# Effects of Beta Ratio and Reynold s Number on Coefficient of Discharge of Orifice Meter

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2 M. M. Rahman et al. Franzini, 977). In principle, orifice meter obstructs the flow in a pipe and creates a pressure difference between upstream and downstream of the plate. The Bernoulli's energy (BE) equation is used to calculate the desired. The actual flow profile downstream of the orifice, is quite complex since it flows as jet having no fixed area like the throat of Venturi meter. However, some researchers suggest to take reading at vena contracta, approximately one half the pipe diameter at downstream. The distance and size of the vena contracta are not constant but vary with beta ratio, β (ratio of diameter of orifice to pipe) and Reynolds s number, R e. The aforementioned problems associated with orifice meter usually overestimate the found from Bernoulli s energy equation. So a coefficient (C d ) is multiplied with theoretical to get actual value. However, C d has no unique value; rather it varies with beta ratio and Reynold's number. For most commercial orifice meters and high Reynold's number (Re > 3), the value of coefficient of (C d ) is nearly constant, whereas for lower Reynold s number C d varies abruptly. On the other hand, C d increases with the beta ratio for a fixed R e, resulting a decrease of differential pressure across the orifice plate. Too much lessening this differential head for extreme beta ratio reduces the accuracy of measurement. For this reason, beta ratio has to be kept to an optimum value. It is suggested that for greatest accuracy the beta ratio should be less than.7 (Florida Cooperative Extension Service, Institute of Food and Agricultural Sciences, 22). Though some researches have been done on the relationship of between C d and R e the combined effects of Re & β on C d are very rare. The objective of this study was to examine the variation of coefficient of (C d ) of locally made orifice meter with different beta ratios and Reynold's numbers. MATERIALS AND METHODS Basic concept of orifice meter Basically an orifice meter is a circular metallic plate having an opening at its centre installed in a pipe to measure flow rate creating pressure drop across it. Due to the simple construction and installation, the thin sharp edged orifice has been adopted as a standard one. As the fluid approaches to the orifice, the pressure increases slightly and then drops suddenly. It continues to drop until the vena contracta and then gradually increases until at approximately 5 to 8 diameters of orifice at downstream. The decrease in pressure as the fluid passes through the orifice is a result of the increased velocity of the fluid passing through the reduced area of the orifice. When the velocity decreases as the fluid leaves the vena contracta the pressure increases and tends to reach to its original level. All of the pressure losses are not recovered because of friction and turbulence losses in the stream. The pressure drop across the orifice increases with the rate of flow. The differential pressure is proportional to the square of the velocity. Applying BE equation at upstream and downstream sections of orifice plate, the following equation can be found to measure the theoretical, Q th 2g h () Q th = A2 4 d 2 d where h is the differential pressure head, d & d 2 are diameters of pipe and jet at vena contracta, respectively where pressure gauges are installed. Since it was difficult to measure the diameter of the jet inside the pipe, the diameter of the orifice was taken d 2 and A 2 was calculated accordingly. For the same reason, location of the vena contracta was approximated at 8 cm downstream of the orifice plate. While fluid passing through orifice, some irreversible energy is lost due to obstruction, friction, contraction, eddies etc. that causes a higher differential pressure head not responsible for at all. Besides, pressure tapping at downstream in stead of at orifice, may overestimate or underestimate the actual. To consider all these losses in account, the theoretical is multiplied by a coefficient of (C d ) to get actual. So, actual can be expressed as Q act = C d Q th (2) 52

3 Effects of Beta Ratio and Reynold s Number on Discharge of Orifice Meter C d depends on the ratio of orifice to pipe diameter called beta ratio (β) and Reynold s number (R e ) that can be expressed as equation (3) 4Qact = πdγ R (3) e Where, ν is kinematics viscosity of water at 3 C (.8 x -6 m 2 /s). Fabrication of orifice meter Five orifice plates having different opening diameter were made by locally available cast-iron steel plate in a local workshop (Fig.). The diameters of the opening were 2.54, 3., 4., 5. and 6. cm where pipe diameter was 8.5 cm. The orifice hole was perfectly round in the center of the circular steel plate and sharp edged. The plate thickness was about 6 mm around the circumference of the hole. Circular rubber gaskets were used on both sides of the orifice plate to prevent any unnecessary leakage from pipe. d: 2.54 cm d: 3. cm d: 4. cm d: 5. cm d: 6. cm Fig.. Orifice meters of different diameter Experimental set-up This study was conducted in the Hydraulic Laboratory of the Department of Irrigation and Water Management of the Faculty of Agricultural Engineering and Technology, BAU. A differential U-tube mercury manometer was tapped at a distance of cm upstream and 8 cm downstream of the orifice to measure differential pressure head. Water was pumped by a pump which was operated by a 7.5 horse power electrical motor and required actual was measured by volumetric method. Pipe flow rate was controlled by a valve. Five separate readings of and its corresponding pressure difference were recorded for each beta ratio. The valve was kept at five fixed opening positions and for each valve opening condition, five orifice plates were installed individually and their corresponding s were recorded also. In this way, the variation of was ensured solely by different orifice plates for each valve opening condition in the pipe. 53

4 M. M. Rahman et al. RESULTS AND DISCUSSION Effects of beta ratio (β) on coefficient of (C d ) Figure 2 shows the relationship between β and C d for different s controlled by varying valve opening in pipe..8.6 Full valve opening R 2 = Four fifth valve opening R 2 = (a) (b).8.6 Three fifth valve opening R 2 = Tw o fifth valve opening R 2 = (c) (d).8.6 One fifth valve opening R 2 = (e) Fig.2. Relationship between beta ratio and coefficient of Since in Fig.2 the relationship is only for a fixed valve opening condition, so definitely the cause behind the changes of C d is the variation of influenced by the degree of obstruction imparted by orifice plate due to different beta ratios. At every case, C d follows a positive linear relationship with β where the coefficients of determination (R 2 ) are differing rather than identical for five valve opening conditions. The value of R 2 is increasing with valve opening except for third case (Fig. 2c). However, the highest R 2 value was found to be.893 for the s corresponding to one fifth valve opening condition (Fig. 2e) i.e. when the flow rate was comparatively lower than other valve opening conditions. It can be said that the coefficient of 54

5 Effects of Beta Ratio and Reynold s Number on Discharge of Orifice Meter is more sensitive to beta ratio in case of lower flow rate. It might be due to the reduction of irreversible losses for lower. Effects of Reynold s Number (R e ) on coefficient of (C d ).3.35 (Cd) R 2 = (Cd).64 R 2 = (a) (b) 7.59 (Cd) R 2 = (Cd) R 2 = Reynold's Numebr (Re) 2 (c) (d) (Cd) R 2 = (e) Fig.3. Relationship between beta ratio and Reynold s number From figure 3, it is shown that for each beta ratio a curvilinear relationship exists between C d and R e. For the all cases except beta ratio 7, firstly C d decreases with increasing R e until it reaches to a minimum value and then starts to rise again. Stronger relationship can be seen for the higher beta ratio (.7) with maximum R 2 value of However, the value of C d changes rapidly when β is.7. Similar results were reported by Smajstrla and Harrison (22), and Danial (997). The curvilinear variation of C d may be explained regarding the dominancy of actual and theoretical s measured from differential head, h. It means that the increment of Q th is 55

6 M. M. Rahman et al. more than that of Q act until C d reaches to a minimum value. Differential head comprises of two components, temporary head loss due to converting potential to kinetic energy at orifice and irreversible losses which is not responsible for actual flow rate at all. It seems, while C d decreasing with R e, head losses h overestimate the theoretical. These findings agree with the result of Daugherty, R. L. and Franzine, J. B. (977), whose results show that the value of C d drops steadily with R e until reaches to a constant figure. However, unlike Daugherty and Franzines experiment, in the present case C d follows a step up trend in stead of remaining constant. Reasons behind this finding might be that the losses do not increase as faster as it could be beyond the minimum value of coefficient of, i.e. there would have a critical value of R e for which C d is minimum for an specific beta ratio. However, the behaviour of C d for β value 7 needs further investigation. Results from figure 2 & 3, reveal that C d depends on both beta ratio and Reynold s Number where former has positive linear relationship and latter has curvilinear relationship. For this reason, there will have an optimum value of beta ratio for maximum C d. (Cd) Reynold's Number Fig.4. Bar diagram of coefficient of (C d ) for different beta (β) ratios Figure 4 represents average pattern of C d found by changing pipe i.e. R e for each β separately. The coefficient of lies within the range of.6-.7 for first three beta ratios while it exceeds.8 for last two β. Interestingly, despite having linear relationship between C d and β (fig.2), the highest average value of C d (about.9) was found at beta ratio.59 (Fig.4) in stead of.7. These research findings confirmed that the coefficient of was affected by both beta ratio and Reynold s Number simultaneously and orifice meter with β equal to.6 may be used effectively for pipe flow measurement. LITERATURE CITED Daniel, 997. Fundamentals of Orifice Meter Measurement. Measurement and Control, Western Hemisphere, Houston, Texas U. S. A. pp. -8. Daugherty, R. L. and Franzini, J. B A text book of Fluid Mechanics with Engineering Application. MeGraw- Hill publication. pp Florida Cooperative Extension Service. Document No. AE 22. Institute of Food and Agricultural Sciences. University of Florida, July 22. < Smajstnla, A. G. and Harrison, D. S. 22. Orifice Meters for Water Flow Measurement. Florida Cooperative Extension Service, Institute of Food and Agriculteral Sciences, University of Florida,Gainesville. < 56

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