Hydraulic Characteristics of a Modified Venturi Section

Transcription

1 Utah State University Reports Utah Water Research Laboratory January 1968 Hydraulic Characteristics of a Modified Venturi Section Muhammad Aslam Rasheed Follow this and additional works at: Part of the Civil and Environmental Engineering Commons, and the Water Resource Management Commons Recommended Citation Rasheed, Muhammad Aslam, "Hydraulic Characteristics of a Modified Venturi Section" (1968). Reports. Paper This Report is brought to you for free and open access by the Utah Water Research Laboratory at DigitalCommons@USU. It has been accepted for inclusion in Reports by an authorized administrator of DigitalCommons@USU. For more information, please contact dylan.burns@usu.edu.

2 HYDRAULIC CHARACTERISTICS OF A MODIFIED VENTURI SECTION by Muharrunad Aslam Rasheed Utah Water Researc h Laboratory College of Engineering Utah State University Logan, Utah 1968 PR WR /3-12T

3 ACKNOWLEDGEMENTS The author wishes to express his gratitude to Professor J. E. Christiansen, his major professor for suggesting the research problem and supervising the experimental work. Without his continuoub assistance and encouragement this study could not be completed. Gratitude is also expressed to Dr. R. W. Jeppson for his help in obtaining the Numerical Solution and to Dr. Glen E. Stringham for correcting the rough draft and making many valuable BuggeStiOns. Thanks are also expressed to the Dean of Graduate Studies for making available the funds for constructing the apparatus, and to Utah Water Research Laboratory for allowing the use of laboratory space. The author also wishes to thank bis friend Mr. Nasfin Ahmad and Mr. Kenneth Steele and his staff of the Utah Water Research Laboratory for their valuable help in conducting this study. Mohammad Aslam Rasheed

4 TABLE OF CONTENTS Page INTRODUCTION , Characteristic,S of a suitable water measuring device Objectives REVIEW OF LITERATURE Water measuring devices in use 3 Modified venturi section 4 Equation of discharge 7 THEORY Numerical solution Dimensional analysis of the problem Head loss PROCEDURE Description of apparatus Experimental procedure... RESULTS AND DISCUSSION Results from numerical solution Equation of discharge Coefficient of discharge Head loss Limitations of the study CONCLUSIONS LITERA TURE CITED APPENDIX It Tables "

5 TABLE OF CONTENTS CONTINUED Page List of symbols in computer program Fortran IV program... III III III 62 VITA

6 LIST OF TABLES Table Page L Dimensionless parameters for the device Coefficient of discharge 3. Head loss for the device -55 :58

7 ,! LIST OF FIGURES Figure Page 1. Shape of modified venturi section as proposed by Yallalee Shape of modified venturi section as proposed by Ferguson Shape of modified venturi section used in this thesis 4. Formulation of flow under the top cover section in potential plane... 13, 5. Test model Laboratory arrangement Details of top cover section Secti_on at A-A Location of points for piezometric connections Flow net from numerical solution Q ffa5/4 as a function of f!,.h t Empirical equation for discharge Relation between discharge and quantity Atl2g 6... " Discharge as a function of differential head '2 15. JAh Q as a function of d : 16. Coefficient of discharge for a 5 inch throat opening Coefficient of discharge fora 6 inch throat opening ! 18. Coefficient of discharge for "a 7 inch throat opening 36"

8 LIST OF FIGURES CONTINUED Figures Page 19. Coefficient of discharge for an 8 inch throat opening., Coefficient of discharge for a 9 inch throat opening Coefficient of discharge CD for various throat openings Coefficient of discharge C for various throat openings 39 q 23. Throat velocity as a function of differential head Relationship of throat velocity to differential head Head loss fora 5 inch throat opening Head loss fora 6 inch throat opening Head loss fora 7 inch throat opening Head loss for an 8 inch throat opening Head loss for a 9 inch throat opening Head loss for various throat openings

9 ABSTRACT Hydraulic Characteristics of a Modified Venturi Section by Muhammad Aslam Rasheed utah State University, 1967 Major Professor: Professor J. E. Christiansen Department: Civil Engineering The measurement of water in open channels on extremely flat grades is difficult because of the head loss that is necessary. A modified ventrui section, the contraction of which is provided by a top cover section mounted in a rectangular flume, is proposed to meet the requirements of a suitable measuring device for use in canals of flat gradients. The method of finite differences has been used for the mathematical solution to the idealized inviscid flow problem. This was followed by a laboratory investigation to ascertain the hydraulic characteristics of the proposed design. The experiments were conducted in a 3 feet by 3 feet by 200 feet concrete flume. The equation for discharge as obtained in this study is as follows: Coefficient of discharge CD has been calculated for throat openings of 5, 6, 7, 8 and 9 inches.

10 The results have been presented in the form of curves and these curves have been compared with the wok done by previous investigators. (78 pages)

11 INTRODUCTION One of the important problems in irrigation is measuring irrigation water that flows in open channels. Several devices have been used for measuring irrigation water including weirs and Parshall flumes. Usually welt's! and Parshall flumes may be used effectively. However, in areas of flat topography, low canal gradients and lack of freeboard may not provide sufficient head to allow for the use of such devices. One of the main disadvantages of weirs and Parshall flumes in canals of flat gradients is the reduction of the capacity of the canal 9wing to the further reduction of the gradient. Characteristics of a suitable water measuring device Following are the desirable characteristics of a suitable water measuring device for use on canals of low gradients. 1. A low resistance to flow i. e. low head loss; 2. Easily measurable head; 3. Small effect on silt and debris deposition; 4. Simple construction and low cost; 5. Freedom from clogging by floating debris.

12 2 Objectives The objective of this study is to develop a modified venturi section. to measure water in open channels on extremely flat grades. The contraction of the venturi section is provided by a movable top. cover section installed in a reotangular flume. The design will be done by 1. Determining the head loss caused by the section. 2. Determining the coeffioient of discharge for the section. 3. Developing an equation for discharge for the section. The development includes both a mathematical solution to the idealized inviscid flow problem by the method of finite differences and a laboratory investigation to ascertain the validity of the design and its hydraulic characteristics. The work done in this thesis is a continuation of the work previously done by graduate students at the University of California and Utah State University.

13 3 REVIEW OF LITERATURE Water measuring devices in use There are many different types of water measuring devices in use in irrigation systems. The most commonly used devices are, the "weir in its different forms, orifices, venturi flumes and Parshall flumes. Each device has its own advantages and disadvantages, depending on canal gradients and other specific requirements. The 'Weir is a simple device, easy to construct and simple to operate. It consists of a small overflow dam constructed across the channel to create a measurable head. The weir is a suitable metering device for open channels with steep grades. One of the main disadvantages of a weir is that the aerated nappe requires an unrecoverable height of fall which increases with the size of the structure. The upstream stilling pool, required for accurate results, causes a reduction in velocity which might cause deposition of sediments. For these reasons a weir is not considered a suitable measurtng device for open channels with small slopes, especially in medium to large canals. A submerged orifice has an advantage over the weir, because the required head loss is smaller than that for a weir, as some of the velocity, head can be recovered. The disadvantages of a submerged orifice are that it requires an operating head which is attained by impounding the water. This makes the installation a little difficult. The range of flow that can be measured

14 4 with an orifice of fixed dimensions is often too low to satisfy the requirements. Orifices collect sediments and floating debris. These limitations reduce their utility for measuring water in open channels with low grades. The venturi tube is one of the best measuring devices because it offers a low resistance to the flow and a high degree of accuracy. It creates an easily measurable differential head, but is expensive to construct when applied to open channels. It must run full to accurately measure the flow. Venturi tubes are not suitable for large channels. The Parshall measuring flume is probably the best measuring device available for use in canals with low gradients. It violates the criterion of low resistance to the flow, because the head loss depends on the total head measurement, which itself is dependent on the size of the structure and flow. Modified venturi section Considering the limitations of the available metering devices the pioneers of a modified venturi section thought of a device that could be used effectively on canals with flat grades. The device was based on the principles of a venturi tube. The principle of a venturi tube as stated by Daugherty and Ingersell is, A converging tube is an efficient device for converting pressure head to velocity head, while the diverging tube converts velocity head to the pressure head. The definite relationship between the pressure differential and the rate of flow affords a basis for flow measurement. (Daugherty and Ingersell, 1954, p. 125) The modified venturi section differs from a venturi tube as in the case of the modified venturi section the area and shape of the channel section remain constant. The required convergence is obtained by introducing a top cover-section

15 5 in a rectangular flume. Thus the channel can be considered a closed conduit for the length of the device. According to Garton (1950) a device utilizing the top cover section.was first tested by the Bureau of Reclamation in The first field application of a similar device was made by I. H. Tielman Chief Engineer of the Consolidated Irrigation District at Selma, California. The first known experimental work to find the best possible shape for a modified venturi section was done by Yallalee (1938) at the University of California. As a result of these investigations Yallalee recommended the shape of the section shown in Figure 1. This shape was obtained by joining two arcs of different radii. The surface which comes in contact with water starts as a plane surface 60 degrees to the horizontal and tangent to the arc of radius R. Another arc of radius 3R is tangential to this and a plane surface at 20 degrees to the horizontal on the other end. The overall length for a radius of 2 feet was 8.84 fee.t. The experiments were performed in a flume 0.5 feet wide. Ferguson (1949) revised the shape of the top cover section in order to facilitate construction. A straight line was used in place of the arc of radius 3R on the downstream portion. The top cover section was made from corrugated sheet metal. Ferguson's (1949) experiments were performed in a plywood flume 48 feet long and 0.5 feet wide. A 3 inch plate orifice made of brass was used to measure the quantity of water actually coming into the flume. Garton (1950) conducted a field study for a modified venturi section by using a device similar to the one used by Ferguson but it had a width of 24 1/2

16 6 1:=2.34' --l 8.84' Figure 1. Shape of modified venturi section as proposed by yallalee. r: 2.3'-1 8' I Figure 2. Shape of modified venturi section as proposed by Ferguson. 3 '6" --I 10'-7" Figure 3. Shape of modified venturi section used in this thesis.

17 7 inches instead of 6 inches. Garton's study was conducted in a trapezoidal concrete canal. The trapezoidal section was built with a bottom width of 2 feet and 1 1/2:1 side slopes. The lined section was 55 feet long. The modified venturi section was installed in a rectangular concrete section 10 feet long, 2.5 feet deep and 24.5 inches wide. The top cover section was supported by a frame of steel angles. A 24 inch rectangular supressed weir was used to measure the flow. The precision of flow measurement is not known. Skjerseth (1951) conducted a study to determine the hydraulic characteristics of a modified venturi section. This work was a further extension of the work done by Ferguson and Garton. He used two different sections. One of these was similar to Ferguson's but was constructed from 16 ounce copper instead of corrugated galvanized steel. The second model was the same as used by Garton. Skjerseth also applied the principles of dimensional analysis to develop a general equation applicable to geometrically similar devices. Equation of discharge The discharge through a section can be calculated by the application of Bernoulli's theorem and the equation of continuity. The appliction of Bernoulli theorem is depenent on the assumptions that the flow is incompressible, steady uniform across a cross section and there is no friction loss through the section. To simplify the application of Bernoulli theorem the flow is also assumed to be turbulent so that:the value of energy coefficient can be taken as unity. Ferguson (1949) applied this equation between two points on the top cover section. Garton (1950) in addition also applied this equation between two points in the side of the

18 8 test channel. Yallalee (1938) applied this equation between a section upstream of the device and the throat section. The expression for discharge used by Ferguson and Garton was one form of the venturi tube equation, i. e., Q= Orl A 2 g (h - h ) qt.. x t, ( 1) Where Q = discharge (). = coefficient of discharge q A 1 = cross sectional area at point x At = cross sectional area at the throat hx' h t = piezometric head at points x and t respectively g = acceleration due to gravity" From equation (1) the expression for coefficient of discharge was written as follows: c = q (2)

19 9 THEORY Numerical solution The solution to the problem of fluid flow under the modified venturi section is based on potential theory, and is obtained by formulating. a boundary value problem whose solution has been obtained using the method of finite differences. The velocity components u and v in the x and y coordinate directions for two_:dimensional inviscid flow can be obtained from either the stream function If or-potential function rj as shown by the following equations and u= a't d y = - - a y (3) (4) Substituting these expressions in the continuity equation and irrotational equation respectively gives partial differential equations forrj and r. From the solution of either of these equations, the velocity, pressure etc. can be obtained anywhere throughout the flowfield. The major difficulty in obtaining the solution for either rj or r is that the position of the upper surface of the flow region in unimown, and consequently the region of the boundary value problem cannot be defined. An approach which has been used by Southwell (1946) and others is to assume a position, relax the field points, and then adjust the position in light of the inconsistencies which appear in the solution. No

20 10 systematic means of adjusting the position of the unknown boundary is available and any change in the position has an influence throughout the entire flow field. In the problem at hand the added difficulty exists as it is desired to find a shape of the top cover section that would result in a velocity distribution with mild enough adverse pressure gradients to prevent separation. In order to overcome these difficulties inherent in obtaining the solution in the physical plane, the proglem has been transformed to a problem in the plane of the potential function and the stream function f. The coordinate directions x and y appear to be the most appropriate dependent variables in the formulation. The governing equation can be obtained by following the rules for transformation of variables as given by Taylor (1955), that is if = F(x, y) andf = G(x, y) then x and yare also some functions of and If orx = f(, 0/) and y = g(, 0/) such that _ 1 d -o\.f- J ax (5) 1 J 1-0 If J (1 x (6) (7) (8)

21 11 Where J is the Jacobian given by J= -'Qf 0 a x d y 2 =v (9) By substituting these last equations into (3) and (4) one arrives at the Cauchy-Rieman condition " x = "0 0' ( 10) (11) Upon integrating (10) and (11) one can obtain the solution of x from the solution of y or the solution of y from the solution of xas indicated below. (12) ( 13) (14) (15) In these last equations the subscripts by the integral sign denote that the integration is to be carried out along either a 0' = constant line or If= constant

22 12 line. Jeppson (1966) in Appendix A describes methods for performing the indicated partial differentiation and integration by numerical ethods which are well adapted to digital computers. From the Cauchy-Rieman conditions either x or y may be eliminated by differentiation giving the following Laplace equations "0 2 x + '0 2 x =0 d (6'2 a'f2 (16) l: + =0 0(6'2 (17) In order to formulate the boundary value problems for either x or y in the (6'/f plane it is necessary to have a relation between x and y and such quantities as the velocity and the direction of flow in the (6''f plane. Jeppson (1966) has developed several of these relations and shown how they can be used for boundary conditions. Two of these relations which have been used here are and V= 1 <' i) 2 + ( P) 2 (18) (19) The last two equations also permit a determination of the velocity and direction of flqw at any point in the flow field. The problem can most readily be solved for y first. The formulation of the boundary value problem for y is shown in Figure 4. In the formulation it is

23 13 1 Y = f (0) v 'f y( 0) 2 2 t (tl)+ () = 1 o 0 I a.::!...x t""' ri)..y () + ( y,\ 1 a f eo \;10 (p a 0 0 \ d 'P) =---z r Upstream from Top cover section Downstream from top cover section top cover section vt Channelbottom y-o Figure 4. Formulation of flow under the top cover section in potential plane.

24 assumed that at some distance upstream and downstream from the device the flow is uniform. It would be just as easy to specify a typical open channel velocity distribution for the end sections as the normal derivative given in Figure 4. Any of the three conditions for the top cover section may be used. The first would be used if one wished to specify the height as a function of the potential. The second would be used if one wished to determine the shape which would give a given velocity or pressure distribution along the top cover section. The third specify the shape by giving local angle for the top cover section. The solution to the boundary value problem formulated in Figure 4 has been obtained by finite differences using the following nine point operator on a square grid network line b. rj = 1:::::.. r" 14 y(i,j) = O. 05 [Y(I + 1, J) + y(i - 1, J) + y(i, J + 1) + y(i, J - 1+ y(i + 1, J + 1) + y(i - 1, J + 1) y(i - 1, J - 1) + y(i + 1, J - (20) in which I increases with rj as given I = and J with r as given tfj by J = L1 '1'. The operators, used for the boundary conditions along the free surfaces and top cover section are the same as those used by Jeppson (1966) in the solution of plane jet and cavity problems. These operators can be had also by examining the Fortran IV Program which has been used to obtain the solutions and which is given in the Appendix.

25 15 Dimensional analysis of the problem Dimensional analysis is based solely on the relationships that must exist among the pertinent variables because of their dimensions. Dimensional analysis gives qualitative rather than quantitative relationships, but when combined with experimental procedures it may be made to supply quantitative results and accurate prediction equations. (Murphy, 1950, p. 17) The technique of dimensional analysis has been applied in this case to find a set of quantities that can be applied to other geometrically similar models. This analysis is also applied to find an expression for discharge through the device. The flow through a system is affected by the dimensions of the system, the properties of the fluid and the applied forces. The variables that might effect this problem are listed below. 1. Dimensions defining the geometry of the system e. g., throat opening, d, bed width, b, and radius of the cover sections, r. 2. Flow characteristics e. g., rate of flow, Q, depth at any section, h, piezometric head at the throat, ht 3. Fluid properties e. g., density,, elastic modulus, e, specific weight, a, surface tension, b, and viscosity,,. The viscous forces become less important at high Reynold's numbers. The scope of this study is limited to a small range of flow and the temperature variation is limited to about 50 F., so viscous forces will be assumed constant for the limited range and, therefore, will be neglected. Surface tension and elastic modulus are- also not expected to affect this problem. The ratio, g, of specific weight to density will be used as an independent variable in place of these -two quantities. Radius, r, is constant for this study so it is assumed that it will

26 16 not affect the discharge for different throat openings. If the effect of radius, r, on discharge is to be considered, it will require several models with different radii so r is eliminated from the list of variables affecting discharge. Throat area At' a product of band d, has been used as an independent variable replacing both b and d. This implies that the velocity of approach is neglected. Differential head A h = hi - h t is used as an independent variable in place of hi and hr This process of elimination of variables results in only four variables which result in the following functional relationship. f (Q, At' g, A h) = 0 (21) 'There are two basic dimensions and four variables so from the principles of dimensional analysis there will be two 11 terms taking At as the repeating variable the functional relationship of these 7{terms can be written as.6h) = 0 At (22) which can also be written as Q (23) Equation (23) is a functional relationship of 7r terms. The determination of this function must be done by experiment.

27 17 Head loss The modified venturi section is designed to be used in canals which have a low allowable head loss, consequently it follows that it should have a low resistance to flow. The converging section changes the pressure head to velocity head while the diverging section converts the velocity head to pressure head. This recovery is not complete and there is a loss of energy in the passage of water through the section. The head loss is considered to be the differential head between point 1, just upstream from the section and point 5, just downstream from the section (see Figure 9), and thus includes the normal friction loss in this length of channel. The velocity head, though small, is added to the depth to obtain the energy at the section. The expression for head loss can be written from Bernoulli's theorem as (24) Where HL = Head loss through the section y l' Y 5 = Depths at points 1 and 5 respectively v12, v5 2 r.::; Velocity head at points 1 and 5 respectively. 2g 2g

28 18 PROCEDURE Description of apparatus The experimental work for this thesis has been done at the utah, Water Research Laboratory in 3 feet x 3 feet x 200 feet flume. The modified venturi section, as shown in Figures a and 5 was built from 1/8 inch thick steel sheet. The semicircular surface that comes in contact with water has a radius of 2. 5 feet and is tangential to a plane surface which makes an angle of 45 degrees with the horizontal. On the downstream side the semicircular surface is tangential to a plane surface that makes an angle of 20 degrees with the horizontal. The overall length of the device is 10 feet 7 inches. The top cover section was supported by a frame 1 1/2 inches x 1 1/2 inches steel angles, with 1 1/4 inches x 1 1/4 inches steel angles to stiffen the frame. The cover section was stiffened by three 1/2 inch pipes running longitudinally (see Figures 5 and 7). Transverse steel plates, 1/8 inch x 1 1/2 inches were welded to provide rigidity to the structure. A rubber gasket was fastened on the sides of the strucutre by screws to minimize the leakage. The top cover section was hinged at the downstream and the upstream end was raised and lowered by a car jack. The top cover section was built in the shop of Utah Water Research Laboratory by Mr. Kenneth Steele and his staff.

29 Figure 5. Test model. 19

30 Figure 6. Laboratory arrangement. 20

31 21 10'-7" "'E---- 3'-6" /2" x 1 1/2" steel angle n Figure 7. Details of top cover section. A /' l/s" steel sheet - - r--... '" ) t 1..", 1/2" diameter pipe '. 1 1/2" 4 ) Figure S. Section at A-A. 't w;:- l/s" x 1 1/2" steel plate \.. Rubber gasket

32 I 10'-7" I I _I L 3' 6" _I r ---. Too cover section 1 3 r //77,,,,# /7 / 7 I 7/ 7 7 I' 777 /" / 7 77 " lela/7- / / /,,'7/7777/1'7 ", 7 7 I I 7 7' 7' 7 7 " 7' 7' 7 7, 7 t anne! bottom 5 Figure 9. Location of points for piezometric connections. t-:l

33 23 The measurements were taken in five stilling wells. Two of the wells were connected to holes in the top cover section (see Figure 9). Inlets to the other three wells were located just upstream of the section, at the throat section in the side of the channel, and just downstream from the section. The inlets were connected to the stilling wells by one-half inch outside diameter poly flo tubing. The water supply consisted of two pipe lines of diameters 12 and 8 inches. The pipe lines were controlled by ya.lves to regulate the flow at the head of the test flume. The supply pipes were connected to the reservoir formed by the dam on Logan River upstream from the Water Research Laboratory. Water from the supply pipes was run in the flume and made to pass through the modified venturi section installed about 150 feet downstream from the supply. The water level in the stilling wens was read by a hook gage. The same gage was used for all the stilling wells. A 12 inch Parshall flume downstream from the device was used to measure the flow. The discharge was obtained from tables in "Measuring Water with Parshall Flumes." The water was discharged back into Logan River. Experimental procedure A certain discharge flowing at a certain depth constituted a series. A series consisted of several trials till the water level in all stilling wells was fairly constant. The discharge was varied by changing the valve opening consequently, eupstream depth also varied. Sufficient time was allowed for water to come to equilibrium. The velocity in the flume could be reduced by installing a number of wooden overflow boards across the flume downstream

34 24 from the device. Thus the device could be tested for very low velocities. The tests were continued by progressively reducing the flow until the differential head was approximately 0.06 feet, then the throat opening was increased, or the velocity was changed by changing the overflow boards. The throat opening was measured with'a point gage. The point gage measured the depth for taps 3 and 4 on the top cover section. The throat opening was obtained from this measurement by allowing for the thiclmess of the cover sheet. To eliminate the wave action, gratings used to cover the flumes in the laboratory were used as baffles. These gratings were placed at three sections 'across the flume and this reduced the waves considerably.

35 25 RESULTS AND DISCUSSION Results from numerical solution Results from the numerical solution are presented as a flownet in Figure 10. This flownet was obtained by plotting the x and y coordinates from the finite differences network. The complete solution as given by the program also includes the velocities and angles of the flow at each of the grid points. The uppermost streamline gives the theoretical shape for the top cover section of the modified venturi section which will result in a velocity distribution with mild enough adverse pressure gradients to prevent separation in an idealized inviscid flow. This shape will offer no resistance to inviscid flow for a particular upstream depth and velocity. Field conditions, however, will be considerably different as the upstream depth and velocity will fluctuate considerably depending on discharge. The top cover section is to be hinged about the downstream section, so the shape of modified venturi section will be different at different throat openings. Considering these limitations the top cover section has not been built according to this design. The shape of modified venturi section used in this thesis is shown in Figure 3. Equation of discharge In the section on dimensional analysis the following functional relationship of 7T terms was obtained.

36 > C) Q) rt.l > 0 C) go E-t 1 0 rt.l -c:\s C)... J.I S a of:: +> I... So

37 Q _C A 5/4 '" g t = 27 (25) for discharge. In this section these 'ff terms have been combined to give an equation When Q 5/4 is plotted against.f7;;, data forll throat At t openings plot on the same parabolic curve (see Figure 11). If Q 5/4 r;- At " 1 tted "t (A h)i/2 " t ad f bh t i ht I" It ( F" IS P 0 agams' A ms e 0 Jj1' a s ra g me resu s see 19ure,! t y.n.t 12). From this plot, the equation relating the 7T terms of equation (25) can be derived. The resulting equation is: (26) or (27) or (28) Equation (28) is the equation for discharge through the modified venturi section and is identical with the equation for discharge through a venturi meter. Figure 12 gives an average value of Cn = From Equation (28) the expression for coefficient of discharge Cn can be written as

38 C = D Q 28 (29) Figure 13 shows the relation between discharge and the quatitity At 2gbh. The data for throat openings of 6, 7, 8 and 9 inches plot on the same curve giving an average value of CD = The data for throat opening of 5 inch do not follow the above graph. This discrepancy might be the result of experimental error in measuring the throat opening, or a difference in downstream control conditions. The relation of discharge Q to h, the ratio of differential head to throat opening, is a family of parabolic curves, one for each throat opening (Figure 14). A plot of Q vs. V h results ina family of straight lines for different throat openings (Figure 15). The latter set of curves can be used to find the discharge for various throat openings for the shape of the modified venturi section used in this study. Coefficient of discharge In the equation for discharge the coefficient of discharge CD accounts for factors which were not considered in the derivation of the equation. The expression for CD can be written as C = D Q At2gAh The expression for coefficient of discharge obtained from Bernoulli's theorem and continuity equation can be written as

39 L V 9' / / ;; c:> , 0.20 o Figure 11. Q 19 A 5/ 4 as a function of V';.. h t

40 I V / t> / " 10.6, r.q'. +> «0.5 (,. v- j e>. AO/ h. d = 5 inches <u d = 6, 7, 8, 9 inches " Q =Ati 2gAh 0.3 f! ( A h )1/2 Ft Figure 12. Empirical equation for discharge.

41 v- a> d = 6, 7, 8 and 9 inches 6 Ad = 5 inches G iii' C> Q = At 2gh 41! A!l (!) I!> - I> I> l) 2 o Figure 13. Relation between discharge and quantity At V 2g 6. h

42 " 10 I tll '+-< o 6 ( 0) II 'b b 81 II / :\. 111?f' 6.-:::-1 I 0. -:::- r:," Ir CD bo f..! til,..!:l 0 tll //1/ / lift 7 0.(. 0.2 J 1'\ AI I 0.,* I V h d Figure 14. Discharge as a function of differential head I:A:J

43 33.. r r r r r t i : r t_ j ex> 0 ltlt s:l CoO !:IS f1l!:is 0-0 It:) C"I 0 Q)... o: l l o CoO SJO a.8.i'8qosla

44 34 ( 80) The comparison of Equations (29) and (30) results in the following relationship between CD and Cq. ( 31) Both of these coefficients have been determined in this study. 'rhe coefficient of discharge has been based on differential head between points 1 and 3 and points 1 and 4 (see Figure 9). Figures 16 through 20 show the coefficient of dishcarge' as a function of discharge and is based on differential head between points 1 and 3. The values of CD and O' based on differential head between points 1 and.4 can be found in Table 2 (see Appendix). The curves in Figures 16 through 20 have been compared with Skjerseth's (1951) data for the model of inches width. The value of coefficient of discharge is comparatively higher in this study than those obtained by Skjerseth. When comparing these curves it should be understood that the Size, shape and the surface of circular transition were different. The locations of the taps where piezometric head was measured is also different and this is a major factor affecting the coefficient of discharge. There is also the question of absolute accuracy of the flow measurements in all tests. For the limited range of flow, the coefficient of discharge tends to decrease with an increase in discharge and differential head for a particular

45 I ". I I g>o rn :a 1. 0 o 1;;... CD (,)... ::::: 0.9 o I J fa G) - _... G A... - I A CD (DC q A - -- e - "" e.. Q '" Skjerseth I o Discharge (cfs) en Figure 16. Coefficient of discharge for a 5 inch throat opening.

46 a... a:i 'lj 'g... C)... ::::: A AC D 0C q A t-etseth --I" Discharge (cfs) '" G>.. 7 Figure 17. Coefficient of discharge for a 6 inch throat opening A CD C) a:i :a 0 "t:; Q)... C)... :::= Q) 0 Q , "- "'....&. (!) " (!)C q A ca Gl - - Skjerseth AI. II> Discharge (cfs) 6 7 Figur 18. CoefOclent of dia,chilrge for a 7 inch tbrqat opening.

47 38 throat opening. Figures 21 and 22 show that the coefficient of discharge tends to increase with an increase in throat opening. The coefficient of discharge is comparatively constant at higher throat openings. The coefficients of discharge, based on observations which had a differential head of less than 0.1 feet, have been plotted in Figures 16 through 20. These observations have not been given much weight in drawing the curves because the possibility of experimental error is a maximum at very small differential heads. A plot of throat velocity, V t, against the differential head.6h, shows that the data for all throat openings plot on the same parabolic curve (see Figure 23). When V t is plotted against YAh, data for all throat openings except the throat opening of 5 inches plot on the same straight line (Figure 24) which results in the following relationship, (32) Head loss Head loss through the section is calculated from the following relationship V1 2 H =y + -- L I 2g (33) Figures 25 through 29 show head loss as a function of differential head between points 1 and 3 (see Figure 9). The relationship of head loss to differential head is dependant on the throat opening and discharge. The maximum head loss

48 39 1.1, o J.I 1. 0 Co> rtl... "t:$... o d Q).... S 0.9 t:: 8 V ---l r- p p Throat openingd inches Figure 21. Coefficient of discharge Cn for various throat openings rtl... 't:l '0 d Co> t:: o / () i i Skjerseth Throat opening d inches Figure 22. Coefficient of discharge C q for various throat openings.

49 ril a 5 4 :> g 1I 2 E-t J, L O.. /. /9... _ 1 I o O Differential head.all Figure 23. Throat velocity as a function of differential head. ti:>o

50 51 > II> 7 c p Q i, /" Ii> : 2 y / L / o JAh Figure 24. Relationship of throat velocity to differential head

51 O. 5 H O. == Ul Ul 0 - ai <U == O. 2 O. 1 o / /, H L =0.425Ah Differential head h Figure 25. Head loss for a 5 inch throat opening i I I t-.)

52 rj.l rj.l 0 -"0 (\I CD HL = O. 36Ah Differential head Ah Figure 26. Head loss for a 6 inch throat opening. rj.l rj.l... o r CD O.04r _ _r o Differential head A h Figure 27. Head loss for a 7 inch throat opening

53 H L =0.296Ah.- o Differential head h Figure 28. Head loss for an 8 inch throat opening. o.12p r r _4..:I ::t: CI.l CI.l o -0.04r_ _ as Q) ::t: HL = 0.28Ah o Differential head A h Figure 29. Head loss for a 9 inch throat opening.

54 occurs at the smallest throat openings and large flows. The head loss is about 43 percent of the differential head at a 5 inch throat opening and reduces to less than 30 percent for a 9 inch throat opening. H Figure 30 shows the ratio of head loss to differential head, L:l as a function of throat opening. This ratio decreases with the increasing throat opening till it tends to be constant, but this observation is difficult to make from the limited range of this study. This head loss is for a rectangular flume with straight approach. In wider canals, the canal has to be constricted for installing the modified venturi section. Trapezoidal canals also require a rectangular flume for the installation of this section. In both of these cases there will be additional head loss in the converging and diverging sections. This head loss is to be added to the head loss determined in this study. 45 Limitations of the study The range of flow was limited by the capacity of the Parshall flume and the depth of the test flume. The maximum flow that could be measured with the Parshall flume was 7.10 cfs. There was no provision for accelerating the flow so the section could not be tested over a wide range of flow. It is believed that this section could be used to measure much larger flows.

55 r... r---. -,:: 1, o Throat opening (inches) Figure 30. Head loss for various throat openings.

56 47 - CONCLUSIONS The modified venturi section studied in this thesis meets the requirements of a suitable water measuring device except for passing floating debris. The provision of a hinge at the downstream side appears to be practical and offers the advantage of ease in changing the throat opening. The provision of the hinge will also reduce 'the cost of the top cover section. The combination of taps 1 and 3 is considered to be the most practical combination for field application. The equation for discharge obtained from dimensional analysis is as follows: The average value of Cn is The actual values of coefficients of discharge for various throat openings are given in Table 2. For the limited range of flow, the coefficient of discharge tends to decrease with an increase in discharge and differential head for a particular throat opening. The coefficient of discharge tends to increase with an increase in throat opening. The coefficient of discharge becomes comparatively constant at higher throat openings. Head loss has been expressed as a function of differential head between points 1 and 3. The head loss varies from about 43 percent of the differential

57 head for a 5 inch throat opening to less than 30 percent of differential head for a 9 inch throat opening. For a given discharge the differential head decreases With the increase in throat openings, consequently the head loss also decreases. The throat opening can be adjusted to control the head loss and keep it below a certain desired limit. The head loss through the section can be reduced if the divergence of the downstream side is more gradual. This can be done by reducing the downstream angle. An angle of about 12 degrees with the horizontal is recommended. This will tend to increase the overall length of the device but it is expected that the head loss will be reduced considerably. 'its

58 49 LITERA TURE CITED America.n Society of Mechanical Engineering Fluid meters, their theory and application, Part I. Report of American Society of Mechanical Engineering. Specialftesearch Committee on Fluid Meters. American Society of Mechanical Engineering, New York. 203 p. Christiansen, J. E Measuring water for irrigation. University of California, Berkeley, California. Bulletin No. 588, 91 p. Christiansen, J. E., and I. H. Tielman A practical venturi meter for irrigation service. Engineering News Record, New York. 2 p. Daugherty, R. L., and A. C. Ingersoll Fluid mechanics with engineering applications. McGraw Hill Book Company, Inc., New York. 472 p. Ferguson, J. E A modified venturi section for measuring irrigation water in open channels. Unpublished Master's thesis in Agricultural Engineering, Utah State University Library, Logan, Utah. 35 p. Garton, J. E A field study of a modified venturi section for measuring irrigation water in open channels. Unpublished Master's thesis in Agricultural Engineering, Utah State University Library, Logan, Utah. 46 p. Jeppson, R. W Technique for solving free streamline, cavity, jet and seepage problems by finite differences. Technical Report No. 68, Department of Civil Engineering, Stanford University, Stanford, California. 296 p. King, H. W., and E. F. Brater Handbook of hydraulics for the solution of hydrostatic and fluid flow problems. McGraw,Hill Book Company, Inc., New York. p Murphy, Glenn Similitude in Engineering. The Ronald Press Company, New York. 296 p. Skjerseth, Torlief. The hydraulic characteristics of a modified venturi section for use as a water measuring device in open channels. Unpublished Master's thesis in Civil Engineering, Utah State University Library, Logan, Utah. 131 p.

59 5-0 Skogerboe, G. V., M. Leon Hyatt, Joe D. England, J. Raymond Johnson, and Richard E. Griffiil;. Year Unknown. Measuring water with Parshall flumes. Bulletin EC-323. Utah Water Research Laboratory and Utah State University Extension Services, Logan, Vtah. 41 p. Southwell, R. V Relaxation methods in theoretical physics. The Clarendon Press, Oxford, England. 248 p. Taylor, A. E Advanced Calculus, Ginn and Company, New York. 786 p. Venard, J. K Elementary Fluid Mechanres, John Wiley and Sons, Inc., New York. 563 p. Yallalee, W. P Modified venturi section for open channels. Unpublished report from files of J. E Christiansen. 3 p.

60 61 APPENDIX '.

61 Tables Table 1. Dimensionless parameters for the device A 5/4 fa; Ah Series d At t Q Ah Ah 1/2 {At C.JA t) {g A /4 A 2g h 1 5" " " " o " " " " " Q.l " " " " Cll 1:'0..

62 Table 1. Continued Ah 1/2 A A 5/4 Q.6.h Series d t t (TAt) \fat Ah fi A74 A t \]2gAh 14 6" " " " " " " " " " " " " Q 01..

63 " Table 1. Continued A 5/4 h Ah 1/2 g Series d At Q fr\ Ah t g t At (VAt ) (i A 5/4 At" 2gAh 27 8" " " " " " " " " " " " f "

64 ,-, , r--,.--' ::: : ft} h4 h 1 -h VL:lh CD C h 1 -h Jb.h CD c" V t q q Q Q

65 Table 2. Continued. Parshall Flume Piezometric hea Series d h Q hi h3 14 6" C 15 6" " " " " " " " " " " "

66 Table 2. Continued. Parshall Flume Piexometric head Series d h Q hi h3 27 8" !' " " " " " " "

67 Table 3. Head loss for the device 2' V2 V 2..-? HL VI V Y1 HI :±Y l y 2 h - H2=Y2+ =I),-H l::p Ah14 2 Series d Y1 2g 2g 2 2g 2g 2 13 V/2g d Y 1 1 5" " " " Jl " " " " " " " " " ' Q ()O

68 Table 3. Continued. V 2 2 V 2 V 2 V 2 HL 1 1 Y1 Ah 2g H2=Y2+ 2g HL=HJ.-H2.6h 13 bh14 H1=Yl 2 2g Y - Series d Y 2 2g V 1 /2g d Y " " " " " p " " " " " ' " " : ft CJJ co