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Unformatted text preview: mam/a Experimental Study of Wall Effect in Viscosity _‘ W3) WM ‘ The falling sphere technique used commonly in flu'a' mechanics involves viscous frictional forces beyond those predicted by traditional ow analysis, commonly referred W to as the wall effect. The Dinsdale and Moore ' model is traditionally accepted to Cornell University \ Grant Corley, Albert Ho Ithaca, NY 14853 Abstract Co(r€C1[(‘<r/f account for this wall effect, but has not been an quately tested for viscosities in the W\‘ “ regime of 1—20 poise. Due to this insufficiency, a different model has been proposed by . «HA/9 Schottenheimer that uses an exponential relationshiwmt viscosities in this range, ’\ P l’ Laboratory measurements of glycerin viscosity for sph es of different densities and radii X” fail to verify the recently proposed Schottenheimer theory. I n r/R (“M e x law”) I In many cases where viscous drag is influential, it is helpfiil to have an accurate W numerical value for fluid viscosity. A common viscosity measurement method is the falling sphere technique. However, this method yields apparent viscosity measurements that are uncorrected for viscous frictional forces beyond those predicted by traditional flow analysis, a result commonly referred to as the wall effect. The Dinsdale and Moore COCO @model is traditionally accepted to account for this wall effect, but has not been 1.16” an equately tested for viscosities in the regime of 1-20 poise. Due to this insufficiency, a My! different model has been proposed by Schottenheimer that uses an exponential ,- ad‘féc relationship to predict viscosities in this range. Using the falling sphere technique to find n IC 6 4 the apparent viscosity of glycerin, a fluid in the regime of 1-20 poise, we can test both H6015 " theories against experimental results. SWAT“a . , The falling sphere technique is based upon finding the terminal velocity of a sphere falling through a liquid and using that velocity to calculate an apparent viscosity W (uncorrected for the wall effect). This is accomplished by measuring the time it takes a sphere of known density and diameter to traverse known lengths of the fluid. The W viscometer apparatus consists of a vertical glass cylinder filled with glycerin, into which 0 spheres of differing densities and diameters are dropped. Steel and glass spheres of various sizes are used in this partilcaiilarcemiflipe’rirnéern}a kThe cylinder is equipped with four ' 9}/ sets of optical sensors placed at eve long its length. These allow for 9,9 detection of an object impeding light transfer between sensors. Each sphere is dropped individually into the cylinder and allowed to disrupt each of the sensors. The sensors’ outputs are then used (by—computer—pregram/ to calculate the time between each sensor trigger. The program calculates velocities of each sphere by dividing the known distance ’ \3‘)’ V between optical sensors by the time it takes to traverse each of those distances. In this A , y particular experiment, four optical sensors and three equally spaced gaps are used. Thus ‘0" l,- a total of three velocities are obtained for each sphere. As it can be shown that terminal a3" “° 0 /\ RY velocity is reached well in advance of the first optical sensor (which is situated well below the glycerin free surface), all three measured velocities can be considered terminal. Thus, the terminal velocity for an individual sphere to be used in further calculations is} (inc, ( 41‘ taken to be the average of its three measured velocities. ' Once the terminal velocities for the different spheres are determined, the corresponding apparent Viscosities of the glycerin, uncorrected for the wall effect, are easily obtained from a simple force balance. Buoyancy (Pb) and drag (Fd) forces act upward on a falling 'h— sphere (of radius r) and, at terminal velocity (Vt), are exactly counteracted by the [r gravitational (Fg) force: Fb + Fd = Fg (1) Concise (4/3)m3pg1ycefin g + 671% = (4/3)1rr3psphmg (2) - ~ The expression for the drag force (Fd) arises from our assumption that all flow is laminar, + not turbulent; that is, all flow falls into the Stokes flow regime. This assumption can be #53le J 7% Qay’” -. easily checked by verifying that the Reynolds number for each sample is below unity, Wni’infl . and indeed our data meets this requirement (with a maximum Reynolds number of 0.54). k W Thus apparent viscosity (u) for each sphere is obtained by simply rearranging equation (2). whgmo/ W2, {’th7 “‘ In order to determine the “true viscosity” of the glycfl average apparent viscosity ([931. over 3 trials is first determined for the smallest sphere diameter (1/8 inch). This value I (u(r/R)) is then used to calculate the “true viscosity” 11(0) by solving for 11(0) in both the traditional Dinsdale and Moore1 expression and the recently proposed Schottenheimerz ‘ 1 expression and averaging the results (where R is the radius of the cylinder): ”3490 Schottenheimer: p.(r/R) = 11(0) exp(3.8 r/R) (3) u)! Dinsdale and Moore: u(r/R) = 11(0) [1 — 2.104(r/R) + 2.09(r/R)3 — O.95(r/R)5]'1 (4) W The average 11(0) from equations (3) and (4) is used as the “true viscosity” of the glycerin a) due to the near—convergence of their results for small values of r/R. [/ 6/6 The apparent Viscosities for the spheres of differing radii and densities must be corrected to account for the additional frictional forces of the wall effect. This effect arises from fluid being displaced under the influence of a falling sphere, bouncin_ off the c linder, 0 and reboundin_ to exert an additional force on the Su-her As expected, this leads to €00 , an narent Viscosities that ., ,- , er than the true fluid Viscosity. Our ana ysis griffin-55 assumes an exponential variance of viscosity with sphere radius, as suggested by the recent work of Schottenheimerzz “(r/R) = M0) 6XP(1< r/R) (5) By working with the logarithm of equation (5) and using a least squares regression approach, k was determined to be 3.02, and error analysis based on the fimdamental / equation of error propagation yielded an uncertainty in k (Ak) of 0.08. l an, I’D Figure 1 depicts our data (apparent viscosity normalized by ”(0) versus r/R) along with our exponential best fit (equation (5) with k = 3.02 i 0.08) and the values predicted by the Dinsdale and Moore equation as well as the Schottenheimer equation. Both the least squares regression and the Dinsdale and Moore equation fit our data well, but the Schottenheimer equation predicts apparent viscosities which are too large for any given sphere size. This stems from the fact that that expression (equation (3)) uses a larger value for the exponential constant k (3.8 versus our 3.02). As a result of our analysis, we conclude that the Dinsdale and Moore theoretical model that accounts for the wall effect can be used to accurately predict apparent viscosity, but that Schottenheimer’s recently proposed theoretical model can be used only if its exponential constant of 3.8 is replaced with a value of 3.02 d: 0.08. References 1. A. Dinsdale and F. Moore, Viscosity and its Measurement, New York, NY, 1962. 2. M. Schottenheimer, Proceedings of the International Conference on Fluid Dynamics, San Francisco, CA, March 2006. Viscometer Laboratory Data: The Wall Effect 5.5 0 Raw Data -—— Linear Regression Best Fit of Log(Schottenheimer) Schottenheimer Equation Fit -—~ Dinsdale and Moore Equation Fit U1 P- m 4:. 3.5 MuirfR) Normalized by Mum) 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 riR 30> ' FIG. 1. Apparent glycerin viscosities showing wall effect for different sphere radii. ' Theoretical plots shown as disconnected lines, best fit of measured values as smooth line. 0?; :5 8 \g. Kl s \“S § 194/ W W1» 5 W” We!) ...
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