Hydrostatic equilibrium in a centrifugal field Ref: McCabe, Smith, Harriott

In a rotating centrifuge a layer of liquid is thrown outward from the axis of rotation and is held against the wall of the bowl by centrifugal force. The free surface of the liquid takes the shape of a paraboloid of revolution, but in industrial centrifuges the rotational speed is so high and the centrifugal force so much greater than the force of gravity that the liquid surface is virtually cylindrical and coaxial ith the axis of rotation. This situation is illustrated in Fig. 1, in which r is the radial distance from the axis of rotation to the free liquid surface and r is the radius if the centrifuge bowl. The entire mass of liquid indicated in Fig. 2.2 is rotating as .t rigid body, with no sliding of one layer of liquid over another. Under these conditions the pressure distribution in the liquid may be found from the principles of fluid statics.

FIG 1. Single liquid in centrifuge bowl

The pressure drop over any ring of rotating liquid is calculated as follows. Consider the ring of liquid shown in Fig. 2.2 and the volume element of thickness dr at a radius r:

dF = ω2 r dm

where dF = centrifugal force

dm = mass of liquid in element; ω = angular velocity, rad/s

If ρ is the density of the liquid and b the breadth of the ring,

dm = 2πr ρ r b dr

Eliminating dm gives

dF = 2πr ρ ω2 r2 dr

The change in pressure over the element is the force exerted by the element of liq uid, divided by the area of the ring:

dp = dF/(2πr r b) = ω2 ρ r dr

The pressure drop over the entire ring is

p2 — p1 = ∫r ω2 ρ r dr

Assuming the density is constant and integrating gives

p2 – p1 = ω2 ρ (r22 –r1

2)/2

Equation (2.8) strictly applies only when r1 and r2 are not greatly different, but for practical systems the error is small.

Centrifugal decanter When the difference between the densities of the two liquids is small, the force of gravity may be too weak to separate the liquids in a reasonable time. The sepa ration may then be accomplished in a liquid-liquid centrifuge, shown diagrammat ically in Fig. 2. It consists of a cylindrical metal bowl, usually mounted vertically, that rotates about its axis at high speed. In Fig. 2a the bowl is at rest and contains a quantity of two immiscible liquids of differing densities. The heavy liquid forms a layer on the floor of the bowl beneath a layer of light liquid. If the bowl is now rotated, as in Fig. 2b, the heavy liquid forms a layer, denoted as zone A in the fig ure, next to the inside wall of the bowl. A layer of light liquid, denoted as zone B, forms inside the layer of heavy liquid. A cylindrical interface of radius r, separates the two layers. Since the force of gravity can be neglected

FIG. 2 Centrifugal separation of inmiiscible liquids: (a) bowl at rest; (b) bowl rotating. Zone A, separation of light liquid from heavy; zone B, separation of heavy liquid from light. (1) Heavy-liquid drawoff. (2) Light-liquid drawoff.

in comparison with the much greater centrifugal force, this interface is vertical. It is called the neutral zone.

In operation of the machine, the feed is admitted continuously near the bottom of the bowl. Light liquid discharges at point 2 through ports near the axis of the bowl; heavy liquid passes under a ring, inward toward the axis of rotation, and dis charges over a dam at point 1. If there is negligible frictional resistance to the flow of the liquids as they leave the bowl, the position of the liquid-liquid interface is established by a hydrostatic balance and the relative “heights” (radial distances from the axis) of the overflow ports at 1 and 2.

Assume that the heavy liquid, of density ρA, overflows the dam at radius rA, and the light liquid, of density ρB, leaves through ports at radius rB. Then if both liquids rotate with the bowl and friction is negligible, the pressure difference in the light liquid between rB and ri must equal that in the heavy liquid between rA and ri The principle is exactly the same as in a continuous gravity decanter.

Thus

pi — pB = pi — pA

where pi = pressure at liquid-liquid interface, pB = pressure at free surface of light liquid at rB, pA = pressure at free surface of heavy liquid at rA

From above

pi – pB = ω2 ρΒ (ri2 –rB

2)/2 and pi – pA = ω2 ρΑ (ri2 –rA

2)/2

Equating these pressure drops and simplifying lead to

pB (ri2 –rB

2) = pA (ri2 –rA

2)

Solving for ri gives

ri = { [ rA2 – (ρΒ /ρΑ)rB

2] / ( 1 - ρΒ /ρΑ)}1/2

This equation is analogous to a similar equation for a gravity settling tank. It shows that r, the radius of the neutral zone, is sensitive to the density ratio, especially when the ratio is nearly unity. If the densities of the fluids are too nearly alike, the neutral zone may be unstable even if the speed of rotation is sufficient to separate the liquids quickly. The difference between ρΑ and ρΒ should not be less than approximately 3 percent for stable operation.

The equation also shows that if rB is held constant and rA, the radius of the discharge lip for the heavier liquid, is increased, then the neutral zone is shifted toward the wall of the bowl. If TA is decreased, the zone is shifted toward the axis; An increase in rB, at constant rA, also shifts the neutral zone toward the axis; and a decrease in r causes a shift toward the wall. The position of the neutral zone is important practically. In zone A, the lighter liquid is being removed from a mass of heavier liquid; and in zone B, heavy liquid is being stripped from a mass of light liquid. If one of the processes is more difficult than the other, more time should be pro vided for the more difficult step. For example, if the separation in zone B is more difficult than that in zone A, zone B should be large and zone A small. This is accomplished by moving the neutral zone toward the wall by increasing rA or decreasing rB. To obtain a larger time factor in zone A, the opposite adjustments would be made. Many centrifugal separators are so constructed that either rA or rB can be varied to control the position of the neutral zone.

Flow through continuous decanters Equations for the interfacial position in continuous decanters are based entirely on hydrostatic balances. As long as there is negligible resistance to flow in the outlet pipes, the position of the interface is the same regardless of the rates of flow of the liquids and of the relative quantities of the two liquids in the feed. The rate of separation is the most important variable, for as mentioned be fore, it fixes the size of a gravity decanter and determines whether a high centrifugal force is needed. The rates of motion of a dispersed phase through a continuous phase are discussed in Chap. 7 of McCabe, Smith and Harriott.