ENES100_0702
Prof. R. Phaneuf
Fall 2002
(Click here to return to list of ENES100 documents)
Water Pumps
Pumps are devices that cause the motion of a fluid, usually by generating a change in
pressure. (An exceptional case is the chain of pots described in section 2.2 of your
textbook) Most pumps use mechanical motion to produce this change in pressure. The
motion is usually repetitive, and can be either reciprocating or rotary.
Reciprocating Pumps
Reciprocating pumps can be broken down into two main subcategories,
bellows/diaphragm type pumps and piston pumps. Both of these types of pumps use a
change in the volume of the pump chamber to produce a lower pressure than that of the
reservoir to draw water in, and a larger pressure than that at the receptacle to force the
water out. Both require inlet and outlet valves to restrict fluid flow to be in one direction
only.
Valves
The most common type of valve used is a check valve, in which a movable element ,
shaped like a cone or ball is positioned opposite a cylindrical tube called a seat.
Normally the movable element held away from the seat either by the force of gravity or
by a spring. However, if the pressure force on the same side of the movable element
exceeds that on the opposite side by more than the weight of spring force, the element is
pushed against the seat, closing the valve, and blocking the flow of fluid through it. If the
pressure force is greater on the opposite side, the element is forced back, and fluid can
flow. Spring valves have the advantage that they can be used in any position, while
gravity valves must be oriented downwards. So called flapper valves are also used in
some cases (for example in toilet tanks), and have the advantage of being very simple. A
disadvantage is that they tend to seal more slowly, and thus allow more fluid to leak
through before sealing.
Provided that the minimum volume of the chamber during the inlet stroke is extremely
small compared to its maximum volume, full retraction of the movable part will provide a
nearly complete vacuum so that the distance through which the water can be lifted from
the reservoir to the pump will be close to the maximum, h=Patm/(g). Providing that the
height of the pump above the reservoir hpr is less than this maximum value the volume
will fill with water until the pressure at the inlet line, balances atmospheric pressure.
When a force is applied to the actuator to decrease the chamber volume it this imparts an
additional pressure to the water causing the inlet valve to close and the outlet valve to
open. This pressure causes water to rise in the outlet line to a maximum height H=(Papp-
Patm)/(g). Water will flow out of the outlet line provided the difference in height
between the catch basin and pump hcp is less than this value. (Note that the overall
maximum height difference is the sum of the two or H=Papp/(g) as it should beatmosphere acts both at the reservoir and at the catch basin.) We’ll discuss the function
of piston pumps first. Much of what we say also applies to diaphragm pumps.
Priming
Piston pumps need to be “primed” at startup. The inlet line is immersed into the
reservoir, forming an effective seal, however initially it will be full of air. Retracting the
piston will increase the volume of the pump chamber decreases the air pressure within it,
causing air to flow from the inlet line into the pump chamber to equilibrate the pressure
in the two. This decreases the pressure at the air/water interface defined by the inlet line,
so that water will be forced up into the tube by the higher pressure acting on the reservoir.
This is spoken of as “suction”, but remember that the water is pushed from the high
pressure side, not pulled from the low pressure side. At the end of the inlet stroke, the
water will rise to a point where the combined weight of the water in the line and the air
pressure within the chamber balance atmospheric pressure. Next the outlet stroke begins,
with the applied force being reversed. This pushes closed the inlet valve and pushes
open the outlet valve, as air is pushed out of the pump chamber into the outlet line. On
the next complete stroke the process is continued, with the level of the water in the inlet
line increasing, due to the smaller amount of air. After multiple strokes the inlet line and
pump housing are completely primed (i.e. purged of air), and the pump begins operating
in steady state, drawing only water from the inlet line to fill the volume defined by the
cylinder wall, head, and retracted piston on the inlet stroke, and forcing a volume of
water equal to the difference in chamber volumes out into the outlet line during the outlet
stroke, independent of the stroke rate. This property is referred to as “positive
displacement”.
Requirements for a Piston Pump
Note that the discussion on priming assumes an air-tight seal between the inlet line and
the inlet valve. A leak here will result in air leaking in, allowing the level of the water in
the inlet line to drop when this valve closes, during the outlet stroke. A leak through the
valve will result in an even more rapid drop, as air is forced back through the valve into
the inlet line.
Requirements on the piston pump include a tight, sliding seal between the piston and the
cylinder. A leak here will result in air infiltrating into the pump housing during the inlet
stroke, and water leaking past the cylinder during the outlet stroke. In a single action
design, this would cause water to collect behind the piston, impeding its motion on the
inlet stroke. A tight seal requires that the inside of the cylinder be smooth, round and
uniform. (Pistons and housings are always chosen to be cylindrical to avoid needing a
fixed relative azimuthal-angle orientation-in addition it is possible to grind cylindrical
surfaces to be precisely round and uniform.) The seal must be made of a resilient
material to allow it to conform closely to the inside of the cylinder, and fill slight
imperfections. It must also slide easily back and forth within the cylinder to limit the loss
of input power due to frictional forces. O-ring seals are often used commercially.
Standard radial O-ring seals rely on an interference fit between the outer diameter of the
O-ring and the inner diameter of the cylinder. Disadvantages of a standard radial O-ring
seal are that it requires precise machining, and that the amount of interference cannot be
adjusted. Some level of adjustability can be added by trapping an O-ring into a groove
whose length can be changed by making a piston consisting of a shouldered body and a
front plate, which can be drawn back against the body, compressing the O-ring along the
axis of the piston, and causing push out more tightly against the cylinder. This results
extra complexity during fabrication. Alternatively, a cup seal can be made using a rubber
or leather gasket which wraps around the edge of the piston and is held in place by a
backup plate.
The materials for the pump housing and piston need to be sufficiently stiff to withstand
the differences in pressure they will be subject to during operation without distorting
significantly. It must also be possible to join the head to the cylinder, and the valves in
such a way that the joints are mechanically strong, and leak free. The piston shaft must
also be quite stiff to avoid deflecting appreciably under the forces that the operator will
be applying. This will also affect the choice of its dimensions.
Diaphragm Pumps
For diaphragm (or bellows or bladder) pumps the deformation of a flexible element
results initially in an increase in a sealed volume defined by the diaphragm and the pump
chamber, and thus a decrease in the gas pressure from that of the surrounding
atmosphere. This type of pump must be primed as is the case for a piston pump.
Some things to note about diaphragm pumps:
(1) The diaphragm must be sufficiently flexible so as to allow it to stretch and conform to
the shape of the chamber without cracking or tearing. It will be cycled back and forth
between two extreme positions many times, and so the material must allow this without
failing due to fatigue.
(2) The maximum pressure applicable with the diaphragm will be limited by the strength
of the diaphragm. The pressure will also cause the diphragm to distort, limiting to what
extent moving the plunger to one or the other extreme position will result in it being
stretched tight about the walls of the container.
(3) Diaphragms are typically made from an elastomer, usually a thick piece of rubber to
allow deformability, and also to act as a seal between the halves of the pump chamber.
(4) The diaphragm must be mounted securely to the plunger, either using an adhesive, or
trapping it between the plunger disk and a backup plate. A backup plate however can
increase the minimum volume attainable with the plunger completely advanced, reducing
the
(4) Fabricating the chamber is difficult, compared to some other pump designs. It must
be made in two parts which bolt together, around the edges of the diaphragm, which
makes the seal.
Differences between piston and diaphragm pumps
Diaphragm pumps are less likely to leak than piston pumps as the seal is static rather than
sliding.
Piston pumps can generally produce much larger differences in chamber volume between
maximum and minimum for a given applied force. That is because the change in the
dimension of a diaphragm is limited by the largest strain beyond which it yields and tears.
By contrast, a piston can have a stroke length much larger than its diameter. To get an
equivalent average volumetric flow from a diaphragm pump at a given area requires that
it be cycled at a higher rate.
Rotary Pumps
The most commonly used rotary pump is the centrifugal type. It consists of a collection
of curved vanes, called an impeller, which rotate at a high angular velocity. Water enters
at the center of the pump, and is directed around and outward by the vanes. The force
applied by a moving vane to a water molecule imparts a momentum to it which after the
interaction has a radially outward component. This is referred to as a “centrifugal force”,
it isn’t really an outward force of course, it’s simply that the force sets the water into
motion along a which at the instant of interaction is along the circumfrential direction,
but as the orientation of the vane continues to change, there is a component of the
velocity which is not parallel to the applied force, and is thus unaffected by the
subsequent interaction.
Although it is possible to build an operational centrifugal pump with vanes which extend
outward radially (the “Boston Pump” was an example), this configuration tends to stir the
water more causing turbulence, and resulting in less efficiency. Having vanes which are
“swept-back” increases the efficiency with which the water is directed outwards from the
center. The outward motion of the water results in a decrease in the pressure at the
center, so that water is forced by atmosphere though the inlet line and into the pump.
The impeller is enclosed within a housing called the volute. Water which is forced out of
the impeller continues around the space between it and the volute wall. The impeller
passes in close proximity to the volute at the point at which the outlet tube, called the
diffuser begins. This forces the water into the diffuser, rather than reentering the impeller
where there is an extra dynamic pressure due to the water which is exiting it.
The space between the impeller and volute wall flares outward, as does the diffuser.
From the incompressibility of water and the resulting constancy of the volumetric flow,
the velocity must decrease inversely with the increasing cross sectional area. From
Bernoulli’s equation, this results in an in crease in the pressure.
Since the motion of the pump is always in the same sense, inlet and outlet valves are not
required to control the water flow direction. A valve is typically used at the base of the
inlet line to keep the water from flowing down into the reservoir if the motion of the
impeller is halted however.
Unlike the reciprocating pumps discussed above, the volume of water pumped per cycle
is not fixed. A throttle valve can be installed after the pump, decreasing the flow without
halting the pump. The excess water merely continues to move around the pump. It is
thus not a positive displacement pump.
The efficiency of a centrifugal pump increases with rotational rate. The measured
efficiency is usually given in terms of a combination of rotational rate, volumetric flow
and head referred to as the “specific speed”
N s  (qv ) / h 0.75 ,
0.5
(1)
with n equal to the impeller rotation rate (in rotations/minute), qv the volumetric flow
rate (in gallons/minute) and h the head (in feet). It clearly does not have the dimensions
of velocity, and so the name speed is deceptive. Its magnitude would be that of the
rotational rate (although in different units!) if the flow rate were 1 gallon/minute and the
head were 1 ft.
The reason for using this combination of quantities comes from a dimensional analysis of
the power which could be supplied by an ideal, frictionless pump. Experimentally, for a
given design (impeller vane shape, number of vanes, volute shape) the power is found to
be a function of the density of the fluid , the rotational rate of the impeller , and the
overall size, measured by the diameter of the impeller D. If it is assumed that the power
can be written as a product of these quantities raised to different powers, dimensional
analysis allows the individual exponents to be determined.
P  C a b D c ,
(2)
where C is a constant of proportionality. Power has units of force times velocity, while
density has units of mass/volume, rotational rate has units of 1/time and head has units of
length, thus:
 [mass][length]  [length]   [mass] 



  
3 
[time] 2

 [time]   [length] 
a
b
 1 
c

 [length]
 [time] 
(3)
By equating the exponents for [mass], a=1. By equating the exponents for [length], 2=c3a, so that c=5. By equating the exponents for [time], -3=-b, or b=3. So that
P  C 3 D 5
(4)
But the total power supplied by the pump is proportional to the volumetric flow rate
times the head. The flow rate should clearly increase both with the impeller rotational
rate and the diameter of the impeller. Again, based on dimensional arguments, and based
on an assumption of a power-law dependence we can work out the exponents:
q v  D 3
(5)
so that comparing these two equations, and using the proportionality of the power and
the product of the flow rate times the total head:
h   2 D 2 ,
(6)
and solving both of these proportionalities for D3:
D 3   3 / h 3 / 2  qv /
(7)
Therefore it is possible to define a constant ratio:
 2 qv
h 3 / 2
 C
(8)
and its square root is also a constant, and independent of the size of the pump (but of
course not of the shape and number of vanes, shape of the volute,…)
 qv1 / 2
h 3 / 4
 Ns
(9)
Requirements for Centrifugal Pumps
High efficiency requires high rotational rates. This means supporting the impeller with a
quality bearing to minimize frictional power losses. It also means using a step up
transmission to change the relatively low rotational rates that a human operator can
generate to rates of several hundred RPM.
The volute must have smooth inner walls.
The space between the impeller and volute should flare out from the “lip” to the
“diffuser”, to decrease the water speed and increase the pressure.
(Click here to return to list of ENES100 documents)