- Question 1 A Falling Cylinder Viscometer Consists Of A Long Vertical Cylindrical Tube Of Radius R Capped At Both Ends 1 (116.95 KiB) Viewed 68 times
- Question 1 A Falling Cylinder Viscometer Consists Of A Long Vertical Cylindrical Tube Of Radius R Capped At Both Ends 2 (76.43 KiB) Viewed 68 times
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Question 1 A falling-cylinder viscometer consists of a long vertical cylindrical tube (of radius R), capped at both ends, and a solid cylindrical slug (of radius KR). The system is sketched in Fig. 1. The slug is equipped with fins so that its axis coincides with that of the tube. One can observe the rate of descent of the slug in the cylindrical container when the latter is filled with fluid. The objective of this exercise is to find an equation that gives the viscosity of the fluid (assumed to be incompressible and Newtonian) in terms of the terminal velocity V of the slug and of the various geometric quantities shown in Fig. 1. COO Cylindrical slug descends with speed through the liquid Liquid tills the cylindrical cavity Figure 1: Sketch of the falling-cylinder viscometer. Assume that the flow is laminar and neglect the entrance and end effects that are present when the fluid enters and leaves the gap between the tube and the slug. Therefore, assume that the only nonzero component of the Aluid velocity vector is v, and that in space this depends solely on r, where r denotes the radial coordinate of a cylindrical coordinate system. Answer all the questions in the reported order. Linear momentum balance equation: = (4*) d(rr) with P(x) = p(2) + Pgz (1.1) dr Here, AP is the (positive) dynamic pressure change over a generic length L, g is the magnitude of the gravitational field and p is the fluid density. (1.2) APR (1–82) - (1 – ?) n(1/8)] _ In(1/8) +1 'In(1/k)] In(1/K) where denotes the fluid viscosity and & = r/R. 4x[V]
In Eq. 1.2, the quantity AP/L is unknown; thus, for now the velocity profile is also unknown. But with a mass balance equation written on a macroscopic control volume appropriately chosen, it can be proved (the passages are quite lengthy) that this relation holds: APRP 1 (1.3) 4uLV (1 – K2) - (1+x2) ln(1/6) so that Eq. 1.2 becomes: (1 - 8) - (1+x2) In(1/8) +1 (1.4) (1 - R2) - (1+x2) In(1/K) 1) Using Eq. 1.4, calculate the frictional force in the axial direction) F, exerted by the fluid on the slug. In particular, show that it is: F = [1 =r=DER"] *](%.-P) 2 where !, -P, is the change in dynamic pressure between the bottom and top of the slug. Then, make a force balance on the slug and obtain an expression relating the fluid viscosity to the terminal velocity V of the slug and to the various geometric quantities characterizing the system. g) Using the expression derived in part f) with: p=950 kg/m ; Pa = 1500 kg/m3 ; R= 1 cm; k=0.95 ; V= : m/s calculate the viscosity of the fluid. Here, p, is the density of the slug •