Orifice Discharge Coefficient

Abstract

The discharge coefficient is required to measure flow using a differential-pressure meter. This chapter considers the discharge-coefficient equation for orifice plates: its history, some older equations, the database collected mostly in the 1980s and the analysis. The orifice discharge-coefficient equation is made up of the following terms: the discharge coefficient using corner tappings for infinite Reynolds number, the slope term which gives the increase in the discharge coefficient (using corner tappings) for lower Reynolds number, the tapping terms, and possibly an additional term for small orifices. The derivation of the Reader-Harris/Gallagher (1998) Equation, which is included in ISO 5167-2:2003, is given. The quality of its performance, both absolutely and relative to the Reader-Harris/Gallagher (RG) Equation in API 14.3:1990 and the Stolz Equation in ISO 5167:1980, is demonstrated and a calculation of its uncertainty provided. Orifice plates with small pipe diameter or with no upstream or with no downstream pipeline or with no upstream and no downstream pipeline are considered in an appendix.

Appendices

Appendix 5.A: Better Options for Tapping Terms

When data were taken by NEL in oil in 2″ pipe in 1990 for inclusion in the European database not only discharge coefficients but also direct measurements of pressure profile were made: the pressure rise to the upstream corner from tappings at distances D, D/2, D/4 and D/8 upstream was measured, where D is the pipe diameter, as well as the pressure drop from the downstream corner to tappings at distances D/8 and D/4 downstream of the downstream face of the orifice plate and to the downstream D/2 tapping. Whereas for high Re _d (greater than approximately 80,000) the tapping terms do not depend on Re _d , the tapping terms at the Reynolds numbers obtained in oil are significantly different.

Moreover, the data collected on tapping terms by NEL in 2″ pipe (NEL 1991) both show that the reality is more complex than the analysis in Sect. 5.4.2.3 and also provide revised tapping terms which correspond much better to the tapping term collapse found in the database as a whole than the Reader-Harris/Gallagher (RG) (API) tapping terms did (Reader-Harris et al. 1992a). Figure 5.A.1 shows all the American discharge-coefficient data for β ≈ 0.735 together with the NEL 2″ data. Because the tapping terms have a strong dependence on β, where there is a wide range of β the data have been divided.

Fig. 5.A.1

American (CEESI and NIST) data (all with flange tappings) and NEL 2″ data for β ≈ 0.735

It can be seen in this figure that most of the data collapse on to one another as Re _D decreases. However, the corner-tapping data are not collapsing on to the other data, or at least only to a very limited extent. The collapse of the flange tapping data on to one another can be seen more clearly in Fig. 5.7, which shows the American data for β ≈ 0.735: the fact that the tapping terms are approximately constant for high Re _d can also be seen. Both Figs. 5.A.1 and 5.7 confirm the need for tapping terms which are functions of Reynolds number, but also show that the simple dependence on Re _d used in the Reader-Harris/Gallagher (1998) Equation is insufficient.

The main features of the tapping term data collected in 2″ pipe (in Figs. 83–100 of NEL (1991)) are as follows: the upstream tapping term for D and for D/2 tappings decreases with decreasing Re _d as expected from the work of Johansen and of Witte and Schröder, although the NEL data decrease a little more slowly with Re _d ; the upstream tapping term for D/4 tappings remains approximately constant; the upstream tapping term for D/8 tappings increases with decreasing Re _d . The dependence of the downstream tapping terms on Re _d depends on β: for β > 0.7 they decrease in magnitude with decreasing Re _d ; otherwise they are constant. At the bottom of the Re _d range the uncertainties in the data become large, especially for the upstream D/2 tapping data because only one differential-pressure transmitter, rather than two covering different ranges, was used for this measurement.

It is important to see the pattern in the tapping term data: to do this it is necessary to do an analysis of the uncertainty of these data. It is then possible to analyse all the upstream tapping term data simultaneously, and, in particular, to verify that the dependence on β ⁴/(1 − β ⁴) which characterizes the data for high Re _d continues to apply for low Re _d . Figure 5.A.2 shows the change in discharge coefficient due to moving the upstream pressure tapping from the upstream corner for the four values of L ₁ for which measurements were made. Where the data are multiplied by (1 − β ⁴)/β ⁴ they fall on to a single curve for each value of L ₁. Data are only plotted if (1 − β ⁴)u/(β ⁴∆p _c) < 0.04, where u is the uncertainty of the pressure measurement between upstream pressure tappings at that point and ∆p _c is the differential pressure across the orifice plate using corner tappings.

Fig. 5.A.2

Upstream tapping term for low Re _d (following ISO 80000-2:2009, log₁₀ is written lg)

Various possible forms of the upstream tapping term, ∆C _up, for low Re _d data were tried, and the best one was found to be the following:

$$ \Updelta C_{\text{up}} = (0.043 + (0.090 - aA^{\prime}){\text{e}}^{{ - 10L_{1} }} - (0.133 - aA^{\prime}){\text{e}}^{{ - 7L_{1} }} )(1 - bA^{\prime})\frac{{\beta^{4} }}{{1 - \beta^{4} }} $$

(5.A.1)

where

$$ A^{\prime} = \left( {\frac{2100\beta }{{Re_{D} }}} \right)^{n} $$

and a, b, and n are to be determined.

This equation is similar to Eq. 5.9a at high Reynolds number, but has a different shape for lower Reynolds number. The best fit of the form of Eq. 5.A.1 to the upstream tapping term data included in Fig. 5.A.2 was obtained, making appropriate allowance for the fact that the small product term is omitted from the final formula for the tapping terms. Allowance was also made for the fact that especially for L ₁ = 0.125 the measured tapping terms (corrected for the product term) even for Re _d  ≈ 100,000 are not equal to the high Re _d values of Eq. 5.A.1; the fitted equation was therefore calculated based on data points shifted so that for each value of L ₁ the mean value of the data for Re _d  > 80,000 (corrected for the product term) agrees with the high Re _d version of Eq. 5.A.1, because the high Reynolds number version of Eq. 5.A.1 was fitted to many sets of data.

The best fit value for n was 0.925, but for simplicity this was rounded to 0.9, and with n = 0.9 the other constants were

$$ a = 0.833 \;\; {\text{and}}\;b = 1. 30 7. $$

However, these values were adjusted to give a better fit to the database: the best fit to the complete database gave a larger value of a than the fit to the upstream tapping term data: a compromise value was obtained as follows: from the Figures in NEL (1991) it appears that the data for L ₁ = 1, those for L ₁ = 0.25 and those for L ₁ = 0.125 meet at Re _d  ≈ 13,000. Since in Eq. 5.A.1 the three curves representing the three values of L ₁ do not intersect at a single point, it was decided that the intersection of the curve for L ₁ = 0.167 (corresponding to flange tappings in 6″ pipe) with the curve for L ₁ = 1 should occur at Re _d  = 13,000: this occurs for a = 1.03. This constant is then rounded to 1. Equation 5.A.1 with a = 1, b = 1.307, and n = 0.9 is then plotted in Fig. 5.A.2 for comparison with the data. This equation describes a change in the pressure profile upstream of the orifice in which, as Re _d decreases, the upstream tapping term at D decreases but the gradient of the tapping term near the corner increases.

It is unnecessarily complicated to construct a downstream tapping term which decreases in magnitude with decreasing Re _d for very large β but is constant for smaller β to fit the data in NEL (1991); since the upstream term is significant for large β, but very small for small β, this downstream Re _d effect is incorporated in the upstream term by reducing b from 1.307 to 1. The optimum upstream tapping term (incorporating a downstream effect) (for use in the PR14 equation) is therefore

$$ \Updelta C_{\text{up}} = (0.043 + (0.090 - A^{\prime}){\text{e}}^{{ - 10L_{1} }} - (0.133 - A^{\prime}){\text{e}}^{{ - 7L_{1} }} )(1 - A^{\prime})\frac{{\beta^{4} }}{{1 - \beta^{4} }} $$

(5.A.2)

where

$$ A^{\prime} = \left( {\frac{2100\beta }{{Re_{D} }}} \right)^{0.9} . $$

With this upstream formula no change in the downstream formula from that in Eq. 5.6 is required for Re _D  > 4000. However, from examination of the data for Re _D  < 4000 in Sattary et al. (1992) it can be seen that over the range of data in the database for Re _D  < 4000 the discharge coefficient using corner tappings becomes increasingly larger than that using flange or D and D/2 tappings as Re _D decreases; since this applies even for small β this can best be represented by the downstream tapping term being modified, although both upstream and downstream tapping terms change with Re _D . The model used was as follows:

$$ \Updelta C_{\text{down}} = - 0.031\,(M_{2}^{\prime} - 0.8M_{2}^{\prime1.1} )\left\{ {1 + c\hbox{max} \left( {\lg \left( {\frac{{Re_{{D,{\text{T}}}} }}{{Re_{D} }}} \right),\;0} \right)} \right\}\beta^{1.3} $$

(5.A.3)

where c is a constant and Re _D,T is the pipe Reynolds number at which transition to fully turbulent flow occurs. Re _D,T varies, as would be expected, from one set of data to another, but a reasonable estimate of the range of values encountered in the database is 3000–5000, and Re _D,T = 3700 has been used for both the tapping term and the slope term. With this value for Re _D,T c is determined by fitting the data in Sattary et al. (1992): using the difference between flange and corner tappings only, c = 8.20; using the difference between D and D/2 and corner tappings only, c = 7.88; using all the data, c = 8.04. The agreement between the values of c obtained using flange and D and D/2 tappings is very good, and the optimum downstream tapping term (for use in the PR14 Equation) is as follows:

$$ \Updelta C_{\text{down}} = - 0.031\,(M_{2}^{\prime} - 0.8M_{2}^{\prime1.1} )\left\{ {1 + 8\hbox{max} \left( {\lg \left( {\frac{3700}{{Re_{D} }}} \right),\;0} \right)} \right\}\beta^{1.3} $$

(5.A.4)

Appendix 5.B: Small Orifice Diameters Within the EEC/API Database

As a result of collecting the NEL 2″ data, which included measurements of edge sharpness, it was appropriate to add an additional term for small orifice diameters to the equation accepted at New Orleans. The problem is that it is extremely difficult to obtain a sufficiently sharp edge where the orifice diameter is small: Fig. 5.B.1, which gives measured edge radii (r) from the plates used in the European tests, in which the pipe size was in the range 2″–24″, shows that for orifice diameter, d, less than about 50 mm the plates rarely met the requirements of ISO 5167-1:1991 (ISO 1991). For pipe sizes other than 2″ the edge radii shown are averages of measurements by several European laboratories; for 2″ pipes the measurements are those of NEL alone since NEL was the only European laboratory at which 2″ tests were performed. It is clear that for d < 25 mm large shifts in C were to be expected. When the edge radii themselves are plotted as in Fig. 5.B.2, it appears that the edge radius, r, increased as d decreased from about 50 mm, whereas to meet the standard it needed to decrease fairly rapidly.

Fig. 5.B.1

r/d as a function of d (for the orifice plates used in the EEC tests)

Fig. 5.B.2

Edge radius as a function of d (for the orifice plates used in the EEC tests)

The change in discharge coefficient due to edge roundness, ΔC_round, was measured by Hobbs (1989) and seen to be a function of change in edge radius, Δr, that can be expressed approximately (see Sect. 2.2.4) as

$$ \Updelta {\text{C}}_{\text{round}} = 3.33 \varDelta r/d. $$

(5.B.1)

It was assumed that the mean value of r, $\bar{r}$, for d < d ₁, where d ₁ ≈ 50 mm, was given by

$$ \bar{r} = 0.0002 d_{ 1} + B(d_{1} - d) $$

(5.B.2)

where B is a constant, since this is linear with d and gives $\bar{r}/d$ equal to 0.0002 where d = d ₁. Given that the discharge-coefficient equation for large d is based on $\bar{r}/d$ being approximately equal to 0.0002, the additional term for d < d ₁ was calculated to be

$$ \Updelta C_{\text{round}} = 3.33\left( {\frac{{\bar{r}}}{d} - 0.0002} \right), $$

(5.B.3)

which on substituting from Eq. 5.B.2 becomes

$$ \Updelta C_{\text{round}} = 3.33(B + 0.0002)\left( {\frac{{d_{1} }}{d} - 1} \right). $$

(5.B.4)

When ΔC _round was determined by fitting the database the minimum standard deviation was obtained for d ₁ = 44 mm, but over the range 40 mm < d ₁ < 50 mm the overall standard deviation was within 0.0002 % of the minimum. It was therefore reasonable to approximate d ₁ by 50 mm since to round d ₁ to 40 mm would mean that one plate would still be too rounded (i.e. it would lie outside the limit corresponding to Eq. 5.B.7). With d ₁ = 50 mm a good approximation to the small orifice diameter term was

$$ \Updelta C = 0.0015\hbox{max} \left( {\frac{50}{d} - 1,\;0} \right) , $$

(5.B.5)

which corresponds to B = 0.00025 and to

$$ \bar{r} = { \hbox{max} }\left( {0.0 2 2 5- 0.000 2 5d,0.000 2d} \right). $$

(5.B.6)

The maximum value of r, r _max, is equal to 2$\bar{r}$, i.e.

$$ r_{ \hbox{max} } = { \hbox{max} }\left( {0.0 4 5- 0.000 5d,\,\,0.000 4d} \right), $$

(5.B.7)

and from Fig. 5.B.2 it can be seen that all the plates lie within this limit. Clearly this term gives rise to an increase in uncertainty for d < 50 mm.

Twenty years after this work, plates with an edge radius of around 9 μm were being made by one manufacturer using spark erosion (see Fig. 2.A.3). If such were normal practice the small orifice diameter term for the orifice equation would be smaller than the one in this Appendix.

Appendix 5.C: The PR14 Equation and an Equation in Terms of Friction Factor

5.3.1 5.C.1 The PR14 Equation

Given the optimum tapping terms in Eqs. 5.A.2 and 5.A.4 and the small orifice diameter term in Eq. 5.B.5, the coefficients in the C _∞ and slope terms were obtained to fit the EEC/API database in Sect. 5.3. The forms of the C _∞ and slope terms were as in Eqs. 5.11 and 5.17. Previously m ₁ had been taken to be equal to 2, but a better fit was obtained with a smaller value, and following a suggestion of Stolz (1991) the same value as the exponent of β in the downstream tapping term (Eq. 5.A.4) was used. The optimum value of m ₁ in terms of the lowest standard deviation of the data about the equation lay between 1.2 and 1.3. The mean Re _D at which the flow becomes fully turbulent was taken to be 3700. This gave c ₁ in terms of c ₂. c ₂ was obtained by trying appropriate values in turn and obtaining the best overall fit: c ₂ = 4800 gave an excellent overall fit. With m ₂ and n ₁ as in earlier work a least-squares fit of the complete database was performed: on rounding the constants, the C _∞ and slope terms became

$$ \begin{aligned} C_{\infty } + C_{s} &= 0.5934 + 0.0232\beta^{1.3} - 0.201\beta^{8} + 0.000515\;(10^{6} \beta /Re_{D} )^{0.7} \\ & \quad + (0.0187 + 0.0400A^{\prime})\beta^{3.5} \hbox{max} \{ (10^{6} /Re_{D} )^{0.3} ,\;\;23.1-4800\;(Re_{D} /10^{6} )\} .\\ \end{aligned} $$

(5.C.1)

The complete PR14 Equation can be brought together from Eqs. 5.A.2, 5.A.4, 5.B.5 and 5.C.1 to give

$$ \begin{aligned} & C = 0.5934 + 0.0232\beta^{1.3} - 0.201\beta^{8} + 0.000515\;(10^{6} \beta /Re_{D} )^{0.7} \\ & \quad + (0.0187 + 0.0400A^{\prime})\beta^{3.5} \hbox{max} \{ (10^{6} /Re_{D} )^{0.3} ,\;\;23.1-4800\;(Re_{D} /10^{6} )\} \\ & \quad + (0.043 + (0.090 - A^{\prime}){\text{e}}^{{ - 10L_{1} }} - (0.133 - A^{\prime}){\text{e}}^{{ - 7L_{1} }} )(1 - A^{\prime})\frac{{\beta^{4} }}{{1 - \beta^{4} }} \\ & \quad - 0.031\,(M_{2}^{\prime} - 0.8M_{2}^{\prime1.1} )\left\{ {1 + 8\hbox{max} \left( {\lg \left( {\frac{3700}{{Re_{D} }}} \right),\;0.0} \right)} \right\}\beta^{1.3} \\ & \quad + 0.0015\,\hbox{max} \left( {\frac{50}{d} - 1\,,\,\;0} \right) \\ \end{aligned} $$

(5.C.2)

where d is in mm,

$$ A^{\prime} = \left( {\frac{2100\beta }{{Re_{D} }}} \right)^{0.9} {\text{and}}\;\,M_{2}^{\prime} = \frac{{2L_{2}^{\prime} }}{1 - \beta }. $$

Although the PR14 Equation has tapping terms that fit the pressure-profile data for low Reynolds number better than those in the Reader-Harris/Gallagher (1998) Equation and a term for small orifice diameter rather than small pipe diameter, nevertheless for Re _d  > 10⁵, d ≥ 50 mm, and 0.3 ≤ β ≤ 0.65 it differs from the Reader-Harris/Gallagher (1998) Equation by less than 0.05 %. Changes in terms for low Reynolds number or small diameter have a very small effect on the equation in the range where it is most importantly used. Using β ^1.3 in C _∞ rather than β ² makes a difference for small β but it is difficult to choose the best exponent because most data for very small β also have small d and thus possible edge rounding.

5.3.2 5.C.2 An Equation in Terms of Friction Factor

Using the tapping terms in Eqs. 5.A.2 and 5.A.4 and the small orifice diameter term in Eq. 5.B.5 but including the effect of friction factor as in Eq. 5.15 the database was refitted with the C _∞ and slope terms of the following form:

$$ \begin{aligned} C_{\infty } + C_{s} &= a_{1} + a_{2} \beta^{1.3} + a_{3} \beta^{8} + b_{1} (10^{6} \beta /Re_{D} )^{0.7} \\ &\quad + (b_{2} + b_{3} A^{\prime})\beta^{3.5} \hbox{max} \{ \lambda ,\;\;c_{1} - c_{2} (Re_{D} /10^{6} )\} . \\ \end{aligned} $$

(5.C.3)

The mean Re _D at which the flow becomes fully turbulent was taken to be 3700. This gave c ₁ in terms of c ₂. c ₂ was obtained by trying appropriate values in turn and obtaining the best overall fit: c ₂ = 4800 gave an excellent overall fit. The C _∞ and slope terms were

$$ \begin{aligned} C_{\infty } + C_{s} & = 0.5945 + 0.0157\beta^{1.3} - 0.2417\beta^{8} + 0.000514\;(10^{6} \beta /Re_{D} )^{0.7} \\ & \quad+ (3.134 + 4.726A^{\prime})\beta^{3.5} \hbox{max} \{ \lambda ,\;0.1704 - 35{\kern 1pt} (Re_{D} /10^{6} )\} .\quad \quad \\ \end{aligned} $$

(5.C.4)

This equation is theoretically desirable and indeed is used to determine roughness limits in Sect. 2.3.3.1. However, it would be difficult to use commercially.

Appendix 5.D: The Effect on the Discharge-Coefficient Equation of Changing the Expansibility -Factor Equation

In the database (Sect. 5.3) the discharge coefficients were calculated using the expansibility-factor equation in ISO 5167-1:1991 (ISO 1991); when this was changed to Eq. 6.13 the discharge coefficients should have been changed, and there would have been an effect on the calculated discharge-coefficient equation. This did not happen; however, it is shown in this appendix that to have recalculated the discharge coefficients would have had an insignificant effect on the discharge-coefficient equation.

In order to prove that recalculating the discharge coefficients would have had an insignificant effect on the discharge-coefficient equation, for each point of the database the value of discharge coefficient which would have been obtained if an alternative equation for expansibility factor had been used was calculated and the equation refitted. This work was done before the actual Eq. 6.13 was determined, but, using other proposed expansibility-factor equations, showed that the discharge-coefficient equation was insensitive to the expansibility-factor equation. If the value of discharge coefficient given in the database is termed C _I , based on the expansibility factor given by the equation in 8.3.2.2 of ISO 5167-1:1991, ε _I , then C _N , the value of discharge coefficient based on an alternative value of expansibility factor, ε _N , is given by

$$ C_{N} \varepsilon_{N} = C_{I} \varepsilon_{I} , $$

(5.D.1)

where

$$ \varepsilon_{I} = 1 - (0.41 + 0.35\beta^{4} )\frac{\Updelta p}{{\kappa p_{1} }}. $$

(6.1)

Where the value of ε _I is given in the discharge-coefficient database it was possible to calculate the isentropic exponent κ using the values of β, Δp and p ₁ and then to calculate ε _N . Where ε _I was not given in the database it was necessary, in the first instance, to estimate what value of κ might have been used on the basis of other data; if it had been shown to be the case that the discharge-coefficient equation fitted to the values of C _N differed significantly from that fitted to the values of C _I it would have been necessary to obtain better values for κ.

The only sets of gas data for which ε _I was not provided in the discharge-coefficient equation database were those from SwRI and Ruhrgas. For SwRI the downstream expansion factor , Y ₂, (see API 1990) was provided but not ε _I . So for SwRI and Ruhrgas values for κ of 1.41 and 1.32, respectively, were used: 1.41 is appropriate for nitrogen; 1.32 is a typical value for natural gas.

Three alternative equations for ε _N were used: they were as follows:

$$ \varepsilon_{N,1} = 1 - (0.35 + 0.38\beta^{4} )\frac{\Updelta p}{{\kappa p_{1} }}, $$

(5.D.2)

$$ \varepsilon_{N,2} = 1 - (0.352 + 0.433\beta^{4} )\frac{\Updelta p}{{\kappa p_{1} }}, $$

(5.D.3)

and

$$ \varepsilon_{N,3} = 1 - (0.357 + 0.557\beta^{4} )\frac{\Updelta p}{{\kappa p_{1} }}. $$

(5.D.4)

ε _N,1 and ε _N,2 were taken from Eqs. 9 (rounded as in the conclusions of the paper) and 10 of Kinghorn (1986) and ε _N,3 was taken from Eq. 8 (the recommended equation) of Seidl (1995).

The absolute value of the coefficient of $\frac{\Updelta p}{{\kappa p_{1} }}$ for the equation in ISO 5167-1:1991 (Eq. 6.1) and for the alternative equation numbers 1, 2 and 3 (Eqs. 5.D.2, 5.D.3 and 5.D.4 respectively) and the absolute value of the coefficient of $1 - \left( {\frac{{p_{2} }}{{p_{1} }}} \right)^{{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle \kappa $}}}}$ for the equation in ISO 5167-2:2003 (Eq. 6.13) are shown in Fig. 5.D.1.

Fig. 5.D.1

Absolute value of slope of expansibility-factor equation for different equations

Calculating C _N,i on the basis of ε _N,i for i = 1,3 and, using the tapping terms given in Eqs. 5.10a, b, the sum of the other terms (the C _∞, C _s and small pipe diameter, ΔC _D , terms), C _Σ, was refitted, assuming it was of the following form:

$$ \begin{aligned} C_{\varSigma } & = C_{\infty } + C_{s} + \Updelta C_{D} = a_{1} + a_{2} \beta^{2} + a_{3} \beta^{8} + b_{1} (10^{6} \beta /Re_{D} )^{0.7} \\ & \quad + (b_{2} + b_{3} A)\beta^{3.5} \hbox{max} \{ (10^{6} /Re_{D} )^{0.3} ,\;\;22.7 - 4700\,(Re_{D} /10^{6} )\} \\ & \quad + h(0.75 - \beta )\hbox{max} (2.8 - D/25.4,0.0). \\ \end{aligned} $$

(5.D.5)

In Eqs. 5.D.5–5.D.9 D is in mm.

In each case the standard deviation of the data in the database about the equation, s, the number of points which are shifted by more than 0.2 %, N _s , and the largest magnitude of shift for a point in the database, S _M , were calculated. The results were as follows:

$$ \begin{aligned} C_{\varSigma ,N,1} & = 0.59590 + 0.02638\,\beta^{2} - 0.21794\,\beta^{8} + 0.0005288\;(10^{6} \beta /Re_{D} )^{0.7} \\ & \quad + (0.01904 + 0.005864\,A)\beta^{3.5} \hbox{max} \{ (10^{6} /Re_{D} )^{0.3} ,\;\;22.7 - 4700\,(Re_{D} /10^{6} )\} \\ & \quad + 0.01135\;(0.75 - \beta )\hbox{max} (2.8 - D/25.4,0.0). \\ s_{N,1} &= 0.0016775\;;\quad N_{s,1} = 191{\kern 1pt} {\kern 1pt} ;\quad S_{M,1} = 0.98\% . \\ \end{aligned} $$

(5.D.6)

$$ \begin{aligned} C_{\varSigma ,N,2} & = 0.59591 + 0.02645\,\beta^{2} - 0.21778\,\beta^{8} + 0.0005286\;(10^{6} \beta /Re_{D} )^{0.7} \\ & \quad + (0.01895 + 0.005894\,A)\beta^{3.5} \hbox{max} \{ (10^{6} /Re_{D} )^{0.3} ,\;\;22.7 - 4700{\kern 1pt} (Re_{D} /10^{6} )\} \\ & \quad + 0.01133\;(0.75 - \beta )\hbox{max} (2.8 - D/25.4,0.0). \\ s_{N,2}& = 0.0016768\,{\kern 1pt} ;\quad N_{s,2} = 177{\kern 1pt} {\kern 1pt} ;\quad S_{M,2} = 0.85\% {\kern 1pt} . \\ \end{aligned} $$

(5.D.7)

$$ \begin{aligned} C_{\varSigma ,N,3} & = 0.59592 + 0.02662\,\beta^{2} - 0.21740\,\beta^{8} + 0.0005281\;(10^{6} \beta /Re_{D} )^{0.7} \\ & \quad + (0.01876 + 0.005965\,A)\beta^{3.5} \hbox{max} \{ (10^{6} /Re_{D} )^{0.3} ,\;\;22.7 - 4700\,(Re_{D} /10^{6} )\} \\ & \quad + 0.01129\;(0.75 - \beta )\hbox{max} (2.8 - D/25.4,0.0). \\ s_{N,3} & = 0.0016784\;;\quad N_{s,3} = 139{\kern 1pt} {\kern 1pt} ;\quad S_{M,3} = 0.76\% {\kern 1pt} . \\ \end{aligned} $$

(5.D.8)

Since small differences between equations were being investigated the constants for C _Σ are also required where ε _I was used:

$$ \begin{aligned} C_{\varSigma ,I} & = 0.59615 + 0.02609{\kern 1pt} \beta^{2} - 0.21675{\kern 1pt} \beta^{8} + 0.0005216\;(10^{6} \beta /Re_{D} )^{0.7} \\ & \quad + (0.01874 + 0.006071\,A)\beta^{3.5} \hbox{max} \{ (10^{6} /Re_{D} )^{0.3} ,\;\;22.7 - 4700\,(Re_{D} /10^{6} )\} \\ & \quad + 0.01101{\kern 1pt} \;(0.75 - \beta )\hbox{max} (2.8 - D/25.4,0.0). \\ s_{I} & = 0.0016747. \\ \end{aligned} $$

(5.D.9)

The constants have been rounded and then rebalanced to ensure that there is no mean deviation between the equation and the database.

It can be seen that the differences in s and thus in overall quality of fit are very small. Moreover the coefficients in Eqs. 5.D.6–5.D.9 are very similar. The largest value of S _M for Eqs. 5.D.6–5.D.8 occurs for Eq. 5.D.6, as expected from Fig. 5.D.1; however, even in this case the largest magnitude of difference between the discharge-coefficient equations (i.e. between C _Σ,N,1 and C _Σ,I ) is 0.04 % for any values of β, D and Re _D except at the very lowest end of the Reynolds number range, below 4000. Therefore the choice of the expansibility-factor equation has very little effect on the discharge-coefficient equation. Equation 6.13 itself generally lies between the ISO 5167-1:1991 Equation and Eq. 5.D.2, and the difference between $\frac{\Updelta p}{{\kappa p_{1}}}$ and $1 - \left({\frac{{p_{2}}}{{p_{1}}}} \right)^{{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle \kappa $}}}}$ is small for the points in the database, most of which were collected at high pressure. Moreover, whereas changing the expansibility-factor equation would reduce the discharge coefficient in the database, at the same time the isenthalpic temperature correction (see Sect. 4.6.2) was introduced, which, if it had been applied to all the points in the database, would have increased the discharge coefficient in the database. Both effects will be larger for small β than for large β. However, although the discharge-coefficient equation is very little affected by the choice of expansibility-factor equation, the choice of expansibility-factor equation has a significant effect on some data points in the field, and it is important that the best equation is used.

Appendix 5.E: Orifice Plates in Pipes of Small Diameter or with No Upstream or with No Downstream Pipeline or with No Upstream and No Downstream Pipeline

5.5.1 5.E.1 Introduction

This area is covered by ISO/TR 15377:2007, and this appendix gives a justification for the appropriate sections in ISO/TR 15377 (see also Reader-Harris et al. 2006). Practical applications of Sects. 5.E.3 to 5.E.5 would be flow out of and/or into a reservoir or tank.

5.5.2 5.E.2 Orifice Plates in Pipes of Small Diameter

ISO/TR 15377:1998 permitted pipe diameter D ≥ 25 mm and orifice diameter d ≥ 5.75 mm and gave an additional uncertainty on the discharge coefficient of 1 %. However, when the data collected in the EEC/API Orifice Project in the 1980s (see Sect. 5.3) are analysed the uncertainty increases rapidly when d < 12.5 mm (only API had data collected for d < 12.5 mm or diameter ratio β < 0.2). For the water data (Whetstone et al. 1989) for d ≈ 9 mm the deviations from the Reader-Harris/Gallagher (1998) Equation can be as large as 1.5 %; for d ≈ 6 mm they can be as large as around 3 %. The deviations are a very weak function of Reynolds number. It appears likely that in most cases the shifts are due to edge sharpness, but that explanation would not suit one set (4″ pipe, β = 0.06, run 2). Therefore in ISO/TR 15377:2007 it states that the uncertainty increases significantly if d < 12.5 mm.

The OSU data for orifice plates (Beitler 1935) in 1″ and 1.5″ pipes were compared with the Reader-Harris/Gallagher (1998) Equation as they form an excellent set of data for D < 50 mm. Provided that the uncertainty derived from 5.3.3.1 of ISO 5167-2:2003 is increased by 0.33, 95 % of those points for which d > 12.5 mm and β < 0.762 lie within the uncertainty. In ISO/TR 15377:2007 the increase in uncertainty was rounded up to 0.5 %.

Only corner tappings are permitted: flange tappings would give M′₂ > 4 (i.e. a tapping in the pressure recovery zone: see Fig. 5.6) for sufficiently high β. Although flange tappings could be used for sufficiently small β it is generally wise to design a system so that β may be easily changed if the flow is higher or lower than the design flow.

5.5.3 5.E.3 Orifice Plates with No Upstream or Downstream Pipeline

ISO/TR 15377 permits orifice plates without either an upstream pipeline or a downstream pipeline or both. Data from Sect. 5.3 (Whetstone et al. 1989; Britton et al. 1988) with d ≥ 12.5 mm and β = 0.1 are plotted in Fig. 5.E.1.

Fig. 5.E.1

Data (all API: NIST and CEESI) from EEC/API orifice project taken for β = 0.1 and d > 12.5 mm

All the oil data (Britton et al. 1988) for d ≥ 9 mm and β ≤ 0.22 are plotted in Fig. 5.E.2. On the basis of Figs. 5.E.1 and 5.E.2 it appears reasonable to suppose that the Reader-Harris/Gallagher (1998) Equation will continue to perform well as β → 0.

Fig. 5.E.2

Data (all CEESI) from EEC/API orifice project taken in oil for β < 0.22 and d > 9 mm

• NOTE In Appendix 2.A it is shown that the Reader-Harris/Gallagher (1998) Equation continues to perform well for β down to 0.016 provided that allowance is made for the roundness of the orifice plates for very small d.

It is also necessary to check over what range of β a lower throat Reynolds number limit of 3500 (chosen for ISO/TR 15377:2007) might be appropriate. From Eq. 5.16 the discharge coefficient C is the sum of two Reynolds number terms, the throat term, which can be seen from Figs. 5.E.1 and 5.E.2 to be applicable down to at least Re _d  = 3500, and the velocity profile term, which is applicable down to the beginning of transition to laminar flow, giving a limit of Re _D  ≥ 5000 in ISO 5167-2:2003. However, if β is sufficiently small, the velocity profile term is small. In Fig. 5.E.2 there appears to be no problem for β = 0.21. In the complete discharge-coefficient equation (5.21) fitted to the complete EEC/API database including points for Re _D  < 4000 there are changes to the velocity-profile and to one other term for Re _D  < 3700. From Eq. 5.21, for β ≤ 0.2 when Re _d  ≥ 3500, the maximum error in C due to just using the equation in ISO 5167-2:2003 is 0.53 %. Provided that β ≤ 0.2, d ≥ 12.5 mm and Re _d  ≥ 3500, it seems reasonable to use the Reader-Harris/Gallagher (1998) Equation as in ISO 5167-2:2003 with an uncertainty of 1 % (for Re _D  < 5000), as stated in NOTE 2 of 5.3.2.2.2 of ISO/TR 15377:2007. The simplified Equation (A) in both Figs. 5.E.1 and 5.E.2 can be used for d ≥ 12.5 mm and Re _d  ≥ 3500 with an uncertainty of 1 % with a large upstream space as in 5.3.2.2.3 of ISO/TR 15377:2007.

5.5.4 5.E.4 Orifice Plates with No Upstream Pipeline

Section 5.E.3 showed the suitability of the Reader-Harris/Gallagher (1998) Equation for use where there is neither an upstream nor a downstream pipe. To investigate the suitability of this equation for the situation where there is no upstream pipe but there is a downstream pipe Computational Fluid Dynamics (CFD) was used. Solutions were obtained using the commercial code Fluent v6.1 with the realizable k − ε turbulence model with d = 100 mm and Re _d  = 2.5 × 10⁵. It was not easy to obtain converged solutions. With some grids converged solutions could only be obtained with first-order interpolation. However, with the same fine grid in the vicinity of the orifice second-order solutions were obtained for both Cases 1 and 2, shown in Figs. 5.E.3 and 5.E.4. 20 square cells were used along the orifice bore. The cells were concentrated around the orifice plate edges, expanding outwards to reduce cell-count. With 35,500 cells in Case 1 and 52,600 cells in Case 2, the discharge coefficients for Cases 1 and 2 with corner tappings were 0.59291 and 0.59324 respectively. So, if corner tappings are used, a flow from a large space into a downstream pipe of diameter 2d can be considered to have the same discharge coefficient as a flow from a large pipe into a large pipe.

Fig. 5.E.3

Flow from a hemispherical inlet into a pipe of diameter 2d (case 1)

Fig. 5.E.4

Flow from large pipe (β = 0.05) into large pipe (case 2)

So since the Equation works well where there is neither an upstream nor a downstream pipe, the CFD implies that it will work well for flow from a plenum into a downstream pipe of diameter 2d, and thus by obvious extension for flow into a pipe of diameter greater than 2d. This confirms 5.3.2.2.2 of ISO/TR 15377:2007.

5.5.5 5.E.5 Orifice Plates with No Downstream Pipeline

In the case of an orifice plate with no downstream pipeline, ISO/TR 15377:1998 implied that an orifice plate discharging into a large space from a pipe of finite bore will have the same discharge coefficient using corner tappings as an orifice plate installed with a downstream pipe of the same diameter as the upstream one. However, the permissible upstream diameter ratio was unclear. In 5.3.3.1 (simul) the upstream pipeline should be no greater than 2.5d (that is, β ≤ 0.4). In 5.3.3.1 (ter) it stated that 0.4 ≤ β ≤ 0.7, which is very different from the previous statement.

A total of four simulations were run for this case: Case 3: pipe of diameter 2.5d upstream, large pipe (diameter 10d) downstream (β = 0.4 upstream); Case 4: pipe of diameter 2.5d upstream and downstream (β = 0.4 upstream); Case 5: pipe of diameter 1.33d upstream, large pipe (diameter 10d) downstream (β = 0.75 upstream); Case 6: pipe of diameter 1.33d upstream and downstream (β = 0.75 upstream), where d = 100 mm, as before. Cases 5 and 6 were run to see whether the diameter of the upstream pipe influences the solution.

As the inlet was always a pipe in these cases, instead of a mass flow inlet a velocity inlet boundary condition was used, in which a fully developed flow profile was superimposed on the inlet. The profiles of velocity magnitude and turbulence were produced by running a separate model with cyclic boundary conditions. Figure 5.E.5 shows the grid nomenclature for Cases 3 and 4, where β = 0.4; the dotted lines show the portion of the grid used to represent the case where the orifice was installed (as normal) with the same pipe diameter upstream and downstream of it. The grids for cases 5 and 6 were similar. As in Sect. 5.E.4 there were 20 square cells along the orifice bore and the cells expanded outwards from there. All of the solutions in this case converged using second order interpolation. Table 5.E.1 summarizes the results for C.

Fig. 5.E.5

Grids for a small upstream pipe issuing into a larger diameter pipe where β = 0.4 (case 3) and for a standard β = 0.4 orifice meter (case 4)

Table 5.E.1 Results for orifice with a large space or a pipe downstream

Provided that corner tappings are used the discharge coefficient is the same with a plenum downstream as it would have been with a pipe downstream of the same diameter as the upstream pipe. Since this is true for β = 0.4 and 0.75 (and trivially for β = 0.1) it is reasonable to suppose that it will be true for all β. On this basis the Reader-Harris/Gallagher (1998) Equation as in ISO 5167-2:2003 can be used as in ISO/TR 15377:2007.

Figure 5.E.6 shows the axial pressure profiles along the pipe wall, with β = 0.4. It is clear that there is no recovery of pressure when the jet issues from the orifice plate into a large diameter pipe, but that although the pressure in Case 4 varies downstream of the orifice it converges in the corner with that in Case 3.

Fig. 5.E.6

Axial pressure profiles along the wall for cases 3 and 4 (β = 0.4)

Appendix 5.F: Lower Reynolds Number Limit for the Reader-Harris/Gallagher (1998) Equation

There are two lower limits on Reynolds number for each pair of tappings in 5.3.1 of ISO 5167-2:2003. Firstly, the Equation should not be used outside the range of the database. Secondly, even where there are data in the database, there is a lower limit of 5000 for Re _D (rather than 4000 as had been expected). The reason for choosing 5000 is that it is about the maximum value at which the change in the slope of the discharge coefficient as the flow becomes fully turbulent upstream of the orifice meter was seen to occur: see the 4″ and 6″ API oil data for β = 0.37 shown in Fig. 5.F.1: for each pipe size Set 1 and Set 2 were taken with different plates in the same pipe.

Fig. 5.F.1

The API oil data (collected by CEESI) in 4″ and 6″ pipes for β = 0.37 with flange tappings

The change in slope may, of course, occur for Re _D well below 5000: see, for example, Fig. 5.F.2: Set 1 and Set 2 were taken with different plates in the same pipe; Set 3 was taken with the same plate as in Set 2 but after the pipe was plated.

Fig. 5.F.2

The API oil data (collected by CEESI) in 2″ pipes for β = 0.36 with flange tappings

A typical value for Re _D at which the change in slope occurs is presumably 3700, since in fitting the data this value was calculated (see Sect. 5.4.3). If β is around 0.1 or below there is no significant change in slope: this is clear by calculation from Eq. 5.21; see also Sect. 5.E.3. Because of the variation in the Reynolds number at which the change of slope occurs and the steep gradient of the discharge coefficient below that Reynolds number for medium or large β, orifice plates are rarely used for Re _D  < 5000 except for small β (for example as in Sect. 5.E.3).