2.10.4. Forced VortexA body of fluid, contained in a vessel, which is rotating about a vertical axis withuniform angular velocity, will eventually reach relative equilibrium and rotate with the sameangular velocity w as the vessel, forming a forced vortex. The acceleration of any particular offluid at radius r due to rotation will be (-w2r) perpendicular to the axis of rotation, taking thedirection of r as positive outward from the axis. Thus, from Equ. (2.34),

dp= -w2 rdr

Rotation | Rotating VesselTags:centrifugal forceangular velocityparaboloidsquared property of parabolacentripetal accelerationslope of paraboloidvolume of paraboloidWhen at rest, the surface of mass of liquid is horizontal at PQ as shown in the figure. When this mass of liquid is rotated about a vertical axis at constant angular velocityradian per second, it will assume the surface ABC which is parabolic. Every particle is subjected to centripetal force or centrifugal forceCF = m2xwhich produces centripetal acceleration towards the center of rotation. Other forces that acts are gravity forceW = mgand normal forceN.

Where tanis the slope at the surface of paraboloid at any distance x from the axis of rotation.From Calculus, y = slope, thus

For cylindrical vessel of radius r revolved about its vertical axis, the height h of paraboloid is

Other FormulasBy squared-property of parabola, the relationship of y, x, h and r is defined by

Volume of paraboloid of revolution

Important conversion factor

.Example 1.8Overflow from a Spinning ContainerA cylindrical container of height H and radius a is initially half-filled with aliquid. The cylinder is then spun steadily around its vertical axis Z-Z, as shownin Fig. E1.8. At what value of the angular velocity _ will the liquid just start tospill over the top of the container? If H = 1 ft and a = 0.25 ft, how many rpm(revolutions per minute) would be needed?

7.94 A rotary mixer consists of two 1-mlonghalf-tubes rotating around a centralarm, as in Fig. P7.94. Using the drag fromTable 7.2, derive an expression for thetorqueT required to drive the mixer atangular velocity in a fluid of density .Suppose that the fluid is water at 20C andthe maximum driving power available is20 kW. What is the maximum rotationspeedr/min?Fig. P7.94Solution: Consider a strip of half-tube of width dr, as shown in Fig. P7.94 above. Thelocal velocity is U r, and the strip frontal area is Ddr. The total torque (2 tubes) is

7.96 A Savonius rotor (see Fig. 6.29b)can be approximated by the two open halftubesin Fig. P7.96 mounted on a centralaxis. If the drag of each tube is similar tothat in Table 7.2, derive an approximateformula for the rotation rate as afunction of U, D, L, and the fluidpropertiesand .Solution: The analysis is similar toProb. 7.91 (the cup anemometer). At anyarbitrary angle as shown, the net torquecaused by the relative velocity on eachhalf-tube is set to zero (assuming africtionless bearing):