WCSM-10-CH04-Force Main and Pumping Hydraulics

WASTEWATER COLLECTION SYSTEM MODELING AND DESIGN

Authors

Thomas M. Walski

Thomas E. Barnard

Eric Harold

LaVere B. Merritt

Noah Walker

Brian E. Whitman

Contributing Authors

Christine Hill, Gordon McKay, Stan Plante, Barbara A. Schmitz

Peer Review Board

Jonathan Gray (Burns and McDonnell), Ken Kerri (Ret.),

Neil Moody (Moods Consulting Pty, Ltd.), Gary Moore (St. Louis Sewer District),

John Reinhardt (Massachusetts Department of Environmental Protection),

Reggie Rowe (CH2M Hill), Burt Van Duin (Westhoff Engineering Resources)

Click here to visit the Bentley Institute Press Web page for more information

http://www.bentley.com/en-US/Training/Bentley+Institute+Press.htm

Recall from Chapter 1 that there are four situations in which pressure flow occurs insewer collection systems:

• Force mains – Sewage is pumped along stretches where gravity flow is notfeasible.

• Pressure sewers – Each customer has a pump that discharges to the pressuresewer.

• Vacuum sewers – Flow is pulled through the system by vacuum pumps.• Surcharged gravity sewers – The depth of flow in a gravity pipe is above the

crown because of downstream control.

Although gravity flow is generally the first choice in sewer networks, pressure flow isfrequently encountered and models must be able to simulate pumping systems andthe flow in pressure systems. Closed-conduit flow is governed by the continuity,energy, and momentum equations, as described in Chapter 2. With closed-conduit flow,pressure terms in the energy equations must be considered. Head losses are caused bypipe friction and also occur at pipe fixtures. Energy is added to the system by pumps.

This chapter reviews the basic principles of pressure hydraulics and pumping, whichare frequently employed in sewer models.

4.1 Friction Losses

In pipe flow, shear stresses develop between the liquid and the pipe wall. The magni-tude of this shear stress is dependent upon the properties of the fluid, its velocity, theinternal roughness of the pipe, and the length and diameter of the pipe.

C H A P T E R

4Force Main and Pumping Hydraulics

114 Force Main and Pumping Hydraulics Chapter 4

Consider, for example, the fluid segment shown in Figure 4.1. Such an element issometimes referred to as a control volume. A force balance on the control volume canbe used to form a general expression describing the head loss due to friction. Note theforces acting upon the element:

• The pressure difference between Sections 1 and 2.• The weight of the fluid volume contained between Sections 1 and 2.• The shear at the pipe walls between Sections 1 and 2.

If the flow has a constant velocity, the sum of all the forces acting on the segment mustbe zero, which is expressed as

(4.1)

where p1 = pressure at section 1 (lb/ft2, N/m2)A1 = cross-sectional area of section 1 (ft2, m2)p2 = pressure at section 2 (lb/ft2, N/m2)A2 = cross-sectional area of section 2 (ft2, m2)A = average area between section 1 and section 2 (ft2, m2)L = distance between section 1 and section 2 (ft, m)g = fluid specific weight (lb/ft3, N/m3)a = angle of the pipe to horizontalt0 = shear stress along the pipe wall (lb/ft2, N/m2)P = average perimeter of pipeline cross section (ft, m)

1 2Datum

V g22/2

(HGL)

V g12/2

p1/�

p2/�

L

Z1

Z2

p A1 1

p A2 2

AL�

�

��PL

Energy Grade Line (EGL)

Hydraulic Grade Line

Figure 4.1 Free-body diagram of water flowing in an inclined pipe.

p1A1 p2A2– ALg asin– t0PL– 0=

Section 4.1 Friction Losses 115

The last term on the left side of Equation 4.1 represents the friction losses along thepipe wall between the two sections. Figure 4.1 shows that the sine of the angle of thepipe, a, is given by

(4.2)

Substituting this result into Equation 4.1, assuming that the cross-sectional area doesnot change, and rearranging gives the head loss due to friction as

(4.3)

where hL = head loss due to friction (ft, m)z1 = elevation of centroid of section 1 (ft, m)z2 = elevation of centroid of section 2 (ft, m)

Note that the velocity head is not considered in this case because the pipe diameter ateach stage is the same, so the areas and the velocity heads are the same.

The shear stress is a function of the following parameters:

t0 = F(r, m, V, D, e) (4.4)

where r = fluid density (slugs/ft3, kg/m3)m = absolute viscosity (lb-s/ft2, N-s/m2)V = average fluid velocity (ft/s, m/s)D = diameter (ft, m)e = index of internal pipe roughness (ft, m)

In turbulent flow, it is not possible to develop an analytical expression for head lossfrom the energy and head loss equations presented in Section 2.5. However, there areseveral commonly used empirical equations, including the Darcy-Weisbach andHazen-Williams equations for closed pipes.

asinz2 z1–

L----------------=

hL t0PLgA-------

p1g----- z1+ç ÷å õ p2

g----- z2+ç ÷å õ–= =


Darcy-Weisbach Equation

The Darcy-Weisbach equation was developed using dimensional analysis and gives thehead loss as

(4.5)

where f = Darcy-Weisbach friction factorg = gravitational acceleration (32.2 ft/s2, 9.81 m/s2)Q = pipeline flow rate (ft3/s, m3/s)

A functional relationship for the Darcy-Weisbach friction factor, f, can be developed inthe form

(4.6)

where Re = the Reynolds number, given by

(4.7)

The pipe roughness factor, e, divided by the pipe diameter, D, is called the relativeroughness. Sometimes e is called the equivalent sand grain roughness of the pipe.Table 4.1 provides values of e for various materials.

Table 4.1 Equivalent sand grain pipe roughnesses (e) for various sewer pipe materials.

Material

Equivalent Sand Grain Roughness, e

ft mm

Wrought iron, steel 1.5 × 10–4—8 x 10-3 0.046—2.4

Asphalted cast iron 4 × 10–4—7 x 10-3 0.1—2.1

Galvanized iron 3.3 x 10-4 — 1.5 x 10-2 0.102—4.6

Cast iron 8 × 10–4—1.8 x 10-2 0.2—5.5

Concrete 10–3–10–2 0.3–3.0

Uncoated cast iron 7.4 × 10–4 0.226

Coated cast iron 3.3 × 10–4 0.102

Coated spun iron 1.8 × 10–4 0.056

Cement 1.3 × 10–3–4 × 10–3 0.4–1.2

Wrought iron 1.7 × 10–4 0.05

Uncoated steel 9.2 × 10–5 0.028

Coated steel 1.8 × 10–4 0.058

PVC 5 × 10–6 0.0015

Sources: Data from Lament, 1981; Moody, 1944; Mays, 1999.

hL f LV2

D2g----------- 8fLQ2

gD5p2-----------------= =

f F Re, eD----ç ÷

å õ=

Re VDrm------------ VD

u--------= =


Colebrook-White Equation and the Moody DiagramThe best known equation relating the friction factor to the Reynolds number and rela-tive roughness is the Colebrook-White equation:

(4.8)

Since the Colebrook-White equation is an implicit function with f on both sides, it isdifficult to use. Typically, it is solved by iterating through assumed values for f untilboth sides are equal.

The Moody diagram (Moody, 1944), shown in Figure 4.2, is a graphical solution for theColebrook-White equation. It is interesting to note that for laminar flow (low Re) onthis log-log plot, the friction factor is a straight-line function of the Reynolds number,while in the fully turbulent range (high e/D and high Re) the friction factor is only afunction of the relative roughness. Most applied water and wastewater pipeline situa-tions fall into the low e/D range near the “smooth” boundary for Re in the range of 105

to 106. The smooth boundary exists as a lower limit for e/D ratios since the laminarsublayer along the wall completely covers the roughness; thus roughness no longeraffects the functional resistance to flow.

1f----- 0.86– e

3.7D------------ 2.51

Re f-------------+ç ÷

å õ=

Values of for water at 60° F (diameter in inches, velocity in ft/s)DV0.1 0.2 0.4 0.6 1 2 4 6 10 20 40 60 100 200 400 600 1000 2000 4000 10,000

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.025

0.020

0.015

0.010

0.009

0.008

Fric

tion

fact

or,

=

fL D

V2

2g

103 2(10 )3 4 6 8 104 2(10 )4 4 6 8 105 2(10 )5 4 6 8 106 2(10 )6 4 6 8 107 2(10 )7 4 6 8 108

Reynolds number, =Re DV�

�/D

Rel

ativ

e ro

ughn

ess,

0.05

0.03

0.02

0.01

0.006

0.004

0.002

0.001

0.0006

0.0004

0.0002

0.00005

Laminar flow, =f 64

Drawn TubingSteel or wrought ironAsphalted cast ironGalvanized ironCast ironWood staveConcreteRiveted steel

0.0000050.000150.00040.00050.000850.0006 0.003–0.001 0.01–0.003–0.03

�, ft. �, mm

0.00150.0450.120.150.250.18 0.9–0.3 3–0.9 9–

Smoothpipes

Re

h L

Figure 4.2 Moody diagram for solving the Colebrook-White equation (Moody, 1944).


Example 4.1 Darcy-Weisbach equation and the Moody diagram

Determine the head loss in 250 feet of 24-in. diameter concrete pipe conveying a dis-charge of 25 ft3/s. Assume that the water temperature is 60° F.

Solution

Table 4.1 gives the equivalent sand grain roughness for the pipe as e = 0.001 ft. The rel-ative roughness is therefore e/D = 0.0005.

The cross-sectional area of the 24-in. pipe is A = 3.14 ft2, and the velocity of flow is V =Q/A = 7.96 ft/s. Table C.1 on page 567 provides the kinematic viscosity for the water asn = 1.217 ³ 10–5 ft2/s, so the Reynolds number is

The Moody diagram gives the friction factor as f = 0.017. The Darcy-Weisbach equationgives the head loss in the pipe as

Hazen-Williams Equation

Another head loss expression that is frequently used, particularly in North America,is the Hazen-Williams equation (Williams and Hazen, 1920; American Society of CivilEngineers, 1992):

(4.9)

where hL = pipe friction head loss (ft, m)

L = pipe length (ft, m)

C = Hazen-Williams C-factor

D = diameter (ft, m)

Q = flow rate (ft3/s, m3/s)

Cf = unit conversion factor (4.73 for US customary units, 10.7 for SI)

The Hazen-Williams equation uses many of the same variables as Darcy-Weisbach,but adds a water-carrying capacity factor, C, that is assumed to be a constant for agiven pipe material. Higher C-factors occur with smoother pipes (with higher carry-ing capacities) and lower C-factors describe rougher pipes. Table 4.2 lists typical C-factors for various pipe materials.

Re VDn-------- 7.96 2.0³

1.217 10 5–³------------------------------ 1.31 106³= = =

hL 0.017 2502.0--------- 7.962

2 32.2³-------------------³³ 2.1 ft= =

hLCf L

C1.852D4.87----------------------------Q1.852=


Table 4.2 Hazen-Williams C-factors.

Type of Pipe

Discrete Pipe Diameter, in. (cm)

3.0(7.6)

6.0(15.2)

12(30)

24(61)

48(122)

Uncoated cast iron, smooth and new 121 125 130 132 134

Coated cast iron, smooth and new 129 133 138 140 141

30 years old

Trend 1 – slight attack 100 106 112 117 120

Trend 2 – moderate attack 83 90 97 102 107

Trend 3 – appreciable attack 59 70 78 83 89

Trend 4 – severe attack 41 50 58 66 73

60 years old





100 years old





Miscellaneous

Newly scraped mains 109 116 121 125 127

Newly brushed mains 97 104 108 112 115

Coated spun iron, smooth and new 137 142 145 148 148

Old – take as coated cast iron of same age

Galvanized iron, smooth and new 129 133

Wrought iron, smooth and new 137 142

Coated steel, smooth and new 137 142 145 148 148

Uncoated steel, smooth and new 142 145 147 150 150

Coated asbestos cement, clean 147 149 150 152

Uncoated asbestos cement, clean 142 145 147 150

Spun cement-lined and spun bitumen-lined, clean 147 149 150 152 153

Smooth pipe (including lead, brass, copper, poly-ethylene, and PVC), clean 147 149 150 152 153

PVC wavy, clean 142 145 147 150 150

Concrete – Scobey

Class 1 – Cs = 0.27; clean 69 79 84 90 95

Class 2 – Cs = 0.31; clean 95 102 106 110 113

Class 3 – Cs = 0.345; clean 109 116 121 125 127

Class 4 – Cs = 0.37; clean 121 125 130 132 134

Best – Cs = 0.40; clean 129 133 138 140 141

Tate relined pipes – clean 109 116 121 125 127

Prestressed concrete pipes – clean 147 150 150

Source: Compiled from Lamont, 1981.


Example 4.2 Hazen-Williams equation

Use the Hazen-Williams formula to determine the head loss in a 400 m section of a 300mm PVC force main. The discharge is 100 L/s.

Solution

Table 4.2 gives the C-factor as 150. Substituting this result into Equation 4.9 gives thehead loss as

Swamee-Jain Equation

The Swamee-Jain equation (Swamee and Jain, 1976) is much easier to solve than theColebrook-White equation. This equation is an explicit function of the Reynolds num-ber and the relative roughness,

(4.10)

It is accurate to within about one percent of the Colebrook-White equation over thefollowing ranges:

and

Manning Equation

The Manning equation is commonly used for open-channel flow, but can still be usedfor fully rough closed pipes in the form

(4.11)

where n = Manning roughness coefficientCf = unit conversion factor (4.66 for customary units, 5.29 SI)

Higher n values correspond to greater internal pipe roughness. Manning’s n is alsocommonly assumed to be constant for a given pipe material, although this is notstrictly the case (see Chapter 2). Values of n for commonly used pipe materials are pre-sented in Table 2.3 on page 42.

hL10.7 400³

1501.852 0.304.87³---------------------------------------------0.101.852 1.98 m= =

f 1.325

ln e3.7D------------ 5.74

Re0.9-------------+ç ÷

å õ 2---------------------------------------------------=

4 103³ Re 1 108³¢ ¢

1 10 6–³ eD---- 1 10 2–³¢ ¢

hLCf L nQ( )2

D5.33--------------------------=


Example 4.3 Manning equationUse the Manning equation to determine head loss in a 500-ft section of an 18-in. diam-eter cast iron force main. The discharge is 3000 gpm.

Solution

The discharge must be first converted to ft3/s, which gives

Substituting this value into Equation 4.11 gives

Pipe Roughness ChangesWall roughness may change over time because of pipe-wall corrosion or scale deposi-tion. In sewage pipelines, the problem is mainly one of corrosion and/or slime coating.This problem may be mitigated through the use of corrosion-resistant pipe materials orpipe coatings, with associated pipe velocities high enough to minimize slime buildup.

Comparison of Friction Loss MethodsMost hydraulic models allow the modeler to select from the Darcy-Weisbach, Hazen-Williams, or Manning head loss formulas, depending on the nature of the problemand the modeler’s preferences.

The Darcy-Weisbach formula is more physically-based than the others. It is derivedfrom the balance of forces acting on flow in pipes (although f is still found empiri-cally). With appropriate fluid viscosities and densities, Darcy-Weisbach can be used tofind the head loss in a pipe for any Newtonian fluid in any flow regime.

The Hazen-Williams and Manning formulas, however, are empirically based and gen-erally only apply to water in turbulent flow.

The Hazen-Williams formula is the predominant equation used in the United States forpressure pipes, while the Darcy-Weisbach formula predominates in Europe. The Man-ning formula is not generally used for pressure flow except with inverted siphons andsurcharged sewers. Table 4.3 presents these three equations in several common unitconfigurations. These equations solve for the friction slope, Sf, which is the head lossper unit length of pipe.

Table 4.3 Pipe friction loss equations.

Equation Q (m3/s); D (m) Q (ft3/s); D (ft) Q (gpm); D (in.)

Darcy-Weisbach

Hazen-Williams

Q 3000 gpm

448.7 gpmft3 s£-------------

----------------------------- 6.69 ft3 s£= =

hL4.66 500³ 0.012 6.69³( )2

1.55.33--------------------------------------------------------------- 1.73 ft= =

Sf0.083fQ2

D5-----------------------= Sf

0.025fQ2

D5-----------------------= Sf

0.031fQ2

D5-----------------------=

Sf10.7D4.87------------- Q

C----ç ÷å õ 1.852

= Sf4.73D4.87------------- Q

C----ç ÷å õ 1.852

= Sf10.5D4.87------------- Q

C----ç ÷å õ 1.852

=


4.2 Minor LossesHead losses occurring at fixtures, such as manholes, valves, tees, bends, reducers, andother appurtenances within the piping system, are called minor losses. These losses arethe result of velocity changes and increased turbulence and eddies caused by the fix-ture. Although head loss might be quite high in a fixture compared to the same lengthin the associated pipe, most pipelines are very long, with only a few minor loss fix-tures. Therefore, overall, the pipe friction losses are large compared to the fixture orminor losses.

Most minor head losses are computed by multiplying a minor loss coefficient by thevelocity head, as given by

(4.12)

where hm = minor head loss (ft, m)KL = minor loss coefficientV2 = average fluid velocity (ft/s, m/s)

g = gravitational acceleration constant (ft/s2, m/s2)Q = pipeline flow rate (ft3/s, m3/s)A = cross-sectional area of pipe (ft2, m2)

Minor loss coefficients are determined experimentally. Table 4.4 is a list of coefficientsassociated with commonly used fittings.

Manning

Table 4.3 (Continued) Pipe friction loss equations.

Equation Q (m3/s); D (m) Q (ft3/s); D (ft) Q (gpm); D (in.)

Sf10.3 nQ( )2

D5.33--------------------------= Sf

4.66 nQ( )2

D5.33--------------------------= Sf

13.2 nQ( )2

D5.33--------------------------=

Table 4.4 Minor loss coefficients.1

Fitting KL Fitting KL

Pipe Entrance 90¯ Smooth Bend

Bellmouth 0.03-0.05 Bend radius/D = 4 0.16–0.18

Rounded 0.12-0.25 Bend radius/D = 2 0.19–0.25

Sharp Edged 0.50 Bend radius/D = 1 0.35–0.40

Projecting 0.80

Mitered Bend

Contraction – Sudden q = 15¯ 0.05

D2/D1 = 0.80 0.18 q = 30¯ 0.10

D2/D1 = 0.50 0.37 q = 45¯ 0.20

D2/D1 = 0.20 0.49 q = 60¯ 0.35

q = 90¯ 0.80

hm KLV2

2g------ KL

Q2

2gA2-------------= =

Section 4.2 Minor Losses 123

1 D = pipe diameter, q = downstream bend angle

Minor Loss Valve CoefficientsMost valve manufacturers provide a chart of percent opening versus valve coefficient,Cv, which is related to the minor loss, KL, as

(4.13)

where D = diameter of the valve (in., m)Cv = valve coefficient (gpm/psi0.5, m3/s/kPa0.5)Cf = unit conversion factor (880 for US customary units, 1.22 for SI

units)

Table 4.5 gives values of minor loss coefficients for valves that are commonly found inwastewater conveyance systems.

Contraction – Conical Tee

D2/D1 = 0.80 0.05 Line Flow 0.30–0.40

D2/D1 = 0.50 0.07 Branch Flow 0.75–1.80

D2/D1 = 0.20 0.08

Cross

Expansion – Sudden Line Flow 0.50

D2/D1 = 0.80 0.16 Branch Flow 0.75

D2/D1 = 0.50 0.57

D2/D1 = 0.20 0.92 45¯ Wye

Expansion – Conical Line Flow 0.30

D2/D1 = 0.80 0.03 Branch Flow 0.50

D2/D1 = 0.50 0.08

D2/D1 = 0.20 0.13

Source: Data from Walski, 1984

Table 4.5 Minor loss coefficients (KL) for various types of fully open valves.

Valve Type KL

Check

Ball 0.9–1.7

Center-guided globe style 2.6

Double door

8 in. or smaller 2.5

10–16 in. 1.2

Foot

Hinged disk 1–1.4

Poppert 5–14

Table 4.4 (Continued) Minor loss coefficients.1

Fitting KL Fitting KL

KLCf D4

Cv2

--------------=


Example 4.4 Minor losses

The flow in an 8-in. force main is 900 gpm. The following fittings are encounteredalong a pipe. What is the total head loss through the junction?

• two 90° smooth bends (bend radius/D = 2)• knife gate valve, resilient seal – open• mitered bend, q = 45°

Solution

The flow rate is

The flow velocity is

The minor loss coefficients for the fittings are found in Table 4.4 and 4.5 as follows:

two 90° smooth bends (bend radius/D = 2) KL = 2 × 0.25 = 0.50

knife gate valve, resilient seal – open KL = 0.30

mitered bend, q = 45° KL = 0.20

The sum of these is the total minor loss coefficient, or

SKL = 1.00

The minor head loss is then given by Equation 4.12 as

Rubber flapper

V < 6 ft/s 2.0

V > 6 ft/s 1.1

Knife gate

Metal seat 0.2

Resilient seat 0.3

Plug

Lubricated 0.5–1.0

Eccentric – rectangular (80%) opening 1.0

Eccentric – full-bore opening 0.5

Slanting disk 0.25–2.0

Swing 0.6–2.2

Diaphragm or pinch 0.2–0.75

Source: Adapted from Sanks, 1998.

Table 4.5 (Continued) Minor loss coefficients (KL) for various types of fully open valves.

Valve Type KL

Q 900 gpm

448.7 gpmft3/s------------

---------------------------- 2.01 ft3/s= =

V QA---- 2.01 ft3/s

p 412------ç ÷å õ 2

------------------------ 5.75 ft/s= = =

hm 1.0 5.752

2 32.2³------------------- 0.51 ft= =

Section 4.3 Energy Addition – Pumps 125

4.3 Energy Addition – Pumps

Sanitary sewage systems sometimes include pumping stations to lift sewage fromdeep gravity sewers to collectors for pumping across drainage divides or for movingsewage in flat terrain. The energy added is called pump head. The following discussionis oriented toward centrifugal pumps, since they are the pump type most frequentlyused in sanitary sewage systems. In centrifugal pumps, a rotating impeller transfersthe energy from the motor shaft to the water. Figure 4.3 shows an impeller for a largewastewater pump and Figure 4.4 shows a centrifugal pump in a dry well. Additionalinformation about pumps can be found in Bosserman (2000), Hydraulic Institute(2000), Karassik et al. (2001), Sanks (1998), and Water Environment Federation (1993).

Pump Head-Discharge Relationship

The relationship between energy added and pump discharge can be shown as a headversus discharge curve (also called a pump head characteristic curve), as in Figure 4.5.Empirical equations are often used to describe the relationships. Pump head (alsocalled total dynamic head) is the difference in head from the suction to the dischargeside of the pump. Data on this relationship are usually available from the pump man-ufacturer. A reasonable fit to experimental data is often obtained with a power func-tion of the form

(4.14)

Figure 4.3 Impeller for a large wastewaterpump.

hp ho cQpm–=


where hp = pump head (ft, m)ho = cutoff (shutoff) head (ft, m)

Qp = pump discharge (ft3/s, m3/s)c, m = coefficients describing pump curve shape

System Head Curves

The purpose of a pump is to add the energy (head) necessary to overcome elevationdifferences and head losses. The head necessary to achieve the elevation difference iscalled static head or static lift. The head needed to compensate for head losses is addedto the static head to obtain the system head curve, as illustrated in Figure 4.6.

The pump head characteristic curve shows the hydraulic capability of a given pump.The system head curve presents how the system responds to a range of flow ratesintroduced into or pumped through the system. The system head curve is continuallysliding up and down as wet well water levels change and other pumps discharginginto the force main or pressure sewer come on or off. The result is actually a family ofsystem head curves forming a band on the graph. The band is narrow for a sewagepump station with only one pump serving the force main, since wet well depth fluctu-ations are not normally very large. As described further in Chapter 12, the band canbe quite large for force mains with multiple pumps or pressure sewer systems.

Figure 4.4 Centrifugal pump in a dry well.


Capacity, L/s

Tota

l Dis

char

ge H

ead,

m

Shutoff Head

Design Point

Maximum Flow

24.4

18.3

12.2

6.1

00 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4

Figure 4.5 Pump head characteristic curve.

Efficiency Curve

Pump Operating Point

Pum

p E

ffici

ency

Head Losses

Flow, gpm

Static Head

Hea

d, ft

System Head Curve

Pump Head Curve

Figure 4.6 Pump curve and system head curve defining the pumpoperating point.


For the case of a single pipeline between two points, the system head curve is given by

(4.15)

where H = total head (ft, m)h1 = static lift (ft, m)Kp = pipe head loss coefficient (sz/ft3z–1, sz/m3z–1)Q = pipe discharge (ft3/s, m3/s)z = coefficient

KM = minor head loss coefficient (s2/ft5, s2/m5)

Thus, the head losses and minor losses associated with each segment of pipe aresummed along the total length of the pipeline, as illustrated in Figure 4.7.

When the pump head discharge curve and the system head curve are plotted on thesame axes, as in Figure 4.6, the intersection defines the pump operating point, whichsimultaneously satisfies both the pump characteristic curve and the system curve.

Other Pump Characteristic Curves

In addition to the pump head discharge curve, other curves representing pumpbehavior describe brake horsepower and efficiency. The brake horsepower is the powerdelivered to the pump for efficient pump operation. Since utilities want to minimizethe amount of energy used, the engineer should select pumps that run as efficiently aspossible. Pump selection and operating costs are discussed further in Chapter 12.

Another issue to consider when designing a pump in a sewage system is the requirednet positive suction head (NPSH), which is the head at the suction side of a pump. Therequired NPSH generally increases with flow through the pump. The available NPSHmust be greater than the required NPSH to ensure that local pressures within thepump do not drop below the vapor pressure of the fluid and cause cavitation. Cavita-tion is essentially a boiling and then collapse of the liquid vapor within the pump,

H h1 KpQzä KMQ2ä+ +=

Wet Well

Wet Well

Pump Valve

Bends

StaticLift

HeadLossHGL

Minor Loss

TDH

Figure 4.7 Schematic of hydraulic grade line for a pumped system.


which can cause damage. The required NPSH is unique for each pump model and is afunction of flow rate. Manufacturers’ specifications should be consulted to ensure thatpumping station design has sufficient NPSH.

In manifolded pumping systems, where many sewage pumping stations may pumpinto a common force main network, primary design consideration is given to the con-ditions in the network during peak wet-weather conditions. However, during verylow flow conditions, the pressure in the force main network can drop low enough tocause cavitation in an individual pump.

Fixed-Speed and Variable-Speed Pumps

A pump characteristic curve is related to the speed of the pump. The motor of a fixed-speed pump spins it at a constant speed. A variable-speed pump has a variable-speedmotor or other device that changes the pump speed.

A variable-speed pump is not really a special type of pump, but rather a pump con-nected to a variable-speed drive or controller. The most common type of variable-speed drive controls the voltage to the pump motor, which in turn changes the speedat which the motor rotates. Differences in speed then shift the pump’s characteristiccurve. Variable-speed pumps are useful in locations where large system head varia-tions occur for a given flow rate, such as in a sewer force main with multiple pumps.

Affinity Laws for Variable-Speed Pumps

A centrifugal pump’s characteristic curve is fixed for a given rotational speed andimpeller diameter. However, for a given model, modified curves can be determined


quite accurately for any speed and any diameter by applying affinity laws. Forvariable-speed pumps, two of these affinity laws are

(4.16)

and

(4.17)

where Qp1, Qp2 = pump flow rate (ft3/s, m3/s)n1, n2 = pump speed (1/s)

hp1, hp = pump discharge head (ft, m)

These relations show that pump discharge rate is directly proportional to pump speedand pump discharge head is proportional to the square of the speed. These equationsallow the designer to establish pump curves for other speeds, if the curve is known fora reference speed. They must be applied to the curves and not directly to the operat-ing point. Figure 4.8 shows how pump head curves change with changing pumpspeed. The line labeled “Best Efficiency Points” is a trace of the best efficiency pointsat each speed. The actual flow that a pump produces at each speed depends on thesystem head curve as well.

Power and EfficiencyThe term power may have one of several meanings when dealing with a pump:

• Input power – the power that is delivered to the motor, usually in electrical form.

Qp1Qp2----------

n1n2-----=

hp1hp2--------

n1n2-----ç ÷å õ 2

=

180

160

140

120

100

80

60

40

20

0

Hea

d, ft

0 100 200 300 400

Flow, gpm

n = 1.00

n = 0.95

n = 0.90

n = 0.85

n = 0.80

BestEfficiencyPoints

n =speed

full speed

Figure 4.8 Relative speed factors (n) for variable-speed pumps.


• Brake power – the power that is delivered to the pump from the motor.

• Water power – the power that is delivered to the water from the pump.

Energy losses occur as energy is converted from one form to another (electrical tomotor, motor to pump, pump to water), and every transfer has an efficiency associatedwith it. The efficiencies associated with these transfers may be expressed either as per-centages (100 percent is perfectly efficient) or as decimal values (1.00 is perfectly effi-cient), and are typically defined as follows:

• Motor efficiency – the ratio of brake power to input power.

• Pump efficiency – the ratio of water power to brake power.

• Wire-to-water (overall) efficiency – the ratio of water power to input power.

Pump efficiency tends to vary significantly with flow, while motor efficiency remainsrelatively constant over the rated range. Note that there may also be an additional effi-ciency associated with a variable-speed drive. Some engineers refer to the combina-tion of the motor and speed controls as the driver.

Figure 4.9 shows input power (horsepower curves), net positive suction head, andwire-to-water efficiency curves overlaid on typical pump head curves. For eachimpeller size, there is a flow rate corresponding to maximum efficiency. At higher orlower flows, the efficiency decreases. This maximum point on the efficiency curve is

P/V 2 x 2 x 8 3500 RPM

US gallons per minute

Tota

l Dyn

amic

Hea

d, ft

280

240

200

160

120

80

40

0

0 40 80 120 160 200

8"

7"

6"

5"

5957

5550

45

60

40

20

0

NPSH 3HP

5HP

7½HP

10HP

15HP

35

35 45 50

55

59EFF

NP

SH

, ft

Curve # 2897840Impeller # V-1728-B

57

Impellerdiameter

Diagram courtesy of Peerless Pump Company

Figure 4.9 Pump curves with wire-to-water efficiency, net positive suction head (NPSH), andhorsepower overlays for different impeller sizes.


called the best efficiency point (BEP) and is the ideal operating point for the pump. Forexample, with a 7-in. impeller, the BEP is at a discharge of 100 gal/min and the wire-to-water efficiency is 59%.

Pump curves are available from suppliers and manufacturers. Every pump differsslightly from its catalog specifications, and normal impeller wear causes a pump’sperformance to change over time. Pumps should be periodically field tested to verifythat the characteristic curves on record are representative of field performance. If apump is found to be operating at relatively low efficiency, an economic analysis canhelp to decide among impeller, motor, or complete pump replacement.

ReferencesAmerican Society of Civil Engineers (ASCE). 1992. Design and Construction of Urban

Stormwater Management Systems. ASCE Manuals and Reports of EngineeringPractice No. 77, WEF Manual of Practice FD-20. Reston, VA: American Society ofCivil Engineers.

Bosserman, B. E. 2000. Pump system hydraulic design. In Water Distribution SystemHandbook, edited by L. W. Mays. New York: McGraw-Hill.

Hydraulic Institute. 2000. Pump Standards. Parsippany, NJ: Hydraulic Institute.

Karassik, I. J., J. P. Messina, P. Cooper, and C. C Heald, eds. 2001. Pump Handbook, 3ded. New York: McGraw-Hill.

Lamont, P. 1981. Common pipe flow formulas compared with the theory ofroughness. Journal of the American Water Works Association 73, No. 5: 274.

Mays, L. W. ed. 1999. Hydraulic Design Handbook. New York: McGraw-Hill.

Moody, L. F. 1944. Friction factors for pipe flow.” Transactions of the American Society ofMechanical Engineers 66.

Sanks, R. L., ed. 1998. Pumping Station Design. 2d ed. London: Butterworth.

Swamee, P. K. and A. K. Jain. 1976. Explicit equations for pipe flow problems. Journalof Hydraulic Engineering, ASCE 102, No. 5: 657.

Walski, T. M. 1984. Analysis of Water Distribution Systems. New York: Van NostrandReinhold.

Water Environment Federation (WEF). 1993. Design of Wastewater and StormwaterPumping Stations. MOP FD-4. Alexandria, VA: Water Environment Federation.

Williams, G. S., and Hazen, A. (1920) Hydraulic Tables. New York: John Wiley andSons.

Problems 133

Problems

4.1 From the following figure, estimate the water pressure (in lb/in.2) entering thepipe at the bottom of the tank.

4.2 For the system in Problem 4.1, estimate the head loss due to friction in the pipefrom the tank to the pressure gauge. The pipe has a sharp edged entrance.

4.3 For the system in Problem 4.1, estimate the Darcy-Weisbach friction factor (f)and the Hazen-Williams C-factor.

4.4 Determine the Reynolds number in the 4-in. pipe in the system in Problem 4.1 ifthe water temperature is 50¯ F.

a. Is the flow laminar, transitional, or turbulent?

b. Estimate the equivalent sand roughness height (e) of the pipe.

4.5 Determine the flow rate in the 150-mm PVC pipe shown in the following figure, using

a. the Darcy-Weisbach equation

b. the Hazen-Williams equation

4.6 Explain why the flow rates found by the two methods in Problem 4.5 are differ-ent. What could be done to either the Darcy-Weisbach or Hazen-Williams equa-tion to force the determined flow rates to be equal?

D = 4 inH = 35 ft

L = 300 ft

Q = 300 gpm

Pgage = 7.5 psi

P = 248 kPagage

P = 214 kPagageL = 125 m

z = 20 m

z = 8.5 mT= 10º C


4.7 Estimate the minor loss coefficient for a partially closed 205-mm gate valve if themeasured flow rate is 0.062 m3/s and the pressure drop is 30.3 kPa.

4.8 Estimate the discharge flow rate from the 6-in. cast iron pipe illustrated in thefollowing figure. Use the Darcy-Weisbach equation to estimate the friction losses.

4.9 For the system in Problem 4.8, determine the gauge pressure in the 6-in. castiron pipe located 20 ft from the discharge end.

4.10 Determine the pump head (hp) for a centrifugal pump lifting water 9.1 m in a 180-m, 205 mm. pipe (f = 0.019) at a flow rate of 0.06 m3/s. The sum of the minor losscoefficients is 5.70.

4.11 Derive the system head curve to lift water 45 ft in 2500 ft of 8-in. new ductile(cast) iron pipe. The sum of the minor loss coefficients is 7.35. Clearly indicatethe units of flow (Q) in the equation.

a. Use the Darcy-Weisbach equation to model friction losses. Clearly indicatehow you determined the friction factor (f). Do you expect the friction factor tobe constant? Why or why not?

b. Use the Hazen-Williams equation to model friction losses.

4.12 A pump system was installed to deliver water with a system curve defined ashp (ft) = 53.4 + 0.000097Q2 with Q in gpm. Use the following pump performancecurve to determine the shutoff head and flow rate, total static head, and totaldynamic head at the operating point.

Q = ?P = ?

z = 29 ft6-in. cast iron pipeTotal length = 40 ft

K = 0.88-in. cast iron pipeTotal length = 150 ft

K = 0.3(each)

Problems 135

0

10

20

30

40

50

60

70

80

90

0 100 200 300 400 500 600 700 800

Hea

d, ft

Flow Rate, gpm

90

80

70

60

50

40

30

20

10

0

0 100 200 300 400 500 600 700 800

Documents

WCSM-10-CH04-Force Main and Pumping Hydraulics