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Relationship between velocity profile and distribution of induced potential for an electromagnetic flow meter
Flow Measurement and Instrumentation Flow Measurement and Instrumentation 18 (2007) 99–105 www.elsevier.com/locate/flowmeasinst
Relationship between velocity profile and distribution of induced potential for an electromagnetic flow meter J.Z. Wang a,∗ , G.Y. Tian b , G.P. Lucas b,∗∗ a Department of Electronic Engineering, Huaihai Institute of Technology, Lianyungang, Jiangsu 222005, PR China b School of Computing and Engineering, University of Huddersfield, Queensgate, Huddersfield HD1 3DH, UK
Received 1 March 2006; accepted 24 March 2006
Abstract This paper investigates the relationship between the induced electric potential and the velocity distribution of the conductive continuous phase in two-phase flows in pipes to which an electromagnetic field is applied, with a view to measuring the continuous phase velocity profile. In order to investigate the characteristics of an electromagnetic flow meter in multiphase flow, an alternating current electromagnetic flow meter was modelled using FEMLAB software. Using the model, electrodes could be placed at any position on the insulating internal surface of the flow meter to satisfy the requirement of measuring the induced potentials at specific locations at the boundary of the flow. The induced electric potential or potential differences from the electrodes were analysed for various simulated flow conditions. The numerical simulation results suggest that electromagnetic flow metering may be an effective novel method for measuring the axial velocity profile of the conducting continuous phase. Furthermore, when combined with the local volume fraction distribution of the continuous phase (obtained, for example, using Electrical Resistance Tomography, also known as ERT), it is expected that the measured continuous phase velocity profile would enable the volumetric flow rate of the continuous phase to be obtained. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Electromagnetic flow meter; Velocity profile; Induced voltage; Electromagnetic inductance tomography; Multiphase flow
1. Introduction There are many potential applications for an electromagnetic flow meter in industrially important multiphase flows in which the continuous phase has a relatively high electrical conductivity (e.g. water) whilst the dispersed phase or phases have a much lower conductivity or may even be insulators (e.g. oil). These include (i) ‘rock particle in water’ flows which occur during the hydraulic transportation of rock in mineral processing applications; (ii) ‘rock cuttings in water based drilling mud’ flows which occur during oil well drilling operations; (iii) oil-in-water flows which occur downhole in offshore oil wells at pressures which are too great to allow dissolved gases to come out of solution; (iv) oil and gas ∗ Corresponding author. Tel.: +86 518 5890107; fax: +86 518 5817480. ∗∗ Corresponding author.
E-mail addresses: [email protected] (J.Z. Wang), [email protected] (G.P. Lucas). c 2007 Elsevier Ltd. All rights reserved. 0955-5986/$ - see front matter doi:10.1016/j.flowmeasinst.2006.03.001
in water flows which can occur at the well-head in oil production operations and (v) solids-in-water flows which occur in wastewater treatment applications. In these, and many other, applications it is frequently important to be able to measure the volumetric flow rates of each of the flowing phases. In the past decade, significant progress has been made in measuring the volumetric flow rate Q d of the dispersed phase in two-phase, water continuous flows using dual-plane electrical resistance tomography (ERT) [1]. At one of the two image planes the distribution of the local mixture conductivity σ is measured via an array of electrodes mounted around the internal circumference of the pipe. The distribution of the local dispersed phase volume fraction α is inferred from the σ distribution using relationships such as that developed by Maxwell [2]. By performing pixel–pixel correlation between the two image planes, the distribution of the local axial velocity u d of the dispersed phase in the flow cross section can also be determined [3]. The volumetric flow rate Q d is then given by
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J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
Z
αu d dA
Qd =
(1)
A
where A represents cross sectional area. Dong et al. [1] put forward a method of dispersed phase transient flow rate measurement based around a dual-plane ERT system and using eigenvalue correlation. They claimed that this kind of eigenvalue extraction method can be used to obtain the dispersed phase velocity accurately. Although significant progress has been made in measuring the properties of the dispersed phase, very little progress has been made in the development of techniques for measuring the volumetric flow rate of the high conductivity continuous phase (which in most cases is water). It is well known that electromagnetic flow meters are very useful for measuring the volumetric flow rate of single phase water. It has also been shown [4] that in vertical, bubbly gas–water flows, electromagnetic flow meters can be used to measure the mean velocity u¯ w of the water. The water volumetric flow rate Q w can then be easily determined since Q w = u¯ w (1 − α)A ¯
(2)
where α¯ is the mean gas volume fraction and is determined using an independent technique such as ERT or by measurement of the mean fluid density. In such vertical, bubbly gas–water flows the water velocity profile is essentially axisymmetric, enabling measurements from the electromagnetic flow meter to be readily interpreted. However, in many other multiphase flow applications, the water velocity profile is not axisymmetric. Such non-axisymmetric velocity profiles have been observed in horizontal and inclined air–water and horizontal oil-in-water flows [5]. For non-axisymmetric water velocity profiles in ‘water continuous’ multiphase flows, the operation of electromagnetic flow meters is poorly understood. Furthermore, there is a demand for investigating novel forms of electromagnetic flow meter based on electromagnetic tomography (or EMT) which could enable the distribution in the flow cross section of the axial velocity u c of the high conductivity continuous phase to be determined for a wide variety of velocity profiles. This would enable the volumetric flow rate Q c of the continuous phase to be determined according to the relationship Z Qc = u c (1 − α)dA (3) A
where the local dispersed phase volume fraction α is determined using ‘conventional’ (e.g. ERT) techniques. In the present study, an electromagnetic flow meter utilising an alternating current (AC) sinusoidal excitation source to excite the magnetic coils of the electromagnetic flow meter is modelled and analysed. The induced potentials and potential differences at various positions in the cross section of the flow meter, under different kinds of axial velocity distribution, are simulated and presented. The numerical simulation uses the following assumptions: (i) The investigation is carried out into the distribution of induced potentials on the insulating internal circumference of the flow meter.
(ii) Different conductivity distributions, corresponding to different local volume fraction distributions of the nonconducting dispersed phase in the conducting continuous phase, are investigated; (iii) A single fluid model (i.e. no slip velocity between the continuous and dispersed phases) is assumed. Different axial velocity profiles are simulated which are similar to those likely to be encountered in horizontal and inclined multiphase flows [6]. Using (i)–(iii) above, the potential on the flow meter boundary is calculated and attempts are made to relate calculated potential differences to the axial velocity distribution of the flow. The model is a valid representation of an electromagnetic flow meter in two-phase flows because the velocity distribution of the discontinuous phase has no effect on the induced potentials. Thus, the induced potentials obtained are those that would be obtained from a two-phase flow in which the continuous phase velocity profile is the same as the ‘single-fluid’ velocity profile used in the model. The rest of the paper is organized as follows. Section 2 introduces the electromagnetic flow meter and its extension for multiphase flow measurement. Section 3 presents different simulation results and a discussion. Section 4 concludes the work and proposes further work. 2. Theory and modelling 2.1. Basic theory of electromagnetic flow meters Electromagnetic flow meters have been used for several decades, while the basic principle dates from the days of Faraday. The theory of electromagnetic flow meters belongs to the subject of magnetohydrodynamics, formed by the combination of the classical disciplines of fluid mechanics and electromagnetism. Fluid moving in a magnetic field experiences an electromotive force acting in a direction perpendicular both to the motion and to the magnetic field. The conventional form of an electromagnetic flow meter with a transverse magnetic field is shown in Fig. 1(a). But more electrode pairs were introduced in the present research as shown in Fig. 1(b). The distribution of the potential U inside the pipe of the electromagnetic flow meter is described by a Poisson type equation [7,8] of the form ∇ 2 U = div(v × B)
(4)
where v is the fluid velocity vector and B is the magnetic flux density vector. Making the assumption of a rectilinear flow and a uniform magnetic field, Shercliff [7] derived a solution to Eq. (4) for an electromagnetic flow meter with a nonconducting inner wall and the assumption of a homogeneous liquid in which the conductivity is uniform: 1U A A0 = Bvm d =
4B Q πd
(5)
where 1U A A0 and d are the potential difference and distance between electrode A and electrode A0 in Fig. 1, respectively.
J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
(a) Single pair electrodes.
101
meter boundary and the conductivity distribution within that boundary. The virtual current j weights the effect at electrodes of elemental B × v generators at every point in the liquid. In principle, the concept of virtual current permits any flow meter problem to be solved and it also clarifies the functioning of electromagnetic flow and velocity measuring devices. In the case of a uniform and isotropic suspension twophase flow, Bernier and Brennen [4] investigated the use of electromagnetic flow meters in two-phase flows. They concluded that a homogeneous two-phase flow, consisting of a conducting, continuous liquid phase and a non-conducting discontinuous phase, would give rise to a potential difference 1UTP of the form given in Eq. (7) 1UTP =
1USP 4B Q L = . πd(1 − α) 1−α
(7)
Here, Q L is the flow rate of the conducting continuous liquid phase, 1UTP is the potential difference between the electrodes for two-phase flow, 1USP is for single phase liquid flow only (at the same flow rate Q L as in the two-phase flow) and α is the volume fraction of the non-conducting phase. Wyatt [9] verified this result further. Eq. (7) shows that for an air-in-water flow the electromagnetic flow meter would respond only to the mean water velocity and its output would be independent of the velocity of the air. However, the presence of the air increases 1 the water velocity by a factor 1−α . (b) Multi electrodes. Fig. 1. Transverse field flow meter geometries.
The term vm is the average velocity in the cross section and Q is the volumetric flow rate. 1U A A0 in Eq. (5) is independent of the electrical conductivity, viscosity, and pressure of the liquid. Shercliff [7] also proposed a weight function that describes the contribution of the velocity, in different parts of the flow meter cross section, to the total output signal. This weight function shows that the effect of velocity is strong near the electrodes and decreases with increasing distance from the electrodes. The weight function therefore describes spatial variations in meter sensitivity which is a function of the magnetic flux density and the size, shape and positions of the electrodes. Bevir [8] extended the weight function concept of Shercliff to a three-dimensional ‘weight vector’. Bevir’s solution of the flow meter equation is given by Z Z Z 1U A A0 = v • Wdτ (6) τ
where W is called the weight vector and weights the contribution to 1U A A0 of the velocity v at every point. W is given by W = B × j. The integral is taken over the flow meter volume τ . The weight vector is dependent on the geometry of the meter and the form of its electrodes. The term j is a hypothetical current density known as the virtual current (j can be thought of as the current density that would be set up in stationary liquid by passing unit current density into one electrode and extracting it from the other). When j is formulated, it mathematically defines the electrodes, the flow
2.2. Model of electromagnetic flow meter In order to calculate the induced potential of an electromagnetic flow meter a geometrical model was designed as shown in Fig. 2. The sub-figure Fig. 2(a) is the x-y view of the flow meter. The symbol θ denotes the angle of an electrode on the circumference of the pipe relative to the x axis. The model assumes a flow meter made from a PTFE pipe and two identical copper excitation coils. The fluid media flow in the pipe. The outermost cylindrical surface shown in Fig. 2(d) is used to limit the computing domain and has a diameter of 0.12 m. The inner diameter of the flow meter is 0.053 m and the outer diameter is 0.065 m. The length of the flow meter is 0.15 m. The inner and outer diameters of the two excitation coils are 0.065 m and 0.075 m respectively. The model is 3-dimensional and Figs. 2(b)–(d) present the y–z, x–z and 3-D views respectively. Since AC excitation of the coils is used, the magnetic potential vector A must satisfy the following equation: jωσ A + ∇ × (µ−1 ∇ × A) = Je
(8)
where ω is the angular frequency, σ is the conductivity, and µ is the permeability. Je denotes the externally applied current density. The relations between fields and potentials are given by B=∇ ×A
(9)
and H = µ−1 B.
(10)
This model uses the following parameter values. The AC source frequency is 50 Hz; the permeability of free space is
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component of B in the x–y plane, x–z plane and y–z plane respectively. Fig. 3(d) denotes distribution of the y component of the magnetic flux density B along the x-axis from x = −0.0265 to 0.0265 m. The mean value of the magnitude of the y component of magnetic flux density B in the flow cross section is about 2.5 × 10−3 T (25 G) according to Fig. 3(d). 3.2. Induced electric potential distribution (a) x-y view.
The next stage is to impose an axial flow velocity distribution on the system (note that in the context of this paper ‘axial flow’ refers to velocity in z direction only). The initial distribution of the axial velocity vz that was investigated is given by the expression vz = 5 + 200y
(b) y-z view.
(c) x-z view.
(d) 3D view. Fig. 2. Geometrical model and coordinate system of flow meter.
µ0 = 4π × 10−7 V s/A m; the conductivity of the PTFE in the flow meter is 1 × 10−15 S m−1 . The two excitation coils, through the imposed external currents, generate the magnetic field. The current in each coil is Je = − √
zJ0 x2
+
z2
i + 0j + √
x J0 x 2 + z2
k
(11)
where J 0 is a constant value of current equal to 1.5 × 107 A. 3. Simulation results Different velocity distributions in combination with different conductivity variations have been simulated. 3.1. Magnetic flux density According to the geometrical model, physical model and the preconditions given in Section 2, the distributions of the y component of the magnetic flux density B on the x–y, x–z and y–z planes that all contain the origin of the coordinate system are presented in Fig. 3. Figs. 3(a)–(c) show the y
(12)
which means that vz varies from −0.3 ms−1 at y = −0.0265 m to 10.3 ms−1 at y = +0.0265 m. (NB: An (almost) linearly increasing axial flow velocity, varying from a slightly negative value at the lower side of an inclined pipe to a large positive value at the upper side of the pipe, has been experimentally observed in a multiphase flow by one of the authors and is reported in [3].) The resultant variation of induced electric potential within the system is shown plotted in Fig. 4. In this figure, induced electrical potential U is plotted for x = −0.0265 to 0.0265 m for various values of y. It should be noted that for values of |y| > 0 some of the potentials shown in Fig. 4 will lie outside of the flowing fluid. Nevertheless there is an obvious relationship, within the fluid, that for a given value of y, the induced electric potential U increases almost linearly with x. Furthermore the quantity dU/dx within the fluid (i.e. the rate that the induced electrical potential increases with increasing x) is larger for higher values of y, where the axial flow velocity vz is higher. Thus from Fig. 4 the general conclusion can be drawn that, for a given value of y, it may be possible to relate the potential difference between points on the internal circumference of the PTFE wall of the flow meter to the axial flow velocity vz for that value of y. For the axial velocity distribution given by Eq. (12) the distribution of induced potential in the planes z = 0 and y = 0 is shown in Figs. 5(a) and (b) respectively. These figures illustrate the relationship between the induced electric potential and the velocity of fluid. 3.3. Comparison of induced electric potentials and potential differences under different flow conditions On the basis of the results described in Section 3.2 above, simulations were performed using the electromagnetic flow meter model described in Section 2, for four different combinations of the distributions of the local axial velocity vz and the local fluid conductivity σ . These combinations are: (i) (ii) (iii) (iv)
vz vz vz vz
= 5 ms−1 and σ = 10−2 S m−1 ; = 5 ms−1 , σ = 10−2 × (1 + 35y) S m−1 ; = 5 + 200y ms−1 , σ = 10−2 S m−1 ; = 5 + 200y ms−1 , σ = 10−2 × (1 + 35y) S m−1 .
J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
(a) Magnetic flux density (T ) on x-y plane.
(c) Magnetic flux density (T ) on y-z plane.
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(b) Magnetic flux density (T ) on x-z plane.
(d) Magnetic flux density (T ) along line y = 0 (in plane z = 0).
Fig. 3. Magnetic flux density distribution of electromagnetic flow meter.
Combinations (i) and (ii) above correspond to a pipe Reynolds number of 2.65 × 105 . Note that fluid conductivities of 10−2 to 2 × 10−2 S m−1 are typical values for mains water in northern England. Note also that the conductivity distributions in (ii) and (iv) above are similar to those that could arise in a horizontal solids-in-water flow where the solids density is greater than that of water. The local volume fraction of the low conductivity solids decreases from the lower side of the horizontal pipe (y = −0.0265 m) to the upper side of the pipe (y = +0.0265 m) causing the mixture conductivity to increase with increasing y [3]. For each of conditions (i)–(iv) above, Fig. 6 shows the simulated, induced electrical potential U at different angular positions on the internal circumference of the PTFE wall of
the flow meter (i.e. at the interface between the wall and the flowing fluid) for the plane defined by z = 0. Angular position is measured anticlockwise from the increasing x-axis. It is clear from Fig. 6 that although the velocity distribution has a marked effect on the distribution of the induced electrical potential, the effect of the conductivity distribution on the induced electrical potential is much less marked. Fig. 7a shows the calculated potential differences between the ends of chords, parallel to the x-axis, which terminate on the internal circumference of the flow pipe. The abscissa in Fig. 7a is the y co-ordinate of each chord. Fig. 7b shows the actual values of the induced potentials at either end of these chords. Positive values of induced potential in Fig. 7b correspond to chord ends for which x > 0 (i.e. for angular positions,
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J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
Fig. 4. Induced electric potential distribution along lines parallel to the x axis at different values of y (z = 0). Axial velocity profile given by Eq. (12).
measured anti-clockwise with respect to the increasing x-axis, from 0◦ to 90◦ and from 270◦ to 360◦ ). Negative values of induced potential correspond to chord ends for which x < 0 (i.e. for angular positions from 90◦ and from 270◦ ). [NB: The results presented in Figs. 7a and 7b are again relevant to the plane z = 0.] It is clear from Fig. 7a that when the axial flow velocity vz is constant at 5 ms−1 the calculated potential differences between the chord ends remain relatively constant irrespective of the y co-ordinate of the chord, except for where |y| is greater than about 0.02 m. This is because chords, parallel to the x-axis, for which |y| is greater than about 0.02 m, are relatively short compared to chords for smaller values of |y|. In the same way that the potential difference across the ends of a metal rod, moving at constant speed through a magnetic field, is proportional to the length of the metal rod, so the potential difference between the chord ends is related to the length of the chord. It is also apparent from Figs. 7a and 7b that the influence of the fluid conductivity σ on the calculated potentials and potential differences is relatively small for constant vz . For the cases where the axial flow velocity vz increases with increasing y, it is clear from Fig. 7a that the induced potential differences between the chord ends also increase with increasing y (except where y greater than about +0.02 m where the chord length is, again, relatively small). The effect of fluid conductivity on the calculated potentials is more marked when vz is not constant, nevertheless the fact that the induced potential differences generally increase with y when the axial velocity vz increases with y suggests that an array of electrodes similar to that shown in Fig. 1(b) may enable the axial velocity profile to be measured. To compensate for the effect of chord length on the induced potential differences, a plot of ‘calculated potential difference divided by chord length’ 1U/C is presented in Fig. 8. It is clear that by plotting 1U/C versus y rather than 1U versus y, the influence of relatively small chord length has been eliminated. Fig. 8 suggests that measurement of 1U/C using an array of
(a) Induced electrical potential (V ) on plane z = 0.
(b) Induced electrical potential (V ) on plane y = 0. Fig. 5. Induced electric potential distribution at planes z = 0 and y = 0 (axial velocity profile given by Eq. (12)).
electrodes similar to that shown in Fig. 1(b) could indeed enable axial flow velocity distributions to be measured. 4. Conclusions and further work Numerical simulations on the use of electromagnetic flow meters to measure axial velocity profiles similar to those encountered in multiphase flows have been performed. The induced electric potentials and potential differences at various positions on the internal circumference of the flow pipe have been calculated for two kinds of axial velocity distribution and two kinds of fluid conductivity distribution. The important conclusion is that induced potential differences, measured using an array of boundary electrodes, could conceivably be used to infer the axial velocity distribution of the flow. The influence of the conductivity distribution of the fluid on the
J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
Fig. 6. Induced potential along the inner circumference of the flow meter versus angular position with respect to the increasing x-axis.
105
Fig. 8. ‘Potential difference over chord length’ versus y coordinate of chord (z = 0).
reconstruct the axial flow velocity distribution from potential differences measured on the internal circumference of the flow pipe. The contribution of each fluid element or ‘pixel’ to induced potential differences between electrode pairs would be investigated. Numerical simulations would provide design parameters for such a tomographic instrument and would also assist in the development of inverse (reconstruction) algorithms to enable axial velocity distributions to be determined from boundary potential difference measurements. Acknowledgements
Fig. 7a. Induced potential difference between ends of chords parallel to the x-axis versus y coordinate of chord (z = 0).
The authors would like to acknowledge the financial support of the Jiangsu Government Scholarship for Overseas Studies and the Natural Science Research Project of Jiangsu Education Department (Grant No. 06KJB510006). Dr. Jingzhuo Wang would also like to thank the University of Huddersfield, UK and Huaihai Institute of Technology for their support of his work in the UK. References
Fig. 7b. Induced potential at each end of chords parallel to the x-axis versus y coordinate of chord (z = 0).
induced potential differences is relatively small, however the conductivity distribution has greater influence when there is spatial variation in the axial velocity. It is suggested that further work could be performed to investigate whether tomographic techniques could be used to
[1] Dong F, Xu YB, Xu LJ, Hua L, Qiao XT. Application of dual-plane ERT system and cross-correlation technique to measure gas–liquid flows in vertical upward pipe. Flow Measurement and Instrumentation 2005; 16(2–3):191–7. [2] Maxwell JC. A treatise on electricity and magnetism, vol. 1. Oxford: Clarendon Press; 1873. [3] Lucas GP, Cory J, Waterfall RC, Loh WW, Dickin FJ. Measurement of the solids volume fraction and velocity distributions in solids–liquid flows using dual-plane ERT. Flow Measurement and Instrumentation 1999;10(1): 249–58. [4] Bernier RN, Brennen CE. Use of the electromagnetic flow meter in a twophase flow. International Journal of Multiphase Flow 1983;9(3):251–7. [5] Vigneaux P, Chenais P, Hulin JP. Liquid–liquid flows in an inclined pipe. AIChE Journal 1988;34(5):781–9. [6] Lucas GP. Gas volume fraction and velocity profiles. Journal of Visualization 2006;9(4):419–26. [7] Shercliff JA. The theory of electromagnetic flow-measurement. Cambridge: Cambridge University Press; 1962. [8] Bevir MK. The theory of induced voltage electromagnetic flow meters. Journal of Fluid Mechanics 1970;43(3):577–90. [9] Wyatt DG. Electromagnetic flow meter sensitivity with two phase flow. International Journal of Multiphase Flow 1986;12(6):1009–17.
Relationship between velocity profile and distribution of induced potential for an electromagnetic flow meter J.Z. Wang a,∗ , G.Y. Tian b , G.P. Lucas b,∗∗ a Department of Electronic Engineering, Huaihai Institute of Technology, Lianyungang, Jiangsu 222005, PR China b School of Computing and Engineering, University of Huddersfield, Queensgate, Huddersfield HD1 3DH, UK
Received 1 March 2006; accepted 24 March 2006
Abstract This paper investigates the relationship between the induced electric potential and the velocity distribution of the conductive continuous phase in two-phase flows in pipes to which an electromagnetic field is applied, with a view to measuring the continuous phase velocity profile. In order to investigate the characteristics of an electromagnetic flow meter in multiphase flow, an alternating current electromagnetic flow meter was modelled using FEMLAB software. Using the model, electrodes could be placed at any position on the insulating internal surface of the flow meter to satisfy the requirement of measuring the induced potentials at specific locations at the boundary of the flow. The induced electric potential or potential differences from the electrodes were analysed for various simulated flow conditions. The numerical simulation results suggest that electromagnetic flow metering may be an effective novel method for measuring the axial velocity profile of the conducting continuous phase. Furthermore, when combined with the local volume fraction distribution of the continuous phase (obtained, for example, using Electrical Resistance Tomography, also known as ERT), it is expected that the measured continuous phase velocity profile would enable the volumetric flow rate of the continuous phase to be obtained. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Electromagnetic flow meter; Velocity profile; Induced voltage; Electromagnetic inductance tomography; Multiphase flow
1. Introduction There are many potential applications for an electromagnetic flow meter in industrially important multiphase flows in which the continuous phase has a relatively high electrical conductivity (e.g. water) whilst the dispersed phase or phases have a much lower conductivity or may even be insulators (e.g. oil). These include (i) ‘rock particle in water’ flows which occur during the hydraulic transportation of rock in mineral processing applications; (ii) ‘rock cuttings in water based drilling mud’ flows which occur during oil well drilling operations; (iii) oil-in-water flows which occur downhole in offshore oil wells at pressures which are too great to allow dissolved gases to come out of solution; (iv) oil and gas ∗ Corresponding author. Tel.: +86 518 5890107; fax: +86 518 5817480. ∗∗ Corresponding author.
E-mail addresses: [email protected] (J.Z. Wang), [email protected] (G.P. Lucas). c 2007 Elsevier Ltd. All rights reserved. 0955-5986/$ - see front matter doi:10.1016/j.flowmeasinst.2006.03.001
in water flows which can occur at the well-head in oil production operations and (v) solids-in-water flows which occur in wastewater treatment applications. In these, and many other, applications it is frequently important to be able to measure the volumetric flow rates of each of the flowing phases. In the past decade, significant progress has been made in measuring the volumetric flow rate Q d of the dispersed phase in two-phase, water continuous flows using dual-plane electrical resistance tomography (ERT) [1]. At one of the two image planes the distribution of the local mixture conductivity σ is measured via an array of electrodes mounted around the internal circumference of the pipe. The distribution of the local dispersed phase volume fraction α is inferred from the σ distribution using relationships such as that developed by Maxwell [2]. By performing pixel–pixel correlation between the two image planes, the distribution of the local axial velocity u d of the dispersed phase in the flow cross section can also be determined [3]. The volumetric flow rate Q d is then given by
100
J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
Z
αu d dA
Qd =
(1)
A
where A represents cross sectional area. Dong et al. [1] put forward a method of dispersed phase transient flow rate measurement based around a dual-plane ERT system and using eigenvalue correlation. They claimed that this kind of eigenvalue extraction method can be used to obtain the dispersed phase velocity accurately. Although significant progress has been made in measuring the properties of the dispersed phase, very little progress has been made in the development of techniques for measuring the volumetric flow rate of the high conductivity continuous phase (which in most cases is water). It is well known that electromagnetic flow meters are very useful for measuring the volumetric flow rate of single phase water. It has also been shown [4] that in vertical, bubbly gas–water flows, electromagnetic flow meters can be used to measure the mean velocity u¯ w of the water. The water volumetric flow rate Q w can then be easily determined since Q w = u¯ w (1 − α)A ¯
(2)
where α¯ is the mean gas volume fraction and is determined using an independent technique such as ERT or by measurement of the mean fluid density. In such vertical, bubbly gas–water flows the water velocity profile is essentially axisymmetric, enabling measurements from the electromagnetic flow meter to be readily interpreted. However, in many other multiphase flow applications, the water velocity profile is not axisymmetric. Such non-axisymmetric velocity profiles have been observed in horizontal and inclined air–water and horizontal oil-in-water flows [5]. For non-axisymmetric water velocity profiles in ‘water continuous’ multiphase flows, the operation of electromagnetic flow meters is poorly understood. Furthermore, there is a demand for investigating novel forms of electromagnetic flow meter based on electromagnetic tomography (or EMT) which could enable the distribution in the flow cross section of the axial velocity u c of the high conductivity continuous phase to be determined for a wide variety of velocity profiles. This would enable the volumetric flow rate Q c of the continuous phase to be determined according to the relationship Z Qc = u c (1 − α)dA (3) A
where the local dispersed phase volume fraction α is determined using ‘conventional’ (e.g. ERT) techniques. In the present study, an electromagnetic flow meter utilising an alternating current (AC) sinusoidal excitation source to excite the magnetic coils of the electromagnetic flow meter is modelled and analysed. The induced potentials and potential differences at various positions in the cross section of the flow meter, under different kinds of axial velocity distribution, are simulated and presented. The numerical simulation uses the following assumptions: (i) The investigation is carried out into the distribution of induced potentials on the insulating internal circumference of the flow meter.
(ii) Different conductivity distributions, corresponding to different local volume fraction distributions of the nonconducting dispersed phase in the conducting continuous phase, are investigated; (iii) A single fluid model (i.e. no slip velocity between the continuous and dispersed phases) is assumed. Different axial velocity profiles are simulated which are similar to those likely to be encountered in horizontal and inclined multiphase flows [6]. Using (i)–(iii) above, the potential on the flow meter boundary is calculated and attempts are made to relate calculated potential differences to the axial velocity distribution of the flow. The model is a valid representation of an electromagnetic flow meter in two-phase flows because the velocity distribution of the discontinuous phase has no effect on the induced potentials. Thus, the induced potentials obtained are those that would be obtained from a two-phase flow in which the continuous phase velocity profile is the same as the ‘single-fluid’ velocity profile used in the model. The rest of the paper is organized as follows. Section 2 introduces the electromagnetic flow meter and its extension for multiphase flow measurement. Section 3 presents different simulation results and a discussion. Section 4 concludes the work and proposes further work. 2. Theory and modelling 2.1. Basic theory of electromagnetic flow meters Electromagnetic flow meters have been used for several decades, while the basic principle dates from the days of Faraday. The theory of electromagnetic flow meters belongs to the subject of magnetohydrodynamics, formed by the combination of the classical disciplines of fluid mechanics and electromagnetism. Fluid moving in a magnetic field experiences an electromotive force acting in a direction perpendicular both to the motion and to the magnetic field. The conventional form of an electromagnetic flow meter with a transverse magnetic field is shown in Fig. 1(a). But more electrode pairs were introduced in the present research as shown in Fig. 1(b). The distribution of the potential U inside the pipe of the electromagnetic flow meter is described by a Poisson type equation [7,8] of the form ∇ 2 U = div(v × B)
(4)
where v is the fluid velocity vector and B is the magnetic flux density vector. Making the assumption of a rectilinear flow and a uniform magnetic field, Shercliff [7] derived a solution to Eq. (4) for an electromagnetic flow meter with a nonconducting inner wall and the assumption of a homogeneous liquid in which the conductivity is uniform: 1U A A0 = Bvm d =
4B Q πd
(5)
where 1U A A0 and d are the potential difference and distance between electrode A and electrode A0 in Fig. 1, respectively.
J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
(a) Single pair electrodes.
101
meter boundary and the conductivity distribution within that boundary. The virtual current j weights the effect at electrodes of elemental B × v generators at every point in the liquid. In principle, the concept of virtual current permits any flow meter problem to be solved and it also clarifies the functioning of electromagnetic flow and velocity measuring devices. In the case of a uniform and isotropic suspension twophase flow, Bernier and Brennen [4] investigated the use of electromagnetic flow meters in two-phase flows. They concluded that a homogeneous two-phase flow, consisting of a conducting, continuous liquid phase and a non-conducting discontinuous phase, would give rise to a potential difference 1UTP of the form given in Eq. (7) 1UTP =
1USP 4B Q L = . πd(1 − α) 1−α
(7)
Here, Q L is the flow rate of the conducting continuous liquid phase, 1UTP is the potential difference between the electrodes for two-phase flow, 1USP is for single phase liquid flow only (at the same flow rate Q L as in the two-phase flow) and α is the volume fraction of the non-conducting phase. Wyatt [9] verified this result further. Eq. (7) shows that for an air-in-water flow the electromagnetic flow meter would respond only to the mean water velocity and its output would be independent of the velocity of the air. However, the presence of the air increases 1 the water velocity by a factor 1−α . (b) Multi electrodes. Fig. 1. Transverse field flow meter geometries.
The term vm is the average velocity in the cross section and Q is the volumetric flow rate. 1U A A0 in Eq. (5) is independent of the electrical conductivity, viscosity, and pressure of the liquid. Shercliff [7] also proposed a weight function that describes the contribution of the velocity, in different parts of the flow meter cross section, to the total output signal. This weight function shows that the effect of velocity is strong near the electrodes and decreases with increasing distance from the electrodes. The weight function therefore describes spatial variations in meter sensitivity which is a function of the magnetic flux density and the size, shape and positions of the electrodes. Bevir [8] extended the weight function concept of Shercliff to a three-dimensional ‘weight vector’. Bevir’s solution of the flow meter equation is given by Z Z Z 1U A A0 = v • Wdτ (6) τ
where W is called the weight vector and weights the contribution to 1U A A0 of the velocity v at every point. W is given by W = B × j. The integral is taken over the flow meter volume τ . The weight vector is dependent on the geometry of the meter and the form of its electrodes. The term j is a hypothetical current density known as the virtual current (j can be thought of as the current density that would be set up in stationary liquid by passing unit current density into one electrode and extracting it from the other). When j is formulated, it mathematically defines the electrodes, the flow
2.2. Model of electromagnetic flow meter In order to calculate the induced potential of an electromagnetic flow meter a geometrical model was designed as shown in Fig. 2. The sub-figure Fig. 2(a) is the x-y view of the flow meter. The symbol θ denotes the angle of an electrode on the circumference of the pipe relative to the x axis. The model assumes a flow meter made from a PTFE pipe and two identical copper excitation coils. The fluid media flow in the pipe. The outermost cylindrical surface shown in Fig. 2(d) is used to limit the computing domain and has a diameter of 0.12 m. The inner diameter of the flow meter is 0.053 m and the outer diameter is 0.065 m. The length of the flow meter is 0.15 m. The inner and outer diameters of the two excitation coils are 0.065 m and 0.075 m respectively. The model is 3-dimensional and Figs. 2(b)–(d) present the y–z, x–z and 3-D views respectively. Since AC excitation of the coils is used, the magnetic potential vector A must satisfy the following equation: jωσ A + ∇ × (µ−1 ∇ × A) = Je
(8)
where ω is the angular frequency, σ is the conductivity, and µ is the permeability. Je denotes the externally applied current density. The relations between fields and potentials are given by B=∇ ×A
(9)
and H = µ−1 B.
(10)
This model uses the following parameter values. The AC source frequency is 50 Hz; the permeability of free space is
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component of B in the x–y plane, x–z plane and y–z plane respectively. Fig. 3(d) denotes distribution of the y component of the magnetic flux density B along the x-axis from x = −0.0265 to 0.0265 m. The mean value of the magnitude of the y component of magnetic flux density B in the flow cross section is about 2.5 × 10−3 T (25 G) according to Fig. 3(d). 3.2. Induced electric potential distribution (a) x-y view.
The next stage is to impose an axial flow velocity distribution on the system (note that in the context of this paper ‘axial flow’ refers to velocity in z direction only). The initial distribution of the axial velocity vz that was investigated is given by the expression vz = 5 + 200y
(b) y-z view.
(c) x-z view.
(d) 3D view. Fig. 2. Geometrical model and coordinate system of flow meter.
µ0 = 4π × 10−7 V s/A m; the conductivity of the PTFE in the flow meter is 1 × 10−15 S m−1 . The two excitation coils, through the imposed external currents, generate the magnetic field. The current in each coil is Je = − √
zJ0 x2
+
z2
i + 0j + √
x J0 x 2 + z2
k
(11)
where J 0 is a constant value of current equal to 1.5 × 107 A. 3. Simulation results Different velocity distributions in combination with different conductivity variations have been simulated. 3.1. Magnetic flux density According to the geometrical model, physical model and the preconditions given in Section 2, the distributions of the y component of the magnetic flux density B on the x–y, x–z and y–z planes that all contain the origin of the coordinate system are presented in Fig. 3. Figs. 3(a)–(c) show the y
(12)
which means that vz varies from −0.3 ms−1 at y = −0.0265 m to 10.3 ms−1 at y = +0.0265 m. (NB: An (almost) linearly increasing axial flow velocity, varying from a slightly negative value at the lower side of an inclined pipe to a large positive value at the upper side of the pipe, has been experimentally observed in a multiphase flow by one of the authors and is reported in [3].) The resultant variation of induced electric potential within the system is shown plotted in Fig. 4. In this figure, induced electrical potential U is plotted for x = −0.0265 to 0.0265 m for various values of y. It should be noted that for values of |y| > 0 some of the potentials shown in Fig. 4 will lie outside of the flowing fluid. Nevertheless there is an obvious relationship, within the fluid, that for a given value of y, the induced electric potential U increases almost linearly with x. Furthermore the quantity dU/dx within the fluid (i.e. the rate that the induced electrical potential increases with increasing x) is larger for higher values of y, where the axial flow velocity vz is higher. Thus from Fig. 4 the general conclusion can be drawn that, for a given value of y, it may be possible to relate the potential difference between points on the internal circumference of the PTFE wall of the flow meter to the axial flow velocity vz for that value of y. For the axial velocity distribution given by Eq. (12) the distribution of induced potential in the planes z = 0 and y = 0 is shown in Figs. 5(a) and (b) respectively. These figures illustrate the relationship between the induced electric potential and the velocity of fluid. 3.3. Comparison of induced electric potentials and potential differences under different flow conditions On the basis of the results described in Section 3.2 above, simulations were performed using the electromagnetic flow meter model described in Section 2, for four different combinations of the distributions of the local axial velocity vz and the local fluid conductivity σ . These combinations are: (i) (ii) (iii) (iv)
vz vz vz vz
= 5 ms−1 and σ = 10−2 S m−1 ; = 5 ms−1 , σ = 10−2 × (1 + 35y) S m−1 ; = 5 + 200y ms−1 , σ = 10−2 S m−1 ; = 5 + 200y ms−1 , σ = 10−2 × (1 + 35y) S m−1 .
J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
(a) Magnetic flux density (T ) on x-y plane.
(c) Magnetic flux density (T ) on y-z plane.
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(b) Magnetic flux density (T ) on x-z plane.
(d) Magnetic flux density (T ) along line y = 0 (in plane z = 0).
Fig. 3. Magnetic flux density distribution of electromagnetic flow meter.
Combinations (i) and (ii) above correspond to a pipe Reynolds number of 2.65 × 105 . Note that fluid conductivities of 10−2 to 2 × 10−2 S m−1 are typical values for mains water in northern England. Note also that the conductivity distributions in (ii) and (iv) above are similar to those that could arise in a horizontal solids-in-water flow where the solids density is greater than that of water. The local volume fraction of the low conductivity solids decreases from the lower side of the horizontal pipe (y = −0.0265 m) to the upper side of the pipe (y = +0.0265 m) causing the mixture conductivity to increase with increasing y [3]. For each of conditions (i)–(iv) above, Fig. 6 shows the simulated, induced electrical potential U at different angular positions on the internal circumference of the PTFE wall of
the flow meter (i.e. at the interface between the wall and the flowing fluid) for the plane defined by z = 0. Angular position is measured anticlockwise from the increasing x-axis. It is clear from Fig. 6 that although the velocity distribution has a marked effect on the distribution of the induced electrical potential, the effect of the conductivity distribution on the induced electrical potential is much less marked. Fig. 7a shows the calculated potential differences between the ends of chords, parallel to the x-axis, which terminate on the internal circumference of the flow pipe. The abscissa in Fig. 7a is the y co-ordinate of each chord. Fig. 7b shows the actual values of the induced potentials at either end of these chords. Positive values of induced potential in Fig. 7b correspond to chord ends for which x > 0 (i.e. for angular positions,
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Fig. 4. Induced electric potential distribution along lines parallel to the x axis at different values of y (z = 0). Axial velocity profile given by Eq. (12).
measured anti-clockwise with respect to the increasing x-axis, from 0◦ to 90◦ and from 270◦ to 360◦ ). Negative values of induced potential correspond to chord ends for which x < 0 (i.e. for angular positions from 90◦ and from 270◦ ). [NB: The results presented in Figs. 7a and 7b are again relevant to the plane z = 0.] It is clear from Fig. 7a that when the axial flow velocity vz is constant at 5 ms−1 the calculated potential differences between the chord ends remain relatively constant irrespective of the y co-ordinate of the chord, except for where |y| is greater than about 0.02 m. This is because chords, parallel to the x-axis, for which |y| is greater than about 0.02 m, are relatively short compared to chords for smaller values of |y|. In the same way that the potential difference across the ends of a metal rod, moving at constant speed through a magnetic field, is proportional to the length of the metal rod, so the potential difference between the chord ends is related to the length of the chord. It is also apparent from Figs. 7a and 7b that the influence of the fluid conductivity σ on the calculated potentials and potential differences is relatively small for constant vz . For the cases where the axial flow velocity vz increases with increasing y, it is clear from Fig. 7a that the induced potential differences between the chord ends also increase with increasing y (except where y greater than about +0.02 m where the chord length is, again, relatively small). The effect of fluid conductivity on the calculated potentials is more marked when vz is not constant, nevertheless the fact that the induced potential differences generally increase with y when the axial velocity vz increases with y suggests that an array of electrodes similar to that shown in Fig. 1(b) may enable the axial velocity profile to be measured. To compensate for the effect of chord length on the induced potential differences, a plot of ‘calculated potential difference divided by chord length’ 1U/C is presented in Fig. 8. It is clear that by plotting 1U/C versus y rather than 1U versus y, the influence of relatively small chord length has been eliminated. Fig. 8 suggests that measurement of 1U/C using an array of
(a) Induced electrical potential (V ) on plane z = 0.
(b) Induced electrical potential (V ) on plane y = 0. Fig. 5. Induced electric potential distribution at planes z = 0 and y = 0 (axial velocity profile given by Eq. (12)).
electrodes similar to that shown in Fig. 1(b) could indeed enable axial flow velocity distributions to be measured. 4. Conclusions and further work Numerical simulations on the use of electromagnetic flow meters to measure axial velocity profiles similar to those encountered in multiphase flows have been performed. The induced electric potentials and potential differences at various positions on the internal circumference of the flow pipe have been calculated for two kinds of axial velocity distribution and two kinds of fluid conductivity distribution. The important conclusion is that induced potential differences, measured using an array of boundary electrodes, could conceivably be used to infer the axial velocity distribution of the flow. The influence of the conductivity distribution of the fluid on the
J.Z. Wang et al. / Flow Measurement and Instrumentation 18 (2007) 99–105
Fig. 6. Induced potential along the inner circumference of the flow meter versus angular position with respect to the increasing x-axis.
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Fig. 8. ‘Potential difference over chord length’ versus y coordinate of chord (z = 0).
reconstruct the axial flow velocity distribution from potential differences measured on the internal circumference of the flow pipe. The contribution of each fluid element or ‘pixel’ to induced potential differences between electrode pairs would be investigated. Numerical simulations would provide design parameters for such a tomographic instrument and would also assist in the development of inverse (reconstruction) algorithms to enable axial velocity distributions to be determined from boundary potential difference measurements. Acknowledgements
Fig. 7a. Induced potential difference between ends of chords parallel to the x-axis versus y coordinate of chord (z = 0).
The authors would like to acknowledge the financial support of the Jiangsu Government Scholarship for Overseas Studies and the Natural Science Research Project of Jiangsu Education Department (Grant No. 06KJB510006). Dr. Jingzhuo Wang would also like to thank the University of Huddersfield, UK and Huaihai Institute of Technology for their support of his work in the UK. References
Fig. 7b. Induced potential at each end of chords parallel to the x-axis versus y coordinate of chord (z = 0).
induced potential differences is relatively small, however the conductivity distribution has greater influence when there is spatial variation in the axial velocity. It is suggested that further work could be performed to investigate whether tomographic techniques could be used to
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