Fuzzy Regulator for Two-Phase Gas–Liquid Pipe Flows Control

Abstract

The paper presents an intelligent module to control dynamic two-phase gas–liquid mixtures pipelines flow processes. The module is intelligent because it uses the algorithm based on AI methods, namely, fuzzy logic inference, to build the fuzzy regulator concept. The developed modification has allowed to design and implement the black-box type regulator. Therefore, it is not required to determine any of the complicated computer models of the flow rig, which is unfortunately necessary when using the classic regulators. The inputs of the regulator are four linguistic variables that are decomposed into two classes and two methods of fuzzification. The first input class describes the current values of gas and liquid pipe flows, which at the same time are the controlled values manipulated to generate desired flow type. The second class of the input signals contains a current flow state, namely, its name and the name preferred by the operator flow type. This approach improves the control accuracy since the given flow type can be generated with different gas and liquid volume fractions. Those values can be optimized by knowing the current flow type. Moreover, the fuzzification algorithm used for the input signals included in the first-class covers the current crisp signal value and its trend making the inference more accurate and resistant to slight measurement system inaccuracy. This approach of defined input signals in such environments is used for the first time. Considering all mentioned methods, it is possible to generate the desired flow type by manipulating the system input signals by minimum required values. Furthermore, a flow type can be changed by adjusting only one of the input signals. As an output of the inference process, two linguistic values are received, which are fuzzified adjustment values of the liquid pump and gas flow meter. The regulator looks to be universal, and it can be adopted by multiple test and production rigs. Moreover, once configured with a dedicated rig, it can be easily operated by the non (domain) technical staff. The usage of fuzzy terms makes understanding both the control strategy working principles and the obtained results easy.

1. Introduction

In the frame of this work, the fuzzy regulator dedicated to flow processes control in the laboratory or industrial rigs was constructed. Some of the significant improvements of the classic fuzzy regulator were developed and tested. Due to the new features, the two-phase flow type can be tuned by adjusting one control parameter. It will be shown within this paper that the application of the fuzzy logic mechanism to control the dynamic processes such as a two-phase gas–liquid mixture pipe flow allows to design and implement the black-box type [1] regulator. It is not required to determine any complicated computer models of the flow rig [2], which, unfortunately, is necessary when using the classic regulators. In addition, it is easily possible to understand the following stages of the control process which seems to be an advantage from the technical point of view. This is because the idea of control reflects an expert method of the control process [3].

2. The Two-Phase Gas–Liquid Mixtures Flow

The two-phase gas–liquid flows belong to the most rapidly growing trend in fluid mechanics research. In recent years, there has been significant but still insufficient progress in broadening knowledge about these industrial processes. The two-phase pipe flows arouse growing interest because of their great practical significance. They are closely related to the rapidly developing research fields in bioprocess engineering, biotechnology, environmental engineering, energy, and many other related areas.

The two-phase gas–liquid pipe flows are components of many industrial processes [4,5]. Some of them are the aeration processes in chemical and bio-reactors, flotation processes, and water and sewage aeration systems. The main task of the aeration system is the production of an adequate fraction of aerated liquid and oxygen. The oxygen injection process is essential since the proper liquid circulation must be achieved. It helps to intensify a mass transfer process. One of the fundamental problems in this matter is an appropriate evaluation of the surface between two phases (i.e., gas and liquid)—so-called inter-phase surface. This crucial parameter from the mass transfer point of view is a subject of the research made by the scientific community. A significant effort has been made in the field of numerical modelling. In particular, the fractional volume-of-fluid (VOF) concept [6] has been adapted to simulate the dynamics of free boundaries. The combined method with the height–function interfacial curvature estimation was presented in [7]. The VOF technique has also been recently applied to simulate gas–liquid flows of scientific and practical interest [8,9,10].

The two-phase flow processes also occur in the bubble columns where the various physical and chemical processes occur. To determine the intensity of these processes, the size of the inter-phase surface should be controlled. For example, in air-lift columns and ejectors [11], the movement of the liquid stream is forced by the gas stream. Such devices are commonly used in extractive industries (e.g., flotation processes) or to precipitate some fraction of liquid in the sedimentation processes, such as degreasing where the size of bubbles is significant.

2.1. Characteristics of the Flow Types

Prediction of the two-phase gas–liquid pipe flows features is one of the biggest challenges in such a mixture domain. It is even more challenging since multiple variables affect such prediction, including pipeline orientation, phases contribution, pipeline geometry, and even gravity (pushing the gas phase to the top of the pipeline in a horizontal direction). In the horizontal pipeline orientation, there are six basic flow types recognized [7]: plug, slug, dispersed, wavy, annular, and stratified. There are also additional hybrid flow types keeping the features of two or more basic types. Those can be defined as transitional flows; however, there is a need to add an additional description since there will be a slight difference between the transitional flow of slug and plug types and transitional flow of slug and wavy flows. The flow regimes with the transition borders are often reported on flow maps, showing how the different two-phase pipe flows can be generated by varying the inflow conditions. Figure 6 presents the map of flow types that have been identified (see details in Section 6) for the flow rig (see details in Section 3) during experiments in the current work. However, in Figure 6 the reader may be familiar with qualitative CCD images of the mentioned flow types. The observed structures of the gas–liquid mixture flow in the pipes of the vertical ascending movement are compatible with those presented by Nicklin and Davidson [12]. These structures have been used to classify these types of flow. Five basic flow patterns known as bubble flow, slug flow, foam flow, ring flow, and dispersion flow are characterized. When the two-phase mixture is flowing from the top to bottom (counter-flow), the flow structures were defined by Oschinowo and Charles [13]. It is worth noting that these structures relate to the two-phase mixtures of liquids with a high viscosity—not exceeding 100 mPa·s. For liquids with higher viscosity, the flow structures have to be defined differently [14].

2.2. Flow Type Recognition

From a variety of industry-oriented imaging solutions’ points of view, the electrical capacitance tomography (ECT) [15,16,17,18,19,20,21,22,23,24,25,26] applied to the two-phase gas–liquid mixtures visualization [27] and the phase distribution calculation [28] is getting popular, especially when flow key parameters are required. Industry demands notably include efficient non-invasive automatic phase fraction calculation and flow structure recognition in the vertical and horizontal pipelines. This can be solved by using non-deterministic fuzzy-logic-based techniques for the analysis of volumetric images. The authors developed the automated two-phase gas–liquid flow type recognition based on a fuzzy evaluation of a series of reconstructed 3D ECT volumetric images [29] and the raw 3D ECT measurement data [30]. The set of fuzzy-based features was calculated for flow substructure classification. As a result of this analysis, the obtained features were used to classify given diagnostic information (reconstructed images or raw measurement data) into one of the known flow regime structures.

Authors demonstrated in their works that recognizing the two-phase vertical and horizontal flow substructures is possible using the information provided by the spatial analysis of the large set of three-dimensional images and measurement data generated by a non-invasive 3D ECT system. The flow substructures recognition was successfully performed using fuzzy logic inference with very high classification accuracy as good as the human expert work. The classification process of the two-phase flow substructures provides valuable information to construct more accurate two-phase flow maps [12] and reliable industrial process control and diagnosis. Due to the efficiency of the proposed flow evaluation technique and fast 3D image reconstruction [31], the CUDA development library was implemented [32]. Therefore, this process was done in online mode and was limited only by the 3D ECT hardware that could provide input measurement data with a speed of 12 fps.

The signal of recognized flow type will be given as an input to the regulator module presented in this article. Moreover, the recognition block, together with the regulator, is closed within one feedback loop.

2.3. Control Techniques for Flow Processes

The task of flow control is of substantial meaning in any branch of industry where these processes are used for production, transport, etc. Because of the applications’ variety in each case, the needs and demands are different. Nevertheless, the important thing is to keep the flow regime on the given level or avoid slugs’ occurrences. In the case of such dynamic, stochastic, and nonlinear processes as flows, one must remember that holding the same level of phases’ streams supply does not guarantee the flow regime stability [33]. Therefore, it is necessary to process continuous monitoring and to react in case of any abnormalities. In the world literature, many research works may be found that use various modelling techniques to provide the tools for control process simulation or even for threats estimation. The example showing the control task’s complexity is the mathematical flow control scheme introduced in [34] to minimize the turbulences based on four different models. Next, in [35] the readers may find out the discussion on the interdisciplinarity of the algorithms development process for flow control purposes. The authors issued requirements and limitations for optimizing process control tasks in the context of various applications. Furthermore, some problems in the flow modelling and control based on the flow pattern and flow map using the specified Eötvös number classification have been demonstrated in [36]. The authors of [37] deal with the results from simulations with the feedback flow control. They show how to avoid slugs and hold on to the stable conditions at the pipeline inlet and outlet. In contrast, without control, severe slug flow was experienced. In [38], in turn, the authors made an exhaustive analysis of control models commonly used by industrial control engineers. A discussion about the advantages and limitations of the single-input single-output (SISO) system in contrast to the multi-input multi-output (MIMO) feedback control system may be found. The case study on the controllability properties of a typical pipeline-riser system of two fluid flow with the PDE-based (partial differential equations) model was described in [39].

Besides the theoretical achievements, some works have been performed to design the feedback control solutions based on PID [40] or PI [41] controllers to demonstrate that this strategy can guarantee the stability of the flows, whereas manual choking appeared to be insufficient. In [42], the auto-tuned PI controller algorithm was introduced, which was developed and implemented for severe slugging control focusing on achieving stable operation and maximizing production. On the other hand, the example of the linear quadratic Gaussian feedback control applied to the separated boundary layer to reduce its separation bubble size has been developed in [43]. Moreover, the feedback linearization based on the riser-base and topside pressure was used in [44] for the control design of the production choke valve to prevent severe slugging flow conditions. Nevertheless, the authors of [45] analyzed the disadvantages of the production choke control system based on feedback PI-controller in the context of their new approach to the nonlinear state-feedback control law based on a first-principles model of the slugging phenomenon. In this case, only one parameter (the gas pressure) was adjusted. Elsewhere, the detailed analysis and discussion about some problems and optimization of the linear controllers were performed in [46] and especially in the case of slug attenuation in [47] or in [48]. Generally, after studying the referred works, one conclusion may be drawn. Due to the complexity of the flow processes, the design of the robust control system based on an active feedback control makes a significant challenge to achieve the desired performance and optimal efficiency.

Some solutions for flow control were developed to increase the system stability and optimize production. One of the techniques which may successfully compete with the linearized feedback systems is fuzzy inference. The fuzzy controllers have already been applied many times, including scheduling and controlling electrical operators [49], the maintenance of a floating level in a tank on top of the atmospheric distillation unit of the refinery [50], or the boiling water reactor as a recirculation flow control system [51]. The automation control based on fuzzy logic was adapted to the water intake area of the pumping station [52]. Many examples of the practical applications of fuzzy logic controllers may be found in the air conditioning systems for efficient energy operation and a comfortable environment [53,54,55]. The total mass flow rate has been divided between all rooms by a certain percentage using a fuzzy-logic system to get the optimum performance for each room. The authors of [56] designed the hybrid of PID and fuzzy controllers for indoor temperature control. The PID part was responsible for the main heat source management and the fuzzy part, in turn, supervised the PID controller.

PID controllers do not exhibit good performance for nonlinear systems such as flow processes. Some research works as comparative studies have been carried out such as [57] or [58] to indicate the differences between the PID and fuzzy controllers. Many more advantages of fuzzy controllers than the PID may be found in [59]. Finally, to overcome problems inherited with the conventional PID control scheme such as handling unpredictable disturbance, non-measurable noise, and further improving the transient or steady-state response performance, the authors of [60] designed the fuzzy controller for flow application in tanks.

Despite many examples for fuzzy logic controllers, the systems are typically dedicated to specific applications each time. A lot of work must be performed to adjust such a controller to the process conditions. In this paper, the universal approach of the fuzzy logic controller will be introduced. The intuitive inference rules and individual sets of parameters ensure the easy adaptation to the work conditions, which can be carried out by the maintenance staff of the flow instrumentations. It is not required to determine any of the complicated computer models of the flow rig, which is unfortunately necessary when using the classic regulators. In Section 4, the universal equations of the fuzzy regulator have been presented, and in Section 5 the example of usage for the experimental rig (Section 3) has been included.

3. Two-Phase Flow Experimental Rig: Facility and Control Methodology

The experimental rig was built at the Institute of Applied Computer Science at the Lodz University of Technology. The flow rig can be supplied with fluid from two storage tanks and by a multi-stage centrifugal pump. In one of the tanks, a propeller stirrer is installed to prepare for research on liquid-like aqueous solutions of the various substances. Due to the installed electromagnetic flow meters, it is possible to precisely set the volumetric flow rate. The maximum pumping stream of liquid (with a viscosity close to the viscosity of water) delivered by Grundfos pump CRN’s 45 is 50 m³/h. In addition, the applied screw compressor DMD 100CR allows obtaining the volume stream of air at 1000 L/min with a pressure of about 7bar. A more detailed technical description of the rig, including the schemas and applied 3D electrical capacitance tomography (3D ECT) diagnosis principles, may be found in the authors’ previous work [30]. Due to the characteristic features of the flow rig, it is possible to obtain all types of two-phase pipe flows commonly found in industrial applications.

In Figure 1, the concept of the diagnostic control system for the experimental flow rig is depicted. The abbreviations used in the schema define the following signals: S_G—set value of gas stream; S_L—set value of liquid stream; f_C—current flow type; S_PT—process tomographic (i.e., 3D ECT) diagnostic signal; f_I—identified flow type; f_G—given flow type; N_SG—a new set of the gas stream; and N_SL—a new set of the liquid stream.

The three parts: rig hardware (i.e., pump, flow meters, pipelines, etc.), diagnostic (i.e., 3D ECT and fuzzy identification module) and control are distinguished. Because the fuzzy regulator is designed to consider three input signals and provide two output signals, it can be classified as the so-called MIMO feedback control model [38]. The following section describes the regulator’s input signals and their division into two classes.

4. Fuzzy Regulator Principles

The classic fuzzy regulators consist of fuzzification, inference, and defuzzification modules. In the inference module, the defined fuzzy rules set is used to determine the behavior of the regulator. The task for the fuzzification module is to change the input signals domain from crisp (i.e., numerical values) into fuzzy (i.e., linguistic terms: small, middle, or large). The defuzzification module, in turn, changes the output fuzzy signal into the crisp one. The inference realized in the fuzzy regulator depends on the type of applied fuzzy rules. In practice, the following types of rules are used most often [61].

if (S_G = t1_SG) and (S_L = t1_SL) then (F₃ = t1_Fc)
if (S_G = t1_SG) or (S_G = t2_SG) and (S_L = t1_SL) then (F_c = t3_Fc)

(1)

Specificity of physical phenomena of two-phase pipe flows requires adapting the classical approach to the fuzzy regulator to enable control of the flow in the most effective and precise way. To achieve that, the authors decided to use two types of the regulator input signals. The first type (class) describes the current state of the flow, namely, the values of the signals that supply the process. Both signals in the first class are provided as linguistic variables, S_G and S_L, which terms are described by experimentally determined membership functions. Both linguistic variables are classically defined as a five-elements set: $\left(V,T,D,G,M\right)$, where:

• V—variable name, e.g., gas stream,
• T—set of variable terms, e.g., small, medium, or big
• D—crisp domain of a linguistic variable, e.g., $D\in \left(0;50\right)\frac{m^3}{h}$,
• G—the syntactic rule, whose grammar generates the derivative terms—the T labels, e.g., T(gas stream) = {…, “medium small”, “medium”, …}
• M—the semantic rule that sets the meaning M(l) to each linguistic value l.

Moreover, M(l) describes the fuzzy set defined in D. The example for l = “medium”:

$$\begin{array}{c}M:\mathrm{medium}\to SR;\\ {\mu }_{SR}\left(x\right); x\in D,\end{array}$$

(2)

where, SR is the fuzzy set defined with the membership function μ_SR in the domain D.

Crisp values of signals for the gas and the liquid streams provided into the mixture (before fuzzification) are provided by the measurement system mounted in the flow rig.

Unfortunately, the nature of crisp values acquired by the measurement system influences the classic fuzzification method and can be imprecise regarding the measurement system’s sensitivity. Retrieved crisp values cannot be treated as correct ones since the measurement system provides the set of temporary values (this situation is strongly connected with the dynamic nature of flow). The first approach was to calculate an average value for the measurements set and use it as an input value for the fuzzification process. Unfortunately, this approach did not reflect the dynamics of the rig state properly.

The next challenge associated with the classic approach for fuzzification was the lack of possibility of changing the shape and type of membership functions to split the linguistic variable representing the input signals into terms [60].

Moreover, the linguistic variables representing the current state of the flow are not the subject of the classic fuzzification approach, and their conclusions always depend only on its value (i.e., current flow is a slug flow).

Since the input signal S is not a classical crisp value, it is provided in the form of a fuzzy number defined with the following membership function:

$${\mu }_{iS}\left(x\right)=\{\begin{array}{c}\begin{array}{c}0\to x\le \overline{x}-\sigma \\ \frac{1}{\sigma }x-\left(\frac{\overline{x}}{\sigma }+1\right)\to x\in (\overline{x}-\sigma ;\overline{x})\end{array}\\ -\frac{1}{\sigma }x+\left(\frac{\overline{x}}{\sigma }+1\right) \to x\in \left(\overline{x};\overline{x}+\sigma \right)\\ 0\to x\ge \overline{x}+\sigma \end{array},$$

(3)

where ${\mu }_{iS}\left(x\right)$ is the value of the membership function for $x \in d$, and:

• d—is the domain of the measurement,
• $\overline{x}$—is the mean measurement value for the measurement data collected in time t,
• $\sigma$—is the standard abbreviation for the measurement data collected in time t,
• $iS$—is the i-th term of input signal S.

The fuzzy number determined by the triangle-shaped function ${\mu }_{is}\left(x\right)$ is then fuzzified according to Zadeh’s extension principle [62,63] (i.e., determining the intersection of two fuzzy sets). Such an approach has not been used in similar scenarios so far. This means that the fuzzified input signal represents the current measurement and covers the dynamics of crisp input change over time. Such usage of fuzzy numbers introduces a possibility of reducing the influence of the measurement system inadequacy. The use of the modified membership functions for the input of linguistic variables increases the system’s sensitivity, especially in the areas close to the flow type boundaries. In this area, usage of the standard/common shape of membership functions gave slightly worse results of the flow type recognition.

The authors introduced the possibility of adjusting the membership functions by manipulating only one parameter in the described regulators. This parameter is within the ten-step regulation scale $a\in \langle 1;10\rangle$ reflecting the linear transition from classic function shape (parameter value 1) to the most enhanced shape, covering the requested area of the measurement domain. As a base function, the trapezoidal-shape function was used, in which the sections of the sides of the trapezoids (legs) were replaced with the exponential functions. Consequently, the non-crisp nature of the border between any of the two-phase flow types is reflected more accurately. It is also better customizable to the user’s subjective assessment. An example of the equation defining one of the membership functions (case Middle Large Gas) is as follows:

$${\mu }_{t_i{}^{\prime }}\left(x\right)=\{\begin{array}{c}\begin{array}{c}0\to x\le x_1\\ {(a_{1}x-b_1)}^{funMLaG1\left(a\right)}\to x\in \left(x_1 ;x_2\right)\end{array}\\ \begin{array}{c}1\to x\in \langle x_2 ;x_3\rangle \\ {(-a_{2}x+b_2)}^{funMLaG2\left(a\right)}\to x\in \left(x_3 ;x_4\right){\mathrm{m}}^3/\mathrm{h}\end{array}\\ 0\to x\ge x_4\end{array},$$

(4)

where:

$$\begin{array}{c}funMLaG1\left(a\right)=a_{3}a+b_3\\ funMLaG2\left(a\right)=a_{4}a+b_4\end{array}$$

(5)

In the second class of the input signals, two linguistic variables express the current flow type in the rig and the flow type required by the user. The linguistic variables in both classes differ because they are decomposed into terms using fuzzy relations regarding terms defined independently within their classes. Moreover, the 2D space in which the mentioned fuzzy relations are characterized determines the dependencies between two fractions streams. This is done the same way as constructing the two-phase flow maps [64] adapted to the particular flow rig. The membership functions describe the fuzzy relations in the set of linguistic variables. These correlate with the crisp boundaries between areas marked on the classic flow maps, which help build the fuzzy set concerning the flow types. It must be remembered that the same membership functions and the same set of values correspond with both linguistic variables.

Equation (6) demonstrates the fuzzy rule applied in the regulator:

if (S_C = t1_SC) and (S_L = t1_SL) and (f_C = t1_fC) and (f_G = t1_fG)
then (N_SG = t2_SG) and (N_SL = t2_SL),

(6)

where:

• S_C, S_L—the signals of the first class: the fuzzified value of the gas stream and the fuzzified values of the liquid stream,
• f_C, f_G—the signals of the second class: the current flow type and the required (given) flow type,
• ti_j—the i-th value (term) of the j-th input signal.

Having the input signals defined, it is possible to introduce the flow type transition with required dynamics. This means that an extended set of fuzzy rules allows changing the flow type without overloading the rig, i.e., changing the value of the input signal by significant value and putting the rig into an undefined state that can occur for a long time. However, generating desired flow type can be done much faster by introducing a progressive input signal value change.

The fuzzified Mamdani implication (7) was implemented to perform the interference process:

$$\varphi \left[{\mu }_{z1}\left(x\right),{\mu }_{z2}\left(y\right),{\mu }_{z3}\left(x,y\right),{\mu }_{z4}\left(x,y\right)\right]={\mu }_{z1}\left(x\right)\wedge {\mu }_{z2}\left(y\right)\wedge {\mu }_{z3}\left(x,y\right)\wedge {\mu }_{z4}\left(x,y\right),$$

(7)

where:

• ${\mu }_{z1}\left(x\right),{\mu }_{z2}\left(y\right)$—the membership functions describing the first class linguistic variables (fuzzy sets—µ_SG(x), µ_SC(x)),
• ${\mu }_{z3}\left(x,y\right),{\mu }_{z4}\left(x,y\right)$—the membership functions describing the second class linguistic variables (fuzzy relations—µ_fC(x,y), µ_fG(x,y)).

Two signals are the results of the inference rules’ conclusions of the control process. The sum of these signals is the output of the interference module that represents the requested flow rig state. This is the sum of the fuzzy sets. Two output sets represent the current state of the gas and liquid stream values. Their sum creates a fuzzy relation that can be visualized as an area on the flow map. Summation is done before defuzzification.

The defuzzification of the final fuzzy sets is done with the well-known center of sums defuzzification [65] method:

$$\overline{y}=\frac{{{\displaystyle \int }}_Y^{}y{{\displaystyle \sum }}_{k=1}^N{\mu }_{{\overline{B}}^k}\left(y\right)dy}{{{\displaystyle \int }}_Y^{}{{\displaystyle \sum }}_{k=1}^N{\mu }_{{\overline{B}}^k}\left(y\right)dy}$$

(8)

5. Implementation

The fuzzy regulator was implemented according to a schema shown in Figure 2.

The regulator consists of:

• The module for input signal fuzzification (1) that contains the first class (measurement data) (2) and second class (flow type) (3) signals fuzzification modules,
• The diagnostic signals (Z₁, Z₂, Z₃) retrieved from the measurement instruments,
• The flow type reference signal set by the user (Z₄),
• The inference rules module (4),
• The inference module based on the modified fuzzified Mamdani implication,
• The defuzzification module that implements the center of sums defuzzification method (5) and which processes the fuzzified control signals (Z′₁ and Z′₂),
• The module of the crisp sets (n₁ and n₂) (6).

The first step of implementation is to determine the membership functions that can transform the domain of the measured signal (i.e., streams of gas and liquid) from crisp into fuzzy form. The most proper solution for this is to use function (3) with the acquisition time interval t equal to 15 s. The data acquisition frequency results from the period for a single complete measurement frame acquisition (about 1 Hz). Next, the trapezoidal-shape membership function must be determined to split the measurement range of gas and liquid streams into the values of the first-class linguistic variables (fuzzified gas stream and fuzzified liquid stream). These membership functions are used in premises and conclusions of the inference rules. To correctly determine the shape of the membership function, it is required to collect a series of test measurement data. This should be done in the presence of an expert who can recognize (classify) the two-phase flow types. The gas and liquid streams intensity was changing sequentially within the discrete measurement range of the pump and compressor to generate all possible flow types. Next, each stabilized state was evaluated. The expert, not being aware of the values of the current settings, assessed the flow type in the pipeline only based on observations. Then, all the flow types should be printed on the chart as depicted in Figure 3. All flows’ series, observed in the whole measurement range, are split into all flow type occurrence areas.

To better clarify this description, the implementation of the control strategy and the obtained results are shown in Section 6 for two specific flow configurations, plus one case which is transitional between the two. The procedure explained here is analogous for the other flow types, for both horizontal and vertical pipelines. For the test data presented in Figure 3 it is possible to determine the following membership functions (using clustering or expert method):

• For the linguistic variable—fuzzified gas stream:

$${\mu }_{small ga{s}^{\prime }}\left(x\right)=\{\begin{array}{c}1\to x\le 5 \\ {(-x+6)}^{funLoG\left(a\right)}\to x\in \left(5;6\right)\\ 0\to x\ge 6 \end{array}$$

(9)

$${\mu }_{middle small ga{s}^{\prime }}\left(x\right)=\{\begin{array}{c}\begin{array}{c}0\to x\le 4 \\ {(0.5x-2)}^{funMLoG1\left(a\right)}\to x\in \left(4;6\right)\end{array}\\ \begin{array}{c}1\to x\in \langle 6;15\rangle \\ {(-x+16)}^{funMLoG2\left(a\right)}\to x\in \left(15;16\right)\end{array}\\ 0\to x\ge 16\end{array}$$

(10)

$${\mu }_{middle large ga{s}^{\prime }}\left(x\right)=\{\begin{array}{c}\begin{array}{c}0\to x\le 14.5 \\ {(0.5x-7.25)}^{funMLaG1\left(a\right)}\to x\in \left(14.5;16,5\right)\end{array}\\ \begin{array}{c}1\to x\in \langle 16.5;36\rangle \\ {(-x+16.5)}^{funMLaG2\left(a\right)}\to x\in \left(36;37\right)\end{array}\\ 0\to x\ge 37\end{array}$$

(11)

$${\mu }_{large ga{s}^{\prime }}\left(x\right)=\{\begin{array}{c}0\to x\ge 35 \\ {(0.5x-17.5)}^{funLaG\left(a\right)}\to x\in \left(35;37\right)\\ 1\to x\le 37 \end{array}$$

(12)

where:

$$funLoG\left(a\right)=0.089a+0.011$$

(13)

$$funMLoG1\left(a\right)=0.056a+0.444$$

(14)

$$funMLoG2\left(a\right)=0.11a+0.389$$

(15)

$$funMLaG1\left(a\right)=0.078a+0.122$$

(16)

$$funMLaG2\left(a\right)=0.078a+0.122$$

(17)

$$funLaG\left(a\right)=0.444a+0.556$$

(18)

where: $1\le a\le 10$

• For the linguistic variable—fuzzified liquid stream:

$${\mu }_{small liqui{d}^{\prime }}\left(x\right)=\{\begin{array}{c}1\to x\le 3.75 \\ {(-2x+8.5)}^{funLoL\left(a\right)}\to x\in \left(3.75;4.25\right)\\ 0\to x\ge 4.25\end{array}$$

(19)

$${\mu }_{middle small liqui{d}^{\prime }}\left(x\right)=\{\begin{array}{c}\begin{array}{c}0\to x\le 3.5 m^3/h\\ {(x-3.5)}^{funMLoL1\left(a\right)}\to x\in \left(3.5;4.5\right)\end{array}\\ \begin{array}{c}1\to x\in \langle 4.5;5.25\rangle \\ {(-2x+11.5)}^{funMLoL2\left(a\right)}\to x\in \left(5.25;5.75\right)\end{array}\\ 0\to x\ge 5.75\end{array}$$

(20)

$${\mu }_{middle large liqui{d}^{\prime }}\left(x\right)=\{\begin{array}{c}\begin{array}{c}0\to x\le 5 \\ {(x-5)}^{funMLaL1\left(a\right)}\to x\in \left(5;6\right)\end{array}\\ \begin{array}{c}1\to x\in \langle 6;6.25\rangle \\ {(-2x+13.5)}^{funMLaL2\left(a\right)}\to x\in \left(6.25;6.75\right)\end{array}\\ 0\to x\ge 6.75\end{array}$$

(21)

$${\mu }_{large liqui{d}^{\prime }}\left(x\right)=\{\begin{array}{c}0\to x\ge 6 \\ {(x-6)}^{funLaL\left(a\right)}\to x\in \left(6;7\right)\\ 0\to x\le 7 \end{array}$$

(22)

where:

$$funLoL\left(a\right)=0.1a$$

(23)

$$funMLoL1\left(a\right)=0.556a+2.444$$

(24)

$$funMLoL2\left(a\right)=0.889a+2.111$$

(25)

$$funMLaL1\left(a\right)=0.556a+2.444$$

(26)

$$funMLaL2\left(a\right)=0.089a+0.011$$

(27)

$$funLaL\left(a\right)=0.778a+0.222$$

(28)

where: $1\le a\le 10$.

When the scaled membership functions and measurement data are combined into one chart, it is possible to achieve a series of consistent areas that include only one flow type (see Figure 4).

The fuzzy relation that represents the flow type to reproduce it within the measurement range of the rig (second class signals) can be defined with previously determined membership functions. This relation should have two-dimensional nature such as $R={\displaystyle \sum }_{UxV}\frac{{\mu }_R\left(U,V\right)}{\left(U,V\right)}$, where U and V are the measurement values of the gas and liquid stream, respectively. The membership function of the relation should be an aggregation of the membership function values calculated according to the previous step for the streams: gas and liquid. The membership function, which represents each of the flow types (i.e., the terms of the second-class input signal), should look like:

$${\mu }_{f{t}_{\left(x,y\right)}=\min \left(\max \left({\mu }_{Gt1}\left(x\right), {\mu }_{Gt2}\left(x\right),{\mu }_{Gt3}\left(x\right),{\mu }_{Gt4}\left(x\right)\right), \max \left({\mu }_{Lt1}\left(y\right), {\mu }_{Lt2}\left(y\right),{\mu }_{Lt3}\left(y\right),{\mu }_{Lt4}\left(y\right)\right)\right), }$$

(29)

where:

• x—is the gas stream value,
• y—is the liquid stream value,
• ${\mu }_{f{t}_{\left(x,y\right)}}$—is the membership level of the flow type achieved for the gas stream x and liquid stream y,
• ${\mu }_{Gt\mathrm{i}}\left(x\right)$—is the membership level determined for i-th term of the fuzzified gas stream for the measured value x of the gas stream,
• ${\mu }_{Lt\mathrm{i}}\left(y\right)$—is the membership level determined for i-th term of the fuzzified liquid stream for the measured value y of the liquid stream.

The considered membership functions must be determined for each flow type that may occur in the entire measurement range of the flow rig. Next, it is required to define the fuzzy rules that consider the fuzzy relations and membership functions for the linguistic variables: fuzzy gas stream and fuzzy liquid stream. The inference structure of each rule should be derived from the Equation (1), for instance:

iffuzzified gas stream is smallandfuzzified liquid stream is small
andflow type is plugandrequired flow type is slug
thenfuzzy gas stream is largeandfuzzy liquid stream is large

(30)

The graphical interpretation of this is depicted in Figure 5.

The premise of the rule includes the following:

• Area #2 represents the current flow type,
• Area #3 represents the required flow type,
• Area #2 corresponds with the fuzzified values of gas and liquid streams that determine for current flow type.

The rule’s conclusion includes knowledge about the required gas and liquid streams: area #3.

The inference process is performed using the modified fuzzified Mamdani implication in the following form:

$$\begin{array}{c}{\mu }_{A,B,C,D\to E}\left(z_1,z_2,z_3,z_4\right)={\mu }_{R\left(z_1,z_2,z_3,z_4\right)}={\mu }_A\left(z_1\right)\wedge {\mu }_B\left(z_2\right)\wedge {\mu }_C\left(z_3\right)\wedge {\mu }_D\left(z_4\right)\\ =\min \left[{\mu }_A\left(z_1\right),{\mu }_B\left(z_2\right),{\mu }_C\left(z_3\right),{\mu }_D\left(z_4\right)\right]\end{array}$$

(31)

As it was assumed, the first-class signals are fuzzified using the non-singleton method. Thus, the measured signals of gas and liquid streams have to be changed into fuzzy numbers (using Equation (3), thereby eliminating the measurement system’s insufficient sensitivity that can occur when applying the non-optimal membership functions of the first-class signals). Next, the fuzzy numbers (measured gas stream and measured liquid stream) are combined with membership functions described by the Equations (9)–(12) and (19)–(22). During the fuzzification process, the maxima of sums of the sets measured gas/liquid stream and fuzzified gas/liquid stream are calculated (for each term of these linguistic variables under the condition that the fuzzy set represented by the term and the sets fuzzified gas/liquid stream are cumulated).

The final fuzzy sets are the aggregation of the conclusions of each fuzzy rule used in the inference process. In addition, the fuzzy sets from the conclusions of the fuzzy rules are multiplied by the result of the implication applied for the rule (i.e., the level of performing the inference rule that results from the level of the input signals fuzzification).

The defuzzification process of the fuzzy sets is performed using Equation (8), yielding the crisp values of the sets for the gas and liquid streams.

6. Results

The regulator described in this article has been verified in the experimental flow rig [26]. Even if the mentioned rig allowed to obtain all basic flow types, the effective control range of the gas flow rate (0–50 m³/h) and the liquid flow rate (3–9 m³/h) allowed to manipulate the flow type between the plug, slug, and transitional (both plug and slug features) flow. The same pipeline was used for algorithm learning and validation (horizontal orientation, 60 mm diameter). The validation covered 30 independent test flows’ examinations by both the algorithm and a human expert in the field of flow dynamics. It is worth mentioning that the flows recognized and categorized by the human expert in most cases matched the corresponding types defined in the flow map. The only differences occurred when the slug flow was classified as a transitional flow. This miscarriage can be explained by the fact that there is an ambiguous border between areas where the basic flows have their specific features. It should be noted that even if the flow is classified as a basic type, it can still represent some additional features that can be included in the other basic flow type. The flow map for the research setup shown in Figure 6.

The experiments were conducted as follows:

• Establishing the initial state,
• Selecting the required flow type by the expert,
• Validation of the stabilized flow type by the expert and the feedback from the diagnostic module [30].

Within the experiments, 30 tests were run. In each case, the final stabilized flow type was verified with that set by the expert.

Table 1. Report of the results of the test for control process.

Test Run

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Human expert evaluation

Plug → Transitional

Transitional → Plug

Plug → Slug

Slug → Plug

Plug → Transitional

Transitional → Slug

Slug → Transitional

Transitional → Slug

Slug → Plug

Plug → Slug

Slug → Transitional

Transitional → Stratified

Stratified → Slug

Slug → Dispersed

Dispersed → Stratified

Presented system evaluation

Plug → Transitional

Transitional → Plug

Plug → Slug

Slug → Plug

Plug → Transitional

Transitional → Slug

Slug → Transitional

Transitional → Slug

Slug → Plug

Plug → Slug

Slug → Transitional

Transitional → Stratified

Stratified → Slug

Slug → Dispersed

Dispersed → Stratified

Test Run

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Human expert evaluation

Stratified → Waved

Waved → Stratified

Stratified → Slug

Slug → Dispersed

Dispersed → Transitional

Transitional → Dispersed

Dispersed → Plug

Plug → Waved

Waved → Plug

Plug → Dispersed

Dispersed → Stratified

Stratified → Dispersed

Dispersed → Waved

Waved → Dispersed

Dispersed → Slug

Presented system evaluation

Stratified → Waved

Waved → Stratified

Stratified → Slug

Slug → Dispersed

Dispersed → Transitional

Transitional → Dispersed

Dispersed → Plug

Plug → Waved

Waved → Plug

Plug → Dispersed

Dispersed → Stratified

Stratified → Dispersed

Dispersed → Waved

Waved → Dispersed

Dispersed → Slug

The notation $T_1\to T_2$ is the transition from the initial flow type $T_1$ to the target flow type $T_2$.

consists of results for the tests that cover most flow types’ transitions that are possible to conduct within the setting range of the flow rig.

It took no more than 0.25 s to obtain the crisp values of the control signals. The flow stabilization, in turn, took significantly longer, up to 30 s. It is related to the initial and target flow types. The computer applied to conduct the computations consists of the i7-3612QM CPU and 8 GB and works under Microsoft Windows 10. The algorithms of the described regulator have been implemented in C# using the NET 4.5 framework. The dedicated communication protocol was designed to transmit (in a UDP multicast mode) the input and output signals (i.e., 3DECT measurements, the pump inverter, flow and pressure meters, and other instrumentation) through the computer network.

7. Conclusions

The fuzzy-logic-based pipe flow regulator presented in this article is characterized by the high operating speed and the universality of its application. It can be utilized in any rig capable of generating two-phase gas–liquid pipe flows where precise flow type control is needed. The mentioned precision was achieved using instrument input signals’ analysis and the current rig state feedback loop. This feedback loop gives the additional valuable algorithm input and allows for auto adjustment after changing the flow type from one to another. What is more, the fuzzy interference used in the algorithm favors smooth flow transitions over the crisp rig input parameter set. Introducing transitional flows into the recognition system avoids over-loop states and deadlocks when the basic type cannot be generated or recognized in the flow rig.

The regulator usage is intuitive and can be adopted efficiently by the rig operating staff. The adaptive method of the system development allows it to be utilized in different environments. Furthermore, it allows it to be extended with a dedicated UI (user interface), which can help with its adoption and suit an organization’s UX (user xperience) recommendation. Modular system design can also be the foundation for future redesign into a microservices ecosystem. This would allow creating a single solution for multiple rigs using a high-tech approach, such as cloud utilization or hybrid environment usage by utilizing the Kubernetes approach for orchestration.