Figure 2.1. Sketch of centrifugal pump.

Perhaps the best way to understand the way centrifugal pumps work is
to recall the mechanical energy balance equation. (In the present analysis,
this will be Bernoulli's equation, with the additional term of shaft, or
pump, work included.)

The mechanical energy balance (MEB) equation is:

(5-1)

Recall that this equation is valid, under steady-state conditions, along
a streamline of the liquid. Since we are dealing with a liquid of relatively
low viscosity, we can ignore the viscous term, lwf, and, with constant
density, the integral term simplifies, so the modified MEB equation (Bernoulli's
equation with shaft work included) becomes:

(5-2)

Referring to Figure 2.1, we see what amounts to a spiral fan, with the
inlet at the fan axis and the outlet in a line tangential to the tip of
the rotating fan. The fan, or impeller, is usually of uniform thickness
or width and is sandwiched in a casing formed by two parallel disks which
are sealed at the edges with a short, squat cylinder with one exit tangential
to the cylinder. In operation, the liquid enters along the axis, into the
"eye" of the impeller and both the pressure and velocity are
relatively low at this entrance. As the liquid travels through the pump,
the rotating impeller (previously, spiral fan) imparts shaft work to the
liquid as the liquid develops higher, mainly -direction,
velocity. As a matter of fact, within the pump,

v= 0er+ (Vr)(r/R)eq+ 0ez(5-3)

where VR is the -direction
speed of the liquid at the tip of the impeller (which should only be slightly
less than the actual tip speed, depending on the design of the pump) and
**e**r, **e**q, and **e**z are the orthogonal unit vectors in
the r, q, and z directions, respectively. The reason that v
varies linearly with r/R is geometric; think of the example of a rock attached
to a string as you swing the rock around your head-the -direction
velocity of the string at any radial position varies as r/R. This means
that the amount of motor power delivered to the impeller is concentrated
towards the outer parts of the impeller, because the shaft work is proportional
to the square of the velocity, from Equation (5-1). Getting the liquid
out to a radial position halfway towards the exit requires only one-fourth
of the power and two-thirds out still less than half (only 4/9) of the
power. At the tip of the rotating impeller, most of the pump power or shaft
work has gone into increasing the velocity of the liquid; the MEB equation
shows that this arises in the kinetic energy term. If the liquid encounters
no resistance downstream of the pump-let's say that it goes directly into
a vertical fountain with no additional change in cross-sectional area of
the exit tube-then very little pressure is "developed" by the
pump. Again, pursuing the fountain example, you will recall calculations
in physics about how the height reached by a ball thrown into the air can
be calculated from the conversion of kinetic energy at the bottom to potential
energy at the top, and you realize that the same thing could be done here.
You could, theoretically, calculate the impeller tip speed from the height
of the vertical fountain. Of course, you quickly realize that problems
with friction inside the pump, changing tube sizes, and friction with the
air would likely render the calculation too inaccurate. But you do realize
that the same principle can be applied here. In the typical setting, the
fluid does encounter resistance downstream of the pump. In the extreme,
the resistance is infinite (the valve is closed!) and now the pump is used
to develop pressure, instead of velocity. If we now apply the MEB equation
from the impeller entrance to a point just past the rotating impeller,
where the liquid velocity is zero, the change in kinetic energy, KE, is
zero, but the change in potential energy, P/r
is equal to the shaft work, Ws (per unit mass of liquid pumped).

Before we move on to some of the design equations with centrifugal pumps, we should briefly mention some other pumps for low-viscosity liquids:

Straight-through pumps: Axial fan and turbine pumps are typical of these units, although they are more commonly used with gases.

Piston/cylinder pumps: Multi-stroke units are used as metering pumps, where a constant flowrate, though pulsating is needed; single-stroke units are typified by infusion pumps, which are useful in medical applications where medicine has to titered into a patient over a sufficiently long time period that replenishing the stroke volume produces no problem.

Air-lift pumps: These produce vertical, two-phase flow and, although they are not energy efficient, are relatively cheap and can be used with highly corrosive liquids. Drip coffee makers work on this principle, moving the hot water to the top by vaporizing a portion of the water and the resulting steam bubbles rise in a tube, dragging and pushing the hot water upwards