1 2
Reconstruction of Velocity Profiles in Axisymmetric and Asymmetric Flows using an Electromagnetic Flow Meter
3
László E. Kollár1, Gary P. Lucas2, Yiqing Meng2
4
1Institute
of Technology, University of West Hungary, Károlyi Gáspár tér 4, Szombathely,
5
H-9700, Hungary
6
Corresponding author:
[email protected]
7
2School
of Computing and Engineering, University of Huddersfield, Queensgate, HD1 3DH, UK
8 9
Abstract
10
An analytical method that was developed formerly for the reconstruction of velocity profiles in
11
asymmetric flows is improved to be applicable for both axisymmetric and asymmetric flows. The
12
method is implemented in Matlab, and predicts the velocity profile from measured electrical potential
13
distributions obtained around the boundary of a multi-electrode electromagnetic flow meter (EMFM).
14
Potential distributions are measured in uniform and non-uniform magnetic fields, and the velocity is
15
assumed as a sum of axisymmetric and polynomial components. The procedure requires three steps.
16
First, the Discrete Fourier Transform (DFT) is applied to the potential distribution obtained in a uniform
17
magnetic field. Since the direction of polynomial components of order greater than two in the plane of
18
the pipe cross section is not unique multiple solutions exist, therefore all possible polynomial velocity
19
profiles are determined. Then, the DFT is applied to the potential distribution obtained in a specific
20
non-uniform magnetic field, and used to calculate the exponent in a power-law representation of the
21
axisymmetric component. Finally, the potential distribution in the non-uniform magnetic field is
22
calculated for all of the possible velocity profile solutions using weight values, and the velocity profile
23
with the calculated potential distribution which is closest to the measured one provides the optimum
24
solution. The method is validated by reconstructing two quartic velocity profiles, one of which includes
25
an axisymmetric component. The potential distributions are obtained from simulations using COMSOL
26
Multiphysics where a model of the EMFM is constructed. The reconstructed velocity profiles show
27
satisfactory agreement with the input velocity profiles. The main benefits of the method described in
28
this paper are that it provides a velocity distribution in the circular cross section of a pipe as an
29
analytical function of the spatial coordinates which is suitable for both axisymmetric and asymmetric
30
flows.
1
1 2
Keywords: axisymmetric flow, Discrete Fourier Transform, electromagnetic flow measurement,
3
asymmetric flow, velocity profile
4 5
1. Introduction
6
Conventional electromagnetic flow meters (EMFMs) can provide accurate measurements of the
7
volumetric flow rate of conducting fluids in axisymmetric flows. However, the accuracy of conventional
8
EMFMs is considerably reduced in flows where the axial velocity profile is not axisymmetric for
9
example: (i) single phase flow just downstream of a pipe bend or (ii) multiphase flow in an inclined
10
pipe consisting of a conducting continuous phase carrying a non-conducting dispersed phase of
11
different density. In many multiphase metering applications involving such multiphase flows it is often
12
required to measure the volumetric flow rate of the conducting continuous phase and this can be
13
performed by integrating the product of the local volume fraction and the local axial velocity of this
14
phase in the flow cross section [Leeungculsatien and Lucas (2013)]. Various techniques, including
15
Electrical Resistance Tomography [Lucas et al (1999)], exist to enable the distribution of the local
16
volume fraction of the conducting continuous phase to be measured but there are very few techniques
17
enabling the local axial velocity distribution of this phase to be measured. A further feature of such
18
inclined multiphase flows is that the velocity profile of the continuous phase exhibits significant
19
stratification which arises from variations of the local fluid mixture density in the cross section. For
20
example in an upward inclined solids-in-water flow in which the density of the solids particles is
21
greater than that of water, the axial water velocity may be in the upward direction at the upper side of
22
the inclined pipe, will change continuously across the pipe and be in the downward direction at the
23
lower side of the inclined pipe [Leeungculsatien and Lucas (2013)].
24
Horner et al. (1996) proposed an approach using EMFMs that is applicable to measure
25
volumetric flow rate with high accuracy in non-uniform single phase flows. They added additional pairs
26
of electrodes to a conventional EMFM that has only one pair of electrodes. Xu et al. (2001) also
27
proposed a multi-electrode EMFM that measures volumetric flow rate accurately in single phase flows
28
and which was claimed to be insensitive to the flow velocity profile. Cao et al. (2014) optimized the
29
shape of excitation coils in order to reduce the impact of the flow profile.
2
1
The drawbacks of the EMFMs listed above is that in multiphase flows they are not applicable to
2
the determination of the local axial velocity distribution of the conducting continuous phase, which is a
3
requirement for measuring the volumetric flow rate of this phase as described above. Therefore
4
further effort had to be made to develop EMFMs where the flow induced potentials depend on the flow
5
pattern, thus enabling the axial flow velocity profile to be reconstructed. Xu et al. (2004) developed a
6
modified filtered backprojection algorithm to reconstruct velocity profiles in non-axisymmetric flows.
7
Sakuratani & Honda (2010) used the weight vector corresponding to water level in the pipe to
8
reconstruct the flow field in partially filled pipes. Leeungculsatien & Lucas (2013) proposed a design
9
and a method that provided the axial velocity in 7 subdomains of a pipe section. The method
10
presented in Kollár et al. (2014) determined velocity profiles as a function of spatial coordinates in the
11
pipe section, however it could not distinguish between two different axisymmetric velocity profiles
12
(including a uniform velocity distribution) with the same mean velocity. Thus, it was most suitable for
13
asymmetric flows exhibiting significant velocity stratification as described above. A novel technique to
14
reconstruct different axisymmetric velocity profiles was presented in Zhang & Lucas (2013). The key
15
idea of the present paper is the addition of this technique to the method of Kollár et al. (2014) so that
16
it becomes applicable for both axisymmetric and asymmetric flows. An important novelty of this new
17
method from the practical point of view is that it makes the user’s decision about the applicability
18
easier, because a preliminary assessment of whether the flow is axisymmetric is not necessary.
19
Furthermore, if the flow is axisymmetric, the method provides a power-law approximation of the
20
velocity profile rather than determining only its mean velocity. The method reconstructs the velocity
21
distribution of a conducting fluid in a single-phase flow, or that of the conducting continuous phase in
22
a two-phase flow in a circular pipe section.
23
As stated above, very few alternative techniques exist for non-invasive, on-line imaging of the
24
axial velocity profile of the conducting continuous phase in multiphase process flows. One of the most
25
promising alternative techniques is Magnetic Resonance Imaging (Flow MRI) [Gladden et al. (2013)],
26
but this requires the use of a large magnet and can be very sensitive to interference from external
27
magnetic fields and so may not be suitable for many process industry applications. Ultrasonic
28
techniques can be used to measure liquid velocity profiles in single phase flows [Nichols et al. (2011)]
29
and ultrasound Doppler methods are very useful when the liquid is seeded with small particles which
30
provide reflective surfaces [Stener et al. (2014)]. However the use of ultrasound for measuring the
3
1
local axial velocity profile of the continuous phase in multiphase flows, in which there is a high in-situ
2
volume fraction of the dispersed phase (e.g. gas or solid particles) and hence a high number of large
3
reflective surfaces, is questionable.
4
The present paper describes how the analytical method proposed in Kollár et al. (2014) can be
5
extended to reconstruct velocity profiles in both axisymmetric and asymmetric flows. The method
6
provides a velocity profile in a circular pipe section as a superposition of polynomials up to 6th order
7
and a power-law representation of an axisymmetric component. The theoretical background of this
8
paper focuses on the calculation of the axisymmetric velocity component in terms of the mean velocity
9
of this component and a power law exponent. The application of the method is demonstrated via
10
examples where two quartic velocity profiles are reconstructed. One of these velocity profiles includes
11
an axisymmetric component whereas the other one has a uniform component instead.
12 13
2. Electromagnetic flow meter and magnetic fields
14
The EMFM and the magnetic fields considered in the simulations are described in detail in Kollár
15
et al. (2014), and summarized briefly in this section. The EMFM consists of a non-conductive flow
16
pipe mounted within a Helmholtz coil (see Fig. 1a). The inner diameter of the pipe is 80 mm, and the
17
inner and outer diameters of the two coils forming the Helmholtz coil are 204.8 mm and 255 mm,
18
respectively. The potential distribution is measured by means of 16 electrodes that are placed at
19
angular intervals of 22.5 degrees on the pipe circumference as shown in Fig. 1b.
nonconductive pipe
conductive fluid
16 electrodes on pipe circumference
y
Helmholtz coils
pipe section
boundary of computational domain
20 21
x
(a)
(b)
4
1
Fig. 1: Electromagnetic flow meter; (a) geometry and computational domain; (b) position of
2
electrodes (adopted from Kollár et al. 2014)
3 4
The uniform magnetic field is generated by applying current of equal magnitude in the same
5
direction in the two coils so that a magnetic flux density of 0.01 T is obtained. The magnetic flux
6
density vector points in the –y direction. The non-uniform magnetic field is generated by switching the
7
direction of the current in one of the coils. Thereby, currents of equal magnitude flow in opposite
8
directions in the two coils, and the maximum of the y component of the magnetic flux density in the
9
created magnetic field is 0.005 T. This maximum magnetic flux density occurs at the position y = –R
10
(i.e. –40 mm in the case considered), and it decreases in the positive y direction so that it reaches 0 T
11
at the x axis, and –0.005 T at the position y = +R. Note that in this magnetic field, the magnetic flux
12
density vector also has a non-negligible x component [Kollár et al. (2014)].
13 14
3. Calculation of axisymmetric velocity component from measured potential distribution
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The reconstruction method proposed in the present paper requires three steps. First, possible
16
velocity profiles are determined as a superposition of polynomial components up to 6th order. These
17
possible solutions do not include an axisymmetric component but assume a uniform component
18
instead, which can be considered as a limiting case. This uniform component is replaced by an
19
axisymmetric component in the second step. Then in the third step an optimum solution is chosen
20
among the possible solutions, each of which consists of the sum of an axisymmetric component and
21
polynomial components of order 1 up to 6. This section describes the second step in detail. The first
22
and third steps are described in Kollár et al. (2014), but Section 4 also summarizes the complete
23
procedure.
24 25 26
3.1 Procedure to determine axisymmetric velocity component The nth order polynomial velocity component is assumed in the form if the order of polynomial n
27
is odd
28
v n (x; y ) = an,n
(x cosθ
(x cosθQ,n + y sinθQ,n ) + y sinθQ,n ) + an,n−2 n R R n −2
n −2
n
Q,n
29
+ K + an,1
x cosθQ,n + y sinθQ,n R (3.1)
5
1
or, if n is even,
2
v n (x; y ) = an,n
3
where R is the pipe radius. The coefficients an,n , an,n −2 , …, are related to the magnitude of the nth
4
order velocity component in the pipe section, and θ Q,n denotes the angle of direction with respect to
5
the x-axis along which this component is defined. The velocity component vn only changes in the
6
direction θQ,n and is constant along lines orthogonal to this direction (see Fig. 2). The coefficients
7
an,n , an,n −2 , …, as well as the angle θ Q,n are determined in the first step of the reconstruction. The
8
reason for the form of the velocity components given in equations (3.1) and (3.2) is that the flow
9
induced potential distribution U n (θ ) associated with the n th order velocity component, obtained in the
10
uniform magnetic field, can be expressed purely in terms of the trigonometric quantities cos(n+1)θ and
11
sin(n+1)θ. Consequently, the nth order velocity component can be determined purely from the (n+1)th
12
DFT component of the potential distribution obtained in the uniform magnetic field as discussed in
13
detail in Kollár et al. (2014). It was also shown in this reference that the maximum order of polynomial
14
that can be reconstructed using a 16-electrode EMFM is 6. The overall velocity profile was then
15
assumed as a superposition of polynomial components as follows
16
v (x; y ) = ∑ v n (x; y )
(x cosθ
(x cosθQ,n + y sinθQ,n ) + y sinθQ,n ) + an,n−2 n R R n −2
n −2
n
Q,n
+ K + an,0
(3.2)
6
(3.3)
n =0
17 18 19
Fig. 2: Definition of the direction θQ,n (adopted from Kollár et al. 2014)
20
6
1
The axisymmetric velocity component is not considered in the above approach. Lucas et al.
2
(2004) proposed a power-law approximation of the axisymmetric velocity component which is a
3
generalisation of the uniform, i.e. 0th order, velocity component and which is applicable to a wide
4
range of single phase and vertical multiphase flows. This component is written in the system of polar
5
coordinates in the following form
6
v (r ) =
7
where the two parameters v and q have to be determined in the reconstruction procedure. The
8
parameter v is the mean value of the axisymmetric velocity component which is identical to the term
9
a00 associated with the uniform (0th order) polynomial component as described in Kollár et al. (2014).
10
a00 is determined in the first step of the reconstruction procedure. This section focuses on the
11
calculation of the exponent q which is 0 if the axisymmetric velocity component is constant in the flow
12
cross section but which increases as the axisymmetric velocity component becomes ‘peakier’ at the
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pipe centre.
q
1 r v 1 − (q + 1)(q + 2) 2 R
(3.4)
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The exponent q can be determined by applying the Discrete Fourier Transform (DFT) to the
15
boundary potential distribution obtained in the non-uniform magnetic field. The imaginary part of the
16
DFT component Im Xˆ (2) of the boundary potential distribution obtained in the non-uniform magnetic
17
field is influenced predominantly by the axisymmetric velocity component as well as by the quadratic
18
velocity component, but the effects of other velocity components are negligible. This will be
19
demonstrated in Section 3.2. [NB: Xˆ (2) is associated with components of the flow induced potential
20
distribution, obtained in the non-uniform magnetic field, which have a wavelength of πR around the
21
inner boundary of the pipe wall]. The effect of the quadratic velocity component on Im Xˆ (2) depends
22
on its angle θQ,2 in the pipe section. The relationship between the DFT component Im Xˆ (2) and the
23
angle θQ,2 is discussed in Section 3.3. Once this relationship is determined, the effect of the quadratic
24
velocity component on the DFT component Im Xˆ (2) can be removed and the remaining part provides
25
information about the exponent q of the axisymmetric velocity component. The calculation of the
26
exponent q from the DFT component Im Xˆ (2) after the removal of the effect of the quadratic velocity
27
component is explained in Section 3.4. Note that the symbol Xˆ will be used for the complex numbers
28
calculated by applying the DFT to the potential distribution obtained in the non-uniform magnetic field.
7
1
This will distinguish them from the complex numbers X that are obtained when a uniform magnetic
2
field is applied.
3 4
3.2 Influence of axisymmetric and polynomial velocity components
5
The influence of the axisymmetric and polynomial velocity components on the imaginary part of
6
the DFT component Im Xˆ (2) will be discussed in this section. First, it will be demonstrated that the
7
DFT component Im Xˆ (2) obtained from the measured potential distribution is not the same for a
8
uniform velocity component and for different power law representations of the axisymmetric velocity
9
component. Then, it will be shown that the only polynomial velocity component whose effect on the
10
DFT component Im Xˆ (2) is not negligible is the quadratic component. The following methodology was
11
applied to achieve this goal. Measurements of potential distribution obtained using the non-uniform
12
magnetic field were simulated using COMSOL Multiphysics [COMSOL (2008)]. The non-uniform
13
magnetic field described in Section 2 and purely axisymmetric or polynomial velocity profiles were
14
defined, and COMSOL determined the potential distribution. The potentials at the 16 electrodes were
15
collected using Matlab and the DFT was applied.
16
In the first series of simulations, the parameters v and q in a purely axisymmetric velocity profile
17
were varied in order to study their effects on the DFT component Im Xˆ (2) . Fig. 3 shows the real and
18
imaginary parts of the DFT components when the mean velocity v is varied and the exponent q =
19
0.5. The imaginary part of the DFT component Im Xˆ (2) is at least 1 to 2 orders of magnitude greater
20
than any other DFT component. Furthermore, it can be observed that this DFT component is linearly
21
proportional to the mean velocity v of the axisymmetric velocity component.
22
23
8
1
Fig. 3: Real and imaginary parts of DFT components for axisymmetric velocity profiles with various
2
values of mean velocity v and for q = 0.5
3 4
Fig. 4 shows the real and imaginary parts of the DFT components when the exponent q is varied
5
and v = 1 m/s. Again, the imaginary part of the DFT component Im Xˆ (2) is dominant, and a decrease
6
in the value of Im Xˆ (2) with increasing q can clearly be observed. This can be explained by the fact
7
that when q increases, the velocity becomes relatively higher at the centre of the pipe where the
8
magnetic flux density is close to 0, whereas it becomes relatively lower close to the pipe wall where
9
the absolute value of magnetic flux density is greater. This relationship will be investigated more fully
10
in Section 3.4. According to the results shown in Figs. 3 and 4, the DFT component Im Xˆ (2) varies
11
significantly with both of the parameters v and q describing the axisymmetric velocity component.
12
Thus, the value of this DFT component obtained from the measured potential distribution cannot be
13
predicted simply by assuming that the axisymmetric velocity component is uniform i.e. q = 0.
14
Moreover, Im Xˆ (2) can be the basis for fully reconstructing the axisymmetric velocity component.
15
16 17
Fig. 4: Real and imaginary parts of DFT components for axisymmetric velocity profiles with various
18
values of exponent q and for v = 1 m/s
19 20
In what follows, it will be demonstrated that apart from the axisymmetric velocity component, only
21
the quadratic velocity component has a significant influence on the DFT component Im Xˆ (2) . In this
22
series of simulations, polynomial velocity components were defined in the form of equation (3.1) or
23
(3.2) in COMSOL. The parameters an,n (n = 1,…,6) were kept constant, and the angles θ Q,n were
9
1
varied between 0 and 360 deg. Fig. 5 presents the variation of the DFT component Im Xˆ (2) with the
2
angle θ Q,n for each of the polynomial velocity components. It can be seen clearly that the effects of the
3
1st, 3rd, 4th, 5th and 6th order polynomial velocity components on the DFT component Im Xˆ (2) are
4
negligible compared to that of the 2nd order polynomial velocity component. Thus, if a relationship
5
between the DFT component Im Xˆ (2) and the parameters describing the quadratic velocity
6
component can be established then the contribution of the quadratic velocity component to the DFT
7
component Im Xˆ (2) can be removed and the remaining part can then be used to determine the
8
values of v and q associated with the axisymmetric velocity component. The remaining part of
9
Section 3 is devoted to this problem.
10
11 12
Fig. 5: The imaginary part of the second DFT component Im Xˆ (2) for various polynomial velocity
13
components with varying angle
14 15
3.3 Influence of Angle of Quadratic Velocity Component
16
A relationship is derived in this section between the contribution of the quadratic velocity
17
component to the DFT component Im Xˆ (2) and the parameters describing the quadratic velocity
18
component. As was shown in Section 3.2, the contribution of velocity components other than the
19
axisymmetric and the quadratic to Im Xˆ (2) is negligible and therefore this DFT component can be
20
considered as being composed only from the contributions from these two velocity components as
21
follows:
22
Im Xˆ (2) = Im Xˆ (2 )0 + Im Xˆ (2)2
(3.5)
10
1
where Im Xˆ (2)0 and Im Xˆ (2)2 refer to the contributions of the axisymmetric and quadratic velocity
2
components respectively. According to equation (3.2), the quadratic velocity component is written in
3
the form:
(x cosθ
+ y sinθ Q,2 ) + a20 R2 2
4
v 2 (x; y ) = a22
5
The parameter a20 is calculated from a22 during the first step of the reconstruction procedure (see
6
also Kollár et al., 2014) thus the influence of the two parameters a22 and θQ,2 on Im Xˆ (2)2 has to be
7
studied. Furthermore, the maximum of the y component of magnetic flux density Bop (which is
8
measured at the position –R as discussed in Section 2) and the pipe radius R also have effect on the
9
potential distribution, and thereby on the calculated DFT components. Therefore, these parameters
10
Q, 2
(3.6)
must also appear in the expression for Im Xˆ (2)2 .
11
Similar to the procedure presented in the previous section, measurements of potential
12
distribution in the non-uniform magnetic field were simulated using COMSOL Multiphysics. Different
13
quadratic velocity profiles were defined by varying the parameters a22 and the angle θQ,2 . The value
14
of Bop describing the maximum flux density of the non-uniform magnetic field was also changed in
15
some of the simulations. Then, the DFT was applied for the potentials obtained at the 16 electrodes
16
using Matlab and a relationship was obtained by fitting a function to the data calculated from the
17
results of the COMSOL simulation.
18
A trigonometric relationship was observed between the DFT component Im Xˆ (2)2 and the angle
19
θQ,2 . Moreover the DFT component Im Xˆ (2)2 was found to be linearly proportional to the term a22 and
20
also to the maximum magnetic flux density Bop and the pipe radius R. The DFT component Im Xˆ (2)2
21
can be normalized by defining a parameter A2 ,θ
22
A2,θ =
23
and a relationship between A2 ,θ and θQ,2 , the direction of the quadratic velocity component (see Fig.
24
2), was established as follows.
25
A2,θ = ac 2 cos 2θ Q,2 + as 2 sin 2θQ,2 + b2
Im Xˆ (2)2 Bop Ra22
(3.7)
(3.8)
11
1
The constants in the relationship (3.8) were determined, after data fitting, as follows: ac 2 = 0.0099; as 2
2
= 0; and b2 = –0.0172. Note that the value of A2 ,θ is independent of the value of the term a22 , the
3
maximum magnetic flux density Bop and the pipe radius R. A2 ,θ is shown plotted against θQ,2 in Fig. 6.
4
It should be noted that this relationship is valid for other flow meters that are geometrically similar to
5
that modelled in the present study but for which the pipe radius is different. However, the constants
6
will be different if the flow meter is not geometrically similar to that considered in this study. Using
7
equations 3.7 and 3.8, and knowing a22 , θQ,2 , Bop and R, the contribution Im Xˆ (2)2 to Im Xˆ (2) can be
8
readily determined [note that a22 and θQ,2 are calculated during the first step of the reconstruction
9
procedure (section 4.1) whilst for a given EMFM Bop and R can easily be measured].
10 11
Fig. 6: The dependence of parameter A2,θ on the angle of the quadratic velocity component θQ,2 for
12
different values of velocity a22 and magnetic flux density Bop
13 14
3.4 Empirical Relationship to Determine the Exponent q in the Power-Law Approximation of the
15
Axisymmetric Velocity Component
16
The previous section explained how to determine the contribution of the quadratic velocity
17
component to the imaginary part of the DFT component Im Xˆ (2)2 . In practice, the imaginary part of
18
the DFT component Im Xˆ (2) is obtained from the measured potential distribution, thus the
19
contribution of the axisymmetric velocity component Im Xˆ (2)0 to Im Xˆ (2) can be calculated from
20
equation (3.5). If a relationship between the exponent q and Im Xˆ (2)0 can now be found then the
21
axisymmetric velocity component can be reconstructed in the form of equation (3.4). Zhang & Lucas
22
(2013) proposed a procedure to determine this relationship. Using COMSOL they simulated
23
axisymmetric velocity profiles with varying exponent q, in the same non-uniform magnetic field
12
1
described in section 2, to obtain the boundary potential distribution in the pipe section. They defined a
2
parameter Aq that can be calculated from the values of potential at the electrode positions, more
3
precisely, from the difference of potentials at electrodes 3 and 5 (for electrode positions see Fig. 1b).
4
Then, they fitted a function to the data representing the relationship between the exponent q and the
5
parameter Aq . The present study follows the same procedure with the exception that the parameter
6
Aq is defined using the DFT component Im Xˆ (2)0 instead of the electrode potentials as follows:
7
Aq =
8
If the calculations are performed with sufficient accuracy then the two methods provide the same
9
values for Aq . The accuracy of the calculation can be improved by increasing the values of potentials,
10
and thereby those of potential differences, and this may be achieved by considering greater pipe
11
radius, magnetic flux density or velocity. Therefore, most of the COMSOL simulations used in the
12
present study for deriving empirical relationships were carried out with mean velocities v of 10 m/s or
13
100 m/s, which are otherwise unrealistically high in such pipe flows.
14
2 Im Xˆ (2)0 Bop Rv
(3.9)
The q versus Aq relationship obtained from the simulations described above were found to be
15
closely approximated by a 2nd order polynomial as given in Fig. 7 and by equation (3.10):
16
q = 10.79 Aq2 − 13.32 Aq + 3.73
17
where Aq,max is the maximum value of Aq which occurs for q = 0. Aq,max can be obtained by solving
18
equation (3.10) after substituting q = 0 giving Aq,max = 0.429 for the flow meter geometry presented
19
here. If an Aq value greater than Aq,max is ever calculated from equation (3.9), then it is assumed that
20
q = 0, i.e. the axisymmetric velocity component is uniform. The lower limit for Aq is theoretically 0 but
21
the relationship in equation 3.10 was only verified for a domain of practical interest i.e. 0 ≤ q ≤ 1 . The
22
value of q = 1 is obtained for Aq = 0.260, which may be considered the value of Aq obtained for the
23
lower limit of the range of practical interest. If Aq takes a lower value, or in other words if q is greater
24
than 1, then the velocity profile forms an unrealistically high peak at the centre of the pipe [Zhang &
25
Lucas (2013)].
0 < Aq ≤ Aq ,max
13
(3.10)
1 2
Fig. 7: The q versus Aq relationship as obtained empirically from COMSOL simulations
3 4
4. Procedure of the proposed reconstruction method
5
This section summarizes the three steps of the procedure for reconstruction of velocity profiles.
6
The first and the third steps are explained in detail in Kollár et al. (2014), whereas the second step is
7
based on the discussion in Section 3.
8 9
4.1 Finding possible velocity profiles with polynomial components
10
In the first step, possible velocity profiles are determined as a superposition of polynomial
11
components up to 6th order. It should be noted that reconstructed velocity profiles containing
12
components higher than 4th order display unrealistic spatial variations. Therefore, the authors have
13
decided to limit the highest order velocity component to 4th order in practical applications [Kollár et al.
14
(2014)].
15 16
• The nth order polynomial component of the velocity profile is assumed in the form of (3.1) if n is odd or in the form of (3.2) if n is even.
17
~ • The overall potential distribution U (θ ) , measured in the uniform magnetic field for the velocity
18
profile that it is required to reconstruct, can be written as the sum of a series of components
19
U n (θ ) given by
20
U n (θ ) = K n [cos nθQ,n cos(n + 1)θ + sin nθQ,n sin(n + 1)θ ]
21
where
22
Kn =
1
(n + 1)2
n
(4.1)
(4.2)
an,n BR
14
1
and 0 ≤ n ≤ n max . As was mentioned in Section 3.1 and in Kollár et al. (2014), the maximum
2
allowable value of n is 6 for a 16-electrode EMFM.
3
~ • The measured potential distribution U (θ ) is discretized in order to obtain the potentials Up (p =
4
0,…,N-1) at the positions of the measurement electrodes. The DFT of the series Up then provides
5
a series of N complex numbers as follows
6
X (n ) =
N −1
∑ U exp(− j (2π np / N ))
1 N
p
n = 0,1,..., N − 1
(4.3)
p =0
7
where N is the number of samples. The complex numbers X(n), n = 1,…,nmax+1, are related to
8
the amplitude and phase of the component U n −1 (θ ) of the potential distribution, which is
9
associated with the (n-1)th velocity component v n −1 (x; y ) .
10
• The coefficient an ,n in equation (3.1) or (3.2) is obtained from one of the following equations:
11
a0,0 = sgn(Re X (1))
12
a1,1 =
13
an ,n = ±
14
8 X (2) BR
(n + 1)2
n +1
BR
X (n + 1)
for the axisymmetric term (n = 0)
(4.4)
for linear term (n = 1)
(4.5)
for quadratic and higher-order terms ( n ≥ 2 )
(4.6)
• The coefficients an,n −2 , …, an,n −2 m in equation (3.1) or (3.2) are calculated from the following
15
equations:
16
an,n −2 = −
n −1 an,n 4
17
an,n −4 = −
n −3 (n + 1)2 4
18
…
19
an,n −2 m = −
20
−
21
2 X (1) BR
(4.7)
n + 1 n−3 an ,n − 2 an ,n −2 2 2
(n + 1 − 2m ) n + 1a (n + 1)2 m 2m
(n + 1 − 2m ) (n + 1 − 2(m − 1))2
2
n ,n
−
(4.8)
(n + 1− 2m ) n − 1 a (n − 1)2 m − 1
n ,n − 2
2 m −2
n + 1 − 2(m − 1) an,n −2 (m −1) 1
where m = n/2 if n is even and m = (n–1)/2 if n is odd.
15
−
(n + 1 − 2m ) n − 3 a (n − 3)2 m − 2 2 m −4
n ,n − 4
− ...
(4.9)
1
• The possible angles of direction of the nth velocity component θ Q,n are determined from the
2
argument ψ n +1 associated with the (n+1)th DFT component X n +1 . If n is odd and an,n > 0:
3
θQ,n = −
4
Note that if n is odd and an,n < 0, then the same solutions are obtained. If n is even and an,n > 0:
5
θQ ,n = −
6
If n is even and an,n < 0:
7
θ Q,n = −
8
Equations 4.10 to 4.12 imply that there exist n possible values of θ Q,n for the n th order
9
polynomial velocity component (where n=1 to a maximum value of 6). Consequently if a velocity
10
profile is composed of the sum of velocity components comprising an axisymmetric component
11
and polynomial components up to a maximum order nmax (where n max ≤ 6 ) there exist n max !
12
possible velocity profile solutions. Section 4.3 explains how the optimum velocity profile solution
13
is selected.
ψ n +1 n
ψ n +1 n
ψ n +1 n
+
+
+
(2k − 1)π (2k + 1)π , n n
2kπ n
θQ,n ∈
(2k − 1)π (2k + 1)π , n n
2k π n
(2k + 1)π n
k = 0,1,…,n–1
(4.10)
θQ,n ∈
k = 0,1,…,n/2–1
(4.11)
k 2π (k + 1)2π θQ,n ∈ , n n
k = 0,1,…,n/2–1
(4.12)
14 15 16
4.2 Finding the axisymmetric velocity component The second step is now to find the axisymmetric velocity component as follows.
17
• The axisymmetric component of the velocity profile is assumed in the form of (3.4), where the
18
mean velocity v is equal to a00 that has already been obtained in the first step (Section 4.1).
19
• The DFT is applied to the potential distribution Uˆ (θ ) , measured in the non-uniform magnetic field
20
for the velocity profile that it is required to reconstruct, to provide the complex numbers Xˆ (n ) , n =
21
0, 1, …, N – 1.
22
• The effect of the quadratic velocity component on the DFT component Im Xˆ (2) is determined
23
from equations (3.7) and (3.8) enabling the calculation of Im Xˆ (2)2 . Im Xˆ (2)2 is then subtracted
24
from Im Xˆ (2) using equation (3.5) to obtain the term Im Xˆ (2)0 .
25
• The parameter Aq is determined from Im Xˆ (2)0 using equation (3.9).
16
1
• The exponent q is determined from Aq using equation (3.10).
2 3
4.3 Choice of optimum velocity profile solution
4
In the third step an optimum velocity profile solution is chosen from all of the possible solutions,
5
which each include an axisymmetric component and polynomial components. First, weight values
6
have to be determined, which can be used in any reconstruction where the same or a geometrically
7
similar EMFM is under consideration. The method to obtain the weight values is described in detail in
8
Kollár et al. (2014). The pipe cross section is divided, for example, into 30 subdomains with areas Ai
9
(i = 1,…,30) and 30 COMSOL simulations are carried out using the non-uniform magnetic field. In
10
each of these simulations the axial velocity in the chosen subdomain (with index i) is set equal to v wt ,i
11
whilst the axial velocity in all of the other subdomains is set equal to zero. The potentials on each of
12
the 16 electrodes are used to generate 15 potential differences Uˆ wt , j (j = 1 to 15) by subtracting the
13
potential at a reference electrode successively from the potentials on each of the remaining
14
electrodes. The weight values w ij associated with the chosen subdomain with index i are then
15
calculated using the expression
16
w ij =
17
This process is repeated for the remaining subdomains, thereby allowing 450 weight values to be
18
calculated. [Note: for a given EMFM geometry these weight values only need to be calculated once,
19
i.e. they don’t have to be recalculated every time the velocity profile reconstruction procedure
20
described herein is used]. The optimum velocity profile is then chosen by the following procedure.
21
πR
⋅
Uˆ wt , j
2Bop Ai v wt ,i
i = 1,...,30;
j = 1,...,15
(4.13)
• The potential distribution in the non-uniform magnetic field is calculated for all of the possible
22
velocity profiles using weight values as follows
23
2Bop Uˆ j = πR
M
v i w ij Ai ∑ i =1
j = 1,..., N − 1
(4.14)
17
1
where Uˆ j is the calculated potential difference on the jth electrode relative to the reference
2
electrode, M is the total number of subdomains and v i is the mean velocity in the ith subdomain
3
as known from the relevant velocity profile under consideration.
4
• Reference potential differences Uˆ j , ref (j = 1 to 15) are determined from the potential distribution
5
Uˆ (θ ) measured in the non-uniform magnetic field for the velocity profile that it is required to
6
reconstruct, by subtracting the value of the potential on the reference electrode from the value of
7
the potential on each of the remaining electrodes. [Note, in practical applications these reference
8
potential differences may be obtained from measurements from a real multi-electrode
9
electromagnetic flow meter – although in this paper they are obtained by using COMSOL to
10
calculate the boundary potential distribution in the non-uniform magnetic field for the reference
11
velocity profile that it is desired to reconstruct, as described in section 5].
12
• Finally, for each of the possible n max ! velocity profile solutions a quantity SU is calculated where 15
13
(
SU = ∑ Uˆ j ,ref − Uˆ j j =1
)
2
(4.15)
14
and the optimum velocity profile from the possible solutions is taken as that for which the quantity
15
SU is a minimum.
16 17
5. Validation of Reconstruction Method
18
The method presented in this paper is validated in this section by reconstructing two quartic
19
velocity profiles and comparing the results to the original velocity profiles. Both of the profiles consist
20
of superposed polynomial components up to 4th order. The difference between them is that one of
21
them includes a uniform component (i.e. an axisymmetric component with exponent q = 0), whereas
22
the other one includes an axisymmetric component with non-zero exponent q. The velocity profiles
23
together with the physical specifications of the EMFM are defined in a COMSOL model that simulates
24
the measurements. The computational domain is shown in Fig. 1a. Note that the pipe was not
25
simulated in its full length in order to reduce computational cost. The simulation produces a potential
26
distribution on the internal circumference of the pipe, which is then used in Matlab where the
27
reconstruction method is implemented. The method is validated by comparing the difference between
28
local velocities in the reconstructed and the original velocity profiles.
18
1
The following 4th-order (quartic) polynomial velocity profiles were used for testing the
2
reconstruction method
3
v (x; y ) = v 0 (x; y ) + v 1 (x; y ) + v 2 (x; y ) + v 3 (x; y ) + v 4 (x; y )
(5.1)
4
v (x; y ) = v axi (x; y ) + v 1 (x; y ) + v 2 (x; y ) + v 3 (x; y ) + v 4 (x; y )
(5.2)
5
where the indices refer to the order of polynomial, and the components are defined as follows
6
v 0 (x; y ) = 1
7
v axi (x; y ) = 1.321 −
8
y x v 1 (x; y ) = 1 cos 90 o + sin 90 o R R
9
y x v 2 (x; y ) = 1 cos 45o + sin 45o − 0.25 R R
x2 + y 2 R
0 .2
2
3
10
y y x x v 3 (x; y ) = 1 cos 20 o + sin 20 o − 0.5 cos 20 o + sin 20 o R R R R
11
y y x x v 4 (x; y ) = 1 cos 0 o + sin 0 o − 0.75 cos 0o + sin 0o + 0.0625 R R R R
12
The axisymmetric component v axi is defined by substituting v = 1 m/s and q = 0.2 into equation (3.4).
13
The velocity profile represented by equation (5.1) was defined in Section 5.1 in Kollár et al. (2014),
14
and the uniform component is replaced by an axisymmetric component in equation (5.2).
4
2
15
The potential distribution in the uniform magnetic field is determined in the COMSOL simulation
16
and presented in Fig. 8a. Theoretically it is equivalent for the two velocity profiles considered, since
17
the potential distribution in the uniform magnetic field is the same for any axisymmetric component
18
with the same mean velocity v . Practically, there may occur some differences in the potential values,
19
i.e. in the range of 1%, due to the numerical simulations where the potential distributions are
20
determined. The potential distributions in the non-uniform magnetic field as determined in the
21
COMSOL simulation are indicated by “COMSOL” in Figs. 8b and 8c for the two velocity profiles.
22
The potential distributions (i.e. those obtained in the uniform and in the non-uniform magnetic
23
fields) are used as inputs to the reconstruction method. The entire procedure is applied for both
24
velocity profiles. First, possible solutions are determined considering polynomial velocity components
25
up to 4th order. Then, the axisymmetric component is calculated, and considered when computing the
19
1
sum of differences SU , as defined by equation (4.15), for each of the 24 possible velocity profile
2
solutions. The optimum velocity profile is the one for which SU has a minimum value. Figs. 8b and 8c
3
also show potential distributions calculated for the chosen optimum velocity profile (indicated by
4
“Closest calculated”), and calculated for the possible solution for which SU has maximum (indicated
5
by “Farthest calculated”). According to these figures, the calculated potential distributions are close to
6
that obtained from COMSOL, but the “Closest calculated” shows the closest agreement.
7 -4
6
x 10
COMSOL 4
U(V)
2
0
-2
-4
-6
0
50
100
150
200
250
300
350
θ (deg)
8 9
(a) -4
2
-4
x 10
2 COMSOL Closest calculated Farthest calculated
1.5
x 10
COMSOL Closest calculated Farthest calculated
1.5
1
0.5
0.5 U (V)
U (V)
1
0
0
-0.5
-0.5
-1
-1
-1.5
0
50
100
150
200
250
300
-1.5
350
0
50
100
150
10
θ (deg)
θ (deg)
11
(b)
(c)
200
250
300
350
12
Fig. 8: Potential distribution of simulated and reconstructed quartic polynomial velocity profiles; (a)
13
uniform magnetic field; (b) non-uniform magnetic field for velocity profile (5.1), adopted from Kollár et
14
al. (2014); (c) non-uniform magnetic field for velocity profile (5.2)
15
20
1
The original and the chosen optimum velocity profiles are shown in Fig. 9. The chosen solutions
2
agree well with the corresponding reference velocity profile in both cases, although some minor
3
differences are visible. Variation of the exponent q in the axisymmetric velocity component may
4
change the velocity profile significantly, but its influence on the potential distribution is relatively minor
5
(or not at all in the uniform magnetic field). It is of interest how precisely the reconstruction method
6
can predict the exponent q. The exponents in the axisymmetric component of the original velocity
7
profiles are q = 0 and q = 0.2 for velocity profiles (5.1) and (5.2), respectively; whereas they are
8
calculated as q = 0 for the velocity profile (5.1), and q = 0.1891 for the velocity profile (5.2). The
9
calculated values represent an error of 0% and 5.5%, respectively.
3
3
2.5
2.5
3
3 2
2
2
2 1.5
1.5
1
1 1
0
1 0
0.5
-1 0.04
-1 0.04
0 0.02
0.04 0.02
0
0 0.02
-0.5
-0.02 -0.04
-0.04
0.04 0.02
0
0
-0.02 y
0.5
-1
-0.02 -0.04
y
x
-0.5
0
-0.02 -0.04
-1 x
10 11
(a)
(b) 3
3
2.5
2.5
3
3
2
2
2
2 1.5
1
1
0
-2 0.04 0.04 0.02
0
-2 0.04
0 0.02
-0.5
0.04
-0.02 -0.04
-0.04
0.02
0
0
-0.02 y (m)
0.5
-1
0 0.02
1
0
0.5
-1
1.5
1
-1
y (m)
x (m)
-0.5
0
-0.02
-0.02 -0.04
-0.04
-1 x (m)
12 13
(c)
(d)
14
Fig. 9: Original and reconstructed velocity profiles, colour bar is in m/s, (a) original, defined by
15
equation (5.1), (b) reconstructed from equation (5.1), (c) original, defined by equation (5.2), (d)
16
reconstructed from equation (5.2)
17 21
A term δv that expresses the mean deviance of local velocities was defined as follows in Kollár et
1 2
al. (2014) in order to evaluate the reliability of the reconstruction method: ~ M
∑ v i − v in ,i
∆v average = ~ i =1 ⋅ 100 % (v in,max − v in,min ) v in ,max − v in ,min M
3
δv =
4
~ Here, M is a number of subregions into which the cross section can be divided and where the
5
velocity is calculated; vin,i is the input velocity in the ith subregion; vin,max and vin,min are, respectively,
6
the maximum and minimum velocities in the input velocity profile. Furthermore, a term δvi that
7
expresses the local error between velocities is defined here as follows:
8
δv i =
9
For the chosen optimum solutions in the two cases, the local error between the velocities is presented
10
in Fig. 10. For the mean deviance δv, the following values were obtained: δv = 2.4% for the velocity
11
profile (5.1), and δv = 2.0% for the velocity profile (5.2). Thus, the value of the term δv and the error in
12
the exponent q in the axisymmetric velocity component are satisfactorily low for the chosen solutions
13
for both velocity profiles. They are well below 10%, which is an acceptable level of error in many
14
industrial multiphase flow applications. A similar conclusion can be drawn for the absolute values of
15
the local errors between velocities in most parts of the pipe section, except in the close proximity of
16
the pipe wall. However, the absolute value of local error does not exceed the range of 10 to 15% even
17
in this region. Consequently, the improved reconstruction method as presented in this paper is
18
applicable for both axisymmetric and asymmetric flows.
(5.3)
v i − v in,i ⋅ 100% v in,max − v in ,min
(5.4)
0.04
5
0.04
0.03
5
0.03
0.02
0.02
0
0 0.01 y (m)
y (m)
0.01 0 -0.01
0 -0.01
-5 -0.02
-0.03
-0.03
-0.04 -0.04
19
-5
-0.02
-0.03
-0.02
-0.01
0 x (m)
0.01
0.02
0.03
0.04
-0.04 -0.04
-10
-0.03
-0.02
-0.01
0 x (m)
0.01
0.02
0.03
0.04
-10
20
Fig. 10: Local percentage error between velocities, colour bar is in %, (a) velocity profile defined by
21
equation (5.1), (b) velocity profile defined by equation (5.2)
22
1 2
In practical applications, measurements are carried out consecutively in the uniform and non-
3
uniform magnetic fields over several seconds in order to reduce the influence of velocity oscillation in
4
time on the measured potential distributions. In steady oil-in-water flows the authors have found that
5
an averaging period of approximately 10 seconds for each of the uniform and non-uniform fields is
6
sufficient to eliminate the effects of random fluctuations in the velocity field on the measured axial
7
water velocity profile. Typical ‘raw’ flow induced signals, obtained in single phase flow using a
8
practical multi-electrode EMFM constructed by the authors and based on Figure 1, are reported in
9
Zhao et al. (2014).
10 11
6. Conclusions
12
A formerly developed analytical method has been improved, which is applicable to reconstruct
13
velocity profiles in axisymmetric and asymmetric flows in a circular pipe section using a multi-
14
electrode EMFM. The method requires two measurements and includes three steps. First, the DFT is
15
applied for a potential distribution measured in a uniform magnetic field, and possible solutions are
16
determined in polynomial form. Then, the DFT is applied for a potential distribution measured in a
17
non-uniform magnetic field to obtain the axisymmetric velocity component. Finally an optimum
18
solution is chosen among the possible solutions using weight values and the potential distribution
19
measured in the non-uniform magnetic field. The peak in the axisymmetric velocity component is
20
expressed mathematically by an exponent in the power-law representation of this component. The
21
imaginary part of the DFT component Xˆ (2) obtained from the measurement in the non-uniform
22
magnetic field provides information about this exponent and therefore the method can distinguish
23
between different axisymmetric velocity components with the same mean velocity. The application of
24
the method for quartic velocity profiles with different axisymmetric velocity components shows that the
25
exponent in the power-law approximation and the mean deviance of local velocities δv are estimated
26
by an error of less than 10% that is acceptable in many industrial multiphase flow applications.
27 28
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25
