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Energy and Buildings 62 (2013) 570–580

Contents lists available at SciVerse ScienceDirect

Energy and Buildings

j ourna l ho me pa g e: www.elsev ier .com/ locate /enbui ld

ssessments of experimental designs in response surface modellingrocess: Estimating ventilation rate in naturally ventilated livestockuildings

iong Shena,b, Guoqiang Zhangb,∗, Bjarne Bjergc

School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, ChinaDepartment of Engineering, Faculty Sciences and Technology, University of Aarhus, Blichers Allé 20, 8830 Tjele, DenmarkDepartment of Large Animal Sciences, University of Copenhagen, Grønnegaardsvej 2, 1870 Frederiksberg C, Denmark

r t i c l e i n f o

rticle history:eceived 7 November 2012eceived in revised form 26 February 2013ccepted 24 March 2013

eywords:FDesign of experiment

a b s t r a c t

Precise modelling the ventilation rate through a naturally ventilated livestock building can benefit thecontrol of indoor climate and reduction of ammonia emission. In terms of agricultural dairy buildings, themodelling of ventilation rates may involve in several variables, including the opening sizes at side wallsand the outdoor wind conditions. A statistical modelling process requires knowing how the experiment isdesigned and what modelling technique is followed. In this paper, several different methods for design ofexperiment (DOE) such as central composite rotation design (CCRD), optimal design (OPD), Box–Behnkendesign (BBD) and space filling design (SFD) were compared for their accuracies of the acquired models

esponse surface methodologyPVDS

and numbers of experimental runs. Response surface methodology (RSM) was applied and discussed formodelling the ventilation rate in relation to those variables. Results demonstrated the BBD had the bestperformance in the model development. The fraction of design space (FDS) tool was also evaluated forits ability in comparing different DOE methods and results showed that this tool performed inadequatelyin comparing between traditional DOE methods such as CCRD, BBD and FFD and modern DOE methods,

such as OPD and SFD.

. Introduction

Ventilation rate is a crucial parameter for indoor climate andir quality control of ventilated livestock buildings [1] and ammo-ia emission from the buildings [2]. However, it is difficult toirectly measure or model the ventilation rate of a naturally venti-

ated livestock building (NVLB), due to the varied wind conditions,he building geometry including the ventilation openings, andhe effects of surrounding structures [3–5]. In practical NVLBs,he parameter such as the opening size of the building is oftenontrolled so as to regulate the ventilation rate following variedutdoor climate conditions. However, the outdoor wind conditionsre very time-dependent and change dynamically that may resultn the ventilation openings being over sensitively regulated [6]. Inrder to achieve an optimal control of the ventilation rates in NVLBs,t is very important to develop a model that can be used to estimatehe ventilation rate according to the varied weather and opening

onditions [7,8].

The statistical data-based models, namely the meta-modelethod [9], can be used for underlying the correlation between the

∗ Corresponding author. Tel.: +45 8715 7735; fax: +45 8999 1619.E-mail address: Guoqiang.Zhang@agrsci.dk (G. Zhang).

378-7788/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.enbuild.2013.03.038

© 2013 Elsevier B.V. All rights reserved.

output variables such as the ventilation rates, and the input vari-ables such as the wind conditions and opening sizes [10–12]. Thesemodels can be developed empirically based on the data collectionfrom series of experiments or simulations. In order to make themodel functional in field applications, the model should be built tohave highly accurate predictions [11]. However, there always liesa challenge in how many cases of experiment should be carriedout and what statistical methods should be followed to developthe model accurately [13]. One of the most well-established andeasy-to-use meta-modelling technique is the Response SurfaceMethodology (RSM) [13–15]. This modelling technique can depictthe interaction relations between variables and has been found tobe mostly suitable for system modelling with no more than tenvariables, which matched very well with the case as the one in ourstudy.

In an optimal statistic modelling process, another crucial issueis to make correct experimental design. In previous researcheson NVLB, the data collections for the model development wereplanned by the one-factor-at-a-time (OFAT) method [16,17]. Thismethod assumed the independence of the design variables and

managed to establish an additive Linear model. However, theempirical model following this method is not statistically signif-icant and may fail to account for the complicated relation betweendesign variables [18]. Instead, several other design of experiment

Buildings 62 (2013) 570–580 571

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a[psftr[mma

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otTtthao

2

2

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Table 1Ranges of design variables in our study.

Design variable Name Limit of variable (coded)

Upper Lower

X1 (V, m/s) Outdoor wind speed (m/s) 2 (−1) 10 (1)

X. Shen et al. / Energy and

DOE) methods can be applied [6,13,15,19]. Furthermore, thoseethods can be applied to control the number of experimental

uns (design points) and meanwhile reduce the prediction variancef the developed model. Among these DOE methods, the tradi-ional design methods such as central composite rotatable designCCRD), full factorial design (FFD) and Box–Behnken design (BBD)

ethod, as well as modern design methods such as optimal designOPD), space filling design (SFD), can be applied [6,13,15,18,19].mong these DOE methods, the traditional ones were firstly devel-ped because they were more simply and conveniently used toetermine the design points compared to the modern DOE. In theraditional DOE methods, the most concern lies in the orthogonal-ty, uniformity or rotatability of the prediction variance [18,20].n the contrary, modern DOE often aims to minimize the pre-iction variance of the DOE methods for model development andchieves widely spread of design points as well [18]. Most modernOE methods such as optimal design methods require computer

terations to generate the design points.However, the DOE method would affect the modelling accuracy

nd there is uncertainty about the best choice of the DOE methods6,15,21–24]. For different DOE methods based on varied statisticalrinciples, the distribution of the design points inside the designpace varies, and would accordingly influence the model accuracyor the RSM method [23,24]. Even though there were several inves-igations conducted previously for the specified case studies, it stillemains unclear to determine which DOE methods are optimal6,21–24], especially when it comes to this study that involving

ore than two variables. Thus, in order to make certain which DOEethod is suitable for modelling the ventilation rate of the NVLB,

test of the above-mentioned DOE methods is required.The developed model by RSM has the form of a Taylor series

olynomial equation, and consisted of a number of model terms.he number of the terms is related to the quality of data fit ofhe observations with the model predictions [6,25]. In previousesearch, the fully developed model that involving all terms showsnadequate quality of fit and reduction of terms can benefit it25]. Transformation of the model response by a function of logr square-root is also a very useful tool to enhance the quality of fit18]. However, there is no evidence that both the terms reductionnd response transformation can be suitable for all DOE methodsn model development and their effects on the model accuracy aretill not validated yet.

Thus, the objective of this study is to assess the performancesf different DOE method in developing the model for the ventila-ion rate with the building opening size, wind speed and direction.he RSM is applied for modelling and CFD simulation is appliedo estimate the ventilation rate. Several commonly used statis-ical tools such as terms reduction and response transformationas been tested and discussed. And tools such as the FDS tool forssessments of DOE methods are also analyzed and discussed inur study.

. Materials and methods

.1. Reponses surface modelling

A full dth-order polynomial model concerning three design vari-bles is expressed by the following Eq. (1):

(X) = ˆ 0 +∑

j

ˆjXj +

∑j

ˆjjX

2j +

∑j

∑k>j

ˆjkXkXj

+∑

j

∑k>j

∑l>k

ˆjklXjXkXl· · · +

∑j

ˆj,...,jX

dj + ε (1)

X2 (�,◦) Outdoor wind direction (◦) 0 (−1) 90 (1)X3 (Hc , m) Sidewall opening size (m) 0.4 (−1) 2.4 (1)

where X1, X2, X3 are the vector of design variables, in this study,they refer to the wind speed, V (m/s), wind direction, � (◦) and theopening size, Hc (m); ε is the vector of random errors.

Firstly the design space is constructed by the range of con-strained variables in order to determine where the design pointsshould be chosen from. The sample, namely the design point, rep-resents a level of each variable that selected from the space forexperiment. Table 1 classifies the limit of three design variables ofthis study, and these variables will construct a three dimensionalspace within their ranges.

The design variables (X1, X2, X3), was normalized by their rangeof the design space, and encoded into code variables (x1, x2, x3), byEq. (2):

xCod = 2(Xnat − X)(Xhigh − Xlow)

(2)

where xCod is the code variables, Xnat is the natural variables. Xhigh,Xlow is the high and low limit of the design space, respectively aslisted in Table 1 and X is the average of Xhigh, Xlow.

2.2. Definition of DOE methods

In order to select the design points for experiment, several dif-ferent experimental design methods, namely design of experiment(DOE) have to be followed, which are discussed below.

The fitted RSM model (Eq. (1)) can be written into the form inEq. (3):

y = Z ˆ (3)

where y is the N × 1 vector of the fitted value; Z = (1, x1 · · · xP−1)is an N × P matrix; ˆ is the P × 1 vector of estimated regressioncoefficients.

As can see above, when N = P, the vector of model coefficient ˆ iscan be exactly determined in Eq. (3) by designing P design points.However, when N > P, ˆ has to be chosen in a way of minimizingthe prediction error, ε. The least squares estimators of vector ˆ arethus solved by minimizing the matrix of errors, ε′ ε, and expressedin Eqs. (4) and (5):

ε = y − y = y − Z ˆ (4)

ε′ · ε = (y − y)′(y − y) = (y − Z ˆ )′(y − Z ˆ ) (5)

y is the vector of the observations. The least squares estimator of ˇis:

ˆ = (Z ′Z)−1Z ′ y (6)

And the variance of predicted y(xi) is given by Eq. (7):

Var y(xi) = �2xi′ (Z′Z)−1xi (7)

where x′i

is 1 × P vector whose elements corresponding to theelements of a row of matrix Z. y(xi) is corresponding estimated

response vector of xi. �2 is the experiment error.

The coefficients and variance involve the matrix:

H = (Z ′Z)−1. (8)

572 X. Shen et al. / Energy and Buildings 62 (2013) 570–580

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upbfeai

2

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ig. 1. 3D distributions of design points: BBD, Box–Behnken design. CCRD, central c are design points. Among the design points by the SFD, © are designed by distan

.3. Full factorial design (FFD) method

The full factorial design (FFD) is the most fundamental and pop-larly used experimental design method. The number of designoints of a full factorial design is the product of the num-er of levels of each variable (Fig. 1). The 23 design is usedor evaluating main effects and interactions and 33 designs forvaluating main, quadratic and interactions effects for three vari-bles. The design points by the full factorial design are shownn Table 2.

.4. Central composite design (CCRD) method

A central composite design (CCD) is a composite of a two level2k) factorial design, augmented by n0 centre points and two addi-ional points positioned at ±� for each variable. n0 centre pointsre repeated to calculate the experimental error. For deterministicxperiment, n0 = 1. In terms of three variables, setting = 1.68179ocates the points on a hypercube as well as a sphere surface, giv-ng a centre-composite rotatable design (CCRD) design. As seen inig. 1, the CCRD method has the design points surrounding a central

oint with equal distances. The variance of the predicted responset all design point xi, Var y(xi), are also equal and locate the sameistance from the centre. The dataset by CCRD method is shown inable 3.

ite design; FFD, full factorial design; OPD, optimal design; SFD, space-filling design.ed design, others are designed by the optimal design method.

2.5. Box–Behnken design (BBD) method

The FFD method may require massive number of design points,and hence, the smallest number of levels of variables in an experi-mental design is desirable. Box–Behnken design is a common classof such designs [26]. They are constructed by combining two-levelfactorial designs with incomplete block designs[18]. The distri-bution of the BBD method is shown in Fig. 1. From that, we cansee that one advantage of BBD method is that it does not con-tain design points at the vertices of the cube defined by the upperand lower limits for each variable. This would benefit the experi-ment conducting because sometimes extreme points are expensiveor impossible to test. Another advantage is that this design canachieve high rotatability and the centre points can be repeated soas to account for the experimental error, similar as CCRD designmethod [13]. Table 4 shows the dataset designed by the BBDmethod.

2.6. Optimal design (OPD) method

From Eqs. (5) and (6), we know that decreasing the (Z′Z)−1, we

can obtain lower Var y. The optimal criterion is developed to mini-mizes the (Z′Z)−1 matrix. Equivalently, it will decrease the varianceof y. In this study, the D-optimality was used to obtain design thatthe determinant of H is minimal and in turn gives the minimal

X. Shen et al. / Energy and Buildings 62 (2013) 570–580 573

Table 2FFD dataset.

X1 (V, m/s) X1 (�,◦) X3 (Hc , m) y (Q, m3/s)

2 0 0.4 20.326 45 0.4 165.97

10 90 0.4 652.796 0 1.4 206.442 45 1.4 181.01

10 45 1.4 913.136 90 1.4 700.326 45 2.4 882.172 90 2.4 938.992 0 1.4 68.306 90 0.4 389.636 0 2.4 295.476 90 2.4 1513.086 45 1.4 546.59

10 0 1.4 343.602 90 1.4 232.24

10 90 1.4 1170.982 45 0.4 109.48

10 45 0.4 555.542 45 2.4 361.47

10 45 2.4 1817.826 0 0.4 122.32

10 0 0.4 121.732 0 2.4 98.00

10 0 2.4 493.3310 90 2.4 2525.80

2 90 0.4 64.06

Table 3CCRD dataset.

X1 (V, m/s) X1 (�,◦) X3 (Hc , m) y (Q, m3/s)

3 20 1 125.258 20 1 336.833 70 1 284.018 70 1 764.433 20 2 238.638 20 2 639.013 70 2 348.768 70 2 1393.321.295518 45 1.5 125.949.704482 45 1.5 953.775.5 2.955179 1.5 213.385.5 87.04482 1.5 694.225.5 45 0.7 246.24

po

tt

TB

Table 5SFD dataset; the dataset consists of design points by two DOE methods: OPD, D-optimal design; DBD, distance-based design.

DOE X1 (V, m/s) X1 (�,◦) X3 (Hc , m) y (Q, m3/s)

OPD 10 0 0.4 121.73OPD 2 32.31101 0.4 46.93OPD 2 90 0.4 64.06OPD 7.2 90 0.4 234.52OPD 6.68 37.35 0.5 200.17OPD 5.2 0 1.2 148.79OPD 10 53.55 1.2 878.03OPD 5.88 70.65 1.4 664.90OPD 2.2 37.8 1.6 199.20OPD 2 90 1.7 292.00OPD 6.08 9.9 2.2 371.38OPD 2 0 2.4 98.00OPD 10 0 2.4 493.33OPD 5.2 54 2.4 1006.39OPD 10 90 2.4 2525.80DBD 3.92 0 0.4 39.71DBD 10 42.3 0.4 270.80DBD 4.24 63 0.6 194.02DBD 10 90 0.9 730.18DBD 9.16 9 1.4 352.00DBD 2 0 1.5 73.86DBD 8.8 85.5 1.6 1201.47DBD 6 90 2.2 1266.95

5.5 45 2.3 999.345.5 45 1.5 539.40

roduct of the eigenvalues of the matrix. In other words, the D-ptimal design is achieved by minimizing the |(Z′Z)−1|.

The design points fulfil this criterion at the maximum level haveo be iteratively selected within the design space. The dataset byhis method can be seen in Table 5.

able 4BD dataset.

X1 (V, m/s) X1 (�,◦) X3 (Hc , m) y (Q, m3/s)

2 0 1.4 68.3010 0 1.4 343.60

2 90 1.4 232.2410 90 1.4 1170.98

2 45 0.4 109.4810 45 0.4 555.54

2 45 2.4 361.4710 45 2.4 1817.82

6 0 0.4 122.326 90 0.4 389.636 0 2.4 295.476 90 2.4 1513.086 45 1.4 546.59

DBD 9.2 45 2.2 1429.73DBD 2 67.95 2.4 438.02

2.7. Space-filling design (SFD)

The optimal design does not consider the spread of design pointswithin the design space, because it just searches the group of designpoints that can efficiently decrease the model variance. For deter-ministic experiments, the use of ‘space filling’ are favoured bymany scientists, in which designs treat all regions of the designspace equally [19]. Through this, more extensive information of theoverall design space can be reached and depicted. Anderson, et al.suggested a way to achieve the space filling using a combination ofthe optimal design and distance-based design [18]. Distance baseddesign chooses design points in a way that achieves maximumspreading throughout the design space [18]. Therefore combinedtwo designs, the shortcoming of the optimal design can be over-come by making design points more space-filling.

Based on Eq. (7), number of the SFD is determined by examininghow large the fraction of design space where Var y(xi) is under theexperimental error �2 by a given design, xi. Consequently, if 90% ofthe design space lies below �2, it indicates that adequate numberof efficient design points has been determined [18].

The distribution of the design points of SFD can be seen in Fig. 1.The dataset designed by the SFD method is shown in Table 5.

2.8. Number of design points for DOE methods

Table 6 shows the amount of design points for different DOEmethods under varied number of design variables. The OPD methodrequires the least amount of design points. This method has itsadvantage in reducing the number of design points and minimiz-ing the prediction error within the design space [18]. The minimumrequired number of design points for OPD method is equal to thenumber of terms of a Quadratic model. To analyze the uncertainty ofthe coefficients, additional design points other than the minimumrequired are needed [24]. The recommenced number of additionalpoints was 20–50% of the minimum required [24]. This method

has been applied in a case study involving six design variables andresulted into 45 design points and the amount is nearly 50% morethan minimum [27]. Similarly, we used fifteen points (50%) for OPDin this study. In order to provide information about all portion of

574 X. Shen et al. / Energy and Build

Table 6Number of design points of DOE methods under different number of design variables.

Num of variables CCRDa BBDa OPDb FFD SFDc

2 9 9(6) 9 123 15 13 15(10) 27 204 25 25 23(15) 81 305 43 41 32 (21) 243 426 77 49 42 (28) 729 567 143 57 54 (36) 2187 728 273 113 68 (45) 6561 909 531 121 83 (55) 19,683 110

10 533 161 99 (66) 59,049 132

dt[

rbwlwetic

2

u(Ftsi

onsbiCl

ibttLta

ea

r

p

waet

a DOEs with single centre point.b In the bracket is minimum required number of design points.c Equivalent to doubled number of design points of minimum required of OPD.

esign space [19], SFD used doubled number of design points ashe minimum required of OPD as recommended by Anderson et al.18].

The rotatable and nearly rotatable design as CCRD and BBD,espectively, share similar number of design points when the num-er of design variables is lower than six as shown in Table 6. CCRDas reported to function well when the amount of variables was

ower than six [28]. BBD shows fewer design points, especiallyhen more than six variables involves in. The size of a full factorial

xperiment increases exponentially with the number of variables;his leads to an unmanageable number of experiments. Therefore,t is not an efficient method when involved with lots of variables asompared with other methods.

.9. Development of RSM models

As described above in Eqs. (4)–(6), the least square method wassed to estimate the model coefficients. The model with Lineard = 1), 2FI (2-fator interaction) (d = 1), second (d = 2), Cubic (d = 3),ourth- (d = 4), and Fifth- (d = 5) order were developed, where d ishe order of the model. The generalized form of the model can beeen in Eq. (1). The 2FI model is only consisted of first order andnteraction terms.

The backward elimination procedure can enhance the data fitf the model by reducing insignificant terms [18]. In the begin-ing, the model is developed by least square method. Then theignificant levels (P-values) of model coefficients were estimatedy ANOVA test. Terms that with P-value higher than 0.10 indicts the

nsufficiently affect the model and were removed from the model.onsequently, the final model consisted only of terms with P-value

ess than 0.10.Transformation of the response is used to enhance the qual-

ty of data fit with the observations [18]. The quality of fitting cane seen from how the errors (residuals), ε, of the model go withhe predicted values. Transformations are needed especially whenhe residuals are not stabilized to present a normal distribution.ots of transfer function can be applied to transform the responseo improve the model fit. In this study, we evaluated commonlypplied function such as square root and natural log [18].

The performance of the model to fit with the observations wasvaluated by adjusted R2 and predicted R2. adjusted R2 is the R2

djusted by the total number of design points and model terms.Predicted R2 is a measure of how good the model predicts a

esponse value, which is computed as:

redicted R2 = 1 − PRESSSStotal

, (9)

here SStotal is the total variance between the model predictionnd observations. PRESS is calculated by systematically removingach observation from the dataset, formulating a new model byhe left dataset, and calculating the variance (squares) between

ings 62 (2013) 570–580

predicted values by the newly established models and the removedobservation. The sum of squares is equal to the value of PRESS.

2.10. Acquire ventilation rate by CFD simulation

For a designated opening sizes and outdoor wind conditions,the ventilation rate can be estimated by either experimental orCFD methods [29]. In terms of the experiment, it is always a chal-lenge to monitor the ventilation rate accurately in field experimentsbecause of the uncontrollable outdoor wind conditions [30]. Thescale model experiment can be conducted alternatively in thewind tunnel facility to account for the real phenomenon. How-ever, the dynamic and dimensional similarities are very difficult toachieve simultaneously by the experiment. Moreover, the exper-iment requires massive time and cost to conduct [31]. Instead,CFD simulation can be applied alternatively, which mathematicallysolve the mass, momentum and energy equations governing thefluid field, hence, can be efficiently used to estimate the ventilationrate [32].

The full-size stand-alone Danish cattle buildings are inves-tigated for this study. The building has the dimension: lengthL = 147.2 m, width W = 39.6 m, height H = 11.8 m and the roof slopewere 20◦, yielding the total volume of the building 40,279 m3. Thebuilding is ventilated through openings at ridge and two sidewalls.The maximum dimensions of the sidewall openings are 2.4 m inheight. The configuration of the ridge top shelf and side-wind breakgives two ridge openings on the ridge with dimension of 0.25 m inwidth.

The detailed dimension of the geometry and simulation setupincluding boundary conditions and CFD turbulence model, etc.can be found in the literature [6]. Fig. 2 shows the computationaldomain and building geometry, in which the cells are condensedas coming close to the envelop walls with a minimum distance of0.1 m The geometry is simplified from the real dairy buildings inisothermal conditions. Non-slip wall are defined at all solid sur-faces with treatments of standard wall functions. The extension ofthe outdoor boundary dimensions is according to the guideline ofFranke et al. [33]. Those setups have yielded nearly 1,700,000 meshcells.

Standard k–ε model is used for the CFD turbulence model. Thecommercial solver from the software ANSYS Fluent 12.0 is appliedfor computation. The convergence of iterations is reached when allterms of velocity, turbulence and energy have no obvious fluctua-tions and the net flow rate through the building lower the thresholdvalue of 10−6. After the simulation, the ventilation rate Q (m3/s) canbe obtained by summing up the partial mass flow rate at all cellfaces of each opening.

3. Results and discussion

3.1. CFD simulation results

The simulation results of different DOE methods can be seenin Tables 2–5. The number of design points is 13, 15, 25 and 27for BBD, CCD, OPD and FFD, respectively. In total there were 63cases without repetition are simulated and belonged to the groupof combined DOE method.

3.2. RSM model development

Table 7 shows the fitted RSM models of varied DOE methods. Themodel with higher adjusted R2 and predicted R2 represents better

quality of data fit and more reliable performance on prediction,respectively as seen from the actual vs. predicted plots.

From Table 7, we can see that Linear models present thelowest adjusted R2 and predicted R2 and thus worst quality

X. Shen et al. / Energy and Buildings 62 (2013) 570–580 575

F , side as

om(m

TMQaa

ig. 2. Computational grid (1,700,000 grids): (a) perspective view of grid at bottomurfaces.

f fit and prediction in general. In contrast, 2FI and Quadraticodels perform much better. Listed number of design points

Tables 2–5) is more than the required to develop Linear and 2FIodels. For Linear and 2FI model development, the orthogonal

able 7odel coefficients of RSM model under varied DOE methods. 2FI is the 2-level factoriauadratic model. T–Q (log and square) is the model after transformation of response varifter transformation of response variable by log and square function, as well as reduction ond predicted R2 of the model, respectively. ˇ0–9 are the model coefficients.

Model DOE ˇ0 ˇ1 ˇ2 ˇ3 ˇ

Linear CCRD −861.37 103.37 6.62 347.92

BBD −806.80 97.39 6.88 351.36

OPD −1107.40 103.15 9.54 421.46

FFD −849.25 90.57 7.92 373.57

SFD −893.58 87.84 7.39 402.61

2FI CCRD 399.45 −91.74 −7.60 −191.33

BBD 304.99 −32.48 −6.04 −265.07

OPD 485.25 −59.27 −11.01 −270.80

FFD 225.47 −25.38 −7.01 −184.96

SFD 416.00 −56.41 −9.94 −277.45

Quadratic CCRD 482.38 −81.52 −2.38 −473.44

BBD 488.67 −46.78 −1.07 −672.24

OPD 687.72 −87.59 −12.20 −550.32

FFD 467.71 −68.07 −4.27 −558.72

SFD 586.78 −93.86 −7.59 −497.13

R–Q CCRD 211.74 −91.74 −3.43 −66.19

BBD 56.20 8.99 −0.51 −265.07

OPD 485.25 −59.27 −11.01 −270.80

FFD 398.12 −25.38 −7.01 −558.72

SFD 416.00 −56.41 −9.94 −277.45

T–Q (log) CCRD 0.98 0.17 0.01 0.42

BBD 2.91 0.39 0.03 0.32

OPD 0.80 0.19 0.01 0.65

FFD 0.89 0.20 0.01 0.45

SFD 0.81 0.18 0.01 0.67

T–Q (square) CCRD 1.27 1.12 0.16 0.42

BBD 5.96 1.39 0.12 −5.07

OPD 2.39 1.11 0.05 1.33

FFD 3.27 1.23 0.08 −1.47

SFD 2.62 0.94 0.05 1.88

T–R–Q (log) CCRD 1.04 0.20 0.01 0.28

BBD 1.22 0.18 0.01 0.20

OPD 0.80 0.19 0.01 0.65

FFD 0.94 0.20 0.01 0.35 −SFD 0.80 0.18 0.01 0.67

T–R–Q (square) CCRD 6.21 −0.37 0.01 1.45

BBD −6.55 2.06 0.15 6.61

OPD 3.43 0.49 0.07 1.26

FFD 2.76 0.83 0.08 1.48

SFD 5.12 0.19 0.06 0.25

nd back face of the computational domain; (b) perspective view of grid at building

design (Taguchi design) and two-level full factorial design wasoften used. This was because both methods required less designpoints and were more efficient in developing such models[13].

l interaction model; R–Q is the model after reduction of insignificant terms fromable by log and square function, respectively. T–R–Q (log and square) is the modelf terms of full according T–Q model, respectively.Adj. R2 and Pred. R2 is the adjusted

4 ˇ5 ˇ6 ˇ7 ˇ8 ˇ9 Adj. R2 Pred. R2

0.829 0.7380.711 0.5220.691 0.3740.729 0.6420.713 0.554

1.83 75.29 2.78 0.930 0.8160.92 63.14 5.28 0.879 0.6631.50 55.24 6.97 0.955 0.7571.08 47.96 6.02 0.933 0.8911.28 60.27 6.63 0.954 0.879

1.83 75.29 2.78 −0.93 −0.06 94.04 0.929 0.7230.92 63.14 5.28 1.19 −0.06 145.42 0.8991.51 49.72 7.27 2.73 0.01 108.30 0.950 0.6781.08 47.96 6.02 3.56 −0.03 133.49 0.946 0.8951.34 60.67 6.58 2.83 −0.03 79.68 0.954 0.846

1.83 75.29 0.930 0.85863.14 5.28 0.847 0.695

1.50 55.24 6.97 0.955 0.7571.08 47.96 6.02 133.49 0.945 0.9051.28 60.27 6.63 0.954 0.879

0.00 0.02 0.00 −0.01 0.00 −0.05 0.961 0.8850.00 0.00 0.00 −0.02 0.00 0.05 0.9940.00 0.00 0.00 −0.01 0.00 −0.12 0.998 0.9920.00 −0.01 0.00 −0.01 0.00 −0.04 0.948 0.8990.00 0.00 0.00 −0.01 0.00 −0.13 0.996 0.992

0.02 1.05 0.00 −0.14 0.00 0.30 0.950 0.8320.01 0.66 0.07 −0.07 0.00 1.60 0.9570.01 0.54 0.09 −0.06 0.00 −0.15 0.974 0.8700.01 0.38 0.08 −0.03 0.00 1.05 0.966 0.9410.02 0.68 0.08 −0.06 0.00 −0.56 0.985 0.958

−0.01 0.00 0.964 0.9380.00 −0.01 0.00 0.996 0.9870.00 −0.01 0.00 −0.12 0.998 0.9960.01 0.00 −0.01 0.00 0.952 0.918

0.00 −0.01 0.00 −0.13 0.996 0.993

0.02 1.05 0.947 0.9090.853 0.759

0.02 0.50 0.09 0.00 0.979 0.9210.01 0.38 0.08 0.00 0.966 0.9460.02 0.69 0.09 0.00 0.984 0.962

5 d Buildings 62 (2013) 570–580

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3

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Fig. 3. Predictions of RSM models (listed in Table 8) and CFD simulations for venti-

76 X. Shen et al. / Energy an

Quadratic, R–Q and 2FI models show small difference in thedjusted R2 and predicted R2. The latter two models are reducedrom the former Quadratic models, indicts that the full model mayontain insignificant terms. The R–Q models are similar to the 2FIodels, indicts the model terms (V2, �2, Hc2) in according with

he coefficients (ˇ7, ˇ8, ˇ9) are insignificant (P < 0.10) in the fulluadratic models.

By comparing T–Q and Quadratic models, one can see that trans-orming the response variable, Q (m3/s), will increase the quality oft of Quadratic models, especially using the log transfer function.–R–Q models shows higher adjusted R2 and predicted R2 than T–Qodels, indicates the T–R–Q models can improve the quality of fit

y reducing the insignificant terms from the full T–Q models.In Table 7, the models by CCRD, BBD, OPD and SFD shows better

uality of fit after using log transfer function. However, FFD doesot; but perform well with square transfer function. It proves thathoosing the proper transfer function of model seems very impor-ant to improve the quality of fit. The Box–Cox plot is widely useds a tool to determine the most appropriate power transforma-ion. The detailed information about this tool can be founded in theiterature [18].

Table 8 shows the best selections of RSM models by differentOE methods. These models are built after being enhanced by

ransformation and terms reduction. We can see that even if con-aining varied number of design points, those varied DOE methodstill achieve similarly high quality of fit.

.3. Verification of DOE methods

In the following sections, different DOE methods were evalu-ted by comparing the coefficient of variation of Root-mean-squareeviations (RMSDs), CV. In order to compare the DOE methods, aandomly selected dataset from the design spare was applied [34],hich presented in Table 9. Moreover, for a given DOE method,

dditional considerations should be taken into not only the num-er of design points but also the complexity of developed model.ollowing this, anew evaluation method was proposed in this study.hen, the comparison results were used to test the FDS tools, whichave been commonly used for examining the performance of DOEethods.

.4. Evaluating the coefficient of variation of root-mean-squareeviations (RMSDs), CV

The parameter CV (RMSD), the coefficient of variation of theMSD, was adopted to analyze the accuracy in model prediction ofaried DOE methods. RMSD is the commonly used statistical toolso test the performance of a model.

MSD

√∑n1(y − y)2

n(10)

V(RMSD) = RMSDy

(11)

here n is the number of design points in the evaluation dataset. ys the actual observation, Q (m3/s), calculated by CFD simulation. ys the predicted values by the model. y is the average value of theimulation of n design points.

From Eqs. (10) and (11), we can see that the model with lower CVRMSD) shows higher model accuracy with less variance betweenrediction and observations.

Table 10 presents the evaluation results of the developed mod-ls by the according CV (RMSD). For a specified type of model, e.g.inear or Quadratic model, varied prediction accuracy exists amongOE methods. For linear and T–Q (log) model establishment, BBD

lation rate, Q (m3/s), by different DOE methods. The actual values in different runsare the CFD predictions of ventilation rate as listed in Table 9.

method shows the most accurate predictions. For 2FI, Quadratic andT–Q (square) model, SFD performs the best.

The Cubic model shows the highest accuracy among all modelsthat constructed by the combined DOE method. The accuracy doesnot increase by turning the model from quadratic to Fourth and Fifthorder. Meanwhile, as for the combined design and SFD, the T–R–Q(square) models show similar accuracy as the Cubic model. Thisindicts that increasing the model order does not necessarily addto the model accuracy. Quadratic models can be improved to havehigh accuracy by model reduction and response transformation.This also can be seen into the CCRD method, after we reduce theinsignificant model terms from the Quadratic model and transformthe response by log function into T–R–Q (log) models, the highestprediction accuracy can be achieved among all the listed models.

Fig. 3 shows the model predictions with observations by theevaluation dataset. The predictions of five DOE methods are quitesimilar and consequently the differences of the CV (RSMD) valuesare little. From the plot it can be seen that the CCRD designs showsthe smallest CV (RSMD) values and presents the closest predictionswith the actual values. However, we can see it show the worst qual-ity of fit as shown in Table 8. On the contrary, OPD shows the bestquality of fit, but does not present the high CV (RSMD) according tothe assessments. This indicts that even through a model was welldeveloped with high quality of fit by a DOE method, the model accu-racy may not be maintained through validation by another groupof dataset. But the developed model with highest quality of fit stillhas very high accuracy. Model reduction and response transforma-tion is shown to be reliable to improve the quality of fit during themodel development.

3.5. Efficiency of different DOE methods

The efficiency of different DOE method should consider com-prehensively the cost time, the model accuracy and the modelcomplexity [18,24]. The time needed to accomplish the experimen-tal runs depends on the number of design points. DOE methodsthat consist of more number of design points, would lead to morework loads and thus costs during the experimenting. Like in ourstudy, CV (RMSD) can be used to depict the accuracy of a developedmodel. The model complexity is related to the number of terms

in the model. The complex model with more terms can includemore extensive related information between responses and vari-ables. Thus, in order to synthesize these of a specified DOE i, a

Build

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TR

X. Shen et al. / Energy and

imensionless parameter ei was proposed in this study to standor the efficiency in Eq. (12):

i = N × CV(RMSD)P

(12)

here N is the number of design points of DOE i. P is the number ofhe model terms.

Table 11 shows the efficiency of different DOE methods for

odel development. BBD shows the lowest ei, indicts it is the most

fficient DOE method to use, followed by the CCRD. Advantagef BBD method was also reported by the research involved withve design variables [22]. CCRD also had better performance than

able 8SM models with best quality of fit by different DOE methods.

DOE Established models Ad

CCRD Log10(Q) = 0.203V + 0.012� + 0.283Hc − 9.513 × 10−3 V2

− 6.938 × 10−�20.9

BBD Log10(Q) = 1.219 + 0.177V + 0.013� + 0.200Hc + 1.146 × 10−3�Hc− 7.401 × 10−3 V2 − 9.220 × 10−5 �2

0.9

OPD Log10(Q) = 0.796 + 0.189V + 0.012� + 0.647Hc + 1.633 ×10−3 VHc − 8.343 × 10−3 V2 − 8.610 × 10−5 �2 − 0.123 Hc2

0.9

FFD Q0.5 = 2.757 + 0.828V + 0.084� + 1.477Hc + 0.0125V�+ 0.383VHc + 0.0811�Hc − 1.222 × 10−3�2

0.9

ings 62 (2013) 570–580 577

OPD and FFD method in dealing with two variables as seen in theliterature[6].

Among all DOE methods, FFD is the worst as shown in Table 11.The efficiency of it may be subjected to the larger amount of designpoints (Table 6). SFD and OPD method cannot achieve higher effi-ciency than CCRD and BBD method, however, the method canbe improved and compensate for its own inadequacy by modelreduction and response transformation. The combined method that

contains the largest number of design points is able to developcomplex model with higher order, however, the efficiency is notshown to be improved.The Quadratic models by different DOE

j. R2 Pred. R2 Actual versus predicted plots

64 0.938

96 0.987

98 0.996

66 0.946

578 X. Shen et al. / Energy and Buildings 62 (2013) 570–580

Table 8 (Continued)

DOE Established models Adj. R2 Pred. R2 Actual versus predicted plots

SFD Q = −224.051 + 11.573V − 5.050� + 928.591Hc + 0.777V�− 20.517VHc + 0.609�Hc + 0.060�2 − 734.712Hc2

+ 0.972V�Hc − 0.0101V�2 + 15.581VHc2 − 0.032�2Hc+ 0.975�Hc2 + 158.459Hc3

0.998 0.990

Combined Q0.5 = 2.618 + 1.300V + 0.071� − 0.491Hc+ 0.0135V� + 0.564VHc + 0.078�Hc − 0.061V2 − 1.227 × 10−3�2

+ 0.527Hc2

0.963 0.950

Table 9Validation dataset.

Run X1 (V, m/s) X1 (�,◦) X3 (Hc , m) y (Q, m3/s)

1 5.0 45.0 1.0 321.782 4.0 41.0 2.0 505.073 6.0 80.0 2.0 1082.434 2.0 4.0 1.0 48.425 2.0 81.0 2.0 359.176 9.0 66.0 2.0 1544.517 9.0 84.0 2.0 1634.63

mtQs(atd

TVTD

SPV = i

�2= N × x′

i(Z′Z)−1xi (13)

8 8.0 16.0 2.0 557.639 8.0 61.0 1.0 615.54

ethods, involve many insignificant terms inside (Table 7); Inerms of all DOE methods, after reduced to the R-Q models fromuadratic model, the model accuracy depicted by CV (RMSD) valueslightly change (Table 10), however, the efficiency decreases largelyTable 11). This is because the R–Q models have fewer model termsnd lower complexity as compared to full models. Known from that,

he use of terms reduction would not be helpful in increasing theesign efficiency of DOE methods.

able 10alidate the established model by the CV (RMSD). The dataset is listed in Table 9he bold labelled values represent the models with the best quality of fit of variedOE methods as shown in Table 8.

Model type CCRD BBD OPD FFD SFD Combined

Linear 0.297 0.291 0.318 0.300 0.312 0.3102FI 0.113 0.113 0.140 0.138 0.095 0.099Quadratic 0.101 0.086 0.136 0.107 0.084 0.080Cubic 0.106 0.062Fourth 0.080Fifth 0.479R–Q 0.123 0.172 0.140 0.110 0.095 0.080T–Q (log) 0.098 0.077 0.140 0.142 0.106 0.088T–Q (square) 0.104 0.062 0.070 0.083 0.059 0.062T–R–Q (log) 0.060 0.100 0.138 0.112 0.101 0.125T–R–Q (square) 0.174 0.207 0.074 0.093 0.075 0.064

3.6. An alternative evaluation method for DOE methods

As presented above, the performance of DOE methods wasevaluated by randomly selected datasets. However, for the mostcommon application of RSM methods, the DOE methods wereoften pre-specified before the experiments have been done[25,27,35–37]. It was not strictly because uncertainty may becaused by different DOE methods. Still, the evaluation process likethe one in our study requires many time for cases of simulation tocompare different DOE methods. In this contest, a ‘prior’ evalua-tion method named fraction of design space (FDS) technique wasintroduced to evaluate the prediction capability of a given DOE [21].In this method, the commonly considered parameter, scaled pre-diction variance (SPV), is to measure the prediction performanceof different DOE methods [21]. SPV can be calculated based on Eq.(13)

N × Var y(x )

where N is the total number of design points.

Table 11Efficiency of DOE methods, ei (Eq. (12)), for the development of models. Lower valueof ei indicates higher efficiency in the model development. The bold labelled valuesrepresent the models with the best quality of fit of varied DOE methods as shownin Table 8.

Model type CCRD BBD OPD FFD SFD Combined

Linear 1.114 0.946 1.193 2.025 1.950 4.8832FI 0.242 0.210 0.300 0.532 0.339 0.891Quadratic 0.152 0.112 0.204 0.289 0.210 0.504Cubic 0.133 0.195Fourth 0.144Fifth 0.539R–Q 0.308 0.373 0.300 0.371 0.339 0.560T–Q (log) 0.147 0.100 0.210 0.383 0.265 0.554T–Q (square) 0.156 0.081 0.105 0.224 0.148 0.391T–R–Q (log) 0.150 0.186 0.259 0.378 0.316 0.984T–R–Q (square) 0.435 0.673 0.139 0.314 0.234 0.448

X. Shen et al. / Energy and Build

Fe

odgsFlim

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[8] H. Wang, Q. Chen, A new empirical model for predicting single-sided, wind-

ig. 4. FDS (faction of design space) with SPV (scaled prediction variance) for thevaluation of different DOE methods in Quadratic model development.

Fig. 4 shows the FDS curve of the investigated five DOE meth-ds. From Eq. (13), we know that for a given DOE method, theistribution of SPV in the design space can be addressed. The FDSraph is a line graph shows what fraction (percentage) of the designpace equals to the averaged SPV. In general, a lower and flatterDS curve is better. Lower SPV means less experimental runs andess prediction variance of the given DOE method. A flatted curvendicts a higher FDS would have useful precision by the chosen DOE

ethods.From this plot, we can see that the FFD method shows the best

erformance among all. However, as shown in Table 11, FFD is theorst method among all. The CCRD method shows the worst per-

ormance (Fig. 4), but it has relative good performance. Besides,he OPD is very close to the SFD and both shows good performance.ven if the BBD method shows large SPV, however, it is still bet-er than the CCRD method, the similar finding related with threeariables was reported by Anderson-Cook and Myers [21]. The FDSethod has shown the incompatible results of the traditional DOEethod such as CCRD, BBD and FFD in our study, e.g. CCRD and BBD

re less efficient as observed by this method but actually are not.

.7. Further investigation on DOE methods

The performance of DOE methods was different among studies.BD method has found to have good performance with three designariables [22]. However, the CCRD method was highlighted and SFDs highly recommended for the computer experiment like the onen our study [19]. The most suitable DOE method still varies amongtudies, indicts that it may depend on different applied subjectsuch as hydrodynamics experiments [22] and other experiments23,24]. This may indict that the performance may be dependent onhe varied subject being studied. Thus more comprehensive inves-igation on the comparison of DOE methods should be addressedith more researches involved.

Several computer experiments use test functions to evaluateOE methods, the functional relationship between the designariables and response are known [21,23,24]. This way can savehe time and cost to conduct the experiments or simulations. How-

ver, in our study, the relationship between response and variablesre unknown. The findings of those researches still have differentrospective on optimal DOE method and thus may be not suitableo apply directly for our study.

ings 62 (2013) 570–580 579

The state-of-art evaluation tool as FDS can be used to test DOEmethods [21]. However, as found in our evaluation for computerexperiments, it may suffer from the inability in evaluating tradi-tional DOE methods such as CCRD, BBD and FFD. These tools arefound to be much reliable for examining modern DOE methodssuch as OPD [21]. Despite this, other graphical tools such as quin-tile plots should also be addressed in further research [20]. Furtherpromoting the evaluation method is needed in future research.

Other DOE methods have been applied for ventilation study,such as uniform design [38], Latin hypercube design [39]. ThoseDOE methods can be further investigated. Other meta-model meth-ods [15], such as artificial neural network (ANN), Kriging andinductive learning are recommended to be compared with RSMmethod for modelling [10,40]. Our study combined the DOE methodwith the RSM modelling. The combination of DOE methods with themodelling methods is required to be further discussed.

4. Conclusion

In this paper, we assessed the performances of using differentDOE methods for constructing Response Surface model based onCFD simulations to estimate the ventilation rate in NVLB. In general,the SFD method shows the best accuracy in developing QuadraticRSM model, followed by the BBD method and the FFD is the lastrank.

For varied DOE methods, the statistical tools such as theresponse transformation and model reduction of insignificantterms can further ensure the RSM model prediction having higherquality of fit with the observations. Results has shown that themodel accuracy does not increase as we refine the Quadratic modelinto Cubic, Fourth- or Fifth-order models, as well as the efficiency ofDOE methods to formulate those models.

A measure of efficiency to compare different DOE methods isproposed. Based on the measure, the BBD method has shown to bethe most efficient DOE methods in formulating Quadratic models.BBD method also shows the least number of design points and ade-quate model accuracy, followed by the CCRD method; FFD methodshows in the last rank. The FDS tool, which has been used as a fasttool to assess DOE methods, is tested in this study. We found thatlarge incompatibility exists in comparing the performance of tradi-tional DOE methods, such as CCRD, BBD and FFD with the modernDOE methods such as OPD and SFD. Further promotion of this toolis needed in future researches.

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