ERT Based Computation of Solid Phase Fraction in Solid-Liquid Flow With Various Object Sizes

Solid phase fraction (SPF) is one of the most important parameters in solid-liquid two-phase flow, and has been increasingly addressed on most of the measuring techniques. As an effective measuring technique, Electrical Resistance Tomography (ERT) has been applied to measure SPF owing to low-cost, fast-response, non-invasive and non- radiation characteristics. The ERT-based SPF estimation is greatly affected by different solid object sizes from the existing methods, but currently there is none efficient method to solve this problem. In this paper, a mathematical model firstly is proposed to generally approximate various object sizes and thereby reconstruct all measurements. Therefore, when all solid objects have unevenly distributed and different sizes, SPF can still be effectively computed. Experiments are implemented in three groups of actual experiments by a building platform, where the solid objects in each group have individual object size. Results show that the new method can compute the value of SPF more accurate than the existing method, and thus provide a more accurate way to SPF computation.

The associate editor coordinating the review of this manuscript and approving it for publication was Mansoor Ahmed . estimation methods can be categorized to hardware refor-31 mulation and algorithm progress. Our research in this paper 32 focuses on the latter. Almost all the SPF estimation algo-33 rithms result from the Maxwell-Garnett (MG) formula [7], 34 but its preliminary form is very inaccurate due to inevitable 35 assumptions and complex application conditions. For exam-36 ple, for the solid-liquid two-phase flow in dredging engineer-37 ing [8], the MG formula remains rather inaccurate. Generally, 38 there are the following three problems at least: 39 1) Solid and liquid objects are assumedly small-size and 40 evenly distributed, and thereby SPF can be computed by the 41 MG equation. But actual sizes generally are various [9], and 42 thereby the computed value of SPF may be very inaccurate. 43 2) Most the existing methods focus on the gas-liquid two-44 phase flow whose natural characteristics are different from 45 those of the solid-liquid two-phase flow. And the conductivity 46 difference between gas and liquid is much larger than that 47 between solid and liquid inclusions. Meanwhile, gas is com-48 pressible but solid is not, leading different SPFs.   Fig.1 (b).

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The ERT measuring process obeys the general Maxwell where I is the exciting current. The ERT process is tightly 103 close to the inverse problem of the Dirichlet boundary condi-104 tions [17], which solves σ in by all boundary value u.

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The actual measurement in is required to use a reference 106 field 0 in which all pixels have the same conductivity value 107 [18]. After individually exciting 0 and , the increments 108 of both σ and u in the two fields are σ and u, and (1) is 109 further expressed as where J is a nonlinear relation from σ to u. Based on 112 finite element method, the linearized and discrete form after 113 neglecting nonlinear item of (2) can be expressed as the 114 following equations, where σ ∈ R n×1 is the vector of σ , U ∈ R m×1 is the vector 117 of measurements, S ∈ R m×n is called as sensitivity matrix in 118 ERT as well, n is the number of pixels in , and m is the 119 number of measurements. For a 16-electride system, m = 120 240; n is typically taken as 812 due to only 240 measurements 121 available. When both U and S are known, to solve σ can be 122 used to compute all parameters in .
where σ 1 is the conductivity of liquid-phase objects (e.g., where a and b are two regression parameters to reconstruct 172 σ mc from σ . Therefore, after taking (9) to (8), it is However, (10) is irrelative to solid object size in solid-liquid 175 two-phase flow due to the following reasons: 176 1) There is none way to effectively express solid object 177 shapes and sizes. In most applications, these objects may have 178 VOLUME 10, 2022 various sizes and uneven distributions which coexist in the same solid-liquid two-phase flow. 180 2) The mechanism is unknown that various object size 181 affects ERT measurement. Even though there is an effective 182 way to express the effect of object size to measurement, how 183 to apply it to improve the accuracy of SPF is unknown as well.

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3) The use of σ mc to compute SPF in MG in fact assumes 185 that solid object size is infinitesimal, but the spatial resolution 186 is very limited. The actual size in ERT is impossible more than 187 the size of pixels. Therefore, the assumption in MG doesn't 188 hold.  Since each exciting current goes through the same field , 214 thus any measurement is determined by their shortest dis-215 tance. In this following, we simulate the change of object size 216 from a large circle to a set of small circles.

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Assume that the object is a large circle O that is located 218 in the center of the field with radius r (see Fig.2 (a)).

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From an exciting electrode C to a measuring electrode E, their 220 connection line CE has angle θ to the horizontal line.

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If CE is not intersected to O, the measurement at E can 222 nearly be computed as where I is the exciting current intensity, γ is the conductivity 225 in , and d E is equal to the length of segment CE.
230 FIGURE 3. The changing tendency of the measuring sum as r or κ is changed.
The value of ϕ E can still be computed after taking (12) to 231 (11), but it is smaller since d E has an increment and becomes 232 larger than the length of CE. 233 Generally, if all objects are m randomly distributed circles 234 with the same radius r in (see Fig.2 (b)), ϕ 1 , ϕ 2 , . . . , 235 ϕ 15 are 15 relative measurments from the same excitation 236 C to 15 other measuring electrodes. They have indiviudal 237 segments lengths from C to 15 measuring electrodes, d 1 , 238 d 2 ,. . . , d 15 , respectively.

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Note that the 15 segments at C are intersected to m circles 240 in at a certain proability. In terms of width 2r, the covering 241 area of ith segment in nearly is 2rd i (see Fig. 2(b)), and thus 242 the proability that anyone of m circles intersects to the line is 243 2rd i /(πR 2 ). Hence, the number that all m circles intersects to 244 the ith segment probabilistically is In practice, the following problems must be considered: 247 1) These actual circle sizes usually are neither identical nor 248 circle-shaped, but it is impossible to construct an accurate 249 model to approximate these sizes and shapes.  mearuing electrode must have an increment whose largest 255 value is (π-2)r, as shown in Fig.2(c). 256 For the first poblem, we assume tha the effect of vari- So the shortest distance of ith measment is added to (d i + s i ), 267 ϕ i in (11) is turned to (18) shows that each measurement ϕ i is a nonlinearly decreas-282 ing function on object size r, whereas the measurement in 283 the MG formula is not related to r, leading to inaccurate 284 estimation of κ.

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Note that the sum of all measurements can reflect their 286 global changing tendency. As r increases from 0 to R in but 287 the value of κ is fixed individually at three different values, 288 Fig.3 (a) shows the three curves of measuring sum from (18). 289 Note that it is impossible that r tends to an infinitely small 290 value due to the limitation of pixel size. For example, the 291 typical number of pixels in is 812, and thereby the minimal 292 pixel size r min is πR 2 by 812. Fig. 3(a) shows that r min is much 293 larger than zero, and the measuring sum decreases globally as 294 r increases. 295 Alternatively, according to (16), (18) is rewritten along κ 296 to Equation (19) shows that each measurement is determined 300 by m and κ after r is fixed. But m is far larger than κ, thus 301 m plays a key role in the computing measurement. As κ 302 increases but m is fixed individually at four different values, 303 Fig.3 (b) shows four curves of measuring sum from (19), 304 which are globally decreasing. Table 1 further shows the effect of the object size r to all 306 measurements in COMSOL Multiphysics [25], where κ is 307 fixed to 25%, 30%, and 40%, respectively. The change of 308 all measurements is evaluated by their sums. For each fixed 309 VOLUME 10, 2022     (20), 325 the range of q can be restricted to a more accurate interval 326 than (0, 1). Table 2 shows a group of solutions of q according to (20) 328 when the background conductivity γ is individually taken 329 as 32.51, 18.52, and 0.055 uS/cm along a group of values 330 of κ. It is seen that the solved range of q is very small, and 331 the resultant relative error between maximal and minimal 332 measurements is small as well. Meanwhile, Table 2 shows 333 that the range of relative error nearly is reduced as γ decreases 334 or κ rises.

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After determining q and r, κ in (18) can be solved from ith 336 measurement ϕ i as In practice, the values of d i , γ , r must be determined in       With a close look on the accuracy of (22) along various 437 sizes of r, Fig.7 shows that computed transient values of κ by 438 MG and EMG when κ(l)=0.15, 0.20, and 0.25, respectively. 439 The mean of EMG is closer to the value of κ(l) than that of 440 MG. And Each EMG curve fluctuates around the correspond-441 ing line of κ(l), whereas MG does not fluctuate along the 442 line. Meanwhile, the amplitude of the computed values using 443 EMG is much smaller than that of MG. Therefore, EMG is 444 more accurate no matter which object size is encountered. 445 However, when the applicable condition of EMG is not meet, 446 it must include errors. There are inevitable noisy measure-447 ments in an ERT process, leading to the error between the 448 computed values of κ and the actual ones.

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In summary, the above experimental results demonstrate 450 that EMG can provide higher accuracy to compute κ along 451 various object sizes if its applicable condition is meet.

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The current SPF estimation methods based on ERT are prob-454 lematic in solid-liquid two-phase flow since these methods 455 have three inevitable limitations. Our proposed method can 456 decrease the negative effect of these limitations. To our 457 knowledge, so far there is none formulas that can effectively 458 compute SPF along various object sizes. The preliminary 459 results presented in this paper show that the proposed method 460 is comprehensive and effective for estimating the SPF values. 461 But the proposed method in this paper sometime is uncom-462 plete and inaccurate under complex flow conditions, such 463 as changeable liquid conductivity and uneven objects distri-464 butions in cross-section in a pipe. Meanwhile, the mean of 465 object size is necessarily known in prior. To overcome these 466 problems is just the work we are focusing on in present.