Venturi Tube Discharge Coefficient in High-Pressure Gas

Abstract

In this chapter the performance of Venturi tubes in high-pressure gas is described: the discharge coefficient tends to increase with Reynolds number largely because of static-hole error, the effect that pressure tappings of finite size do not measure the pressure which would have been measured using an infinitely small hole. There are also effects of throat velocity, including humps and dips in the calibration curve, sometimes an audible tone. The discharge coefficient depends on the shape of the Venturi tube: of the different shapes tested the best results were obtained with a Venturi tube identical to the standard Venturi tube with a machined convergent except that the convergent angle was 10.5° (instead of 21°).

Appendices

Appendix 7.A: Shape of Venturi Tubes: Tests at NEL

7.1.1 7.A.1 Design

With the intention of reducing the uncertainty in flow measurement using Venturi tubes, Venturi tubes of different shapes (ones with longer convergents, ones without sharp corners or even ones with curved convergents) were tested (Reader-Harris et al. 2001, 2002).

The first design was of a Venturi tube with a machined convergent of angle 10.5° with sharp corners. Data for β = 0.4, 0.6 and 0.75 were collected first, but are presented below where appropriate. 4 mm pressure tappings connected in triple-T arrangements were used.

It had been suggested at meetings of ISO/TC 30/SC 2 (see Sect. 12.3.2) that better results would be achieved if the sharp corners in a Venturi tube with a machined convergent were rounded so that it were machined with the profile of an ‘as cast’ convergent. One Venturi tube of this type was lent by Atelier Pochet, of Ransart, Belgium, and is identified as AP1 and referred to subsequently as “rounded corners”. Moreover, the first tests had shown that a convergent with an included angle of 10.5° gave particularly good results. So it was decided that the following three Venturi tubes should be manufactured:

• A Venturi tube with a machined convergent of angle 10.5° with rounded corners whose radii of curvature are those required by ISO 5167-1:1991 (ISO 1991) for a Venturi tube with an ‘as cast’ convergent. This Venturi tube is identified as 29478 and is referred to subsequently as “rounded corners, long”.

• A Venturi tube with the distance between upstream and throat tappings equal to that for a Venturi tube with convergent angle 10.5° but with a machined convergent whose wall profile has continuous second derivatives. This Venturi tube is identified as 29479 and is referred to subsequently as “curved, long”.

• A Venturi tube with the distance between upstream and throat tappings equal to that for a standard classical Venturi tube but with a machined convergent whose wall profile has continuous second derivatives. This Venturi tube is identified as 29480 and is referred to subsequently as “curved”.

AP1 and 29478–29480 were 4″ Schedule 40 with β = 0.6: the desired values of the diameter of the entrance cylinder, D, and the throat diameter, d, were 102.26 and 61.36 mm respectively.

Much literature was reviewed and many calculations undertaken to determine the detailed form of Venturi tubes 29479 and 29480. Profiles by Witoshinsky and by Spencer (1956) were considered, but since wall profiles with continuous second derivatives were available it was felt that they would be desirable. The details of the wall profiles are given in Reader-Harris and Hodges (2001).

In the case of 29478–29480 the throat is parallel for d/3 upstream of the throat tapping and the upstream cylinder is parallel for at least D/3 downstream of the upstream tapping. In practice the throat is close to parallel over a longer distance. These parallel sections are significantly longer than those required by ISO 5167-1:1991. These three Venturi tubes were manufactured by ISA Controls Ltd, Shildon, Co. Durham. Drawings of these Venturi tubes are given in Fig. 7.A.1. The profiles which would have been obtained if sharp corners had been used are shown with dotted lines for comparison. In the case of the long convergent the profile with sharp corners is almost indistinguishable on the scale of the graph from that with rounded corners. Figure 7.A.1 also shows the shape of the Venturi tube AP1 manufactured by Atelier Pochet.

Fig. 7.A.1

Profile from the upstream tapping to the throat tapping for Venturi tubes AP1 and 29478–29480

The Venturi tubes were manufactured to drawings with tight tolerances designed to ensure that where possible the results were not affected by uncontrolled variables. Details are given in Reader-Harris et al. (2001, 2002). They were made of stainless steel and were suitable for use at a design pressure of 70 barg with ANSI Class 600 flanges. A divergent angle of 7½° was specified for 29478–29480. AP1 had a divergent angle of 7°; it also had a divergent truncated by 26 % of its length.

In the case of AP1 the Venturi tube was to be used elsewhere and the diameter values supplied by the manufacturer were used. The Venturi tube was subsequently measured at NEL, and if the NEL measurements were used the discharge coefficients for AP1 would increase by 0.12 %.

The Standard then in force (ISO 5167-1:1991) required that the surface finish of the entrance cylinder, the convergent section and the throat be such that R _a /d is less than 10⁻⁵, where R _a is the arithmetical mean deviation of the roughness profile. The measured values generally exceeded the permitted values by a factor of approximately 2.4; the typical surface finish was R _a  ≈ 1.5 μm. All the Venturi tubes had 10⁻⁵ < R _a /d < 10⁻⁴. The project wished to use Venturi tubes with surface roughnesses typical of those used in the field. All the Venturi tubes met the roughness requirements of ISO 5167-4:2003.

It was decided that 29478–29480 should have four 4 mm tappings in each tapping plane so that data collected in triple-T arrangements would be comparable with all the data described in Sect. 7.2. It had been observed that the data obtained in gas with 8 mm tappings in previous work had larger humps than the data obtained with 4 mm tappings. So 29478–29480 were each designed to have 1, 2, 6 and 8 mm single tappings in addition to the 4 mm tappings, with the exception of 29480, which did not have 1 mm tappings. For the 4, 6 and 8 mm tappings, the throat pressure tappings were of constant diameter for a depth of 94 mm and the upstream tappings for a depth of 53 mm. As actually manufactured the 1 and 2 mm tappings were much shorter; over the distance between the interior of the Venturi tube and the ¼ BSP fitting there could be up to two changes in tapping diameter. AP1 had four 4 mm tappings upstream and four 2 mm tappings in the throat. In both cases the tappings were connected by annular chambers. The 4 mm tappings were of depth 10 mm and the 2 mm tappings were described on the drawing as being of minimum depth 8 mm.

7.1.2 7.A.2 Calibration in Water

The Venturi tubes were calibrated first in water at NEL. In order to investigate whether the performance of tappings connected in triple-T arrangements differed from that of pairs of single tappings, data were collected with a single pair of 4 mm tappings as well as with triple-T tappings in the case of 29478 and 29479. For each Venturi tube the data in water lay on a straight line as a function of pipe Reynolds number, Re _D , and with a small scatter, provided that Re _D was above a critical value. The pipe Reynolds number below which C was not on the straight line varied, but was typically about 3 × 10⁵. In the case of 29480 there was a significant hump in the data at Re _D  ≈ 2.5 × 10⁵.

The differences between data taken with single tappings of different diameters were surprisingly large. Some of the variation is due to the fact that the tappings are of different diameters, but there is also significant scatter due to variation between tappings of the same diameter (see Appendix 7.B for examples of two Venturi tubes each originally with four pairs of single tappings of the same diameter and depth). Therefore if a Venturi tube is used uncalibrated the use of single tappings instead of triple-T tappings (or other multiple pressure tappings) results in increased uncertainty, although there may be other practical reasons for using single tappings (see also Sect. 4.3.2). This problem was reduced in later work by using spark erosion to insert the tappings into the tube (see Sect. 7.A.7).

7.1.3 7.A.3 Calibration in Gas

The Venturi tubes were calibrated in nitrogen at NEL at two static pressures, 20 and 60 barg. Although discharge coefficients are presented largely in terms of Reynolds number there is an effect of throat velocity in many of the data sets, where peaks and troughs of discharge coefficient occur for both static pressures at the same throat velocity.

7.1.4 7.A.4 Analysis

Since the effect of static hole error at the upstream tapping was much smaller than at the throat tapping it is possible simply to correlate the data with the throat tapping Reynolds number; the simplest presentation of this is to define the Venturi-throat-tapping Reynolds number as in Eq. 7.11:

$$ Re^{*} = \frac{{d_{tap} }}{d}Re_{d} , $$

(7.11)

where d _tap is the diameter of the throat tapping and Re _d is the throat Reynolds number, which is equal to Re _D /β. Re* is used rather than Re _tap , because a measurement of friction factor in the throat is not available; however, Re* does not have as much physical significance as Re _tap .

Then for each set of data it was found that

$$ C - C_{\text{water}} = a - b\,e^{{ - 0.4(Re*/10^{5} )}} , $$

(7.A.1)

where a and b are coefficients to be determined. For each data set it was possible to determine both the coefficients in Eq. 7.A.1 and the scatter about the resulting equation. In Figs. 7.A.2, 7.A.3, 7.A.4, 7.A.5 and 7.A.6 the data from Venturi tubes of revised shape taken with 4 mm tappings connected in triple-T arrangements are presented. Data for two standard Venturi tubes (included in Sect. 7.2: their numbers are 28909 and 28909C) and for Venturi tube 98491, which was standard except for convergent angle 31.5°, have been included as Figs. 7.A.7, 7.A.8 and 7.A.9. In each case data for Re _D  < 2 × 10⁵ were excluded as the discharge coefficient tends to decrease as the Reynolds number decreases below this point. Data for Re* < 20000 have also been excluded as the static hole error decreases rapidly as Re* decreases below this point. Where data in water start to depart from a linear fit to the data as the Reynolds number decreases, for instance to form a hump, all the data below an appropriate Reynolds number were excluded.

Fig. 7.A.2

Discharge coefficient for Venturi tube 29478 (rounded corners, long) [4 mm (triple-T) tappings]

Fig. 7.A.3

Discharge coefficient for Venturi tube 29479 (curved, long) [4 mm (triple-T) tappings]

Fig. 7.A.4

Discharge coefficient for Venturi tube 29480 (curved) [4 mm (triple-T) tappings]

Fig. 7.A.5

Discharge coefficient for Venturi tube AP1 (rounded corners)

Fig. 7.A.6

Discharge coefficient for Venturi tube 98488 (10.5° convergent angle) [4 mm (triple-T) tappings]

Fig. 7.A.7

Discharge coefficient for Venturi tube 28909 (standard) [4 mm (triple-T) tappings]

Fig. 7.A.8

Discharge coefficient for Venturi tube 28909C (standard) [4 mm (triple-T) tappings]

Fig. 7.A.9

Discharge coefficient for Venturi tube 98491 (31.5° convergent angle) [4 mm (triple-T) tappings]

From this work the good agreement between water and gas data is clear. The line fits from Figs. 7.A.2, 7.A.3, 7.A.4, 7.A.5, 7.A.6, 7.A.7, 7.A.8 and 7.A.9 and other data that were collected but are not presented here graphically are included in Table 7.A.1, which shows the line fits in terms both of C and of C − C _water and the standard deviation of the data about the line fits.

Table 7.A.1 Equations fitted to water and gas data collected with Venturi tubes of different shapes

Given that the standard deviation depends on the tapping diameter it was desirable to see how for 4″ β = 0.6 Venturi tubes with 4 mm triple-T tappings the standard deviation depends on the shape of the Venturi tube. On the basis of Table 7.A.2 it appeared that the best choice for subsequent work was the Venturi tube with sharp corners which is standard except for a convergent angle of 10.5°. Since two other Venturi tubes of this shape had already been manufactured and tested, numbers 98487 and 98489 of diameter ratio 0.4 and 0.75 respectively, their water and gas data are shown in Figs. 7.A.10 and 7.A.11 for comparison. The equations for 4″ Venturi tubes with convergent angle 10.5° and 4 mm triple-T tappings are presented in Table 7.A.2.

Fig. 7.A.10

Discharge coefficient for Venturi tube 98487 [4 mm (triple-T) tappings, 10.5° convergent angle, β = 0.4]

Fig. 7.A.11

Discharge coefficient for Venturi tube 98489 [4 mm (triple-T) tappings, 10.5° convergent angle, β = 0.75]

Table 7.A.2 Equations fitted to water and gas data for 4″ Venturi tubes with convergent angle 10.5° and 4 mm triple-T tappings

On the basis of Table 7.A.2 it was clear that the relationship between water and gas data as expressed by the equations for C − C _water is remarkably similar for the three Venturi tubes. The standard deviation is very low. The next stage was to manufacture and test a larger population of such Venturi tubes.

One other important area is the effect of tapping diameter on Venturi tube discharge coefficients. It is clear from Table 7.A.1 that for three Venturi tubes the standard deviation increased for tapping diameters greater than 4 mm. It was obviously important to establish whether the same is true for standard Venturi tubes. Within earlier work (Reader-Harris et al. 1999) two Venturi tubes, 28908 and 28911, of diameter ratio 0.5 and 0.7 respectively, had tapping diameters modified. Given that the tappings for smaller tapping diameters were achieved using inserts those data have not been reanalysed here. However, the 8 mm tappings were obtained by drilling, and those data together with the 4 mm tapping data have been reanalysed and presented in Table 7.A.3, although they are not plotted here.

Table 7.A.3 Equations fitted to water and gas data for 4″ Venturi tubes with β = 0.5 and 0.7

On the basis of Tables 7.A.1 and 7.A.3 it is possible to see the effect of the diameter of pressure tappings on the standard deviation. Data from Venturi tubes with more than one diameter of pressure tapping are given in Fig. 7.A.12. The 4 mm single tapping and triple-T tapping data are slightly offset from 4 mm so that they can be distinguished. It is interesting to note that the standard deviation increased rapidly as the tapping diameter increased from 4 to 6 mm. Whether 4 mm triple-T tappings actually perform significantly better as regards standard deviation than 4 mm single tappings is not clear. The main advantage of triple-T tappings is that when a Venturi tube is used uncalibrated the discharge coefficient is likely to be closer to the expected value for that type of Venturi tube than if a single tapping were used. They must also require shorter upstream lengths downstream of some fittings than would be required with the worst choice of circumferential location for single tappings.

Fig. 7.A.12

The standard deviation of the water and gas discharge-coefficient data about the line fit versus exp(−0.4Re*/10⁵), as a function of tapping diameter (d was between 51 and 72 mm)

The slopes of the different fits versus exp(−0.4Re*/10⁵) are plotted in Fig. 7.A.13 as a function of the tapping diameter for those Venturi tubes with more than one diameter of pressure tapping. It is noticeable that the slopes increase in absolute value with tapping diameter and that the spread of slope values is smaller for tapping diameters in the range from 2 to 6 mm than it is for tapping diameters of 1 or 8 mm.

Fig. 7.A.13

The slope of the water and gas discharge-coefficient data about the line fit versus exp(−0.4Re*/10⁵), as a function of tapping diameter (d was between 51 and 72 mm)

7.1.5 7.A.5 Conclusions on Shape from the 4″ Venturi Tubes

The standard deviation of the data about the best-fit line was determined for each set of data with 4″ diameter and β = 0.6 but different convergent profiles. The best convergent profile was determined to be one with a 10.5° included angle with sharp corners on the basis that it gives the lowest standard deviation of the data about the fitted line. Moreover, two Venturi tubes with this particular profile and the same diameter but β = 0.4 and 0.75 were calibrated in water and gas; when they were evaluated in the same way the standard deviations of the data about best-fit lines were low. Moreover, for all three diameter ratios the slope of the line fit was very similar.

The three Venturi tubes manufactured to look for the best profile had single pairs of tappings of diameter 1 mm (except for one Venturi tube), 2, 6, and 8 mm and four pairs of tappings of diameter 4 mm. Data could be collected with a single pair of 4 mm tappings or using all four pairs connected in triple-T arrangements. All the work to determine the best profile used 4 mm tappings in triple-T arrangements. The same process of evaluation used to determine the best profile showed that 8 mm tappings and, to a lesser extent, 6 mm tappings gave worse performance than smaller diameters. 4 mm appeared to be the best choice: uncertainty increased with tapping diameter, but anything smaller than 4 mm was too likely to block.

• NOTE All these Venturi tubes had similar throat diameters; so it is not clear from these data that in general performance is diminished by increasing d _tap ; best performance might, in fact, depend on avoiding too high a value of d _tap /d.

7.1.6 7.A.6 Manufacture of Additional Venturi Tubes with 10.5° Convergent Angle and Sharp Corners

Following the work described in Sect. 7.A.5 it was decided that additional Venturi tubes of diameter 2″ and 6″ with diameter ratios of 0.4, 0.6 and 0.75 should be manufactured with a 10.5° convergent angle and sharp corners where the convergent meets the throat and the upstream cylinder. They should have four pairs of tappings of diameter 4 mm and a single pair of tappings of diameter 2 mm.

As with the standard Venturi tubes the revised Venturi tubes were manufactured to drawings with tight tolerances designed to ensure that where possible the results were not affected by uncontrolled variables. A description is given in Reader-Harris et al. (2005).

For the 4 mm tappings the throat pressure tappings were of constant diameter for a depth of 94 mm and the upstream tappings for a depth of 53 mm; for the 2 mm tappings the throat pressure tappings were of constant diameter for a depth of 37 mm and the upstream tappings for a depth of 27 mm. To give consistent tapping quality at least the final portion of each tapping (near the pipe internal wall) was produced using Electrical Discharge Machining (spark erosion).

7.1.7 7.A.7 Calibration of Additional Venturi Tubes in Water and in Gas

The Venturi tubes were calibrated in gas at two static pressures and in water. The 6″ gas data were collected in nitrogen in the NEL high-pressure recirculating loop. The 2″ gas data were collected in air in the NEL gravimetric facility. In order to check the quality of the individual tappings, data were collected in water with each pair of tappings separately. The data from each pair of tappings were then compared with those from each other pair and with the triple-T data. The agreement between the data from different pairs of single tappings was excellent, with a typical spread of line fits to the data of 0.2 %. In the case of the 6″ Venturi tubes the triple-T data were very close to the mean of the single-tapping data, differing by only 0.07 % at most in terms of line fits. In the case of the 2″ Venturi tubes the triple-T data were in one case higher and in the other case lower than the mean of the single-tapping data, differing by about 0.3–0.4 %. For each Venturi tube the data in water lay on a straight line as a function of pipe Reynolds number, Re _D , and with a small scatter, provided that Re _D was above a critical value.

For the sets of data it is possible to determine both the coefficients in Eq. 7.A.1 and the scatter about the resulting equation. In Figs. 7.A.14, 7.A.15, 7.A.16 and 7.A.17 triple-T data for 2″ and 6″ Venturi tubes are presented; the equations are given in Table 7.A.4. Where data in water start to depart from a linear fit to the data as the Reynolds number decreases, all the data below an appropriate Reynolds number (usually Re _d  ≈ 5 × 10⁵) have been excluded.

Fig. 7.A.14

Discharge coefficient for Venturi tube 29624 with convergent angle 10.5°: 6″, β = 0.6, 4 mm triple-T tappings

Fig. 7.A.15

Discharge coefficient for Venturi tube 29623 with convergent angle 10.5°: 6″, β = 0.4, 4 mm triple-T tappings

Fig. 7.A.16

Discharge coefficient for Venturi tube 29622 with convergent angle 10.5°: 2″, β = 0.75, 4 mm triple-T tappings

Fig. 7.A.17

Discharge coefficient for Venturi tube 29621 with convergent angle 10.5°: 2″, β = 0.6, 4 mm triple-T tappings

Table 7.A.4 Equations fitted to water and gas data for 6″ and 2″ Venturi tubes with convergent angle 10.5° and 4 mm triple-T and 2 mm tappings

The 2″ data appear significantly different from the 4″ and 6″ data. This is probably due to the fact that d _tap /d is not negligible:

• As already stated above, in the case of the 2″ Venturi tubes (for which d _tap /d was 0.10 and 0.13) the triple-T data were in one case higher and in the other case lower than the mean of the single-tapping data, differing by about 0.3–0.4 %, whereas for the 6″ Venturi tubes (for which d _tap /d was 0.043 and 0.065) the triple-T data were very close to the mean of the single-tapping data, differing by only 0.07 % at most in terms of line fits.

• In the data of Shaw (1960) the static-hole errors in a straight-pipe are consistent for d _tap /d up to and including 0.0625 but differ for d _tap /d equal to 0.075 and 0.0875, where d here represents the diameter of the pipe in which the tapping is located.

• In Chap. 9 on nozzles the static-hole errors are consistent for d _tap /d up to and including 0.06 but differ for d _tap /d equal to 0.08.

The data from Figs. 7.A.12 and 7.A.13 (except those with single 4 mm tappings) together with all the data from Tables 7.A.2 and 7.A.4 (i.e. those for Venturi tubes with convergent angle 10.5°) have therefore been replotted in Figs. 7.A.18 and 7.A.19 as a function of d _tap /d.

Fig. 7.A.18

The standard deviation of the water and gas discharge-coefficient data (shown by tapping diameter) about the line fit versus exp(−0.4Re*/10⁵), as a function of d _tap /d

Fig. 7.A.19

The slope of the water and gas discharge-coefficient data (shown by tapping diameter) about the line fit versus exp(−0.4Re*/10⁵), as a function of d _tap /d

If only data with tapping diameters of 4 mm or greater were considered, the standard deviation of the data would increase with d _tap /d and the slope would increase in magnitude for d _tap /d greater than about 0.08. However, some of the 2 mm points are inconsistent with the other points.

For all the 2 mm data there is a sudden expansion in tapping diameter from 2 mm to a higher value at some depth h. For the two 2 mm points with the smallest standard deviation, which are also the two points with the smallest magnitude of slope, h/d _tap  = 3.5 in one case and 8.5 in the other, both at the throat tappings; in both cases c/d _tap  = 2. For the other five 2 mm points h/d _tap is either 18.5 or 19.5 at the throat tappings and the tapping does not just expand, but enters a ¼ BSP fitting; the four 2 mm points with the largest standard deviation and the largest magnitude of slope have h/d _tap  = 18.5: it appears that this tapping arrangement can give surprising results.

Until an explanation is available it seems best to have d _tap  ≥ 4 mm, to avoid change in tapping diameter over as long a distance as possible and, where possible, to have d _tap /d ≤ 0.07.

The analysis for Venturi tubes with convergent angle 10.5° which led to the text in ISO/TR 15377:2007 is presented in Sect. 7.4.2. All the 4 mm triple-T tapping data were included, with no restrictions due to d _tap /d.

An alternative analysis whose aim is to determine a curve for static-hole error is given in Appendix 7.C.

Appendix 7.B: Depth of Tappings: Tests at NEL

To investigate the effect of changing the depth of the tappings two 4″ Venturi tubes were used, one of which had a diameter ratio of 0.4, the other a diameter ratio of 0.65. Calibrations were carried out in gas at 20 barg. On each Venturi tube the tapping diameter was 4 mm, and the throat pressure tappings were of constant diameter for a depth of 94 mm and the upstream tappings for a depth of 53 mm. First the two Venturi tubes were calibrated with each pair of tappings considered separately. Then one of the throat tappings (A) was left unmodified while three of the throat tappings were drilled with a 9 mm drill so that for the first one (B) the remaining depth of 4 mm diameter tapping became 40 mm (expanding at that point to 9 mm), for the second one (C) the depth of 4 mm tapping became 20 mm and for the third one (D) the depth of 4 mm tapping became 10 mm. The Venturi tubes as modified were then recalibrated. The data from the unmodified tappings (A) are shown in Fig. 7.B.1: the first sets were taken when all the tappings were unmodified, the second sets when all the other tappings had been modified. The fact that the results with the unmodified tappings changed little suggests that the modifications to the other tappings did not have a significant effect on the flow as a whole. As the 4 mm depth was reduced to 40 mm so the discharge coefficient reduced by approximately 0.4 % (see Fig. 7.B.2, which shows the B tappings). As the 4 mm depth was reduced to 20 mm so the plotted data became less linear and the discharge coefficient increased from the initial value by approximately 0.15 % (see Fig. 7.B.3, which shows the C tappings). As the 4 mm depth was reduced to 10 mm so the plotted data became less linear and the discharge coefficient reduced from the initial value by approximately 0.55 % (see Fig. 7.B.4, which shows the D tappings). It is clear that tapping depth has an effect on the discharge coefficient, but it is not easy to determine its exact effect. See the final paragraph of Appendix 7.C for further comments.

NOTE As stated in Sect. 7.3.4 Livesey et al. (1962) considered tappings of depth between 0.3 and 7.4d _tap , which beyond the parallel section of stated depth entered a chamber of diameter 14d _tap : they found that holes with a depth/diameter ratio less than 5 were sensitive to variations of tapping depth: if the depth/diameter ratio were 3 or less the static hole error was very different from that in a deep hole. Shaw (1960) had a tapping on the wall entering a tapping of width 2d _tap : provided the depth was greater than or equal to 1.5d _tap his results were generally unaffected by depth. In both cases the tapping-hole Reynolds number was much lower than in the Venturi tube data taken in gas and described here.

Fig. 7.B.1

Discharge coefficient: 4″ Venturi tubes: unmodified throat tappings (d _tap  = 4 mm) of depth 94 mm

Fig. 7.B.2

Discharge coefficient: 4″ Venturi tubes: throat tappings (d _tap  = 4 mm) modified from depth 94 mm to depth 40 mm

Fig. 7.B.3

Discharge coefficient: 4″ Venturi tubes: throat tappings (d _tap  = 4 mm) modified from depth 94 mm to depth 20 mm

Fig. 7.B.4

Discharge coefficient: 4″ Venturi tubes: throat tappings (d _tap  = 4 mm) modified from depth 94 mm to depth 10 mm

Appendix 7.C: Refitting the Data With Convergent Angle 10.5°

The aim of the work described in this appendix was to determine the static-hole error for Venturi tubes with convergent angle 10.5°. Accordingly all the data from the Venturi tubes considered in Sect. 7.4.2 were refitted, that is the Venturi tubes with convergent angle 10.5° and sharp corners together with the one which has a 10.5° convergent but with rounded corners of the radius required by an ‘as cast’ Venturi tube.

The data were fitted on the assumption that the data taken without tappings would have the discharge coefficient given by Eq. 7.14:

$$ C = 1.0039 - 1.66\lambda_{th} . $$

Then it was assumed from Eq. 7.5 that the tapping terms were given by

$$ \frac{{\lambda_{th} f(Re_{tap,th} ) - \lambda_{up} \beta^{4} f(Re_{tap,up} )}}{{8(1 - \beta^{4} )}} $$

where

$$ f = \left\{ {\begin{array}{*{20}c} {a - be^{{ - nRe{\kern 1pt}_{tap} }} } & {{\text{for}}\;\;Re_{tap} > 800} \\ {\left( {a - be^{ - 800\,n} } \right)\frac{{Re_{tap} }}{800}} & {{\text{for}}\;\,Re_{tap} \le 800.} \\ \end{array} } \right. $$

(7.C.1)

and to fit Fig. 7.9 it was prescribed that a − be ⁻²⁰⁰⁰ⁿ = 3.7

From Eqs. 7.2 and 7.3

$$ Re_{tap,up} = \sqrt {\frac{{\lambda_{up} }}{8}} \frac{{d_{tap} }}{D}Re_{D} \quad {\text{and}}\quad Re_{tap,th} = \sqrt {\frac{{\lambda_{th} }}{8}} \frac{{d_{tap} }}{d}Re_{d} $$

From the computational work

$$ \lambda_{th} = (0.711 + 0.0624\;\lg (Re_{d} ))\lambda_{sp} $$

(7.C.2)

where λ _sp was obtained from the Colebrook-White Equation (Schlichting 1960) (see also Sect. 1.5) in the following form:

$$ \frac{1}{{\sqrt {\lambda_{sp} } }} = 1.74 - 2\lg \left( {\frac{{2\pi R_{a} }}{d} + \frac{18.7}{{Re_{d} \sqrt {\lambda_{sp} } }}} \right) $$

(7.C.3)

For the upstream tapping

$$ \frac{1}{{\sqrt {\lambda_{up} } }} = 1.74 - 2\lg \left( {\frac{{2\pi R_{a,up} }}{D} + \frac{18.7}{{Re_{D} \sqrt {\lambda_{up} } }}} \right), $$

(7.C.4)

where the roughness R _a,up was taken as 1.6 µm. The assumed upstream pipe roughness has little effect on the values of a and b.

It seems reasonable to suppose that in practice Venturi tube i will differ from the theoretical value by its own constant, ∆C _i , due to any defects in the tappings. So C was taken as

$$ C = 1.0039 - 1.66\lambda_{th} + \frac{{\lambda_{th} f(Re_{tap,th} ) - \lambda_{up} \beta^{4} f(Re_{tap,up} )}}{{8(1 - \beta^{4} )}} + \varDelta C_{i} $$

(7.C.5)

Considering only the data with d _tap /d ≤ 0.07 and requiring that for each Venturi tube the mean deviation over the water data (where the tapping terms are most well known) is equal to 0, the best fit to f is given by

$$ f = \left\{ {\begin{array}{*{20}l} {10.09 - 7.06e^{{ - 0.00005\;Re_{tap} }} } & {{\text{for}}\;\;Re_{tap} > 800} \\ {0.00413\,Re_{tap} } & {{\text{for}}\;\,Re_{tap} \le 800.} \\ \end{array} } \right. $$

(7.C.6)

∆C _i varies from −0.0054 to 0.0006, and the deviations of the data from Eq. 7.C.5 are shown in Fig. 7.C.1.

Fig. 7.C.1

Percentage deviation of the data with d _tap /d ≤ 0.07 for Venturi tubes with convergent angle 10.5° from Eq. 7.C.5 with f as in Eq. 7.C.6 and ∆C _i chosen so that the mean deviation over the water data is 0

The effect of increasing the coefficient of λ _th by 30 % and recalculating f to obtain a revision to Eq. 7.C.6 is a change in f by 5 % in magnitude at most.

Considering the data with d _tap  = 2 mm and requiring that for each Venturi tube the mean deviation over the water data (where the tapping terms are most well known) is equal to 0, ∆C _i varies from −0.0015 to 0.0130, and the deviations of the data from Eq. 7.C.5 are shown in Fig. 7.C.2. The data are plotted against throat velocity. It should be noted that for the 6″ β = 0.6 Venturi tube the edge radius of the 2 mm throat tapping was about 0.4 mm.

Fig. 7.C.2

Percentage deviation of the data with d _tap  = 2 mm for Venturi tubes with convergent angle 10.5° from Eq. 7.C.5 with f as in Eq. 7.C.6 and ∆C _i chosen so that the mean deviation over the water data is 0 (values of h/d _tap are for the throat tappings)

The data with h/d _tap  = 18.5 at the throat tappings are very different from the data in Fig. 7.C.1: they give a surprisingly large positive deviation. However, the data with h/d _tap  = 8.5 at the throat tappings are in good agreement with the data in Fig. 7.C.1: moreover, they are consistent with what might have been expected from Figs. 7.B.2 and 7.B.3.