- 1.1 Fluid Mechanics in Chemical Engineering
- 1.2 General Concepts of a Fluid
- 1.3 Stresses, Pressure, Velocity, and the Basic Laws
- 1.4 Physical Properties-Density, Viscosity, and Surface Tension
- 1.5 Units and Systems of Units
- 1.6 Hydrostatics
- 1.7 Pressure Change Caused by Rotation
- Problems for Chapter 1

## 1.7 Pressure Change Caused by Rotation

Finally, consider the shape of the free surface for the situation shown in Fig. 1.20(a), in which a cylindrical container, partly filled with liquid, is rotated with an angular velocity *ω*—that is, at *N* = *ω/*2*π* revolutions per unit time. The analysis has applications in fuel tanks of spinning rockets, centrifugal filters, and liquid mirrors.

*Fig. 1.20 Pressure changes for rotating cylinder: (a) elevation, (b) plan.*

Point O denotes the origin, where *r* = 0 and *z* = 0. After a sufficiently long time, the rotation of the container will be transmitted by viscous action to the liquid, whose rotation is called a *forced vortex.* In fact, the liquid spins as if it were a *solid body*, rotating with a uniform angular velocity *ω*, so that the velocity in the direction of rotation at a radial location *r* is given by *v _{θ}* =

*rω*. It is therefore appropriate to treat the situation similar to the hydrostatic investigations already made.

Suppose that the liquid element P is essentially a rectangular box with cross-sectional area *dA* and radial extent *dr*. (In reality, the element has slightly tapering sides, but a more elaborate treatment taking this into account will yield identical results to those derived here.) The pressure on the inner face is *p*, whereas that on the outer face is *p* + (*∂p/∂r*)*dr*. Also, for uniform rotation in a circular path of radius *r*, the acceleration toward the center O of the circle is *rω*^{2}. Newton’s second law of motion is then used for equating the net pressure force toward O to the mass of the element times its acceleration:

Note that the use of a *partial* derivative is essential, since the pressure now varies in both the horizontal (radial) *and* vertical directions. Simplification yields the variation of pressure in the radial direction:

so that pressure *increases* in the radially outward direction.

Observe that the gauge pressure at all points on the interface is zero; in particular, *p*_{O} = *p*_{Q} = 0. Integrating from points O to P (at constant *z*):

However, the pressure at P can also be obtained by considering the usual hydrostatic increase in traversing the path QP:

Elimination of the intermediate pressure *p*_{P} between Eqns. (1.45) and (1.46) relates the elevation of the free surface to the radial location:

Thus, the free surface is *parabolic* in shape; observe also that the density is not a factor, having been canceled from the equations.

There is another type of vortex—the *free* vortex—that is also important, in cyclone dust collectors and tornadoes, for example, as discussed in Chapters 4 and 7. There, the velocity in the angular direction is given by *v _{θ}* =

*c/r*, where

*c*is a constant, so that

*v*is inversely proportional to the radial position.

_{θ}**Example 1.8—Overflow from a Spinning Container**

A cylindrical container of height *H* and radius *a* is initially half-filled with a liquid. The cylinder is then spun steadily around its vertical axis Z-Z, as shown in Fig. E1.8. At what value of the angular velocity *ω* will the liquid just start to spill over the top of the container? If *H* = 1 ft and *a* = 0.25 ft, how many rpm (revolutions per minute) would be needed?

*Fig. E1.8 Geometry of a spinning container: (a) at rest, (b) on the point of overflowing.*

*Solution*

From Eqn. (1.47), the shape of the free surface is a parabola. Therefore, the air inside the rotating cylinder forms a paraboloid of revolution, whose volume is known from calculus to be exactly one-half of the volume of the “circumscribing cylinder,” namely, the container.^{8} Hence, the liquid at the center reaches the bottom of the cylinder *just* as the liquid at the curved wall reaches the top of the cylinder. In Eqn. (1.47), therefore, set *z* = *H* and *r* = *a*, giving the required angular velocity:

For the stated values: