Response surface strategies in constructing statistical bubble flow models for the development of a novel bubble column simulation approach

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Computers and Chemical Engineering 36 (2012) 22– 34

Contents lists available at ScienceDirect

Computers and Chemical Engineering

j ourna l ho me pag e: w ww.elsev ier .com/ locate /compchemeng

esponse surface strategies in constructing statistical bubble flow models for theevelopment of a novel bubble column simulation approach

aldo Coetzeea, Roelof L.J. Coetzerb, Randhir Rawatlal a,∗

Department of Chemical Engineering, University of Cape Town, Rondebosch, 7701, South AfricaSasol Technology Research and Development, PO Box 1, Sasolburg, 1947, South Africa

r t i c l e i n f o

rticle history:eceived 29 March 2010eceived in revised form 12 June 2011ccepted 25 July 2011vailable online 30 July 2011

a b s t r a c t

Bubble columns represent a substantial contribution to chemical industrial equipment and due to theircomplex hydrodynamics are difficult and computationally expensive to simulate. A model of the flowstructure around a single bubble is required for the development of a computationally efficient alterna-tive model. The flow structure data obtained from CFD is highly non-linear, which lends itself to a Designand Analysis of Computer Experiments (DACE) approach. The situation differs from conventional DACE

eywords:ubble columnFDomputer experimentriging

applications since high resolution data is summarized, which allows different sampling criteria, corre-lation functions and optimization criteria to be evaluated in an assessable manner. A modified poweredexponential correlation function is introduced, which in conjunction with a gradient based space-fillingdesign resulted in an averaged 8 times smaller mean squared error compared to other correlation designcombinations. Furthermore, sequentially fitting the data is shown to produce the best overall fits, whenused in combination with space-filling designs.

. Introduction

Bubble columns are popular multiphase reactors that have beensed in various industries for decades because of their high heat andass transfer rates coupled with low operating and maintenance

osts. Despite their popularity, the behaviour of bubble columnsan be difficult to predict due to the complex nature of the hydro-ynamics. With recent advances in computer processing capability,omputational fluid dynamics (CFD) approaches have been devel-ped for simulating bubble columns. This has resulted in thevolution of three distinct approaches, i.e. Direct Numerical Sim-lation (DNS), the Euler–Lagrange approach and the Euler–Eulerpproach.

The DNS approach is the most fundamentally based and can besed to study very small scale bubble flow detail (Deen, van Sintnnaland, & Kuipers, 2004). It has successfully been applied to sim-late the shape and behaviour of bubbles rising (Delnoij, Kuipers,

Van der Swaaij, 1997; Dijkhuizen, van den Hengel, & Kuipers,005; Hua & Lou, 2007; Krishna, van Baten, & Ellenberger, 1999),owever, its computational expense severely restricts its usage for
he simulation of any more than a few bubbles. The Euler–Lagrangepproach tracks each bubble in a Lagrangian frame which is coupledo a continuum model for the liquid phase (Buwa, Deo, & Ranade,
∗ Corresponding author. Tel.: +27 21 650 4431; fax: +27 21 650 5501.E-mail address: [email protected] (R. Rawatlal).

098-1354/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2011.07.014

© 2011 Elsevier Ltd. All rights reserved.

2006; Delnoij, Lammers, Kuipers, & van Swaaij, 1997; Hu & Celik,2008; Laín, Bröder, & Sommerveld, 1999; Webb, Que, & Senior,1992). It operates at larger time and length scales and allows for thestraight forward incorporation of bubble size distributions. At thelargest time and length scales, the Euler–Euler treats both phasesas continua, resulting in a model appropriate at larger geometriesand higher superficial gas velocities. This model has been shown toprovide realistic predictions of the flow field inside bubble columns(Becker, Sokolichin, & Eigenberger, 1994; Gasche, Edinger, Kömpel,& Hofmann, 1990; Sokolichin & Eigenberger, 1999) and is mostsuited at high gas hold-ups. More recently, the incorporation ofa population balance framework to take account of bubble sizegroups has been investigated (Bhole, Joshi, & Ramkrishna, 2008;Lo, 1996), where a set of momentum and continuity equations aresolved for each size group, however this greatly increases the com-putational expense of the approach.

The Euler–Lagrange and Euler–Euler approaches have beenshown to provide reasonable to good agreement with experimen-tal results for bubble columns, however, the question of whichphysical effects to include in the models is still under debate(Sokolichin, Eigenberger, & Lapin, 2004). Furthermore, accuratesimulation of industrial scale equipment can be very time consum-ing and remains a major design inhibitor. A more computationally
efficient model of bubble columns which is capable of solvingflow behaviour while providing bubble scale flow informationcould be a valuable tool for optimizing bubble column operation(Fig. 1).
dx.doi.org/10.1016/j.compchemeng.2011.07.014

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W. Coetzee et al. / Computers and Chem

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procedures have been developed.

Ft

ig. 1. A column of liquid where a gas is sparged through known as a bubble column.

A method involving the tracking of each bubble as well as itsmmediate flow field is proposed as it is this region where theteepest gradients, and hence, the greatest contributors to compu-ational expense, will occur. Accounting for these flow regions from

cheap-to-evaluate surrogate model could drastically reduce theomputation time in solving the overall reactor model, while keynformation about the bubble microflow structures are available in

model which could be useful for building further complexities intohe simulation. The flow structures around the bubble can be gen-rated by solving the Navier–Stokes equations for flow over a fixedphere. This can be thought of as the computer experiment which isvaluated for different velocities. The surrogate model in the con-ext of the macro reactor model is referred to as the cell model. The
ell model will therefore be used in a macro reactor model to updatehe flow field around each bubble for given velocities determinedt each reactor timestep (Fig. 2). It is therefore the fundamental
ig. 2. The macro model uses the information provided by the cell model to update the flhe analogous situation of flow over a sphere which constitutes as the experiment.

ical Engineering 36 (2012) 22– 34 23

building block upon which the bubble column simulation is builtand its accuracy is imperative.

The cell model approach could also be constructed for dropsand particles and has the potential to be applied to variousheterogeneous systems in many fields such as flotation, rain-fall, heterogeneous reactors etc. The information available in thecell model can in addition be used to optimize the systemsperformance, which is computationally impractical with currentmultiphase CFD models. Furthermore, having a tool capable of sum-marizing high resolution complex data with a small number ofpoints could be extremely valuable.

The development of the cell model essentially translates to astudy of fitting complex surfaces. The Design and Analysis of Com-puter Experiments (DACE) is a statistical approach to develop anuseful understanding of how the outputs are related to the inputs(Santner, Williams, & Notz, 2003). Jones, Schonlau, and Welch(1998) have shown how the DACE approach can be used to opti-mize expensive black-box functions. It has also been successfullyapplied in aeronautical shape optimization based upon CFD simu-lations (Jouhad, Sagaut, Montagnac, & Laurenceau, 2007; Marsden,Wang, Dennis, & Moin, 2004). Some key advantages to using theDACE model are its flexibility to any type of data, its ability todescribe non-linearity and the potential to work with high dimen-sional data. In the context of the cell model, the DACE approachis effectively a regression of high resolution data. Since the inten-tion of the present study is merely to establish the usefulness ofthis approach (Fig. 2), we will restrict attention to flow in twodimensions. It must be noted that bubble columns operating in het-erogeneous regimes (different sized bubbles at higher velocities)are inherently 3 dimensional (Joshi, 2001), however this approachcould easily be extended to such a domain once the construction

The generation of the bubble flow fields to which the surrogatemodel will be fitted is described in Section 2. This is followed by adiscussion on the surrogate model in Section 3, and an evaluation

ow field around each bubble in the column. The cell model is generated by solving

24 W. Coetzee et al. / Computers and Chemical Engineering 36 (2012) 22– 34

Fig. 3. Experimental geometry and mesh, the geometry is set up according to the diameter (D) of the bubble and ratio’s chosen to allow flow dynamics to sufficiently developa

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(wflmiAel

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cross the domain. (a) Geometry; (b) mesh.

f the different sampling strategies used to construct the surrogateodel in Section 4. The results from the investigation are presented

nd analyzed in Section 5 with conclusions drawn in Section 6.

. Flow field data generation

The Navier–Stokes equations describe the motion of fluidsBatchelor, 1967) and their solution constitute the data which weill approximate. The equations are derived by assuming that theuid is a continuum and applying the principles of conservation ofass and momentum. To complete the equations, the mathemat-

cal expression describing the fluid viscosity needs to be assumed.ssuming an incompressible Newtonian fluid at steady state, thequations can be expressed in Cartesian coordinates in dimension-ess vector form:

∇ · u = 0

u · ∇u︸︷︷︸Non-Linear Advection

= −∇p︸︷︷︸Pressure Gradient

+ 1Re

∇2u︸︷︷︸Diffusive Terms

(1)

here u represents the velocity vector, p the pressure and Re theeynolds number. The Reynolds number is a dimensionless quan-ity which is expressed as Re = (�Durel)/�, where �, D, urel and �epresent the fluid density, characteristic length (in this case theubble diameter), relative velocity and fluid density respectively.s a result, various flow situations with respect to the bubble size,elative velocity and fluid properties can be summarized with theeynolds number. At low Reynolds numbers the viscous termsominate resulting in diffusive behaviour, while with increas-

ng Reynolds number the non-linear advection terms’ influenceecomes more pronounced. Although it may seem that at higheynolds numbers the viscous terms become entirely negligible,hey are important in the thin layers adjacent to boundaries, i.e. theoundary layers where velocity gradients are high and their influ-nce propagates through the rest of the fluid (Clift, Grace, & Weber,
978). Furthermore, since the low Reynolds numbers required forhe viscous only terms to be a valid approximation is rarely expe-ienced in bubble columns, the full Navier–Stokes equations areolved in generating the data.
It should be noted that the boundary condition on a bubble isdifferent to that of a rigid object, in that the liquid slips along thesurface of the bubble as opposed to the no-slip condition whichis appropriate for a rigid object (Magnaudet & Eames, 2000). Inaddition the flow behaviour of bubbles is very sensitive to con-tamination of the system. Surfactants tend to accumulate on thegas–liquid interface and retard bubble motion to that of a no-slipboundary condition. For a contaminant-free system the gas–liquidinterface can be described as a free surface for which a zero-shear-stress of the tangential component of the liquid velocity isappropriate (Batchelor, 1967).

The ranges for experimentation should follow from the phys-ical properties of bubbles. This method could be applicable to awide range of bubble properties, however, for this investigationwe will confine ourselves to clean spherical bubbles in an air-water system. From physical experimental data (Gaudin, 1957),it follows that a range of Re ≤ 270 for these conditions would beappropriate.

The equations are solved utilizing the Finite-Volume methodwith the SIMPLE pressure-velocity coupling scheme and 2nd orderUPWIND spatial discretization in the FLUENT 12 solver. The meshis set up according to the geometry displayed in Fig. 3. To allow foraccurate flow structures to develop, the resolution of the mesh hasto be fine enough. This was determined by computing the force onthe bubble on consecutively finer grids until no change was found.Triangular elements were used since they provide less skewed ele-ments for the given geometry. The velocity and pressure data istherefore available on the grid, which results in 164,983 data pointsin the sampling region.

In order to establish the validity of the simulation results forthe flow field approximation of a physical bubble, the resultingdrag coefficients resulting from the velocity flow field solutionshave been compared to that of experimental results (Haberman& Morton, 1953) and established drag correlations for a contam-inant free system (Hamielec, Storey, & Whitehead, 1963; Moore,
1963) at various Reynolds numbers within the operating range.The simulated drag coefficients are shown to correspond well to thepublished data in Fig. 4, although slightly lower at the low Reynoldsnumbers.


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ig. 4. Comparison of the simulated drag coefficient with experimental resultsHaberman & Morton, 1953) and correlation models (Hamielec et al., 1963; Moore,963).

. Surrogate model

.1. Computer experiments

The DACE method was introduced by Sacks, Welch, Mitchell, andynn (1989) as a way of approximating complex computer mod-

ls with more efficient predictors by modelling the deterministicutput as the realization of a stochastic process Y(x) that includes

regression model of n chosen functions fi with regression coef-cients ˇi and a random process Z(x) (see Eq. (2)). The method

ocusses on modelling the error as opposed to the functional formf the regression model.

From a modelling perspective, the key difference betweenhysical experimental data and that produced by a computerxperiment is that the latter is effectively free from random errorexcept for truncation errors, or if this is simulated), and the erroran be attributed due to lack of fit (Jones et al., 1998). It is thisistinction that allows computer experiments to be approximatedore accurately in a way that predicts non-linear responses and

ther complex behaviour, which in many cases becomes a futileask with physical data due to random noise.

Since the error in the model is due to the lack of fit of the regres-ion function, spatial dependence of the error may be assumed,hich implies that when two points are close to each other, their

rrors should be close as well. The form and the weight of thisependence will depend on the behaviour of the data being fitted.herefore, DACE is aimed at estimating how a function typicallyehaves in contrast to regression, which estimates the weighting onn assumed functional form, thereby estimating what the functions (Jones et al., 1998).

The response is treated as

(x) =n∑

i=1

fi(x)ˇi + Z(x) (2)

here

[Z(x)] = 0, E[Z(x)Z(x)T ] = �2R (3)

ere �2 represents the process variance, and R the covarianceatrix. In the case of simple least squares, R will correspond

o the identity matrix. This model is a generalized least squares


implementation, where the off-diagonal entries of R is modelledby a correlation function.

For a given set of sample points, S = [s1, . . . , snsamp ], a methodknown as Kriging has been established which describes the pre-diction at some point x as the linear combination of samplingresponses YS = [Y(s1), . . . , Y(snsamp )]:

y(x) = cT (x)YS. (4)

Minimizing the mean squared error (MSE) of the predictor sub-ject to the unbiasedness constraint and solving for c(x) results inthe Best Linear Unbiased Predictor

y(x) = f T (x) ˆ + rT (x)R−1(YS − F ˆ ) (5)

where ˆ = FT R−1YS(FT R−1F)−1 is the generalized least squares esti-

mate of (Santner et al., 2003). Here F corresponds to the regressionfunction evaluated at S, and r the correlation between the sam-ple points, S, and the prediction point x. Please refer to Sacks et al.(1989) for a detailed derivation of model. The Best Linear UnbiasedPredictor will be referred to as the DACE model.

It follows that the DACE model predicts the response as the lin-ear combination of the sampling responses, where the influenceof each sample response is determined by a distance dependantcorrelation function. The DACE model can therefore be seen as anoptimal interpolator, whose parameters are represented by those ofthe correlation function and are estimated using the Nelder–Meadoptimization scheme.

3.2. Correlation function

The form of the correlation is extremely important as it has to beable to capture the behaviour of the given data. The requirementis generally that the correlation approaches zero as the distancebetween two points approaches infinity, and approaches one as thedistance approaches zero. This behaviour is given by the flexiblefunctional form of the powered exponential

Rij = exp

(−

d∑k=1

�k|xik − xjk|pk

)(6)

for d dimensional data. Eq. (6) is also referred to as the Krigingbasis function (Jones, 2001), and will here be referred to as EXP.The � parameters correspond to the activity while p represents thesmoothness of the correlation. Different activities can be used foreach dimension which is known as anisotropic behaviour, whichallows the form of the correlation function to vary depending uponthe approaching direction. For p = 2 the correlation function willcorrespond to the Gaussian correlation, which is smooth, while forp = 1 the correlation will be linear.

Another popular correlation is known as the Matérn covariancefunction:

Rij = 12p−1�(p)

(d∑

k=1

�k|xik − xjk|)p

Kp

(d∑

k=1

�k|xik − xjk|)

(7)

where Kp is a modified Bessel function of the second kind of orderp, and � and p carry the same significance as in Eq. (6). It is alsoknown as the Whittle–Matérn function since it was introduced byMatérn (1960) but was deduced earlier by Whittle (1954). The cor-relation can produce ill-conditioned covariance matrices for certainp parameters and therefore it might be preferred to optimize adiscrete set of p (Diggle, 2007). Therefore, when dealing with thiscorrelation function, the initial conditions of the correlation param-
eters are adjusted when mostly ill-conditioned covariance matricesare produced.
We introduce a modification to the powered exponential (Eq.(6)) which allows the correlation to be dependent on the distance

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45

6 W. Coetzee et al. / Computers and

o some point of interest. This point could potentially representhe point of maximum gradient (optimum) or mark an importanteature in the data. This correlation function will be referred to asXP-C.

ij = exp

(−

d∑k=1

�∗k |xik − xjk|pk

)(8)

ith �∗k

= �k + �k1m(w) and m(w) is the desirability function of theorm:

(w) = exp(s) − exp(s|w|)exp(s) − 1

. (9)

ere s is the importance or activity of the critical point xc, while wan be defined for each dimension as

1 =d∑

k=1

(xik − xck)2

(xmax,k − xmin,k)2(10)

2 =d∑

k=1

(xjk − xck)2

(xmax,k − xmin,k)2(11)

The desirability can now be specified as a function of theeighted contribution of w1 and w2:

(w) = m(�w1 + (1 − �)w2) (12)

For this investigation, the point of interest was chosen as theenter of the bubble, the weighting as � = 0.5 and the activity, s, sets an optimization variable. Desirability functions of the form ofq. (9) is commonly applied to dual response surface optimizationroblems (Coetzer, Rossouw, & Lin, 2008).

.3. Correlation parameter estimation

To close the model, the parameters of the correlation functioneed to be defined. A popular technique is to assume that thenderlying distribution of Z(·) is Gaussian which allows the �s toe estimated such that they maximize the corresponding log likeli-ood function in terms of the sampled data (Koehler & Owen, 1996).his ensures that the data being predicted is most likely to haveome from the same model as the data from the sample points whenhe underlying error distribution is Gaussian. This method is knowns the Maximum Likelihood Estimation (MLE) of the parameters.

Since we are summarizing high resolution data, there are how-ver data points available which could be used for a more accuratestimate of � and allows for the estimation of other correlationarameters such as the smoothness parameter, p, and the desir-bility parameter s, which are generally chosen as discrete values.his provides potential for exploring creative correlation functionsith the only constraint being the effectiveness of the optimization

outine in the light of potential non-linear multi-parameter corre-ation functions. This investigation also serves as an test case forsing the DACE model to summarize high resolution data.

The procedure for constructing the DACE model and estimatingts parameters is given:

. Obtain the velocity field, Y, for a given Reynolds number usingthe CFD experiment discussed in Section 2.

. Determine the sampling points and corresponding sampleresponses YS (discussed in Section 4).

. Construct the covariance matrix, R, using a correlation function
as discussed in Section 3.2 with parameters � and p (for EXP-Cinclude s).
. Compute the Cholesky decomposition of R such that CCT = R.

. Compute the generalized least squares solution for ˆ using C.

cal Engineering 36 (2012) 22– 34

6. Evaluate the DACE model, y, at all the grid points of the CFDsolution using Eq. (5) with R−1 = C−TC.

7. Compute the mean squared error for the fit, i.e. MSE = mean((Y −y)2).

An optimization problem is solved for the estimation of parameters� and p (for EXP-C include s), where steps 3–7 are repeated untilthe convergence criteria is met.

3.4. Development of a bubble column simulator

To arrive at a complete bubble column simulator, there are fur-ther aspects that need to be considered, which form part of ongoingresearch being carried out by the authors. These aspects will beexplored briefly to provide the context of the work presented inthis paper. The subject of this paper is the evaluation of the DACEmodel as a tool for approximating the spatially dependent flowfields in the direct vicinity of individual bubbles. Further researchwill focus on describing the Reynolds number dependence of thisspatial model, such that the completed cell model can describe theflow field around a bubble for a given Reynolds number. Since theflow structure evolves smoothly as a function of Reynolds number,it is considered reasonable that the parameters to the spatial modelcould be described using functional forms.

The next step would be the integration of the cell model with amacro fluid model. As discussed in Section 1, the cell model regionswill be tracked in a Lagrangian frame, while the continuum outsideof these regions will be resolved by solving the Navier–Stokes equa-tions in an Eularian frame. The computational cells in the cell modelregions will have to be identified in a computationally efficientmanner, i.e. without looping over the entire grid, and their influenceremoved from the linear set of equations resulting from the dis-cretization of the Navier–Stokes equations. This could be achievedthrough developing code that can be integrated with existing CFDsolvers. The final goal is to present a validated simulator which,through the use of a cheap to evaluate cell model, can reduce someof the considerable computational expense associated with bubblecolumn simulation.

4. Sampling design

The sampling design, together with the error correlation func-tion, have to represent the entire domain of the non-linear model.Furthermore, a different design may result in an entirely differentcovariance matrix on the same data set and therefore the best errorcorrelation function will also be dependent on where the points aresampled. It is theoretically possible to regress for the positions ofall the points together with the correlation parameters accordingto the MSE of predicted response, but this would result in a veryhigh dimensional optimization exercise which is extremely com-putationally expensive and unreliable. Therefore, some criteria forgenerating designs are evaluated to establish what defines a goodsampling design for the current problem.

Since we are dealing with deterministic data, Fisher’s (Fisher,1935) design of experiments methodology of randomization, repli-cation and blocking become irrelevant. Instead, all the attention canbe focused on the location of the sample points. Sampling designsdeveloped specifically for computer experiments tend to fill outthe design space (space-filling) rather than being concentratedon a specific boundary. This allows information from across thedesign space to be taken into account and is especially useful when
no information about the functional behaviour of the response isknown.
The design criteria compared are summarized in Table 1 andillustrated in Fig. 6. Many more space-filling designs are available


Table 1Sampling designs.

Design Description Data dependent nsamp

RND Randomly selected points No 30, 60, and 90LHD Latin hypercube design

(with maximin criteria)No 30, 60, and 90

UDC Uniform design inCartesian coordinates

No 24, 56, and 84

UDP Uniform design in polarcoordinates

No 25, 64, and 81

UTM Uniform triangulated mesh No 30, 60, and 90CPD Critical point design Yes 30, 60, and 90

iepDcTaoe

adtTnacin

hisnpthEc

uses a high minimum distance (Fig. 6f and g).

SRD Sequential residual design Yes 30, 60, and 90

n literature (Santner et al., 2003). However, the best or optimalxperimental design for constructing approximate models of com-utational fluid dynamic simulations, and specifically for usingACE as a data summarizing tool of these simulations have not beenommunicated previously in literature, and thus are not known yet.herefore, the designs in this paper were selected to evaluate thepplication of DACE for the purpose of creating a surrogate modelf the micro-flow structure around a bubble, and the list is notxhaustive.

The designs were chosen to establish which design objectivesre best suited to provide the most accurate DACE model. The RNDesign consists of randomly chosen points from a uniform dis-ribution where points outside the constraint space are rejected.he number of design points is satisfied by iteratively generatingew designs where the amount of points in the square domain isdjusted by a ratio of the required number of sample points to theurrent number of sample points inside the sampling domain. Thiss repeated until the current number of points equal the requiredumber of points.

McKay, Beckman, and Conover (1979) introduced the Latinypercube design as an improvement over random sampling for

nput variables to computer code. It ensures that the design ispace-filling to a certain extent by dividing the sample space intosamp rows and columns and requiring that there is only one sam-le point in each row and column. The space-filling measure ofhe design can be further improved by generating a series of Latin
ypercubes and choosing the one which maximizes the minimumuclidian distance between points. This is known as the maximinriteria (Santner et al., 2003).
Fig. 5. Distribution of points in the Cartesian space of the UD


The UDC is a full factorial design in Cartesian coordinates withthe points outside of the constraint space rejected. Therefore thenumber of sample points can only assume certain discrete valuesand the resolution is chosen to minimize the difference between therequired number of sample points and that available. The UDP is afull factorial design as well, but in polar coordinates, and thereforethe number of points are chosen as (round(

√nsamp))2. Due to the

points becoming more spread out as the radius increases, moreare concentrated in the center region which results in points beingroughly normally distributed in the Cartesian coordinate system(see Fig. 5).

The UTM makes use of an unstructured mesh generator (Persson& Strang, 2004) which utilizes an iterative procedure to triangulatethe data with Delaunay triangulation together with solving for theposition of the points by treating the mesh as a truss and solv-ing for a force equilibrium. This leads to approximately equilateraltriangles, resulting in a fairly evenly spread out design which isadaptable to complex geometries. The meshing scheme is appli-cable to n-dimensions but is largely untested for higher than 3dimensions.

The CPD is a novel sampling design introduced in this paperwhich makes use of the high resolution data available to computethe first and second order derivatives of the model with the centraldifference formula. The first derivative is used to determine wherelocal extrema in both directions in the data occur. These extremamay occur along ridges in the data and therefore may include manypoints. The magnitude of the second order derivative is then usedto rank the importance of these extrema in order to identify theregions where the gradients are steepest. The required number ofsample points can then be picked as the highest ranked points. Sincethese gradients tend to be steepest in certain regions, the top rankedpoints tend to be clustered together which is not good for the space-filling requirement of DACE models. Therefore a minimum distancecriterion is also incorporated which starts at the highest rankedpoint and adds a penalty value to the all points not conformingto its minimum distance criterion. This procedure is repeated as itmoves through all the points, ensuring that points are spread outto the degree required. The CPD-L design represents a low valuewhere the points are closer together whereas the CPD-H design

Sequential experimental designs are an efficient alternative tofixed-point designs (all the above designs) and allow the DACEmodel to be optimized to a certain criterion by sequentially

P with nsamp = 120. (a) x distribution; (b) y distribution.


F 56); ((

atJM

ig. 6. Samplings designs. (a) RND (nsamp = 60); (b) LHD (nsamp = 60); (c) UDC (nsamp =g) CPD-H mind = 0.2 (nsamp = 60, Re = 270); (h) SRD (nsamp = 60, Re = 270).

dding sampling points accordingly. Often the goal is to findhe global optimum and an efficient improvement algorithm byones et al. (1998) has been shown to produce good results.

ore recently Lam (2008) introduced the Expected Improvement

d) UDP (nsamp = 64); (e) UTM (nsamp = 60); (f) CPD-L mind = 0.05 (nsamp = 60, Re = 270);

Criterion for Global Fit (EIGF) which was also shown to performwell compared to other criteria. These sequential design criteriaare for the selection of a new point under the absence of data,however since we have data available it is possible to optimize

W. Coetzee et al. / Computers and Chemical Engineering 36 (2012) 22– 34 29

Fig. 7. Scaled y-velocity contours for different Reynolds numbers, the blue tones denote higher relative velocities while the red represent the lower flow rates. (a) Re = 0.1;(

teS

o(ttiab(dsceo

5

fpSptaod

b) Re = 15; (c) Re = 75; (d) Re = 135; (e) Re = 270.

he accuracy of the overall fit according to the mean squaredrror of the DACE predictor, which is the criterion used for theRD.

The SRD is the most computationally expensive design. Initiallynly a few points are present on which a DACE fit is performedthe optimization process uses higher stopping criteria otherwisehe process would be too computationally expensive), a point ishen sequentially added at the location where the absolute residuals maximum. This is repeated until the desired amount of pointsre obtained. Since it is computationally demanding, it will onlye examined with the powered exponential correlation functionEq. (6)). This method can also be used in combination with otheresigns where the initial design is augmented with a number ofequential points, which accounts for features the original designriterion do not take account of. For this investigation this has beenxamined with the UDC, UDP, UTM and CPD-H designs, with a thirdf the points sequentially added.

. Results and discussion

The aim is to assess the performance of the different correlationunctions and sampling designs at different numbers of samplingoints on velocity fields across the operating range of Re ≤ 270.ince the optimization of the correlation parameters can be com-utationally expensive, five y-velocity surfaces will be evaluated in
his investigation, i.e. at Re = 0.1, 15, 75, 135 and 270. These surfacesre generated as discussed in Section 2, producing the y-componentf the liquid velocity over a bubble. The contours of these fields areepicted in Fig. 7.
The velocity field for Re = 0.1 consists of smooth features whichare symmetrical across the y = 0 line. As the Reynolds numberincreases to 15, the symmetry across y = 0 is lost and a tailfeature begins to develop. At Re = 75, the flow features concen-trate into smaller areas in the domain and become steeper. Thetransition from Re = 75 to 270 is characterized by further increasesin the steepness of the gradients as the advection terms (seeEq. (1)) become even more prominent. At higher Reynolds num-bers the shape of the bubble changes while the flow becomesunsteady and the tail will begin to oscillate, resulting in more com-plex flow structures. These surfaces will be the subject of futureinvestigations.

When summarizing the high resolution data such as in this casethe sample points could be viewed as parameters to the DACEmodel and therefore it is favourable to summarize the data intoas few sample points as possible. For this examination we evalu-ated 3 levels, i.e. 30, 60, and 90, for the number of samples in thedesign (nsamp). Before commencing with the sampling designs andcorrelation functions, the estimation of the correlation parametersis to be considered.

5.1. Estimation criteria

As mentioned in Section 3.3 the residual with respect to the highresolution data can be used to optimize the correlation functionparameters instead of using the MLE. This is evaluated by compar-
ing the results from the different estimation methods for the fivesurfaces using a UDC design with 84 sample points. In order to pre-vent bias towards the residual criterion, the error comparison wasperformed on a grid of 91,055 points.


Table 2Comparison of the maximum likelihood (MLE) and residual based parameter esti-mation criteria.

Method Re = 0.1 Re = 15 Re = 75 Re = 135 Re = 270

scu

5

etfEt

5

5

ctId

tmdb

Table 3Generation times of DACE models with the EXP, EXP-C and Matérn correlation func-tions with varying number of sample points, nsamp . The times are averaged withrespect to the sampling design and given in minutes.

nsamp EXP EXP-C Matérn SRD (EXP)

30 1.2 2.3 7.3 5.860 2.3 29.3 18.1 20.7

Fc

MLE 6.91E−12 1.29E−05 2.45E−04 6.55E−04 2.10E−03Residual 3.75E−13 1.24E−07 8.11E−06 4.65E−05 2.87E−04

It is seen from Table 2 that in all cases, minimizing the sum ofquares of the residual produced much better results. It is thereforeoncluded that this method will be the most efficient for the datander consideration.

.2. Investigation

The different DACE fits are compared based on the mean squaredrror of the prediction at the original data points. All the results areabulated in Tables A4–A9. Some of the results for the MSE as aunction of the nsamp for the different designs are depicted in Fig. 8.xamples of good fits for each surface is also depicted together withheir designs in Fig. 9.

.3. Discussion

.3.1. Correlation functionThe general powered exponential correlation, EXP, was the most

omputationally efficient to optimize, as evident from Table 3, andhe least prone to producing ill-conditioned covariance matrices.t does not require assumptions about the data and can representifferent types of behaviours due to its flexible form.

The Matérn correlation function produced the largest predic-
ion errors for all the different designs and surfaces evaluated. This
ay be a result of the correlation function being prone to pro-ucing ill-conditioned covariance matrices, which results in “NaN”eing passed to the optimization scheme for certain combinations

ig. 8. Designs and nsamp compared based upon the MSE for different surfaces and corrorr = EXP-C; (d) Re = 270, corr = EXP, designs = SRD.

90 2.6 36.0 31.0 42.1

of correlation function parameters. This influences the optimiza-tion scheme to move away from potential regions of global minimaand therefore results in difficulty in finding the global minima. Sub-sequently, the Matérn correlation function is not recommended foruse with computer experiments of CFD code since the increasedcomputational effort does not yield a better fit.

The modified exponential correlation, EXP-C, consistently pro-duced the lowest MSE. It shows that the behaviour of the data isnot the same throughout the domain and varies with distance tothe center of the bubble. Therefore, including this feature rendersthe correlation function more flexible. However, it may producemore ill-conditioned covariance matrices than the EXP correlationand was the most expensive to optimize as shown in Table 3. Theincreased computational expense could partly be attributed to theextra parameter, s, that is estimated and is justified by the improvedfits. It should be noted that the way we included the point of interestin the EXP-C correlation function, i.e. by modifying the �s, does notimpact the conditioning of the covariance matrix as negatively asdirectly including the point of interest in the correlation equationwould have.

There is potential to further modify and improve the correlation
function to be dependent on the angle from the point of interest.This could be useful when there are certain slices in reference tothe critical point which exhibit distinct behaviour.
elation functions. (a) Re = 0.1, corr = EXP-C; (b) Re = 75, corr = Matérn; (c) Re = 270,


Fig. 9. Examples of good fits; marker dots indicate sample points. (a) DACE surface for Re = 0.1 with corr = EXP-C, Design = CPD-H, nsamp = 30; (b) residual of (a); (c) DACEsurface for Re = 135 with corr = EXP-C, Design = CPD-H, nsamp = 60; (d) residual of (c); (e) DACE surface for Re = 270 with corr = EXP, Design = UTM + SRD, nsamp = 90; (f) residualof (e).

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.3.2. Sampling designsThe LHD was found to generally perform slightly better than the

ND although this was not always the case. It would be preferredver random points since it guarantees some degree of uniformityn univariate dimensions, whereas in some cases randomly cho-en points can be very close together resulting in ill-conditionedovariance matrices.

The UDC performed reasonably well for the Re = 0.1 and 15 sur-aces (Tables A4 and A5), but performed poorly with nsamp = 56, andelow average with nsamp = 84 for the Re = 75, 135 and 270 surfacesTables A6–A8). This was due to the spacing of the points whichesulted in no points directly on the region where the tail features present. Therefore certain design criteria might be inefficient toescribe certain features in the data, even though the design is uni-orm in some sense. Alternatively a maximin or sequential designriterion may be employed together with the uniformity criterionor constructing an improved design. Specifically, from Fig. 8d it cane observed that the UDC + SRD performs reasonably well, which

llustrates how the addition of another criterion can vastly improvehe DACE model.

In contrast to the UDC, the UDP places points directly on theail feature where the high residuals for these surfaces occur, andherefore yielded a smaller MSE. The UDP also inherently places

ore points close to its reference point while maintaining someniform scatter of points, which was advantageous for the data

nvestigated. However it must be noted that it also performed wellor the Re = 0.1 case (see Fig. 8a) where no tail feature is present,hich implies that this design could be useful for any data whichas some point of interest or region where the data is most non-

inear.The UTM resulted in excellent fits for Re = 0.1 and comparable

ts for the other surfaces. This again is due to the non-linear taileature which dominates the errors for those surfaces. It did how-ver perform better than the UDC in all cases which makes it aood choice for non-rectangular geometries in 2 and 3 dimensions.ophisticated meshing algorithms are freely available which spreadoints out evenly subject to very complex geometries (e.g., Perssonnd Strang, 2004; Schöberl, 1997).

The CPD performed best when a high minimum distanceas specified between the points. Therefore, in addition to

electing points in the critical areas, these points should beubjected to some space-filling criterion in order to obtain accu-ate DACE models. A design whose only criterion is to spreadut points can better represent the data than one where pointsre clustered together even when a good criterion is used.he consistently best fixed point design was found as the onehich identifies critical regions and then places the points inside

ased on a maximum minimum distance which the number ofamples would allow. The CPD-H combination with the EXP-

correlation function resulted in excellent fits and producedn MSE which was on average 8 times smaller than the otherombinations.

The SRD designs performed well (Table A9), with the UTM+SRDesign having consistently produced the lowest mean squaredrrors. It can be concluded that a design which aims to spread outoints evenly, in combination with a sequential design criterion,roduces the most accurate DACE models. Even the UDC which per-
ormed poorly for higher Reynolds numbers, performed even betterhan the UDP+SRD for Re = 270, since it has better space-filling prop-rties than the UDP, whose design points are distributed towardshe center (see Fig. 5). Therefore it can be concluded that a design
cal Engineering 36 (2012) 22– 34

that is aimed at filling the design space would be the preferredstarting design for a sequential strategy. Furthermore, it is shownthat the rate of improvement from 60 to 90 points is higher withthe sequential addition of points compared to the designs based ona single criterion.

6. Conclusion

An accurate model of the micro-flow structure around a bubbleis required for a new approach to simulating bubble columns. Theflow model is generated by solving a set of non-linear partial dif-ferential equations (Navier–Stokes equations) on a high resolutiongrid which is required for accurate computation. A DACE approachhas been presented to summarize this high resolution data withat least 3 orders of magnitude less sampling points together withcovariance correlation parameters. Furthermore, since high resolu-tion data is available the covariance function is optimized subjectto the mean squared error of the prediction. This also provided agood opportunity to evaluate the performance of different sam-pling criteria and covariance correlation functions in an assessablemanner.

The powered exponential correlation proved to be the mostcomputationally efficient to optimize. However, we introduced amodified powered exponential function, which is distance depen-dent on a point of interest (center of the bubble) and yielded thebest overall results. We believe that this function could be appliedsuccessfully to a wider suite of industrial modelling problems.

Seven different design of experiment approaches have beenevaluated for developing DACE models. It is recommended thata design with good space-filling properties be used for perform-ing computer experiments. Combining design criterion can vastlyimprove the accuracy of the design as shown with the UDC + SRDcase. Specifically, compound or sequential design strategies can beused to add design points where the model is highly non-linear withsteep gradients. The critical point design (CPD) has been introducedand shown to be superior compared to the other designs for many ofthe cases evaluated. Specifically, the CPD combined with a space-filling criterion was shown to be the best non-sequential design.However, further research is required in optimal space-filling andcompound design criteria for experiments on computer code withhighly non-linear outputs.

This paper shows how the DACE approach can be appliedeffectively to summarizing non-linear data. It also illustrates theperformance of different designs and correlation functions in aassessable manner. These type of models could be applied to manydifferent applications where a surrogate model of complex datais required. In future research some effort will be made to fur-ther improve the accuracy of the surrogate model, and describethe Reynolds number dependency. Furthermore, the integrationof the surrogate cell model with a macro fluid model will beinvestigated.

Acknowledgement

Waldo Coetzee would like to thank Sasol Technology (Pty) Ltd.for financial support.

Appendix A.

See Tables A4–A9.


Table A4Fitting results for surface Re = 0.1.

corr:nsamp RND LHD UDC UPD UTM CPD-L CPD-H

exp:30 1.47E−011 1.64E−011 5.08E−012 8.91E−012 7.60E−012 7.70E−012 2.08E−012exp:60 2.26E−012 2.68E−012 2.89E−012 7.70E−013 1.45E−012 4.50E−012 1.41E−012exp:90 1.10E−012 8.74E−013 7.39E−013 1.06E−012 4.96E−013 1.43E−012 1.18E−012exp-c:30 1.41E−011 1.46E−011 5.08E−012 3.71E−012 6.77E−012 1.17E−012 1.11E−012exp-c:60 2.14E−012 3.05E−012 1.81E−012 1.45E−013 8.22E−013 5.98E−013 2.35E−013exp-c:90 4.31E−013 3.15E−013 5.47E−013 1.32E−013 3.37E−013 2.55E−013 2.04E−013matern:30 2.10E−011 1.39E−011 1.29E−011 3.07E−011 1.97E−011 1.90E−011 9.69E−012matern:60 9.82E−012 5.79E−012 6.80E−012 –a 2.40E−012 1.63E−011 3.91E−012matern:90 4.37E−012 2.32E−012 2.33E−012 6.82E−012 1.61E−012 5.12E−012 3.11E−012

a Design and correlation function are not compatible.

Table A5Fitting results for surface Re = 15.














exp:30 9.78E−004 1.24E−003 7.44E−004 4.23E−004 1.26E−003 5.31E−004 3.49E−004exp:60 4.99E−004 5.30E−004 1.02E−003 1.50E−004 5.21E−004 4.17E−004 1.17E−004exp:90 5.02E−004 2.84E−004 4.47E−004 1.49E−004 3.36E−004 2.49E−004 6.50E−005exp-c:30 8.17E−004 7.13E−004 7.41E−004 3.80E−004 1.19E−003 1.55E−003 3.48E−004exp-c:60 7.34E−004 7.32E−004 9.79E−004 1.29E−004 4.42E−004 3.26E−004 7.28E−005exp-c:90 2.81E−004 2.15E−004 3.99E−004 9.30E−005 2.76E−004 3.39E−004 5.11E−005matern:30 1.36E−003 1.61E−003 7.91E−004 4.12E−004 2.06E−003 4.91E−004 3.67E−004matern:60 1.30E−003 7.20E−004 1.15E−003 -a 6.43E−004 4.41E−004 1.34E−004matern:90 4.43E−004 3.27E−004 5.49E−004 1.86E−004 4.28E−004 3.28E−004 1.13E−004



Table A9Fitting results for surface SRD.

Re:nsamp SRD SRD + UDC SRD + UDP SRD + UTM SRD + CPD-H

0.1:30 9.87E−013 2.10E−012 3.48E−012 8.30E−013 1.41E−0120.1:60 2.24E−013 3.98E−013 8.39E−013 2.19E−013 3.03E−0130.1:90 7.79E−014 1.62E−013 1.55E−013 7.89E−014 2.26E−01315:30 5.68E−007 2.84E−007 6.27E−007 2.52E−007 3.80E−00715:60 3.72E−008 4.90E−008 9.32E−008 6.94E−008 1.89E−00715:90 1.33E−008 1.81E−008 3.08E−008 1.87E−008 5.28E−00875:30 1.24E−005 9.73E−006 1.53E−005 1.19E−005 2.73E−00575:60 1.91E−005 4.01E−006 3.94E−006 3.96E−006 6.31E−00675:90 1.80E−005 1.60E−006 1.93E−006 1.46E−006 1.40E−006135:30 4.55E−005 4.04E−005 4.09E−005 3.59E−005 1.48E−004135:60 4.43E−005 1.50E−005 1.85E−005 1.28E−005 1.17E−005135:90 8.58E−006 9.74E−006 8.05E−006 4.29E−006 6.35E−006

R

BB

B

B

C

C

D

D

D

DD

FG

GHH

H

H

J

J

270:30 1.85E−004 1.82E−004

270:60 6.51E−005 5.56E−005270:90 3.27E−005 2.48E−005

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