Distance traveled (mm) Velocity (m/s) Trial 2 Ball Drop Time (sec) Distance traveled (mm) Velocity (m/s) Trial 3 Ball Drop Time (sec) Distance traveled (mm) Velocity (m/s) 1 2 3 4 5 1. Plot the quantity in brackets from the absolute viscosity formula versus velocity. 2. Find the slope of the line to find the absolute viscosity of each sample. 3. Compute the kinematic viscosity of each sample. Questions/Deliverables 1. What characteristics are associated with a fluid that has a high-viscosity? Ones with a lowviscosity? 2. What are five occupations that have direct applications with fluid viscosity? 3. List three common fluids used every day in increasing order of viscosity. 4. In other experiments, it has been found that an increase of temperature in a liquid will decrease the viscosity. Oppositely, as the temperature of a gas increases, the viscosity also increases. Please give an explanation for these observations. References [1] http://www.coleparmer.com/TechLibraryArticle/933 [2] http://enterprise.astm.org/filtrexx40.cgi?+REDLINE_PAGES/D1545.htm [3] http://www.britannica.com/EBchecked/topic/630428/viscosity