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An electrical impedance monitoring method of water-lubricated oil transportation
Flow Measurement and Instrumentation 46 (2015) 327–333
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
An electrical impedance monitoring method of water-lubricated oil transportation Hyeuknam Kwon, Jung-Il Choi n, Jin Keun Seo Department of Computational Science & Engineering, Yonsei University, Republic of Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 3 December 2014 Received in revised form 17 June 2015 Accepted 24 June 2015 Available online 16 July 2015
We propose a simple and efficient monitoring system based on a single-drive interleaved electrode impedance technique that provides cross-sectional images for real-time detection of fouling in oil–water flows. The simple monitoring method is proposed using a data-interface formula relating between voltage differences measured at sensing electrodes and interfaces of the two immiscible fluids in a pipeline. We estimate the minimum distance between the interface and pipe wall at the images, that is an indicator for monitoring the fouling, by using a voltage–distance map. The robustness of the proposed method is validated through various numerical simulations including oil–water flows in an U-bend return pipe. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Process tomography Electrical impedance tomography Electric potential Core-annular flows Water-lubricated flows
1. Introduction Oil–water flows with high-viscosity ratio have recently attracted much attention owing to the increase of exhaustion of light oil reserves and the depletion of on-shore oil fields. For heavy and extra-heavy crude oil transportation, water-lubricated transport re-emerges as an energy efficient technology in the last few decades in order to reduce pressure drops as well as pumping powers. Since annular water-film along the pipe wall prevents the oil from a direct contact with the wall, the wall friction mainly arises due to flow of water only [1,2]. This flow is known as a core-annular flow (CAF) that consists of a core of highly viscous fluid surrounded by a thin layer of lower viscous fluid [3,4]. One of the major challenging issues in the use of CAF based heavy oil transportation is to retain the water film at the pipe wall or to avoid fouling of the oil [1]. A removal process of high viscous oil that sticks to the pipe wall during shut down is needed for mitigating undesirable higher pressure drops in the pipeline when the fouling occurs. Pre-detection or real-time detection of the fouling is very important for operating the heavy oil transportation using CAF technique without an intervention for restarting the process. Moreover, the recent experimental and numerical studies [5–8] reported that the chances of fouling in oil–water flows through a return bend such as U or Π bend for pipefitting become higher unless proper operation conditions for the CAF are provided. Therefore, monitoring n
Corresponding author. E-mail address: [email protected] (J.-I. Choi).
http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.029 0955-5986/& 2015 Elsevier Ltd. All rights reserved.
of the fouling is necessary in CAF technique to optimize and control its operation conditions. Process tomography has been widely used in visualizing multiphase flows such as oil–gas or oil–water flows in industrial pipelines. It provides real-time cross-sectional images of the distribution of materials [9]. Electric capacitance tomography (ECT) has been used for process tomography where external capacitance measurements are used to produce the cross-sectional images of permittivity distribution inside the pipeline [10–17]. This technique enables to visualize the contents of a process vessel or pipeline that contains dielectric materials and non-conducting continuous phases. ECT has several advantages such as non-intrusiveness, simple manufacturing, and low cost. However, in the case of conducting working fluid being in contact with the boundary of the pipeline, ECT may not be appropriate to visualize the water-lubricated oil flows. This is because the insulated oil flow may be poorly detected by the capacitance sensors due to the conducting water. On the other hand, electrical impedance tomography (EIT) can be a promising technique for monitoring the distributions of the non-conducting materials inside pipelines. EIT provides images of both conductivity and permittivity distributions of internal objects from current–voltage measurements at the boundary of pipeline. EIT method uses multiple current injection pattern to produce electrical current density distribution in the entire region so that Ohm's law can be used to provide tomographic image [18–20]. However, the standard EIT method is not suitable for the waterlubricated oil flows, because the injected electrical current flows only near the boundary of the pipeline and the current cannot
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H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
a
b
Fig. 1. (a) Schematic of a core-annular flow configuration in cross-sectional view. The region D is initially filled with heavy oil, while the rest of domain Ω is with conducting water. (b) Schematics of electrode configurations for the proposed EIT system. Note that black rectangles on the boundary indicate sensing electrodes while red and blue rectangles represent driving electrodes of inward and outward electric current, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
penetrate into the insulated oil region. Indeed, we are only interested in detecting the water–oil interface near the boundary of the pipeline. For monitoring CAF, we design a simple and efficient monitoring system which can be viewed as a modified version of EIT system. The proposed system uses a single-drive interleaved electrode system, as shown in Fig. 1 which measures the voltage difference through sensing electrodes to probe the interface of D. We found a data-interface formula relating between the interface of D and measured data, which provides essential information of detecting the interface. Based on the data-interface formula, we develop a simple method of monitoring the interface of a continuous two phase flow in the pipeline. The minimum distance for the interface from the pipe walls is estimated for monitoring the fouling based on a voltage–distance map. Various numerical simulations validate the robustness of the proposed method in visualizing the interface and estimating the minimum distance.
2. Method 2.1. Mathematical model In this section, we describe a mathematical model for the proposed EIT system shown in Fig. 1. Let Ω denote a cross-sectional region in the pipeline. For water-lubricated oil flows, the oil and water regions are defined as D and Ω⧹D , respectively, as in Fig. 1(a). The boundaries of the domain Ω and D are denoted as ∂Ω and ∂D , respectively. We assume that the oil region is insulated while the water region is highly conductive. As shown in Fig. 1(b), we attach N driving electrodes for injecting current and 2N sensing electrodes for measuring voltage along the circumferential direction of the pipe in such a way that the two sensing electrodes are placed between the driving electrodes. We inject currents of I at the driving electrodes so that the induced voltage u inside the cross-sectional domain Ω is governed by
⎧− ∇2u = 0 in Ω⧹D, ⎪ ⎪ if j is even, ⎪ , d n·∇u ds = I ⎪ j ⎪ ⎪ , d n·∇u ds = − I if j is odd, ⎪ j ⎪ ⎞ ⎛ 2N ⎞ ⎛ N ⎪ on ⎜⎜ ⋃ , dj ⎟⎟ ∪ ⎜⎜ ⋃ , ks ⎟⎟, ⎨ n × ∇u = 0 ⎝ j=1 ⎠ ⎝k=1 ⎠ ⎪ ⎪ ⎪ s n·∇u ds = 0 for k ∈ {1, 2, …, 2N} , ⎪ ,k ⎪ ⎛⎛ N ⎞ ⎛ 2N ⎞ ⎞ ⎪ on ∂Ω⧹ ⎜⎜ ⎜⎜ ⋃ , dj ⎟⎟ ∪ ⎜⎜ ⋃ , ks ⎟⎟ ⎟⎟, ⎪ n·∇u = 0 ⎝⎝ j=1 ⎠ ⎝k=1 ⎠⎠ ⎪ ⎪ on ∂D, ⎩ n·∇u = 0
∫ ∫
∫
(1)
where , dj is a driving electrode for j = 1, 2, … , N and , ks is a sensing electrode for k = 1, 2, … , 2N . One pair of sensing electrodes {, 2s j−1, , 2s j } are placed between one pair of driving electrodes {, dj , , dj+1}, as shown in Fig. 1(b). We measure voltage difference Vk = u ,ks+1 − u ,ks between two electrodes , ks+1 and , ks . The measured data set {V1, V2, … , V2N } is used to reconstruct the shape of ∂D .
2.2. The relation between the data and the interface In order to estimate the interface of D from the data {V1, V2, … , V2N }, we introduce the computed reference data Uk0 defined as
⎛ Uk0≔I ⎜⎜ ∑ uk0 ⎝ j = even
, dj
−
∑ j = odd
uk0
⎞ ⎟, ⎠
, dj ⎟
(2)
where uk0 is the solution of the following equations for k = 1, 2, … , 2N :
H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
329
integral term in right hand side of Eq. (4) can be viewed as a functional Φ k (ϕ) with the smooth function ϕ (θ ). Defining Φ k (ϕ)≔ ∫∂D u (∂uk0 /∂n) ds , the inverse problem of determining the
shape of D is reduced to find ϕ minimizing the cost functional Ψ (ϕ) that is the sum of differences between Φ k (ϕ) and IVk − Uk0 : N
(
Ψ (ϕ)≔ ∑ Φ k (ϕ) − IVk − Uk0
)
2
(6)
k=1
Fig. 2. Electric potentials u0k for k = 1, 2, …, 2N . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
⎧ 2 0 ⎪− ∇ uk = 0 ⎪ 0 ⎪ , s n·∇u k ds = I , ⎪ k +1 ⎪ 0 ⎪ , s n·∇u k ds = − I , ⎪ k ⎪ ⎪ 0 ⎪ n × ∇u k = 0 ⎪ ⎨ ⎪ 0 ⎪ , d n·∇u k ds = 0 ⎪ j ⎪ ⎪ n·∇u k0 ds = 0 ⎪ ,s ⎪ k′ ⎪ ⎪ 0 ⎪ n·∇u k = 0 ⎪ ⎩
in Ω,
∫
Solving the above minimization problem requires for an iterative procedure such as steepest descent method. In addition, we usually add a regularization term to the cost functional in order to obtain a stable solution ϕ [21]. For monitoring in the waterlubricated oil flows, one of the critical issues is precisely to find the minimum distance between ∂D and ∂Ω in order to determine whether the fouling occurs or to predict chances of the fouling. The present reconstruction method based on the minimization problem may accurately visualize the shape of ∂D and measure the minimum distance. However, due to the computation-intensive iterative procedure in finding the minima of Ψ (ϕ), we propose a fast direct method in the next section.
∫
2.3. Simple direct reconstruction
⎞ ⎞ ⎛ 2N ⎛ N on ⎜ ⋃ , dj ⎟ ∪ ⎜ ⋃ , s ⎟ , ⎜ j =1 ⎟ ⎜ k′⎟ ⎠ ⎝ k′= 1 ⎠ ⎝
∫
for j ∈ {1, 2, …, N} ,
∫
for k′ ∈ {1, 2, …, 2N}⧹{k, k + 1} , ⎛⎛ N ⎞⎞ ⎞ ⎛ 2N ⎟ ⎜ on ∂Ω⧹ ⎜ ⎜ ⋃ , dj ⎟ ∪ ⎜ ⋃ , s ⎟ ⎟ . ⎟ ⎟ ⎜ ⎜ ⎝ ⎝ j = 1 ⎠ ⎝ k′= 1 k′⎠ ⎠
(3) 0
It is worthy to note that the electrical potential uk as an eigenmode, that is independent of the interface of D, can be numerically constructed by solving Eq. (3) for computing the reference data Uk0. In Fig. 2, the colors indicate the electric potentials uk0 while the black lines represent paths of electric current density. Taking an advantage of the reference data, the quantity IVk − Uk0 is related to D through the following key identity:
IVk − Uk0 =
∂u 0
∫∂D u ∂nk ds.
(4)
In this paper, the above identity (4) is referred as a data-interface formula that relates the measurement data with the interface. The proof of the identity is as follows:
⎞ ⎛ IVk = I ⎜⎜u , ks+1 − u , ks ⎟⎟ = ⎠ ⎝
0
∫∂Ω u ∂∂unk
ds
=
∫Ω ∇u·∇uk0 dr = ∫Ω ⧹D ∇·(uk0 ∇u) dr + ∫D ∇·(u∇uk0 ) dr
=
∫∂Ω uk0 ∂∂nu ds + ∫∂D u ∂∂unk
0
= Uk0 +
ds
p j ≔max {p: (p cos θ j, p sin θ j ) ∈ D, 0 < p < 1}
for j = 1, 2
, …, N ,
(7)
where θ j = π /N + (j − 1) × 2π /N represents an angle between a mid-point of the sensing electrodes , sj and , sj+1 and the reference driving electrode. In the circular domain with a radius R, the distance dj from the domain boundary ∂Ω to the interface ∂D is defined as d j≔R − p j . If ∂D is closer to ∂Ω , i.e., d j /R⪡1, the electric potential distribution is mostly affected by currents injected from the electrodes , 2j − 1 and , 2j . Hence, the measured voltage difference Vj can be approximated by the electric potential generated by the currents from the electrodes. Moreover, the electric potential in the circular domain can be estimated by that in a square domain Ξ [d j ] due to a thin layer approximation (d j /R⪡1). The square domain is defined as
Ξ [h]≔{square domain with height h and width |∂Ω| /3} ,
0
∫∂D u ∂∂unk
For a real-time monitoring, a fast direct method is more attractive than the minimization problem based iterative method described in Section 2.2. Since a measured voltage difference on a sensing electrode may not be significantly affected by driving electrodes that are not adjacent to the sensing electrode, it is possible to obtain an explicit data-interface formula that provides a reasonably accurate estimation of the minimum distance. In addition, when the conductivity distribution is homogeneous, the resulting electric potential u can be approximated in a circular domain as u (r ) ≈ r 8, where r is a radial distance from the center of the pipe. Based on the assumptions, we propose a simple and direct method to reconstruct ∂D . We only use the measured data {V1, V3, … , V2k − 1, … , V2N − 1} at the sensing electrodes that are in between two driving electrodes as shown in Fig. 3. In the direct method, the target object D is approximated by N points on ∂D . As shown in Fig. 3, a radial distance pj at each angle θj from the origin to ∂D is defined as
(8)
ds .
Assuming that D is a star-shaped smooth domain given by a smooth function ϕ (θ ), 0 ≤ θ ≤ 2π :
D = {(x, y) = (r cos θ , r sin θ ):
x2 + y2 = r < ϕ (θ ), 0 ≤ θ ≤ 2π} , (5)
where (x,y) represents Cartesian coordinates for a point in D. The
where the corresponding electrodes system in the square domain is illustrated in Fig. 3(c). The voltage difference W between two sensing electrodes ,±s can be computed by an explicit data-interface formula
W = g (h)≔
∫,
s +
vh ds −
∫,
s −
vh ds,
(9)
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H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
a
b
c
Fig. 3. Schematic configurations of (a) electrodes, (b) unit segment for electrodes in the proposed EIT system with the direct reconstruction method, and (c) unit segment for electrodes in a thin layer region Ξ with conducting fluid.
where the electric potential vh is the solution of the following equation:
⎧∇2v = 0 h ⎪ ⎪ ⎪ , d+ n·∇vh ds ⎪ ⎪ n·∇vh ds ⎪ ,−d ⎪ ⎪ ⎨ n × ∇vh = 0 ⎪ ⎪ d n·∇vh ds ⎪ ,± ⎪ ⎪ s n·∇vh ds ⎪ ,± ⎪ ⎪ ⎩ n·∇vh = 0
We performed numerical simulations to demonstrate the feasibility of the direct method described in Section 2.3. The direct method is applied in the following steps.
in Ξ [h],
∫
= I,
∫
= − I,
1. 2. 3. 4. 5.
on , ±s ∪ , ±d ,
∫
= 0,
∫
= 0,
(
3. Simulation results
)
on ∂Ω⧹ , ±s ∪ , ±d .
Measure voltage difference data {V1, V3, … , V2k − 1, … , V2N − 1}. Obtain the solution set {vh }h ∈(0,0.5] of Eq. (10). Construct the explicit data-interface formula W = g (h). Compute the estimated distance h j = g −1 (V2j − 1) at each angle θj. Visualize ∂D with the estimated nodes (h j cos θ j, h j sin θ j ).
(10)
For a given distance h, the electric potential vh is computed from Eq. (10) and the voltage difference W is then obtained from the explicit data-interface formula in Eq. (9). The corresponding voltage–distance map is shown in Fig. 4. Note that the amplitude of injected current is 2 mA with 10 kHz frequency, which is within the range of experimental conditions in [22]. We assume R = 1 m for simplicity. The voltage difference W monotonically decreases as the distance h increases. Moreover, the voltage difference is sensitive to the distance for h/R < 0.1. This implies that the resulting voltage differences may estimate the minimum distance of the interface ∂D accurately when the insulated material is closer to the boundary of the electrodes. Furthermore, the distance h j = R − d j in a radial direction at each angle θj can be explicitly estimated by using the computed g −1 with the measured voltage difference data V2j − 1:
h j = g −1 (V2j − 1)
for j = 1, 2, …, N .
(11) Fig. 4. Distance (h) induced voltage difference (W ) .
H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
a
b
331
c
Fig. 5. (a) Regions for the two immiscible fluids, (b) electric voltage distributions and current patterns, and (c) the reconstructed interface of the two fluids indicated by the red line in the model problem. Note that a grey region indicates the insulated oil while a region outside the oil contains conducting water. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.) Table 1 Minimum distances and reconstruction errors of the distance at each case for the model problem. Case
dmin /R
∼ dmin /R
∼ hmin /R
e/R
e∼/R
1 2
0.2500 0.0833
0.2569 0.1083
0.2500 0.1000
0.0000 0.0167
0.0069 0.0083
To quantify the proximity of ∂D and ∂Ω , a parameter dmin is defined as
{
}
dmin = min |p1 − p2 | : p1 ∈ ∂D, p2 ∈ ∂Ω , which implies the exact minimum distance between ∂D and ∂Ω in the domain Ω . Due to the limitation of the number of sensing
∼ electrodes, a pseudo-minimum distance dmin is defined as
∼ dmin
{
}
= min |R − r| : (r cos θ j, r sin θ j ) ∈ ∂D, 0 < r < R, j = 1, 2, …, N
where θ j = π /N + (j − 1) × 2π /N . This indicates the pseudo-minimum distance from the mid-point of sensing electrodes to a point on ∂D aligned with the radial direction to the mid-point, which is the best measurable minimum distance. The estimated minimum distance based on the proposed method is defined as
∼ hmin = min h j : j = 1, 2, …, N .
{
}
The corresponding reconstruction errors for the minimum dis∼ ∼ ∼ tances are defined as e = |dmin − hmin | and e∼ = |dmin − hmin |. We consider model problems that are numerically defined for a
Fig. 6. The flow configuration of core-annular flow with U-bend return pipe in [7].
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H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
a
b
Fig. 7. (a) The cross-sectional images of the oil–water flow at each downstream location, which are adopted from the CFD results in [7] and (b) the present reconstructed interfaces (red line) with the reference interfaces from [7] (black line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Table 2 Minimum distances and reconstruction errors of the distance at each downstream location for the oil–water flow. Location
dmin /RU
∼ dmin /RU
∼ hmin /RU
e/RU
e∼/RU
1 2 3 4 5
0.3627 0.2543 0.1666 0.1007 0.0000
0.3627 0.2546 0.1965 0.1276 0.0000
0.3250 0.2500 0.1875 0.1250 0.0022
0.0377 0.0043 0.0209 0.0243 0.0022
0.0377 0.0046 0.0090 0.0026 0.0022
Fig. 8. Minimum distances at each downstream location for the oil–water flow.
region D in the domain Ω . The domain Ω is the unit circle centered at (0, 0) and D is also a circle with radius R D = 2/3R . The center position for D is (0, − 1/12R ) and (0, − 1/4R ) for the cases 1 and 2, respectively. In both cases, the conductivity and permittivity of Ω⧹D are set to unity while those are 10 6 for D. Eight driving electrodes and sixteen sensing electrodes are used for the reconstruction of ∂D . Fig. 5 shows numerical simulation results for both cases. As D moves toward ∂Ω , it is observed for the case 2 in Fig. 5(b) that electrical potential is higher near the region where the distance between ∂D and ∂Ω is smaller. The proposed method reconstructs the interfaces accurately only on the region where the distance is small, as shown in Fig. 5(c). Table 1 shows quantitative errors for reconstructing the interface using the proposed method for the model problems, which shows less than 2% errors for
finding the minimum distances in both cases. As an example of the water-lubricated oil flows, we consider a core-annular flow in an U-bend return pipeline, which was numerically studied in [7]. The flow configuration in Fig. 6 shows that the U-bend geometry is a tube of RU = 0.006 m radius having 0.15 m straight length at the upstream and downstream. We choose the case for core-annular downflow with oil superficial velocity vso = 0.15 m s−1 and water superficial velocity − vsw = 0.3 m s 1. The distributions of oil and water phases from the CFD results in [7] are used for testing the proposed method. Fig. 7 shows the reconstruction images of the interfaces between oil and water phases compared with reference data [7] at different downstream locations. The initial shape of oil phase is maintained up to the straight tube region as shown in the image of cross section ①. However, the images of cross sections ②–④ show that the oil phase moves toward the outer portion of bend curvature due to the impact of centrifugal forces. At the cross section ⑤, the oil phase sticks to the outer portion of bend curvature. The proposed model clearly captures the interfaces near the region where the interfaces are closer to the pipe wall while the interfaces are less accurately reconstructed where they are far from the wall. It is worthy to note that the proposed model is able to capture the fouling region at the cross section ⑤ as shown in Fig. 7(b). In overall, Table 2 shows that the reconstruction errors of the minimum distances that relate with monitoring of the fouling are less than 4% for all test images. Comparisons of the estimated ∼ minimum distance hmin with the exact minimum distance dmin and ∼ the pseudo-minimum distance dmin are illustrated in Fig. 8. As expected, the estimated distances show in a good agreement with the pseudo-minimum distances. Interestingly enough, as the interface is closer to the pipe wall, the reconstruction error for the minimum distance becomes smaller. This implies that the proposed method might be more accurate in estimating the minimum distance just before fouling occurs.
4. Conclusion We proposed a simple and efficient monitoring method for detecting fouling in water-lubricated oil flows. Based on a singledrive interleaved electrode system which measures the voltage difference through sensing electrodes, we derived a data–interface relation between oil–water interfaces and measured data at each cross section of a pipeline. By introducing a thin-layer approximation for the electric potential near the pipe wall, the data–interface formula was simplified as the voltage–distance map for reconstructing the interfaces. We found that, for the given current
H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
injection with the constant value at fixed electrodes, voltage differences monotonically increases as insulated materials are closer to the boundary of the electrodes. The results for the model problem showed that the proposed direct reconstruction method accurately estimates the minimum distance of the interface from the boundary while it provides a rough shape of the insulated region. Finally, we demonstrated the feasibility of the proposed method by reconstructing the oil–water interfaces and analyzing the minimum distance as well as rough shape of the oil phase from cross-sectional images of CFD results for the U-bend return pipeline in [7].
Acknowledgment This work is supported by the National Research Foundation of Korea (NRF) Grant (NRF-20151009350) funded by the Korean government (MEST).
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Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
An electrical impedance monitoring method of water-lubricated oil transportation Hyeuknam Kwon, Jung-Il Choi n, Jin Keun Seo Department of Computational Science & Engineering, Yonsei University, Republic of Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 3 December 2014 Received in revised form 17 June 2015 Accepted 24 June 2015 Available online 16 July 2015
We propose a simple and efficient monitoring system based on a single-drive interleaved electrode impedance technique that provides cross-sectional images for real-time detection of fouling in oil–water flows. The simple monitoring method is proposed using a data-interface formula relating between voltage differences measured at sensing electrodes and interfaces of the two immiscible fluids in a pipeline. We estimate the minimum distance between the interface and pipe wall at the images, that is an indicator for monitoring the fouling, by using a voltage–distance map. The robustness of the proposed method is validated through various numerical simulations including oil–water flows in an U-bend return pipe. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Process tomography Electrical impedance tomography Electric potential Core-annular flows Water-lubricated flows
1. Introduction Oil–water flows with high-viscosity ratio have recently attracted much attention owing to the increase of exhaustion of light oil reserves and the depletion of on-shore oil fields. For heavy and extra-heavy crude oil transportation, water-lubricated transport re-emerges as an energy efficient technology in the last few decades in order to reduce pressure drops as well as pumping powers. Since annular water-film along the pipe wall prevents the oil from a direct contact with the wall, the wall friction mainly arises due to flow of water only [1,2]. This flow is known as a core-annular flow (CAF) that consists of a core of highly viscous fluid surrounded by a thin layer of lower viscous fluid [3,4]. One of the major challenging issues in the use of CAF based heavy oil transportation is to retain the water film at the pipe wall or to avoid fouling of the oil [1]. A removal process of high viscous oil that sticks to the pipe wall during shut down is needed for mitigating undesirable higher pressure drops in the pipeline when the fouling occurs. Pre-detection or real-time detection of the fouling is very important for operating the heavy oil transportation using CAF technique without an intervention for restarting the process. Moreover, the recent experimental and numerical studies [5–8] reported that the chances of fouling in oil–water flows through a return bend such as U or Π bend for pipefitting become higher unless proper operation conditions for the CAF are provided. Therefore, monitoring n
Corresponding author. E-mail address: [email protected] (J.-I. Choi).
http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.029 0955-5986/& 2015 Elsevier Ltd. All rights reserved.
of the fouling is necessary in CAF technique to optimize and control its operation conditions. Process tomography has been widely used in visualizing multiphase flows such as oil–gas or oil–water flows in industrial pipelines. It provides real-time cross-sectional images of the distribution of materials [9]. Electric capacitance tomography (ECT) has been used for process tomography where external capacitance measurements are used to produce the cross-sectional images of permittivity distribution inside the pipeline [10–17]. This technique enables to visualize the contents of a process vessel or pipeline that contains dielectric materials and non-conducting continuous phases. ECT has several advantages such as non-intrusiveness, simple manufacturing, and low cost. However, in the case of conducting working fluid being in contact with the boundary of the pipeline, ECT may not be appropriate to visualize the water-lubricated oil flows. This is because the insulated oil flow may be poorly detected by the capacitance sensors due to the conducting water. On the other hand, electrical impedance tomography (EIT) can be a promising technique for monitoring the distributions of the non-conducting materials inside pipelines. EIT provides images of both conductivity and permittivity distributions of internal objects from current–voltage measurements at the boundary of pipeline. EIT method uses multiple current injection pattern to produce electrical current density distribution in the entire region so that Ohm's law can be used to provide tomographic image [18–20]. However, the standard EIT method is not suitable for the waterlubricated oil flows, because the injected electrical current flows only near the boundary of the pipeline and the current cannot
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a
b
Fig. 1. (a) Schematic of a core-annular flow configuration in cross-sectional view. The region D is initially filled with heavy oil, while the rest of domain Ω is with conducting water. (b) Schematics of electrode configurations for the proposed EIT system. Note that black rectangles on the boundary indicate sensing electrodes while red and blue rectangles represent driving electrodes of inward and outward electric current, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
penetrate into the insulated oil region. Indeed, we are only interested in detecting the water–oil interface near the boundary of the pipeline. For monitoring CAF, we design a simple and efficient monitoring system which can be viewed as a modified version of EIT system. The proposed system uses a single-drive interleaved electrode system, as shown in Fig. 1 which measures the voltage difference through sensing electrodes to probe the interface of D. We found a data-interface formula relating between the interface of D and measured data, which provides essential information of detecting the interface. Based on the data-interface formula, we develop a simple method of monitoring the interface of a continuous two phase flow in the pipeline. The minimum distance for the interface from the pipe walls is estimated for monitoring the fouling based on a voltage–distance map. Various numerical simulations validate the robustness of the proposed method in visualizing the interface and estimating the minimum distance.
2. Method 2.1. Mathematical model In this section, we describe a mathematical model for the proposed EIT system shown in Fig. 1. Let Ω denote a cross-sectional region in the pipeline. For water-lubricated oil flows, the oil and water regions are defined as D and Ω⧹D , respectively, as in Fig. 1(a). The boundaries of the domain Ω and D are denoted as ∂Ω and ∂D , respectively. We assume that the oil region is insulated while the water region is highly conductive. As shown in Fig. 1(b), we attach N driving electrodes for injecting current and 2N sensing electrodes for measuring voltage along the circumferential direction of the pipe in such a way that the two sensing electrodes are placed between the driving electrodes. We inject currents of I at the driving electrodes so that the induced voltage u inside the cross-sectional domain Ω is governed by
⎧− ∇2u = 0 in Ω⧹D, ⎪ ⎪ if j is even, ⎪ , d n·∇u ds = I ⎪ j ⎪ ⎪ , d n·∇u ds = − I if j is odd, ⎪ j ⎪ ⎞ ⎛ 2N ⎞ ⎛ N ⎪ on ⎜⎜ ⋃ , dj ⎟⎟ ∪ ⎜⎜ ⋃ , ks ⎟⎟, ⎨ n × ∇u = 0 ⎝ j=1 ⎠ ⎝k=1 ⎠ ⎪ ⎪ ⎪ s n·∇u ds = 0 for k ∈ {1, 2, …, 2N} , ⎪ ,k ⎪ ⎛⎛ N ⎞ ⎛ 2N ⎞ ⎞ ⎪ on ∂Ω⧹ ⎜⎜ ⎜⎜ ⋃ , dj ⎟⎟ ∪ ⎜⎜ ⋃ , ks ⎟⎟ ⎟⎟, ⎪ n·∇u = 0 ⎝⎝ j=1 ⎠ ⎝k=1 ⎠⎠ ⎪ ⎪ on ∂D, ⎩ n·∇u = 0
∫ ∫
∫
(1)
where , dj is a driving electrode for j = 1, 2, … , N and , ks is a sensing electrode for k = 1, 2, … , 2N . One pair of sensing electrodes {, 2s j−1, , 2s j } are placed between one pair of driving electrodes {, dj , , dj+1}, as shown in Fig. 1(b). We measure voltage difference Vk = u ,ks+1 − u ,ks between two electrodes , ks+1 and , ks . The measured data set {V1, V2, … , V2N } is used to reconstruct the shape of ∂D .
2.2. The relation between the data and the interface In order to estimate the interface of D from the data {V1, V2, … , V2N }, we introduce the computed reference data Uk0 defined as
⎛ Uk0≔I ⎜⎜ ∑ uk0 ⎝ j = even
, dj
−
∑ j = odd
uk0
⎞ ⎟, ⎠
, dj ⎟
(2)
where uk0 is the solution of the following equations for k = 1, 2, … , 2N :
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integral term in right hand side of Eq. (4) can be viewed as a functional Φ k (ϕ) with the smooth function ϕ (θ ). Defining Φ k (ϕ)≔ ∫∂D u (∂uk0 /∂n) ds , the inverse problem of determining the
shape of D is reduced to find ϕ minimizing the cost functional Ψ (ϕ) that is the sum of differences between Φ k (ϕ) and IVk − Uk0 : N
(
Ψ (ϕ)≔ ∑ Φ k (ϕ) − IVk − Uk0
)
2
(6)
k=1
Fig. 2. Electric potentials u0k for k = 1, 2, …, 2N . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
⎧ 2 0 ⎪− ∇ uk = 0 ⎪ 0 ⎪ , s n·∇u k ds = I , ⎪ k +1 ⎪ 0 ⎪ , s n·∇u k ds = − I , ⎪ k ⎪ ⎪ 0 ⎪ n × ∇u k = 0 ⎪ ⎨ ⎪ 0 ⎪ , d n·∇u k ds = 0 ⎪ j ⎪ ⎪ n·∇u k0 ds = 0 ⎪ ,s ⎪ k′ ⎪ ⎪ 0 ⎪ n·∇u k = 0 ⎪ ⎩
in Ω,
∫
Solving the above minimization problem requires for an iterative procedure such as steepest descent method. In addition, we usually add a regularization term to the cost functional in order to obtain a stable solution ϕ [21]. For monitoring in the waterlubricated oil flows, one of the critical issues is precisely to find the minimum distance between ∂D and ∂Ω in order to determine whether the fouling occurs or to predict chances of the fouling. The present reconstruction method based on the minimization problem may accurately visualize the shape of ∂D and measure the minimum distance. However, due to the computation-intensive iterative procedure in finding the minima of Ψ (ϕ), we propose a fast direct method in the next section.
∫
2.3. Simple direct reconstruction
⎞ ⎞ ⎛ 2N ⎛ N on ⎜ ⋃ , dj ⎟ ∪ ⎜ ⋃ , s ⎟ , ⎜ j =1 ⎟ ⎜ k′⎟ ⎠ ⎝ k′= 1 ⎠ ⎝
∫
for j ∈ {1, 2, …, N} ,
∫
for k′ ∈ {1, 2, …, 2N}⧹{k, k + 1} , ⎛⎛ N ⎞⎞ ⎞ ⎛ 2N ⎟ ⎜ on ∂Ω⧹ ⎜ ⎜ ⋃ , dj ⎟ ∪ ⎜ ⋃ , s ⎟ ⎟ . ⎟ ⎟ ⎜ ⎜ ⎝ ⎝ j = 1 ⎠ ⎝ k′= 1 k′⎠ ⎠
(3) 0
It is worthy to note that the electrical potential uk as an eigenmode, that is independent of the interface of D, can be numerically constructed by solving Eq. (3) for computing the reference data Uk0. In Fig. 2, the colors indicate the electric potentials uk0 while the black lines represent paths of electric current density. Taking an advantage of the reference data, the quantity IVk − Uk0 is related to D through the following key identity:
IVk − Uk0 =
∂u 0
∫∂D u ∂nk ds.
(4)
In this paper, the above identity (4) is referred as a data-interface formula that relates the measurement data with the interface. The proof of the identity is as follows:
⎞ ⎛ IVk = I ⎜⎜u , ks+1 − u , ks ⎟⎟ = ⎠ ⎝
0
∫∂Ω u ∂∂unk
ds
=
∫Ω ∇u·∇uk0 dr = ∫Ω ⧹D ∇·(uk0 ∇u) dr + ∫D ∇·(u∇uk0 ) dr
=
∫∂Ω uk0 ∂∂nu ds + ∫∂D u ∂∂unk
0
= Uk0 +
ds
p j ≔max {p: (p cos θ j, p sin θ j ) ∈ D, 0 < p < 1}
for j = 1, 2
, …, N ,
(7)
where θ j = π /N + (j − 1) × 2π /N represents an angle between a mid-point of the sensing electrodes , sj and , sj+1 and the reference driving electrode. In the circular domain with a radius R, the distance dj from the domain boundary ∂Ω to the interface ∂D is defined as d j≔R − p j . If ∂D is closer to ∂Ω , i.e., d j /R⪡1, the electric potential distribution is mostly affected by currents injected from the electrodes , 2j − 1 and , 2j . Hence, the measured voltage difference Vj can be approximated by the electric potential generated by the currents from the electrodes. Moreover, the electric potential in the circular domain can be estimated by that in a square domain Ξ [d j ] due to a thin layer approximation (d j /R⪡1). The square domain is defined as
Ξ [h]≔{square domain with height h and width |∂Ω| /3} ,
0
∫∂D u ∂∂unk
For a real-time monitoring, a fast direct method is more attractive than the minimization problem based iterative method described in Section 2.2. Since a measured voltage difference on a sensing electrode may not be significantly affected by driving electrodes that are not adjacent to the sensing electrode, it is possible to obtain an explicit data-interface formula that provides a reasonably accurate estimation of the minimum distance. In addition, when the conductivity distribution is homogeneous, the resulting electric potential u can be approximated in a circular domain as u (r ) ≈ r 8, where r is a radial distance from the center of the pipe. Based on the assumptions, we propose a simple and direct method to reconstruct ∂D . We only use the measured data {V1, V3, … , V2k − 1, … , V2N − 1} at the sensing electrodes that are in between two driving electrodes as shown in Fig. 3. In the direct method, the target object D is approximated by N points on ∂D . As shown in Fig. 3, a radial distance pj at each angle θj from the origin to ∂D is defined as
(8)
ds .
Assuming that D is a star-shaped smooth domain given by a smooth function ϕ (θ ), 0 ≤ θ ≤ 2π :
D = {(x, y) = (r cos θ , r sin θ ):
x2 + y2 = r < ϕ (θ ), 0 ≤ θ ≤ 2π} , (5)
where (x,y) represents Cartesian coordinates for a point in D. The
where the corresponding electrodes system in the square domain is illustrated in Fig. 3(c). The voltage difference W between two sensing electrodes ,±s can be computed by an explicit data-interface formula
W = g (h)≔
∫,
s +
vh ds −
∫,
s −
vh ds,
(9)
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a
b
c
Fig. 3. Schematic configurations of (a) electrodes, (b) unit segment for electrodes in the proposed EIT system with the direct reconstruction method, and (c) unit segment for electrodes in a thin layer region Ξ with conducting fluid.
where the electric potential vh is the solution of the following equation:
⎧∇2v = 0 h ⎪ ⎪ ⎪ , d+ n·∇vh ds ⎪ ⎪ n·∇vh ds ⎪ ,−d ⎪ ⎪ ⎨ n × ∇vh = 0 ⎪ ⎪ d n·∇vh ds ⎪ ,± ⎪ ⎪ s n·∇vh ds ⎪ ,± ⎪ ⎪ ⎩ n·∇vh = 0
We performed numerical simulations to demonstrate the feasibility of the direct method described in Section 2.3. The direct method is applied in the following steps.
in Ξ [h],
∫
= I,
∫
= − I,
1. 2. 3. 4. 5.
on , ±s ∪ , ±d ,
∫
= 0,
∫
= 0,
(
3. Simulation results
)
on ∂Ω⧹ , ±s ∪ , ±d .
Measure voltage difference data {V1, V3, … , V2k − 1, … , V2N − 1}. Obtain the solution set {vh }h ∈(0,0.5] of Eq. (10). Construct the explicit data-interface formula W = g (h). Compute the estimated distance h j = g −1 (V2j − 1) at each angle θj. Visualize ∂D with the estimated nodes (h j cos θ j, h j sin θ j ).
(10)
For a given distance h, the electric potential vh is computed from Eq. (10) and the voltage difference W is then obtained from the explicit data-interface formula in Eq. (9). The corresponding voltage–distance map is shown in Fig. 4. Note that the amplitude of injected current is 2 mA with 10 kHz frequency, which is within the range of experimental conditions in [22]. We assume R = 1 m for simplicity. The voltage difference W monotonically decreases as the distance h increases. Moreover, the voltage difference is sensitive to the distance for h/R < 0.1. This implies that the resulting voltage differences may estimate the minimum distance of the interface ∂D accurately when the insulated material is closer to the boundary of the electrodes. Furthermore, the distance h j = R − d j in a radial direction at each angle θj can be explicitly estimated by using the computed g −1 with the measured voltage difference data V2j − 1:
h j = g −1 (V2j − 1)
for j = 1, 2, …, N .
(11) Fig. 4. Distance (h) induced voltage difference (W ) .
H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
a
b
331
c
Fig. 5. (a) Regions for the two immiscible fluids, (b) electric voltage distributions and current patterns, and (c) the reconstructed interface of the two fluids indicated by the red line in the model problem. Note that a grey region indicates the insulated oil while a region outside the oil contains conducting water. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.) Table 1 Minimum distances and reconstruction errors of the distance at each case for the model problem. Case
dmin /R
∼ dmin /R
∼ hmin /R
e/R
e∼/R
1 2
0.2500 0.0833
0.2569 0.1083
0.2500 0.1000
0.0000 0.0167
0.0069 0.0083
To quantify the proximity of ∂D and ∂Ω , a parameter dmin is defined as
{
}
dmin = min |p1 − p2 | : p1 ∈ ∂D, p2 ∈ ∂Ω , which implies the exact minimum distance between ∂D and ∂Ω in the domain Ω . Due to the limitation of the number of sensing
∼ electrodes, a pseudo-minimum distance dmin is defined as
∼ dmin
{
}
= min |R − r| : (r cos θ j, r sin θ j ) ∈ ∂D, 0 < r < R, j = 1, 2, …, N
where θ j = π /N + (j − 1) × 2π /N . This indicates the pseudo-minimum distance from the mid-point of sensing electrodes to a point on ∂D aligned with the radial direction to the mid-point, which is the best measurable minimum distance. The estimated minimum distance based on the proposed method is defined as
∼ hmin = min h j : j = 1, 2, …, N .
{
}
The corresponding reconstruction errors for the minimum dis∼ ∼ ∼ tances are defined as e = |dmin − hmin | and e∼ = |dmin − hmin |. We consider model problems that are numerically defined for a
Fig. 6. The flow configuration of core-annular flow with U-bend return pipe in [7].
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H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
a
b
Fig. 7. (a) The cross-sectional images of the oil–water flow at each downstream location, which are adopted from the CFD results in [7] and (b) the present reconstructed interfaces (red line) with the reference interfaces from [7] (black line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Table 2 Minimum distances and reconstruction errors of the distance at each downstream location for the oil–water flow. Location
dmin /RU
∼ dmin /RU
∼ hmin /RU
e/RU
e∼/RU
1 2 3 4 5
0.3627 0.2543 0.1666 0.1007 0.0000
0.3627 0.2546 0.1965 0.1276 0.0000
0.3250 0.2500 0.1875 0.1250 0.0022
0.0377 0.0043 0.0209 0.0243 0.0022
0.0377 0.0046 0.0090 0.0026 0.0022
Fig. 8. Minimum distances at each downstream location for the oil–water flow.
region D in the domain Ω . The domain Ω is the unit circle centered at (0, 0) and D is also a circle with radius R D = 2/3R . The center position for D is (0, − 1/12R ) and (0, − 1/4R ) for the cases 1 and 2, respectively. In both cases, the conductivity and permittivity of Ω⧹D are set to unity while those are 10 6 for D. Eight driving electrodes and sixteen sensing electrodes are used for the reconstruction of ∂D . Fig. 5 shows numerical simulation results for both cases. As D moves toward ∂Ω , it is observed for the case 2 in Fig. 5(b) that electrical potential is higher near the region where the distance between ∂D and ∂Ω is smaller. The proposed method reconstructs the interfaces accurately only on the region where the distance is small, as shown in Fig. 5(c). Table 1 shows quantitative errors for reconstructing the interface using the proposed method for the model problems, which shows less than 2% errors for
finding the minimum distances in both cases. As an example of the water-lubricated oil flows, we consider a core-annular flow in an U-bend return pipeline, which was numerically studied in [7]. The flow configuration in Fig. 6 shows that the U-bend geometry is a tube of RU = 0.006 m radius having 0.15 m straight length at the upstream and downstream. We choose the case for core-annular downflow with oil superficial velocity vso = 0.15 m s−1 and water superficial velocity − vsw = 0.3 m s 1. The distributions of oil and water phases from the CFD results in [7] are used for testing the proposed method. Fig. 7 shows the reconstruction images of the interfaces between oil and water phases compared with reference data [7] at different downstream locations. The initial shape of oil phase is maintained up to the straight tube region as shown in the image of cross section ①. However, the images of cross sections ②–④ show that the oil phase moves toward the outer portion of bend curvature due to the impact of centrifugal forces. At the cross section ⑤, the oil phase sticks to the outer portion of bend curvature. The proposed model clearly captures the interfaces near the region where the interfaces are closer to the pipe wall while the interfaces are less accurately reconstructed where they are far from the wall. It is worthy to note that the proposed model is able to capture the fouling region at the cross section ⑤ as shown in Fig. 7(b). In overall, Table 2 shows that the reconstruction errors of the minimum distances that relate with monitoring of the fouling are less than 4% for all test images. Comparisons of the estimated ∼ minimum distance hmin with the exact minimum distance dmin and ∼ the pseudo-minimum distance dmin are illustrated in Fig. 8. As expected, the estimated distances show in a good agreement with the pseudo-minimum distances. Interestingly enough, as the interface is closer to the pipe wall, the reconstruction error for the minimum distance becomes smaller. This implies that the proposed method might be more accurate in estimating the minimum distance just before fouling occurs.
4. Conclusion We proposed a simple and efficient monitoring method for detecting fouling in water-lubricated oil flows. Based on a singledrive interleaved electrode system which measures the voltage difference through sensing electrodes, we derived a data–interface relation between oil–water interfaces and measured data at each cross section of a pipeline. By introducing a thin-layer approximation for the electric potential near the pipe wall, the data–interface formula was simplified as the voltage–distance map for reconstructing the interfaces. We found that, for the given current
H. Kwon et al. / Flow Measurement and Instrumentation 46 (2015) 327–333
injection with the constant value at fixed electrodes, voltage differences monotonically increases as insulated materials are closer to the boundary of the electrodes. The results for the model problem showed that the proposed direct reconstruction method accurately estimates the minimum distance of the interface from the boundary while it provides a rough shape of the insulated region. Finally, we demonstrated the feasibility of the proposed method by reconstructing the oil–water interfaces and analyzing the minimum distance as well as rough shape of the oil phase from cross-sectional images of CFD results for the U-bend return pipeline in [7].
Acknowledgment This work is supported by the National Research Foundation of Korea (NRF) Grant (NRF-20151009350) funded by the Korean government (MEST).
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