NO20240077A1 - Differential pressure-based mass flow meter for fluid and fluid mixture metering - Google Patents

Description

TITLE: Differential pressure-based mass flow meter for fluid and fluid mixture metering

Field of the invention

[1001] The present invention relates to measuring the flow rate of a fluid. More 5 particularly, the invention relates to measuring the flow rate of a fluid flowing through a conduit/ a pipe by utilizing differential pressures measured at two specified positions along two parallel vertical pipes of uniform cross-section and two specified positions along a pipe with a varying cross-sectional area.

Background and state of the art

10 [1002] Measurement of fluid flow rate is an important aspect of every process industry including gas processing, food and beverage, oil and gas, water treatment, and power generation. There are different flow meter technologies with different operating principles based on measurement of temperature, pressure, vibration, suspended particles and process noise signals. The degree of variation of these signals also 15 determines the accuracy expected in the applied technology within the limit of measurement range of the signals.

[1003] Flow rate technology can be grouped into electromagnetic, vortex, paddle wheel, thermal dispersion, floating element, ultrasonic, Coriolis and differential pressure types. Each of these flow meter types or technologies, has some peculiar 20 advantages and disadvantages over the other. For example, the thermal dispersion meters are not suitable for liquid phase and for gas phase at temperature above 50<o>C. The ultrasonic and Coriolis flow meters are considered very expensive but relatively accurate for measurement of clean single-phase fluids including gas, oil and water. However, the accuracy of a Coriolis meter is affected by the process noise such as 25 pipe vibration and pulsation of fluid pump or compressor. Likewise, the ultrasonic meter is affected by presence of suspension in the fluid which deteriorates the ultrasonic frequency arriving the signal receiver. Differential pressure flow meters on the other hand, is widely applied and they are immune against process noise and suspended particles in the main flow. The major disadvantage is that differential 30 pressure (DP) meters, especially orifice plates, cause a permanent pressure loss in the flow, and may not be applicable where low pressure is available. It is also impossible to apply the DP meters for flow rate measurement over a wide range without a prior calibration across the range of application.

[1004] The differential pressure flow meters are well standardized as documented in 5 ISO 5167 technical documents for those based on venturi, nozzles and orifice plates.

Other structures such as V-cone and wedge are also in use for metering of fluid stream. All DP meters operate in the same principle, in which the primary element, that is venturi, orifice, cone, etc., installed along a conduit conveying the fluid, introduces a restriction in the flow, thereby causes the fluid pressure to drop across 10 the restriction. The indicated pressure drop is used to derive the fluid flow rate following the classical relation governed by the Bernoulli’s principle as reported in many fluid mechanics textbooks. The flow rate established by the governing principle is an approximation and generally regarded as the theoretical value because other phenomena such as frictions and further contraction of flow below the physical size of 15 the meter restriction are not considered. By multiplying the theoretical flow rate with the so-called discharge coefficient, the actual flow rate can be obtained.

[1005] Till date, the discharge coefficient is considered across a certain range of Reynolds number above 2x10<5 >as suggested by the standard, ISO 5167 for venturi, or is based on an empirical expression correlated against the Reynolds number and beta 20 ratio (equivalent diameter of the minimum constricted cross section to the pipe diameter) for orifice plate. Whether the discharge coefficient is fixed or obtained by correlations with meter geometry, fluid properties or both, there is always associated uncertainty in the measurement accuracy. Since the flow condition is dynamics, the condition at which the calibration or correlation is obtained may not always hold true 25 when the meter is in operation. Because of the high cost of meter calibration in flow laboratories, the calibration is often carried out only within the range of flow rate or Reynolds number intended to be covered in operation, limiting the meter turn down ratio (i.e., the ratio of the maximum to the minimum flow rate that can be achieved over the calibration uncertainty level). For most venturi and orifice meters, the turn down 30 ratios are usually 3:1. In addition to the limited flow rate range, there are also uncertainties in metering the actual flow rates even within the calibration zone. The presence of suspended particles (either solids, bubbles or foam), process noise and flow entering disturbances due to surge and ramping of process control valve may cause significant shift in the flow rate accuracy. Moreover, due to influence of entering condition in the DP signals, there is always a length requirement for DP meters, which sometimes may not be fully available or may be compromised due to space limitation.

5 In operation where the Reynolds number is low, for example due to low flow rate or high fluid viscosity, the discharge coefficient strongly depends on flow properties. This implies that any slight shift in the Reynolds number will result in inaccurate discharge coefficient when their empirical correlation is used; hence, accurate prediction of the Reynolds number is necessary to be able to determine accurate discharge coefficient.

10 Based on this issue, DP flow meters are generally not recommended for high viscous fluid.

[1006] In conventional DP flow meters, only one differential pressure sensor is applied. The sensor is usually connected on the meter body to measure the pressure drop between the entrance of the meter and the constriction caused by the primary element 15 where the lowest flow area is determined. With only one DP measurement, the actual flow rate obtained from the established flow rate model will always depend on the empirically fitted discharge coefficient equation. However, an improvement in the meter can incorporate measurement of the recovered pressure drop due to expansion of the fluid downstream of the constriction.

20 [1007] The amount of pressure drop across the meter constriction may not be fully recovered owing to frictional loss, flow separation and turbulence as the flow expands into the section with larger cross-sectional area, usually the same size as the conduit on which the meter is installed. The amount of pressure drops recovered can be measured by using additional DP sensor installed between the section of the meter 25 with minimum cross section and downstream after the flow has emerged into the full size of the conduit. In alternative, the permanent pressure loss (that is the amount of pressure drop that cannot be recovered) can be measured using a DP sensor installed between the upstream and downstream of the meter. By energy balance, it is obvious that the sum of the recovered pressure drops, and permanent pressure loss is the 30 same as the pressure drop between the entrance and the minimum constricted area of the meter. Since the Bernoulli’s principle can be applied at any given position along a flow line, each of this pressure drop measurements, i.e. the primary pressure drop, the recovery pressure and the permanent pressure loss, can be used to establish the fluid flow rate through the conduit with appropriate coefficient.

[1008] As reported in many published research articles, a relationship exists between the discharge coefficient and a ratio of permanent pressure loss to primary pressure 5 drop for a given fluid through a DP meter. However, the method proposed in literature for extraction of discharge coefficient for the primary pressure drop also depends on calibration of the meter, where the pressure loss ratio has to be recorded, correlated or tabulated against the corresponding discharge coefficient obtained from a set of known flow rates for the same meter. This method is also related to fitting of discharge 10 coefficient against Reynolds number or flow rate, introducing the associated problems as highlighted above. Recently, measurements of pressure recovery or permanent pressure loss or both in addition to the primary pressure drop have been incorporated in the DP flow meters. The recent applications of multiple DP measurement as introduced in US Patent Application Publication No. 2020/0132535 are only to 15 diagnose the state of the meter for quality checks, and never used to improve the measurement accuracy or applied to make the discharge coefficient independent of calibration and empirical correlations.

[1009] For heavy oil flow measurement, the US Patent No. 11,150,121 describes a method and apparatus that includes a double differential pressure measurement in 20 calculating different variables including Reynolds number, discharge coefficient, density, viscosity and flow rate. In the invention, the first differential pressure is measured across a DP flow meter, a second differential pressure relies on frictional pressure loss over a length in a horizontal pipe, and the flow rate or velocity is measured using a flow meter whose operating principle and technology is different 25 from a DP flow meter. The challenge in the described method is that a correlation of discharge coefficient with Reynolds number must be known prior to measurement, and the length of the meter has to be excessively long in order to measure a significant frictional pressure loss. In further applications, particularly in multiphase flow measurement, systems using double or multiple differential pressures measured 30 across one or more devices that introduces constrain in the flow have been patented.

[1010] A European Patent No. 2,192,391 describes an apparatus and method for determining the mass flow rate of a multiphase fluid flowing through a conduit, where it is used a double differential pressure sensor installed across a venturi. A ratio of the two DP readings is used to calculate a characteristics Reynolds number which is then 5 used to calculate the discharge coefficient using pre-known correlations corresponding to each step of calculation.

[1011] A US Patent No.7,293,471 describes a flow meter for measuring individual flow rates of gas, liquid hydrocarbons and water in a flowing fluid mixture containing a very high gas volume fraction. One of the elements of the flow meter comprises a 10 measuring structure generating a double differential pressure. Using a calibration model, the ratio of the two differential pressure measurements is used to determine the Lockhart-Martinelli parameter which expresses the liquid fraction of a flowing fluid.

[1012] Another US Patent No.6,502,467 describes a system for measuring multiphase flow using multiple differential pressure readings obtained at three or more positions 15 in a venturi tube with an extended throat, which defines two or more pressure differentials in the flow conduit. The differential pressures are then used to calculate the mass flow of the gas phase, the total mass flow rate, and then the liquid phase flow rate.

[1013] Yet another US Patent No. 6,935,189 describes an apparatus and method for 20 measuring a multiphase flow using two or more venturi tubes and a means of measuring the associated pressure drops located between the two venturi tubes. The two venturis in combination with a water fraction meter are used to determine the mass flow rates of the gas phase, the oil phase, and the water phase. All the methods disclosed in the cited references share a common problem, arising on their 25 dependencies on empirical correlation for the discharge coefficient.

[1014] Further, a PCT Publication No. 1995033980 describes an apparatus for measuring density and volumetric flow rate of a multiphase fluid mixture of gas and liquid. The fluid density is determined by a combination of a hydrostatic pressure difference measured along a vertical conduit leg with another measured along a 30 horizontal conduit leg of equal length of span, such that calculation of pressure drop due to friction is eliminated, while the flow rate is measured by any possible available means. The method described can lead to high pressure loss in the flow due to inclusion of a static mixer at the entrance of the meter, and it is applicable only where the gas volume fraction is 20% or less.

5 [1015] To improve the performance of differential pressure flow meters, thereby enhancing their range of applications, reliability, and competitiveness, while maintaining their advantage as low cost meters, the DP meters should be selfcalibrating. Hence, a means and method by which a differential pressure meter like venturi and cone can be used without prior knowledge of flow rate before and during 10 operation, is described in this present invention.

Summary of the invention

[1016] Aspects of the present invention provide measuring the flow rate of a fluid flowing through a conduit/ a pipe by utilizing differential pressures measured at two specified positions along two parallel vertical pipes of uniform cross-section and two 15 specified positions along a pipe with a varying cross-sectional area.

[1017] In one aspect of the present invention, the pipe with a varying cross-sectional area can be either a venturi tube or a double cone structure, and adapted for its installation in various orientations, including horizontal and vertical positions.

[1018] In one aspect of the present invention, the fluid may be in a single phase, such 20 as a gas with a homogeneous mixture of two or more different gases or a liquid with a homogeneous mixture of two or more different liquids. Alternatively, the fluid may be multiphase, constituting a mixture of gas and liquid.

[1019] In one aspect the present invention is applied to a gas-liquid or a two-phase flow to provide gas flow rate, liquid flow rate and liquid density where only the gas 25 density or a means of calculating it is available for the flow rate computation of the fluid mixture in the pipe.

[1020] In one aspect the present invention is applied to a gas-oil-water or a multiphase flow to provide gas flow rate, oil flow rate and water flow rate where the gas density, oil density and water density or a means of calculating them are available for the flow rate computation of the fluid mixture in the pipe.

[1021] In one aspect the present invention is applied to a CO2 flow in a transport pipeline to provide mass flow rate, fluid density, fluid viscosity and possible two-phase 5 flow composition for the fluid mixture flow in a pipe.

[1022] In one aspect the present invention is applied to a liquid flow with entrained gas to provide total mass flow rate, fluid density, fluid viscosity and possible two-phase flow composition for the fluid mixture flow in a pipe.

[1023] In one aspect the present invention is applied to a heavy oil or high viscous 10 liquid flow to provide mass flow rate, fluid density and fluid viscosity for the fluid flow in a pipe.

[1024] In one aspect the present invention provides a flow meter device or an apparatus for metering mass flow rate of a fluid and a fluid mixture in a pipe based on differential pressure measurements techniques.

15 [1025] In one aspect the present invention provides a flow meter device or an apparatus for metering mass flow rate of a fluid and a fluid mixture, the device or apparatus comprising means of accelerating a flow by reducing a flow cross-sectional area in a pipe, thereby creating a differential pressure between a section of the pipe with a largest flow cross-section and a section of the pipe with a smallest flow cross-20 sectional area; means of deaccelerating a flow by increasing a flow cross-sectional area in a pipe, thereby creating a differential pressure between a section of the pipe with a smallest flow cross-section and a section with a largest flow cross-sectional area; connecting two differential pressure sensors, where a first differential pressure sensor between an upstream section where the flow cross-sectional area is largest 25 and a throat section where the flow cross-sectional area is smallest, and a second differential pressure sensor between the throat section where the flow cross-sectional area is smallest and a downstream section where the flow cross-sectional area is largest along a flow direction; where the first differential pressure sensor is configured for providing a first differential pressure reading between the upstream section where 30 the flow cross-sectional area is largest and the throat section where the flow crosssectional area is smallest; where the second differential pressure sensor is configured for providing a second differential pressure reading between the throat section where the flow cross-sectional area is smallest and the downstream section where the flow cross-sectional area is largest; where the two differential pressures are configuratively 5 related to a permanent pressure loss which can be calculated by subtracting the second differential pressure reading from the first differential pressure reading, or measured with a differential pressure sensor connected between the upstream section and the downstream section with the same cross-sectional area as the pipe.

[1026] In one aspect the present invention provides means of accelerating including 10 a venturi tube structure, comprising a hollow entrance cylindrical pipe, a hollow converging truncated conical pipe, a hollow short cylindrical pipe, a hollow diverging truncated conical pipe and a hollow exit cylindrical pipe through which the fluid flows, where in such configuration two differential pressure signals are measured, further a discharge or a flow coefficient is approximated and then the approximated discharge 15 coefficient is further updated to the actual discharge or flow coefficient by ensuring that any two simulated differential pressure across the differential pressure measurement positions, correspond to the two differential pressures measured, thereby preventing the use of empirical correlation for determining parameters in the total flow rate calculation.

20 [1027] In another aspect the present invention provides means of accelerating including a double cone structure situated inside a hollow cylindrical pipe comprising a diverging solid conical block, a short solid cylindrical ring, and a converging solid conical block over which the fluid flows, where in such configuration two differential pressure signals are measured, further a discharge or flow coefficient is first 25 approximated and then the approximated discharge coefficient is further updated to the actual discharge or flow coefficient by ensuring that any two simulated differential pressure across the differential pressure measurement positions, correspond to the two differential pressures measured, thereby preventing the use of empirical correlation for determining parameters in the total flow rate calculation.

30 [1028] In one aspect the present invention provides measuring total flow rate of a fluid stream flowing in a pipe.

[1029] In one aspect of the present invention, the permanently lost differential pressure is used to calculate the liquid density in the fluid stream.

[1030] In one aspect the present invention provides measuring a total flow rate of a gas-liquid flow or a multiphase flow stream in a pipe.

5 [1031] In one aspect the present invention provides measuring density of a fluid stream flowing in a pipe.

[1032] In one aspect the present invention provides measuring flow rates of all the fluid phases or components including gas, liquid, oil and water, in a fluid stream flowing through a pipe.

10 [1033] In one aspect the present invention is applied in flow of liquids, especially a heavy oil or high viscous liquid to provide accurate measurement of mass flow rate, fluid density and fluid viscosity.

[1034] In one aspect the present invention is applied in pipeline transporting CO2 in liquid phase, supercritical phase or in a combination of liquid and gas phases, or in a 15 combination of supercritical and gas phases, or in a combination of liquid and supercritical phases, to provide accurate measurement of mass flow rate, fluid density and fluid viscosity whether with or without impurities.

[1035] In another aspect the present invention provides measuring density of the fluid stream, comprising two vertical and parallel pipes such that the fluid stream goes in 20 opposite direction at each time in the different pipes, where two differential pressure sensors are installed to read differential pressures created along each pipe at a specified distance where the two differential pressure readings are combined to obtain density and viscosity of the fluid or fluid mixture.

[1036] In another aspect the present invention is applied to wet gas stream or a flow 25 stream with extreme high gas volume fraction to provide gas and entrained liquid flow rates where both the gas and liquid densities are available for the flow rate computation of the fluid mixture in the pipe.

[1037] Further details and specification of the invention are described in the detailed description and accompanying figures, providing comprehensive insights into the operation and functionalities of the present invention.

Brief description of the drawing

[1038] The foregoing aspects and many of the advantages of the present invention will be more appreciated and better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:

5 [1039] Fig.1 shows a structure of a venturi tube fitted with two differential pressure sensors for measurement of fluid flow rate in accordance with a first embodiment of the present invention;

[1040] Fig.2 is an alternative structure comprising of a double cone fitted with two differential pressure sensors for measurement of fluid flow rate in accordance with a 10 second embodiment of the present invention;

[1041] Fig.3 illustrates a horizontal configuration of a venturi structure for measurement of total flow rate with a pair of vertical flow tubes for fluid density measurement in accordance with a third embodiment of the present invention;

[1042] Fig.4 illustrates a horizontal configuration of a double cone structure for 15 measurement of total flow rate with a pair of vertical flow tubes for fluid density measurement in accordance with a fourth embodiment of the present invention;

[1043] Fig.5 shows an example of positioning the primary element being it a venturi tube or a double cone structure in a vertical orientation for measurement of total flow rate with a pair of vertical flow tubes for fluid density measurement in accordance with 20 a fifth embodiment of the present invention;

[1044] Fig.6 illustrates the required geometry, sensor connection positions and measurement principle of the device used to measure the fluid density in accordance with the third, the fourth and the fifth embodiments of the present invention;

[1045] Fig.7 illustrates the required geometry, sensor connection positions and 25 measurement principle of the device used to measure a venturi discharge coefficient and total fluid flow rate in accordance with the first, the third and the fifth embodiments of the present invention, where (I) is the complete venturi tube structure with pressure measurement positions, and (II) the isolated upstream and downstream cones showing the geometrical properties required for derivation of the discharge coefficient;

[1046] Fig.8 illustrates the required geometry, sensor connection positions and measurement principle of the device used to measure a double cone discharge 5 coefficient and total fluid flow rate in accordance with the second, the fourth and the fifth embodiments of the present invention, where (I) is the complete cone structure with pressure measurement positions and geometrical properties required for derivation of the discharge coefficient, and (II) the meter cross section showing the restriction structure and the available area opened for flow rate calculation; and

10 [1047] Fig.9 illustrates a pressure distribution plot along either a venturi tube or a double cone structure in accordance with all embodiments of the present invention showing the measured pressures at the measurement positions and the simulated pressures along the meter primary structure for self-calibration of the discharge coefficient.

15 Detailed description of the invention

[1048] The present invention will now be described more fully hereinafter with reference to the accompanying drawings in which a preferred embodiment of the invention is shown. This invention may, however, be embodied in many different forms and should not be construed as being limited to the embodiments set forth herein.

20 Rather, the embodiments are provided so that this disclosure will be thorough, and will fully convey the scope of the invention to those skilled in the art.

[1049] Embodiments of the present invention will be described herein with reference to the figures. Referring to Fig.1, the present invention is a measuring unit or a meter for measuring flow rate of a single phase fluid such as water, oil or gas, or a 25 homogeneous combination of any two or more different liquids such as water and oil, or a combination of liquid and gas, provided that properties such as density of the fluid or individual fluid components are known for a single phase fluid or gas-liquid two phase fluid measurement. The invention here described utilizes a combination of two differential pressure measurements , ∆?1? and ∆?2? to measure the fluid flow rate in a pipe flow without recalibration of the primary element, being in this case a venturi tube 500 (A – E), and with a high repeatability as the flow condition changes.

[1050] As shown in the Fig.1, the fluid stream 100 enters a flow pipe or conduit 200 in the indicated direction, and accelerates through a converging cone 500B due to 5 continuous reduction in the flow area from that corresponding to the conduit at the entrance cylinder 500A, until it reaches a throat section 500C which constitutes the lowest flow area through the meter. Leaving the throat section 500C, the fluid then decelerates through a diverging cone 500D due to increasing cross sectional area, until the flow area corresponds to that of an exit cylinder 500E connected to the flow 10 pipe 200 downstream of the meter. A differential pressure sensor connected between the two pressure tappings 400C and 400D measures a first differential pressure, ∆?1? resulted due to acceleration of the fluid, and a second differential pressure sensor connected between the pressure tappings 400D and 400E measures a second differential pressure, ∆?2? caused by fluid deceleration.

15 [1051] Any non-uniform section of a pipe will cause a change in the fluid velocity, and consequently a change in the fluid pressure within the affected sections of the pipe. Therefore, the same corresponding first ∆?1? and second ∆?2? will be measured when a double cone structure 600 (B – D) included as second embodiment of the present invention, is placed in a pipe tube 600A as illustrated in Fig.2. A double cone structure 20 600 (B – D) is inserted in flow chamber 600A, and as first part it contains a conical element 600B which increases in diameter in the flow direction, thereby reducing the flow area as the fluid flows externally over it. Downstream of the first conical element 600B is a second conical element 600D whose diameter decreases in the flow direction, causing the flow area to increase as the fluid flows externally over it.

25 Between the first 600B and the second 600D conical elements is a third part comprising a short cylindrical ring 600C connecting the two conical elements.

[1052] Besides measurement of a second differential pressure, ∆?2?, a permanent pressure loss, ∆???? can be measured in alternative using a differential pressure sensor connected between a first pressure tapping 400C and a third pressure tapping 400E 30 as shown in Fig.1. By the energy balance, the three differential pressures can be related as given in Eq. (1).

∆?1? = ∆?2? ∆𝑝𝐿 (1)

[1053] In the present invention, a first differential pressure and either a second differential pressure, ∆?2? or a permanent pressure loss, ∆???? are the preferred differential pressures for measurement of a flow rate in a pipe. When the fluid in the 5 pipe 200 has a low viscosity, e.g. water or gas, the permanent pressure loss, ∆???? is low; then, the second differential pressure, ∆?2? becomes the preferred additional differential pressure to be measured. In alternative, when the fluid has a high viscosity, e.g. oil or when the fluid contains a mixture of liquid and significant volume of gas, the permanent pressure loss, ∆???? becomes the preferred differential pressure to be 10 measured in addition to the first differential pressure, ∆?1? since the recovered pressure difference, ∆?2? is low. The means by which the present invention is novel for accurate measurement of total flow rate of a fluid in a pipe without recalibration of the flow meter are described in detail in the following, by first referring to Fig.7.

[1054] Fig.7 (I) describes geometry of a venturi structure with entrance cylinder 15 diameter denoted by ???? and throat diameter denoted by ????, corresponding to the largest and smallest cross-sectional area of the flow, respectively. ?1? is the distance between the first pressure tapping located at position 1 where the first fluid pressure ?1? is measured, and the second pressure tapping located at position 2 where the second fluid pressure ?2? is measured, and?2? is the distance between the second 20 pressure tapping and the third pressure tapping located at position 3 where the third fluid pressure?3? is measured. The length of the converging cone is denoted by𝐿𝑐 and the length of the diverging cone is given as𝐿𝑑. In Fig.7(II),𝑥 is any length along the venturi structure starting from the entrance cylinder 500A and𝑑 is the diameter of the 1

flow area,𝐴 = 4𝜋?<2>? at any position𝑥 along the flow path. References are then made 25 to point 1 corresponding to a position of the first pressure tapping, point 2 corresponding to a position of the second pressure tapping and point 3 corresponding to a position of the third pressure tapping. Following the Bernoulli’s principle, the energy balance between point 1 and point 2 can be written as in Eq. (2).

?1? ?2

1?𝑡ℎ ?2? ?2?𝑡ℎ

𝜌𝑔+ 2𝑔= 𝜌𝑔+ 2

2𝑔 (2) where 𝜌 is the fluid density assumed invariant across the flow,?1??ℎ? is the theoretical flow velocity at point 1,?2??ℎ? is the theoretical velocity at point 2 and𝑔 = 9.81 m/s<2 >is

1

the acceleration due to gravity. With𝐴𝑝 = 4𝜋????<2 >the flow area of pipe at point 1 and 1

???? = 4𝜋????<2 >the flow area of throat at point 2, the conservation of mass flow rate,?̇? 5 satisfying the flow continuity is expressed thus as given in Eq. (3):

?̇? =𝜌?1????? =𝜌?2????? (3)

giving ?1? =𝛽<2>?2?, where𝛽 = √????/𝐴𝑝 and?1? and?2? are actual flow velocities at points 1 and 2, respectively. With this velocity relationship, Eq. (2) and Eq. (3) are simplified further:

2∆?

10 ?2? =? 1?

???√?<(>?1−?<4>?<) >(4)

2?∆??

?̇? =? 1?

???????√ <(>1−?<4>?<) >(5)

[1055] Here, ∆?1? =?1? −?2? is the difference in fluid pressure between points 1 and 2, which is the quantity measured by a differential pressure sensor connected at the first pressure tapping 400C and the second pressure tapping 400D in accordance with all 15 the embodiments of the present invention. ???? is a flow coefficient of the venturi tube accounting for deviation of the flow from an ideal case described by Eq. (2) where frictional losses and other effects including fluid expansion, flow separation, turbulent eddies and further contraction of the flow beyond the throat section are neglected. Note in many fluid mechanics textbooks and published articles, the correction 20 coefficient, ???? is usually referred to as “coefficient of discharge” and often obtained through series of experiments as being introduced; hence, its accuracy is often restricted to the flow condition at which its value is derived. However, in this invention, the term flow coefficient is preferred. The method by which the flow coefficient,???? of a venturi tube is obtained analytically without knowledge of experimental data for 25 accurate flow rate measurement is disclosed in the present invention. The derivation of this analytical means of calculating ???? for a given venturi tube equipped with two sensors to measure a first differential pressure, ∆?1? between the upstream cylinder 500A and throat 500C, and a second differential pressure, ∆?2? between the downstream cylinder 500E and throat 500C or ∆???? between the upstream cylinder 500A and downstream cylinder 500E, is further described in details.

[1056] Considering all other effects on the fluid flow through a horizontally positioned 5 venturi tube, the fluid mechanical energy balance between point 1 and point 2 in Fig.7 can be represented by

?1? ?12? ? ?2?2

𝜌𝑔+ 2𝑔= 2?

𝜌𝑔+ 2𝑔+ ℎ?1? (6)

?

ℎ?1? =?1? 1??<2>?

̅ ∫ 0 2𝑔𝑑𝑑𝑥 (7)

where ℎ?1? is a term used to account for the total losses in the flow between point 1 10 and point 2 for all elemental fluid displacements, 𝑑𝑥 under assumption that frictional loss dominates other losses in the flow, and𝑣 = ?̇?/(𝜌?)? is the fluid velocity at any position in the tube.?1?̅ is the average Fanning friction factor, which can be obtained from a suitable correlation such as the well-known Darcy equation. Again, taking ∆?1? = ?1? −?2?, Eq. (6) and Eq. (4) can be written as

∆? 1−?<4>?

15 1?

𝜌𝑔= ( 2𝑔) ?2?<2 >+ ℎ?1? (8)

<∆? >1−?<4>?

? 1

??2 <?>

?𝜌𝑔= ( 2𝑔) ?2?<2 >(9)

[1057] A combination of Eq. (8) and Eq. (9) with further simplification gives

?2

?<2 >2?

??? = 2𝑔 (10) ?2?<2>+( )ℎ𝐿11−𝛽4

[1058] Neglecting any head loss between the tapping points at the entrance and 20 downstream of the converging cone while ensuring flow continuity, Eq. (7) is approximated.

?

ℎ ???𝑑𝑥

?1? ≈?1?2?2

̅ 2𝑔?<4 >?

??? ∫ 0 ?<5>? (11)

[1059] By geometrical relationship, the diameter,𝑑 of the available flow area at any position, 𝑥 along the converging upstream cone is expressed as

?

𝑑 = ???? − ( ???−????

???? )𝑥 (12)

[1060] Substituting Eq. (12) in Eq. (11) and integrating over the length of the cone,𝐿𝑐; 5 then with further simplification yields

1 ? 𝑣2

ℎ?1? ≈ 4 (1 −𝛽<4>) ??? (????−???)?1?2

̅? 2𝑔 (13)

Further, substituting Eq. (13) in Eq. (10) results in

1

?<2>??? ≈ 1 𝐿? (14)

1+ 4𝑓̅1( ?

𝐷 )

𝑝−𝐷𝑡

[1061] Similarly, the energy balance between point 2 and point 3 for a fluid flowing 10 through a venturi diverging cones as depicted in Fig.7 can be expressed in the following two forms,

∆?2? 1−?<4>? <2>

𝜌𝑔= ( 2𝑔) ?2? − ℎ?2? (15)

<∆? >1−?<4>?

? 2 2<?>

???𝑑𝜌𝑔= ( 2𝑔)?2?<2 >(16)

𝐿 ?<2>?

ℎ?2? =?2?̅ ∫ 2

0 2𝑔𝑑𝑑𝑥 (17)

15 where ∆?2? =?3? −?2?,????𝑑 is a flow coefficient accounting for flow deviation from an ideal behavior represented by the Bernoulli’s principle across a diverging cone downstream of a venturi tube, and ℎ?2? accounts for a frictional loss as being the dominant of all the losses for an elemental fluid displacement,𝑑𝑥 in a flow through the diverging cone. With the available flow area diameter, 𝑑 expressed as in Eq. (18) at 20 any position 𝑥 along the diverging cone based on a geometrical relationship, the average head loss, ℎ?2? over the length,?2? of the cone, neglecting any head loss between the tapping points at the entrance and downstream of the diverging cone while ensuring flow continuity, is approximately given by Eq. (19).

?

𝑑 = ???? + ( ???−????

???? )𝑥 (18)

1 ? ?2?2

ℎ?2? ≈ 4 (1 −𝛽<4>) ( ???

????−???)?1?̅? 2𝑔 (19)

[1062] A combination of Eq. (15), Eq. (16) and Eq. (19) gives the venturi flow coefficient, ????𝑑 in relation to the downstream cone as

5 ? <2 >1

???𝑑 ≈ 1 𝐿 (20)

1− 4𝑓̅2( 𝑑

𝐷𝑝−𝐷)

𝑡

Further, by mass balance,

2?∆?? 2𝜌

?̇? =????????√ 1? ∆?2?

<(>1−?<4>?<) >=????𝑑????√ <(>1−?<4>?<) >(21)

? 2 ∆?

( ??? ?

???) = 2

?𝑑 ∆?1? (22)

1 1

Assuming 4?1?̅ = 4?2?̅ =????̅, and substituting Eq. (14) and Eq. (20) in Eq. (22) gives

1−𝑓̅ ∆?

10 𝑣????𝑟2 2?

1+𝑓̅𝑣????𝑟1 = ∆?1? (23)

where 𝐿𝑣?1? =?1?/(???? −????) and𝐿𝑣?2? =?2?/(???? −????) are effective hydraulic length ratio for a venturi converging cone between a first pressure measurement at point 1 and a second pressure measurement at point 2, and for a venturi diverging cone between the second pressure measurement at point 2 and a third pressure measurement at 15 point 3, respectively. Solving Eq. (23) for ????̅ results in Eq. (24), and substituting Eq.

1

(24) with????̅ = 4?1?̅ in Eq. (14) gives an expression, Eq. (25) for a venturi flow coefficient, ????, also known as discharge coefficient based on the primary differential pressure signal, ∆?1? obtained between an upstream cylinder 500A and a throat section 500C and across a converging cone 500B in accordance with the first, third and fifth 20 embodiments disclosed in this invention, to enable a total mass flow rate, ?̇? of a fluid going through the venturi metering structure to be accurately measured using the expression discussed under Eq. (5).

∆𝑝

1− 2

? ∆𝑝1

???̅ =????𝑟2+(∆𝑝2)??? (24)

∆𝑝 ?𝑟1

1

∆𝑝2 𝐿

∆𝑝+𝛼 ( 𝑣𝑟2

? 𝐿 )

1 𝑣𝑟1

??? = √( 𝐿

1+?(?𝑣𝑟2 (25)

𝐿 ) )

𝑣𝑟1

[1063] In practice, the average friction coefficient, ?1?̅ due to a flow between the upstream pressure tapping 400C and the throat pressure tapping 400D may not be 5 same as the average friction coefficient,?2?̅ between the throat pressure tapping 400D and the downstream pressure tapping 400E along the venturi tube, due to turbulent effect and possible flow separation across the diverging cone 500D. Therefore, a correction factor, 𝛼 is included in Eq. (25) to account for these unknown effects. Assuming no correction is required, 𝛼 = 1. The actual value of 𝛼 in a given flow is 10 obtained by solving a steady-state momentum balance across the venturi tube using an equation as described by Eq. (26) or its similar such that the predicted pressures, 𝑝 at point 1, point 2 and point 3 correspond to the measured pressures at point 1, point 2 and point 3, respectively as illustrated in Fig.9.

𝑑𝑝 1?(??̇?.?)? 𝑓

𝑑𝑥= −𝐴 𝑑𝑥 − 2𝑑𝜌𝑣|𝑣| 𝜌𝒈 (26)

15 [1064] Here, 𝑝 is the local pressure at any position,𝑥 along the venturi tube,𝑣 the local velocity and 𝑓 the local friction factor.𝒈 is the gravity constant defined in the flow direction, where its value is 0 for a horizontally installed pipe, 9.81 m/s<2 >for a vertically downward flow and -9.81 m/s<2 >for a vertically upwards flow. The local friction factor,𝑓 depends on the fluid viscosity or the Reynolds number. In the present invention, the 20 value of 𝑓 = 4????̅ is applied, where????̅ is calculated from Eq. (24), thereby making a venturi flow coefficient,???? or discharge coefficient as it is widely called to be calculated accurately without knowledge of fluid viscosity or flow Reynolds number.

[1065] Alternatively, the primary element for measuring a fluid flow rate in a pipe is a double cone structure 600(B – D). In Fig.8, the fluid is indicated flowing externally over 25 a double cone structure 600(B – D) placed in a pipe. Three pressure sensors are located at positions 1, 2 and 3 to provide readings of the fluid pressures, ?1?,?2? and?3? at the three different locations, respectively. Similar to geometrical description of a venturi tube illustrated in Fig.7,?1? is the distance between a pressure tapping at point 1 located before the first cone and a pressure tapping at point 2 located on a ring joining the first and second cones and with diameter, ???? corresponding to base diameters of the two cones. ?2? is the distance between the pressure tapping at point 5 2 and a pressure tapping at point 3 located after the second cone. As given in Fig. 8(I), the length of the first cone is denoted as 𝐿𝑐, the length of the second cone as𝐿𝑑 and the length of the joining ring as ????. Fig. 8(II) illustrates an area open for a flow where the cones are positioned. The position of the smallest flow area is at point 2 over the connection ring, and similar to a venturi tube, this position can be regarded 𝜋

10 as a double cone throat, with flow cross-sectional area given as ???? = <2>

4 (???? −??<2>??). At 𝜋 any position,𝑥 along the first and second cones, the local flow area, 𝐴 = <2>

4 (???? −𝑑<2>) is described, where the diameter, 𝑑 is given by Eq. (27), respectively.

? ?

𝑑 = ???

??;? 𝑑 = ???

???? ???(𝐿?

? − ?)? (27)

[1066] Applying the Bernoulli’s principle and mass conservation law between points 1 15 and 2 in Fig.8, Eq. (28) and Eq. (29) can be derived for fluid velocity at point 2 with flow area,????, and for fluid mass flow rate,?̇? measured by a double cone for any flow in a pipe.

2∆?

?2? =????√ 1?

?<(>?1−?<4>?<) >(28)

2?∆??

?̇? =? 1?

???????√ <(>1−?<4>?<) >(29)

20 where 𝛽 = √????/𝐴𝑝, ∆?1? =?1? −?2? and???? a flow coefficient or discharge coefficient for the cone, correcting flow deviations from an ideal assumption. Following the steps described by Eq. (6) – Eq. (10) for deriving a venturi flow coefficient, the first cone flow coefficient, ???? can be similarly derived as given by Eq. (30).

?2

?<2 >2?

??? =

?2?<2>+( 2𝑔 (30)

1−𝛽4)ℎ𝐿1 insufficientOCRQuality for page 23

1

[1069] Here, ?2? = 2????. The integral in Eq. (37) is developed based on the generic definition described in Eq. (7) and substitution of flow area across the second cone based on the local diameter,𝑑 of the obstruction cross section defined in Eq. (27). With evaluation of the integral,

? ?

ℎ ??? <4 >2?2

5 ?2? = (? 𝛽?2?̅𝐿𝑐?2????) 2𝑔 (38)

? 1 1−<(>? 1

𝐿 𝑑<)>2 <2>

𝑐?2? = − ln ( ??? ??? −?2

0?

? n [ 2?𝑚

0?) 2 l 1−<(>?2??0?<)2>] − 2?2?<2 >?

[ (1−(?2?????)<2>)(1−(?2??0?)<2>)] (39)

𝐷 ?2? ?

𝑘 𝑐

2 = 𝐷?; ?0? ? ≈ 1 −? 1 − ???+?2?

??2?; ?? ≈ ?2? (40)

Substituting Eq. (38) in Eq. (36), then

1

??? <2>

?𝑑 = 𝛽4 𝐿 (

1−( 𝑑 41)

1−𝛽4)(𝐷)?

𝑐 ???𝑟2𝑓̅2

10 From mass balance across the double cone structure, Eq. (42) is obtained, and further substitution of Eq. (35) and Eq. (41) gives Eq. (43).

? 2 ∆?

( ???

? ) = 2?

???𝑑 ∆?1? (42)

𝐿

1−𝑓̅𝑐( 𝑑

𝐷)? ?2

𝑐 ???? ∆?

𝐿

𝑓̅𝑐( 𝑐

𝐷)? = 2?

1 ∆? (43) 𝑐 ???𝑟1 1?

where ???̅? is the average frictional factor across the double cone structure, and is 15 defined by

?<4>? ?<4>?

???̅? = ( 1−?<4>?)?1?̅ = ( 1−?<4>?)?2?̅ (44)

[1070] Solving for ???̅? from Eq. (43) and substituting result in Eq. (35) provides corresponding expressions, Eq. (45) and Eq. (46) for the average frictional factor, ???̅? and flow coefficient,???? based on the primary differential pressure signal, ∆?1? between 20 a first pressure tapping 400C and a second pressure tapping 400D and across a first cone 600B in accordance with the second, fourth and fifth embodiments disclosed in this invention, to enable a total mass flow rate,?̇? of a fluid going over a double cone metering structure to be accurately measured using the expression discussed under Eq. (29).

1−∆𝑝2

𝑓 ∆𝑝1

?̅? = 𝐿

( 𝑑? 𝐿𝑐 (45)

𝐷)

? ???𝑟2+(∆𝑝2

∆𝑝)(

1 𝐷)?

𝑐 ???𝑟1

?

∆𝑝2 𝐿 𝐿

? ( 𝑑

5 ? 𝐿)( 𝑐𝑟2

𝐿 )

𝑐 𝑐𝑟1

??? = √( ∆𝑝+?

1

𝐿 𝐿𝑟2 ) (46)

1+?(?𝑑

𝐿)( 𝑐 )

𝑐 𝐿𝑐𝑟1

[1071] In Eq. (46), a correction factor,𝛼 is introduced to account for ideal behavior assumption in the method derivation, particularly in practice where the local average friction coefficient,?1?̅ due to a flow between an upstream pressure tapping 400C and a throat pressure tapping 400D may not be same as the local average friction 10 coefficient, ?2?̅ between the throat pressure tapping 400D and a downstream pressure tapping 400E along a double cone structure, due to turbulent effect and possible flow separation across the second cone 600D. Theoretically, a value𝛼 = 1 is applied. The actual value of 𝛼 is obtained by solving a momentum balance equation as described in Eq. (47) between the first pressure tapping at point 1 and the third pressure tapping 15 at point 3 in Fig.8, and across the double cone structure 600(B – D) such that the simulated pressures,𝑝 at points 1, 2 and 3 correspond to the measured pressures at points 1, 2 and 3 as illustrated in Fig.9.

𝑑𝑝 1?(??̇?.?)? 𝑓

𝑑𝑥= −𝐴 𝑑𝑥 − 2?ℎ?𝜌𝑣|𝑣| 𝜌𝒈 (47)

?ℎ? =???? −𝑑 (48)

1−?<4>?

20 𝑓 = ?<4>????̅? (49)

[1072] Here, ?ℎ? is the meter hydraulic diameter defined as a ratio of the flow cross sectional area to wetted perimeter of the flow boundary. The local flow cross-sectional 𝜋

area is obtained from 𝐴 = 4 (??<2>?? −𝑑<2>), where𝑑 is the obstruction cross sectional diameter as described in Eq. (27). The local friction factor,𝑓 is estimated from Eq. (49) 25 where the average friction factor,???̅? across the double cone structure and between the first pressure tapping 400C and the third pressure tapping 400D, is calculated from Eq. (45), thereby making a cone flow coefficient,???? or discharge coefficient as it is widely called, to be calculated accurately without knowledge of fluid viscosity or flow Reynolds number.

5 [1073] As shown in Fig.3, a venturi tube installed in a horizontal position above two adjoining vertical pipes or tubes is described. A fluid stream 100 in a horizontal pipe 200A makes a 90<o >turn and flows through the metering device. The fluid flow direction is vertically upward as it enters a first vertical tube 300A having a uniform diameter and cross-sectional area. Then, a first pressure difference, ∆?3? is measured between 10 the position at a first pressure tapping 400A and the position at a second pressure tapping 400B. Further, the fluid changes direction back to horizontal by means of a device 200B which can be a 90<o >elbow, and then flows through a venturi tube beginning from an entrance cylinder 500A where a third pressure tapping 400C is positioned. The fluid further enters into a converging cone 500B of the venturi tube 15 where due to continuous reduction in the flow cross-sectional area the fluid velocity increases and accelerates downstream of the cone. At the exit of the first truncated cone 500B where the cross-sectional area is lowest, a venturi throat is established and runs through a short cylindrical pipe 500C of cross-sectional area the same as the exit of the upstream converging cone. On the venturi throat, a fourth pressure tapping 20 400D is installed. The venturi throat also connects a first cone 500B to a second cone 500D whose cross-sectional area increases downstream until it reaches a value similar to a cylindrical pipe 500E connecting the venturi to a second vertical tube 300B through a second 90<o >elbow 200C. The diverging cone 500D is a fourth part of the venturi tube while the exit cylinder 500E is a fifth part of the tube structure used to 25 measure the total flow rate of a fluid stream 100. A differential pressure sensor connected at a pressure tapping 400C located on the inlet cylinder and a pressure tapping 400D located on the venturi throat measures a second pressure difference, ∆?1? created due to acceleration of fluid through the first conical part 500B of the venturi. Between the venturi throat 500C and the exit cylinder 500E, a third pressure 30 difference, ∆?2? is measured using a differential pressure sensor connected at a pressure tapping 400D located on the throat and a pressure tapping 400E located on the exit cylinder. The pressure differential, ∆?2? is created in the venturi as the fluid flows through the diverging cone 500D which by increasing cross-sectional area causes the fluid velocity to decrease downstream, thereby increasing the fluid pressure at location 400E above the fluid pressure at location 400D following the energy conservation principle. Further, the fluid moves downward in a vertical direction 5 through a second vertical tube 300B, and then flows through a 90<o >elbow 200D into a horizontal pipe conduit as it leaves the metering section. On the vertical tube 300B, a differential pressure sensor is connected by means of a sixth pressure tapping 400F and a seventh pressure tapping 400G to measure the reading, ∆?4? being a difference in pressure generated due to weight of the fluid, flow velocity and frictional losses 10 along the measurement positions between pressure tappings 400F and 400G.

[1074] The apparatus in Fig.3 herein disclosed can be used to measure flow rate of a single-phase fluid or a homogeneous mixture of different fluids of the same phase or a homogeneous mixture of different phases, where the fluid or fluid mixture density is not known. By definition and in accordance of this disclosure, a fluid phase means 15 either a gas or a liquid. A homogeneous mixture of the same phase can be a fluid mixture of oil and water, and a homogeneous mixture of fluid of different phases is a fluid containing gas and liquid of any kind. In particular, the apparatus can be used to measure a flow rate of heavy oil whose properties such as density and viscosity can easily change with changes in the operating conditions, making it difficult to obtain 20 accurate measurements with conventional flow meters whose operating principles rely on accurate values of fluid density, viscosity and Reynolds number. Further, the apparatus in Fig.3 can be designated for CO2 flow rate measurement, and then can be regarded as a state of advancement into measurement of CO2 for carbon capture & storage as well as oil & gas applications. In a pipeline, CO2 can either be transported 25 in a gas phase, dense liquid or supercritical phase. Temperature and pressure at which these CO2 phases co-exist are usually close to the operating condition in the transport pipeline. Therefore, any slight change in operating pressure, temperature or both can cause the CO2 stream to move from one phase to another or to contain a mixture of any two dominant phases such as gas CO2 and liquid CO2. In addition, 30 traces of any unwanted material called impurities in CO2 stream can also lead to transition of CO2 from one phase to another, and then to difficulty in obtaining the essential fluid properties such as density and viscosity. The listed number of issues makes measurement of CO2 stream challenging for conventional meters which are designed for measuring only a distinct CO2 phase such as dense liquid, gas and supercritical, or whose measurement principle requires accurate knowledge of fluid density and viscosity to provide flow rate of CO2 stream in a pipe. Further, the 5 apparatus can be used to measure gas and liquid flow rates in a two-phase flow or gas, oil and water in a three-phase fluid flow. Specifically, the apparatus is a wet gas flow meter installed to provide readings of gas and liquid flow rates for a fluid stream where the gas volume fraction is above 90%.

[1075] The total flow rate of a fluid stream is measured with a venturi tube 500 10 described in this embodiment. The mathematical equations used for flow rate calculation are a combination of Eq. (5) and Eq. (25). The method for calculating the fluid density,𝜌 required in Eq. (5) for the total flow rate calculation is further disclosed. A first vertical tube 300A and a second vertical tube 300B with the attached differential pressure sensors to provide the required differential pressure readings, ∆?3? and ∆?4? 15 are a combined system used as a unit for measuring fluid density in accordance with third, fourth and fifth embodiments of this present invention.

[1076] Fig.6 illustrates the principle of measuring fluid density by means of two vertical flow tubes, each having a uniform cross-sectional area across the pressure measurement positions. In an upward flow direction, a pressure ?4? is measured by a 20 pressure sensor connected to a pressure tapping located at point 4, and a pressure, ?5? is measured by a pressure sensor connected to a pressure tapping positioned at point 5. When there is no flow, but the vertical tube is filled with a fluid across the pressure measurement points 4 and 5, the difference, ∆?3? =?4? −?5? between the two pressure readings is the same as the weight of the fluid within points 4 and 5 divided 25 by the cross sectional area of the tube. When the fluid flows upwards, the difference in the pressure sensor readings is larger than the weight per unit area of the fluid owing to a flow resistance due to frictions and or flow separation due to turbulence acting in the same downward direction as the fluid weight. Flowing downward through a second flow tube 300B, a pressure sensor connected at a pressure tapping at position 6 30 provides a pressure reading, ?6?, and another pressure sensor connected to another pressure tapping at point 7 provides a fluid pressure reading,?7?. Again, when there is no flow, the difference between the two pressure readings, i.e. ∆?4? =?7? −?6? corresponds to the weight of the fluid contained within a volume bounded by the fluid positions at points 6 and 7 divided by a cross-sectional area of the tube. With a downward flow through the tube 300B, the pressure difference, ∆?4? is lower than the 5 weight per unit area of the fluid due to frictional pressure loses in opposite direction to the flow and to the weight of fluid in the tube. As a means of measuring fluid density, the two vertical tubes 300A and 300B are similar in that both have the same total length, the same internal diameter and are located in the same plane along any horizontal axis. The distance between a pressure tapping 400A at point 4 and a 10 pressure tapping 400B at point 5 is the same distance between a pressure tapping 400F at point 6 and a pressure tapping 400G at point 7, and it is denoted by 𝐿 according to Fig. 6. Since the cross-sectional area of the vertical tubes is the same across the measurement positions, the relationships between the difference in pressure readings, ∆?3? =?4? −?5? in the upward flow tube, and pressure readings, 15 ∆?4? =?7? −?6? in the downward flow tube, are given respectively by Eq. (50) and Eq.

(51) according to the principle of mechanical energy balance.

∆?3?

𝜌𝑔= 𝐿 ℎ𝑓 (50)

∆?4?

𝜌𝑔= 𝐿 − ℎ𝑓 (51)

𝐿 ?̇? 2

ℎ𝑓 =???? 2𝑔??? (? 𝜌? (52)??)?

1

20 [1077] Here, ???? = 4𝜋??<2>?? is the cross-sectional area of the vertical tube where???? is the pipe tube internal diameter. ℎ𝑓 is the frictional head loss across the pressure tapping positions 4 and 5 or 6 and 7 and is modeled as in Eq. (52) according to Fanning frictional loss. The frictional head loss terms, ℎ𝑓 in Eq. (50) and Eq. (51) are considered the same since the pipe diameter,????, the flow velocity in the pipe, the friction factor, 25 ????, and the distance𝐿 over which the frictional losses are recorded are the same in the measurement of the different differential pressures, ∆?3? and ∆?4?. Combining Eq. (50) and Eq. (51) gives an expression for fluid density as

∆?

𝜌 = 3?+∆?4?

2𝑔𝐿 (53)

[1078] Further, Eq. (50) and Eq. (51) can be combined to give an expression for the frictional loss as in Eq. (54). Comparing Eq. (52), Eq. (53) and Eq. (54), an expression for the pipe friction factor,???? can be derived as described by Eq. (55).

∆?

5 ℎ𝑓 = 3?−∆?4?

2𝜌𝑔 (54)

𝜌? 2 ∆?

???? = 2𝑔???? ( ??? ∆?4?

?̇? ) ( 3?−

∆?3?+∆?4?) (55)

[1079] The friction factor,???? calculated from Eq. (55) can be compared with its value calculated from established correlations available in literature, for example the Darcy friction factor described in Eq. (56)

64

?

10 ???=𝑅𝑒; 𝑅?≤?2100

1 .51 (56)

=−2 log( 𝜀

√𝑓𝐷 3.7𝐷+ 2

𝑣 𝑅𝑒√𝑓); 𝑅?>?2100

𝐷

?

𝑅𝑒 = ????̇?

????𝜇 (57)

where 𝑅𝑒 is Reynolds number characterizing the flow regime behavior and as being expressed in Eq. (57), it depends on the fluid viscosity,𝜇. The vertical flow tubes 300A and 300B equipped with two pressure tappings 400A and 400B for tube 300A where 15 the flow transverses in upward direction, and two pressure tappings 400F and 400G for tube 300B where the flow returns in a downward direction, can further be used to measure the fluid viscosity,𝜇 using Eq. (58).

?

𝜇 = ????̇?

????𝑅𝑒 (58)

[1080] When the friction factor,???? is calculated from Eq. (55) based on the mass flow 20 rate, ?̇? calculated from Eq. (5) for a venturi tube or Eq. (29) for a double cone structure, and fluid density,𝜌 calculated from Eq. (53) or provided by any other means, for a fluid flow across a distance, 𝐿 where a first pressure difference ∆?3? associated with a flow tube with an upward flow direction, or a second pressure difference ∆?4? associated with a flow tube with a downward flow direction, the flow Reynolds number,𝑅𝑒 can be calculated from Eq. (56). Then the fluid viscosity,𝜇 can be calculated from Eq. (58) using the flow Reynolds number,𝑅𝑒 calculated as described above, and also the fluid mass flow rate, ?̇? calculated using a venturi tube 500 or a double cone 600 as the primary element where a first and a second differential pressure sensors are 5 connected to provide a first differential pressure reading, ∆?1? and a second differential pressure reading, ∆ ?2? in accordance with all the embodiments of this present invention.

[1081] Alternatively, the primary element introduced in this present invention for measuring flow rate of a fluid in a pipe using a device disclosed in Fig.4 is a double 10 cone structure installed in a horizontal position above two adjoining vertical pipes or tubes. A fluid stream 100 in a horizontal pipe 200A makes a 90<o >turn and flows through the metering device. The fluid flow direction is vertically upward as it enters a first vertical tube 300A having a uniform cross-sectional area. A first pressure difference, ∆?3? is measured between the position at a first pressure tapping 400A and the position 15 at a second pressure tapping 400B. Further, the fluid changes direction back to horizontal by means of a device 200B which can be a 90<o >elbow, and then flows through a double cone structure situated inside a hollow pipe 600A that conveys the fluid in the metering section, and where a third pressure tapping 400C is positioned. The fluid encounters a first cone 600B standing as an obstruction to the flow where 20 due to continuous reduction in flow cross-sectional area as the obstruction expands in a cross-section, the fluid velocity increases and accelerates downstream of the cone. At the outermost part of the obstruction cone 600B where the flow cross-sectional area is lowest, a throat is established and runs through a short cylindrical ring 600C of uniform cross-sectional area the same as the base of the upstream obstruction cone.

25 On the double cone throat, a fourth pressure tapping 400D is installed. The double cone throat also connects a first obstruction cone 600B to a second obstruction cone 600D whose cross-sectional area decreases downstream until it vanishes in the tube 600A housing the cone structure and also connecting the flow stream to a second vertical tube 300B through a second 90<o >elbow 200C. The downstream obstruction 30 cone 600D is a fourth part of a double cone structure used to measure total flow rate of a fluid stream 100. A differential pressure sensor connected at a pressure tapping 400C located on the upstream of the first cone 600B and a pressure tapping 400D located on the double cone throat measures a second pressure difference, ∆?1? created due to acceleration of fluid over the first obstruction cone. Between the cone throat 600C and exit of the housing cylinder 600A, a third pressure difference, ∆?2? is measured using a differential pressure sensor connected at a pressure tapping 400D 5 located on the throat and a pressure tapping 400E located by the exit of the housing cylinder 600A. The pressure differential, ∆?2? is created as the fluid flows over the downstream obstruction cone 600D which by increasing flow cross-sectional area causes the fluid velocity to decrease downstream, thereby increasing the fluid pressure at location 400E above the fluid pressure at location 400D following the 10 energy conservation principle. Further, the fluid moves downward in a vertical direction through a second vertical tube 300B, and then flows through a 90<o >elbow 200D into a horizontal pipe conduit as it leaves the metering section. On the vertical tube 300B, a differential pressure sensor is connected by means of a sixth pressure tapping 400F and a seventh pressure tapping 400G to measure the reading, ∆?4? being a difference 15 in pressure generated due to weight of the fluid, flow velocity and frictional losses along the measurement positions between the pressure tappings 400F and 400G.

[1082] Further, the primary element either a venturi tube or a double cone structure introduced in this present invention is installed vertically or any other orientation different from a horizontal position for measuring flow rate of a fluid in a pipe. Fig.5.

20 Illustrates the apparatus disclosed to measure fluid flow rate where the primary element is positioned in a vertical direction. The figure describes a venturi tube as the primary element. A double cone structure similar to that described in Fig. 2 and Fig. 4 can also be used as the primary element in accordance with the embodiment disclosed in Fig.5 in the present invention. In Fig.5, a fluid stream 100 flows through 25 the metering device. The fluid flow direction is vertically upward as it enters a first vertical tube 300A having a uniform cross-sectional area. Then, a first pressure difference, ∆?3? is measured between the position at a first pressure tapping 400A and the position at a second pressure tapping 400B. Further, the fluid flows through a primary element (venturi tube 500(A – E) in this case or a double cone structure 600(A 30 – D) in another case) beginning from the entrance cylinder where a third pressure tapping 400C is positioned. The fluid then flows through the primary element structure where due to continuous reduction in flow cross-sectional area across a first part of the structure, the fluid velocity increases downstream. At exit of the first flow constraining part where the flow cross-sectional area is smallest, a throat is established and runs through a short cylindrical pipe or ring having a uniform crosssectional area the same as the exit of the upstream constraining structure. On the 5 throat of the measuring element, a fourth pressure tapping 400D is installed. The throat also connects a first constraining structure to a second constraining structure by whose configuration leads to increasing flow cross-sectional area downstream until it reaches a value similar to a cylindrical pipe connecting the primary element structure to a second vertical tube 300B through an inverted U-tube 700 and then through a 10 flow tube 800 where any sensor device 900 if required can be installed for additional measurement in the flow. On the primary element, a differential pressure sensor connected at a third pressure tapping 400C located on the entrance cylinder and a fourth pressure tapping 400D located on the throat measures a second pressure difference, ∆?1? created due to acceleration of the fluid through or over the first conical 15 part of the structure. Between the throat and exit part of the fluid cylinder, a third pressure difference, ∆?2? is measured using a differential pressure sensor connected at a pressure tapping 400D located on the throat and a pressure tapping 400E located on the exit cylinder. The pressure differential, ∆?2? is created due to increasing flow cross-sectional area that causes the fluid velocity to decrease downstream, thereby 20 increasing the fluid pressure at location 400E above the fluid pressure at location 400D following the energy conservation principle. Further, the fluid moves downward in a vertical direction through a vertical flow section 800 where it is possible to measure for example water fraction in the liquid component of the fluid using a device 900 mainly a water cut meter, electrical impedance sensor and microwave sensor. On the vertical 25 tube 300B, a differential pressure sensor is connected by means of a sixth pressure tapping 400F and a seventh pressure tapping 400G to measure the reading, ∆?4? being a difference in pressure generated due to weight of the fluid, flow velocity and frictional losses along the measurement positions between pressure tappings 400F and 400G. The device demonstrated in Fig. 5 is also a multiphase meter and can be used to 30 measure gas and liquid flow rates in both onshore and subsea oil & gas applications as well as other application areas as disclosed for embodiments in Fig. 3 and Fig.4.

[1083] 3, Fig. 4 and Fig. 5. Without knowledge of any of the individual flow rate or composition fraction of gas and liquid in the fluid mixture, the apparatus is used to provide the total mass flow rate,?̇? and fluid density,?.? The fluid density calculated from Eq. (53) is a single-phase fluid density when the fluid is either liquid or gas, or a 5 multiphase fluid density when the fluid is a mixture of gas and liquid. For a multiphase flow, the fluid density,𝜌 is related to the individual gas density, 𝜌𝐺 and liquid density, ???? as in Eq. (59) derived based on conservation of the total mass flow.

𝜌 = ????𝐺𝑉𝐹 ????(1 −𝐺𝑉?)? (59)

[1084] (59) as expressed in Eq. (60) once the individual gas density, 𝜌𝐺 and liquid 10 density, ???? are provided.

?

𝐺𝑉𝐹 = ???−𝜌

????−???? (60)

?

? ???

??? = (𝜌)𝐺𝑉𝐹 (61)

[1085] In the flow mixture, mass fraction of gas component, 𝑦𝐺 is related to 𝐺𝑉𝐹 as shown in Eq. (61). Using the gas mass fraction,???? calculated from Eq. (61), the mass 15 flow rate of gas, ?̇?𝐺 and mass flow rate of liquid,?̇?𝐿 in a gas-liquid flow system are then obtained.

?̇?𝐺 =𝑦𝐺?̇? (62)

?̇?𝐿 =?̇? −?̇?𝐺 (63)

[1086] The liquid density,???? in Eq. (59) can be provided as oil density,𝜌𝑂 if the liquid 20 is only oil phase or as water density, 𝜌𝑊 if the liquid is only water phase. When the liquid is a mixture of oil and water such that the volume fraction of water in the liquid, ? 𝑊?𝑅 (i.e. water-liquid volume ratio) is known, the liquid density,???? can be represented by

???? =𝜌𝑊𝑊𝐿𝑅 𝜌𝑂(1 −𝑊𝐿?)? (64)

25 [1087] The water-liquid ratio, ?𝑊?𝑅 can be obtained by a liquid sampling and analysis or by using a physical device such as water cut meter for direct measurement or sensor such as micro-wave, capacitance, conductive impedance sensor whose signals are related to water-liquid ratio, or any other means which can be used to detect a water fraction in a fluid stream in a pipe.

[1088] is estimated by solving another energy balance equation across a venturi tube 5 500 or a double cone structure 600. The energy balance from which the liquid density, ???? can be derived is related to the permanent pressure loss, ∆𝑝𝐿 recorded across any two positions between upstream and downstream of the primary element. The energy balance is first established to provide a total mass flow rate, ?̇? based on the reading, ∆????, which is then compared with the mass flow rate calculated based on the first 10 pressure differential, ∆?1? recorded between upstream and throat of a venturi tube or a double cone structure. Consider the energy balance between point 1 and point 3 in Fig.7 or Fig.8,

?1? ?3? ?1?2

𝜌𝑔=? ??𝑔+𝑓? 2𝑔? 𝐿???𝐿 (65)

where 𝐿𝐿 =?1? ?2? and???? is the factor accounting for frictional losses and flow 15 contraction and expansion between point 1 and point 3 over the length, 𝐿𝐿 across which the permanent pressure loss, ∆𝑝𝐿 =?1? −?3? is measured. Further, Eq. (65) can be simplified.

1 ?

∆???? = <2>

2 ( ???

? ???𝜌?1???)? ? (66)

𝐷

?̇? =????𝐴𝑝√2?∆?𝑝 𝑝

𝐿 (????) (67)

20 [1089] The flow coefficient,???? = √1/???? can be obtained from experiment. But since ∆???? is related to ∆?1? and ∆?2? according to Eq. (1), the flow coefficient,???? can also be related to ∆?2?/∆?1? analytically similar to???? and???? as given in Eq. (25) and Eq. (46), respectively.

[1090] Alternatively, since the pressure loss, ∆𝑝𝐿 increases with increasing fluid 25 density as indicated in Eq. (66), and where?̇?𝐿 >?̇?𝐺, Eq. (67) can be rewritten in terms of liquid flow rate,?̇?𝐿 and density,????.

?

?̇?𝐿 =????????√2𝜌𝐿∆𝑝 ???

𝐿 (? < 1 (68)??); ?????

[1091] is the flow coefficient dependent on the flow characteristics, and therefore can only be derived from experimental data and has no analytical solution in relation to???? or ????. Setting the liquid mass flow rate,?̇?𝐿 = (1 −????)?̇?, Eq. (68) can be solved to give 5 the total mass flow rate,?̇? or liquid density,𝜌𝐿 as expressed in Eq. (69) and Eq. (70), respectively.

1 𝐷

?̇? = ( 1−? ????????√2𝜌𝐿∆𝑝 ?) ?

𝐿 ( 69)??? ???) (?

2

<(>1−? 1 ????? ???<)>?̇?

??? = [ ???????? ] [ 2∆???(?????)] (70)

[1092] If liquid density, ???? is calculated explicitly from Eq. (70) using appropriate value 10 of the flow coefficient,????, or implicitly from Eq. (67) using appropriate value of the flow coefficient, ????, then Eq. (71) can be used to estimate the water-liquid ratio,𝑊𝐿?.?

?

? 𝑊?𝑅 = ???−????

????−???? (71)

[1093] Alternatively, the fluid mixture density,𝜌 is calculated from Eq. (59) where the liquid density,????, gas density,???? and gas volume fraction,𝐺𝑉𝐹 or gas mass fraction, 15 ???? are provided directly or indirectly by any available method or means. In this case, the vertical tubes 300A & 300B are not required to provide the fluid density. A venturi tube 500 or a double cone structure 600 is then used to provide the total mass flow rate, ?̇? of the fluid mixture in accordance with the calculation formulae disclosed in Eq. (5) and Eq. (29), respectively. Therefore, a venturi tube 500 or a double cone 20 structure 600 equipped with two differential pressure sensors to provide two pressure difference readings, ∆?1? and ∆?2? or ∆????, is a multiphase meter or a measuring device used to provide gas and liquid flow rates in a fluid stream with extreme high gas volume fraction, also known as a wet gas stream. When a parameter, ????𝑚 called Lockhart-Martinelli number and calculated from Eq. (72), has a value in the range 0 < ????𝑚 < 25 0.35, the gas volume fraction,𝐺𝑉𝐹 and total mass flow rate,?̇? of the fluid stream can be obtained where the mass flow rate, ?̇? calculated from Eq. (67) or Eq. (69) corresponds to the mass flow rate, ?̇? calculated from Eq. (5) for a venturi tube or Eq. (29) for a double cone structure.

1−𝐺𝑉𝐹 ?

????𝑚 = 𝐺𝑉𝐹 √ ???

??? (72)?

[1094] The foregoing description of embodiments of the invention has been presented 5 for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of the invention without departing from the scope of the invention. The embodiments were chosen and described in order to explain the principles of the invention and its practical 10 application to enable one skilled in the art to utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated.

15

Claims (20)

CLAIMS What is claimed is:

1. A flow meter device, comprising:

means of accelerating a flow by reducing a flow cross-sectional area in a pipe, 5 thereby creating a differential pressure between a section of the pipe with a largest flow cross-section and a section of the pipe with a smallest flow cross-sectional area;

means of deaccelerating a flow by increasing a flow cross-sectional area in a pipe, thereby creating a differential pressure between a section of the pipe with a smallest flow cross-section and a section with a largest flow cross-sectional area;

10 means of connecting two differential pressure sensors, where a first differential pressure sensor between an upstream section where the flow cross-sectional area is largest and a throat section where the flow cross-sectional area is smallest, and a second differential pressure sensor between the throat section where the flow crosssectional area is smallest and a downstream section where the flow cross-sectional 15 area is largest along a flow direction;

where the first differential pressure sensor is configured for providing a first differential pressure reading between the upstream section where the flow crosssectional area is largest and the throat section where the flow cross-sectional area is smallest;

20 where the second differential pressure sensor is configured for providing a second differential pressure reading between the throat section where the flow crosssectional area is smallest and the downstream section where the flow cross-sectional area is largest;

where the two differential pressures are configuratively related to a permanent 25 pressure loss which can be calculated by subtracting the second differential pressure reading from the first differential pressure reading, or measured with a differential pressure sensor connected between the upstream section and the downstream section with the same cross-sectional area as the pipe.

2. The flow meter device of claim 1, wherein measures total flow rate of a fluid stream flowing in the pipe.

3. The flow meter device of claim 1, wherein measures a total flow rate of a gasliquid flow or a multiphase flow stream in the pipe.

5 4. The flow meter device of claim 1, wherein measures density of a fluid stream flowing in the pipe.

5. The flow meter device of claim 1, wherein measures flow rates of all the fluid phases or components including gas, liquid, oil and water, in a fluid stream flowing through the pipe.

10 6. The flow meter device of claim 1, wherein applied in flow of liquids, especially a heavy oil or high viscous liquid to provide accurate measurement of mass flow rate, fluid density and fluid viscosity.

7. The flow meter device of claim 1, wherein applied in pipeline transporting CO2 in liquid phase, supercritical phase or in a combination of liquid and gas phases, or in 15 a combination of supercritical and gas phases, or in a combination of liquid and supercritical phases, to provide accurate measurement of mass flow rate, fluid density and fluid viscosity whether with or without impurities.

8. The flow meter device of claim 1, wherein consists of means of accelerating a flow including a venturi tube structure (500A – 500E);

20 a. comprising a hollow entrance cylindrical pipe (500A), a hollow converging truncated conical pipe (500B), a hollow short cylindrical pipe (500C), a hollow diverging truncated conical pipe (500D) and a hollow exit cylindrical pipe (500E) through which the fluid flows,

b. such that two differential pressure signals are measured, where a 25 discharge or a flow coefficient is first approximated by using a calculating formula

∆𝑝2 𝐿

𝛼 ( 𝑣𝑟2

? ∆𝑝+

1 𝐿 )

𝑣𝑟1

??? = √( 𝐿

1+?(?𝑣𝑟2

𝐿 ) )

𝑣𝑟1

when all correction factor or coefficient,𝛼 = 1,

c. such that the approximated discharge coefficient is updated to the actual discharge or flow coefficient by ensuring that any two simulated 5 differential pressure using a one-dimensional momentum equation similar to

𝑑𝑝 1?(??̇?.?)? 𝑓

𝑑𝑥= − 𝐴 𝑑𝑥 − 2𝑑𝜌𝑣|𝑣| 𝜌𝒈

across the differential pressure measurement positions, correspond to the two differential pressures measured, thereby preventing the use of 10 empirical correlation for determining parameters in the total flow rate calculation.

9. The flow meter device of claim 1, wherein consists of means of accelerating a flow including a double cone structure (600 B – 600 D) situated inside a hollow cylindrical pipe (600A);

15 a. where the double cone structure (600 B – 600 D) comprises a diverging solid conical block (600B), a short solid cylindrical ring (600C), and a converging solid conical block (600D) over which the fluid flows,

b. such that two differential pressure signals are measured, where a 20 discharge or flow coefficient is first approximated by using a calculating formula

∆𝑝2 𝐿 𝐿

( 𝑑

? 𝐿)( 𝑐𝑟2

𝑐 𝐿 )

𝑐𝑟1

??? = √( ∆𝑝+𝛼

1

𝐿 𝐿?2 )

1+?(?𝑑

𝐿)( 𝑐?

𝑐 𝐿 )

𝑐𝑟1

when all correction factor or coefficient,𝛼 = 1,

c. such that the approximated discharge coefficient is updated to the actual discharge or flow coefficient by ensuring that any two simulated differential pressure using a one-dimensional momentum equation similar to

𝑑𝑝 1 ?(??̇?.?)? 𝑓

5 𝑑𝑥= −𝐴 𝑑𝑥 − 2?ℎ?𝜌𝑣|𝑣| 𝜌𝒈

across the differential pressure measurement positions, correspond to the two differential pressures measured, thereby preventing the use of empirical correlation for determining parameters in the total flow rate calculation.

10 10. The flow meter device of claim 1, wherein the permanently lost differential pressure is used to calculate the liquid density in the fluid stream.

11. The flow meter device of claim 4, wherein applied in measuring density of the fluid stream flowing in the pipe;

a. consists of two vertical and parallel pipes (300A, 300B) such that the fluid 15 stream goes in opposite direction at each time in the different pipes,

b. such that two differential pressure sensors are installed to read differential pressures created along each pipe at a specified distance where the two differential pressure readings are combined to obtain density and viscosity of the fluid or fluid mixture.

20

12. The flow meter device of claim 1, wherein applied to wet gas stream or a flow stream with extreme high gas volume fraction to provide gas and entrained liquid flow rates where both the gas and liquid densities are available for the flow rate computation of the fluid mixture in the pipe.

13. The flow meter device of claim 1, wherein applied to a gas-liquid or a two-phase 25 flow to provide gas flow rate, liquid flow rate and liquid density where only the gas density or a means of calculating it is available for the flow rate computation of the fluid mixture in the pipe.

14. The flow meter device of claim 1, wherein applied to a gas-oil-water or a multiphase flow to provide gas flow rate, oil flow rate and water flow rate where the gas density, oil density and water density or a means of calculating them are available for the flow rate computation of the fluid mixture in the pipe.

5 15. The flow meter device of claim 1, wherein applied to a CO2 flow in a transport pipeline to provide mass flow rate, fluid density, fluid viscosity and possible twophase flow composition for the fluid mixture flow in a pipe.

16. The flow meter device of claim 1, wherein applied to a liquid flow with entrained gas to provide total mass flow rate, fluid density, fluid viscosity and possible 10 two-phase flow composition for the fluid mixture flow in a pipe.

17. The flow meter device of claim 1, wherein applied to a heavy oil or high viscous liquid flow to provide mass flow rate, fluid density and fluid viscosity for the fluid flow in a pipe.

18. A method for metering mass flow rate of fluid and fluid mixture in a pipe, the 15 method comprising:

accelerating a flow by reducing a flow cross-sectional area in the pipe, thereby creating a differential pressure between a section of the pipe with a largest flow crosssection and a section of the pipe with a smallest flow cross-sectional area;

deaccelerating a flow by increasing a flow cross-sectional area in a pipe, 20 thereby creating a differential pressure between a section of the pipe with a smallest flow cross-section and a section with a largest flow cross-sectional area;

connecting two differential pressure sensors, where a first differential pressure sensor between an upstream section where the flow cross-sectional area is largest and a throat section where the flow cross-sectional area is smallest, and a second 25 differential pressure sensor between the throat section where the flow cross-sectional area is smallest and a downstream section where the flow cross-sectional area is largest along a flow direction;

where the first differential pressure sensor is configured for providing a first differential pressure reading between the upstream section where the flow crosssectional area is largest and the throat section where the flow cross-sectional area is smallest;

5 where the second differential pressure sensor is configured for providing a second differential pressure reading between the throat section where the flow crosssectional area is smallest and the downstream section where the flow cross-sectional area is largest;

where the two differential pressures are configuratively related to a permanent 10 pressure loss which can be calculated by subtracting the second differential pressure reading from the first differential pressure reading, or measured with a differential pressure sensor connected between the upstream section and the downstream section with the same cross-sectional area as the pipe.

19. The method of claim 18, wherein the accelerating a flow including a venturi tube 15 structure (500A – 500E);

a. comprising a hollow entrance cylindrical pipe (500A), a hollow converging truncated conical pipe (500B), a hollow short cylindrical pipe (500C), a hollow diverging truncated conical pipe (500D) and a hollow exit cylindrical pipe (500E) through which the fluid flows,

20 b. such that two differential pressure signals are measured, where a discharge or a flow coefficient is first approximated by using a calculating

∆𝑝2 𝐿𝑣𝑟2

formula ? 𝑝+𝛼 (𝐿 )

1 𝑣𝑟1

??? = √( ∆

𝐿

1+?(?𝑣𝑟2 )

𝐿 )

𝑣𝑟1

when all correction factor or coefficient,𝛼 = 1,

c. such that the approximated discharge coefficient is updated to the actual 25 discharge or flow coefficient by ensuring that any two simulated differential pressure using a one-dimensional momentum equation similar to

𝑑𝑝 1?(??̇?.?)? 𝑓

𝑑𝑥= − 𝐴 𝑑𝑥 − 2𝑑𝜌𝑣|𝑣| 𝜌𝒈

across the differential pressure measurement positions, correspond to the two differential pressures measured, thereby preventing the use of empirical correlation for determining parameters in the total flow rate 5 calculation.

20. The method of claim 18, wherein the accelerating a flow including a double cone structure (600 B – 600 D) situated inside a hollow cylindrical pipe (600A);

a. where the double cone structure (600 B – 600 D) comprises a 10 diverging solid conical block (600B), a short solid cylindrical ring (600C), and a converging solid conical block (600D) over which the fluid flows,

b. such that two differential pressure signals are measured, where a discharge or flow coefficient is first approximated by using a calculating formula

∆𝑝2 𝐿𝑑𝐿?𝑟2

15 ? 𝐿)( ?

𝑐 𝐿 )

𝑐𝑟1

?? 𝑝+𝛼 (

1

? = √( ∆

𝐿 )

1+?(?𝑑 2

𝐿)( 𝐿𝑐𝑟

𝑐 𝐿 )

𝑐𝑟1

when all correction factor or coefficient,𝛼 = 1,

c. such that the approximated discharge coefficient is updated to the actual discharge or flow coefficient by ensuring that any two simulated differential pressure using a one-dimensional momentum equation 20 similar to

𝑑𝑝 1 ?(??̇?.?)? 𝑓

𝑑𝑥= −𝐴 𝑑𝑥 − 2?ℎ?𝜌𝑣|𝑣| 𝜌𝒈

across the differential pressure measurement positions, correspond to the two differential pressures measured, thereby preventing the use of empirical correlation for determining parameters in the total 25 flow rate calculation.