Numerical modeling and experimental verification of a low fluid flow inductive flowmeter

Flow Measurement and Instrumentation 78 (2021) 101876 Available online 6 January 2021 0955-5986/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Numerical modeling and experimental verification of a low fluid flow inductive flowmeter P. Drexler a , 1 , P. Fiala b , 2 , * , R. Kadlec a , 3 , P. Londak a , 4 , T. Madrova a , 5 , M. Klima a , 6 , J. Zukal a , 7 a Dept. of Theoretical and Experimental Electrical Engineering, Brno University of Technology, Technick´ a 12, 616 00, Brno, Czech Republic b SIX Research Center, Brno University of Technology, Technick´ a 12, 616 00, Brno, Czech Republic ARTICLE INFO Keywords: Magnetohydrodynamics Fluid flow measurement Finite element methods Turbulent media Coupled modeling Plasma ABSTRACT The article discusses modeling procedures to capture the physical and chemical processes present in operational measurement with a high-precision inductive flowmeter. In this context, a theoretical model and a numerical solution are proposed. Exploiting the combined finite element method (FEM) and the finite volume method (FVM), we prepared numerical models of multiple variants of the flowmeter and computed the output voltage on the device’s electrodes. The model joins together the magnetic, electric, and current fields; a flow field; and a physical-chemical nonlinear ion model. The results were obtained by means of the FEM/FVM in ANSYS software and verified via experimental testing. 1. Introduction THE current trend of implementing various precise industrial measurement techniques [1–3] involves multiple activities, such as performing parametric measurement or special monitoring of liquid, semi-liquid, and solid commodities before transportation, and designing methods or systems to facilitate flow velocity measurement [4–6]. This group of substances comprises, for example, organic and inorganic liquids characterized by different degrees of corrosivity [7], low temperature fluids (−180 ◦C to - 196 ◦C), and fluid gases (including, for instance, argon and nitrogen). The actual liquid flow can be advantageously measured with inductive flowmeters [8], whose design and parametric description embody relatively complex tasks in terms of the analysis of the modeling results and its accuracy. The equivalent models are both lumped parameter ones [11] and those based on solving partial differential equations [9,10]. We designed and verified an applicable model, utilizing precise flow evaluation at a mean speed of v =0.1–10 m/s. The full electro-magneto-hydro-dynamical (EMHD) model of an inductive flowmeter is a coupled problem, comprising coupled electric, magnetic, and fluid flow fields together with electric circuit and physical-chemical (ion) models; the relevant geometrical structure is demonstrated in Fig. 1. The designed physical and mathematical models are solvable from different perspectives. Several methods for measuring the flow rate of small amounts or low velocities of liquids are presently being developed and characterized. An interesting approach was proposed by Prasan [12], who measured fluid flow by compensating the pressure drop across the ends of the measuring unit using a compensation pump. The flow-induced pressure drop is balanced by a feedback control loop. This is the type of measurement being balanced to zero deviation. In small flow measurements, for example, Chen et al. [13] outlined an application for space research; in this context, it is important to stress the necessity of managing the measurement of fuel flow in microgravity conditions. Ultrasonic measurements offer the advantages of non-invasive and immobile component structures as well as fast responses to the detection of bidirectional flow, whose applications in space exploration have already been described. To avoid the drawbacks of pulsed ultrasonic measuring * Corresponding author. Tel.: +420541146276. E-mail address: [email protected] (P. Fiala). 1 [email protected], www.utee.feec.vutbr.cz. 2 [email protected], www.utee.feec.vutbr.cz. 3 [email protected], www.utee.feec.vutbr.cz. 4 [email protected], www.utee.feec.vutbr.cz. 5 [email protected], www.utee.feec.vutbr.cz. 6 [email protected], www.utee.feec.vutbr.cz. 7 [email protected], www.utee.feec.vutbr.cz. Contents lists available at ScienceDirect Flow Measurement and Instrumentation journal homepage: http://www.elsevier.com/locate/flowmeasinst https://doi.org/10.1016/j.flowmeasinst.2020.101876 Received 30 April 2020; Received in revised form 20 November 2020; Accepted 12 December 2020 Flow Measurement and Instrumentation 78 (2021) 101876 2 configurations, the study presents flow measurement via continuous propagation of ultrasonic waves to meet the requirements of a large measurement range and high accuracy. Article [14] then compares two methods for runoff measurement in chimneys to determine the errors exhibited by the techniques in the presence of cyclone flow (a turbulent environment). One of these procedures, based on speed measurements with a Pitot tube in a grid of points, is the standard reference method according to EN ISO 16911–1. The other approach, ultrasonic measurement, is often employed as an automated measuring system pursuant to EN ISO 16911–2. The methods involve the analysis of several typical reservoir configurations, and the flow field in the reservoirs is obtained by using validated computational fluid dynamics (CFD) modeling exploiting OpenFoam software. The researchers emphasize that possible errors in the standard reference method occurring in the presence of a cyclonic flow are significant compared to the requirements of the EU ETS. Thus, procedures associated with numerical modeling are shown as indispensable in solving non-trivial problems. Magneto-inductive flowmeters play an important role in the measurement of conductive, partially conductive, and dielectric liquids [15]. Approximately 30–40% of all applications require flow sensors that utilize alternating EMG field, due to the physical properties of the material being monitored or the hydraulic effects. This aspect affects the measurement of the flow rate of a liquid with a high solid content or a very low conductivity. Magnetic inductive flow sensors have proved reliable in most sectors on a long-term basis; in the given segment, however, magnetic inductive flow sensors without alternating EMG fields often reach their very limits, which leads to uneven or noisy initial signals and inaccurate and/or inadequately reproducible processing of the measurement evaluation data in the ongoing process. An inductive flow sensor has certain drawbacks, including that the application of point electrodes (IFS-SE) is sensitive to the shape of the flow profile and that the device’s usability remains limited to measuring the axially symmetric single-phase flows in a circular tube. The sensor therefore must align with not fully developed flow profiles. To improve the accuracy, the authors of paper [16] designed an inductive flow sensor with a pair of arcuate electrodes embedded in the inner surface of the insulating part of the pipe. In [17], an electric charge induced on a ring sensor with different geometric sizes from a single particle having a single charge was described and mathematically modeled. The spatial sensitivity of the sensor was then derived from a numerical solution obtained via ANSYS finite element analysis software. The influence of the geometric size of the sensor on the spatial sensitivity was also investigated, and the relevant basic theory and effect of spatial filtering were quantitatively analyzed. The time-frequency response characteristics were obtained and derived. In the discussed context, a suitable mathematical model and its detailed analysis have proved necessary to achieve extreme flowmeter requirements. Selecting and using an optimum non-destructive flow measurement method (range of measured mean velocity v =0.01–1.0 mm/s) within dendrology and identifying the issues to be resolved in connection with such activities are the central problems characterized in a dedicated article by ˇ Cerm´ ak [18]. The paper describes methods for measuring several variables, including sap flow rates, which are commonly utilized to investigate transpiration. The research in this subdomain of flow measurement has expanded further also through the advanced procedure outlined in Ref. [19], namely, the “Linear Heat Balance (LHB) method”; the approach exploits an exact (physically correct) equation analytically derived from the basic convection heat transfer equation. More concretely, the paper defines a semi-destructive method for measuring the flow rate of liquids in a tree xylem by using heat balance and the measured temperature differences around a continuously heated point source of a heat-needle. The novel formula is analyzed and verified via numerical simulation (the finite element method), and the LHB method is demonstrated on experimentally measured data. The procedure described below brings a significant shift in the accuracy of the non-destructive measurement of the flow rate in the range of v =0.01–1 mm/s. The research presented above indicates the need to characterize in detail the electrodynamics of the fluid flow, including the chemical composition and effects on the measured quantity. This problem is further discussed in relation to the physical-mathematical model formulated and tested for use with the finite element method (FEM). A viable approach to modeling the physical properties rests in the analytical solution of the flowmeter model, exploiting formulation via concentrated parameters [11,20] and containing the necessary electronics [20]. An alternative analysis then relies on detailed formulation and assembly variants of the numerical bound model [21]. Furthermore, Fig. 1. The structure of an inductive flowmeter. P. Drexler et al. Flow Measurement and Instrumentation 78 (2021) 101876 3 the path following the finite element plus finite volume methods to formulate, describe, and solve the inductive flowmeter model with distributed parameters is selected. 2. Mathematical-physical model The electromagnetic part of the flowmeter is derived from the reduced Maxwell equations curl H=0,(1) div B=0,(2) where H is the magnetic field intensity vector and B denotes the magnetic flux density vector B= μ 0H,(3) J=γE,(4) curl E =0,(5) divJ =0,(6) where E is the electric field intensity vector, J denotes the current density vector, μ 0 stands for the permeability of vacuum, and γ represents the specific conductance of the measured liquid. The vector functions of the electric and the magnetic fields are expressed by means of the scalar electric φ e and magnetic φ m potentials E= − grad φe,(7) H= − grad φm.(8) The final current density J from (4) is influenced by the velocity v of the flowing ion solution and the outer magnetic field J=γ(E+v×B).(9) If the electrodes E 1 and E 2 exhibit different electric potentials (Fig. 2), the current density J is formed in the Ω area, according to (9), and the current I L flows in the ion solution, as follows: IL=∫∫ Se J⋅dS, IL=∫∫ Se γ(E+v×B)⋅dS,(10) where S e is the oriented area of the electrodes E 1 and E 2 into the space Ω. Equation (10) comprises the electric field intensity E for the ion solution |E|< < |v×B|;(11) we therefore disregard the influence of the electric field intensity. The specific force f affecting the moving charge q is f=J×B,(12) and the force in the whole area Ω is F=∫∫∫ Ω J×B dV.(13) The voltage between the flowmeter electrodes E 1 , E 2 is obtained from UL=∫ E2 E1 E⋅d ℓ ,(14) where the electric field intensity E is derived out of the force F, which affects a charge q; the voltage then equals UL=∫ E2 E1 F q⋅d ℓ .(15) If a charge is substituted for the equation of the steady state current I L, we have UL=∫ E2 E1 ∫∫∫ Ω J(v) × B dV ILΔ ℓ ⋅(vio +v)⋅d ℓ ,(16) where Δ ℓ is the length element in the direction between the electrodes E 1 and E 2 ; and v io denotes the ion velocity in the lengthwise direction between E 1 and E 2 . The current density J(v) depends on the immediate ion velocity between E 1 and E 2 . After reduction, the voltage on the flowmeter electrodes equals UL=∫∫∫ Ω(J(v) IL ×B)⋅(vio +v)dV (17) The fluid flow velocity distribution model is derived for an incompressible fluid as divv =0;(18) with a stable flow, we have div ρ v=0(19) from the law of conservation of energy, where ρ is the specific density. We assume the turbulent flow curl  ν =2 ω ,(20) where ω denotes the angular velocity of fluid. By utilizing Stokes’ theorem and the Helmholtz theorem for the moving particles to obtain the continuity equation, we can formulate from the balance of forces the Navier-Stokes equation for the fluid element, ∂ v ∂ t+v⋅gradv =A−1 ρ grad p + υ ⋅Δv,(21) where A is the external acceleration, ν the kinematic viscosity, and p the pressure. In formula (21), we can substitute the pressure losses Fig. 2. The principal configuration of an inductive flowmeter. P. Drexler et al. Flow Measurement and Instrumentation 78 (2021) 101876 4 grad p = − (Kx ρ vx|v| + fr Dh ρ vx|v| + Cx μ pvx)ux −(Ky ρ vy|v| + fr Dh ρ vy|v| + Cy μ pvy)uy, −(Kz ρ vz|v| + fr Dh ρ vz|v| + Cz μ pvz)uz (22) where K denotes the suppressed pressure losses, f r the resistance coefficient, D h the hydraulic diameter, C the air permeability of the system, μ p the dynamic viscosity, and u the unit vector of the Cartesian coordinate system. The resistance coefficient is obtained from the Boussinesq theorem [22–28] as fr=aR−b e(23) where a, b are the coefficients from Ref. [20]. The magnetic field, expressed in (17) by the flux density B, is gained from the Biot-Savart law via the differential scalar magnetic potential (DSP), T=1 4 π ∫ Ω Jc×R |R|3dV (24) where R is the positional vector between the point where we seek the magnetic field intensity T and the point where the magnetic field source is I c (Fig. 2), characterized by the current density J c . The magnetic field intensity H in the area Ω can be expressed as H=T−grad φm,(25) where T is the above (estimated) magnetic field intensity. The boundary conditions are n⋅ μ 0 μ r(T−gradφm) = 0,on the boundary ΓFe−0,(26) where n is the normal vector and Γ Fe-0 denotes the interface between the areas Ω Fe and Ω in Fig. 3. The continuity of the tangential components of the magnetic field intensity on the interface between the area and a ferromagnetic material is n× (T−gradφm) = 0,on the boundary ΓFe−0.(27) With the help of formulas (1), (2), and (5), we yield div μ 0 μ rT−div μ 0 μ rgradφm=0(28) Equation (28) is discretized by means of an approximation of the scalar magnetic potential φm=∑ Nφ k=1 ϕmkWk(x,y,z),∀(x,y,z)⊂Ω,(29) where ϕ m is the nodal value of the scalar magnetic potential and W k denotes the base function. We obtain the semidiscrete solution via approximating (29) within formula (28) and through the use of the Galerkin method: ∑ Nφ j=1 −∫ Ω μ tj⋅gradWi+ μ grad ϕmj ⋅gradWidΩ =0,i=1, .., Nϕ(30) where t j represents the nodal value of the known magnetic field intensity in the mesh. It is possible to write the above equation (30) briefly as −[kTij]+[kij]{ϕ} = 0i,j=1, .., Nϕ(31) The coefficients for (31) are expressed as kem Tij = − ∫ Ωe μ etj⋅gradWidΩ i,j=1, .., Ne,(32) kem ij = − ∫ Ωe μ egrad ϕj⋅gradWidΩ (33) where μ e is the material permeability of a selected element and N e denotes the number of mesh elements. The equation system changes into the formula −[kem Tij ]+[kem ij ]{ϕ} = 0e=1, .., Ne;(34) this system can then be solved (34) by utilizing standard algorithms. The solution exploiting the DSP comprises two processes: First, we express the distribution of the magnetic field intensity T from the current sources according to (24), with respect to the boundary conditions (26) and (27) in the area Ω Fe . Second, based on the previous step, we solve the distribution of the magnetic intensity H according to (25). Assuming that the cross-section of the wire S of the exciting coil’s winding is constant, we can write for the current density in the coil winding J=I Sn,(35) where n is the normal component related to the cross-section of a wire having the area S. The model of the electric or current field is formulated from equations (3), (4), (6) and (7): γ div gradφe=0.(36) The boundary conditions are n⋅γ(gradφe) = 0,at ΓE−k,(37) where n is the normal vector of the surface of an electrode E, and Γ E-k denotes the interface between the fluid and an electrode E. The continuity of the tangential components of the electric field intensity on the interface is n× (gradφe) = 0 at ΓE−k.(38) The scalar electric potential can be approximated similarly to the procedure in formula (29); by exploiting (29), (36), and the Galerkin method, we then obtain the semidiscrete solution ∑ Nφ j=1 −∫ Ω γ grad ϕej ⋅gradWidΩ =0,i=1, .., Nϕ(39) where ϕ ej is the nodal value of the scalar electric potential in a mesh. equation (39) can be written with the help of [kJij]{ϕ} = 0i,j=1, .., Nϕ.(40) The coefficients for (40) are Fig. 3. A geometric model of the flowmeter body. P. Drexler et al. Flow Measurement and Instrumentation 78 (2021) 101876 5 kJe ij = − ∫ Ωe γegrad ϕej ⋅gradWidΩ (41) where γ e is the specific conductivity of fluid in the static state of the selected element. The equation system changes into [kJe ij ]{ϕ} = 0e=1, .., Ne(42) where N e is the number of mesh elements. The model of the velocity field is formulated from the steady-state stability condition, expressed as ∫ Ω fdΩ +∫tdΓ=0(43) where f denotes the specific forces in the area Ω, and t represents the pressures, tensions, and shear stresses on the interface of the area Γ. By utilizing transformation into local coordinates, we obtain the differential form for the static equilibrium, f+div2Tv=0,(44) where div 2 stands for the tensor quantity operator div, and T v denotes the internal tension tensor 2Tv=⎡ ⎣ XxXyXz YxYyYz ZxZyZz⎤ ⎦(45) where X, Y, Z are the stress components acting on the elements of the area Ω. A form of the specific force from (18)-(21) can be added to the static equilibrium condition; this form is obtained via the external acceleration A, via the pressure losses and shear stresses τ . We have ρ ( ∂ v ∂ t+v⋅gradv)− ρ A−∑ Ns ℓ =1 F ℓ +div2Tv=0(46) where F ℓ are the discrete forces. The model covering the forces, viscosity, and pressure losses is expressed as ρ ( ∂ v ∂ t+v⋅gradv)− ρ A−∑ Ns ℓ =1 F ℓ +grad p − υ ⋅Δv =0(47) Equation (21) can be discretized through an approximation of the velocity v and acceleration a: v=∑ Nφ k=1 ν vkWk(x,y,z),∀(x,y,z)⊂Ω,∀(x,y,z)⊂Ω, a=∑ Nφ k=1 α vkWk(x,y,z),∀(x,y,z)⊂Ω,(48) where v v , a v are the immediate node values, W denotes the base function, and N φ represents the number of mesh nodes. By approximating (47) and utilizing the Galerkin principle from (48), we obtain the semidiscrete solution ∫ Ω Wj[ ρ (∑ Nv i=1 Wi ∂ vvi ∂ t+∑ Nv i=1 Wivvi⋅gradvvi)+∑ Nv i=1 Wigrad pi − ρ ∑ Nv i=1 WiAvi −∑ Nv i=1 Wi∑ Ns ℓ =1 F ℓ i]dΩ −∫ Ω Wj[∑ Nv i=1 Widiv υ ⋅gradvvi]dΩ −∫ Γ Wj[∑ Nv i=1 WiXi]dΓ=0j=1, .., Nv, (49) where X denotes the known conditions on the interface of the area. We substitute the pressure losses in (49) for the function (22), yielding the flowing medium model ρ ∫ Ω WjWidΩ dvvi dt + ρ ∫ Ω Wjvvi ⋅(dWi dx ux+dWi dy uy+dWi dz uz)dΩ vvi +∫ Ω Wj⋅(dWi dx ux+dWi dy uy+dWi dz uz)dΩ pi − ρ ∫ Ω WjdΩ Ai−∫ Ω WjdΩ F ℓ i −∫ Ω(dWj dx +dWj dy +dWj dz ) υ i⋅(dWi dx ux+dWi dy uy+dWi dz uz)i dΩ vvi. −∫ Γ WjdΓ Xi=0i,j=1, .., Nv (50) The boundary conditions are n⋅(v) = 0 at Γvr1,(51) where n is the normal vector to the direction of the fluid flow, and Γ vr1 ⊂ Γ vr denotes the interface between the fluid and the solid parts of the flowmeter. We have n⋅(p) = 0 on the boundary Γvr,(52) where Γ vr2 ⊂ Γ vr is the interface between the liquid and the solid parts of the flowmeter. The system of equation (50) can now be written as [Cf ij]{dvvi dt }+[Ksx ij −KFx ij ]{vvi} + [Kcx ij ]pi=[Kgx ij ]{Ai} + [Fbx ij ]{F ℓ i} +[Fsx ij ]{Xi}i,j=1, .., Nv(53) The matrices C ij f , K ij sx , K ij Fx , K ij cx , K ij gx , F ij bx , K ij Fx , F ij sx relate to the coefficients of the system of equation (49); we can rewrite the form for an element of mesh into [Cf e]{dvvi dt }+[Ksx e−KFx e]{vvi} + [Kcx e]pi=[Kgx e]{Ai} + [Fbx e]{F ℓ i} +[Fsx e]{Xi}e=1, .., Ne v(54) The above physical-mathematical model describing the effects of the electromagnetic field depending on the fluid flow is formulated for FEM processing [33], utilizing eqs. ((4) and (29)–(34) and (39)–(42)9-50), by the Galerkin method. An open system and the ANSYS environment were applied to deliver the model. The dependencies and functions outlined below were tested, solved, and analyzed in the ANSYS system, and the finite volume method was employed for their formulation in some cases [34]. 3. Evaluating the desired functions The functionality of the numerical model was verified on a DN-100 standard flow meter, with multiple quantities evaluated, compared, and measured. In the numerical model, we evaluate the voltage on the terminals of E 1 a E 2 UL≅∑ NΩ e=1(Je(v) IL ×Be)⋅(vio +v)eΔVe,(55) where ΔV e is the element volume of the discretized area Ω, and N Ω denotes the number of elements of the area Ω. Assuming that the current density is given by the motion of the positive and negative ions which do not recombine, we can express the current density as J=NqΔq S Δt n,(56) where N q stands for the number of charges q, Δq represents the change of P. Drexler et al. Flow Measurement and Instrumentation 78 (2021) 101876 6 an elementary charge q, Δt is the time instant, and S denotes the crosssection where the charges q are in motion. Considering the number of ions with a positive or a negative charge, we can express the immediate velocity of the motion v ok , obtaining the current density J=∑ N+ i=1 q+ i+∑ N− j=1 q− j ΔV vok ,(57) where ΔV is the volume element, N + represents the number of positive charges q + , and N − denotes the number of negative charges q − [8,20,21, 29] (Fig. 4). In Fig. 4, v m refers to the immediate velocity of the measured fluid, v -ok is the immediate velocity of the negative charges, and v +ok denotes the immediate velocity of the positive charges. The current density (56) is rewritten as J=∑ N+ i=1 q+ i⎛ ⎜ ⎝ v+ ok,i  (|vm,i|2+v+ ok,i 2) √⎞ ⎟ ⎠ 2ΔV +∑ N− j=1 q− j⎛ ⎜ ⎝ v− ok,j  (|vm,j|2+v− ok,j 2) √⎞ ⎟ ⎠ 2ΔV .(58) The formula for the voltage on the terminals of the electrodes from (55) reads UL≅∑ NΩ e=1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2IL∑ N+ i=1 q+ i⎛ ⎜ ⎝ v+ ok,i  (vm,i 2+v+ ok,i 2) √⎞ ⎟ ⎠×Be +1 2IL∑ N− j=1 q− j⎛ ⎜ ⎝ v− ok,j  (vm,j 2+v− ok,j 2) √⎞ ⎟ ⎠e ×Be ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⋅((v+ ok +v− ok) 2+vm)e . (59) If we consider the effect of a moving electric charge and the impact of a magnetic field on a charge in motion, then the voltage on the electrodes E 1 , E 2 from (17) equals UL=∫∫∫ Ω(J(v) IL ×B)⋅(vio +vm)dV +UH,(60) where U H is the electric voltage. After evaluating the U H in equation (60), we get UL=∫∫∫ Ω(J(v) IL ×B)⋅(vio +vm)dV +∫∫∫ Ω(J(v) IL ×B)⋅(vio)dV UL=∫∫∫ Ω(J(v) IL ×B)⋅(2vio +vm)dV (61) The voltage is rewritten for the numerical solution according to (58), yielding Fig. 4. A geometric model of the flowmeter body. Fig. 5. The principal configuration of the modified inductive flowmeter. P. Drexler et al. Flow Measurement and Instrumentation 78 (2021) 101876 7 UL≅∑ NΩ e=1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2IL∑ N+ i=1 q+ i⎛ ⎜ ⎝(v+ ok,i)2  (vm,i 2+v+ ok,i 2) √⎞ ⎟ ⎠×Be +1 2IL∑ N− j=1 q− j⎛ ⎜ ⎝(v− ok,j)2  (vm,j 2+v− ok,j 2) √⎞ ⎟ ⎠e ×Be ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ⋅(v+ ok +v− ok +vm)e (62) Possible spurious effects can be suppressed by means of a differential measurement method (Fig. 5). In the lengthwise direction of the electrodes E 1 , E 2 , we measure the impedance of the fluid as ZL=f(v,B,γ),(63) and in the cross-direction of the electrodes E 3 , E 4 then ZT=f(v,γ).(64) 4. Geometric model The geometric model of the flowmeter is characterized in Fig. 3 above. Three functional variants were considered: a solution with a circular inlet pipe (Fig. 6) causing a turbulent flow in the area of the flowmeter; an embodiment utilizing the angle of 0◦, 45◦(or 90◦) between the inlet pipe and the lengthwise direction of the flowmeter; and a straight pipe version. The ANSYS-BASED FEM/FVM model, described in detail above, was designed for the dynamic condition (via the APDL language) to facilitate the analyses of the current and magnetic fields and fluid velocity distribution by means of the finite element method. In the areas Ω, we evaluated equation (61), which is substituted for the numerical model of the mesh elements. The main APDL-based model was formulated according to equations (34), (42) and (55), by using tools such as SOLID236 (237), SOLID122 (123), SOLID96, SOURCE36, and FLUID142. The obtained solution relates to the steady state, with respect to the boundary conditions (26), Fig. 6. An inductive flowmeter geometric model: a pipe with the angle of 90 ◦. Fig. 7. The magnetic field intensity H [A/m] distribution in the flowmeter body. Fig. 8. The distribution of the module of velocity v [m/s] in the fluid: the variant with 90 ◦. Fig. 9. The pressure distribution p [Pa] in the fluid: the variant with 90 ◦. Fig. 10. The dynamic viscosity μ p distribution in the fluid: the variant with 90◦. P. Drexler et al. Flow Measurement and Instrumentation 78 (2021) 101876 8 (27), (37), (38), (51), and (52). The coupled model, solved via the sequential method, was initialized and controlled with the help of the APDL language in ANSYS [10]. We computed the flowmeter models for the velocity distribution within the ranges v =0.5–12 m/s and v =0.01–0.5 m/s. The pipe walls were simulated such that the pressure losses and other parameters exhibited the following values: K =0.003 m -1 , C = 0.999–0.989, D h =0.05 m, a =0.013, b =0.25, ρ =998 kg m −3 , μ p = 0.001 kg⋅m −1 s −1 , ν =1⋅10 −6 m 2 s −1 , T 0 =20 ◦C. To analyze the magnetic field, we set I c =100 Az. Further, for the given purpose, μ r1 =8000 was used in the magnetic yoke; μ r2 =1000 in the external magnetic shielding; and μ r3 =1 in air and other diamagnetic materials. The models enabling us to analyze the current field were solved at I Lstatic =171 mA, γ =1 S m −1 . 5. Model evaluation In the magnetic field analysis, we evaluated the flux density B and the field intensity H; the module H is shown in Fig. 7. The main characteristics of the turbulent flow and the distribution of the velocity, pressure, and dynamic viscosity are displayed in Figs. 8–10; all of the results were evaluated for the mean velocity value of v =0.2 m/s. As regards the evaluation of the voltage on the terminals of the electrodes E1, E2, the procedure depends on the mean value of the fluid velocity v and the pressure losses in the pipe, without considering the impact of the velocity on the current density distribution J [29]. The voltage is given as UL≅∑ NΩ e=1(Je IL ×Be)⋅veΔVe,or (65) UL≅1 2IL∑ NΩ e=1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∑ N+ i=1 q+ i⎛ ⎜ ⎝v+ ok,iv+ ok,i  (vm,i 2+v+ ok,i 2) √⎞ ⎟ ⎠ +∑ N− j=1 q− j⎛ ⎜ ⎝v− ok,iv− ok,i  (vm,j 2+v− ok,j 2) √⎞ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦e ×Be ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ⋅(v+ ok +v− ok +vm)e (66) The last formula (66) is required to respect the physical-chemical composition of the fluid. The ion composition of potable water is indicated in Table I (volume density m io ) and Table II (molar mass M mo ). According to Refs. [24–36], we can modify equation (66) upon considering the condition of neutrality and regular distribution of ions in the fluid. We have UL≅1 2IL∑ NΩ e=1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ q+ e⎛ ⎜ ⎝v+ ok,iv+ ok,i  (vm,e 2+v+ ok,e 2) √⎞ ⎟ ⎠+ q− e⎛ ⎜ ⎝v− ok,iv− ok,i  (vm,e 2+v− ok,e 2) √⎞ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ×Be ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⋅(v+ ok,e+v− ok,e+vm,e) (67) v+ ok,e=Eeγ FcΔVe∑ Nion+ k=1 c+ kN+ion k ,v− ok,e=Eeγ FcΔVe∑ Nion− k=1 c− kN−ion k ,(68) q+ e=FcΔVe∑ Nion+ k=1 c+ kN+ion k,(69) q− e=FcΔVe∑ Nion− k=1 c− kN−ion k,(70) where F c is the Faraday constant, F c =96,484 C mol −1 ; E e denotes the electric field intensity in the direction of the ions’ motion in an element of mesh; c + represents the positive ions’ concentration; c - stands for the negative ions’ concentration; ΔV e is the element volume; N k +ion denotes the integer multiple of the electron charge for a specific positive ion; N k - ion is the integer multiple of the electron charge for a specific negative ion; q e − represents the whole charge of negative ions in one element; q e + is the whole charge of positive ions in one element; N ion+ stands for the number of different positive charge carriers (elements, compounds); and Table 1 The physical-chemical composition of the fluid: the volume density. Substance Volume density m io [mg⋅dm −3 ] Positive ions Na I+ 32.71 K I+ 1.53 Mg II+ 43.81 Ca II+ 157.70 Negative ions F I− 1.58 Cl I− 5.35 SO 4 II− 13.08 NO 3 I− 0.54 HCO 3 I− 762.40 Neutral substances CO 2 4063 H 2 O 1,000,000 Table 2 The physical-chemical composition of the fluid: the molar mass. Substance Molar mass M mo [g⋅mol −1 ] Positive ions Na I+ 22.990 K I+ 39.102 Mg II+ 24.312 Ca II+ 40.080 Negative ions F I− 19.998 Cl I− 35.453 SO 4 II− 32.064 +4⋅15.999 NO 3 I− 14.007 +3⋅15.999 HCO 3 I− 1.008 +12.011+3⋅15.999 Neutral substances CO 2 12.011 +2⋅15.999 H 2 O 2⋅1.008 +15.999 Table 3 The physical-chemical composition of the fluid: the substance concentration. Substance Concentration c k [mol⋅dm −3 ] Positive ions c 1,Na 1.39192⋅10 −3 c 2,K 3.83612⋅10 −5 c 3,Mg 1.80199⋅10 −3 c 4,Ca 3.93463⋅10 −3 Negative ions c 5,F 8.31648⋅10 −5 c 6,Cl - 1.80904⋅10 −4 c 7,SO4 − 2.04382⋅10 −4 c 8,NO3 8.708989⋅10 −6 c 9,HCO3 − 12.494000⋅10 −3 Neutral substances c CO2 92.3200⋅10 −3 c H2O 2.47038⋅10 5 P. Drexler et al. Flow Measurement and Instrumentation 78 (2021) 101876 9 N ionis the number of different negative charge carriers (elements, compounds). The concentration of ions in the fluid is given by ck=mio,k Mmo,k k=1, .., Nion.(71) The values of the concentration c k in the fluid are summarized in Table III. The above formulas (67) - (71) yield the voltage Fig. 11. The relationship between the flowmeter voltage and the flow velocity: the 90 ◦variant (the results obtained from the numerical model). Fig. 12. The relationships between the flowmeter voltage and the flow velocity compared: the 0 ◦, 45◦, and 90◦versions (the results obtained from the numerical model). Fig. 13. The experimental testing of the flowmeter body: a) open; b) magnetic field verification. Fig. 14. The experimental flowmeter measurement: a) the magnetic field; b) the testing of the impact exerted by the voltage U L on the flow velocity v. P. Drexler et al.