Two-Phase Flow Regime Identification with a ... - ACS Publications

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Ind. Eng. Chem. Res. 2005, 44, 4414-4426

Two-Phase Flow Regime Identification with a Multiclassification Support Vector Machine (SVM) Model Theodore B. Trafalis,† Olutayo Oladunni,† and Dimitrios V. Papavassiliou*,‡ The School of Industrial Engineering and The School of Chemical, Biological, and Materials Engineering, The University of Oklahoma, Norman, Oklahoma 73019

This paper presents a novel method for the classification of vertical and horizontal two-phase flow regimes in pipes based on a multiclass support vector machine model. Using previously published experimental data for gas-liquid vertical and horizontal two-phase flows, the goal of the model is to predict the transition region between the flow regimes. The transition region is determined with respect to pipe diameter, superficial gas velocity, and superficial liquid velocity. The support vectors of these data are identified and used to determine the transition zone between the multiphase flow patterns. The model proved to be an accurate classification tool for the identification of two-phase flow regimes in pipes. Our computational results show that flow regime predictions from the MSVM models are generally more accurate than predictions based on theoretical correlations. 1. Introduction Multiphase flow is the simultaneous flow of two or more phases in a conduit. The simultaneous flow causes certain flow patterns to evolve depending on the pipe size, the flow rates, the fluid properties, and the pipe inclination angle (when appropriate). Accurate determination of the flow regime is critical in the design of multiphase flow systems, which are used in various industrial processes, including boiling and condensation, oil and gas pipelines, and cooling systems for nuclear reactors. The problem of identifying flow regimes is the result of a lack of universal delineation criteria for the transition zones from one pattern to the other. Considerable progress has been made in defining flow patterns (Mandhane et al.,1 McQuillan and Whalley,2 Weisman and Kang,3 Taitel et al.,4 Taitel and Dukler5); however, there is no exact theory for the characterization of these patterns. Furthermore, the subjective character of the flow pattern identification often causes disagreements between researchers. While there is agreement on the existence of several flow patterns, there is often disagreement about the delineation point/transition boundaries for each flow pattern. Such disagreements make the selection of an appropriate flow correlation a complicated issue. Mechanistic models have also been used to study multiphase flow behavior. These mechanistic models are theoretical models that incorporate important variables coupled with state-of-the-art laboratory facilities for experiments.5,6 While mechanistic models offer an improvement in the understanding of multiphase flow systems, they are limited by the unavailability of precise solutions for the identification of different flow regimes. For most of the flow patterns observed, one or more * Corresponding author. 100 East Boyd Street, EC-T335, Norman, OK 73019. Tel.: (405) 325-5811. Fax: (405) 325-5813. E-mail: [email protected]. † The School of Industrial Engineering. ‡ The School of Chemical, Biological, and Materials Engineering.

empirical, closed-form relationships are required, even when a mechanistic approach is used. Therefore, it is important to develop a flow pattern model which minimizes the rate of misclassification errors (i.e., errors of predicting the wrong flow regime for a given set of flow data) as well as extends the applicability of any new multiphase flow correlation to different pipe sizes, flow rates, and fluid properties. More recently, attempts have been made to identify multiphase flow regimes using nonlinear classifiers derived from machine learning algorithms (Osman,7 Mi et al.,8 Ternyik et al.9) based strictly on data obtained from laboratory experiments. This new methodology for flow pattern identification has been successful, with researchers reporting promising accuracy rates. The implementation of support vector machine (SVM) for characterizing the multiphase flow regime/patterns in a pipe has not yet been explored. The idea of the SVM is to determine support vectors of experimental data with given attributes (i.e., the independent variables) and dichotomous labels (i.e., the dependent variable) and to use the support vectors to derive the weights that define the separating surface between the different patterns of the experimental data. These weights represent the level of influence of the independent variables on each dependent variable. For two-phase flow in a pipe, the weights represent the level of influence of pipe diameter, liquid superficial velocity, and gas superficial velocity on the flow regime. A considerable amount of data is integral to the development of multiphase flow models based on SVM. In this work, data have been gathered from several previous publications: 424 data points for the vertical upward flow and 2272 data points for the horizontal flow (see Tables 1 and 2, respectively). These experimental data were used to facilitate the modeling effort of the multiclassification support vector machine (MSVM) models to characterize the flow pattern of water-air in pipes. The primary objective of this paper is to identify and characterize vertical and horizontal two-phase flow patterns using MSVM, a data-driven methodology. The MSVM is an extension of support vector machines

10.1021/ie048973l CCC: $30.25 © 2005 American Chemical Society Published on Web 05/12/2005

Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005 4415 Table 1. Vertical Flow Data Sourcesa reference

pipe ID (m) no. of data points

Adsani (2002)26 Govier et al. (1957)27 Govier et al. (1957)27 Govier et al. (1957)27 Govier et al. (1957)27 McQuillan and Whalley (1985)2 McQuillan and Whalley (1985)2 a

0.0254 0.0160 0.0260 0.0381 0.0635 0.0508 0.0920

165 11 44 13 11 76 104

Gas-liquid is air-water.

Table 2. Horizontal Flow Data Sourcesa reference (1986)28

Andritsos Andritsos (1986)28 Barnea et al. (1985)29 Barnea et al. (1985)29 Chen (2001)30 Chen (2001)30 Chen (2001)30 Chen (2001)30 Crowley et al. (1986)31 Govier & Omer (1962)32 Green (1959)33 Jenkins (1947)34 Lamari (2001)35 White (1954)36 a

pipe ID (m)

no. of data points

0.0251 0.0953 0.0250 0.0510 0.0254 0.0455 0.0508 0.0513 0.1715 0.0261 0.0538 0.0254 0.0254 0.0381

175 149 205 193 210 249 154 158 16 15 33 516 90 109

Gas-liquid is air-water.

(SVMs), a machine learning tool involving three or more classes as opposed to the binary or two-class SVM. The foundations of SVM have been developed by Vapnik,10 and the SVM method can be applied in various disciplines such as science,11 finance,12 and engineering,13 because of its promising empirical and theoretical performance. The formulation embodies the structural risk minimization (SRM) principle,10 which equips SVM with a greater potential to generalize than that of the empirical risk minimization (ERM) principle10 that has been integrated in other statistical learning methods. An MSVM model seeks to determine the complex, hidden (and usually unknown) relationships between variables, by invoking its ability to handle ill-defined problems, i.e., discovering the hidden relationships of the inputs and outputs. In the present case, it performs a mapping of the variables in a high-dimensional (possibly infinite) feature space and returns weights used to formulate linear or nonlinear classifiers for transition boundaries of multiphase flow patterns. The objectives of the paper are as follows: (a) to model the transition between the flow regimes and develop MSVM equations with which the occurrence of the transitions may be predicted, (b) to develop MSVM multiphase flow regime maps using superficial velocities as coordinates, and (c) to compare existing flow regime correlations with MSVM flow regime classifiers. 2. Background 2.1. Flow Patterns in Two-Phase Flow. Simultaneous flow of several fluids with different fluid flow properties is more complex than single-phase flow. The influence of one phase over the other permits a flow regime (i.e., a specific distribution of each phase in the pipe relative to the other phase) to develop. Such a pattern may become unstable when the flow conditions change, consequently transitioning to another pattern, which at some point can also become unstable. By changing the flow rates of the gas and the liquid, this

transition from one pattern to another can go on until all possible flow regimes can be observed. Some of the conventionally identified flow regimes are as follows: annular flows, bubble flows, churn flows, stratified smooth flows, stratified wavy flows, and dispersed bubble flows.14 2.1.1. Vertical Flow. The vertical upward flow has four primary flow patterns, as accepted by most researchers.14 Their characteristics are described below. 2.1.1.a. Bubble Flow. In the bubble flow regime, the uniformly distributed gas phase flows as discrete bubbles in a continuous liquid phase. Bubble flow can be further divided into two types of flow, bubbly and dispersed bubble (DB) flow. In bubbly flow, the presence of slippage in the bubble flow allows relatively fewer and larger bubbles to move faster than the liquid phase, while in DB, numerous small bubbles are transported by the liquid phase due to the absence of slippage in the bubble flow, causing no relative motion between the two phases. 2.1.1.b. Slug Flow. In slug flow (which appears upon increasing the gas flow rate in bubble flow), the bubble concentration becomes high, coalescence occurs, and the largest bubbles are of the same cross-section as that of the pipe. Slug flow is characterized by a series of slug units composed of bullet-shaped gas pockets called gas plugs or Taylor bubbles, plugs of liquid called slugs, and a film of liquid around the Taylor bubble flowing vertically downward (there are also some gas bubbles distributed throughout the liquid). The liquid slugs carrying the gas bubbles bridge the pipe and separate two consecutive Taylor bubbles. 2.1.1.c. Churn Flow. Churn flow is a highly disorganized flow of a gas-liquid mixture, in which the vertical motion of the liquid is oscillatory and alternating. There are similarities with slug flow in that both phases do not exhibit any dominance over the other, i.e., neither phase appears to be continuous. The difference from slug flow is that the gas plugs become narrower and more irregular; the continuity of the liquid in the slugs is repeatedly destroyed by regions of high gas concentration, and the thin film of liquid surrounding the gas plugs is absent. Both slug and churn flow can be considered intermittent flow. Some researchers also define a subregion of the churn flow as froth flow,15 which occurs at higher gas velocities and exhibits a frothy mixture consisting of large bubbles. 2.1.1.d. Annular Flow. In annular flow, gas flows along the center of the pipe. The liquid flows upward, both as film and as dispersed droplets in the gas core. At high gas velocities, liquid becomes dispersed in the gas core, leaving a very thin film of liquid flowing along the pipe wall. Vertical flow pattern maps are used to predict the flow pattern in a vertical upward pipe that will occur for a given set of parameters, namely, flow rates, fluid properties, and pipe diameter. Taitel et al.4 developed a theoretical model for gas-liquid flows in vertical tubes. They identified the four distinct flow patterns mentioned above: bubble, slug, churn, and annular flow. They studied the physical mechanisms, taking into account the influence of fluid properties, pipe size, and flow rates by which regime transitions occur, and developed models for transition criteria. Weisman and Kang,3 using experimental data, proposed a theoretical model for vertical and upwardly inclined lines, with the exception of the vertical bubble-

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Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005

intermittent transition, which the authors contended can be described as the relationship between the gas phase and the total Froude number. The model also incorporated the effect of fluid properties and pipe diameters. They concluded that the behavior pattern is consistent with that seen in horizontal lines for the 2.5 cm and above pipes, both for the dispersed and annular transitions. McQuillan and Whalley2 developed a theoretical approach for flow patterns in vertical two-phase flow. A correlation was developed for the plug flow-churn flow transition based on the assumption that the gas flow rate in the plugs increases until it causes the flooding of the falling liquid film around the plug. Modifications were also made on the Taitel et al.4 analysis for determining the stability of bubble flow and annular flow, respectively. Mi et al.8 developed a neural network (NN) model for vertical flow regime identification from the impedance signal of laboratory instruments during a two-phase flow experiment. The NN model was based on two-phase flow models, such as the drift-flux model and a slugflow model, and a two-phase flow experimental database to obtain the impedance used as the input data for training and testing of the NN model. Mi et al.8 have chosen the impedance as input to the NN model. Impedance signals were measured by an impedance voidmeter. In our work, we adopted the superficial phase velocities (as suggested by Mandhane et al.1) and the pipe diameter as inputs for the MSVM model, instead of using the impedance as a model input, to develop a generally useful tool that can be used when a database already exists. 2.1.2. Horizontal Flow. Horizontal flow patterns are more complex than vertical flows due to gravitational forces. Gravity causes an asymmetric distribution of the phases by forcing the liquid phase to progress toward the bottom of the pipe. Described briefly below are the main patterns of horizontal flow that are widely accepted.5,14 2.1.2.a. Stratified Smooth Flow (SS). For the SS flow, the gravitational separation of the liquid and gas phase is complete. Liquid flows at the bottom of the pipe, and gas flows at the top. 2.1.2.b. Stratified Wavy Flow (SW). At increasing gas velocity in the SS flow, large waves start to develop on the liquid stratum giving the Stratified Wavy flow regime. Both SS and SW can be considered stratified flows. 2.1.2.c. Slug Flow. At increasing gas velocity in the SW flow, the waves of the liquid phase become large enough to reach the upper surface of the pipe. The liquid wets the whole pipe surface, allowing liquid film to cover the surface between the bridging waves or slugs. 2.1.2.d. Plug Flow. This is similar to the vertical upward slug flow, with bullet-shaped bubbles that tend to move along in a position closer to the top of the pipe. The liquid layer separating the gas bubble from the wall also tends to be thicker at the bottom of the pipe than at the top. Both slug and plug flows can be considered as intermittent flows. Also part of intermittent flows is the elongated bubble flow, which is considered a limiting case of slug flow free of entrained gas bubbles.16 2.1.2.e. Dispersed Bubble Flow (DB). Such flow occurs at high liquid rates and low gas rates. The gas phase is distributed as discrete bubbles within a continuous liquid phase. It can be characterized as a pipe

full with a liquid that has small bubbles dispersed uniformly throughout. 2.1.2.f. Annular Flow. This is similar to the vertical annular flow, which occurs at higher gas velocities, except that the liquid is much thicker at the bottom of the pipe than at the top. Horizontal flow pattern maps are used to predict the flow pattern in a horizontal pipe that will occur for a given set of parameters, namely, flow rates, fluid properties, and pipe diameter. Taitel and Dukler5 proposed a theoretical approach, perhaps the most significant contribution to the prediction of flow patterns in horizontal and near-horizontal gas-liquid flow. The regimes identified were: dispersed bubble, intermittent, stratified smooth and wavy, and annular flow. They showed that transitions between flow regimes were controlled by the fluid properties and the pipe size and can be represented by a set of two dimensionless groups. Experimental data from Barnea et al.16 were compared to the theoretical models with satisfactory results. Weisman et al.17 also developed a theoretical model for horizontal flow patterns, by investigating the effects of fluid properties and pipe diameter. The test cases for flow pattern covered a wide range of fluid properties in pipes varying in diameter from 1.2 to 5 cm. Comparisons were made with available literature data, and a revised dimensionless correlation was presented. Mandhane et al.,1 using extensive data from the University of Calgary multiphase flow databank, proposed an empirical gas-liquid flow pattern map in horizontal pipes with gas and liquid superficial velocity as its coordinates. Coordinates for the transition boundaries were presented. It was reported that there is no significant improvement in the accuracy of the flow pattern model by including the effects of the fluid properties. Osman7 and Ternyik et al.9 developed a NN model for horizontal flows using the gas and liquid fluid properties and flow rates, the liquid holdup, the pressure, the pipe diameter, and the temperature as inputs into the NN model. The output was a horizontal flow regime map. The NN models were successful, reporting better predictions and higher accuracy than the empirical correlations for the group of data used. 3. Support Vector Machines In this section, we consider the two-class classification problem. The SVM avoids overfitting by maximizing the margin between two classes of training data, i.e., maximizing the distance between the separating hyperplane of the training data on either side of it (see Figure 1). The formulation can be written in primal form10,18-20 as follows

min w,b,ξ

1 2

l

ξi ∑ i)1

|w|2 + C s.t.

yi(w‚xi + b) + ξi g 1 ξi g 0

i ) 1, ..., l

(1)

where xi ∈ Rd are the input training vectors, yi ∈ {+1, -1} are the corresponding labels, |w|2 ) wTw is the

Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005 4417

The decision function is given by l

Riyi〈xi,xj〉 + b) ∑ i)1

f(x) ) sign(

Figure 1. Support vector machine classification problem; the optimal hyperplane is orthogonal to the shortest line connecting the two classes and intersects it halfway.

square of the 2-norm of the weight vector defining the separating hyperplane, and ξi is a nonnegative slack term (penalty term) that measures the degree of violation of the constraints. The parameter C is a constant, called the regularization parameter, which controls the tradeoff between minimizing training errors and minimizing the norm of the weight vector (generalization ability). This parameter is to be chosen by the modeler, keeping in mind that a larger C corresponds to a higher penalty to the errors with less generalization ability and can result in a complex model. A smaller C corresponds to fewer penalties with a higher generalization ability and can result in a less complex model. Thus, there is a need to obtain a suitable tolerance level for errors, and this can be done by finding an optimal value of C that minimizes misclassification errors while training the SVM. This was accomplished in our work by crossvalidation techniques.20 Kuhn and Tucker21,22 have developed the necessary and sufficient optimality conditions by using the concept of the Lagrangian function. For the SVM, the Lagrangian is given by

(4)

Our optimal weight vector is then given by w ) ∑i Riyixi. The free coefficient b then can be deduced from Ri(yi(w‚xi + b) - 1) ) 0, for any i such that Ri is not zero (see Appendix A). In the case of nonlinear separation, the input vector x of the SVM can be transformed into a higher dimensional space, called the feature space, using the function φ:x ∈ Rd (input space) f φ(x) ∈ Rm (feature space). This transformation allows the solution of the classification problem in feature space with linear techniques. Note that the function φ(x) might not be available or cannot be computed. However, while φ(x) might not be available, one can still compute the inner product 〈φ(x1),φ(x2)〉 in feature space implicitly through a kernel function. This function can be viewed as determining the similarity or distance between the input vectors. The inner product in feature space can be expressed by the kernel function18 as follows:

k:Rd × Rd f R:k(x1,x2) ) 〈O(x1),O(x2)〉 Therefore, in the case of nonlinear separability, we can replace 〈xi,x〉 in the dual problem (eq 3) and the decision function (eq 4) by the kernel function, k(xi,x) (see Appendix A). Then the dual problem (eq 3) becomes l

maxw(R) ) max R

R

1

l

l

Ri - ∑∑RiRjyiyjk(xi,xj) ∑ 2i)1j)1 i)1 s.t.

0 e Ri e C, i ) 1, ..., l l

Riyi ) 0 ∑ i)1

(5)

and the decision function becomes

L(w,b,ξ,R,β) ) l

C

1 2

2

|w| +

l

f(x) ) sign(

l

l

ξi - ∑Ri(yi(w‚xi + b) - 1 + ξi) - ∑βiξi ∑ i)1 i)1 i)1

(2)

At the global saddle point, L should be minimized with respect to w, b, and ξ and maximized with respect to Ri, βi g 0, where Ri and βi are positive Lagrange multipliers. From the Kuhn-Tucker conditions, we use the necessary conditions and substitute those conditions in the L equation (eq 2) to formulate the dual problem (see Appendix A): l

maxw(R) ) max R

R

1

l

l

(6)

The kernel function used in this study is the polynomial kernel given as

k(xi,x) ) (xTi x + 1)p where p is the degree of the polynomial for the polynomial kernel function. There are several other kernel functions18 that can be used (such as the Gaussian radial basis kernel function). 4. Multiclassification Support Vector Machines

Ri - ∑∑RiRjyiyj〈xi,xj〉 ∑ 2i)1j)1 i)1 s.t.

0 e Ri e C, i ) 1, ..., l l

Riyi ) 0 ∑ i)1

Riyik(xi,x) + b) ∑ i)1

(3)

MSVM is an extension of support vector machines (SVMs), involving three or more classes. There is active research in this area, aiming at the reduction of the computational effort needed to solve the resulting large scale optimization problem. Earlier attempts involved solving k SVM models, where k is the number of classes and k(k-1)/2 is the number of SVM classifiers.11,23 Other attempts involved the solution of a single opti-

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mization problem using all the data at once.23-25 The method implemented in this study is from the earlier attempts, namely, the one-against-one (OAO) method.23 4.1. One-Against-One (OAO) Method. In the OAO method, one constructs k(k-1)/2 SVM classifiers, each one of which is trained on data for two classes. For example, if we have a three-class classification problem, we have to construct 3 classifiers P12, P13, and P23. When we train P12, all the points from class 1 are labeled with +1 and all the points from class 2 are labeled with -1; similarly for P13, class 1 is labeled with +1 class 3 is labeled with -1; and for P23, class 2 has the label +1 and class 3 has the label -1. For training data from the ith and the jth classes, we solve the following binary classification problem:11,23

min wij,bij,ξijt

1 2

|wij|2 + C

ξtij ∑ t)1

s.t. (wij)Tφ(xt) + bij g 1 - ξijt , if yt ) i (wij)Tφ(xt) + bij e 1 + ξijt , if yt ) j ξijt g 0

(7)

There are different strategies for performing the testing for points not in the training set after all k(k1)/2 classifiers have been constructed. The strategy employed is called the “Max Wins” strategy. This strategy is a voting approach. For example, if sign[(wij)Tφ(x) + bij] says that x is in the ith class, then the vote for the ith class is increased by one. Otherwise, the jth is increased by one. Then we predict x as being in the class with the largest vote. In the case that those two classes have identical votes, the one with the smallest index is selected. Practically, we solve the dual problem of eq 7, whose number of variables is the same as the number of data in the two classes. Hence, if on average each class has l/k data points, we have to solve k(k-1)/2 quadratic programming problems where each of them has 2l/k data points.23 5. Development of the Models 5.1. Dataset Description. An extensive collection of available data for which a flow pattern has been observed and recorded was used for this paper. There are two data sets, since we are dealing with two types of flow. 5.1.1. The Vertical Flow Data Set. There are 424 test cases (data points) obtained from several authors/ publications. The test case comprises bubble, intermittent, and annular flow. Table 1 presents the data sources. 5.1.2. The Horizontal Flow Data Set. There are 2272 test cases obtained from several authors/publications. The test case comprises dispersed bubble, intermittent, stratified smooth, stratified wavy, and annular flow. Table 2 presents the data sources. 5.2. Attribute Definition. The attributes of interest are (i) the gas superficial velocity (m/s); (ii) the liquid superficial velocity (m/s); and (iii) the pipe diameter (m). Two models (2D and 3D classification) were developed to investigate the transitions and flow maps. Table 3 illustrates the combinations of attributes used and the

Table 3. Model Data Combination attributes

2D MSVM classification

3D MSVM classification

gas superficial velocity liquid superficial velocity pipe diameter dimensions of vertical flow data set dimensions of horizontal flow data set

yes yes no 209 data points by 2 attributes 1195 data points by 2 attributes

yes yes yes 424 data points by 3 attributes 2272 data points by 3 attributes

dimensions of the data sets. Note that the fluid properties are not presently being considered. To make the computational effort favorable, the data were scaled by taking the natural logarithm of each test case. Both the vertical and horizontal data sets were divided as follows: 50% of the data were drawn randomly from the data sets and used as training data, while the remaining 50% was used as testing data. Both data sets were also randomized to produce four random sets for 3D (all three attributes) classification and three random sets for 2D classification (using as attributes the two superficial velocities). To reduce the computational time required for developing the model and also because of the larger data set for the 3D case, it was necessary to have more random samples in the 3D case than in the 2D case. 5.3. Multiclassification Support Vector Machines. MSVM models were developed for vertical and horizontal flow regimes using a polynomial kernel with degree p ) 1, 2, 3, and 4 and tradeoff cost, C, within the interval 0.001-1000. The responses considered were as follows: (a) Vertical flows: 1-bubble, 2-intermittent, and 3-annular (slug and churn flows were grouped together as intermittent, because for both flows neither the gas nor the liquid phase is dominant); (b) Horizontal flows: 1-annular, 2-dispersed bubble, 3-intermittent, 4-stratified smooth, and 5-stratified wavy. Once the flow regime responses were completely coded and random samples were obtained, we performed experiments for four sets of training and testing samples for the 3D case and three sets of training and testing samples for the 2D case. The one-against-one (OAO) Matlab Toolbox developed at Ohio State University37 (OSU SVM classifiers) was used to implement SVMs for multiclassification.11 5.4. Multiphase Flow Regime Correlations. To compare the results of the MSVM model with previous results, the performance of theoretical transition model equations when compared with the datasets was obtained. Among the many correlations available in the literature, the adopted ones were selected based on the consensus between multiphase flow researchers in more recent publications. Sections 5.4.1 and 5.4.2 include the transition equations for multiphase flow regime identification. 5.4.1. Vertical Flow Regime Correlations. The transition equations used for the prediction of the vertical flow regime map were based on the work of McQuillan and Whalley2 and are as follows (the reader can refer to the nomenclature section for the definition of the symbols appearing below):

Bubble-intermittent flow transition vLS ) 3.0vGS - 1.15

[

]

gσ(FL - FG) FL2

1/4

(8)


Bubble-dispersed bubble flow transition vLS g

6.8 0.444

FL

{gσ(FL - FG)}0.278

Transition to annular flow

[

D µL

0.112

(9)

]

[gD(FL - FG)] FG

vGS g

()

1/2

(10)

5.4.2. Horizontal Flow Regime Correlations. The transition equations used for the prediction of the horizontal flow regime map were based on the work of Weisman et al.17

Stratified-intermittent transition

( )

vGS ) 0.25 1/2 vLS (gD) vGS

1.1

(11)

Stratified wavy-stratified smooth transition

(

σ gD2∆F

) ( ) DGG µG

0.20

0.45

( )

)8

vGS vLS

0.16

(12)

∆F ) FL - FG Transition to annular flow 1.9

( ) vGS vLS

1/8

) Ku0.2Fr0.18 )

(

)( )

(13)

) 9.7

(14)

0.2

vGSFL1/2

[g(FL - FG)σ]1/4

vGS2 gD

0.18

Transition to dispersed bubble flow

(

)(

(dp/dx)L g∆F

1/2

σ ∆FDg2

(

)

-0.25

)( )

4CL FLvLSD (dp/dx)LS ) D µL

-n

FLvLS2 2

CL ) 0.046, n ) 0.2 6. Results and Discussion To demonstrate the uniqueness of the MSVM performance, a comparison between the MSVM and multiphase flow regime correlations was conducted, by introducing the two performance parameters defined below:

R)

(

)

number of correctly classified points in each flow regime × 100 number of observed points in each flow regime (15)

The accuracy is represented by R, and it is the percentage of success for a given flow pattern model with respect to a particular flow regime.

β)

(

)

total number of correctly classified points in the flow regime model × 100 total number of observed points in the flow regime model (16)

The overall accuracy, i.e., the percentage of success for a given flow pattern model, is represented by β. 6.1. Vertical Flows Results of the MSVM model, in comparison with those of the vertical theoretical model of McQuillan and Whalley2 (M&W), are shown in Tables 4 and 5. It should be noted that intermittent flows include froth, slug, and churn flow. Table 4 presents the results for the accuracy of the test set and the whole data set for the vertical flow 2D classification (gas and liquid velocities as variables) for 1 in. diameter pipes with fluid properties at atmospheric conditions. From Table 4, based on the accuracy metrics (R and β), we can conclude that the MSVM polynomial kernel shows great potential in identifying vertical multiphase flow regimes. Irrespective of the degree of nonlinearity, p, and tradeoff cost, C, the MSVM classifiers appear to beat or at least display an equal level of performance with the M&W theoretical correlations. For the whole data set (i.e., combined trained data set and test data set) accuracy, with respect to R, the MSVM displays a better performance for most flow patterns with the exception of the intermittent flow, where the M&W correlations show a 2-5% increase over the MSVM accuracy. With respect to the overall accuracy, β, the MSVM classifiers are the most favorable candidates. Table 5 presents the vertical flow 3D classification (pipe size and gas and liquid velocities as variables) accuracy for the test set and whole data set for all available pipe diameters with fluid properties at atmospheric conditions (the data were obtained from the data sources described in Table 1). On the basis of the accuracy rates (R and β), it is clear that the MSVM polynomial kernel outperforms the M&W vertical flow theoretical correlations. The annular flow accuracy is low compared to the 2D case; this might suggest that the correlations are inconsistent when identifying flow regimes with different pipe sizes. For the whole data set accuracy with respect to the accuracy percentage R, the MSVM displays a better performance for all flow patterns. With respect to the overall accuracy, β, the MSVM model clearly is the best for the identification of flow regimes. 6.2. Horizontal Flows. Results of the MSVM model, in comparison with the horizontal theoretical model of Weisman et al.17, are shown in Tables 6 and 7. It should be noted that bubble, elongated bubble, plug, slug, and pseudoslug are grouped together as intermittent flows. Furthermore, the annular flow consists of annular, annular mist, semi annular, film, and wavy annular. Table 6 presents the results of the test set and the whole data set for the horizontal flow 2D classification (gas and liquid velocities as variables) for ∼1 in. diameter pipes with fluid properties at atmospheric conditions. From Table 6, based on the accuracy rates (R and β), we can judge that the MSVM polynomial kernel shows great potential in identifying horizontal multiphase flow regimes over the Weisman correlations. For the whole data set accuracy, with respect to R, the MSVM displays a better performance for most flow patterns with the exception of the annular flow, where the Weisman correlations show a 2-5% increase over the MSVM accuracy. With respect to the overall accuracy, β, the MSVM classifiers are the most favorable candidates. Table 7 presents the horizontal flow 3D classification (pipe size and gas and liquid velocities as variables) for the test set of data and for all available pipe diameters

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Table 4. Vertical Flow 2D Classification Test Set and Whole Data Set (Training & Testing Set) Accuracya accuracy (%) MSVM, polynomial kernel p)1

1 in. pipes flow pattern

no. of obs.

C)1

C)3

bubble intermittent annular

18 50 34

100.00 100.00 97.06

100.00 100.00 97.06

total

102

99.02

99.02


44 102 63

93.94 95.10 93.12

94.70 94.77 92.06

total

209

94.26

93.94

p)2 C ) 0.09 C ) 0.1 Test Set 100.00 100.00 97.06 99.02 Train/Test Set 90.91 97.71 92.06 94.58

C ) 50

p)3 C ) 100

M&W2 vertical flow correlationsb

100.00 100.00 97.06

100.00 100.00 97.06

100.00 100.00 97.06

96.30 100.00 97.06

99.02

99.02

99.02

98.37

90.91 97.71 92.06

100.00 98.04 93.65

100.00 98.04 93.65

84.09 100.00 87.30

94.58

97.13

97.13

92.82

The accuracy reported for specific flow patterns is R (see eq 15). The accuracy reported for the total number of observations is β (see eq 16). bM&W refers to the results of McQuillan and Whalley.2 a

Table 5. Vertical Flow 3D Classification Test Set and Whole Data Set (Training & Testing Set) Accuracya accuracy (%) MSVM, polynomial kernel p)1

p)2 C ) 10

p)3 C ) 0.03

C ) 0.05

M&W2 vertical flow correlationsb

100.00 98.92 93.53

97.16 98.92 93.53

97.73 98.71 93.53

96.59 95.04 66.38

97.71

97.13

97.13

87.73

Train/Test Set 95.41 95.92 92.05 92.17 88.76 88.76

93.37 94.12 87.61

93.88 93.66 87.84

89.03 91.13 56.42

92.28

92.22

81.72

varying pipe sizes flow pattern

no. of obs.

C)5

C)7

C)7


44 116 58

94.89 94.89 93.10

94.89 94.89 93.10

Test Set 99.43 99.14 93.97

total

218

96.33

96.33

97.82


98 217 109

90.05 91.24 85.32

90.05 91.13 85.32

total

424

89.45

89.39

91.98

92.16

The accuracy reported for specific flow patterns is R (see eq 15). The accuracy reported for the total number of observations is β (see eq 16). b M&W refers to the results of McQuillan and Whalley.2 a

Table 6. Horizontal Flow 2D Classification Test Set and Whole Data Set (Training & Testing Set) Accuracya accuracy (%) MSVM, polynomial kernel p)1 C ) 50 C ) 100

p)2 C ) 0.5 C ) 0.7

219 3 199 59 118

99.54 100.00 97.82 97.18 88.70

99.54 100.00 97.82 97.18 88.70

99.39 100.00 98.16 98.87 88.98

598

96.60

96.60

96.88

1 in. pipes flow pattern

no. of obs.

annular disp. bubble intermittent SS SW total annular disp. bubble intermittent SS SW total

415 25 407 110 238

93.65 100.00 91.32 89.39 82.35

93.65 100.00 91.56 89.39 82.35

93.73 100.00 92.47 91.52 81.51

1195

90.35

90.43

90.79

p)3 C)7 C ) 10

Test Set 99.39 99.39 100.00 100.00 97.99 96.48 97.18 94.35 89.83 90.68

p)4 C ) 0.3 C ) 0.5

Weisman17 horizontal flow correlationsb

99.39 100.00 96.65 94.35 90.11

97.26 100.00 97.32 98.87 90.68

97.41 100.00 97.15 98.31 90.96

100.00 66.67 80.07 100.00 85.31

96.21

96.15

96.15

96.15

90.30

Train/Test Set 93.65 94.46 100.00 100.00 92.63 91.40 91.82 92.73 81.09 84.17

94.46 100.00 91.48 93.33 83.33

92.85 100.00 92.06 95.45 85.15

93.01 100.00 91.89 95.15 86.27

97.35 44.00 75.84 95.45 79.83

91.24

91.44

91.63

85.24

96.82

90.77

91.32

The accuracy reported for specific flow patterns is R (see eq 15). The accuracy reported for the total number of observations is β (see eq 16). b Weisman refers to the results of Weisman et al.17 a

with fluid properties at atmospheric conditions (the data sources for horizontal flow are shown in Table 2). From Table 7, based on the accuracy rates (R and β), it is clear that the MSVM polynomial kernel outperforms the Weisman theoretical correlations. The low accuracies of the dispersed bubble flow might suggest that the correlations are inadequate for such a flow pattern, even when several pipe sizes are considered.

For the whole data set accuracy with respect to R rates, the MSVM displays a better performance for most flow patterns with the exception of the stratified flows (SS & SW), in which case the Weisman correlation outperforms the MSVM model by at least 4%. However, with respect to the overall accuracy, β, the MSVM model clearly is more reliable for the identification of flow regimes.

Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005 4421 Table 7. Horizontal Flow 3D Classification Test Set and Whole Data Set (Training & Testing Set) Accuracya accuracy (%) MSVM, polynomial kernel p)1 C ) 100 C ) 1000

p)2 C ) 50 C ) 100

varying pipe size flow pattern

no. of obs.

annular disp. bubble intermittent SS SW

374 20 349 138 260

96.59 85.00 95.85 92.93 80.48

96.66 85.00 95.99 92.93 80.29

98.46 92.50 95.49 94.57 89.23

1141

92.05

92.07

94.87

total annular disp. bubble intermittent SS SW total

699 81 745 250 497

88.95 80.86 88.15 79.50 72.74

88.98 80.86 88.26 79.30 72.84

91.81 83.33 87.42 83.30 80.78

2272

83.81

83.86

86.72

p)3 C ) 0.5 C ) 0.7

p)4 C ) 0.03 C ) 0.05

Weisman17 horizontal flow correlationsb

Test Set 98.53 92.50 95.49 94.20 89.13

98.53 90.00 95.34 91.49 89.13

98.20 88.75 95.70 93.12 88.46

98.20 91.25 95.70 88.95 92.21

98.33 88.75 95.49 90.58 92.50

96.52 42.50 84.53 95.65 92.21

94.83

94.41

94.43

94.83

95.03

90.82

Train/Test Set 91.81 93.92 82.72 82.41 87.25 87.25 83.40 83.70 80.63 80.99

93.81 81.48 87.48 85.00 80.38

93.24 83.02 87.85 82.20 84.15

93.24 81.48 88.05 83.30 84.71

90.41 29.63 77.82 88.00 88.73

87.39

87.91

88.16

83.48

86.62

87.37

The accuracy reported for specific flow patterns is R (see eq 15). The accuracy reported for the total number of observations is β (see eq 16). b Weisman refers to the results of Weisman et al.17 a

Figure 2. Vertical two-phase flow regime map that results from the 2D multiclass SVM model using a polynomial kernel (p ) 1, C ) 1) and the McQuillan & Whalley2 correlation (blue lines ) MSVM, red lines ) theoretical correlation).

6.3. MSVM Flow Pattern Maps. Flow patterns are often displayed in the form of a 2-dimensional map illustrating the different flow regime transitions between the two-phase flow patterns. For any given system with flow rates or superficial velocities specified, a respective flow pattern will occur. We adopted the superficial phase velocities as coordinates that can serve as reasonable discrimination criteria, as indicated by Mandhane et al.1 While the superficial velocities are a reasonable choice of coordinates, the natural logarithm of the velocities seems to be more appropriate, because in SVM models we need to scale the data; unscaled data could bias the training network. The superimposed MSVM maps and theoretical maps are illustrated in Figures 2-5. The transition boundaries of the MSVM and theoretical maps were determined on the basis of the ln(VSG) vs ln(VSL) plot of the 2D vertical and horizontal data sets for air-water systems. The maps were created using the polynomial kernel, with degree p ) 1 and p ) 2, for both vertical and horizontal flows. Note that the figures were devel-

oped using 1 in. pipe diameter data. Please refer to Appendix B for MSVM transition classifiers. There is a noticeable difference between the maps with polynomial kernels of degrees 1 and 2. A polynomial of degree 1 is an indication that the flow patterns might actually be linearly separable, i.e., separation of flow patterns can be accomplished with linear equations, while degree 2 clearly implies a nonlinear transition boundary between flow patterns. The MSVM maps are qualitatively similar to the theoretical maps available in the literature (i.e., the flow regimes appear at the same region of the map), but they differ in the exact position of the transition boundaries. The transition zones of the MSVM maps are based strictly on the support vectors of the training data. These support vectors are the most critical points of each flow pattern, and they are used to define the transition boundaries from one flow pattern to another. The transition zones of the theoretical maps are based on analytical models and/or dimensionless correlations which explore relationships between fluid properties, pipe size, and flow

4422


Figure 3. Vertical two-phase flow regime map that results from a 2D multiclass SVM model using a polynomial kernel (p ) 2, C ) 0.1, blue lines) and the McQuillan & Whalley2 correlation (red lines).

Figure 4. Horizontal two-phase flow regime map that results from a 2D multiclass SVM model using a polynomial kernel (p ) 1, C ) 100, blue lines) and the Weisman et al.17 correlation (red lines).

rates. The presented maps have very good prediction capability, outperforming the theoretical model maps (as seen in Tables 4-7), but the MSVM maps are better suited for a system that has a 1 in. diameter pipe. As indicated by Mandhane et al.,1 the fluid properties and pipe diameter, however, have only a moderate influence on the flow patterns. To test this claim, the pipe diameter and fluid properties were omitted and all the available data were tested (424 vertical points and 2272 horizontal pointts) on the MSVM 2D classifiers developed for the 1 in. pipes, and an overall accuracy rate of 87% and 84%, respectively, was obtained. The accuracy rates seem high enough to support the claims of Mandhane et al. and also to surpass the overall accuracy rates of the theoretical models in comparison. This remarkable result demonstrates the exceptional capabilities of the MSVM model. The MSVM models that were constructed for identifying patterns across several pipe sizes have an additional

dimension, making them 3D classification models, which can better account for the pipe diameter effect on the flow pattern. The performance of the 3D classification is reported in Tables 5 and 7. 7. Conclusions In this paper, we investigated and developed twophase flow regime models for vertical upward flows and horizontal flows using a multiclassification support vector machine discriminative model. We applied the one-against-one method and made comparisons with the theoretical models of McQuillan and Whalley2 and Weisman et al.17 The results of sections 6.1 and 6.2 (Tables 4-7) demonstrate that the MSVM models using pipe size and superficial velocities as input training vector attributes of the flow regime produce a better overall misclassification error in comparison with theoretical models. For


Figure 5. Horizontal two-phase flow regime map that results from a 2D multiclass SVM model using a polynomial kernel (p ) 2, C ) 0.5, blue lines) and the Weisman et al.17 correlation (red lines).

the 2D vertical (horizontal) classification test and whole data set (i.e., combined training and testing data sets), the MSVM accuracies are between 93 and 99% (9097%). The theoretical model accuracies are between 93 and 98% (85-90%). Increasing the dimensionality of the classification problem to 3D still produces better error rates for the MSVM models. The MSVM accuracies for the 3D vertical (horizontal) test and whole data set are between 89 and 97% (83-95%). The theoretical model accuracies are between 81 and 87% (83-91%). Selection of MSVM and kernel parameters is important and is an ingredient to the success and potential improvement of the MSVM model; however, an efficient approach for the selection of parameters is still a topic for further research. It is found that both polynomials with kernel degree 1 and with kernel degree 2 perform very well. While a polynomial kernel with degree 1 indicates linear transition equations between flow patterns, a polynomial degree 2 indicates nonlinearity between flow patterns, and that is in agreement with most theoretical models available. Since the polynomial kernel clearly presents the most accurate results, further studies could include the use of other nonlinear kernels. The predictions of the MSVM models have been compared with a range of experimental observations, and in most cases, the MSVM model accuracies for vertical (horizontal) flows have exceeded the theoretical model accuracies. The use of the MSVM model in this paper, therefore, yields more reliable flow pattern information than other theoretical correlations; it is possibly more reliable than most correlations available. It is suggested that this model can be confidently used for flow pattern delineation when a set of data that can be used to train the MSVM is available. Appendix A This appendix contains the primal and dual formulation of the SVM optimization problem.18

Primal Problem.

min w,b,ξ

1 2

l

ξi ∑ i)1

|w|2 + C

s.t. yi(w‚xi + b) + ξi g 1 i ) 1, ..., l

ξi g 0

(A1)

We need to construct the Lagrangian of the primal problem and write out the Kuhn-Tucker conditions (KTC)21,22

L(w,b,ξ,R,β) )

1 2

l

|w|2 + C

ξi ∑ i)1

l

∑ i)1

l

Ri(yi(w‚xi + b) - 1 + ξi) -

βiξi ∑ i)1

(A2)

where Ri and βi are the Lagrangian multipliers introduced to enforce positivity of the constraints. The KTC for the primal problem are given below

∂L ∂b ∂L ∂w

l

Riyi ) 0 ∑ i)1

)0f-

l

)0fw-

∑ i)1

(A3)

l

Riyixi ) 0 f w )

Riyixi ∑ i)1

(A4)

∂L ) 0 f C - Ri - βi ) 0 f Ri + βi ) C ∂ξi

(A5)

yi(w‚xi + b) - 1 + ξi g 0

(A6)

ξi g 0

(A7)

Ri g 0

(A8)

βi g 0

(A9)

βiξi ) 0

(A10)

Ri[yi(w‚xi + b) - 1 + ξi] ) 0

(A11)

4424


Equation A5 combined with eq A10 indicates that ξi ) 0 if Ri < C so, substituting the necessary KTC (eqs A3 and A4) in the Lagrangian (eq A2), we obtain the Wolfe dual problem below l

maxw(R) ) max R

R

1

l

s.t.

0.0933 ln(vSG) ln(vSL) - 0.3962 ln(vSG)2 +

Riyi ) 0 ∑ i)1

(A12)

The solution (optimal point) is given by

R/i yixi ∑ i)1

(A13)

where a training vector for which R/i > 0 is called a support vector. We can use the KTC complementary conditions, eqs A10 and A11, to determine the threshold value b. Kernel Matrix Computation. Given x, the input data matrix, and yi ((1), the class of a data point,

(

) ()

y1 y2 ,y) ‚‚‚ yN

y1 y2 ‚‚‚ yN

We can project the training data inputs into the kernel function space by defining the variable kij as kij ) k(xi,xj), where xi ) (xi1, ..., xid) is training instance i and xj ) (xj1, ..., xjd) is training instance j, for all i,j pairs in {1, ..., N}. Computing k(xi,xj) for N2 pairs gives the Gram Matrix:

(

k11 ‚‚‚ k1N K ) ‚‚‚ l l kN1 ‚‚‚ kNN

0.0475 ln(vSL)2 + 2.1867) (B4) MSVM P ) 1, C ) 100 Classifiers for Horizontal Flow Regimes.

Annular (+1) dispersed bubble (-1) flow transition

l

x1d x2d ‚‚‚ xNd

0.1725 ln(vSL)2 - 0.8166) (B3) f(x) ) sign(-0.0673 ln(vSG) - 0.0701 ln(vSL) +

l

‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚

0.0596 ln(vSG) ln(vSL) + 0.0516 ln(vSG)2 + Intermittent (+1) - annular (-1) flow transition

0 e Ri e C, i ) 1, ..., l

x11 x12 21 x 22 X) x ‚‚‚ ‚‚‚ xN1 xN2

Bubble (+1) - intermittent (-1) flow transition f(x) ) sign(-0.8382 ln(vSG) + 0.8919 ln(vSL) -

l

Ri - ∑∑RiRjyiyj〈xi,xj〉 ∑ 2 i)1j)1 i)1

w* )

MSVM P ) 2, C ) 0.1 Classifiers for Vertical Flow Regimes.

)

Appendix B This appendix contains the MSVM flow pattern transition classifiers for vertical and horizontal flow regime maps. The flow pattern class is in parentheses (+1 or -1). MSVM P ) 1, C ) 1 Classifiers for Vertical Flow Regimes.

f(x) ) sign(0.7271 ln(vSG) - 0.9522 ln(vSL) - 1.0029) (B5) Annular (+1) - intermittent (-1) flow transition f(x) ) sign(2.4171 ln(vSG) - 1.4604 ln(vSL) - 8.0712) (B6) Annular (+1) - stratified wavy (-1) flow transition f(x) ) sign(2.2580 ln(vSG) + 1.2680 ln(vSL) - 1.6984) (B7) Dispersed bubble (+1) intermittent (-1) flow transition f(x) ) sign(-1.9840 ln(vSG) + 15.4087 ln(vSL) 15.9717) (B8) Intermittent (+1) stratified smooth (-1) flow transition f(x) ) sign(2.8682 ln(vSG) + 11.6113 ln(vSL) + 31.2659) (B9) Intermittent (+1) stratified wavy (-1) flow transition f(x) ) sign(-0.4652 ln(vSG) + 4.3801 ln(vSL) + 11.1838) (B10) Stratified smooth (+1) stratified wavy (-1) flow transition f(x) ) sign(-1.9590 ln(vSG) - 0.9028 ln(vSL) 2.2585) (B11)

Bubble (+1) - intermittent (-1) flow transition

MSVM P ) 2, C ) 0.5, Classifiers for Horizontal Flow Regimes.

f(x) ) sign(-1.6546 ln(vSG) + 1.3053 ln(vSL) - 0.4277) (B1)

Annular (+1) dispersed bubble (-1) flow transition

Intermittent (+1) - annular (-1) flow transition

f(x) ) sign(0.4432 ln(vSG) - 0.3071 ln(vSL) -

f(x) ) sign(-1.8081 ln(vSG) + 0.0798 ln(vSL) + 4.1069) (B2)

0.2331 ln(vSG) ln(vSL) + 0.1029 ln(vSG)2 0.0484 ln(vSL)2 - 1.0333) (B12)


Annular (+1) - intermittent (-1) flow transition

Greek Symbols

f(x) ) sign(0.4387 ln(vSG) - 0.4755 ln(vSL) -

R ) specific flow pattern accuracy rate Ri ) Lagrange multipliers β ) overall flow regime model accuracy rate ∆F ) (FL - FG), M/L3 µG, µL ) gas and liquid viscosities, respectively, M/LT ξi ) measure of constraint violation FG, FL ) gas and liquid densities, respectively, M/L3 σ ) surface tension, F/L

2

0.5065 ln(vSG) ln(vSL) + 0.2228 ln(vSG) 0.1552 ln(vSL)2 - 4.4507) (B13) Annular (+1) - stratified wavy (-1) flow transition f(x) ) sign(1.2065 ln(vSG) + 0.4548 ln(vSL) + 0.2444 ln(vSG) ln(vSL) + 0.3409 ln(vSG)2 0.0177 ln(vSL)2 - 1.6092) (B14) Dispersed bubble (+1) intermittent (-1) flow transition f(x) ) sign(-0.5739 ln(vSG) + 2.6988 ln(vSL) 0.2957 ln(vSG) ln(vSL) - 0.0816 ln(vSG)2 + 1.3556 ln(vSL)2 - 4.2072) (B15) Intermittent (+1) stratified smooth (-1) flow transition f(x) ) sign(0.1696 ln(vSG) + 0.4668 ln(vSL) 0.8285 ln(vSG) ln(vSL) + 0.1440 ln(vSG)2 1.1477 ln(vSL)2 + 9.5372) (B16) Intermittent (+1) stratified wavy (-1) flow transition f(x) ) sign(-0.7822 ln(vSG) + 0.1222 ln(vSL) 0.1122 ln(vSG) ln(vSL) + 0.0351 ln(vSG)2 0.9414 ln(vSL)2 + 6.3808) (B17) Stratified smooth (+1) stratified wavy (-1) flow transition f(x) ) sign(-1.8362 ln(vSG) - 1.1213 ln(vSL) 0.1048 ln(vSG) ln(vSL) - 0.3659 ln(vSG)2 0.0367 ln(vSL)2 - 2.5957) (B18) Nomenclature b ) bias or threshold value C ) regularization parameter, tradeoff cost CL ) liquid coefficient constant in friction factor correlation (dp/dx)LS ) pressure drop of liquid phase flowing alone in pipe, F/L3 D ) pipe diameter, L Fr ) Froude number (proportional to the inertial force/ gravitational force) f(x) ) output of decision function, +1 or -1 g ) acceleration due to gravity, L/T2 Ku ) Kutdelaze number, vGSFL0.5/[g(FL - FG)σ]0.25 vGS, vLS ) gas and liquid superficial velocities, respectively, based on one phase flowing separately, L/T n ) exponent, eq 14 P, p ) degree of polynomial kernel function w ) weight vector representing the level of influence on the input x ) input attributes yi ) class of data point, +1 or -1 〈*,*〉 or *.* ) dot product between two vectors

Subscripts and Superscripts d ) number of attributes G ) gas i, j, t ) index k ) number of classes l ) data point L ) liquid m ) higher dimension S ) superficial

Literature Cited (1) Mandhane, J. M.; Gregory, G. A.; Aziz, K. A Flow Pattern Map for Gas-Liquid Flow in Horizontal Pipes. Int. J. Multiphase Flow 1974, 1, 537-553. (2) McQuillan, K. W.; Whalley, P. B. Flow Patterns in Vertical Two-Phase Flow. Int. J. Multiphase Flow 1985, 11, 161-175. (3) Weisman, J.; Kang, S. Y. Flow Pattern Transitions in Vertical and Upwardly Inclined Lines. Int. J. Multiphase Flow 1981, 7, 271-291. (4) Taitel, Y.; Bornea, D.; Dukler, A. E. Modeling Flow Pattern Transitions for Steady Upward Gas-Liquid Flow in Vertical Tubes. AIChE J. 1980, 26, 345. (5) Taitel, Y.; Dukler, A. E. A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas-Liquid Flow. AIChE J. 1976, 22, 47. (6) Petalas, N.; Aziz, K. A Mechanistic Model for Multiphase Flow in Pipes. Proceedings of the 49th Annual Technical Meeting of the Petroleum Society of the CIM, Calgary, Alberta, Canada, June 8-10, 1998; CIM98-39. (7) Osman, E. A. Artificial Neural Networks Models for Identifying Flow Regimes and Predicting Liquid Holdup in Horizontal Multiphase Flow. SPE Middle East Oil Show, Bahrain, March 1720, 2000; Paper SPE 68219. (8) Mi, Y.; Ishii, M.; Tsoukalas, L. H. Flow Regime Identification Methodology with Neural Networks and Two-Phase Flow Models. Nucl. Eng. Des. 2001, 204, 87-100. (9) Ternyik, J.; Bilgesu, H. I.; Mohaghegh, S. Virtual Measurement in Pipes; Part 2: Liquid Holdup and Flow Pattern Correlations. SPE Eastern Regional Conference & Exhibition, Morgantown, WV, September 18-20, 1995; Paper SPE 30976. (10) Vapnik, V. Statistical Learning Theory; John Wiley & Sons: New York, 1998. (11) Santosa, B.; Conway, T.; Trafalis, T. B. Knowledge BaseClustering and Application of Multi-Class SVM for Genes Expression Analysis. Intelligent Engineering Systems through Artificial Neural Networks, Proceedings of the Artificial Neural Networks in Engineering Conference, St. Louis, MO, Nov 2002; Dagli, C. H., Burke, L. I., Ghosh, J., Fernandez, B., Eds.; ASME Press: Fairfield, NJ, 2002; Vol. 12, pp 391-395. (12) Trafalis, T. B.; Ince, H. Support Vector Machine for Regression and Applications to Financial Forecasting. IEEEINNS-ENNS International Joint Conference on Neural Networks, Como, Italy, July 24-27, 2000. (13) Trafalis, T. B.; Oladunni, O. Single Phase Fluid Flow Classification via Neural Networks & Support Vector Machine. Intelligent Engineering Systems Through Artificial Neural Networks, Proceedings of the Artificial Neural Networks in Engineering Conference, St. Louis, MO, Nov 2004; Dagli, C. H., Buczak, A. L., Enke, D. L., Embrechts, M. J., Ersoy, O., Eds.; ASME Press: Fairfield, NJ, 2004; Vol. 14, pp 427-432. (14) Hewitt, G. F.; Hall-Taylor, N. S. Annular Two-Phase Flow; Pergamon Press: Elmsford, NY, 1970. (15) Kokal, S. L.; Stanislav, J. F. An Experimental Study of Two-Phase Flow in Slightly Inclined Pipes. I. Flow Patterns. Chem. Eng. Sci. 1989, 40 (3), 665-679.

4426


(16) Barnea, D.; Shoham, O.; Taitel, Y.; Dukler, A. E. Flow Pattern Transition for Gas-Liquid Flow in Horizontal and Inclined Pipes. Int. J. Multiphase Flow 1980, 6, 217-225. (17) Weisman, J.; Duncan, D.; Gibson, J.; Crawford, T. Effects of Fluid Properties and Pipe Diameter on Two-Phase Flow Patterns in Horizontal Lines. Int. J. Multiphase Flow 1979, 5, 437-462. (18) Cristianini, N.; Shawe-Taylor, J. Support Vector Machines and other kernel-based learning methods; Cambridge University Press: New York, 2000. (19) Chang, C.-C.; Lin, C.-J. LIBSVM: A Library for Support Vector Machines. http://www.csie.ntu.edu.tw/∼cjlin/libsvm (accessed 2001). (20) Hsu, C.-W.; Chang, C.-C.; Lin, C.-J. A Practical Guide to Support Vector Classification, a manual for students; Department of Computer Science and Information Engineering, National Taiwan University: Taipei 106, Taiwan, 2003; http:// www.csie.ntu.edu.tw/∼cjlin/papers.html (accessed May 2005). (21) Bazaraa, M. S.; Sherali, H. D.; Shetty, C. M. Nonlinear Programming - Theory and Algorithms; John Wiley & Sons: New York, 1993. (22) Reklaitis, G. V.; Ravindran, A.; Ragsdell, K. M. B. Engineering Optimization: Methods and Applications; John Wiley & Sons: New York, 1983. (23) Hsu, C.-W.; Lin, C.-J. A Comparison of Methods for Multiclass Support Vector Machines. IEEE Trans. Neural Networks 2002, 13, 415-425. (24) Bredensteiner, E. J.; Bennet, K. P. Multicategory Classification by Support Vector Machines. Comput. Opt. Appl. 1999, 12, 53-79. (25) Weston, J.; Watkins, C. Multi-class Support Vector Machines; Technical Report CSD-TR-98-04; University of London: London, 1998. (26) Adsani, E. Mass Transfer of Corrosion Species in Vertical Multiphase Flow: A Mechanistic Approach. Ph.D. Thesis, University of Tulsa, Tulsa, OK, 2002. (27) Govier, G. W.; Radford, B. A.; Dunn, J. S. C. The Upward Vertical Flow of Air-Water Mixtures I. Can. J. Chem. Eng. 1957, 58.

(28) Andritsos, N. Effect of Pipe Diameter and Liquid Viscosity on Horizontal Stratified Flow. Ph.D. Thesis, University of Illinois, Urbana, IL, 1986. (29) Barnea, D.; Shoham, O.; Taitel, Y.; Dukler, A. E. GasLiquid Flow in Inclined Tubes: Flow Pattern Transitions for Upward Flow. Chem. Eng. Sci. 1985, 40, 131-136. (30) Chen, Y. Modeling Gas-Liquid Flow in Pipes: Flow Pattern Transitions and Drift-Flux Modeling. M.S. Thesis, Stanford University, Stanford, CA, 2001. (31) Crowley, C. J.; Sam, R. G.; Rothe, P. H. Investigation of Two-Phase Flow in Horizontal and Inclined Pipes at Large Pipe Size and High Gas Density; Project PR-172-507; Creare Inc.: Hanover, NH, 1986. (32) Govier, G. W.; Omer, M. M. The Horizontal Pipeline Flow of Air-Water Mixtures. Can. J. Chem. Eng. 1962, 93104. (33) Green, D W. Concurrent Two-Phase Flow of Liquid and Air through Inclined Pipe. M.S. Thesis, University of Oklahoma, Norman, OK, 1959. (34) Jenkins, R. Two-Phase Two-Component Flow of Water and Air. M.S. Thesis, University of Delaware, DE, 1947. (35) Lamari, M. L. An Experimental Investigation of Two-Phase (Air-Water) Flow Regimes in a Horizontal Tube at Near Atmospheric Conditions. Ph.D. Thesis, Carleton University, Ottawa, Canada, 2001. (36) White, P. D. Horizontal Co-Current Two-Phase TwoComponent Flow. Ph.D. Thesis, University of Oklahoma, Norman, OK, 1954. (37) Ma, J.; Zhao, Y.; Ahalt, S. OSU SVM Classifier Matlab Toolbox; Ohio State University: Columbus, OH, 2002; http:// www.ece.osu.edu/∼maj/osu_svm.

Received for review October 21, 2004 Revised manuscript received April 4, 2005 Accepted April 13, 2005 IE048973L

Two-Phase Flow Regime Identification with a ... - ACS Publications

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