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Measurement of velocity profiles in transient single and multiphase flows using inductive flow tomography
Author’s Accepted Manuscript Measurement Of Velocity Profiles In Transient Single And Multiphase Flows Using Inductive Flow Tomography Michael O. Agolom, Gary Lucas, Raymond O. Webilor www.elsevier.com/locate/flowmeasinst
PII: DOI: Reference:
S0955-5986(17)30029-8 http://dx.doi.org/10.1016/j.flowmeasinst.2017.08.010 JFMI1349
To appear in: Flow Measurement and Instrumentation Received date: 26 January 2017 Revised date: 5 June 2017 Accepted date: 16 August 2017 Cite this article as: Michael O. Agolom, Gary Lucas and Raymond O. Webilor, Measurement Of Velocity Profiles In Transient Single And Multiphase Flows Using Inductive Flow Tomography, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2017.08.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Measurement Of Velocity Profiles In Transient Single And Multiphase Flows Using Inductive Flow Tomography Michael O. Agolom1*, Gary Lucas1 and Raymond O. Webilor1 School of Computing and Engineering, University of Huddersfield, Queensgate, HD1 3DH, United Kingdom * Corresponding author:[email protected] 1
ABSTRACT This paper reports on the use of inductive flow tomography (IFT) to study the dynamic flow behaviour that occurs when a control valve on a pipeline is suddenly opened or closed with the flow initially at steady state. Single phase (water) and two phase (oil-in-water) vertical flow conditions were investigated. An electromagnetic flow meter (EMFM) having a 16 electrode array was installed downstream of a control valve. The EMFM generated, sequentially, both uniform and anti-Helmholtz magnetic fields and flow induced potentials proportional to the flow rates of water were measured at the electrode array. A novel IFT image reconstruction algorithm was used to reconstruct the water velocity profile in the pipe cross-section at 2-second time intervals. Velocity profile reconstructions from the EMFM device, both in single phase and multiphase flow, show that when the valve is suddenly opened or closed, the flow downstream of the valve oscillates - with the velocity profile successively changing between a very peaky profile with a much higher than expected velocity at the pipe centre to a velocity profile where the velocity at the pipe centre is much lower than expected. This oscillation occurs until steady state conditions are again reached. It is believed that these novel measurements of transient velocity profiles demonstrate hitherto unseen flow behaviour which may explain some of the damaging effects associated with the phenomenon of ‘water hammer’. Keywords: Transient flow, inductive flow tomography, image reconstruction, electromagnetic flow meter. 1 Introduction Flow variables such as local pressure and velocity remain constant under steady state conditions in a pipeline system. However, events such as starting or stopping a pump, or opening or closing a valve cause the local velocity and pressure to vary with time. When a valve is suddenly closed, the kinetic energy in the fluid can give rise to ‘hydraulic transients’ such as ‘water hammer’ [1] which is often accompanied by a loud banging or hammering noise in the pipeline. Resultant pressure variations accompanying the transient flow can exceed design limits of pipes and fittings and result in noise, vibration and collapse of the pipe and its fittings or even total system failure [1, 2]. The pressure can also drop below the vapour pressure of the liquid and can cause cavitation which is detrimental to the pipe system. A comprehensive understanding of flow velocity profiles under transient conditions can give insight into flow behaviour in transient flows. This can help in the development of transient flow mathematical models which are critical for commercial design and safety of fluid distribution systems in various industrial applications. In [3] and [4] details were published of velocity profiles of fluids in transient horizontal flows which were observed by means of particle image velocimetry (PIV) and ultrasonic velocimetry systems respectively. Important differences in the velocity distributions between the steady state and the transient (unsteady) flow conditions were reported and it was shown that the process of deceleration of flowing fluid when a valve is suddenly closed is highly non-uniform. One of the observations in transient flow conditions was the reversal of velocity within a single profile, with the flow near the pipe wall moving in the opposite direction to the flow around the pipe centre. 1
This paper investigates velocity profiles in transient single phase and multiphase flows using the relatively novel measurement technique of inductive flow tomography (IFT) which requires the use of a multi-electrode electromagnetic flow meter (EMFM). Conventional electromagnetic flow meters have been successfully used in various industrial applications for measuring the volumetric flow rates of conducting fluids in both single and multiphase pipe flows [5] but unfortunately they cannot be used to determine the local velocity distribution across the pipe section. However, recently it has been shown that IFT, using multi-electrode EMFMs, can be used to determine the velocity profile and the mean velocity of the conducting phase in both single phase water flows and water –continuous multiphase flows (e.g. see [6-8]). This paper presents the results of an investigation of transient single phase and multiphase flows using IFT. Section 2 gives a brief summary of the mathematical techniques that were used to reconstruct the water velocity profile from measurements of the flow induced potential distributions obtained from a multi-electrode EMFM in both uniform and anti-Helmholtz magnetic fields. Section 3 presents a description of the IFT system that was used in the present investigation and also describes a multiphase flow facility in which the transient flow experiments were carried out. Section 4 presents sequences of velocity profiles that were measured using the IFT system under three different transient flow conditions. Section 5 presents a simple fluid mechanics model that suggests that the observed velocity profiles measured using the IFT system were plausible.
2. Velocity Profile Reconstruction Technique The multi-electrode EMFM used in this investigation contains a pair of coils forming a Helmholtz coil, a non-conducting flow tube and an array of 16 electrodes (Fig 1). The Helmholtz coil is assumed to generate a uniform magnetic field of flux density B in the y direction in the cross section of the flow tube at the plane of the electrode array. It is further assumed that flow of either, (i) an electrically conducting single phase fluid or (ii) a multiphase mixture where the continuous phase is electrically conducting, occurs in the z direction parallel to the axis of the flow tube. Note that the axial velocity distribution of the conducting phase may be non-uniform in the flow cross section. Flow induced electrical potentials appear at the internal boundary of the flow tube and can be measured using the array of sixteen electrodes shown in figure 1. A method for reconstructing the axial velocity profile of the conducting phase using these flow induced potentials was given in [6, 7] and the key points of this reconstruction technique are summarised below. (i) The axial velocity profile is assumed to be the sum of a series of polynomial velocity components vn ( x , y ) where n 0 to nmax and where if n is odd v n x , y a n ,n ... a n ,1
x cos Q ,n y sin Q ,n n Rn
a n ,n2
x cos Q ,n y sin Q ,n R
x cos Q ,n y sin Q ,n n2 R n2
(1)
and if n is even, 2
v n x , y a n ,n
x cos Q,n y sin Q,n n Rn
a n ,n2
x cos Q,n y sin Q,n n2 R n2
a n ,0
.
(2)
In equations (4) and (5) R is the internal radius of the flow tube and Q ,n is an angle defining the direction of the nth polynomial velocity component, that is to say, vn ( x , y ) changes in the direction of Q ,n but is constant along any line normal to the direction of
Q ,n . The terms an ,n 2 … an ,1 , an ,0 can all be expressed in terms of an ,n as shown in [6]. The overall velocity profile in the cross section of the flow tube is equal to the sum of the velocity components and is given by
v x , y
n max
vn ( x , y )
n0
(3)
where nmax represents the highest order of polynomial velocity component under consideration. It was shown in [6] that for a 16 electrode system the maximum allowable ~ value for nmax is equal to 6. The flow induced potential distribution U ( ) at the inner boundary of the flow tube caused by the interaction of vx , y with the uniform magnetic
field can be discretized into a series U p ( p 0 to N 1 ) where U p represents the measured flow induced potential at the ( p 1 )th electrode in the array (using the electrode numbering convention shown in figure 1) and where N represents the total number of electrodes in the array [note: in this study N 16 ]. If the discrete Fourier transform (DFT) of the series U p is now taken it results in a series of N complex numbers X ( n ) where
1 X n N
N -1
∑U pexp j2 np / N
n 0,1,..., N - 1 .
(4)
p0
It is shown in [6] that the term an ,n in equations (1) or (2), associated with the nth polynomial velocity component, is related to the ( n 1 )th term of the DFT of U p by the following expressions: a0,0 sgn ReX (1)
a1,1
2 X (1 ) BR
8 X( 2) BR
an ,n
n 12n 1 X n 1 . BR
(for n 0 )
(5)
(for n 1)
(6)
(for n 2 ).
(7)
3
It is clear from equation (7) that for n 2 two possible values of an ,n can be obtained which are consistent with the measured value of X ( n 1 ) obtained from the boundary potential distribution at the electrode array. It is also shown in [6] (i) how the terms an ,n 2. ... an ,1 , an ,0 may be calculated from a given value of an ,n ; and (ii) how several possible values for the term Q ,n associated with the nth polynomial velocity component may be obtained from the argument of X ( n 1 ) . Using all possible valid combinations of an ,n and Q ,n it is further shown that, for n 1 , n possible solutions exist for the nth polynomial velocity component which are consistent with the measured value of X ( n 1 ) . This in turn means that nmax ! possible solutions exist for the overall velocity profile v( x , y ) which are consistent with the boundary potential distribution U p measured at the electrode array in the uniform magnetic field. In order to select the optimum solution for v( x , y ) from the nmax ! possible solutions the following procedure was adopted in [6]. A computational model was designed using ‘weight values’ [5, 6] which enables the boundary potential distribution Uˆ at the p , pred
electrode array to be predicted for each of the nmax ! possible solutions of v( x , y ) in the presence of a specific ‘anti-Helmholtz’ magnetic field in the flow cross section. This antiHelmholtz magnetic field is generated by allowing dc electrical currents of equal magnitude to flow in coils 1 and 2 (figure 1) in opposite directions. The optimum solution for v( x , y ) is then taken to be that solution for which the measured flow induced potential distribution Uˆ p
at the electrode array in the anti-Helmholtz magnetic field gives the best agreement with Uˆ p , pred .
Y
R
θ
B X
Fig. 1: The geometry of IFT device in the x-y plane Using the technique described above, the zero’th order polynomial velocity component v0 ( x , y ) is constant and equal to the mean axial flow velocity in the cross section of the flow tube. Furthermore, v0 ( x , y ) is numerically equal to the coefficient a0,0 which is calculated 4
using equation (5). However with reference to [7] all axisymmetric velocity profiles with a mean velocity equal to a0,0 must give rise to the same fundamental component X ( 1 ) when the DFT is taken of the of the discretized boundary potential distribution U p obtained in the same uniform magnetic field. Consequently, the zero’th order velocity component v0 ( x , y ) could, with equal validity, be represented by an expression of the form v0 ( x , y )
a0 ,0 2
(1
r q ) ( 1 q )( 2 q ) . R
(8)
Equation (8) represents an axisymmetric velocity profile component with mean velocity a0,0 and with a power law exponent equal to q . The term r in equation (8) represents radial position in the flow cross section and is given by r ( x 2 y 2 )0.5 . [Note: a0,0 represents the ‘area weighted’ mean value of the velocity distribution defined by equation (8). For a single phase flow a0,0 corresponds to the mean phase velocity. However for a multiphase flow the mean velocity of the conducting continuous phase is dependent upon both v( x , y ) (equation (3)) and the local volume fraction distribution of this phase]. Using an expression for v0 ( x , y ) of the form given by equation (8) in the reconstruction algorithm is likely to yield a much more accurate representation of the true velocity profile, particularly in flows where axisymmetric velocity components are known to exist such as ‘well developed’ single phase flows and vertical multiphase flows. Whereas it is impossible to distinguish between different values of the power law exponent q from the flow induced potential distribution obtained in a uniform field this is not the case for an anti-Helmholtz magnetic field. An anti-Helmholtz magnetic field is relatively strong close to the coils but approaches zero at the pipe centre - and consequently, for a given value of the mean flow velocity, the amplitude of the flow induced potentials is relatively high for low values of q (where the axisymmetric velocity profile is relatively flat) and relatively low for high values of q where the velocity is relatively much higher at the pipe centre than close to the pipe walls. Indeed it is shown in [7] that, for a power law velocity component v0 ( x , y ) of the form given in equation (8), a term Aq may be calculated as follows which is uniquely related to q Aq
2 Im( Xˆ ( 2)) Bop Ra0 ,0
.
(9)
In equation (9) Xˆ ( 2 ) is the first harmonic of the DFT of the discretized flow induced potential distribution Uˆ p measured at the electrode array in an anti-Helmholtz magnetic field, Bop is the magnitude of the y component of the anti-Helmholtz field at electrode e13 (figure 1) and a0,0 is the mean velocity associated with v0 ( x , y ) and is obtained from the 5
flow induced potential distribution measured in the uniform magnetic field as described above. With reference to [9] for a particular electromagnetic flow meter geometry the relationship between q and Aq can be readily determined using magneto-hydrodynamic simulation software such as COMSOL [10]. For the flow metering device used in the present study this relationship was calculated to be
q 48.1Aq3 60.61Aq2 30.13 Aq 5.449 .
(10)
Thus, using the value of a0,0 calculated using equation (5) from the DFT coefficient X ( 1 ) obtained in the uniform magnetic field and the value of q obtained from equation (10), the zero’th order component v0 ( x , y ) of the overall velocity profile can now be expressed in the form given in equation (8). Equation (10) is valid for q 0 corresponding to values of Aq 0.429. This range of values for q is satisfactory for flows where the maximum velocity of the conducting continuous phase occurs at, or close to, the pipe centre. However, in the present study, preliminary results suggested the presence of velocity profiles for which the velocity is a minimum at the pipe centre. If the zero’th order component of such a velocity profile is represented using equation (8) with a negative value of q it is clear that as r / R 1 then v0 ( x , y ) . Values of velocity approaching infinity were found likely to crash the computational algorithm, mentioned above, for selecting the optimum velocity profile. Consequently, for values of Aq 0.429 the zero’th order velocity component v0 ( x , y ) was represented by the expression q' r v0 ( x , y ) a0 ,0 1 n1 0.5( q' 1 )( q' 2 )1 R
(11)
where q' takes the fixed value of 0.5. The ‘area weighted’ mean value of the velocity distribution v0 ( x , y ) as defined in equation (11) is still equal to a0,0 and it is clear, with reference to figure 2, that as the term n increases so the depth of the velocity minimum increases at the pipe centre where r / R 0 . However, it is also clear from figure 2 that at r / R 1 (i.e. at the pipe wall) the value of v0 ( x , y ) remains finite.
6
Figure 2: Variation of v0 ( x , y ) with r/R for a0,0 1 , q' 0.5 and values of n in the range 0.1 to 1. By using simulations performed in COMSOL [10] magneto-hydrodynamic software the following relationship between n and Aq was established for the electromagnetic flow meter geometry used in the present study: n 10.627 Aq 4.464 .
(12)
Consequently when Aq calculated using equation (11) was found to be greater than 0.429, the value of n obtained from equation (12) was combined with the ‘area weighted’ mean velocity term a0,0 to give the zero’th order component v0 ( x , y ) of the overall velocity profile in the form shown in equation (11). As should be apparent to the reader from the previous discussion when a value of Aq 0.429 was observed experimentally, the velocity profile v( x , y ), reconstructed using the techniques outlined above, displayed a velocity at the pipe centre which was lower than the velocity closer to the pipe walls. Although the velocity profiles considered in this paper are mainly axisymmetric (see section 4) the authors have shown that the reconstruction technique described above is also capable of accurately reconstructing highly asymmetric velocity profiles with no axisymmetric component present.
3. Experimental Apparatus 3.1 EMFM Flow Meter Body
7
Figure 3: EMFM flow meter body and flow meter casing [11]. Figure 3 shows the EMFM body used in this research work. The non-conducting flow pipe was made from Delrin and had an internal diameter of 80mm. Two identical circular coils were attached on opposite sides of the pipe wall to form a Helmholtz coil. Sixteen stainless steel electrodes were flush mounted at angular intervals of 22.5o on the internal circumference of the pipe, in the plane orthogonal to the pipe axis which contains the axis of the Helmholtz coil. The casing of the flow meter was made from aluminium to avoid distortions to the magnetic field in the pipe cross section and a cable guide was used to help prevent quadrature voltage in the electrode wires. Each coil forming the Helmholtz coil had a mean radius of 120mm and contained 1024 turns of wire. When a working dc voltage of 48.0V was applied to each coil, the coil current was measured to be 1.41315A. With reference to Figure 1, when the working dc current of 1.41315A flowed in both coils, in the clockwise direction when viewed from above, a uniform magnetic field in the y direction was created in the flow cross section with a magnetic flux density of 106.9297G. Measurements with a gaussmeter showed that, at the plane of the electrode array, there were no significant variations in either the magnitude or direction of the magnetic flux density of this uniform field. Again, with reference to Figure 1, when the working dc current of 1.41315A flowed in coil 1 in the clockwise direction (when viewed from above) and in the anti-clockwise direction in coil 2 an anti-Helmholtz magnetic field was created in the flow cross section for which the magnitude Bop of the y component at electrode e13 was measured to be 38.024G. A further description of the EMFM flow meter body is given by [11]. 3.2 Coil Excitation The selection of the coil excitation sequence is important for an EMFM because (i) an excitation sequence involving magnetic field reversal ensures that electrochemical effects at the electrodes are minimised and (ii) the excitation sequence determines the frequency of operation of the device. A hybrid pulse excitation sequence was applied to the EMFM used in this research, which combines the advantages of both ac and dc coil excitation. A control circuit was used for switching a 48.0V power supply to the Helmholtz coil in such a way as to successively generate both the uniform and anti-Helmholtz magnetic fields described 8
above. The control circuit could also reverse the directions of both the uniform and antiHelmholtz magnetic fields. The uniform magnetic field is generated when the coils are fed with current in the same direction while the anti-Helmholtz field is generated when the coil currents are fed in opposite directions. In this research work, a complete excitation cycle consists of (i) the application of a uniform magnetic field in the y direction (stage in Figure 4); (ii) application of a uniform magnetic field in the y direction (stage
); (iii)
application of the anti-Helmholtz field such that the flux density at electrode e13 is in the y direction (stage ); (iv) application of the anti-Helmholtz field such that the flux density at electrode e13 is in the y direction (stage ). After each of the stages described above, the magnetic field is briefly switched off (stages Ic (A)
,
,
in figure 4).
S 2*
S4
S2
and
S 4*
1.41
S3
S1
(a)
S 1*
S3* Time (s)
-1.41
Uniform magnetic field
Anti-Helmholtz magnetic field
B (G) 107
38
(b) Time (s) -38
τc -107
Figure 4: (a) The variation of current , flowing in the top coil (coil 1) with time over one excitation cycle. is positive when current flows in the clockwise direction in coil 1, when viewed from above. (b) The variation of magnetic flux density in the +y direction at e13 with time over one excitation cycle. The coil control circuit functioned by determining the direction of the electrical current in coils 1 and 2 of the Helmholtz coil (figure 1). This was achieved with a bank of eight solid state relays (SSRs) which were used to connect the terminals of each coil to either the 0V side or the +48V side of the dc power supply. The state of each SSR (open or closed) was set using digital outputs from a National Instruments (NI) data acquisition and control card, interfaced to a PC. By opening and closing the appropriate combinations of SSRs both the uniform and anti-Helmholtz magnetic fields could be successively obtained in either the forward and reverse directions.
9
It should be noted, with reference to Figure 4, that after the SSRs were switched between states a transient time c , due to the coil inductance, elapsed before the coil currents and the magnetic field stabilised. In the present study c was equal to . Table 1 shows the duration of the various stages of magnetic field excitation cycle used in the work reported in this paper. It is clear from this table that a complete excitation cycle had a period of . Flow induced potential difference measurements, averaged over the relevant stages of one excitation cycle, were used to reconstruct a single velocity profile image - thereby giving an effective frame rate of images per second. [Note that flow induced potential difference measurements made with the relevant magnetic field reversed had to be multiplied by -1 in order to be useable in the velocity profile reconstruction technique described in section 2]. Table 1: Duration of stages of magnetic field excitation cycle Excitation Period Time (s) = = = = = = 3.3 Measurement of Flow Induced Potentials During all stages of the magnetic field excitation cycle described above the electrical potential at each electrode, relative to the electrical potential at reference electrode e5 (see Fig. 1) was measured using a 15-channel potential difference measurement circuit. The flow induced potential difference between a given electrode and e5 for the mth stage of magnetic field excitation (m 1 or 3) was obtained by subtracting the mean measured potential difference for the (m 1)th stage (when the magnetic field was switched off) from the mean potential difference measured during the mth stage (when the relevant magnetic field was active). The potentials U and Uˆ required for the velocity profile reconstruction technique p
p
described in section 2 were readily derived from the flow induced potential differences measured as described above, with the proviso that the values of U and Uˆ associated with p
p
reference electrode e5 were always set equal to zero. The 15-channel potential difference measurement circuit used to obtain the experimental results presented in section 4 is described in detail in [11] but the main features of each channel were: (i) very low frequency signals, due to electrochemical effects at the electrode-water interface, were attenuated; (ii) high frequency ‘quadrature’ noise, due to inductive coupling between the magnetic fields and the electrode cabling, was significantly attenuated and (iii) the gain of each channel was set to 1000 to ensure that the low level (of the order of tens of V ) potential difference between each electrode and e5 could be measured with sufficient accuracy by the analogue to digital convertors of the NI data acquisition card mentioned above. 3.4 The Oil Water Flow Loop The experiments described in section 4 were performed in either single phase water flows or water continuous, oil-in-water flows using a multiphase flow loop based at the University of Huddersfield.
10
Figure 5: Flow loop schematic diagram. Symbols used in the diagram are described in the text, except ‘VL’ which denotes a manual valve. The flow loop comprises a 2m long, 80mm internal diameter, transparent Acrylic working section (Fig. 5) which can be positioned vertically or which can be inclined to the vertical, although in the present study all of the flows that were investigated were vertical upward. Flow to the working section is supplied by vertical multistage centrifugal pumps (denoted P1 and P2 in Fig. 5) on the oil and water lines. Immediately downstream of the pumps are pneumatic control valves, denoted CV1 and CV2, which are each connected to a separate Proportional + Integral (PI) controller, enabling the oil and water flow rates to be set independently. Downstream of the control valves are turbine flow meters, denoted FM1 and FM2. Signals from FM1 and FM2 are fed back to the PI controllers to enable the set-point oil and water flow rates to be accurately maintained. The water and oil lines meet at the pipe manifold junction (MJ) where the oil and water mix prior to passing into the flow loop working section. The tank system (TS) comprises a wire mesh separator and storage reservoirs for the oil and water. The wire mesh separator is at the inlet of the tank system and separates the water and oil phases before they flow back into their respective storage reservoirs. In the experiments described in section 4 the EMFM was installed on the flow loop working section at a distance of 1.6m from the working section inlet. 4. Experimentally Observed Water Velocity Profiles at Transient Flow Conditions 4.1 Experimental Procedure With the EMFM installed in the flow loop working section as described above, flow velocity profiles were measured, using the reconstruction technique described in section 2, under three sets of flow conditions case(i) to case(iii) as described below. For each case, the flow was allowed to stabilise to steady state conditions and then transient flow was initiated. Transient flow was initiated when the set point of the PI controller of the relevant flow line was suddenly changed. 11
Case(i): With the water pump running, the water line controller set point was equal to
2m 3 hr 1 (corresponding to a mean water velocity at the EMFM of 0.1105ms 1 ) and the flow rate in the flow loop working section was allowed to attain steady state. Then EMFM measurements were initiated at time . The water line controller set point was then instantly set to 10m 3hr 1 (corresponding to a mean water velocity of 0.5526ms 1 ) at t 23s. Case(ii): With the water pump running, the water line controller set point was 10m 3hr 1 (corresponding to a mean water velocity of 0.5526ms 1 ) and the flow rate in the flow loop working section was allowed to attain steady state. EMFM measurements were initiated at time t 0s . The water line controller set point was instantly changed to 0m 3 hr 1 at t 23s . Case(iii): With both the water and oil pumps running, the water line controller set point was equal to 6m 3 hr 1 , the oil line controller set point was equal to 4m 3 hr 1 (corresponding to a mixture superficial velocity of 0.5526ms 1 ). The fluid flow rates in the flow loop working section were allowed to attain steady state. EMFM measurements were initiated at time
t 0s . The oil controller set point was then instantly changed to 0m 3 hr 1 at t 23s , (corresponding to a mean oil velocity of zero and a mean water velocity of 0.3833ms 1 ). 4.2 Experimental Results The experimentally observed velocity profiles obtained for cases (i) to (iii) using the Inductive Flow Tomography System are presented in Figure 6. It should be noted that velocity profiles for t 22s are not shown because they were always very similar to the velocity profile shown for t 22s (for the relevant case). Similarly, velocity profiles for t 32s are not shown because there was very little change in the shape of the velocity profile after this time. The values of vmean shown in figure 6 represent the mean water velocity as measured by the IFT system. vmean is equal to the term a0,0 given by equation (5). Note that for case(iii) the oil and water were always fairly well mixed, giving a relatively homogeneous oil-water mixture. Consequently, for case(iii), vmean also represents a reasonable estimate of the mean oil velocity. The most noteworthy features of the experimental results shown for cases(i), (ii) and (iii) are summarised below. Case(i). For case (i), for t 0 to 22s, the velocity profile was typical of that which would be expected for a well-developed, turbulent, steady state, single phase flow - namely the velocity profile was relatively flat with the velocity at the pipe centre only slightly greater than its value closer to the pipe wall. However, at t 24s , as transient conditions started to occur, the water velocity close to the pipe centre was observed to be much lower than close to the pipe walls – with reverse flow occurring at the pipe centre. For all of the transient flow conditions investigated, similarly shaped velocity profiles were observed to occur when the mean flow velocity was rapidly increasing; note that at t 26s , vmean 0.5407ms 1 which is considerably higher than at t 24s when vmean 0.1190ms 1 .
12
At t 26s , the water velocity at the pipe centre was observed to be much higher than expected relative to the velocity close to the pipe wall. For all of the transient flow conditions investigated in the present study, this shape of velocity profile was observed to occur when the mean flow velocity was undergoing a rapid decrease; note that at t 28s , vmean 0.3670ms 1 which is considerably lower than at t 26s when vmean 0.5407ms 1 .
For t 30s the velocity profile again became typical for that which would be expected for a well-developed turbulent flow, although the value of vmean was now in excess of 0.4ms 1 compared to its initial steady value (for t 22s ) of about 0.12ms 1 . Case(ii). For case (ii) for t 0 to 22s, the velocity profile was again typical of that which would be expected for a well-developed, turbulent, steady state, single phase flow. However at t 24s , as transient conditions started to occur, the water velocity at the pipe centre was observed to be increase from its previous value of about 0.65ms 1 to about 1.25ms 1 . In a similar manner to the situation described for case(i) above, this velocity profile shape - with a relatively high central velocity - occurred when the mean flow velocity vmean was decreasing from a value of 0.4216ms 1 at t 24s to a value of 0.2958ms 1 at t 26s . [Note: it may be tentatively conjectured that the very high flow velocity observed at the pipe centre when the mean flow velocity was reducing may explain some of the effects of ‘water hammer’ which are observed in process systems when valves in pipelines are rapidly closed]. Again, similar to case(i) described above, when vmean was increasing from 0.2958ms 1 at t 26s to 0.3638ms 1 at t 28s the flow velocity at the pipe centre was observed to be much lower than close to the pipe walls (see diagram for case (ii), t 26s on Fig 6). Case(iii). For case(iii), in which the oil flow rate underwent a rapid reduction in an oil-water multiphase flow, the evolution of the water velocity profile with time was very similar to case(ii) (where, in a single phase flow, the water flow rate underwent a rapid reduction). That is to say, when the mean value of the mixture velocity vmean was undergoing a rapid decrease, the associated water velocity profile exhibited a very high value at the pipe centre; and when vmean was undergoing a rapid increase, the measured water velocity at the pipe centre was lower than close to the pipe walls. Time
Case(i)
Case(ii)
13
Case(iii)
Time
Case (i)
Case (ii)
14
Case(iii)
Figure 6: Reconstructed Velocity profiles for cases(i) to (iii) for 22s t 32s . 5. Simple Fluid Mechanic Model of Transient Velocity Profiles An insight into the plausibility of the experimentally observed velocity profiles in single phase flow may be obtained by considering a simplified 2D fluid mechanics model of flow in a pipe of circular cross section, as described below. In this model the following assumptions are made: (i) The axial velocity profile is always axisymmetric and is constrained to have a ‘power law’ shape of the form q
r v vmx 1 R where R is the pipe radius, vmx is the maximum flow velocity at r R and q is the
(13)
appropriate power law exponent. (ii) Changes in the local axial velocity v at radial position r as a result of variations in the P applied axial pressure gradient are subject to the following highly simplified form of the z Navier-Stokes [12] equation
1 v 1 P v r t r r r z
(14)
where is the kinematic viscosity and is the fluid density. (iii) Changes in the mean axial flow velocity v arising from changes in the applied axial P pressure gradient are defined by the following expression z v 1 P 2v 2 f t z D
(15)
where f is a an appropriate value of the ‘Fanning friction factor’ and where D is the pipe diameter. Note that, as shown in [13] for a power law velocity profile of the form shown in 15
equation (1), the maximum velocity vmx and the mean velocity v are related by the expression
vmx 0.5v q 1q 2 .
(16)
Let us suppose that at time t the local axial velocity v0 at radial position r is given by
r v0 vmx ,0 1 R
q0
(17)
but, as a result of changes to the applied pressure gradient, at time t t the local axial velocity at r changes to v1 as given by;
r v1 vmx ,11 R
q1
(18)
Steps 1 and 2 below show how the terms vmx ,1 and q1 in equation 18 can be obtained from the terms vmx ,0 and q0 in equation (17) using the relationships given in equations (14) and (15). Step 1: As a result of changes in the applied pressure gradient, the initial axial velocity v0 at radial position r changes by an amount v after time t where, from equations (14) and (13), v is given by q0 2 q 1 q0 vmx ,0 1 r 0 q0 1 r t P v t 1 1 . R R r R R z
(19)
At each radial position r a new, ‘interim’ local power law exponent q~1 can now be defined whereby
r vmx ,0 1 R
q~1
v0 v .
(20)
From equation (20), the value of q~1 at each radial position can be calculated using
log v0 v log vmx ,0 q~1 . r log1 R
(21)
16
The new value q1 of the power law exponent in equation (18) is now simply obtained by averaging the values of q~1 for all values of radial position r in the flow cross section. Step 2: Let us further suppose that after time t , as a result of changes in the applied pressure gradient, the mean flow axial velocity in the cross section changes from v0 to v1 where
v1 v0 v
(22)
where, from equation (15), v is given by
1 P 2v02 f v t D z
(23)
and v0 is obtained from vmx ,0 by manipulating the relationship given in equation (16). Finally, and again using the relationship given in equation (16), vmx ,1 can now be obtained from the value of v1 given by equation (22). Using the value of q1 obtained in Step 1 above and the value of vmx ,1 obtained in Step 2 above, the new power law velocity profile after time t as defined in equation (18) can now be obtained. By successively applying this procedure for consecutive time intervals t the evolution with time of the axial velocity profile as a result of changes in the applied pressure gradient can be calculated. Using the model described above, the evolution of the calculated velocity profile with time is highly dependent upon the initial velocity profile that is assumed, the properties of the flowing fluid and the imposed pressure gradient. Nevertheless, by assuming plausible values for these variables it is relatively straightforward to reproduce velocity profiles with features that are very similar to those observed in the experiments described in section 4 of this paper. In the example given below water is assumed to flow in a horizontal 80mm diameter pipe. The initial velocity profile is assumed to be ‘power law’ and of the form given in equation (13), with the exponent q equal to 0.15 and the maximum velocity vmx at the pipe centre equal to 0.68ms 1 , corresponding to a mean flow velocity v of 0.55ms 1 . The water density is taken as 1000kgm 3 and its kinematic viscosity is taken as the commonly accepted value of 1 106 m 2s 1 . The imposed pressure gradient is of the form P / z 60cos( t / 2 ) and has a maximum value of 60Pam 1 and a period of 4s. This form
of pressure gradient is intended to simulate the pressure gradient that might be associated with a control valve that is ‘hunting’ for its set-point position.
17
Results from the model using these assumed conditions are shown in Figure 7 from which it is clear that at time t 0 the water velocity profile is comparatively flat and similar to that which might be observed for a fully developed turbulent single phase flow. Between t 0 and 1s the applied pressure gradient is positive (adverse) and the mean flow velocity v steadily decreases. During this period, the velocity at the pipe centre gradually increases relative to its value at the pipe walls culminating in the velocity profile at t 1s shown in figure 7, for which v 0.4788ms 1 and q 0.342 . Between t 1 and 3s the applied pressure gradient is negative (favourable), v gradually increases and the velocity profile gradually reduces at the pipe centre relative to its value close to the pipe walls, culminating in the velocity profile at t 3s shown in figure 7 for which v 0.5ms 1 and q 0.756 .
Whilst not intended to directly replicate any of the flow conditions associated with experimental cases(i) to (iii) given in section 4.2, similarities between the behaviour of the velocity profiles from the model and those observed experimentally suggest that the results obtained from the EMFM, as shown in figure 6, are indeed plausible. This, in turn, increases confidence that inductive flow tomography system described in this paper is capable of accurately measuring transient velocity profiles in both single phase and water continuous multiphase flows.
Figure 7: Axial velocity distribution at t = 0, 1s and 3s for an imposed co-sinusoidal pressure gradient and for the flow conditions given in the text. From the experimental results presented in section 4 it is apparent that the large amplitude oscillatory transients in velocity profile, which occur just after a change is made to the flow controller set point, have a period of no more than 4 seconds. Successive images from the IFT system are only obtained every 2 seconds and so, to examine the velocity profile transients in greater temporal detail, a much faster IFT frame rate will be required. This is the subject of ongoing work currently being undertaken by the authors. 6. Conclusions
18
A novel inductive flow tomography (IFT) system with a frame rate of 0.5 frames per second, has been designed and built and used to experimentally investigate water velocity profiles in steady state and transient single phase flows and multiphase flows in which water was the continuous phase. Measurements from the IFT system in steady state flows were consistent with the velocity profiles that would be expected for fully-developed turbulent flows. In transient flows it was experimentally observed that when the mean flow velocity was decreasing, the water velocity at the pipe centre became relatively much higher than the water velocity close to the pipe walls. Conversely when the mean flow velocity was increasing, the water velocity at the pipe centre became lower than at the pipe walls – even to the extent that reverse flow sometimes occurred at the pipe centre. Comparison with a simple fluid mechanics model of transient flow suggested that these experimentally observed transient velocity profiles were plausible. It is further suggested that the very high flow velocities observed at the pipe centre when the mean flow velocity is reducing may be at least partially responsible for the some of the damaging effects of ‘water hammer’ which are observed when valves in pipelines are rapidly closed. References [1] C. Schmitt, G. Pluvinage, E. Hadj‐Taieb, R. Akid, Water pipeline failure due to water hammer effects, Fatigue & Fracture of Engineering Materials and Structures 29(12) (2006) 1075-1082. doi:10.1111/j.1460-2695.2006.01071.x [2] P.F. Boulos, B.W. Karney, D.J. Wood, S. Lingireddy, Hydraulic Transient Guidelines for Protecting Water Distribution Systems, Journal American Water Works Association 97(5) (2005) 111-124 [3] M. Brito, P. Sanches, R.M. Ferreira, D.I.C. Covas, PIV characterization of transient flow in pipe coils, Procedia Engineering 89(C) (2014) 1358-1365. doi:10.1016/j.proeng.2014.11.458 [4] B. Brunone, B.W. Karney, M. Mecarelli, M. Ferrante, Velocity Profiles and Unsteady Pipe Friction in Transient Flow, Journal of Water Resources Planning and Management 126(4) (2000) 236-244. https://dx.dio.org/10.1061/(ASCE)0733-9496(2000)126:4(236). [5] J.A. Shercliff, The Theory of Electromagnetic Flow-Measurement, Cambridge University Press, Cambridge, UK, 1962. [6] L.E. Kollar, G. Lucas, Z. Zhang, Proposed method for reconstructing velocity profiles using a multi-electrode electromagnetic flow meter, (2014). doi:10.1088/09570233/25/7/075301 [7] L.E. Kollár, G.P. Lucas, Y. Meng, Reconstruction of velocity profiles in axisymmetric and asymmetric flows using an electromagnetic flow meter, Measurement Science and Technology 26(5) (2015) 1-12. doi:10.1088/0957-0233/26/5/055301. [8] O. Lehtikangas, K. Karhunen, M. Vauhkonen, Reconstruction of velocity fields in electromagnetic flow tomography, Philosophical Transactions of the Royal Society of 19
London A: Mathematical, Physical and Engineering Sciences 374(2070) (2016). doi:10.1098/rsta.2015.0334. [9] Y. Meng. Imaging of the water velocity distribution in water continuous multiphase flows using inductive flow tomography (IFT) Ph.D Thesis, University of Huddersfield. UK (2016). [10] COMSOL, AC/DC module user's Guide, http://hpc.mtech.edu/comsol/pdf/ACDC_Module/ACDCModuleUsersGuide.pdf.
2013.
[11] M. Agolom, G. Lucas, Optimisation of the design of the signal processing circuitry for an inductive flow tomography system, Paper presented at the 7th International Symposium on Process Tomography. Dresden, Germany: 2015. [12] R.H. Sabersky, A.J. Acosta, E.G. Hauptmann, A First Course in Fluid Mechanics, 3rd ed., Macmillan Publishing Company, New York, USA, 1989. [13] G.P. Lucas, R. Mishra, N. Panayotopoulos, Power law approximations to gas volume fraction and velocity profiles in low void fraction vertical gas–liquid flows, Flow Measurement and Instrumentation 15(5–6) (2004) 271-283. http://dx.doi.org/10.1016/j.flowmeasinst.2004.06.004.
Highlights
Novel method for measuring velocity profiles using inductive flow tomography Transient velocity profiles measured in single and multiphase flow Large velocity measured at pipe centre when flow rate increases Reverse flow measured at pipe centre when flow rate is decreasing High velocity measured at pipe centre when the flow rate is increasing
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PII: DOI: Reference:
S0955-5986(17)30029-8 http://dx.doi.org/10.1016/j.flowmeasinst.2017.08.010 JFMI1349
To appear in: Flow Measurement and Instrumentation Received date: 26 January 2017 Revised date: 5 June 2017 Accepted date: 16 August 2017 Cite this article as: Michael O. Agolom, Gary Lucas and Raymond O. Webilor, Measurement Of Velocity Profiles In Transient Single And Multiphase Flows Using Inductive Flow Tomography, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2017.08.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Measurement Of Velocity Profiles In Transient Single And Multiphase Flows Using Inductive Flow Tomography Michael O. Agolom1*, Gary Lucas1 and Raymond O. Webilor1 School of Computing and Engineering, University of Huddersfield, Queensgate, HD1 3DH, United Kingdom * Corresponding author:[email protected] 1
ABSTRACT This paper reports on the use of inductive flow tomography (IFT) to study the dynamic flow behaviour that occurs when a control valve on a pipeline is suddenly opened or closed with the flow initially at steady state. Single phase (water) and two phase (oil-in-water) vertical flow conditions were investigated. An electromagnetic flow meter (EMFM) having a 16 electrode array was installed downstream of a control valve. The EMFM generated, sequentially, both uniform and anti-Helmholtz magnetic fields and flow induced potentials proportional to the flow rates of water were measured at the electrode array. A novel IFT image reconstruction algorithm was used to reconstruct the water velocity profile in the pipe cross-section at 2-second time intervals. Velocity profile reconstructions from the EMFM device, both in single phase and multiphase flow, show that when the valve is suddenly opened or closed, the flow downstream of the valve oscillates - with the velocity profile successively changing between a very peaky profile with a much higher than expected velocity at the pipe centre to a velocity profile where the velocity at the pipe centre is much lower than expected. This oscillation occurs until steady state conditions are again reached. It is believed that these novel measurements of transient velocity profiles demonstrate hitherto unseen flow behaviour which may explain some of the damaging effects associated with the phenomenon of ‘water hammer’. Keywords: Transient flow, inductive flow tomography, image reconstruction, electromagnetic flow meter. 1 Introduction Flow variables such as local pressure and velocity remain constant under steady state conditions in a pipeline system. However, events such as starting or stopping a pump, or opening or closing a valve cause the local velocity and pressure to vary with time. When a valve is suddenly closed, the kinetic energy in the fluid can give rise to ‘hydraulic transients’ such as ‘water hammer’ [1] which is often accompanied by a loud banging or hammering noise in the pipeline. Resultant pressure variations accompanying the transient flow can exceed design limits of pipes and fittings and result in noise, vibration and collapse of the pipe and its fittings or even total system failure [1, 2]. The pressure can also drop below the vapour pressure of the liquid and can cause cavitation which is detrimental to the pipe system. A comprehensive understanding of flow velocity profiles under transient conditions can give insight into flow behaviour in transient flows. This can help in the development of transient flow mathematical models which are critical for commercial design and safety of fluid distribution systems in various industrial applications. In [3] and [4] details were published of velocity profiles of fluids in transient horizontal flows which were observed by means of particle image velocimetry (PIV) and ultrasonic velocimetry systems respectively. Important differences in the velocity distributions between the steady state and the transient (unsteady) flow conditions were reported and it was shown that the process of deceleration of flowing fluid when a valve is suddenly closed is highly non-uniform. One of the observations in transient flow conditions was the reversal of velocity within a single profile, with the flow near the pipe wall moving in the opposite direction to the flow around the pipe centre. 1
This paper investigates velocity profiles in transient single phase and multiphase flows using the relatively novel measurement technique of inductive flow tomography (IFT) which requires the use of a multi-electrode electromagnetic flow meter (EMFM). Conventional electromagnetic flow meters have been successfully used in various industrial applications for measuring the volumetric flow rates of conducting fluids in both single and multiphase pipe flows [5] but unfortunately they cannot be used to determine the local velocity distribution across the pipe section. However, recently it has been shown that IFT, using multi-electrode EMFMs, can be used to determine the velocity profile and the mean velocity of the conducting phase in both single phase water flows and water –continuous multiphase flows (e.g. see [6-8]). This paper presents the results of an investigation of transient single phase and multiphase flows using IFT. Section 2 gives a brief summary of the mathematical techniques that were used to reconstruct the water velocity profile from measurements of the flow induced potential distributions obtained from a multi-electrode EMFM in both uniform and anti-Helmholtz magnetic fields. Section 3 presents a description of the IFT system that was used in the present investigation and also describes a multiphase flow facility in which the transient flow experiments were carried out. Section 4 presents sequences of velocity profiles that were measured using the IFT system under three different transient flow conditions. Section 5 presents a simple fluid mechanics model that suggests that the observed velocity profiles measured using the IFT system were plausible.
2. Velocity Profile Reconstruction Technique The multi-electrode EMFM used in this investigation contains a pair of coils forming a Helmholtz coil, a non-conducting flow tube and an array of 16 electrodes (Fig 1). The Helmholtz coil is assumed to generate a uniform magnetic field of flux density B in the y direction in the cross section of the flow tube at the plane of the electrode array. It is further assumed that flow of either, (i) an electrically conducting single phase fluid or (ii) a multiphase mixture where the continuous phase is electrically conducting, occurs in the z direction parallel to the axis of the flow tube. Note that the axial velocity distribution of the conducting phase may be non-uniform in the flow cross section. Flow induced electrical potentials appear at the internal boundary of the flow tube and can be measured using the array of sixteen electrodes shown in figure 1. A method for reconstructing the axial velocity profile of the conducting phase using these flow induced potentials was given in [6, 7] and the key points of this reconstruction technique are summarised below. (i) The axial velocity profile is assumed to be the sum of a series of polynomial velocity components vn ( x , y ) where n 0 to nmax and where if n is odd v n x , y a n ,n ... a n ,1
x cos Q ,n y sin Q ,n n Rn
a n ,n2
x cos Q ,n y sin Q ,n R
x cos Q ,n y sin Q ,n n2 R n2
(1)
and if n is even, 2
v n x , y a n ,n
x cos Q,n y sin Q,n n Rn
a n ,n2
x cos Q,n y sin Q,n n2 R n2
a n ,0
.
(2)
In equations (4) and (5) R is the internal radius of the flow tube and Q ,n is an angle defining the direction of the nth polynomial velocity component, that is to say, vn ( x , y ) changes in the direction of Q ,n but is constant along any line normal to the direction of
Q ,n . The terms an ,n 2 … an ,1 , an ,0 can all be expressed in terms of an ,n as shown in [6]. The overall velocity profile in the cross section of the flow tube is equal to the sum of the velocity components and is given by
v x , y
n max
vn ( x , y )
n0
(3)
where nmax represents the highest order of polynomial velocity component under consideration. It was shown in [6] that for a 16 electrode system the maximum allowable ~ value for nmax is equal to 6. The flow induced potential distribution U ( ) at the inner boundary of the flow tube caused by the interaction of vx , y with the uniform magnetic
field can be discretized into a series U p ( p 0 to N 1 ) where U p represents the measured flow induced potential at the ( p 1 )th electrode in the array (using the electrode numbering convention shown in figure 1) and where N represents the total number of electrodes in the array [note: in this study N 16 ]. If the discrete Fourier transform (DFT) of the series U p is now taken it results in a series of N complex numbers X ( n ) where
1 X n N
N -1
∑U pexp j2 np / N
n 0,1,..., N - 1 .
(4)
p0
It is shown in [6] that the term an ,n in equations (1) or (2), associated with the nth polynomial velocity component, is related to the ( n 1 )th term of the DFT of U p by the following expressions: a0,0 sgn ReX (1)
a1,1
2 X (1 ) BR
8 X( 2) BR
an ,n
n 12n 1 X n 1 . BR
(for n 0 )
(5)
(for n 1)
(6)
(for n 2 ).
(7)
3
It is clear from equation (7) that for n 2 two possible values of an ,n can be obtained which are consistent with the measured value of X ( n 1 ) obtained from the boundary potential distribution at the electrode array. It is also shown in [6] (i) how the terms an ,n 2. ... an ,1 , an ,0 may be calculated from a given value of an ,n ; and (ii) how several possible values for the term Q ,n associated with the nth polynomial velocity component may be obtained from the argument of X ( n 1 ) . Using all possible valid combinations of an ,n and Q ,n it is further shown that, for n 1 , n possible solutions exist for the nth polynomial velocity component which are consistent with the measured value of X ( n 1 ) . This in turn means that nmax ! possible solutions exist for the overall velocity profile v( x , y ) which are consistent with the boundary potential distribution U p measured at the electrode array in the uniform magnetic field. In order to select the optimum solution for v( x , y ) from the nmax ! possible solutions the following procedure was adopted in [6]. A computational model was designed using ‘weight values’ [5, 6] which enables the boundary potential distribution Uˆ at the p , pred
electrode array to be predicted for each of the nmax ! possible solutions of v( x , y ) in the presence of a specific ‘anti-Helmholtz’ magnetic field in the flow cross section. This antiHelmholtz magnetic field is generated by allowing dc electrical currents of equal magnitude to flow in coils 1 and 2 (figure 1) in opposite directions. The optimum solution for v( x , y ) is then taken to be that solution for which the measured flow induced potential distribution Uˆ p
at the electrode array in the anti-Helmholtz magnetic field gives the best agreement with Uˆ p , pred .
Y
R
θ
B X
Fig. 1: The geometry of IFT device in the x-y plane Using the technique described above, the zero’th order polynomial velocity component v0 ( x , y ) is constant and equal to the mean axial flow velocity in the cross section of the flow tube. Furthermore, v0 ( x , y ) is numerically equal to the coefficient a0,0 which is calculated 4
using equation (5). However with reference to [7] all axisymmetric velocity profiles with a mean velocity equal to a0,0 must give rise to the same fundamental component X ( 1 ) when the DFT is taken of the of the discretized boundary potential distribution U p obtained in the same uniform magnetic field. Consequently, the zero’th order velocity component v0 ( x , y ) could, with equal validity, be represented by an expression of the form v0 ( x , y )
a0 ,0 2
(1
r q ) ( 1 q )( 2 q ) . R
(8)
Equation (8) represents an axisymmetric velocity profile component with mean velocity a0,0 and with a power law exponent equal to q . The term r in equation (8) represents radial position in the flow cross section and is given by r ( x 2 y 2 )0.5 . [Note: a0,0 represents the ‘area weighted’ mean value of the velocity distribution defined by equation (8). For a single phase flow a0,0 corresponds to the mean phase velocity. However for a multiphase flow the mean velocity of the conducting continuous phase is dependent upon both v( x , y ) (equation (3)) and the local volume fraction distribution of this phase]. Using an expression for v0 ( x , y ) of the form given by equation (8) in the reconstruction algorithm is likely to yield a much more accurate representation of the true velocity profile, particularly in flows where axisymmetric velocity components are known to exist such as ‘well developed’ single phase flows and vertical multiphase flows. Whereas it is impossible to distinguish between different values of the power law exponent q from the flow induced potential distribution obtained in a uniform field this is not the case for an anti-Helmholtz magnetic field. An anti-Helmholtz magnetic field is relatively strong close to the coils but approaches zero at the pipe centre - and consequently, for a given value of the mean flow velocity, the amplitude of the flow induced potentials is relatively high for low values of q (where the axisymmetric velocity profile is relatively flat) and relatively low for high values of q where the velocity is relatively much higher at the pipe centre than close to the pipe walls. Indeed it is shown in [7] that, for a power law velocity component v0 ( x , y ) of the form given in equation (8), a term Aq may be calculated as follows which is uniquely related to q Aq
2 Im( Xˆ ( 2)) Bop Ra0 ,0
.
(9)
In equation (9) Xˆ ( 2 ) is the first harmonic of the DFT of the discretized flow induced potential distribution Uˆ p measured at the electrode array in an anti-Helmholtz magnetic field, Bop is the magnitude of the y component of the anti-Helmholtz field at electrode e13 (figure 1) and a0,0 is the mean velocity associated with v0 ( x , y ) and is obtained from the 5
flow induced potential distribution measured in the uniform magnetic field as described above. With reference to [9] for a particular electromagnetic flow meter geometry the relationship between q and Aq can be readily determined using magneto-hydrodynamic simulation software such as COMSOL [10]. For the flow metering device used in the present study this relationship was calculated to be
q 48.1Aq3 60.61Aq2 30.13 Aq 5.449 .
(10)
Thus, using the value of a0,0 calculated using equation (5) from the DFT coefficient X ( 1 ) obtained in the uniform magnetic field and the value of q obtained from equation (10), the zero’th order component v0 ( x , y ) of the overall velocity profile can now be expressed in the form given in equation (8). Equation (10) is valid for q 0 corresponding to values of Aq 0.429. This range of values for q is satisfactory for flows where the maximum velocity of the conducting continuous phase occurs at, or close to, the pipe centre. However, in the present study, preliminary results suggested the presence of velocity profiles for which the velocity is a minimum at the pipe centre. If the zero’th order component of such a velocity profile is represented using equation (8) with a negative value of q it is clear that as r / R 1 then v0 ( x , y ) . Values of velocity approaching infinity were found likely to crash the computational algorithm, mentioned above, for selecting the optimum velocity profile. Consequently, for values of Aq 0.429 the zero’th order velocity component v0 ( x , y ) was represented by the expression q' r v0 ( x , y ) a0 ,0 1 n1 0.5( q' 1 )( q' 2 )1 R
(11)
where q' takes the fixed value of 0.5. The ‘area weighted’ mean value of the velocity distribution v0 ( x , y ) as defined in equation (11) is still equal to a0,0 and it is clear, with reference to figure 2, that as the term n increases so the depth of the velocity minimum increases at the pipe centre where r / R 0 . However, it is also clear from figure 2 that at r / R 1 (i.e. at the pipe wall) the value of v0 ( x , y ) remains finite.
6
Figure 2: Variation of v0 ( x , y ) with r/R for a0,0 1 , q' 0.5 and values of n in the range 0.1 to 1. By using simulations performed in COMSOL [10] magneto-hydrodynamic software the following relationship between n and Aq was established for the electromagnetic flow meter geometry used in the present study: n 10.627 Aq 4.464 .
(12)
Consequently when Aq calculated using equation (11) was found to be greater than 0.429, the value of n obtained from equation (12) was combined with the ‘area weighted’ mean velocity term a0,0 to give the zero’th order component v0 ( x , y ) of the overall velocity profile in the form shown in equation (11). As should be apparent to the reader from the previous discussion when a value of Aq 0.429 was observed experimentally, the velocity profile v( x , y ), reconstructed using the techniques outlined above, displayed a velocity at the pipe centre which was lower than the velocity closer to the pipe walls. Although the velocity profiles considered in this paper are mainly axisymmetric (see section 4) the authors have shown that the reconstruction technique described above is also capable of accurately reconstructing highly asymmetric velocity profiles with no axisymmetric component present.
3. Experimental Apparatus 3.1 EMFM Flow Meter Body
7
Figure 3: EMFM flow meter body and flow meter casing [11]. Figure 3 shows the EMFM body used in this research work. The non-conducting flow pipe was made from Delrin and had an internal diameter of 80mm. Two identical circular coils were attached on opposite sides of the pipe wall to form a Helmholtz coil. Sixteen stainless steel electrodes were flush mounted at angular intervals of 22.5o on the internal circumference of the pipe, in the plane orthogonal to the pipe axis which contains the axis of the Helmholtz coil. The casing of the flow meter was made from aluminium to avoid distortions to the magnetic field in the pipe cross section and a cable guide was used to help prevent quadrature voltage in the electrode wires. Each coil forming the Helmholtz coil had a mean radius of 120mm and contained 1024 turns of wire. When a working dc voltage of 48.0V was applied to each coil, the coil current was measured to be 1.41315A. With reference to Figure 1, when the working dc current of 1.41315A flowed in both coils, in the clockwise direction when viewed from above, a uniform magnetic field in the y direction was created in the flow cross section with a magnetic flux density of 106.9297G. Measurements with a gaussmeter showed that, at the plane of the electrode array, there were no significant variations in either the magnitude or direction of the magnetic flux density of this uniform field. Again, with reference to Figure 1, when the working dc current of 1.41315A flowed in coil 1 in the clockwise direction (when viewed from above) and in the anti-clockwise direction in coil 2 an anti-Helmholtz magnetic field was created in the flow cross section for which the magnitude Bop of the y component at electrode e13 was measured to be 38.024G. A further description of the EMFM flow meter body is given by [11]. 3.2 Coil Excitation The selection of the coil excitation sequence is important for an EMFM because (i) an excitation sequence involving magnetic field reversal ensures that electrochemical effects at the electrodes are minimised and (ii) the excitation sequence determines the frequency of operation of the device. A hybrid pulse excitation sequence was applied to the EMFM used in this research, which combines the advantages of both ac and dc coil excitation. A control circuit was used for switching a 48.0V power supply to the Helmholtz coil in such a way as to successively generate both the uniform and anti-Helmholtz magnetic fields described 8
above. The control circuit could also reverse the directions of both the uniform and antiHelmholtz magnetic fields. The uniform magnetic field is generated when the coils are fed with current in the same direction while the anti-Helmholtz field is generated when the coil currents are fed in opposite directions. In this research work, a complete excitation cycle consists of (i) the application of a uniform magnetic field in the y direction (stage in Figure 4); (ii) application of a uniform magnetic field in the y direction (stage
); (iii)
application of the anti-Helmholtz field such that the flux density at electrode e13 is in the y direction (stage ); (iv) application of the anti-Helmholtz field such that the flux density at electrode e13 is in the y direction (stage ). After each of the stages described above, the magnetic field is briefly switched off (stages Ic (A)
,
,
in figure 4).
S 2*
S4
S2
and
S 4*
1.41
S3
S1
(a)
S 1*
S3* Time (s)
-1.41
Uniform magnetic field
Anti-Helmholtz magnetic field
B (G) 107
38
(b) Time (s) -38
τc -107
Figure 4: (a) The variation of current , flowing in the top coil (coil 1) with time over one excitation cycle. is positive when current flows in the clockwise direction in coil 1, when viewed from above. (b) The variation of magnetic flux density in the +y direction at e13 with time over one excitation cycle. The coil control circuit functioned by determining the direction of the electrical current in coils 1 and 2 of the Helmholtz coil (figure 1). This was achieved with a bank of eight solid state relays (SSRs) which were used to connect the terminals of each coil to either the 0V side or the +48V side of the dc power supply. The state of each SSR (open or closed) was set using digital outputs from a National Instruments (NI) data acquisition and control card, interfaced to a PC. By opening and closing the appropriate combinations of SSRs both the uniform and anti-Helmholtz magnetic fields could be successively obtained in either the forward and reverse directions.
9
It should be noted, with reference to Figure 4, that after the SSRs were switched between states a transient time c , due to the coil inductance, elapsed before the coil currents and the magnetic field stabilised. In the present study c was equal to . Table 1 shows the duration of the various stages of magnetic field excitation cycle used in the work reported in this paper. It is clear from this table that a complete excitation cycle had a period of . Flow induced potential difference measurements, averaged over the relevant stages of one excitation cycle, were used to reconstruct a single velocity profile image - thereby giving an effective frame rate of images per second. [Note that flow induced potential difference measurements made with the relevant magnetic field reversed had to be multiplied by -1 in order to be useable in the velocity profile reconstruction technique described in section 2]. Table 1: Duration of stages of magnetic field excitation cycle Excitation Period Time (s) = = = = = = 3.3 Measurement of Flow Induced Potentials During all stages of the magnetic field excitation cycle described above the electrical potential at each electrode, relative to the electrical potential at reference electrode e5 (see Fig. 1) was measured using a 15-channel potential difference measurement circuit. The flow induced potential difference between a given electrode and e5 for the mth stage of magnetic field excitation (m 1 or 3) was obtained by subtracting the mean measured potential difference for the (m 1)th stage (when the magnetic field was switched off) from the mean potential difference measured during the mth stage (when the relevant magnetic field was active). The potentials U and Uˆ required for the velocity profile reconstruction technique p
p
described in section 2 were readily derived from the flow induced potential differences measured as described above, with the proviso that the values of U and Uˆ associated with p
p
reference electrode e5 were always set equal to zero. The 15-channel potential difference measurement circuit used to obtain the experimental results presented in section 4 is described in detail in [11] but the main features of each channel were: (i) very low frequency signals, due to electrochemical effects at the electrode-water interface, were attenuated; (ii) high frequency ‘quadrature’ noise, due to inductive coupling between the magnetic fields and the electrode cabling, was significantly attenuated and (iii) the gain of each channel was set to 1000 to ensure that the low level (of the order of tens of V ) potential difference between each electrode and e5 could be measured with sufficient accuracy by the analogue to digital convertors of the NI data acquisition card mentioned above. 3.4 The Oil Water Flow Loop The experiments described in section 4 were performed in either single phase water flows or water continuous, oil-in-water flows using a multiphase flow loop based at the University of Huddersfield.
10
Figure 5: Flow loop schematic diagram. Symbols used in the diagram are described in the text, except ‘VL’ which denotes a manual valve. The flow loop comprises a 2m long, 80mm internal diameter, transparent Acrylic working section (Fig. 5) which can be positioned vertically or which can be inclined to the vertical, although in the present study all of the flows that were investigated were vertical upward. Flow to the working section is supplied by vertical multistage centrifugal pumps (denoted P1 and P2 in Fig. 5) on the oil and water lines. Immediately downstream of the pumps are pneumatic control valves, denoted CV1 and CV2, which are each connected to a separate Proportional + Integral (PI) controller, enabling the oil and water flow rates to be set independently. Downstream of the control valves are turbine flow meters, denoted FM1 and FM2. Signals from FM1 and FM2 are fed back to the PI controllers to enable the set-point oil and water flow rates to be accurately maintained. The water and oil lines meet at the pipe manifold junction (MJ) where the oil and water mix prior to passing into the flow loop working section. The tank system (TS) comprises a wire mesh separator and storage reservoirs for the oil and water. The wire mesh separator is at the inlet of the tank system and separates the water and oil phases before they flow back into their respective storage reservoirs. In the experiments described in section 4 the EMFM was installed on the flow loop working section at a distance of 1.6m from the working section inlet. 4. Experimentally Observed Water Velocity Profiles at Transient Flow Conditions 4.1 Experimental Procedure With the EMFM installed in the flow loop working section as described above, flow velocity profiles were measured, using the reconstruction technique described in section 2, under three sets of flow conditions case(i) to case(iii) as described below. For each case, the flow was allowed to stabilise to steady state conditions and then transient flow was initiated. Transient flow was initiated when the set point of the PI controller of the relevant flow line was suddenly changed. 11
Case(i): With the water pump running, the water line controller set point was equal to
2m 3 hr 1 (corresponding to a mean water velocity at the EMFM of 0.1105ms 1 ) and the flow rate in the flow loop working section was allowed to attain steady state. Then EMFM measurements were initiated at time . The water line controller set point was then instantly set to 10m 3hr 1 (corresponding to a mean water velocity of 0.5526ms 1 ) at t 23s. Case(ii): With the water pump running, the water line controller set point was 10m 3hr 1 (corresponding to a mean water velocity of 0.5526ms 1 ) and the flow rate in the flow loop working section was allowed to attain steady state. EMFM measurements were initiated at time t 0s . The water line controller set point was instantly changed to 0m 3 hr 1 at t 23s . Case(iii): With both the water and oil pumps running, the water line controller set point was equal to 6m 3 hr 1 , the oil line controller set point was equal to 4m 3 hr 1 (corresponding to a mixture superficial velocity of 0.5526ms 1 ). The fluid flow rates in the flow loop working section were allowed to attain steady state. EMFM measurements were initiated at time
t 0s . The oil controller set point was then instantly changed to 0m 3 hr 1 at t 23s , (corresponding to a mean oil velocity of zero and a mean water velocity of 0.3833ms 1 ). 4.2 Experimental Results The experimentally observed velocity profiles obtained for cases (i) to (iii) using the Inductive Flow Tomography System are presented in Figure 6. It should be noted that velocity profiles for t 22s are not shown because they were always very similar to the velocity profile shown for t 22s (for the relevant case). Similarly, velocity profiles for t 32s are not shown because there was very little change in the shape of the velocity profile after this time. The values of vmean shown in figure 6 represent the mean water velocity as measured by the IFT system. vmean is equal to the term a0,0 given by equation (5). Note that for case(iii) the oil and water were always fairly well mixed, giving a relatively homogeneous oil-water mixture. Consequently, for case(iii), vmean also represents a reasonable estimate of the mean oil velocity. The most noteworthy features of the experimental results shown for cases(i), (ii) and (iii) are summarised below. Case(i). For case (i), for t 0 to 22s, the velocity profile was typical of that which would be expected for a well-developed, turbulent, steady state, single phase flow - namely the velocity profile was relatively flat with the velocity at the pipe centre only slightly greater than its value closer to the pipe wall. However, at t 24s , as transient conditions started to occur, the water velocity close to the pipe centre was observed to be much lower than close to the pipe walls – with reverse flow occurring at the pipe centre. For all of the transient flow conditions investigated, similarly shaped velocity profiles were observed to occur when the mean flow velocity was rapidly increasing; note that at t 26s , vmean 0.5407ms 1 which is considerably higher than at t 24s when vmean 0.1190ms 1 .
12
At t 26s , the water velocity at the pipe centre was observed to be much higher than expected relative to the velocity close to the pipe wall. For all of the transient flow conditions investigated in the present study, this shape of velocity profile was observed to occur when the mean flow velocity was undergoing a rapid decrease; note that at t 28s , vmean 0.3670ms 1 which is considerably lower than at t 26s when vmean 0.5407ms 1 .
For t 30s the velocity profile again became typical for that which would be expected for a well-developed turbulent flow, although the value of vmean was now in excess of 0.4ms 1 compared to its initial steady value (for t 22s ) of about 0.12ms 1 . Case(ii). For case (ii) for t 0 to 22s, the velocity profile was again typical of that which would be expected for a well-developed, turbulent, steady state, single phase flow. However at t 24s , as transient conditions started to occur, the water velocity at the pipe centre was observed to be increase from its previous value of about 0.65ms 1 to about 1.25ms 1 . In a similar manner to the situation described for case(i) above, this velocity profile shape - with a relatively high central velocity - occurred when the mean flow velocity vmean was decreasing from a value of 0.4216ms 1 at t 24s to a value of 0.2958ms 1 at t 26s . [Note: it may be tentatively conjectured that the very high flow velocity observed at the pipe centre when the mean flow velocity was reducing may explain some of the effects of ‘water hammer’ which are observed in process systems when valves in pipelines are rapidly closed]. Again, similar to case(i) described above, when vmean was increasing from 0.2958ms 1 at t 26s to 0.3638ms 1 at t 28s the flow velocity at the pipe centre was observed to be much lower than close to the pipe walls (see diagram for case (ii), t 26s on Fig 6). Case(iii). For case(iii), in which the oil flow rate underwent a rapid reduction in an oil-water multiphase flow, the evolution of the water velocity profile with time was very similar to case(ii) (where, in a single phase flow, the water flow rate underwent a rapid reduction). That is to say, when the mean value of the mixture velocity vmean was undergoing a rapid decrease, the associated water velocity profile exhibited a very high value at the pipe centre; and when vmean was undergoing a rapid increase, the measured water velocity at the pipe centre was lower than close to the pipe walls. Time
Case(i)
Case(ii)
13
Case(iii)
Time
Case (i)
Case (ii)
14
Case(iii)
Figure 6: Reconstructed Velocity profiles for cases(i) to (iii) for 22s t 32s . 5. Simple Fluid Mechanic Model of Transient Velocity Profiles An insight into the plausibility of the experimentally observed velocity profiles in single phase flow may be obtained by considering a simplified 2D fluid mechanics model of flow in a pipe of circular cross section, as described below. In this model the following assumptions are made: (i) The axial velocity profile is always axisymmetric and is constrained to have a ‘power law’ shape of the form q
r v vmx 1 R where R is the pipe radius, vmx is the maximum flow velocity at r R and q is the
(13)
appropriate power law exponent. (ii) Changes in the local axial velocity v at radial position r as a result of variations in the P applied axial pressure gradient are subject to the following highly simplified form of the z Navier-Stokes [12] equation
1 v 1 P v r t r r r z
(14)
where is the kinematic viscosity and is the fluid density. (iii) Changes in the mean axial flow velocity v arising from changes in the applied axial P pressure gradient are defined by the following expression z v 1 P 2v 2 f t z D
(15)
where f is a an appropriate value of the ‘Fanning friction factor’ and where D is the pipe diameter. Note that, as shown in [13] for a power law velocity profile of the form shown in 15
equation (1), the maximum velocity vmx and the mean velocity v are related by the expression
vmx 0.5v q 1q 2 .
(16)
Let us suppose that at time t the local axial velocity v0 at radial position r is given by
r v0 vmx ,0 1 R
q0
(17)
but, as a result of changes to the applied pressure gradient, at time t t the local axial velocity at r changes to v1 as given by;
r v1 vmx ,11 R
q1
(18)
Steps 1 and 2 below show how the terms vmx ,1 and q1 in equation 18 can be obtained from the terms vmx ,0 and q0 in equation (17) using the relationships given in equations (14) and (15). Step 1: As a result of changes in the applied pressure gradient, the initial axial velocity v0 at radial position r changes by an amount v after time t where, from equations (14) and (13), v is given by q0 2 q 1 q0 vmx ,0 1 r 0 q0 1 r t P v t 1 1 . R R r R R z
(19)
At each radial position r a new, ‘interim’ local power law exponent q~1 can now be defined whereby
r vmx ,0 1 R
q~1
v0 v .
(20)
From equation (20), the value of q~1 at each radial position can be calculated using
log v0 v log vmx ,0 q~1 . r log1 R
(21)
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The new value q1 of the power law exponent in equation (18) is now simply obtained by averaging the values of q~1 for all values of radial position r in the flow cross section. Step 2: Let us further suppose that after time t , as a result of changes in the applied pressure gradient, the mean flow axial velocity in the cross section changes from v0 to v1 where
v1 v0 v
(22)
where, from equation (15), v is given by
1 P 2v02 f v t D z
(23)
and v0 is obtained from vmx ,0 by manipulating the relationship given in equation (16). Finally, and again using the relationship given in equation (16), vmx ,1 can now be obtained from the value of v1 given by equation (22). Using the value of q1 obtained in Step 1 above and the value of vmx ,1 obtained in Step 2 above, the new power law velocity profile after time t as defined in equation (18) can now be obtained. By successively applying this procedure for consecutive time intervals t the evolution with time of the axial velocity profile as a result of changes in the applied pressure gradient can be calculated. Using the model described above, the evolution of the calculated velocity profile with time is highly dependent upon the initial velocity profile that is assumed, the properties of the flowing fluid and the imposed pressure gradient. Nevertheless, by assuming plausible values for these variables it is relatively straightforward to reproduce velocity profiles with features that are very similar to those observed in the experiments described in section 4 of this paper. In the example given below water is assumed to flow in a horizontal 80mm diameter pipe. The initial velocity profile is assumed to be ‘power law’ and of the form given in equation (13), with the exponent q equal to 0.15 and the maximum velocity vmx at the pipe centre equal to 0.68ms 1 , corresponding to a mean flow velocity v of 0.55ms 1 . The water density is taken as 1000kgm 3 and its kinematic viscosity is taken as the commonly accepted value of 1 106 m 2s 1 . The imposed pressure gradient is of the form P / z 60cos( t / 2 ) and has a maximum value of 60Pam 1 and a period of 4s. This form
of pressure gradient is intended to simulate the pressure gradient that might be associated with a control valve that is ‘hunting’ for its set-point position.
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Results from the model using these assumed conditions are shown in Figure 7 from which it is clear that at time t 0 the water velocity profile is comparatively flat and similar to that which might be observed for a fully developed turbulent single phase flow. Between t 0 and 1s the applied pressure gradient is positive (adverse) and the mean flow velocity v steadily decreases. During this period, the velocity at the pipe centre gradually increases relative to its value at the pipe walls culminating in the velocity profile at t 1s shown in figure 7, for which v 0.4788ms 1 and q 0.342 . Between t 1 and 3s the applied pressure gradient is negative (favourable), v gradually increases and the velocity profile gradually reduces at the pipe centre relative to its value close to the pipe walls, culminating in the velocity profile at t 3s shown in figure 7 for which v 0.5ms 1 and q 0.756 .
Whilst not intended to directly replicate any of the flow conditions associated with experimental cases(i) to (iii) given in section 4.2, similarities between the behaviour of the velocity profiles from the model and those observed experimentally suggest that the results obtained from the EMFM, as shown in figure 6, are indeed plausible. This, in turn, increases confidence that inductive flow tomography system described in this paper is capable of accurately measuring transient velocity profiles in both single phase and water continuous multiphase flows.
Figure 7: Axial velocity distribution at t = 0, 1s and 3s for an imposed co-sinusoidal pressure gradient and for the flow conditions given in the text. From the experimental results presented in section 4 it is apparent that the large amplitude oscillatory transients in velocity profile, which occur just after a change is made to the flow controller set point, have a period of no more than 4 seconds. Successive images from the IFT system are only obtained every 2 seconds and so, to examine the velocity profile transients in greater temporal detail, a much faster IFT frame rate will be required. This is the subject of ongoing work currently being undertaken by the authors. 6. Conclusions
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A novel inductive flow tomography (IFT) system with a frame rate of 0.5 frames per second, has been designed and built and used to experimentally investigate water velocity profiles in steady state and transient single phase flows and multiphase flows in which water was the continuous phase. Measurements from the IFT system in steady state flows were consistent with the velocity profiles that would be expected for fully-developed turbulent flows. In transient flows it was experimentally observed that when the mean flow velocity was decreasing, the water velocity at the pipe centre became relatively much higher than the water velocity close to the pipe walls. Conversely when the mean flow velocity was increasing, the water velocity at the pipe centre became lower than at the pipe walls – even to the extent that reverse flow sometimes occurred at the pipe centre. Comparison with a simple fluid mechanics model of transient flow suggested that these experimentally observed transient velocity profiles were plausible. It is further suggested that the very high flow velocities observed at the pipe centre when the mean flow velocity is reducing may be at least partially responsible for the some of the damaging effects of ‘water hammer’ which are observed when valves in pipelines are rapidly closed. References [1] C. Schmitt, G. Pluvinage, E. Hadj‐Taieb, R. Akid, Water pipeline failure due to water hammer effects, Fatigue & Fracture of Engineering Materials and Structures 29(12) (2006) 1075-1082. doi:10.1111/j.1460-2695.2006.01071.x [2] P.F. Boulos, B.W. Karney, D.J. Wood, S. Lingireddy, Hydraulic Transient Guidelines for Protecting Water Distribution Systems, Journal American Water Works Association 97(5) (2005) 111-124 [3] M. Brito, P. Sanches, R.M. Ferreira, D.I.C. Covas, PIV characterization of transient flow in pipe coils, Procedia Engineering 89(C) (2014) 1358-1365. doi:10.1016/j.proeng.2014.11.458 [4] B. Brunone, B.W. Karney, M. Mecarelli, M. Ferrante, Velocity Profiles and Unsteady Pipe Friction in Transient Flow, Journal of Water Resources Planning and Management 126(4) (2000) 236-244. https://dx.dio.org/10.1061/(ASCE)0733-9496(2000)126:4(236). [5] J.A. Shercliff, The Theory of Electromagnetic Flow-Measurement, Cambridge University Press, Cambridge, UK, 1962. [6] L.E. Kollar, G. Lucas, Z. Zhang, Proposed method for reconstructing velocity profiles using a multi-electrode electromagnetic flow meter, (2014). doi:10.1088/09570233/25/7/075301 [7] L.E. Kollár, G.P. Lucas, Y. Meng, Reconstruction of velocity profiles in axisymmetric and asymmetric flows using an electromagnetic flow meter, Measurement Science and Technology 26(5) (2015) 1-12. doi:10.1088/0957-0233/26/5/055301. [8] O. Lehtikangas, K. Karhunen, M. Vauhkonen, Reconstruction of velocity fields in electromagnetic flow tomography, Philosophical Transactions of the Royal Society of 19
London A: Mathematical, Physical and Engineering Sciences 374(2070) (2016). doi:10.1098/rsta.2015.0334. [9] Y. Meng. Imaging of the water velocity distribution in water continuous multiphase flows using inductive flow tomography (IFT) Ph.D Thesis, University of Huddersfield. UK (2016). [10] COMSOL, AC/DC module user's Guide, http://hpc.mtech.edu/comsol/pdf/ACDC_Module/ACDCModuleUsersGuide.pdf.
2013.
[11] M. Agolom, G. Lucas, Optimisation of the design of the signal processing circuitry for an inductive flow tomography system, Paper presented at the 7th International Symposium on Process Tomography. Dresden, Germany: 2015. [12] R.H. Sabersky, A.J. Acosta, E.G. Hauptmann, A First Course in Fluid Mechanics, 3rd ed., Macmillan Publishing Company, New York, USA, 1989. [13] G.P. Lucas, R. Mishra, N. Panayotopoulos, Power law approximations to gas volume fraction and velocity profiles in low void fraction vertical gas–liquid flows, Flow Measurement and Instrumentation 15(5–6) (2004) 271-283. http://dx.doi.org/10.1016/j.flowmeasinst.2004.06.004.
Highlights
Novel method for measuring velocity profiles using inductive flow tomography Transient velocity profiles measured in single and multiphase flow Large velocity measured at pipe centre when flow rate increases Reverse flow measured at pipe centre when flow rate is decreasing High velocity measured at pipe centre when the flow rate is increasing
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