## Abstract

The established method of calculating the pressure loss and the flow rate of flows through metering orifices requires empirical correction coefficients. In this paper, a completely momentum-based approach is presented to predict the flow rate and the pressure loss. It is shown that the proposed approach is as precise as the established method but without any empirical coefficient. The required physical coefficients solely base on physics. They pinpoint the sources of the inaccuracy that effects the employment of an empirical correction coefficient to correct the established method—and confirm the empirical correction coefficient on a physical foundation. Ultimately, the physical coefficients evade the effort to experimentally calibrate the theory with the experiments. The integral momentum balance is applied to a control volume comprising the orifice reaching up to the respective pressure metering cross section. Depending on the location of the control volume's open boundaries, the pressure loss or the flow rate can be obtained. To substitute the integral with averaged expressions, the introduced physical coefficients are deduced directly from numerical simulations and parametrized by simple analytical fit functions to complete the momentum-based approach. The numerical model and the proposed approach are both verified by the comparison with the results obtained by the standard ISO 5167:2003. The agreement with the classical calculation method as per ISO 5167:2003 is very high and within the specified limits, which shows that no empirical coefficient is necessary for high-accuracy flow metering using the momentum balance.

## 1 Introduction

The metering orifice is used in differential pressure-based flow meters to indirectly meter the flow rate of gas, steam, or liquid flows. It is based on the effect that a fluid flowing through an orifice induces a pressure drop which can be obtained with the aid of a differential pressure sensor. The so-called differential pressure-based flowmeter is an assembly essentially comprising a specific flow structure that induces a pressure drop and a differential pressure sensor. When the pressure drop is induced by a metering orifice, the assembly is also called orifice meter. Due to its simple handling and installation, ruggedness, and the absence of moving parts, the metering orifice is a widely used device for differential pressure-based flow metering, especially in the petrochemical and the petroleum industry. Despite its simple geometry, the fluid mechanics behind the flow through an orifice are challenging since the orifice strongly affects the flow.

Figure 1 shows the velocity distribution of the center sectional plane obtained by the numerical simulations detailed in the following. One can see that the upstream flow is affected just shortly before the orifice whereas the downstream flow is affected significantly and also far-reaching. This means that the flow downstream of an orifice is highly disturbed. The contraction of the jet reaches the maximum shortly downstream of the orifice in the vena contracta. Therefore, the established methods based on the Bernoulli principle use the vena contracta as the downstream location, detailed in the following chapter.

But even the downstream pressure can not be assumed evenly distributed over cross sections close to the orifice as shown by the pressure distribution of the center sectional plane in Fig. 2. This results in a nonuniform radial pressure distribution in the vena contracta plane. Therefore, the calculation results of established methods diverge from the measured pressure loss which is bridged by introducing correction coefficients, as shown in the following chapter.

As a result, no satisfyingly accurate theory is available up to now to describe the hydraulics of an orifice without empirically determined correction coefficients. But besides the sole application of the Bernoulli principle to the flow through an orifice, also momentum-based approaches are reported. Literature provides some approaches that apply the momentum balance to the orifice since the orifice's downstream region can be understood as a sudden expansion where the Borda–Carnot equation applies.

Martin and Pabbi [1] employed a blended method based on the application of the momentum balance and the energy conservation to the orifice flow of gas. The authors introduced an empirical coefficient to account for the nonuniform pressure distribution over the orifice plate, correcting the total pressure force of the flow. The simplifications made by applying the momentum balance and the pressure distribution in the vena contracta are not considered further, which results in a more or less reasonable agreement with the references.

Benedict [2] presented a momentum-based application of the stationary momentum balance to calculate the pressure loss of orifices between an inflow cross section and the vena contracta plane. The author applied the inflow pressure also to the upstream orifice plate face but deducted an integral term from the counteracting force term that considers the nonuniform pressure distribution over the orifice plate—called force defect. With an empirical substitution of the force defect and not considering momentum coefficients, the theory deviates non-negligibly from the measurements.

Another blended momentum-based approach to calculating the pressure loss of compressible flows through an orifice is presented by Reader-Harris et al. [3]. To obtain a solution, the momentum balance, the energy balance, and additional equations must be solved iteratively, resulting in a deviation of less than 0.1% from experimental results.

All approaches have in common not to consider the velocity distribution over the specific cross section. The momentum fluxes of the inflow and the outflow cross section are introduced as averaged expressions instead of the area integrals of the velocity over the cross section. Only with the aid of detailed information on the velocity distributions, the integral momentum flux terms can be substituted by averaged expressions.

In contrast to the presented approaches, Malcherek and Müller [4] have shown that the pressure drop of a flow through a sudden contraction can be precisely predicted by applying a purely momentum-based approach without empirical coefficients. In addition, Müller and Malcherek [5] showed that the same approach with the same physical coefficients can be applied to a sudden expansion. By combining the findings of the sudden contraction and the sudden expansion, the momentum balance will be applied to the orifice in the present publication.

## 2 State of the Art

with *p* as the pressure, *Q* as the volumetric flow rate, and *A* as the respective cross-sectional area. Subscript 1 represents the upstream location but subscript 2 is the downstream location. Since the area of the vena contracta is unknown, the orifice bore area *A _{O}* is used as the downstream cross-sectional area

*A*

_{2}as proposed by Benedict [6] and Reader-Harris [7]. Unfortunately, the calculation result is in contradiction to observations.

*K*which includes the dependency of the pressure loss on the area ratio $\sigma =AO/A$, represented by the bracketed term in Eq. (1)

The pressure loss coefficient *K* is determined empirically by comparing the calculation with the experimental results. In consequence, different authors investigating the pressure loss of an orifice obtained different results for the loss coefficient. Figure 3 shows the loss coefficient *K* referred to the mean velocity of the pipe *Q*/*A* versus the area ratio *σ* of various publications, namely, Weisbach 1855 [8], Idelchik 1966 [9], Miller 1978 [10], Alvi et al. 1978 [11], Benedict 1980 [2], Benedict 1984 [6], Westaway and Loomis 1984 [12], Rennels and Hudson 2012 [13], and the international standard ISO 5167 [14].

Most of the experimental data agree with one another, solely the data of Westaway and Loomis [12] deviates noticeable from all other results for medium area ratios. The proposed expression for the loss coefficient by Benedict 1980 [2], represented by the continuous black line, is only for valid $\sigma \u22640.6$. The loss coefficient according to the international standard ISO 5167:2003 [14] is evaluated for a pipe Reynolds number where *K* approaches approximately constant values ($ReD>1\xb7106$).

The loss coefficient *K* is often derived from the so-called discharge coefficient *C _{D}* employed for calculating the flow rate from the measured differential pressure, see Eq. (3). The loss coefficient

*K*proposed by Refs. [2,6,11,12,14] is based on

*C*, so

_{D}*K*is representing the remaining or irreversible loss in the fully (re-)developed flow. This is shown schematically in Fig. 4 by the diagram of the pipe wall pressure over the flow length. Here, the pressure loss (named irreversible pressure drop in the following) $\Delta pirrev$ can be obtained by extrapolating the lines of the linear pressure gradients. In contrast, the pressure drop measured at the defined pressure tapping points $\Delta p$ contains a reversible share and is recovered partially to the irreversible pressure drop with increasing distance to the orifice.

## 3 Metering Orifice

*C*according to the standard ISO 5167-2:2003 [14]. The discharge coefficient

_{D}*C*introduces the ratio between the actual and the theoretical flow rate and is obtained empirically. Analogously to the loss coefficient

_{D}*K*, many investigations made the determination of a precise discharge coefficient

*C*a subject of discussion, see Refs. [2,6,11,12,14,15]. The basic standard for differential pressure-based flow metering is the International Organization for Standardization (ISO) standard ISO 5167:2003. Nearly every commercially manufactured orifice meter device is standardized according to the ISO 5167:2003. Therefore, the present publication is solely focusing on the pressure tapping point couples and the calculation of

_{D}*C*provided by the standard ISO 5167:2003 for orifices [14]. Among other specifications, the ISO 5167:2003 [14] defines the permitted pressure tapping point locations for flow rate measurements. These are the corner, the flange, and the D-D/2 tapping points. The corner tapping points are located directly at the orifice plate surface, which is in the corner of the pipe wall and the orifice plate. The flange tapping points are always 25.4 mm (1 in) distant to the nearest orifice plate surface, whereas the D-D/2 tapping points are located 1 pipe diameter

_{D}*D*upstream and 0.5

*D*downstream of the upstream orifice plate surface. The calculation procedure of the standard ISO 5167:2003 is substantially based on the research of Reader-Harris and Gallagher, compendiously given in Ref. [7]. The so-called Reader-Harris/Gallagher equation for the discharge coefficient

*C*is given by

_{D}*d*and pipe diameter

*D*with Re

*as the pipe Reynolds number. The coefficient $M\u20322$ is defined as*

_{D}Depending on the pressure tapping point locations, the length portions *L*_{1} and $L\u20322$ have to be parameterized as shown in Table 1.

Parameter | Corner | Flange | D − D/2 |
---|---|---|---|

L_{1} | 0 | $25.4D$ | 1 |

$L\u20322$ | 0 | $25.4D$ | 0.47 |

Parameter | Corner | Flange | D − D/2 |
---|---|---|---|

L_{1} | 0 | $25.4D$ | 1 |

$L\u20322$ | 0 | $25.4D$ | 0.47 |

Standard ISO 5167:2003 determines the limits of application for Eq. (4), see Ref. [14] for details. The main relevant limits are:

the orifice diameter: d $\u226512.5\u2009mm$,

the diameter ratio $\sigma $: $0.10\u2264\sigma \u22640.75$,

the pipe diameter D:$\u200950\u2009mm\u2264D\u22641000\u2009mm$,

and the pipe Reynolds number $ReD\u22655000$ (in general).

However, the main benefit of applying the discharge coefficient is that *C _{D}* can be calculated with easily measurable flow and geometry characteristics.

The calculation of *C _{D}* as per Eq. (4) can be assumed well checked against experimental results and, therefore, validated as the reference for the current investigation. Consequentially, own experiments are not necessary while the momentum-based approach is being compared to the results obtained with Eq. (4). These results are also used to validate the numerical simulation.

## 4 The Integral Momentum Balance

### 4.1 From Integral to Averaged.

**I**in an arbitrary control volume Ω equals the sum of the acting forces, which are (1) the gravitational force on the mass

*M*within Ω, (2) pressure forces on the boundary of the control volume $\u2202\Omega $, (3) momentum fluxes, and (4) frictional forces $FF$

The shaded area in Fig. 5 represents the control volume and shows the pressure and velocity distributions for a control volume of short reach upstream and downstream of the orifice. Per convention, the normal unit vector **n** is always directed out of the control volume.

For stationary flow, the left side of Eq. (6) is zero. Viscous friction losses can be neglected in good approximation since the control volume covers only a short reach upstream and downstream of the orifice and the pipe's wall roughness is restricted to low values according to the ISO 5167:2003 [14]. This assumption is in line with the sudden expansion where considerable frictional loss reoccurs when the flow is reattached to the wall, see Ref. [5] and Reader-Harris [7].

Equation (7) physically means that the pressure forces equal the momentum fluxes.

*p*

_{1}with a positive sign because the negative sign of the pressure force term and the negative sign of the normal unit vector

**n**cancel out. The average pressure of the outflow cross section is

*p*

_{2}but negative because of the positive normal unit vector. The third pressure force term considers the counteracting pressure of the upstream-facing orifice plate's area ($A1\u2212AO$) named $pR,1$ with a negative sign analogously to

*p*

_{2}. The fourth and final pressure force term takes into account the counteracting pressure of the downstream facing orifice plate's area ($A1\u2212AO$) named $pR,2$ and is positive in analogy to

*p*

_{1}. The substitution of the integral pressure force term by averaged expressions implies that the pressure is uniformly distributed over the whole cross section, so Eq. (7) yields

*p*

_{1}but the momentum flux leaving the control volume is negative in analogy to

*p*

_{2}. To substitute the integral momentum fluxes by averaged expressions, the nonuniform velocity distribution over the cross section must be considered by the momentum coefficient $\beta =(\u222bAv2dA)/(Q2/A2\u2009A)$. With $A=A1=A2$ where

*A*represents the pipe's cross-sectional area (see Fig. 5), Eq. (8) gives

### 4.2 Irreversible Pressure Drop.

To determine the irreversible pressure drop $\Delta pirrev$, the linear pressure gradients upstream and downstream of the orifice are extrapolated up to the intersection point with the orifice plane. The vertical distance between both intersection points represents the pressure drop which is solely induced by the orifice and which can not be recovered, see Fig. 4. In consequence, the control volume can be defined in a way that its open boundaries are sufficiently far away from the orifice so the flow is fully (re-)developed and the velocity profile is ideally turbulent again.

*r*/

*R*in Fig. 6 for $d/D=0.714$ as a representative example. When the radial flow is attached to the orifice plate, the reacting pressure decreases rapidly up to the orifice bore edge. Constant values of the normalized pressure indicate dead-water zones. However, the counteracting upstream pressure can be parameterized by

*c*accounts for the nonuniform pressure distribution over the orifice plate faces plus the dependency of the differential reacting pressure on the area ratio

_{P}*σ*. The performed numerical simulation shows, that $cP,1$ can be parameterized very precisely with the function of

*c*for the sudden contraction (see Ref. [4]) which yields

_{P}Figure 7 shows the upstream pressure coefficient $cP,1$ referred to the pressure 1*D* upstream as a function of the area ratio *σ*, represented by markers. The numerical simulation reveals that $cP,1$ is not dependent on the flow rate *Q* in analogy to the sudden contraction (Ref. [4]). The parametrization of the pressure coefficient for the sudden contraction is added by a line to Fig. 7, describing $cP,1$ of the metering orifice very precisely.

*v*represents the maximum flow velocity that occurs in the vena contracta where the pressure is minimal. But the low-pressure zone in the vena contracta extends up to the downstream orifice plate, as shown in Fig. 2. Equation (12) matches the downstream reacting pressure for all investigated configurations within a deviation of under 0.5% as shown by the dotted line “Bernoulli approx.” in Fig. 6. This finding confirms that the Bernoulli principle is valid when applied appropriately on a streamline. However, the upstream centerline velocity

_{C}*v*

_{1}is substituted by the average velocity

*Q*/

*A*for turbulent flow in good approximation. Since the shape and the exact location of the vena contracta are unknown, $vC=Q/AC$ with the vena contracta cross section

*A*referenced to the known cross section of the orifice bore

_{C}*A*multiplied by the contraction coefficient

_{O}*μ*to account for the reduced flow area, hence $vC=Q/(\mu C\u2009AO)$. With the substitutions of the unknown velocities

_{C}*v*

_{1}and

*v*, Eq. (12) gives

_{C}*μ*versus the area ratio

_{C}*σ*represented by markers. It is found that

*μ*is not dependent on the flow rate

_{C}*Q*and therefore,

*μ*is parameterized by

_{C}The momentum-based loss coefficient $KM,I$ exhibits a very high agreement with the literature, represented by the bold black line in Fig. 9. In addition, the simulated irreversible pressure drop is predicted very precisely by Eq. (15), which is shown in chapter 5.

*p*

_{1}, $pR,1$, and $pR,2$ can be combined to the differential reacting pressure $\Delta pR$

*p*

_{1}cancels out and therefore, $\Delta pR$ is not dependent on the defined control volume. Using Eq. (18) into Eq. (9) and flipped to $p1\u2212p2$, the irreversible pressure drop $\Delta pirrev,II=p1\u2212p2$ of an orifice for the second approach gives

In contrast to $KM,I$, the loss coefficient $KM,II$ is dependent on the pipe Reynolds number Re* _{D}*.

$KM,II$ is represented by the bold dark gray line and $KM,I$ by the bold black line in Fig. 9, which extends Fig. 3 by the momentum-based loss coefficients *K _{M}*. It can be seen that the parameterization of $KM,II$ for a pipe Reynolds number of $1\xb7106$ (chosen in compliance with the graph of the ISO 5167:2003) is almost equivalent to $KM,I$. Only for area ratios greater than 0.7, the deviation to the literature numbers as well as to $KM,I$ increases considerably. However, the standard ISO 5167:2003 [14] defines the upper applicability limit of

*σ*as 0.56. For lower area ratios, the loss coefficient $KM,II$ shows very high agreement with the values of the standard ISO 5167:2003 [14] as well as with other references. Up to $\sigma =0.56$, the impact of Re

*on $KM,II$ is negligible in good approximation.*

_{D}It shall be mentioned that both momentum-based loss coefficients *K _{M}* are solely based on physical coefficients without any empirically determined coefficient. It is therefore justified to state that both parametrizations of

*K*confirm most of the empirical coefficients by a physical foundation.

_{M}### 4.3 Metering Orifice.

*β*

_{1}and

*β*

_{2}have to be taken into account. Since both momentum coefficients are required for every tapping point couple, the total momentum coefficient

*β*is defined as $\beta 2\u2212\beta 1=\beta $ in the following. In addition, numerical simulations (detailed in the following chapter) indicate that even the pressure is not uniformly distributed over the cross section in the vicinity of the orifice, especially for the corner and flange tapping points. Figure 10 shows the pressure profiles over the pipe radius at the considered tapping point locations. Continuous lines represent the upstream locations and dashed lines the downstream ones. Only for the tapping points at a flow distance of 1

*D*upstream and $D/2$ downstream of the orifice, the pressure can be assumed uniformly distributed over the cross section in good approximation. For the corner and the flange pressure tapping points, the pressure deviates considerably from the pressure at the wall since the flow is highly disturbed in this region. Therefore, the momentum balance as per Eq. (19) applied to the control volume of a metering orifice with the exact integral expression of the pressure over the cross section instead of the averaged pressure terms (compare Eq. (19)) gives

For an analytical approach, the pressure integrals must be replaced by the pressure as obtained at the pressure tapping points at the wall. In consequence, an additional coefficient must be taken into consideration, accounting for the deviation of the average differential pressure of the respective cross section from the differential pressure obtained at the wall $\Delta p$.

*γ*is introduced, depending on the specific pressure tapping point couple. Flipping Eq. (21) to the volume flow rate

*Q*, the final formula for the flow rate through orifices gives

which is completely momentum-based and requires no empirical coefficients.

## 5 Numerical Simulation

### 5.1 Numerical Model.

To close the momentum-based approach, the parametrizations of the required coefficients *β*, *γ*, and *c _{P}* must be obtained by a three-dimensional numerical simulation. Therefore, the CFX-Tool of Ansys 20 is employed to solve the Reynolds-averaged Navier–Stokes (RANS) equations. The governing equations are solved simultaneously for steady-state conditions by a direct method, so no pressure–velocity coupling algorithm is required. The applied high-resolution solver setting introduces a blended method of the second-order central differences scheme and first-order upwind scheme near discontinuities to solve the governing equations. According to the ANSYS CFX-Solver Theory Guide [16], the high-resolution advection scheme is second-order accurate. The solution is assumed fully converged when all residuals are below $1\xb710\u22124$.

The investigation of Morrison et al. [17] reveals that the pressure drop of a metering orifice, and consequently the discharge coefficient *C _{D}*, is very sensitive to the upstream flow condition. Based on these findings, the simulation domain starts 11

*D*upstream of the orifice and ends 8

*D*downstream of the orifice. This ensures a fully developed inlet flow entering the control volume and prevents boundary conditions from diffusing into the region of interest.

The simulation domain's pipe diameter *D* is 105 mm with an orifice bore diameter *d* of 61.75 mm used to approach the final settings of the numerical simulation. The verified simulation is performed for water at $25\u2009\xb0C$.

The entire simulation domain is meshed with tetrahedrons of 2 mm edge length at maximum including an automatic size reduction when approaching edges or faces of the simulation domain. At the pipe walls, the mesh is additionally refined by ten layers with a total thickness corresponding to the maximum edge length of the elements (2 mm) to appropriately resolve the boundary layer. The height of every layer increases by a factor of 1.1 with increasing distance to the pipe wall. This ensures that the first layers are within the recommended dimensionless wall distance *y*^{+} for the chosen turbulence model and the applied wall treatment approach (wall function), detailed in Ref. [16]. The mesh in the vicinity of the orifice is shown in Fig. 11 with an additional mesh refinement around the orifice edges plus the orifice plate faces where the pressure coefficient *c _{P}* is obtained. At the orifice plate faces, the edge length of a mesh element is 0.5 mm increasing smoothly to 2 mm toward the inside. According to the standard ISO 5167:2003 [14], the metering orifice's inlet edge must be sharp, defining the mesh resolution of the edge. Its radius must be less than 4/10000 of the smallest investigated orifice bore diameter

*d*. With a growth rate of 1.1, this refinement also propagates into the surrounding mesh as a beneficial side effect.

For a solvable set of equations, a proper turbulence model must be employed for RANS simulations. In analogy to the numerical simulation of the sudden contraction [4] and the sudden expansion [5], the *k*–*ω* shear stress transport (SST) turbulence model [18] is employed for the metering orifice as well. One of its major assets is the capability of accurately modeling adverse pressure gradients and flow separation, see Ref. [19], which is a meaningful quality for the application to the numerical simulation of an orifice. A detailed evaluation of the performance of commonly used turbulence models compared with experimental results is given in Ref. [4]. Imada et al. [20] numerically investigated the flow through an orifice, showing that a higher agreement between numerical simulations and the results according to ISO 5167:2003 [14] is obtained employing the *k*–*ω* SST turbulence model compared with the k-*ε* turbulence model. This finding is in line with own investigations of the orifice where the deviation of the computed differential pressure from the ISO 5167:2003 result decreases from 2 to 3% with the k-*ε* to 0.6–1.6% with the *k*–*ω* SST turbulence model. In addition, the profound study of Bardina et al. [21] confirms that the *k*–*ω* SST turbulence model provides the best overall performance for simple and complex flows. As a result of the authors' experiences and investigations as well as the findings of other researchers, the *k*–*ω* SST turbulence model is applied to the numerical simulation of the metering orifice.

At the inlet of the simulation domain, the mass flow rate and medium turbulent intensity (5%) are defined as the boundary condition. Average static pressure with pressure averaging over the whole area is defined as the boundary condition at the outlet. The domain's reference pressure is set to the standard pressure of 101 325 Pa. An equivalent sand grain roughness *k _{s}* of 1$\xb7$10

^{– 5}m is applied to the closed boundaries of the simulation domain. This value represents the upper limit according to the standard ISO 5167:2003 for the investigated Reynolds numbers and area ratios.

In compliance with the method proposed by Roache [22], a mesh independence study is performed for an orifice bore diameter *d* of 61.75 mm and a mass flow rate $m\u02d9$ of 20 kg/s. The differential pressure $\Delta p$, the combined momentum coefficient $\beta =\beta 2\u2212\beta 1$, the pressure exaggeration coefficient *γ*, and the pressure coefficient *c _{P}* are chosen as relevant quantities for evaluating the mesh independence. All chosen parameters except the pressure tapping point independent

*c*are obtained at the corner tapping points (represented by the subscript corner) since the parameters in the region very close to the orifice are most sensitive to mesh adaptions. Table 2 shows the results of these quantities as obtained for the given number of mesh elements. No considerable change of the relevant quantities can be found when the mesh is refined to a fine mesh. The numerical uncertainty of the coefficients obtained with the fine grid, represented by GCI fine in Table 2, is 1% for

_{P}*β*

_{corner}but far below 1% for

*γ*

_{corner},

*c*, and $\Delta pcorner$ compared with the respective values for the medium mesh. Calculating the pressure drop $\Delta p$ with the coefficients obtained for a medium mesh and the extrapolated values results in a deviation of less than 0.3%, which means that the applied medium mesh yields sufficiently accurate results.

_{P}Mesh (elements) | β_{corner} | γ_{corner} | c_{P} | $\Delta pcorner$ (Pa) |
---|---|---|---|---|

Coarse 10 392 079 | 0.482 | 1.472 | −19.367 | 53832 |

Medium 18 550 778 | 0.479 | 1.470 | −19.230 | 53337 |

Fine 27 839 559 | 0.477 | 1.472 | −19.243 | 53397 |

Extrapolated value | 0.475 | 1.473 | −19.253 | 53445 |

GCI fine [1] | 0.011 | 0.003 | 0.002 | 0.003 |

Mesh (elements) | β_{corner} | γ_{corner} | c_{P} | $\Delta pcorner$ (Pa) |
---|---|---|---|---|

Coarse 10 392 079 | 0.482 | 1.472 | −19.367 | 53832 |

Medium 18 550 778 | 0.479 | 1.470 | −19.230 | 53337 |

Fine 27 839 559 | 0.477 | 1.472 | −19.243 | 53397 |

Extrapolated value | 0.475 | 1.473 | −19.253 | 53445 |

GCI fine [1] | 0.011 | 0.003 | 0.002 | 0.003 |

In addition, Fig. 12 shows the axial pressure profile along the wall for the coarse (dotted line), medium (dashed line), and fine mesh. The maximum deviation between the pressure profiles of the medium and fine grid amounts to less than 1%. The maximum deviation reaches 2% comparing the medium with the coarse mesh pressure profile. The maximum pressure recovery is predicted at a downstream flow length of approximately 6*D* by all mesh configurations at a very similar magnitude.

Since the numerical simulation is the key element of the present investigation, the numerical model is also checked against the literature. Lebedev et al. [23] investigated the deviation of the loss coefficient occurring when the orifice meter installation does not meet the requirements of the specific standards as well as the effect of manufacturing tolerances. In this study, the previously given diameters and cooling water at a temperature of $70\xb0C$ with a density ϱ of 986 kg/m^{3} and a dynamic viscosity *η* of 4.09$\xb7$10^{– 4 }Pa·s are used to simulate the differential pressure at different mass flow rates. Although Lebedev et al. takes advantage of the simulations domain's axial symmetry, the present three-dimensional numerical simulation can be compared with the numerical solutions of Lebedev et al. The numerical investigation of Lebedev et al. [23] provides a set of parameters that are used for a self-sufficient verification of the numerical model. Lebedev et al. performed the axisymmetric two-dimensional simulation with the Spalart– Allmaras turbulence model [24], which is a one-equation model designed and optimized specifically for aeronautic applications. As a reference, Lebedev et al. employed the k-*ε* turbulence model. The authors obtain higher agreement with the results according to the standard ISO 5167:2003 using the Spalart–Allmaras turbulence model. For all investigated mass flow rates $m\u02d9$, the deviation to the standard results is less compared with the k-*ε* turbulence model.

For additional verification, the Spalart–Allmaras turbulence model is applied to the present numerical model for the medium mesh as well as water at $70\u2009\xb0C$ with the given values for ϱ and *η* diverging from the actual values of pure water at $70\u2009\xb0C$. The results of the numerical simulation with the Spalart–Allmaras turbulence model $\Delta pSA$ and the *k*–*ω* SST turbulence model $\Delta pSST$ are shown in Table 3 together with the results of Lebedev et al. $\Delta pLebedev$ and the standard ISO 5167:2003 $\Delta pISO$. All differential pressures are obtained or calculated for the D-D/2 pressure tapping points.

$m\u02d9\u2009$($kg/s$) | 4.11 | 10.96 | 16.44 | 17.26 |
---|---|---|---|---|

$\Delta pISO\u2009(kPa)$ | 2.256 | 16.17 | 36.46 | 40.20 |

$\Delta pLebedev(kPa)$ | 2.261 | 16.20 | 36.53 | 40.27 |

$\Delta pSA(kPa)$ | 2.229 | 16.30 | 37.27 | 38.75 |

$\Delta pSST(kPa)$ | 2.227 | 15.99 | 36.17 | 39.77 |

$m\u02d9\u2009$($kg/s$) | 4.11 | 10.96 | 16.44 | 17.26 |
---|---|---|---|---|

$\Delta pISO\u2009(kPa)$ | 2.256 | 16.17 | 36.46 | 40.20 |

$\Delta pLebedev(kPa)$ | 2.261 | 16.20 | 36.53 | 40.27 |

$\Delta pSA(kPa)$ | 2.229 | 16.30 | 37.27 | 38.75 |

$\Delta pSST(kPa)$ | 2.227 | 15.99 | 36.17 | 39.77 |

Table 3 shows that applying the Spalart–Allmaras turbulence model, the results of Lebedev et al. $\Delta pLebedev$ can only be reproduced within a deviation of 4%. When the *k*–*ω* SST turbulence model is applied to the numerical model, the agreement between the simulation results $\Delta pSST$, the results according to Lebedev et al. [23], and the standard ISO 5167:2003 is significantly increased to a maximum deviation of 1.3%. Possibly, the one-equation Spalart–Allmaras turbulence model performs better without three-dimensional effects where its major flaw of inaccurately modeling jet spreading rates (see Ref. [25]) is not coming into effect considerably. However, the results show once more that the *k*–*ω* SST turbulence model is an appropriate choice.

### 5.2 Pressure Drop for Flow Metering.

Since the first verification with the *k*–*ω* SST turbulence model shows high agreement with the numerical results of Lebedev et al. [23], the simulation is now checked against the results of the standard ISO 5167:2003 for a set of parameters, given in Table 4. An orifice bore diameter *d* of 12.5 mm indicates the lower applicability limit of the standard ISO 5167:2003 and the orifice bore diameter of 75 mm the upper one. With a combination of every mass flow rate $m\u02d9$ and area ratio $d2/D2$, the numerically obtained differential pressures $\Delta psim$ represent the final verification of the numerical model. Therefore, solely the differential pressure values at the pressure tapping points are used since these values are most crucial to a reasonable accuracy of the proposed approach.

Parameter | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

$d(mm)$ | 12.5 | 21 | 30 | 40.5 | 51 | 61.75 | 75 |

$m\u02d9(kg/s)$ | 2 | 4.11 | 10.96 | 16.44 | 17.26 | 20 | |

Re_{D} | 27,253 | 56,004 | 149,340 | 224,020 | 235,190 | 272,525 |

Parameter | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

$d(mm)$ | 12.5 | 21 | 30 | 40.5 | 51 | 61.75 | 75 |

$m\u02d9(kg/s)$ | 2 | 4.11 | 10.96 | 16.44 | 17.26 | 20 | |

Re_{D} | 27,253 | 56,004 | 149,340 | 224,020 | 235,190 | 272,525 |

Compared with the applicability limits given by the standard ISO 5167:2003 [14], the range of Re* _{D}* of the parameter set (Table 4) is very narrow. As shown in the following, the parametrization of the coefficients and the resulting fit functions are very sensitive to low Reynolds numbers. In contrast, the profiles are not considerably shaped by higher Reynolds numbers. Therefore, the flow rate values employed by Lebedev et al. [23] will mainly also be used for the present numerical study. Only water at a single temperature is numerically investigated to obtain the results discussed in the following. Still, the fit functions and figures are referring to the pipe Reynolds number Re

*instead of the flow rate $m\u02d9$ for the sake of comparability with the literature.*

_{D}With the *k*–*ω* SST turbulence model and a fluid temperature of 25$\u2009\xb0C$, the simulated differential pressures match the ones calculated according to the standard ISO 5167:2003 within −1% to +0.8% deviation for all investigated mass flow rates and area ratios, except for the outliers for $d=$ 75 mm at the corner tapping points. This is within the range of the measurement uncertainty for the discharge coefficient *C _{D}* given by the standard ISO 5167:2003 [14] when referred to the differential pressure. The maximum deviation amounts to 1.6% for the largest area ratio at the corner tapping points, whereas a mean deviation of approx. 0.13% is achieved. In general, the maximum deviation is reached for the corner tapping points, whereas the deviation for the D-D/2 tapping points is minimum. This is shown in Fig. 13, presenting the ratio of the simulated differential pressure $\Delta psim$ to the calculated differential pressure according to ISO 5167:2003 $\Delta pISO$ versus the mass flow rate for the corner, flange, and D-D/2 tapping point couples. It also shows that the deviation decreases with increasing mass flow rate for all tapping point couples. In conclusion, the results of the standard ISO 5167:2003 plus the results of Lebedev et al. [23] for a different fluid than water can be reproduced by the employed numerical model within a small deviation range. This means that the numerical model is entirely verified, so it can be used to investigate the flow through an orifice and quantify the required coefficients.

*c*as a function of the pipe Reynolds number Re

_{P}*in the shape of $f(ReD)=b1ReDb2+b3$ approximates the results with very high agreement but only for a single area ratio. To obtain an expression including the dependency on the area ratio as well, the fit coefficients $b1,3$ are expressed as functions of the area ratio with the aid of the parameter set. Again, the fit function $f(d/D)=b4\u2009(dD)b5+b6$ is applied to*

_{D}*b*

_{1}and

*b*

_{3}to account for the area ratio dependency. This results in a two-parameter fit function of the total pressure coefficient

*c*given as a function of the pipe Reynolds number Re

_{P}*and the diameter ratio $d/D=\sigma $ for better practical applicability*

_{D}Equation (23) reproduces the numerical results with a mean deviation of less than 1% and a maximum deviation of 2%. This confirms the capability and suitability of the proposed simple fit function. Figure 14 shows the fit function as per Eq. (23) and the numerical results in a three-dimensional plot with the z-axis in a logarithmic scale.

*β*

_{1}and the downstream momentum coefficient

*β*

_{2}must be obtained for every pressure tapping point couple, the combined momentum coefficient $\beta tp=\beta 2,tp\u2212\beta 1,tp$ is parametrized with the subscript tp representing the respective tapping point couple: corner, flange, or D-D/2. A representative example for the combined momentum coefficient

*β*

_{tp}as a function of the pipe Reynolds number Re

*is shown in Fig. 15 for the area ratio*

_{D}*σ*of 0.3459 (

*d*/

*D*=

*0.588). For increasing flow rates,*

*β*

_{tp}increases and approaches a final value, depending on the tapping point couple. This is because the difference between the downstream and the upstream momentum coefficient becomes constant for flow rates to increase further. Only for very small flow rates, the difference decreases. As expected, the lowest values are obtained for

*β*

_{corner}where the flow represents the conditions within the orifice since now flow length is covered in the pipe. The deviation between the flange

*β*

_{flange}and the D-D/2 momentum coefficient $\beta D\u2212D/2$ is significantly reduced compared with

*β*

_{corner}. However,

*β*

_{tp}shows a dependency on the flow rate regardless of the tapping point couple. Again, a fit function in the shape of $f(ReD)=b1ReDb2+b3$ can be employed to approximate the results for every

*β*

_{tp}with high agreement at the given area ratio. Analogously to the total pressure coefficient

*c*, the fit coefficients

_{P}*b*

_{1}and

*b*

_{3}are expressed as functions of the diameter ratio as well since

*β*

_{tp}is also a function of the diameter ratio

*d*/

*D*. The fit coefficients' dependency on the area ratio is also approximated very well by the same fit function $f(d/D)=b4\u2009(dD)b5+b6$. Eventually, the final fit functions accounting for the dependency on the flow rate and the diameter ratio are obtained for the momentum coefficient of every tapping point couple, shown for

*β*

_{corner}in Fig. 16.

*γ*. Figure 10 shows that the deviation of the radial pressure profile from the respective average radial pressure is almost 0 for the D-D/2 tapping points but strongly increasing for the flange and especially the corner tapping points. Therefore, the pressure exaggeration coefficient varies with the tapping point couple as well as the combined momentum coefficient

*β*

_{tp}. The numerical simulation shows that

*γ*can be assumed independent of the flow rate for every tapping point couple since the spread around the mean of

*γ*

_{tp}for all flow rates at a single area ratio is less than 0.1%. But the pressure exaggeration coefficient

*γ*

_{tp}exhibits a dependency on the area ratio. In Fig. 17, the numerical results of

*γ*

_{tp}for all tapping point couples are shown by markers together with the fit functions. For

*d*/

*D*approaching zero

*γ*approaches unity since $d/D=0$ means no orifice opening, thus no flow. For

*d*/

*D*approaching unity, which represents a straight pipe,

*γ*should approach unity as well, but this case is not covered by the present approach as it exceeds the limits of

*d*/

*D*given by the standard ISO 5167:2003 [14]. The value for $\gamma D\u2212D/2$ at $d/D=0.588$ is regarded as an outlier and therefore excluded from the current considerations, knowing that the deviation of the radial pressure distribution for $d/D=0.588$ is also 0 at the D-D/2 tapping points. In analogy to the total pressure coefficient

*c*and the combined momentum coefficient

_{P}*β*

_{tp}, the same fit function can be employed, parametrizing the results with very high agreement. In case of the D-D/2 tapping points, the flow length upstream and downstream of the orifice is sufficient for the pressure to be distributed uniformly over the radius, which means the pressure at the wall is equal to the pressure at any point over the radius—resulting in $\gamma D\u2212D/2=1$. The following fit functions are obtained for

*γ*

_{corner}and

*γ*

_{flange}

*Q*substituted by $m\u02d9=\u03f1Q$, Eq. (22) gives

The mass flow rate can now be determined for a given pressure difference at a specific tapping point couple. Using the universal total pressure coefficient *c _{P}* as per Eq. (23), the respective combined momentum coefficient

*β*

_{tp}as per Eqs. ((24),(25),(26)) plus the respective pressure exaggeration coefficient

*γ*

_{tp}as per Eqs. ((27),(28)) into Eq. (29), the momentum-based approach for flow metering is complete.

## 6 Results and Discussion

Figure 18 shows the ratio of both momentum-based irreversible pressure drops $\Delta pirrev,I$ and $\Delta pirrev,II$ to the reference irreversible pressure drop $\Delta pISO,irrev$ according to the standard ISO 5167:2003 [14] versus the mass flow rate $m\u02d9$. The deviation is within 2% for all investigated mass flow rates and area ratios except for an orifice diameter *d* of 75 mm. In this case, the deviation increases to approximately 6% due to the small pressure drop. Nonetheless, Fig. 18 shows that both approaches of the momentum balance are also capable of accurately predicting the irreversible pressure drop of a flow through an orifice.

The standard ISO 5167:2003 specifies a range of approximately ±0.6% for the uncertainty of the discharge coefficient *C _{D}* as per Eq. (3) for metering the mass flow rate $m\u02d9$ (Eq. (4)). Figure 19 shows the momentum-based mass flow rate $m\u02d9$ divided by the mass flow rate as obtained applying the ISO 5167:2003 $m\u02d9ISO$ versus the mass flow rate used as the inlet boundary condition for the numerical simulation.

Figure 19 confirms that the results of the momentum-based approach are within the specified uncertainty range of ±0.6%. Only at the corner tapping points, the deviation increases to ±1%. In conclusion, the proposed momentum-based approach predicts the flow rate of water for a measured differential pressure as precisely as the standard approach given in Ref. [14]. The major asset of the proposed approach is that no empirical coefficient is required but solely coefficients found physically to account for a nonuniform pressure or velocity distribution over specific cross section or walls.

As previously mentioned, the range of the investigated flow rates (and corresponding pipe Reynolds numbers Re* _{D}*) is very limited compared with the range of applicability defined by the standard ISO 5167:2003 [14]. To confirm the validity of the proposed approach for the whole range of Re

*, the calculation of the mass flow rate $m\u02d9$ is extended up to $ReD\u22482.3\xb7107$ employing the given parametrizations. It appears that the deviation for all tapping point couples is reduced with increasing mass flow rate, as already indicated by the profiles of the correlation in Fig. 19. For $m\u02d9>20\u2009kg/s$, the predicted mass flow rates for all pressure-tapping point couples converge to correlation values that correspond to a deviation within a range of ±0.6%.*

_{D}In summary, it is proven that the mass flow rate and the irreversible pressure drop of a flow through a metering orifice can be predicted with very high agreement by employing an approach based on the momentum balance. Analogies to the sudden contraction and the sudden expansion are also drawn that outline the similarities of the mentioned hydraulic structures as well as the modularity of the momentum balance. The present investigation is solely focused on water to demonstrate the proposed method and its applicability. The practicality of the proposed method for different fluids like oil and compressible fluids needs to be examined in subsequent investigations.

## 7 Conclusions

Using the energy conservation principle to predict the flow rate of differential pressure-based flow meters requires an additional coefficient to account for the discrepancy between empirical and calculated results. Therefore, the so-called discharge coefficient *C _{D}* is introduced, whose empirically obtained parametrization results in a remarkable fit function. This fit function accounts for the dependency of

*C*on the pipe Reynolds number Re

_{D}*, the diameter ratio*

_{D}*d*/

*D*, and the distance of the pressure-tapping couple to the orifice. Employing the established calculation method, an uncertainty range of ±0.6% for the calculated values is obtained.

The present investigation shows that the same average deviation can be maintained when the flow rate is predicted by a purely momentum-based approach and without any empirical coefficient. The momentum-based theory accounts for the inhomogeneous velocity and pressure distributions over the cross section and the orifice walls by separate coefficients. But here, the coefficients capture the relevant physical effects since *c _{P}*,

*β*

_{tp}, and

*γ*

_{tp}arise from physically founded substitutions. With the aid of numerical simulations, parametrizations of the required coefficients are obtained by simple analytical fit functions. The obtained parametrizations account for the dependency of the pipe Reynolds number Re

*, the diameter ratio*

_{D}*d*/

*D*, and the distance of the pressure tapping couple to the orifice. As a result, very high agreement between the empirical formulation as per standard ISO 5167:2003 and the formulation of the momentum-based approach is achieved. Additionally, the irreversible pressure drop of an orifice can be calculated in the same way simply by dropping two coefficients.

The presented approaches evolved from the investigations of the sudden contraction [4] and the sudden expansion [5], which gave fundamental insights into the flow through hydraulic structures and its impacts on the flow itself. Deriving the momentum balance for the orifice shows various analogies to the derivations for the sudden contraction and the sudden expansion. This speaks for the universality and applicability of the momentum balance describing hydraulic problems. In addition, it is already proven that the momentum balance is also capable to describe the outflow problem [26], the overflow over a weir [27], and the flow under a sluice gate by Malcherek [28] detailed further by Steppert et al. [29].

The results confirm that a momentum-based approach for practical applications performs very well compared with established empirical methods. Thus, the description of every hydraulic structure should be based on the integral momentum balance instead of the energy conservation principle. Applying the momentum balance leads to a better understanding of the physics behind the problem as well as to new hydraulic formulations without artificial parameters.

## Acknowledgment

The authors thank the anonymous reviewers and the editor-in-chief for their helpful comments and constructive suggestions that improved this paper. The authors acknowledge financial support from the University of the Bundeswehr Munich.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## Nomenclature

*A*=cross-sectional area, m

^{2}- $b1\u20268$ =
fit parameter, 1

*c*=coefficient, 1

*C*=_{D}discharge coefficient, 1

*D*=pipe diameter, m

*d*=orifice bore diameter, m

*F*=force, N

*I*=momentum, kgm/s

*g*=gravitational acceleration, m/s

^{2}*k*=_{s}equivalent sand grain roughness, m

*K*=pressure loss coefficient, 1

*L*=flow length, m

*M*=mass, kg

- $m\u02d9$ =
mass flow rate, kg/s

*n*=normal unit vector

*p*=pressure, Pa

*Q*=volume flow rate, m

^{3}/s*R*=radius, m

*r*=orifice wall radius, m

- Re =
Reynolds number, 1

*v*=velocity, m/s

*y*=distance from the wall, m

*z*=height, m

### Greek Symbols

*β*=momentum coefficient, 1

*γ*=pressure exaggeration coefficient, 1

- Δ =
difference

*η*=dynamic viscosity, kg/sm

*μ*=_{C}contraction coefficient, 1

*ν*=kinematic viscosity, m

^{2}/s- ϱ =
density, kg/m

^{3}*σ*=area ratio, 1

- Ω =
control volume, m

^{3}- $\u2202\Omega $ =
boundary surface of control volume, m

^{2}

### Subscripts or Superscripts

- + =
dimensionless

- 1 =
upstream

- 2 =
downstream

- corner =
corner tapping point couple

*D*=pipe diameter

- $D\u2212D/2$ =
D and D/2 tapping point couple

*F*=friction

- flange =
flange tapping point couple

- ISO =
ISO 5167:2003

- irrev =
irreversible

*M*=momentum

- meas =
measurement

*O*=orifice

*P*=pressure

*R*=reacting

- sim =
simulated

- tp =
tapping point couple

*v*=flow direction

- I =
first approach

- II =
second approach

## References

_{2}

*British Hydromechanics Research Association. Fluid Engineering. Series Engineering*,