ALTERNATIVE LINEAR WEIR DESIGN In different ...

ALTERNATIVE LINEAR W E I R DESIGN By P r a b h a t a K . Swamee, 1 Santosli K. Patfaak, 2 Mahesh Agarwal, 3 A n w a r S. Ansari 4 ABSTRACT: Weirs are devices installed in channels by means of which flow may be measured within a given degree of accuracy. Linear weir flow-measuring devices introduce less relative error in discharge measurement. The design of a linear weir is such that the base approaches infinity; this makes its theoretical design physically unrealizable. Various attempts have been made to achieve a practical weir profile by cutting off the infinite bottom wings and providing base weirs of different shapes. Furthermore, all these designs use a constant discharge coefficient, whereas there is a considerable variation in the discharge coefficient with the head/weir-height ratio. A linear weir profile that includes the variation of discharge coefficient and does not require base and datum corrections is presented herein. A design example has also been included. INTRODUCTION

In different fields, such as hydraulic, irrigation, environmental, and chemical engineering, weirs are used as flow-measuring devices. Among the oldest are rectangular, triangular, and trapezoidal weirs. Linear or proportional weirs, however, may have greater potential than conventional weirs in some cases. A linear weir normally consists of a base, weir, and linear portion whose profile is designed to achieve a linear discharge head relationship. For a linear weir, the proportionate error in discharge computation is equal to the proportionate error in head measurement. Thus, the resulting error in the discharge measurement is independent of the head over the weir. On the other hand, there is a large variation in the proportionate error for other shapes of weirs. In the past decades, several investigations have been carried out in the field of linear weirs (Banks et al. 1968; Chandrasekharan and Lakshmana Rao 1976; Hlavek 1961; Lakshmana Rao and Purushottama 1979; Lakshmana Rao and Chandrasekaran 1970; Lakshmana Rao and Abdul Bhukari 1971; Lakshmana Rao 1975; Murthy and Pillai 1978a, 1978b; Srinivasulu and Raghavendran 1970). Literature reviews indicate that further research work is needed for the design of a linear weir without base and datum corrections. Furthermore, in all of these investigations, the discharge coefficient was assumed to be constant. Since there is considerable variation in the discharge coefficient, the linear weir profiles investigated to date will not exhibit linearity. The present study investigated a linear weir profile and was carried out in two phases. In phase 1, the weir profile was developed assuming the discharge coefficient as constant. Then, experiments were per'Prof. of Civ. Engrg., Univ. of Roorkee, Roorkee-247 667, India. 2 Prof. of Civ. Engrg., Univ. of Roorkee, Roorkee-247 667, India. 'Formerly, Postgrad. Student, Dept. of Civ. Engrg., Univ. of Roorkee, Roorkee247 667, India. 4 Formerly, Postgrad. Student, Dept. of Civ. Engrg., Univ. of Roorkee, Roorkee247 667, India. Note. Discussion open until November 1, 1991. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on March 27, 1990. This paper is part of the Journal of Irrigation and Drainage Engineering, Vol. 117, No. 3, May/June, 1991. ©ASCE, ISSN 0733-9437/91/O003-0311/$1.0O + $.15 per page. Paper No. 25870. 311

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formed with this linear weir to investigate the variation of its discharge coefficient with the head/weir-height ratio. In phase II, this variation of the discharge coefficient was used to derive the profile of a practical linear weir. Performance of the proposed weir was tested experimentally. ANALYTICAL FORMULATION: PHASE I

For the design of a linear weir profile without base and datum corrections, it is proposed that the lower portion has the following discharge equation: Q = KhlVgih: • y— I

h«h*

(i)

in which Q = weir discharge; K = nondimensional constant; h* = length parameter; and g = gravitational acceleration. The upper portion has the following discharge equation: Q = KhWW* (—)

h»h*

(2)

The problem of determining the shape of a linear weir may be solved by assuming a single function of discharge to describe the shape of the weir over the entire range of head. Based on (1) and (2), a discharge equation of the following type may be chosen:

KhWgh*

*»

•

hi

'

\h

»

(3)

in which p = a transition parameter. Considering an elementary strip of length 2x(y) and width dy at height y from the weir crest (Fig. 1), the discharge Q through the weir is given by:

J

Q = 2Cdo

x{yW2QQi - y)dy

(4)

xi o

in which Cd0 = a discharge coefficient. Equating Q from (3) and (4) and introducing nondimensional terms H = h/h*, Y = y/h*, and X(Y) = x(y)/ /z* gives

I

H If

X(Y)VH - YdY = — - — (H-1S/P + H~Up)~p (5) 2\/2Cd0 Jo Eq. (5) is a Volterra integral equation of first kind. Differentiating (5) with respect to H gives H

X(Y)dY_

!0 Jo

K

H°\l.5

H - Y ~ \/2Cd0

+H°-5/p)

(1 + H°-5/p)p+1

(6)

Eq. (6) is in the form of Abel's integral equation and its solution is given by

_ X(H)

K

A . C r°,5(l-5 + Y°5/P)

~ V2irCd0 dH J0

(1 + Y051-)^1

dY VH^Y

(?)

312

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FIG. 1. Definition Sketch

Assuming the integrand to be linear in the range 0 to H and using the modified midpoint rule of product integration approximation, (7) becomes m-lr

/ - A 0.5/p-,

5. w

(17a)

bx = 1.15 - 0.146 W . . . . . . .

(17*)

in which

Combining (16a) and (17a) in such a manner so as to fit the point in the intermediate range of h/w, the following full-range equation for the discharge coefficient was obtained: 316 Downloaded 14 Aug 2009 to 124.124.247.56. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

0.5

*^do

*Y

«i

0.45 + bx

(18)

ANALYTICAL FORMULATION: PHASE II

In this section, the variation of Cdo with h/w, as obtained from phase I experiments, was incorporated in the design of a practical linear weir. Substituting (18) into (5) gives:

r

X(Y)VH - YdY =

Jo

K 2V2

0.5 + ax w

Hl (1 +H°-5/p)"'

+ \ 0.45 + bx -

(19)

Following the procedure used in the analytical formulation phase I by which the weir profile with constant discharge coefficient [(9) and (14)] were obtained, namely, differentiating (19) with respect to H and solving the resulting Abel's integral equation by using product integration equation, gives K(c\ + Y5d\t7 X(Y) = IT

0.25F + ( 1.75 - - L -j(0.5y)°- 5 / p + 1.5

CxdX

K4bxW2YAd\ 2

-flic,

5

5 0

-nc xdsW(c + Yd ) * ~Y(4biW2Y4di

- axc\)

5

_ 2cidiW(Y dl

--c\)

1+

0.5F [1 + (0.5Y)°-5/pf

Y(48bjW2Y3d] + T,a\c\) Wc1d1(.4bW2Y4d61 - aict)

1.5 + (0.5r)0-5/" : + + 0.5 4bxW2Y*d\ - axc\ 1 + (0.5y)°-5/p 2 A

%b{W Y d\

(20)

where c, = 0.45y

2bxW

(21a)

and * = 0.5(1+^

(21b)

Eq. (20) gives the profile of a practical weir without base and datum corrections, which incorporates the variation of Cdo with h/w. The discharge through this linear weir is given by:

f

(22)

CoX^VH^YdY

Jo

in which Cio and X(Y) are defined by (18) and (20), respectively. In this study, (22) has been evaluated by Gauss-Chebyshev quadrature method. Percentage departure e has been calculated for different values of p and W. It was found that the percentage departure is minimum for p = 0.4 and W = 1.5, that is, for these values, the weir discharge for different heads is 317

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FIG. 5. Practical Linear Weir

close to that for the theoretical linear weir. Putting Y = 0, W = 1.5, and p = 0.4 in (20) gives the following nondimensional base width B0: B0 = 1.79K.

(23)

Eliminating K between (20) and (23) and putting p = 0.4 and W = 1.5 gives: / 5 i

X(Y) = 0.18fi„ + 0.18B,

I,5\0.2

,,f

c2d22A

[0.5(0.5r>2-5 + 1.125(0.5y)125 + 1.5]

5.5867 4 ^ - 0.047^ f CM

+ bir

1 + 0.5577

t ^

Y(2.793Y*b62 - 0.023a£) c2(a\ + b\)

3 7

6Y(l5.6Y b 2 + O.OOla^) \\.\12Y%\ c2{5.5S6Y%62 - 0.047af) ~ 5.586K4Z;| - 0.047K I I

1'0

I 1 0'9

(28)

-

0 0 o «

o

o o °

Jr

°

0o

^

o°D

CO 1

o 0^8

o 0

0'7

1 I

o

|

0

'

0-6

1

0'5

1

0-4

0-3

0-2

1

0.1

0 0

I 2

1 4

1

1

FIG. 6. Variation of Crf, with h/h* 320 Downloaded 14 Aug 2009 to 124.124.247.56. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

DESIGN EXAMPLE

The problem is to design a linear weir for a rectangular channel having the following data: Bed slope: S„ = 0.002. Manning's roughness coefficient: n = 0.014. Maximum discharge: g m a x = 0.3 m 3 / s . Minimum discharge: g min = 0.05 m 3 / s . Channel width: bc = 1.5 m. Solution: Let the base width of the weir be equal to the channel width, i.e. b0 = bc = 1.5 m

(29)

Combining (27) and (28) one gets e m i n = Q.664bMVgh^

(30)

Substituting the values of Qmin and ba in (30) one obtains A* = 0.030 m

(31)

Substituting /i* from (31) into (27), yields hmin = 0.12 m

(32)

Similarly 2 m a x = (0A66^boVghi)hmm

(33)

Substituting the values of Qmm, b0, and h*, in (33) gives / W = 0.708 m

(34)

Since w/h* = 1.5, the height of the weir crest is w = 1.5fc* = 0.045 m

(35)

Therefore, for g max the height of the water level is fcmax + w = 0.0708 + 0.045 = 0.753 m

(36)

Given Manning's equation Q = - ARV3Sln

(37)

n For a rectangular channel, (37) may be rearranged as: y5Jl

_

Q„

(bc + 2v„)2'3 " « '

2

(38) 3

in which y„ = the normal depth. When Q = g m a x = 0.3 m /s v 5/3 — = 0.04778 (1.5 + 2v„)

(39)

2/3

Solving 39 by trial and error gives 321

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1

y„ = 0.21 m

(40)

Hence, one of the limitations of a linear weir is that the water level near the weir increases considerably and the channel banks will have to be raised near the weir. CONCLUSIONS Based on the following study, the following conclusions are drawn: 1. Considering the variation of discharge coefficient with head over the weir, a practical linear weir profile was developed. This weir profile does not require base or datum corrections. 2. Weirs designed by the methods presented behave linearly for heads over the crest greater than four times the vertical scaling length. APPENDIX I.

REFERENCES

Banks, W. H. H., Burch, C. R., and Shaw, T. L. (1968). "The design of proportional and logarithmic thin plate weirs." J. Hydr. Res., 6(2), 75-105. Chandrasekharan, D., and Lakshmana Rao, N. S. (1976). "Characteristics of Proportional Weirs." J. Hydr. Div., ASCE, 102(11), 1677-1692. Hlavek, R. (1961). "A contribution to theoretical and experimental research on linear weirs." La Houille Blanche, Grenable, France, 16(4), 469-494. Lakshmana Rao, N. S. (1975). "Theory of weirs." Advances in hydrosciences. Academic Press, New York, N.Y., 10, 346-390. Lakshmana Rao, N. S., and Abdul Bhukari, C. H. (1971). "Linear proportional weirs with trapezoidal bottom." J. Hydr. Res., 9(3). Lakshmana Rao, N. S., and Chandrasekaran, D. (1970). "A new 'baseless' proportional weir." J. Hydr. Res., 8(3), 341-355. Lakshmana Rao, N. S., and Purushottama, G. (1970). "Experimental studies on linear proportional weirs with triangular bottom." J. Hydr. Res., 8(4), 449-455. Murthy, K. K., and Pillai, K. G. (1978a). "Design of constant accuracy linear proportional weir." J. Hydr. Div., ASCE, 104(4), 527-541. Murthy, K. K., and Pillai, K. G. (1978b). "Modified proportional V-notch weirs." J. Hydr. Div., ASCE, 104(5), 775-791. Srinivasulu, P., and Raghavendran, R. (1970). "Linear proportional weirs." J. Hydr. Div., ASCE, 96(2), 379-389. APPENDIX II.

NOTATION

The following

aua2 B0 b„ bc bub2 doyCji CX,C2

dud2

9

H h

= = = = = = = = = = =

symbols are used in this paper:

functions;

b0/K; weir base width; channel width; function; discharge coefficient; function; function; gravitational acceleration; h/h*; head over weir crest; 322

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h* K P Q

w w X X

Y y e

= = = = = = = = = = =

scaling length; proportionality constant; transition exponent; weir discharge; w/7i*; weir height; x/h*; horizontal coordinate; y/h*; vertical coordinate; and percentage departure.

323

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