Standard Methods for Examination of Water & Wastewater_3 potx

These topics include pumping stationsand various types of pumps; total developed head; pump scaling laws; pump char-acteristics; best operating efficiency; pump specific speed; pumping s

Pumping is a unit operation that is used to move fluid from one point to another.This chapter discusses various topics of this important unit operation relevant to thephysical treatment of water and wastewater These topics include pumping stationsand various types of pumps; total developed head; pump scaling laws; pump char-acteristics; best operating efficiency; pump specific speed; pumping station heads;net positive suction head and deep-well pumps; and pumping station head analysis

4.1 PUMPING STATIONS AND TYPES OF PUMPS

The location where pumps are installed is a pumping station There may be only onepump, or several pumps Depending on the desired results, the pumps may be con-nected in parallel or in series In parallel connection, the discharges of all the pumpsare combined into one Thus, pumps connected in parallel increases the dischargefrom the pumping station On the other hand, in series connection, the discharge ofthe first pump becomes the input of the second pump, and the discharge of the secondpump becomes the input of the third pump and so on Clearly, in this mode ofoperation, the head built up by the first pump is added to the head built up by thesecond pump, and the head built up by the second pump is added to the head built

by the third pump and so on to obtain the total head developed in the system Thus,pumps connected in series increase the total head output from a pumping station byadding the heads of all pumps Although the total head output is increased, the totaloutput discharge from the whole assembly is just the same input to the first pump Figure 4.1 shows section and plan views of a sewage pumping station, indicatingthe parallel type of connection The discharges from each of the three pumps areconveyed into a common manifold pipe In the manifold, the discharges are added

As indicated in the drawing, a manifold pipe has one or more pipes connected to it.Figure 4.2 shows a schematic of pumps connected in series As indicated, thedischarge flow introduced into the first pump is the same discharge flow coming out

of the last pump

The word pump is a general term used to designate the unit used to move a fluidfrom one point to another The fluid may be contaminated by air conveying fugitivedusts or water conveying sludge solids Pumps are separated into two general classes:the centrifugal and the positive-displacement pumps Centrifugal pumps are thosethat move fluids by imparting the tangential force of a rotating blade called an

impeller to the fluid The motion of the fluid is a result of the indirect action of theimpeller Displacement pumps, on the other hand, literally push the fluid in order

to move it Thus, the action is direct, positively moving the fluid, thus the name

positive-displacement pumps In centrifugal pumps, flows are introduced into the

4

unit through the eye of the impeller This is indicated in Figure 4.2 where the “Q in”line meets the “eye.” In positive-displacement pumps, no eye exists.

The left-hand side of Figure 4.3 shows an example of a positive-displacementpump Note that the screw pump literally pushes the wastewater in order to move

it The right-hand side shows a cutaway view of a deep-well pump This pump is acentrifugal pump having two impellers connected in series through a single shaftforming a two-stage arrangement Thus, the head developed by the first stage isadded to that of the second stage producing a much larger head developed for thewhole assembly As discussed later in this chapter, this series type of connection isnecessary for deep wells, because there is a limit to the depth that a single pumpcan handle

Figure 4.4 shows various types of impellers that are used in centrifugal pumps.The one in a is used for axial-flow pump Axial-flow pumps are pumps that transmitthe fluid pumped in the axial direction They are also called propeller pumps, becausethe impeller simply propels the fluid forward like the movement of a ship with propellers.The impeller in d has a shroud or cover over it This kind of design can develop more

FIGURE 4.1 Plan and section of a pumping station showing parallel connections.

FIGURE 4.2 Pumps connected in series.

Screen

Float tube

Float tube

Suction well

Rising main

Centrifugal pumps

Sluice valve

Sluice valve

Reflux valve

A

Q in

Impeller eye

Q out

(a) (b)

(c) (d)

Shroud

head as compared to the one without a shroud The disadvantage, however, is that it

is not suited for pumping liquids containing solids in it, such as rugs, stone, and thelike, because these materials may easily clog the impeller

In general, a centrifugal impeller can discharge its flow in three ways: by directlythrowing the flow radially into the side of the chamber circumscribing it, by con-veying the flow forward by proper design of the impeller, and by a mix of forwardand radial throw of the flow The pump that uses the first impeller is called a radial- flow pump; the second, as mentioned previously, is called the axial-flow pump; andthe third pump that uses the third type of impeller is called a mixed-flow pump Theimpeller in c is used for mixed-flow pumps

Figure 4.5 shows various impellers used for positive-displacement pumps andfor centrifugal pumps The figures in d and e are used for centrifugal pumps Thefigure in e shows the impeller throwing its flow into a discharge chamber thatcircumscribes a circular geometry as a result of the impeller rotating This chamber

is shaped like a spiral and is expanding in cross section as the flow moves into theoutlet of the pump Because it is shaped into a spiral, this expanding chamber iscalled a volute—another word for spiral In centrifugal pumps, the kinetic energythat the flow possesses while in the confines of the impeller is transformed intopressure energy when discharged into the volute This progressive expansion of thecross section of the volute helps in transforming the kinetic energy into pressureenergy without much loss of energy Using diffusers to guide the flow as it exits

FIGURE 4.5 Various types of pump impellers, continued: (a) lobe type; (b) internal gear type; (c) vane type; (d) impeller with stationary guiding diffuser vanes; (e) impeller with volute discharge; and (f) external gear type impeller.

Outlet seal hereInternal

Internal seal here Inlet

from the tip of the impeller into the volute is another way of avoiding loss of energy.This type of design is indicated in d, showing stationary diffusers as the guide.The figure in a is a lobe pump, which uses the lobe impeller A lobe pump is apositive-displacement pump As indicated, there is a pair of lobes, each one havingthree lobes; thus, this is a three-lobe pump The turning of the pair is synchronizedusing external gearings The clearance between lobes is only a few thousandths of

a centimeter, thus only a small amount of leakage passes the lobes As the pair turns,the water is trapped in the “concavity” between two adjacent lobes and along withthe side of the casing is positively moved forward into the outlet The figures in b

and f are gear pumps They basically operate on the same principle as the lobepumps, except that the “lobes” are many, which, actually, are now called gears.Adjacent gear teeth traps the water which, then, along with the side of the casing,moves the water to the outlet The gear pump in b is an internal gear pump, socalled because a smaller gear rolls around the inside of a larger gear (The smallergear is internal and inside the larger gear.) As the smaller gear rolls, the larger gearalso rolls dragging with it the water trapped between its teeth The smaller gear alsotraps water between its teeth and carries it over to the crescent The smaller and thelarger gears eventually throw their trapped waters into the discharge outlet The gearpump in f is an external gear pump, because the two gears are contacting each other

at their peripheries (external) The pump in c is called a vane pump, so called because

a vane pushes the water forward as it is being trapped between the vane and the side

of the casing The vane pushes firmly against the casing side, preventing leakage backinto the inlet A rotor, as indicated in the figure, turns the vane

Fluid machines that turn or tend to turn about an axis are called turbomachines.Thus, centrifugal pumps are turbomachines Other examples of turbomachines areturbines, lawn sprinklers, ceiling fans, lawn mower blades, and turbine engines Theblower used to exhaust contaminated air in waste-air works is a turbomachine

4.2 PUMPING STATION HEADS

In the design of pumping stations, the engineer must ensure that the pumping systemcan deliver the fluid to the desired height For this reason, energies are convenientlyexpressed in terms of heights or heads The various terminologies of heads are defined

in Figure 4.6 Note that two pumping systems are portrayed in the figure: pumps nected in series and pumps connected in parallel Also, two sources of the water arepumped: the first is the source tank above the elevation of the eye of the impeller orcenterline of the pump system; the second is the source tank below the eye of the impeller

con-or centerline of the pump system The flow in flow pipes fcon-or the first case is shown bydashed lines In addition, the pumps used in this pumping station are of the centrifugal type.The terms suction and discharge in the context of heads refer to portions of thesystem before and after the pumping station, respectively Static suction lift h is thevertical distance from the elevation of the inflow liquid level below the pump inlet

to the elevation of the pump centerline or eye of the impeller A lift is a negativehead Static suction head h s is the vertical distance from the elevation of the inflowliquid level above the pump inlet to the elevation of the pump centerline Static discharge head h d is the vertical distance from the centerline elevation of the pump

to the elevation of the discharge liquid level Total static head h st is the verticaldistance from the elevation of the inflow liquid level to the elevation of the dischargeliquid level Suction velocity head h vs is the entering velocity head at the suctionside of the pump hydraulic system This is not the velocity head at the inlet to apump such as points a, c, e, etc in the figure In the figure, because the velocity inthe wet well is practically zero, h vs will also be practically zero Discharge velocity head h vd is the outgoing velocity head at the discharge side of the pump hydraulicsystem Again, this is not the velocity head at the discharge end of any particularpump In the figure, it is the velocity head at the water level in the discharge tank,which is also practically zero

4.2.1 T OTAL D EVELOPED H EAD

The literature has used two names for this subject: total dynamic head or total oped head (H or TDH) Let us derive TDH first by considering the system connected

devel-in parallel between podevel-ints 1 and 2 Sdevel-ince the connection is parallel, the head lossesacross each of the pumps are equal and the head given to the fluid in each of the pumpsare also equal Thus, for our analysis, let us choose any pump such as the one withinlet g From fluid mechanics, the energy equation between the points is

(4.1)

where P, V, and z are the pressure, velocity, and elevation head at the indicated points;

g is the acceleration due to gravity; h f is the head equivalent of the resistance loss(friction) between the points; h q is the head equivalent of the heat added to the flow;

Pumping station

Pump centerline

Pumping station

A 1

1

2 B

hd

hd hd

P1

γ - V1

and h p is the head given to the fluid by the pump impeller Using the level at point 1

as the reference datum, z1equals zero and z2 equals h st It is practically certain that

there will be no h q in the physical–chemical treatment of water and wastewater, and

will therefore be neglected Let h f be composed of the head loss inside the pump h lp,

plus the head loss in the suction side of the pumping system h fs and the head loss in

the discharge side of the pumping system h fd Thus, the energy equation becomes

(4.2)

The equation may now be solved for −h lp+h p This term is composed of the

head added to the fluid by the pump impeller, h p, and the losses expended by the

fluid inside the pump, h lp As soon as the fluid gets the h p, part of this will have

to be expended to overcome frictional resistance inside the pump casing The fluid

that is actually receiving the energy will drag along those that are not This

dragging along is brought about because of the inherent viscosity that any fluid

possesses The process causes slippage among fluid planes, resulting in friction

and turbulent mixing This friction and turbulent mixing is the h lp The net result

is that between the inlet and the outlet of the pump is a head that has been

developed This head is called the total developed head or total dynamic head

(TDH) and is equal to −h lp + h p

Solving Equation (4.2) for −h lp + h p, considering that the tanks are open to the

atmosphere and that the velocities at points 1 and 2 at the surfaces are practically

zero, produces

(4.3)

When the two tanks are open to the atmosphere, P1 and P2 are equal; they, therefore,

cancel out of the equation Thus, as shown in the equation, TDH is referred to as

TDH0sd In this designation, 0 stands for the fact that the pressures cancel out.

The s and d signify that the suction and discharge losses are used in calculating

TDH

The sum h fs + h fd may be computed as the loss due to friction in straight runs

of pipe, h fr , and the minor losses of transitions and fittings, h fm Thus, calling the

corresponding TDH as TDH0rm (rm for run and minor, respectively),

(4.4)From fluid mechanics,

(4.5)

(4.6)

P1

γ - V1

where f is Fanning’s friction factor, l is the length of the pipe, D is the diameter of the pipe, V is the velocity through the pipe, g is the acceleration due to gravity

(equals 9.81 m/s2) and K is the head loss coefficient is called the velocity

head, h v That is,

(4.7)

If the points of application of the energy equation is between points 1 and B,instead of between points 1 and 2, the pressure terms and the velocity heads will

remain intact at point B In this situation, refering to the TDH as TDHfullsd ( full

because velocities and pressure are not zeroed out),

(4.8)

where z2 is the elevational head of point B, referred to the chosen datum at point 1

Note that P atm is the pressure at point 1, the atmospheric pressure When the friction

losses are expressed in terms of h fr + h fm and calling the TDH as TDH fullrm, the

4.2.2 I NLET AND O UTLET M ANOMETRIC H EADS ; I NLET

AND O UTLET D YNAMIC H EADS

Applying the energy equation between an inlet i and outlet o of any pump produces

(4.10)

where TDHmano (mano for manometric) is the name given to this TDH h fs + h fd is

2

2g

+

γ V i

2

2g

+

manometric height absolute; is also called either the outlet manometric

head or the outlet manometric height absolute The subscripts i and o denote “inlet”

and “outlet,” respectively

Manometric head or level is the height to which the liquid will rise whensubjected to the value of the gage pressure; on the other hand, manometric heightabsolute is the height to which the liquid will rise when subjected to the true orabsolute pressure in a vacuum environment The liquid rising that results in themanometric head is under a gage pressure environment, which means that the liquid

is exposed to the atmosphere The liquid rising, on the other hand, that results inthe manometric height absolute is not exposed to the atmosphere but under a

complete vacuum Retain h as the symbol for manometric head and, for specificity, use h abs as the symbol for manometric height absolute Thus, the respective formulasare

(4.11)

(4.12)

P g is the gage pressure and P is the absolute pressure Unless otherwise specified,

P is always the absolute pressure.

h vo for the pump inlet and outlet velocity heads, respectively, TDH, designated asTDHabs, may also be expressed as

(4.13)

Note that h abs is used rather than h h is merely a relative value and would be a

mistake if substituted into the above equation

For static suction lift conditions, h i is always negative since gage pressure isused to express its corresponding pressure, and its theoretical limit is the negative

of the difference between the prevailing atmospheric pressure and the vapor pressure

of the liquid being pumped If the pressure is expressed in terms of absolute pressure,

then h abs has as its theoretical limit the vapor pressure of the liquid being pumped Because of the suction action of the impeller and because the fluid is beinglifted, the fluid column becomes “rubber-banded.” Just like a rubber band, it becomesstretched as the pressure due to suction is progressively reduced; eventually, theliquid column ruptures As the rupture occurs, the inlet suction pressure will actuallyhave gone down to equal the vapor pressure, thus, vaporizing the liquid and forming

bubbles This process is called cavitation.

Cavitation can destroy hydraulic structures As the bubbles which have beenformed at a partial vacuum at the inlet gradually progress along the impeller towardthe outlet, the sudden increase in pressure causes an impact force Continuous action

of this force shortens the life of the impeller

P o/γ = h o

h P g

γ -

=

h abs

P

γ -

The sum of the inlet manometric height absolute and the inlet velocity head is

called the inlet dynamic head, idh (dynamic because this value is obtained with fluid

in motion) The sum of the outlet manometric height absolute and the outlet velocity

head is called the outlet dynamic head, odh Of course, the TDH is also equal to

the outlet dynamic head minus the inlet dynamic head

of Celsius and Fahrenheit are expedient or relative measures This is unfortunate,

since oftentimes, it causes too much confusion; however, these relative measureshave their own use, and how they are used must be fully understood, and the results

of the calculations resulting from their use should be correctly interpreted If fusion results, it is much better to use the absolute measures

con-Example 4.1 It is desired to pump a wastewater to an elevation of 30 m above

a sump Friction losses and velocity at the discharge side of the pump system areestimated to be 20 m and 1.30 m/s, respectively The operating drive is to be 1200 rpm.Suction friction loss is 1.03 m; the diameter of the suction and discharge lines are

250 and 225 mm, respectively The vertical distance from the sump pool level to

the pump centerline is 2 m (a) If the temperature is 20°C, has cavitation occurred?

(b) What are the inlet and outlet manometric heads? (c) What are the inlet and outlet

total dynamic heads? From the values of the idh and odh, calculate TDH

2g

+

-V12

2g

- z1 P1

γ - h q–h f h p

=

4 - π 0.225 2

4 - 0.040 m2

0.049 - 1.059 m/s

Therefore, 0+ + +0 0 0–1.03+0 2 9.81 -1.059( 2) 2 P2

γ -+ +

=

= −3.087 m; because the pressure used in the equation is 0, this valuerepresents the manometric head to the pump

At 20°C, P v (vapor pressure of water) = 2.34 kN/m2

= 0.239 m of waterAssume standard atmosphere of 1 atm = 10.34 m of water

Therefore, theoretical limit of pump cavitation = −(10.34 − 0.239) = −10.05 m

−3.087

Cavitation has not been reached Ans

(b) Inlet manometric head = −3.087 m of water Ans

Apply the energy to the equation between the sump level and the discharge 30 mabove

Between the inlet and outlet of pump:

Therefore, outlet manometric head = 57.31 − 10.34 = 46.97 m of water Ans (c)

4.3 PUMP CHARACTERISTICS AND BEST

OPERATING EFFICIENCY

It is important that a method be developed to enable the proper selection of pumps

to meet specific pumping requirements Thus, before selecting any particular pump,the designer must consult the characteristics of this pump in order to make anintelligent selection In fact, manufacturers develop these characteristics for their

particular pump Thus, pump characteristics are a set of curves that depict the

γ -

2

2g

- V1 2

2 9.81( ) -+

TDH=odh–idh= 57.39–7.31 =50.086 m of water Ans

performance of a given particular pump Figure 4.7 illustrates the setup used fordeveloping pump characteristics (Hammer, 1986) and Figure 4.8 shows an example

of characteristic curves of one particular pump

Apply the equation for TDHfullsd to the figure For convenience, it is reproduced

below

With point 1 as the datum, z2 is equal to 0 h fs is the head loss in the suction side

from point 1 to the inlet of the pump and h fd is the head loss in the discharge sidefrom the outlet of the pump to point 2 Because the distances are very short, they

can be neglected compared to the other terms in the equation P B is equal to the

gage pressure at point 2, P gB , plus the barometric pressure In terms of P gB , P B is

then equal to P gB + P atm The most complete treatment will also include the vaporpressure of water Neglecting vapor pressure since it is negligible, however, the previous

FIGURE 4.7 Setup for developing pump characteristics curves.

FIGURE 4.8 Pump characteristic curves for a 375-mm impeller (Courtesy of Smith and

Loveless With permission.)

Pressure gage

Throttle valve

Flow meter Pump

1170 1100 1000 875 800 700 585

equation simply becomes

(4.16)

Note that TDH fullsd has been changed to TDHsetup This equation demonstrates

that the above setup of the unit is a convenient arrangement for determining theTDH As shown in this equation, TDH can simply be calculated using the pressuregage reading and a measured velocity at point 2

As depicted in Figure 4.8, the performance of this particular pump has beencharacterized in terms of total developed head on the ordinate and discharge on theabscissa The other characteristics are the parameters rpm, power, and efficiency Toillustrate how this chart was developed, consider one curve: when the curve for the1,170 rpm was developed, the setup in Figure 4.7 was adjusted to 1,170 rpm andthe discharge was varied from 0, the shut-off flow, up to the abscissa value depicted

in Figure 4.8 The reading of the pressure gage was then taken This readingconverted to head, along with the velocity head obtained from the flow meter readingand the cross-sectional of the discharge pipe, gives the TDH This was repeated forthe other rpm’s as well as for the powers consumed (which are indicated in kW).The relationship of discharge versus total developed head at the shut-off flowcannot be developed for the positive displacement pump operating under a cylinder,without breaking the cylinder head or the cylinder, itself For pumps operating under

a cylinder, the element that pushes the fluid is either the piston or the plunger Thispiston or plunger moves inside the cylinder and pushes the fluid inside into thecylinder head located at the end of the forward travel This pushing creates a tremen-dous amount of pressure that can rupture the cylinder head or the cylinder body itself,

if the piston or plunger has not given way first In the case of centrifugal pumps, thissituation would not be a problem since the fluid will just be churned inside theimpeller casing, and testing at shut-off flow is possible

The activity inside the pump volute incurs several losses: first is the backflow ofthe flow that had already been acted upon but is slipping back into the suction eye ofthe impeller or, in general, toward the suction side of the pump Because energy hadalready been expended on this flow but failed to exit into the discharge, this backflowrepresents a loss The other loss is the turbulence induced as the impeller acts on theflow and swirls it around Turbulence is a loss of energy As the impeller rotates, its

tips and sides shear off the fluid; this also causes what is called disk friction and is a loss of energy All these losses cause the inefficiency of the pump; h lp is these losses.During the testing, the power to the motor or prime mover driving the pump isrecorded Multiplying this input power by the prime mover efficiency produces the

brake or shaft power In the figure, the powers are measured in terms of kilowatts

(or kW) Call the head corresponding to the brake power as h brake The brake efficiency

of a pump is defined as the ratio of TDH to the brake input power to the pump.Therefore, brake efficiency η is

(4.17)

TDH=TDHsetup=–h lp+h p P gB

γ - V B

2

2g

+

But,

(4.18)

as far as the arrangement in Figure 4.7 is concerned With this equation substitutedinto Equation (4.17), the efficiency during a trial run can be determined This newequation for efficiency is

(4.19)

As shown in the figure, along a certain curve there are several values of ciencies determined Among these efficiencies is one that is the highest of all Thisparticular value of the efficiency corresponds to the best operating performance of

effi-the pump; hence, this point is called effi-the best operating efficiency For example, for

the characteristic curve determined at a brake kilowatt input of 40 kW, the bestoperating efficiency is approximately 67% This corresponds to a TDH of approxi-mately 16 m and approximately a discharge of 0.18 m3/s To operate this pump, itsdischarge should be set at 0.18 m3/s to take advantage of the best operating efficiency

In practice, the operating performance is normally set anywhere from 60 to 120%

of the best operating efficiency

Note that the brake power has been given in terms of its head equivalent h brake

To obtain h brake from a given brake power expressed in horsepower, h p, use the

equivalent that h p = 745.7 N· m/s If Q is the rate of flow in m3/s, and γ is the specificweight in N/m3, then h brake in meters is

(4.20)

Example 4.2 Pump characteristics curves are developed in accordance with

the setup of Figure 4.7 The pressure at the outlet of the pump is found to be 196kN/m2 gage The discharge flow is 0.15 m3/s and the outlet diameter of the dischargepipe is 375 mm The motor driving the pump is 50 hp Calculate TDH

Solution:

Assume temperature of water = 25°C; therefore, density of water = 997 kg/m3

γ -

2

2g

+

h brake

- P gBγ - V B

2

2g

+

=

γ -

2

2g

+

-V B

0.15cross sectional area of pipe - 0.15

π 0.375( )2

/4 - 1.36 m/s

997 9.81( ) - 1.36

2

2 9.81( ) -

4.4 PUMP SCALING LAWS

When designing a pumping station or specifying sizes of pumps, the engineer refers

to a pump characteristic curve that defines the performance of a pump Severaldifferent sizes of pumps are used, so theoretically, there should also be a number ofthese curves to correspond to each pump In practice, however, this is not done Thecharacteristic performance of any other pump can be obtained from the curves ofany one pump by the use of pump scaling laws, provided the pumps are similar

The word similar will become clear later.

The following dependent variables are produced as a result of independent variableseither applied to a pump or are characteristics of the pump, itself: the pressure developed

∆P (corresponding to TDH), the power given to the fluid P, and the efficiency η The independent variables applied to the pump are the discharge Q, the viscosity of the

fluid µ, and the mass density of the fluid ρ These are variables applied, since theycame from outside of and are introduced (applied) to the pump The independentvariables that are characteristics of the pump are the diameter of impeller or length of

stroke D, the rotational speed or stroking speed ω, some roughness  of the chamber,and some characteristic length  of the chamber space There may still be otherindependent variables, but experience has shown that the forgoing items are the majorones Although they are considered major, however, some of them may still be con-sidered redundant and can be eliminated as will be shown in the succeeding analysis For ∆P the functional relationship may be written as

(4.21)

At large Reynolds numbers the effect of viscosity µ is constant For example,consider the Moody diagram At large Reynolds numbers, the plot of the friction

factor f and the Reynolds number, with f as the ordinate, is horizontal Both µ and f

are measures of resistance to flow; thus, they are directly related Because the effect

of f at large Reynolds numbers is constant, the effect of µ at large Reynolds numbersmust also be constant The rotation of the impeller or the movement of the strokeoccurs at an extremely rapid rate; consequently, the flow conditions inside the pumpcasing are turbulent or are at large Reynolds numbers Hence, since µ is constant athigh Reynolds numbers, it does not have any functional relationship with ∆P and

may be removed from Equation (4.21)  as a measure of the pump chamber space

is already included in D It may also be dropped Lastly, since the casing is too

short, the effect of roughness  is too small compared to the other causes of the ∆P.

It may therefore be also dropped Equation (4.21) now takes the form

(4.22)Applying dimensional analysis, let [x] be read “the dimensions of x.” Thus,[∆P] = F/L2

∆P = φ ρ, ω, D, Q, µ, , ( )

∆P = φ ρ, ω, D, Q( )

To eliminate the dimension F, divide ∆P by ρ Thus,

(4.28)

But, ∆P = γH, where Η is TDH, the total developed head Because γ = ρg,

substituting in Equation (4.28) produces

(4.29)

Hg/(ω2

D2) is called the head coefficient, C H , while Q/( ωD3

) is called the flow

coefficient, C Q Because no one pump was chosen in the derivation, the equation is

general For any number of pumps a, b, c, …, n and using Equations (4.28) and

(4.29), the relationships next follow:

(4.30)

(4.31)

∆P

ρ - ∆P

ρ -