DISCHARGE COEFFICIENT FOR FULL-WIDTH

SHARP-CRESTED HIGH WEIRS

Masao OSHIMA1, Toru ISHIDO2 and Wubbo BOITEN3

1Retired Professor, Dept. Mechanical Engineering, Kanagawa Inst. Tech. ([Home Address] 431 Bukko-cho, Hodogaya-ku, Yokohama 240-0044, Japan)

E-mail: mosim@nifty.com 2Technical Superintendent, Custom Pump Division, Haneda Plant, Ebara Corp.

(11-1 Haneda-Asahi-cho, Ota-ku, Tokyo 144-8510, Japan) E-mail: ishido.toru@ebara.com

3Retired Lecturer, Hydrology and Quantitative Water Management Group, Wageningen Univ. ([Home Address] Gasthuisbouwing 2, Bennekom, 6721 XK, The Netherlands)

E-mail: wboit14@hetnet.nl

The use of the Rehbock formula for measuring flow rates with full-width sharp-crested weirs, specified in ISO 1438, is restricted to weirs with a weir plate height of less than 1 m. Previous studies have shown that deviations in measured discharge coefficient from the Rehbock coefficient increase with increasing weir height at heights of over 1 m, owing to the scale effect of weir height. In this paper, a new discharge coefficient is proposed for use when applying the Rehbock formula to weir plate heights over 1 m, as discharge flow rates of large-size pumps are usually measured using full-width weirs with a plate height as high as 2 m. Its validity is examined and verified, based on test results in studies by Schoder-Turner and later researchers. The paper also discusses the effect of scale on discharge coefficient in relation to the boundary layer on the weir plate. Key Words : discharge coefficient, flow measurement, full-width weir, Rehbock formula, high-crest

height, high-volume flow

1. INTRODUCTION

A high volume flow – such as 10 m3/s discharged by large size pumps – can only be measured using full-width weirs due to the size limitation of the ori-fice plates and Venturi tubes, where full-width weirs with a weir plate as high as 2 m are commonly used. Failure to use high weirs can lead to high channel velocity causing rough surface flow, resulting in a highly fluctuating head and higher uncertainty.

Rehbock’s full-width weir formula1), specified in ISO 1438: 20082), was verified and determined using test data obtained from weirs with a weir plate height of less than 1 m. However, data for plate heights higher than 1 m show increasing deviation from the formula with increasing plate height. This has led to the field of discharge flow measurement of large-size pumps now requiring new equations for a discharge coefficient for use with full-width weirs with a plate height higher than 1 m, as such pumps become in-creasingly adopted and weirs are the only apparatus

suitable for measuring large flow rate. In this paper, new equations are proposed and

examined for use with full-width weirs with a weir plate height higher than 1 m, based on the original Rehbock formula and test results from studies by Schoder and Turner3), Kindsvater and Carter4) and Kurokawa et al.5). The reasons for the changes in the discharge coefficient with increasing weir height are also discussed, together with the results obtained by using the computational fluid dynamics (CFD) code. 2. FORMULAE FOR DISCHARGE COEF-

FICIENT FOR FULL-WIDTH WEIRS (1) Rehbock formula

Rehbock1) published the following formula for measuring open channel flow using full-width weirs:

2/312

32

ed hbgCQ = (1)

Journal of JSCE, Vol. 1, 360-365, 2013

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Fig. 1 Thin plate weir.

in which

phCd

10813.06035.0 += (2)

0011.011 += hh e (3) where Q denotes flow rate [m3/s]; Cd is discharge coefficient; g is acceleration of free fall [m/s2]; b is width of weir [m]; h1e is effective weir head [m]; h1 is measured weir head [m]; and p is weir plate height [m] (see Fig.1). The results reported by Schoder-Turner3) and other data revealed that the deviation in the measured values of the discharge coefficient from the values by Eq. (2) was less than 1%, provided that the weir plate height was less than 1 m. Although Schoder-Turner3) included test results obtained with weir plate height over 1m, the results were not discussed in Rehbock’s paper1), as the data showed increasing deviations with increasing weir height, owing to the scale effect of the weir height.

The idea of the scale effect was not popular when Rehbock published his paper in 1929. He, therefore, took the deviations as caused by some unexplainable reasons, and excluded the weir plate height over 1m from the application range. (2) JIS formula

For the purpose of providing a formula for full- width weirs with a weir plate height higher than 1 m for measuring high-volume flows discharged from large-scale pumps, Ishihara and Ida6) proposed a new formula for discharge coefficient as shown below, based solely on the test results of Schoder and Turner3).

( )ε+⎟⎟⎠

⎞⎜⎜⎝

⎛++= 1

08.01000

1605.0 1

1 ph

hCd (4)

in which ε = 0 for p ≤ 1 m (5) ε = 0.55 ( p − 1) for p > 1 m

where ε is coefficient adjustment. This is essentially the same as the 1913 Rehbock

formula1), which was limited to application to weirs with a weir plate height of 1 m or less. Expansion of

the limitation range by applying a modification term was attempted, and showed that the term (1+ε) shown above corresponded best to the test results. The term adjusts the flow coefficient only in the range of p above 1 m, and increases with increasing p.

The formula was first adopted in JIS B 83027) in 1953. However, the equation was not non-dimen- sional, and was therefore not appropriate for the equation to be specified in an ISO standard, which led to the present proposal for new discharge coeffi-cient equations stipulated in this paper. 3. NEW EQUATIONS FOR DISCHARGE

COEFFICIENT FOR p OVER 1 m

International Standard ISO 1438:20082) adopted the 1929 Rehbock formula1), Eq. (2) and (3), for flow calculation using full-width weirs, in which the dis-charge coefficient and the effective head were mod-ified slightly as follows:

ph

C d1083.0602.0 += (6)

h1e = h1 + 0.0012 (7)

with the limitation on the weir plate height set to 1 m. For weir plate height over 1m, the measured values

of coefficient by Schoder and Turner3) and others, as referred to later, showed deviations from the values given by Eq. (6), which increased with increasing weir plate height.

The authors tried to devise new equations covering the range of p higher than 1 m, and propose here the following equations for calculating the discharge coefficient, hereinafter referred to as the Boiten co-efficient.

m 1.0for 083.0602.0 1 =+= pph

Cd

[same as Eq. (6)]

m5.2for138.0608.0 1 =+= pph

Cd (8)

For other values of p between 1.0 m and 2.5 m, the values of both constant and coefficient are obtained by linear interpolation, that is

{ }ph

p

pCd1)1(0367.0083.0

)1(004.0602.0

−++

−+= (9)

Practical limitations are: (a) h1 shall be between 0.03 m and 0.8 m but

not greater than b/4. (b) b shall be not less than 0.5 m. (c) p shall be less than 2.5 m.

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The fundamental concepts underlying the new equations are: (1) The new equations maintain the ISO Rehbock

coefficient at p = 1 m, so that the equations have the same construction of terms.

(2) Equation (8) is defined for the maximum height of weir plate by an equation of the type similar to the ISO Rehbock coefficient.

(3) For intermediate height, the equation is defined by linear interpolation for simplicity.

(4) Strictly speaking, both terms in the right-hand side of the discharge coefficient, being equal to Eq. (6) at p = 1 m and equal to Eq. (8) at p = 2.5 m, should be expressed by a function; for example, by an exponential function of a Reynolds Number composed of the weir plate height and the channel velocity. However, no other fluid except water is involved in the measurement, the channel flow is always a gravitational flow, and the maximum head of weir is limited to 0.8 m. Accordingly, it is considered practical to represent the values of both constant and coefficient by a linear-inter- polation-based formula as shown above.

(5) Limitations on the application regarding weir dimensions and measurement ranges should be determined by the particular range subject to testing and from the practical viewpoint. A weir head h1 exceeding 0.8 m causes rough approach channel flow surface, and yields unreliable measurement results. A weir plate height p ex-ceeding 2.5 m is substantially unsuitable for practical use.

4. EXAMINATION OF THE COEFFI-

CIENT WITH TEST RESULTS

This section examines the proposed equations for the discharge coefficient, based on the test results reported by Schoder and Turner3), by Kindsvater and Carter4) and by Kurokawa et al.5), where the values of measured coefficients are calculated by

( ) ( ) 2/31

123/2 ahgb

QCd+

= (10)

with a = 0.0012 m, where a is head adjustment. (1) Schoder and Turner

A study by Schoder and Turner3) examined the results from extensive tests on full-width weirs car-ried out at the Hydraulic Laboratory of Cornell University, New York. The height of the weirs ranged from 0.152 to 2.286 m, the heads from 0.004 to 0.838 m, and the width of channel from 0.274 to 1.280 m. Flow measurement was carried out using

Fig. 2 Test results by Schoder and Turner3), p = 1.676 m.

Fig. 3 Test results by Schoder and Turner3), p = 2.286 m.

the volumetric method.

Figures 2 and 3 show comparisons of the pro-posed coefficient with the test results for p of 1.676 and 2.286 m.

All the test results show very good agreement with the Boiten coefficient, with any deviations within ±1.5%, except for the range of h1 higher than 0.8 m, which corresponds to

h1/p = 0.48 for p = 1.676 m, and = 0.35 for p = 2.286 m,

and except for the range of small values of h1 lower than 0.024m, which corresponds to

h1/p = 0.014 for p = 1.676 m, and = 0.010 for p = 2.286 m. When h1 ≅ 0.610 m for p = 1.676 m, or 0.688 m for

p = 2.286 m, wide scattering is found in the values of the measured coefficients in both cases. This could be caused by the performance features of circulating pumps, such as the unstable head-capacity charac-teristics sometimes observed during parallel opera-tion of centrifugal pumps.

(2) Kindsvater and Carter

A comprehensive solution was reported by Kinds- vater and Carter4) for grasping the discharge charac-

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Fig. 4 Test results by Kindsvater and Carter4), BOR tests,

p = 0.174~1.524 m. teristics of rectangular, thin-plate weirs, including full-width weirs. Their solution was based on the results on a wide range of tests. For full-width weirs, two cases wherein the weir plate height exceeded 1 m ̶ namely p = 1.524 m and 1.402 m ̶ were tested with channel width of 0.612 m at weir head h1 of max. 0.384 m (h1/p = 0.252 or 0.274).

The results of these cases are shown in Fig. 4, along with the results of cases of lower weir plate height. In the figure, lines of the Boiten coefficient for p = 1.402 and 1.524 m and ISO Rehbock coef-ficient, Eq. (6), are indicated for reference. Measured points for cases of p = 1.402 and 1.524 m were not identified in the original figure4), but a group of points in line, shown by solid circles, could be esti-mated to be those for p = 1.402 and 1.524 m, as the maximum head for these cases was 0.384 m (h1/p = 0.274 or 0.252). The measured points run quite well on the Boiten coefficient lines, indicating that the effect of weir plate height had been revealed in their tests. Unfor-tunately, they performed tests only on two cases where the weir plate height exceeded 1 m but not excessively. Therefore, the authors of the paper over- looked the effect of the plate height by not comparatively examining their results versus those by Schoder and Turner. (3) Kurokawa et al.

Kurokawa et al.5) carried out extensive tests on large scale full width weirs. The study aimed to verify Schoder and Turner’s findings that deviations in measured discharge coefficient from the ISO Re-hbock coefficient increase with an increase in weir plate height.

The study used weirs with a width of 1.5 m, in which the weir plate height was changed to 0.8 m, 1.4 m, and 2 m by sliding the weir plate. The flow rate was measured with an orifice plate calibrated by the weighing method, in which the upstream and down-

Fig. 5 Test results by Kurokawa et al.5), p = 1.4 m.

Fig. 6 Test results by Kurokawa et al. 5), p = 2 m.

stream test pipes ̶ along with the test orifice plate ̶ were used in the calibration.

Figures 5 and 6 show a comparison of the pro- posed coefficient with the test results. The test re-sults also show very good agreement with the Boiten coefficient, rather than with the ISO Rehbock coef-ficient, with any deviations within ±1.5%, except for the range of h1 lower than 0.11 m, which corresponds to

h1/p = 0.079 for p = 1.4 m, and = 0.055 for p = 2 m. As seen in the figures, relatively large deviations

are observed when the values of h1/p are small. These are thought to be caused by insufficient sharpness of the crest edge. The effect of the edge is limited to the coefficient in the range of small h1/p, and is consid-ered to be eventually negligible in the range of larger h1/p.

(4) Summary

As shown in Figs. 2 through 6, the Boiten coeffi-cient adequately represents the behavior of the dis-charge coefficient of the full-width weirs with weir plate height exceeding 1 m. It is recommended that the Boiten coefficient expressed by Eqs. (6) and (8) be specified as an ISO formula for full-width weirs

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with the limitations stipulated in the former section.

5. DISCUSSIONS

(1) Causes of change in discharge coefficient for weir plates with a height exceeding 1 m

Flow coming to the weir plate through the channel bottom region goes up along the weir plate (see Fig.1). A boundary layer is formed on the plate, and at the top of the plate, the boundary layer joins the lowest part of the overflow (nappe) from the crest, giving non-negligible effect on the value of the discharge coefficient, as discussed later.

The boundary layer thickness δ (l) on a flat plate with a length of l is shown by the following equation (see Daugherty, R.L. et al.8)):

5/1

377.0)(−

⎟⎠⎞

⎜⎝⎛=

νδ

lUll (11)

where U denotes velocity of flow around the plate, and ν denotes kinematic viscosity [m2/s].

The boundary layer thickness at the top of the weir plate formed by upward flow along the weir plate can be roughly estimated by substituting the upward velocity wy for U and weir plate height p for l in the above equation, where, furthermore for the purpose of simplification, the velocity wy is substituted by the mean channel velocity u0, as the velocity of upward flow has about the same magnitude as the mean channel velocity (see Fig. 1).

Between two weirs with different plate height p but with the same h1/p, the channel velocity, given by weir flow devided by channel depth and width, is proportional to h1e

1/2 and thus to p1/2, as the weir flow per unit width is proportional to h1e

3/2, and accordingly to p3/2.

When two cases of different plate heights p1 and p2

with the same h1/p ratio are considered, the ratio of boundary layer thickness δ(p2) / δ(p1) between both plates is expressed by

7.0

1

2

5/1

2/31

2/32

1

2

1

2

)()(

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛=

−

pp

pp

pp

pp

δδ (12)

As seen in the above equation, the thickness is not proportional to the plate height, but varies in pro-portion to p0.7. This means the boundary layer thickness becomes relatively thinner along an in-crease in the plate height. This is the cause of the scale effect.

Although the difference in thicknesses at the top between low and high plates is small, the boundary layer joins the lowest side of the nappe, where the velocity is highest, causing some effect on the con-traction of the nappe flow, which is closely related

with the discharge coefficient. Thus, the effect of the difference cannot be neglected.

(2) Computational flow analysis on the boundary

layer on weir plate Channel flow in a full-width weir with a width of 1

m and a 10 m approach length was analyzed using a CFD code for two cases of h1/p, 0.15 and 0.3 and for three cases of p, 0.4 m, 0.8 m, and 2 m. A commer-cial CFD code, Fluent (version 6.3.26), was used here employing the Reynolds Stress Model for turbulence and the VOF method for free surface modeling.

The velocity distribution in the nappe section should be analyzed, when the computational analysis is applied to investigate directly the effect of scale on the weir discharge coefficient. Unfortunately, limi-tations of existing technology prohibit a precise flow analysis at the boundary between water and free air. That is, due to the mixture of water and air bubbles changing its density only gradually in the process of the CFD at the lower surface and upper channel flow surface, a clear-cut boundary cannot be established, whereby change in the discharge coefficient due to

Fig. 7 Non-dimensional velocity distributions at 90% of weir

height p (d : distance from weir plate, u0 : mean channel velocity).

Fig. 8 Non-dimensional velocity distribution at 90% of weir

height p.

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the weir plate height cannot be clarified solely by the CFD. In this paper, therefore, the behavior of the boundary layer was studied at a section of 90% weir plate height, or vertical velocity distribution in that section, although this gives only a qualitative result.

The distributions of upward velocity vy along the weir plate at 90% of weir plate height are shown in Figs. 7 and 8. Figure 7 displays the velocity distri-butions in a wide range of distance d from the plate, and Fig. 8 shows a detailed representation in a region close to the plate within d/p < 0.015.

The dimensionless velocity distribution shows very good similarity to each other for both cases of h1/p, as shown in Fig. 7. When observed in detail, however, the distributions show a distinct difference in the boundary layers, that is, in the zone of d/p < 0.005 (d/h1 < 0.02 ∼ 0.03) (see Fig. 8). Although the boundary layer is very thin, its ratio to the weir head amounts to a few percent, and accordingly the change of the ratio brings about not a small effect on the discharge coefficient. 6. CONCLUSION

In this study, new equations for the discharge co-efficient for full-width weirs with weir plate height over 1 m were proposed and examined based on the test results presented by Schoder and Turner3), Kindsvater and Carter4) and Kurokawa et al.5). All data verified the validity of the new equations with their application limited to the practical range of weir plate height and head.

The reason for the change in discharge coefficient

in weirs with a weir plate height exceeding 1 m was explained by the behavior of the boundary layer on the weir plate, using the boundary layer equation on a flat plate, along with the results of the CFD code on the weir plate for three cases of plate height with identical h1/p. ACKNOWLEDGMENTS: The authors wish to express their deep thanks to the members of the Japan Pumps Standards Committee for their encourage-ment and cooperation in preparing this paper. They also sincerely appreciate the work of Mr. M. Ohbuchi of Ebara Corporation, who carried out the flow analysis on full-width weir channels. REFERENCES 1) Rehbock, Th.: Wassermessung mit scharfkantigen Ueber-

fallwehren, Z. VDI, Bd. 73, Nr. 27, pp. 817-823, 1929. 2) ISO 1438, Hydrometry—Open channel flow measurement

using thin-plate weirs, 2008. 3) Schoder, E. W. and Turner, K. B.: Precise weir measure-

ments, Trans. ASCE, Vol. 93 (1711), pp. 999-1190, 1927. 4) Kindsvater, C. E. and Carter, R. W.: Discharge character-

istics of rectangular thin-plate weirs, Journ. ASCE, Vol. 83 (HY-6), pp. 1-35, 1957.

5) Kurokawa, J., Oshima, M., Saito, S. and Ishido, T.: Flow measurement using full width weir, comparison of ISO, JIS and HIS formulae, Trans. Flucome '91 (San Franc.), pp. 669-675, 1991.

6) Ishihara, T. and Ida, T.: Supplemental formulas for rec-tangular weirs without end contractions, Proc. 1st Japan National Congress for App. Mech., pp. 381-384, 1951.

7) JIS B 8302, Measurement methods of pump discharge, 2011.

8) Daugherty, R. L., Franzini, J. B. and Finnemore, E. J.: Fluid Mechanics with Engineering Applications, McGraw-Hill, New York, p. 304, 1989.

(Received March 29, 2013)

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