Standard Methods for Examination of Water & Wastewater_2 ppt

A rectangular weir is a thin plate where the plate is being cut such that a rectangular opening is formed in which the flow in the channel that is being measured passes through.. As indi

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Flow Measurements and Flow and Quality Equalizations

This chapter discusses the unit operations of flow measurements and flow and qualityequalizations Flow meters discussed include rectangular weirs, triangular weirs,trapezoidal weirs, venturi meters, and one of the critical-flow flumes, the Parshallflume Miscellaneous flow meters including the magnetic flow meter, turbine flow meter,nutating disk meter, and the rotameter are also discussed These meters are classified

as miscellaneous, because they will not be treated analytically but simply described

In addition, liquid level recorders are also briefly discussed

3.1 FLOW METERS

Flow meters are devices that are used to measure the rate of flow of fluids Inwastewater treatment, the choice of flow meters is especially critical because of thesolids that are transported by the wastewater flow In all cases, the possibility ofsolids being lodged onto the metering device should be investigated If the flow hasenough energy to be self-cleaning or if solids have been removed from the waste-water, weirs may be employed Venturi meters and critical-flow flumes are wellsuited for measurement of flows that contain floating solids in them All these flow-measuring devices are suitable for measuring flows of water

Flow meters fall into the broad category of meters for open-channel flow surements and meters for closed-channel flow measurements Venturi meters areclosed-channel flow measuring devices, whereas weirs and critical-flow flumes areopen-channel flow measuring devices

mea-3.1.1 R ECTANGULAR W EIRS

A weir is an obstruction that is used to back up a flowing stream of liquid It may

be of a thick structure or of a thin structure such as a plate A rectangular weir is

a thin plate where the plate is being cut such that a rectangular opening is formed

in which the flow in the channel that is being measured passes through The rectangularopening is composed of two vertical sides, one bottom called the crest, and no topside There are two types of rectangular weirs: the suppressed and the fully contractedweir Figure 3.1 shows a fully contracted weir As indicated, a fully contracted rectangular weir is a rectangular weir where the flow in the channel being measuredcontracts as it passes through the rectangular opening On the other hand, a sup- pressed rectangular weir is a rectangular weir where the contraction is absent, that

3

182 Physical–Chemical Treatment of Water and Wastewater

is, the contraction is suppressed This happens when the weir is extended fully acrossthe width of the channel, making the vertical sides of the channel as the two verticalsides of the rectangular weir To ensure an accurate measurement of flow, the crestand the vertical sides (in the case of the fully contracted weir) should be beveledinto a sharp edge (see Figures 3.2 and 3.3)

To derive the equation that is used to calculate the flow in rectangular weirs,refer to Figure 3.2 As shown, the weir height is P The vertical distance from thetip of the crest to the surface well upstream of the weir at point 1 is designated as

H H is called the head over the weir

Recording drum

Indicator scale

Float

Connecting pipe Float well Rectangular weir

Weir

Fully contracted flow Crest Top view

Flow Measurements and Flow and Quality Equalizations 183

From fluid mechanics, any open channel flow value possesses one and only onecritical depth Since there is a one-to-one correspondence between this depth andflow, any structure that can produce a critical flow condition can be used to measurethe rate of flow passing through the structure This is the principle in using therectangular weir as a flow measuring device Referring to Figure 3.2, for this structure

to be useful as a measuring device, a depth must be made critical somewhere Fromexperiment, this depth occurs just in the vicinity of the weir This is designated as

y c at point 2 A one-to-one relationship exists between flow and depth, so this section

is called a control section In addition, to ensure the formation of the critical depth,the underside of the nappe as shown should be well ventilated; otherwise, the weirbecomes submerged and the result will be inaccurate

Between any points 1 and 2 of any flowing fluid in an open channel, the energyequation reads

(3.1)

where V, P, y, z, and h l refer to the average velocity at section containing the point,pressure at point, height of point above bottom of channel, height of bottom ofchannel from a chosen datum, and head loss between points 1 and 2, respectively.The subscripts 1 and 2 refer to points 1 and 2 g is the gravitational constant and γ

is the specific weight of water Referring to Figure 3.2, the two values of z are zero

V1 called the approach velocity is negligible compared to V2, the average velocity

at section at point 2 The two Ps are all at atmospheric and will cancel out Thefriction loss h l may be neglected for the moment y1 is equal to H+P and y2 is veryclosely equal to y c+P Applying all this information to Equation (3.1), and changing

Triangular weir

Trapezoidal weir

L a

2

2g

- P2γ - y2 z2

184 Physical–Chemical Treatment of Water and Wastewater

The critical depth y c may be derived from the equation of the specific energy E

Using y as the depth of flow, the specific energy is defined as

(3.3)

From fluid mechanics, the critical depth occurs at the minimum specific energy

Thus, the previous equation may be differentiated for E with respect to y and equated

to zero Convert V in terms of the flow Q and cross-sectional area of flow A using

the equation of continuity, then differentiate and equate to zero This will produce

(3.4)

where T is equal to dA/dy, a derivative of A with respect to y T is the top width of

the flow A/T is called the hydraulic depth D The expression V/ is called the

Froude number The flow over the weir is rectangular, so D is simply equal to y c,

thus Equation (3.4) becomes

(3.5)

where V has been changed to V c, because V is now really the critical velocity V c

Equation (3.4) shows that the Froude number at critical flow is equal to 1 Equation

(3.5) may be combined with Equation (3.2) to eliminate y c producing

(3.6)

The cross-sectional area of flow at the control section is y c L, where L is the

length of the weir This will be multiplied by V c to obtain the discharge flow Q at

the control section, which, by the equation of continuity, is also the discharge flow

in the channel Using Equation (3.5) for the expression of y c and Equation (3.6) for

the expression for V c, the discharge flow equation for the rectangular weir becomes

(3.7)Two things must be addressed with respect to Equation (3.7) First, remember

that h l and the approach velocity were neglected and y2 was made equal to y c+P

Second, the L must be corrected depending upon whether the above equation is to

be used for a fully contracted rectangular weir or the suppressed weir

The coefficient of Equation (3.7) is merely theoretical, so we will make it more

general and practical by using a general coefficient K as follows

Now, based on experimental evidence Kindsvater and Carter (1959) found that for

H /P up to a value of 10, K is

(3.9)

Due to the contraction of the flow for the fully contracted rectangular weir, the

cross-sectional of flow is reduced due to the shortening of the length L From experimental evidence, for L /H > 3, the contraction is 0.1H per side being contracted Two sides are being contracted, so the total correction is 0.2H, and the length to be

used for fully contracted weir is

Lfully contracted weir = L − 0.2H (3.10)

In operation, the previous flow formulas are automated using control devices.This is illustrated in Figure 3.1 As derived, the flow Q is a function of H For a given installation, all the other variables influencing Q are constant Thus, Q can be found through the use of H only As shown in the figure, this is implemented by communicating the value of H through the connecting pipe between the channel,

where the flow is to be measured, and the float chamber The communicated value

of H is sensed by the float which moves up and down to correspond to the value

communicated The system is then calibrated so that the reading will be directly interms of rate of discharge

From the previous discussion, it can be gleaned that the meter measures rates

of flow proportional to the cross-sectional area of flow Rectangular weirs are therefore

area meters In addition, when measuring flow, the unit obstructs the flow, so the

meter is also called an intrusive flow meter.

Example 3.1 The system in Figure 3.1 indicates a flow of 0.31 m3/s To

inves-tigate if the system is still in calibration, H, L, and P were measured and found to

be 0.2 m, 2 m, 1 m, respectively Is the system reading correctly?

Solution: To find if the system is reading correctly, the above values will besubstituted into the formula to see if the result is close to 0.3 m3/s

P

+

=

Lfully contracted weir = L–0.2H = 2–0.2 0.2( ) = 1.96 m

1 -

=

186 Physical–Chemical Treatment of Water and Wastewater

Example 3.2 Using the data in the above example, calculate the discharge

through a suppressed weir

Solution:

Therefore,

Example 3.3 To measure the rate of flow of raw water into a water treatment

plant, management has decided to use a rectangular weir The flow rate is 0.33 m3/s.Design the rectangular weir The width of the upstream rectangular channel to be

connected to the weir is 2.0 m and the available head H is 0.2 m.

Solution: Use a fully suppressed weir and assume length L = 0.2 m Thus,

Therefore,

Therefore,

3.1.2 T RIANGULAR W EIRS

Triangle weirs are weirs in which the cross-sectional area where the flow passes

through is in the form of a triangle As shown in Figure 3.3, the vertex of this triangle

is designated as the angle θ When discharge flows are smaller, the H registered by

rectangular weirs are shorter, hence, reading inaccurately In the case of triangular

weirs, because of the notching, the H read is longer and hence more accurate for comparable low flows Triangular weirs are also called V-notch weirs As in the case

of rectangular weirs, triangular weirs measure rates of flow proportional to the sectional area of flow Thus, they are also area meters In addition, they obstructflows, so triangular weirs are also intrusive flow meters

cross-The longitudinal hydraulic profile in channels measured by triangular weirs isexactly similar to that measured by rectangular weirs Thus, Figure 3.2 can be usedfor deriving the formula for triangular weirs The difference this time is that thecross-sectional area at the critical depth is triangular instead of rectangular From

Figure 3.3, the cross-sectional area, A, of the triangle is

(3.11)

Multiplying this area by V c produces the discharge flow Q.

Now, the Froude number is equal to V c/ For the triangular weir to be ameasuring device, the flow must be critical near the weir Thus, near the weir, the

Froude number must be equal to 1 D, in turn, is A /T, where T = 2y c tan Along

with the expression for A in Equation (3.11), this will produce D = y c/2 and,consequently, for the Froude number of 1 With Equation (3.2), this

substituted for y c in the expression for A and the result multiplied by to produce the flow Q The result is

(3.12)

where 16/ has been replaced by K to consider the nonideality of the flow The value of the discharge coefficient K may be obtained using Figure 3.4 Thecoefficient value obtained from the figure needs to be multiplied by 8/15 before

using it as the value of K in Equation (3.12) The reason for this indirect substitution

is that the coefficient in the figure was obtained using a different coefficient derivation

from the K derivation of Equation (3.12) (Munson et al., 1994).

Example 3.4 A 90-degree V-notch weir has a head H of 0.5 m What is the

flow, Q, through the notch?

FIGURE 3.4 Coefficient for sharp-crested triangular weirs (From Lenz, A.T (1943) Trans.

AICHE, 108, 759–820 With permission.)

=

gD.

θ 2

188 Physical–Chemical Treatment of Water and Wastewater Solution:

From Figure 3.4, for an H = 0.5 m, and θ = 90°, K = 0.58

Therefore,

Example 3.5 To measure the rate of flow of raw water into a water treatment

plant, an engineer decided to use a triangular weir The flow rate is 0.33 m3/s Designthe weir The width of the upstream rectangular channel to be connected to the weir

is 2.0 m and the available head H is 0.2 m.

Solution: Because the available head and Q are given, from Q = K(8/15)tan ×

, Ktanθ/2, can be solved The value of the notch angle θ may then

be determined from Figure 3.4

From Figure 3.4, for H = 0.2 m, we produce the following table:

Figure 3.4, however, the value of K for θ greater than 90° is 0.58 Therefore,

Given available head of 0.2 m, provide a freeboard of 0.3 m; therefore, sions: notch angle = 171°, length = 2 m, and crest at notch angle = 0.2 m + 0.3 m

dimen-= 0.5 m below top elevation of approach channel Ans

 

2 -tan 2 9.81( ) 0.2( )5/2

15 -

 

2 -tan

8 15

θθθθ

2

2tan

15 -

3.1.3 T RAPEZOIDAL W EIRS

As shown in Figure 3.3, trapezoidal weirs are weirs in which the cross-sectionalarea where the flow passes through is in the form of a trapezoid As the flow passesthrough the trapezoid, it is being contracted; hence, the formula to be used ought to

be the contracted weir formula; however, compensation for the contraction may bemade by proper inclination of the angle θ If this is done, then the formula forsuppressed rectangular weirs, Equation (3.8), applies to trapezoidal weirs, using the

bottom length as the length L The value of the angle θ for this equivalence to be so

is 28° In this situation, the reduction of flow caused by the contraction is balanced by the increase in flow in the notches provided by the angles θ This type

counter-of weir is now called the Cipolleti weir (Roberson et al., 1988) As in the case counter-of the

rectangular and triangular weirs, trapezoidal weirs are area and intrusive flow meters

flow can be measured if a pressure difference can be induced in the path of flow

The venturi meter is one of the pressure-difference meters As shown in b of

Figure 3.5, a venturi meter is inserted in the path of flow and provided with astreamlined constriction at point 2, the throat This constriction causes the velocity

to increase at the throat which, by the energy equation, results in a decrease inpressure there The difference in pressure between points 1 and 2 is then takenadvantage of to measure the rate of flow in the pipe Additionally, as gleaned fromthese descriptions, venturi meters are intrusive and area meters

The pressure sensing holes form a concentric circle around the center of the

pipe at the respective points; thus, the arrangement is called a piezometric ring Each

of these holes communicates the pressure it senses from inside the flowing liquid

to the piezometer tubes Points 1 and 2 refer to the center of the piezometric rings,respectively The figure indicates a deflection of ∆h Another method of connecting

piezometer tubes are the tappings shown in d of Figure 3.5 This method of tapping

is used when the indicator fluid used to measure the deflection, ∆h, is lighter than

water such as the case when air is used as the indicator The tapping in b is used if

the indicator fluid used such as mercury is heavier than water

The energy equation written between points 1 and 2 in a pipe is

(3.13)

where P is the pressure at a point at the center of cross-section and y is the elevation

at point referred to some datum V is the average velocity at the cross-section and h l

is the head loss between points 1 and 2 γ is the specific weight of water The subscripts

P1

γ - V1

2

γ - y1–h l

γ - V2

2

γ - y2

=

190 Physical–Chemical Treatment of Water and Wastewater

1 and 2 refer to points 1 and 2, respectively Neglecting the friction loss for themoment and since the orientation is horizontal in the figure, the energy equationapplied between points 1 and 2 reduces to

(3.14)

Using the equation of continuity in the form of (πD2/4)V1= (πd2/4)V2, where

D is the diameter of the pipe and d is diameter of the throat, the above equationmay be solved for V2 to produce

(3.15)

where β=d/D.Let us now express P1−P2 in terms of the indicator deflection, ∆h Apply themanometric equation in b in the sequence 1, 4, 3′, 3, 2 Thus,

P1+∆h14γ−∆h3′3γind−∆h32γ =P2 (3.16)

FIGURE 3.5 Venturi meter system: (a) flushing system; (b) Venturi meter; (c) coefficient of discharge (From ASME (1959) Fluid Meters—Their Theory and Application, Fairfield, NJ; Johansen, F C (1930) Proc R Soc London, Series A, 125 With permission.) (d) Piezometer taps for lighter indicator fluid.

(d)

Indicator deflection, Dh

Indicator deflection, Dh

Pressure sensing holes

3 3' 4

(b) VENTURI METER 1.1

1.0 0.9 0.8

Re = dYp/m (c)

d/D = 0.6 d/D = 0.4

d/D = 0.5 Flow indicator

Water supply AUTOMATIC FLUSHING SYSTEM

where ∆h14, ∆h3′3 (=∆h), ∆h32, and γind refer to the head difference between points 1and 4, points 3′ and 3, and points 3 and 2, respectively γind is the specific weight ofthe indicator fluid used to indicate the deflection of manometer levels (i.e., the twolevels of the indicator fluid in the manometer tube) Equation (3.16) may be solved

for P1 − P2 producing P1 − P2 = ∆h(γ ind − γ) However, in terms of an equivalent

(3.17)

and

(3.18)

If the tapping in d is used where the indicator fluid is lighter than water and the

above derivation is repeated, γind − γ in Equation (3.18) would be replaced by γ − γind.Note that is not the manometer deflection; it is the water equivalent of themanometer deflection

may be substituted for P1 − P2 in Equation (3.15) and both sides of the

equation multiplied by the cross-sectional area at the throat, A t, to obtain the

dis-charge, Q The equation obtained by this multiplication is simply theoretical, ever; thus, a discharge coefficient, K, is again used to account for the nonideality of actual discharge flows and to acknowledge the fact that the head loss, h l, wasoriginally neglected in the derivation The corrected equation follows:

how-(3.19)

where values of K may be obtained from c of Figure 3.5 and A t = πd2/4 Because

P1 − P2 = Equation (3.19) may also be written in terms of P1 − P2 as follows

(3.20)

Equation (3.20) may be used if the venturi meter is not oriented horizontally This

is done by calculating the pressures at the points directly and substituting them intothe equation

When measuring sewage flows, debris may collect on the pressure sensing holes.Hence, these holes must be cleaned periodically to ensure accurate sensing of pressure

at all times In a of Figure 3.5, an automatic cleaning arrangement is designed using

an external supply of water Water from the supply is introduced into the pipingsystem through flow indicator, pipes, valves, and fittings, and into the venturi meter.The design would be such that water jets at high pressure are directed to the pressuresensing holes These jets can then be released at predetermined intervals of time towash out any cloggings on the holes Of course, at the time that the jet is released,

P1–P2 = ∆h(γind–γ) = ∆hH2Oγ

∆hH2O ∆h(γind–γ)

γ -

=

192 Physical–Chemical Treatment of Water and Wastewater

erratic readings of the pressure will occur and the corresponding Q should not be

used Line pressure of 70 kN/m2 in excess over source water supply pressure issatisfactory

Example 3.6 The flow to a water treatment plant is 0.031 cubic meters per

second The engineer has decided to meter this flow using a venturi meter Designthe meter if the approach pipe to the meter is 150 mm in diameter

Solution: The designer has the liberty to choose values for the design

param-eters, provided it can be shown that the design works Provide a pressure differential

of 26 kN/m2 between the approach to the tube and the throat Initially assume athroat diameter of 75 mm

From the appendix, ρ = 997 kg/m3

and µ = 8.8(10−4

) kg/m⋅s (25°C); therefore,

From c of Figure 3.5, at d /D = 75/150 = 0.5, K = 1.02; therefore,

Therefore, design values: approach diameter = 150 mm, throat diameter = 75 mm,pressure differential = 26 kN/m2

Ans

3.1.5 P ARSHALL F LUMES

Figure 3.6 shows the plan and elevation of a Parshall flume As indicated, the flowenters the flume through a converging zone, then passes through the throat, and outinto the diverging zone For the flume to be a measuring flume, the depth somewhere

at the throat must be critical The converging and the subsequent diverging as wellthe downward sloping of the throat make this happen The invert at the entrance tothe flume is sloped upward at 1 vertical to 4 horizontal or 25% Parshall flumesmeasure the rate of flow proportional to the cross-sectional area of flow Thus, theyare area meters They also present obstruction to the flow by making the constriction

at the throat; thus, they are intrusive meters

2g P( 1–P2)γ -

4 -

997 9.81( ) - 0.032 m3/s  0.031 m3

/s

As defined by Chow (1959), the letter designations for the dimensions are described

as follows:

W = size of flume (in terms of throat width)

A = length of side wall of converging section

2/3A = distance back from end of crest to gage point

B = axial length of converging section

C = width of downstream end of flume

D = width of upstream end of flume

E = depth of flume

F = length of throat

G = length of diverging section

K = difference in elevation between lower end of flume and crest of floor level = 3 in

M = length of approach floor

N = depth of depression in throat below crest at level floor

P = width between ends of curve wing walls

R = radius of curved wing walls

X = horizontal distance to Hb gage point from low point in throat

Y = vertical distance to Hb gage point from low point in throat

The standard dimensions of the Parshall flume are shown in Table 3.1

If the steps used in deriving the equation for rectangular weirs are applied to the

FIGURE 3.6 Plan and sectional view of the Parshall flume.

PLAN Throat zone Converging zone Divergingzone

Water surface

K = 3”

2/3A A

R

194 Physical–Chemical Treatment of Water and Wastewater

Parshall flume between any point upstream of the flume and its throat, Equation (3.7)will also be obtained, namely:

H may be replaced by H a, the water surface elevation above flume floor level in the

converging zone, and L may also be replaced by W, the throat width Using a coefficient K, as was done with rectangular weirs, and making the replacements

produce

(3.21)

The value of K may be obtained from Figure 3.7 (Roberson et al., 1988) All unitsshould be in MKS (meter-kilogram-second) system

This equation applies only if the flow is not submerged Notice in Figure 3.6

that there are two measuring points for water surface elevations: one is labeled H a,

in the converging zone, and the other is labeled H b, in the throat These points

actually measure the elevations H a and H b The submergence criterion is given by

the ratio H b /H a If these ratio is greater than 0.70, then the flume is considered to

be submerged and the equation does not apply