From the SelectedWorks of Dr. Sandip Kumar Lahiri
December 2008
Prediction Prediction of Pressure Pressure drop of Slurry Slurr y Flow in Pipeline by Hybrid
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Chinese Journal of Chemical Engineering , Engineering , 16(6)
(2008)
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Chinese Journal of Chemical Engineering , Engineering , 16(6)
(2008)
Prediction of Pressure drop of Slurry Flow in Pipeline by Hybrid Support Vector Vector Regression and Genetic Algorithm Model Mo del S.K. Lahiri* and K.C. Ghanta Department of Chemical Engineering, NIT, Durgapur, West Bengal, India Abstract This paper describes a robust support vector regression (SVR) methodology, which can offer superior performance performance for important important process engineerin engineering g problems. problems. The method incorporates incorporates hybrid support support vector regression regression and genetic algorithm technique (SVR-GA) for efficient tuning of SVR meta-parameters. The algorithm has been applied for prediction of pressure drop of solid liquid slurry flow. A comparison with selected correlations in the literature showed that the developed SVR correlation noticeably improved the prediction of pressure drop over a wide range of operating conditions, physical properties, and pipe diameters. Keywords support vector regression, genetic algorithm, slurry pressure drop
1
INTRODUCTION
Conventionally, two approaches namely phenomenological (first principles) and empirical, are employed for slurry flow modeling. The advantages of a phenomenological model are: (i) it represents physicochemical phenomenon underlying the process explicitly and (ii) it possesses extrapolation ability. Owing to the complex nature of many multiphase phase slurry processes, the underlying physico-chemical phenomenon is seldom fully understood. Also, collection of the requisite phenomenological information is costly, time-consuming and tedious, and therefore development and solution of phenomenological process models poses considerable practical difficulties. These difficulties necessitate exploration of alternative modeling formalisms. In the last decade, artificial neural networks (ANNs) and more recently support vector regression (SVR) have emerged as two attractive tools for nonlinear modeling especially in situations where the development of phenomenological or conventional regression models becomes impractical or cumbersome. In recent years, support vector regression (SVR) [1-3] which is a statistical learning theory based machine learning formalism is gaining popularity over ANN due to its many attractive features and promising empirical performance. The main differenc di fferencee between conventional ANNs and support vector machines (SVM) lies in the risk minimization principle. Conventional ANNs implement the empirical risk minimization (ERM) principle to minimize the error on the training data, while SVM adheres to the Structural Risk Minimization (SRM) principle seeking to set up an upper bound of the generalization error [1]. This study is motivated by a growing popularity of support vector machines (SVM) for regression problems. Although the foundation of the SVR paradigm was laid down in mid 1990s, its chemical engineering applications such as fault detection [4, 5] have emerged only recently.
However, many SVM regression application studies are performed by ‘expert’ users having good understanding of SVM methodology. Since the quality of SVM models depends on a proper setting of SVM meta-parameters, the main issue for practitioners trying to apply SVM regression is how to set these parameter values (to ensure good generalization performance) for a given data set. Whereas existing sources on SVM regression give some recommendations on appropriate setting of SVM parameters, there is clearly no consensus and (plenty of) contradictory opinions. Conventionally, various deterministic gradient based methods [6] are used for optimizing a process model. Most of these methods however require that the objective function should be smooth, continuous, and differentiable. The SVR models can not be guaranteed to be smooth especially in regions wherein the input-output data (training set) used in model building is located sparsely. In such situations, an efficient optimization formalism known as Genetic algorithm (GA), which is lenient towards the form of the objective function, can be used. The GA was originally developed as the genetic engineering models mimicking population evolution in natural systems and has been extensively used in chemical engineering [7-11]. In the present paper, paper, SVR formalism is integrated with genetic algorithm to automatically determine the optimal meta-parameters of SVR with the highest predictive accuracy and generalization ability simultaneously. taneously. The strategy strategy (henceforth (henceforth referred to as “SVR-GA”) uses an SVR as the nonlinear process modeling paradigm, and the GA for optimizing the meta-parameters of the SVR model such that an im proved prediction performance is realized. To our knowledge, the hybrid involving SVR and GA is being used for the first time for chemical process modeling and optimization. In the present work, we pro pose a hybrid support vector regression- genetic algorithm (SVR-GA) approach for tuning the SVR meta-parameters and illustrate it by applying it for predicting the pressure drop of solid liquid slurry flow. flow.
Received 2008-04-18, accepted 2008-08-16. * To whom correspondence should be addressed. E-mail:
[email protected]
2
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
2 SUPPORT VECTOR REGRESSION (SVR) MODELING 2.1
tion object becomes: Minimize
Support vector regression: at a glance
Consider a training data set g = {( x 1 , y1 ), ( x2 , y2 ), ( x P , yP )} , such that x i ∈υ N is a vector of input variables and yi ∈υ is the corresponding scalar output (target) value. Here, the modeling objective is to find find a regres regressio sion n functi function, on, y = f ( x ) , such that it accurately predicts the outputs { y} corresponding to a new set of input-output examples, {( x x , y)}, which are drawn from the same underlying joint probability distribution as the training set. To fulfill the stated goal, SVR considers following linear estimation function. f ( x) = w , x + b
(1)
where w denotes the weight vector; b refers to a constant known as “bias”; f ( x) x) denotes a function termed feature, and w , x represents the dot product in the feature space, l , such that : x → l , w ∈ l . The basic concept of support vector regression is to map nonlinearly the original data x into a higher dimensional feature space and solve a linear regression problem in this feature space space [1-3]. The regression problem is equivalent to minimize the following regularized risk function: R( f ) =
1
n
1
∑ L ( f ( x ) − y ) + 2 n i
i =1
i
w
2
(2)
where
⎧ f ( x ) − y − ε , for f ( x) − y x ≥ ε
L( f ( x ) − y ) = ⎨
⎩
0,
otherwise
(3) Equation (3) is also called ε-insensitive loss function. This function defines a ε-tube. If the predicted value is within the ε-tube, the loss is zero. If the predicted value is outside the tube, the loss is equal to the magnitude of the difference between the predicted value and the radius ε of the tube. ε is a precision parameter representing the radius of the tube located around the regression function (see Fig. 1); the region enclosed by the tube is known as “ε-intensive zone”.
1
n
* w + C ∑ (ξ + ξ i ) 2
2
(4)
i =1
⎧ yi − w , x i − b ≤ ε + ξ i ⎪ subject to ⎨ w , x i + b − yi ≤ ε + ξ i* (5) ⎪ ξi , ξ i* ≥ 0 ⎩ where the constant C > > 0 stands for the penalty degree of the sample with error exceeding ε. Two positive slack variables ξi , ξ i* represent the distance from actual values to the corresponding boundary values of x ) to the data in a manner such fits f ( x ε-tube. The SVR fits f that: (i) the training error is minimized by minimizing 2 ξi , ξ i* and, (ii) w is minimized to increase the flatness x ) or to penalize over complexity of the fitting of f ( x function. A dual problem can then be derived by using the optimization method to maximize the function: maximize −
n
1
∑ (α 2 i , j =1
i
− α i* ) (α j − α *j ) ( xi , x j ) −
i
+
n
∑ (α
ε
i =1
n
subject to
∑ (α
i =1 * α i ,α i are
i
n
α i*
) +∑ yi (α i − α i* )
(6)
i =1
* − α i* ) = 0 and 0 ≤ α i ,α i ≤ C (7)
where Lagrange multipliers. Owing to the specific character of the above-described quadratic programming problem, only some of the coefficients, ( α i* − α i ) are non-zero and the corresponding input vectors, x i, are called support vectors (SVs). The SVs can be thought of as the most informative data points that compress the information content of the training * set. The coefficients α and α have an intuitive inter pretation as forces pushing and pulling the regression x i) towards the measurements, estimate f estimate f ( x measurements, y yi. The SVM for function fitting obtained by using the above-mentioned maximization function is then given by f ( x ) =
n
∑ (α i =1
i
− α i* ) xi , x + b
(8)
As for the nonlinear cases, the solution can be found by mapping the original problems to the linear ones in a characteristic space of high dimension, in which dot product manipulation can be substituted by a kernel function, i.e. K ( xi , x j ) = ϕ ( xi )ϕ ( x j ) . In this work, different kernel function is used in the SVR. Substituting K ( xi , x j ) = ϕ ( xi )ϕ ( x j ) in Eq. (6) allows us to reformulate the SVM algorithm in a nonlinear paradigm. Finally, we have Figure 1 A schematic diagram of support vector regresregression using ε-sensitive loss function [4] ○ data points; points outside tube; support vectors; ● fitted by SVR
The SVR algorithm attempts to position the tube around the data as shown in Fig. 1. By substituting the ε-insensitive loss function into Eq. (2), the optimiza-
f ( x ) =
n
∑ (α i =1
2.2
i
− α i* ) K xi , x + b
(9)
Performance Performance estimation of SVR
It is well known that SVM generalization performance (estimation accuracy) depends on a good
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
setting of meta-parameters meta-parameters parameters C , ε, and kernel parameters such as kernel type, loss function type and the kernel parameters. The problem of optimal parameter selection is further complicated by the fact that SVM model complexity (and hence its generalization performance) depends on all five parameters. Selecting a particular kernel type and kernel function parameters is usually based on application-domain knowledge and also should reflect distri x ) values of the training data. Pa bution of input ( x rameter C determines determines the trade off between the model complexity (flatness) and the degree to which deviations larger than ε are tolerated in optimization formulation. For example, if C is is too large (infinity), then the objective is to minimize the empirical risk only, without regard to model complexity part in the optimization formulation. Parameter ε controls the width of the ε-insensitive zone, used to fit the training data [1, 3, 12]. The value of ε can affect the number of support vectors used to construct the regression function. The bigger ε, the fewer support vectors are selected. On the other hand, bigger ε-values result in more ‘flat’ estimates. Hence, both C and and ε-values affect model complexity (but in a different way). To minimize the generalization error, these parameters should be properly optimized. As evident from the literature, there is no shortage of (conflicting) opinions on optimal setting of SVM regression parameters and many such approaches to the choice of C and ε are summarized in Cherkassky et al [13]. All the practical approaches rely on noise estimation in training data which may possible for electrical signals but very difficult for chemical engineering data or process parameters. Existing software implementations of SVM regression usually treat SVM meta-parameters as user-defined inputs. For a non-expert user it is very difficult task to choose these parameters as he has no prior knowledge for these parameters for his data. In such a situation, user normally relies on trial and error method. Such an approach apart from consuming enormous time may not really obtain the best possible performance. In this paper we present a hybrid SVR-GA approach, which not only relieve the user from choosing these meta-parameters but also find out the optimum values of these parameters to minimize the generalization error. error. 3
3
Owing to these attractive features, GAs is being used for solving diverse optimization problems in chemical engineering (see e g .g ., ., [7, 8, 10, 11, 14] ) . In the GA procedure, the search for an optimal * solution (decision) vector, vector, x , begins from a randomly initialized population of probable (candidate) solutions. The solutions, usually coded in the form of binary strings (chromosomes), are then tested to measure their fitness in fulfilling the optimization objective. Subsequently, a main loop comprising following operations is performed: (i) selection of better (fitter) parent chromosomes, (ii) production of an offspring solution population by crossing over the genetic material between pairs of the fitter parent chromosomes, and (iii) mutation (bit-flipping) of the offspring strings. Implementation of this loop generates a new population of candidate solutions, which as compared to the previous population, usually fares better at fulfilling the optimization objective. The best string that evolves after repeating the above described loop till convergence, forms the solution to the optimization problem. 4 GA-BASED OPTIMIZATION OPTIMIZATION OF SVR MODELS
There are different measures by which SVM performance formance is assessed, but validation and leave-one-out error estimates are the most commonly used one. Here we divide the total available data as training data (75% of data) and test data (25 % data chosen randomly) on which the SVR performance is assessed. The statistical analysis of SVR prediction is based on the following performance performance criteria: criteria: (1) The average absolute relative error (A) on test data should be minimized: A =
⎛ y pred − yexp ⎞ ∑ ⎜ yexp ⎟⎟ N 1 ⎜⎝ ⎠ 1
N
(2) The standard deviation of error (σ ( σ ) on test data should be minimum: σ =
1
N
∑ ⎡ ( y N − 1 ⎣
pred,i
1
2
(3) The cross-correlation coefficient (R) between input and output should be around unity:
GENETIC ALGORITHM: AT A GLANCE
N
∑ ( y
exp,i
Genetic Algorithms [14] combine the “survival of the fittest” principle of natural evolution with the genetic propagation of characteristics, to arrive at a ro bust search and optimization technique. Principal features possessed by the GAs are: (i) they are zeroth order optimization methods requiring only the scalar values of the objective function, (ii) capability to handle nonlinear, complex and noisy objective functions, (iii) they perform global search and thus are more likely to arrive at or near the global optimum, and (iv) their search procedure being stochastic, GAs do not impose pre-conditions, such as smoothness, differentiability and continuity on the objective function form.
− yexp,i ) / yexp,i − ⎤ ⎦
R =
i =1
N
∑ i =1
− yexp ) ( ypred ,i − ypred )
( yexp,i − yexp )
2
N
∑( y i =1
pred ,i
− ypred )
2
SVR learning is considered successful only if the system can perform well on test data on which the system is not trained. The above five parameters of SVR are optimized by GA algorithm stated below. below. The objective function and the optimization problem of SVR model of the present study are represented as min ⎡⎣ ( x)o ) on test set, X ∈ { x1 , x2 , x3 , x4 , x5 }⎤⎦
4
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
Table 1 Case
Name of kernel
Case 1
linear
Case 2
polynomial
Different kernel type Equation k = uv′ k = (uv′ + 1) p1 ; p1 is degree of polynomial polynomial
Case 3
Gaussian radial basis function
Case 4
exponential radial basis function
⎡ (u − v)(u −v )′ ⎤ −⎢ ⎥ 2
1 ⎦ k =e ⎣ ; p1 is width of radial basis function (σ (σ )
2 p
⎡ (u − v)(u −v )′ ⎤ − ⎢ ⎥ 2 p12 ⎣ ⎦
k =e ; p1 is width of exponential radial basis function (σ (σ ) Case 5
spline
z = 1 + uv +
1
2 k = prod( z ) ;
Case 6
B spline
uv min(u, v) −
1 6
3
min(u , v ) ;
= 0 ; z = = 0 : 2 * ( p1 + 1) for r = r
2 ( p1+1)
z = z + (−1) r [ 2( p1 + 1)] [ max(0, u − v + p1 + 1− r )]
;
end k = prod( z ) ; p1 is degree of B spline
Note: u, v are kernel arguments
subject to x1 ∈ {0,10+5 } , x2 = {0 t o 1} , x3 = 1, 2 , x4 = {1, 2, ⋅ ⋅ ⋅, 6} , x5 = {1, 2, ⋅ ⋅ ⋅, 6} . The objective is minimization of average absolute relative error (AARE) A on test set and X is a solution string representing a design configuration. The design variable x1 takes any values for C in in the range of 0.0 to 10000. x2 represents the ε (epsilon) taking any values in the range of 0.0 to 1. x3 represents the loss function types: ε-insensitive loss function and Huber loss function represented by the numbers 1 and 2 respectively (Table 1). x4 represents the kernel types: all 6 kernels given in Table 2 represented by numbers 1 to 6 respectively. The variable x5 takes six values of the kernel parameters ( represents degree of polynomials, etc.) etc.) in the range 1 to 6. The total num ber of design combinations with these variables is 100×100×2×6×6 = 720000. This means that if an exhaustive search is to be performed it will take at the maximum 720000 function evaluations before arriving at the global minimum A for the test set ( assuming 100 trials for each to arrive optimum C and and ε ). So the strategy which takes few function evaluations is the best one. For minimization of A of A as as the objective function, genetic algorithm technique is applied to find the optimum design configuration of SVR model.
Table 2
Different loss function
Case
Name of loss function
Case 1
ε-insensitive loss function
Case 2
quadratic loss function
The stepwise procedure for the GA-based optimization of a SVR model is provided in the flowchart in Fig. 2.
5 CASE STUDY: STUDY: PREDICTION OF PRESSURE DROP IN SOLID LIQUID SLURRY FLOW 5.1
Background and importance of pressure drop
Compared to a mechanical transport of slurries, the use of a pipeline ensures a dust free environment, demands substantially less space, makes possible full automation and requires a minimum of operating staff. Power consumption represents a substantial portion of the overall pipeline transport operational costs. For that reason great attention was paid to reduction of the hydraulic losses. The prediction of pressure drop of slurries and the understanding of rhelogical behavior makes possible to optimize energy energy and water requirements. requirements. Various investigations [15-19] have tried to show and propose correlation relating pressure drop with other parameters of solid-liquid flow namely solids density, liquid density, particle size, concentration, pipe diameter, viscosity of flowing media, velocity of suspension, etc. etc. They find that pressure drop curve passes through through minima minima (point (point 3 in Fig. Fig. 3) and operati operation on at this velocity will ensure lowest power consumption. Despite the large area of application, the available models describing the suspension mechanism do not completely satisfy engineering needs. To facilitate the design and scale up of pipelines and slurry pumps, there is a need for a correlation that can predict slurry pressure drop over a wide range of operating conditions, physical properties and particle size distributions. Industry needs quick and easily implementing solutions. The model derived from the first principle is no doubt the best solution. But in the scenario, where the basic principles for pressure drop modeling accounting all the interactions for slurry flow is absent, the numerical model may be promising to give some quick, easy solutions for slurry pressure drop prediction. This paper presents a systematic approach using
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
Table 3
5
Literature sources for pressure drop correlations
No
Author
No
Author
1
Wilson (1942) [15]
11
G il illies et al . (1983) [28]
2 Durand Durand & Condol Condolios ios (1952) (1952) [20] [20]
12
Roco & Shook(1984) [29]
3
Newitt et al . (1955) [21]
13
Roco & Shook(1985) [30]
4
Zandi Zandi & Govatos Govatos (1967) (1967) [22]
14
Ma (1987) (1987) [31]
5
Shook et al .(1968) .(1968) [23]
15
Hsu (1987) [17]
6
Schriek et al . (1973) [24]
16
Doron et al . (1987) [32]
7
Scarlett Scarlett & Grimley Grimley (1974) (1974) [25] [25]
17
Ghanta Ghanta (1996) (1996) [18]
8
Turia Turian n & Yuan (1977 (1977)) [26] [26]
18
Gillie Gilliess et al. (1999) (1999) [19] [19]
9
Wasp et al . (1977) [16]
19
Schaan et al .(2000) .(2000) [33]
10
Govier Govier & Aziz Aziz (1982 (1982)) [27] [27]
20 Kaush Kaushal al and and Tomita(2002) [34]
have amply quantified the pressure drop of slurry flow in pipeline. In this work, about 800 experimental points have been collected from 20 sources spanning the years 1942-2002 [refer to Table 3]. This wide range of database includes experimental information from different physical systems to provide a unified correlation for pressure drop. Table 4 suggests the wide range of the collected databank for pressure drop.
Figure 2
Schematic for SVR algorithm implementation
Figure 3 Plot of transitional mixture velocity with pressure drop
robust hybrid SVR-GA techniques to build a pressure drop correlation from available experimental data. This correlation has been derived from a broad ex perimental data bank collected from the open literature [refer to Table 3] (800 measurements covering a wide range of pipe dimensions, operating conditions and physical properties). 5.2
Development of the SVR-based correlation
5.2.1 Collection of data As mentioned earlier, over the years researchers
parameters 5.2.2 Identification of input parameters After extensive literature survey all physical parameters that influence pressure drop are put in a so-called ‘wish list’. Out of the number of inputs in ‘wish list’, we used support vector regression to establish the best set of chosen inputs, which describes pressure drop. The following criteria guide the choice of the set of inputs: The inputs should be as few as possible. Each input should be highly cross-correlated to the output parameter; These inputs should be weakly cross-correlated to each other; The selected input set should give the best output prediction, which is checked by using statistical analysis [e [e g . g . average absolute relative error A, standard deviation]. While choosing the most expressive inputs, there is a compromise between the number of inputs and prediction. Based on different combinations of inputs, trial and error method was used to finalize the input set which gives reasonable low prediction error A when exposed to support vector regression. Based on the above analysis, the input variables such as pipe diameter, particle diameter, solids concentration, solid and liquid density and viscosity of liquid have been finalized to predict pressure drop in slurry pipeline. Table 4 shows the typical range of data selected for support vector regression. 6
RESULTS RESULTS AND DISCUSSION
As the magnitude of inputs and outputs greatly differ from each other, they are normalized in -1 to +1 scale by the formula
6
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
Table 4
Slurry system
and parameter range from the literature data
Pipe diameter D/m D/m
Particle diameter d p/m
Liquid density － ρL/kg·m 3
Solids density － ρS/kg·m 3
Liquid viscosity μ/mPa·s μ/mPa·s
Velocity U － /m·s 1
Solids concentration (volume fraction) f fraction) f
0.019-0.495
38.3-13000
1000-1250
1370-2844
0.12-4
0.86-4.81
0.014-0.333
① Slurry system: coal/water, copper ore/water, sand/water, gypsum/water, glass/water, gravel/water.
xnormal =
2 x − xmax − xmin
Table 5 (Continued)
xmax − xmin
Input
75% of total dataset was chosen randomly for training and rest 25% was selected for validation and testing. Seven parameters were identified as input (Table 5) Table 5
Typical input and output data for SVR training Input U / － m·s 1
f
Output － d p/d p/d z /Pa·m /Pa·m 1
0.85 0.85
1.06 1.06
0.07 0.07
560.00 560.00
2.84 2.84
0.85 0.85
1.13 1.13
0.07 0.07
600.00 600.00
1.00 1.00
2.84 2.84
0.85 0.85
1.15 1.15
0.07 0.07
600.00 600.00
40.1 40.1
1.00 1.00
2.84 2.84
0.85 0.85
1.10 1.10
0.07 0.07
1250.0 1250.00 0
1.90 1.90
40.1 40.1
1.00 1.00
2.84 2.84
0.85 0.85
1.50 1.50
0.07 0.07
1500.0 1500.00 0
1.90 1.90
40.1 40.1
1.00 1.00
2.84 2.84
0.85 0.85
1.70 1.70
0.07 0.07
1700.0 1700.00 0
5.26 5.26
38.3 38.3
1.00 1.00
2.33 2.33
1.00 1.00
1.11 1.11
0.11 0.11
294.10 294.10
5.26 5.26
38.3 38.3
1.00 1.00
2.33 2.33
1.00 1.00
3.01 3.01
0.11 0.11
1651.3 1651.30 0
5.26 5.26
38.3 38.3
1.00 1.00
2.33 2.33
1.00 1.00
4.81 4.81
0.11 0.11
3822.9 3822.90 0
5.26 5.26
38.3 38.3
1.00 1.00
2.33 2.33
1.00 1.00
1.33 1.33
0.31 0.31
542.90 542.90
5.26 5.26
38.3 38.3
1.00 1.00
2.33 2.33
1.00 1.00
3.12 3.12
0.31 0.31
2352.6 2352.60 0
5.26 5.26
38.3 38.3
1.00 1.00
2.33 2.33
1.00 1.00
4.70 4.70
0.31 0.31
4727.7 4727.70 0
20.85 20.85
190.0 190.0
1.00 1.00
1.37 1.37
1.14 1.14
2.59 2.59
0.33 0.33
266.50 266.50
20.85 20.85
190.0 190.0
1.00 1.00
1.37 1.37
1.14 1.14
2.34 2.34
0.33 0.33
226.30 226.30
20.85 20.85
190.0 190.0
1.00 1.00
1.37 1.37
1.14 1.14
2.01 2.01
0.33 0.33
177.30 177.30
20.85 20.85
190.0 190.0
1.00 1.00
1.37 1.37
1.14 1.14
1.78 1.78
0.33 0.33
147.00 147.00
20.85 20.85
190.0 190.0
1.00 1.00
1.37 1.37
1.14 1.14
1.59 1.59
0.32 0.32
123.40 123.40
20.85 20.85
190.0 190.0
1.00 1.00
1.37 1.37
1.14 1.14
1.37 1.37
0.33 0.33
99.90 99.90
5.15 5.15
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
1.66 1.66
0.07 0.07
666.20 666.20
5.15 5.15
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
3.78 3.78
0.09 0.09
2449.2 2449.20 0
5.15 5.15
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
1.66 1.66
0.17 0.17
901.30 901.30
5.15 5.15
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
4.17 4.17
0.19 0.19
3428.9 3428.90 0
5.15 5.15
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
1.66 1.66
0.27 0.27
1136. 1136.40 40
5.15 5.15
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
4.33 4.33
0.29 0.29
4408.1 4408.10 0
26.30 26.30
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
2.90 2.90
0.09 0.09
261.60 261.60
26.30 26.30
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
3.50 3.50
0.09 0.09
334.10 334.10
26.30 26.30
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
2.90 2.90
0.18 0.18
305.70 305.70
26.30 26.30
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
3.50 3.50
0.17 0.17
382.10 382.10
26.30 26.30
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
2.90 2.90
0.26 0.26
355.60 355.60
26.30 26.30
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
3.50 3.50
0.26 0.26
453.60 453.60
26.30 26.30
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
2.90 2.90
0.33 0.33
414.40 414.40
26.30 26.30
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
3.50 3.50
0.32 0.32
526.10 526.10
49.50 49.50
165.0 165.0
1.00 1.00
2.65 2.65
1.00 1.00
3.16 3.16
0.09 0.09
143.00 143.00
D/ D/ cm
μm
d p/
ρL/ ρS/ μ/ μ/ － － kg·L 1 kg·L 1 mPa·s
2.54 2.54
40.1 40.1
1.00 1.00
2.84 2.84
2.54 2.54
40.1 40.1
1.00 1.00
2.54 2.54
40.1 40.1
1.90 1.90
d p/
ρL/ ρS/ μ/ μ/ U / － － － kg·L 1 kg·L 1 mPa·s m·s 1
Output － d p/d p/d z /Pa·m /Pa·m
D/ D/ cm
μm
49.50
165.0
1.00
2.65
1.00
3.76
0.09
186.10
49.50
165.0
1.00
2.65
1.00
3.07
0.17
157.70
49.50
165.0
1.00
2.65
1.00
3.76
0.26
254.70
15.85
190.0
1.00
2.65
1.30
2.50
0.14
475.20
15.85
190.0
1.00
2.65
1.30
2.50
0.29
630.90
15.85
190.0
1.00
2.65
0.12
3.00
0.13
648.90
15.85
190.0
1.00
2.65
0.12
2.90
0.28
866.70
5.07
520.0
1.00
2.65
1.00
1.90
0.09
1175.60
5.07
520.0
1.00
2.65
1.00
2.00
0.21
1763.40
4.00
580.0
1.25
2.27
4.00
2.88
0.16
3926.00
4.00
580.0
1.25
2.27
4.00
2.70
0.14
3580.00
4.00
580.0
1.25
2.27
4.00
2.01
0.10
2217.00
4.00
580.0
1.25
2.27
4.00
1.05
0.06
845.00
26.30 13000.0
1.00
2.65
1.00
3.20
0.04
842.50
26.30 13000.0
1.00
2.65
1.00
4.00
0.04
989.50
f
1
for SVR and the pressure drop is put as target. These data then exposed to hybrid SVR-GA model described described above. After optimization of five SVR parameters described above, the model output was summarized in Table Table 6. The parity plot for experimental and predicted pressure drop is shown in Fig. 4. The low A (12.7%) may be considered as an excellent prediction performance considering the poor understanding of slurry flow phenomena and large databank for training com prising various systems. Fig. 5 represents a comparitive study of present model and Kaushal model [34] against experimental data of Ghanta [31]. The figure clearly represents the improvement of model prediction as compared to Kaushal model especially at low velocity range. The optimum value of SVR meta-parameters were summarized in Table 7, clearly showing that there exists almost 4 different feasible solutions which lead t o same prediction error. Table 6
Prediction error by SVR based model A
σ
R
training
0.125
0.163
0.978
testing
0.127
0.169
0.977
The final equation is as follows: Pressure drop = K (i, j ) * B + b
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
Experimental vs predicted vs predicted pressure drop for exponential radial basis function (erbf) kernel
Figure 4
Figure 5 Comparison of SVR-GA model, Kaushal Kaushal model and experimental pressure drop [31] —— experimental; SVR-GA model; Kaushal [34]
where K where K (i, j) j) is the kernel function t
−
7
( Ist ( i ,7 ) −Trn ( j ,7 )*)(Tst ( i ,7 ) −T rn ( j ,7 ) ) 2 p12
e
for i = 1 to n and j = 1 to m and n = number of test record, m = number of training record (600 in this case), T stst is a n×7 matrix for testing input (which user want to test for calculating pressure drop) and should be arrange in a sequence similar to Table 5, T rn rn is a 600×7 matrix for training input data which is arranged in a sequence similar to Table 5, B is a 600×1 matrix calculated during training
Table 7
and b is zero for this case, p1 is width of rbfs (which # can be found as kernel parameter in Table 7) . In a separate study, we exposed the same dataset to SVR algorithm only (without the GA algorithm) and try to optimize the different parameters based on exhaustive search. We We found that it was not possible to reach the best solutions starting from arbitrary initial conditions. Especially the optimum choice of C and and ε is very difficult to arrive after starting with some discrete value. Many times the solutions got stuck up in sub optimal local minima. These experiments justified the use of a hybrid technique for SVR parameter tuning. The best prediction after exhaustive search along with SVR parameters was summarized in Table 8. From the table it is clear that even after 720000 runs, the SVR algorithm is unable to locate the global minima and the time of execution is 4 h in Pentium 4 processor. processor. On the other hand, the hybrid SVR-GA technique is able to locate the global minima with 2000 runs within 1 h. The prediction accuracy is also much better. Moreover it relieves the non expert users to choose the different parameters and find optimum SVR meta-parameters with a good accuracy. All the 800 experimental data collected from open literature was also exposed to different formulas and correlations for pressure drop available in open literature and AARE were calculated for each of them (Table 9). From Table 9, it is evident that prediction error of pressure drop has reduced considerably in the present work.
Optimum parameters obtained by hybrid SVR- GA algorithm
No
C
ε
Kernel type
Type of loss function
Kernel parameter
A
1
5043.82
0.50
erbf
ε-insensitive
2
0.127
2
6698.16
0.36
erbf
ε-insensitive
3
0.127
3
7954.35
0.48
erbf
ε-insensitive
4
0.127
4
8380.98
0.54
erbf
ε-insensitive
5
0.127
5
9005.8
0.59
erbf
ε-insensitive
6
0.127
Table 8 Comparison of performance of SVR-GA hybrid model Vs SVR model Performance criteria A
σ
R
Execution time/h
prediction performance performance by hybrid SVR-GA model
0.127
0.169
0.977
1
prediction performance performance by SVR model without without GA tuning tuning after exhaustive exhaustive search
0.132
0.170
0.975
4
8
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
Table 9 Per formance of different correlations to predict pressure drop
7
No
Author
A/% A/%
1
Wilson (1942) [15]
49.51
2
Durand & Condolios (1952) [16]
36.53
9
3
Newitt et al . (1955) [17]
93.43
10
4
Zandi & Govatos (1967) [18]
50.02
5
Shook et al . (1968) [19]
34.50
6
Turian & Yuan (1977) [22]
39.97
7
Wasp et al . (1977) [23]
26.68
8
Gillies et al . (1999) [32]
22.31
9
Kaushal & Tomita (2002) [34]
22.01
10
present work
12.70
7
CONCLUSIONS
Support Vector Machines regression methodology with a robust parameter tuning procedure has been described in this work. The method employs a hybrid SVR- GA approach for minimizing the generalization error. Superior prediction performances were obtained for the case study of pressure drop and a comparison with selected correlations in the literature showed that the developed SVR correlation noticeably improved prediction of pressure drop over a wide range of operating conditions, physical properties, and pipe diameters. The proposed hybrid technique ( SVR-GA) also relieve the non-expert users to choose the meta-parameters of SVR algorithm for his case study and find out optimum value of these meta-parameters on its own. The results indicate that SVR based technique with the GA based parameters tuning approach described in this work can yield excellent generalization and can be advantageously em ployed for a large class of regression problems encountered in process engineering.
8
11
12 13
14
15 16
17
18 19
20
21
22 23 24 25
NOMENCLATURE 26 A α , α* ε λ ξ ,ξ * σ
average absolute relative error vectors of Lagrange’s multiplier precision parameter regularization constant slack variables width of kernel of radial basis function
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2 3 4
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6
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