ARTICLE IN PRESS Ocean Engineering 36 (2009) 237–247 Contents lists available at ScienceDirect Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng Combined use of dimensional analysis and modern experimental design methodologies in hydrodynamics experiments Mohammed F. Islam , L.M. Lye Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1B 3X5 a r t i c l e in fo abstract Article history: Received 23 June 2008 Accepted 18 November 2008 Available online 6 December 2008 In this paper, a combined use of dimensional analysis (DA) and modern statistical design of experiment (DOE) methodologies is proposed for a hydrodynamics experiment where there are a large number of variables. While DA is well-known, DOE is still unfamiliar to most ocean engineers although it has been shown to be useful in many engineering and non-engineering applications. To introduce and illustrate the method, a study concerning the thrust of a propeller is considered. Fourteen variables are involved in the problem and after dimensional analysis this reduces to 11 dimensionless parameters. Then, a two-level fractional factorial design was used to screen out parameters that do not significantly contribute to explaining the dependent dimensionless parameter. With the remaining five statistically significant dimensionless parameters, various response surface methodologies (RSM) were used to obtain a functional relationship between the dependent dimensionless thrust coefficient, and the five dimensionless parameters. The final model was found to be of reasonable accuracy when tested against results not used to develop the model. The methodologies presented in the paper can be similarly applied to systems with a large number of control variables to systematically derive approximate mathematical models to predict the responses of the system economically and accurately. & 2008 Elsevier Ltd. All rights reserved. Keywords: Dimensional analysis Design of experiments Fractional factorial design Response surface methodology Space-filling designs Propellers Propulsive performance 1. Introduction Variables used in engineering are usually expressed in terms of a limited number of basic dimensions namely mass, length, time, and sometimes temperature. For certain phenomenon, a large number of variables may be needed to describe or explain the phenomenon. However, by the methods of dimensional or partial analysis, the separate variables involved in the problem can be reduced to a smaller set of independent dimensionless groups or dimensionless parameters. Dimensional analysis (DA) has been used in engineering particularly, fluid mechanics and hydraulics, for about a hundred years. Many methods have been developed for dimensional analysis and these are normally covered as part of an undergraduate course in fluid mechanics or hydraulics. Dimensional analysis is also called partial analysis because the problem is only partially solved when the variables have been suitably combined into independent dimensionless parameters. To obtain a functional relationship between the parameters, a series of experiments must be conducted to obtain data to relate the dependent dimensionless parameter to the other dimensionless parameters. When there are several independent dimensional parameters, a series of experiments coupled with statistical analysis is usually required. How to conduct the experiment such  Corresponding author. Tel.: +1709 743 5627. E-mail addresses: islam@engr.mun.ca (M.F. Islam), llye@mun.ca (L.M. Lye). 0029-8018/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2008.11.004 that experimental runs are minimized and yet obtain meaningful results is often a challenge. In this paper, a combined use of dimensional analysis and modern statistical design of experiments (DOE) methodologies is proposed for a hydrodynamics experiment where there are a large number of variables. While DA is well-known, DOE is still unfamiliar to most ocean engineers although it has been shown to be useful in many engineering and non-engineering areas. To illustrate the combined method, a study concerning the thrust of a propeller is used as a case study. Fourteen potential variables are involved in this problem. The methodology presented in the paper can potentially be applied to most system with a large number of variables to derive approximate mathematical models to predict the responses of the system economically and accurately. In the following section, dimensional analysis will be briefly introduced and performed for the problem at hand followed by a description of statistical design of experiment methodologies. The application of a fractional factorial design followed by applications of various response surface designs to the problem at hand will then be presented. The response surface models developed are then validated against data not used to develop the models. Finally, conclusions will follow. 2. Dimensional analysis There are many methods of dimensional analysis and practically every book in fluid mechanics has a chapter or two devoted ARTICLE IN PRESS 238 M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 to the topic. For example, Munson et al. (1994) provided a list of about 15 books on the subject. More recently, a 970-page book on the subject was written by Szirtes (1997). Popular methods of dimensional analysis include Rayleigh’s method, Buckingham P theorem, matrix method, and method of synthesis. All these methods are described in Sharp (1981), among others. Dimensional analysis is essentially a means of utilizing partial knowledge of a problem when the details are too obscure to permit an exact analysis. For example, the thrust producing phenomenon of a propeller is very complicated and is dependent upon many parameters that it is quite difficult to derive a complete analytical relationship expressing it in terms of the variables of control. For the case study under consideration, the objective is to derive a functional relationship between propulsive performance parameter of a propeller, expressed as the thrust coefficient, KT, and the variables upon which this parameter depends. A detailed derivation and formulation of the problem under consideration is as follows: 2.1. Physical knowledge of the phenomenon The thrust generated by the propeller is dependent on the propeller’s operating environment, the geometrical characteristics of the propeller, and the relative motion between water and propeller. Based on experience, the dependent variable of the phenomenon may be described in terms of relevant fluid properties, blade and pod geometry, and operating conditions as given in (1) (assuming that the blade sectional shapes are fixed). T ¼ f ðD; N; n; V A ; g; m; rw ; p; P 0:7R ; c; ys ; yr ; Dh ; aÞ. (1) The variables that are involved in this phenomenon are summarized in Table 1. There are some terms having length or angle dimensions, which can be excluded from the equation and can be added to the final non-dimensional equation directly. Thus simplifying the above equation results in f ðT; n; V A ; g; m; p; rw ; DÞ ¼ 0. (2) 2.2. Application of Barr’s method of dimensional analysis To develop the non-dimensional equations for each of the variables of interest, a dimensional analysis method, ‘‘the method of synthesis’’ developed by Barr (1969) was used. The details of Barr’s method are also given in Sharp (1981). According to Barr’s Table 1 List of the variables affecting propulsive performance of a propeller. Symbol Dimensions Size of the propeller (represented by diameter) Number of blades Rotational speed (RPM) Speed of advance Gravitational acceleration Dynamic viscosity Mass density of water Pressure of the fluid Pitch at 0.7R (R is the radius of the propeller) Mean chord length Mean skew Mean rake Hub diameter Hub taper angle D N n VA g a L – 1/T L/T L/T2 M/(LT) M/L3 M/(LT2) L L Deg. Deg. L Deg. Response variables Propeller thrust Symbol T Dimensions ML2/T2 p P0.7R c ys yr Dh T 1=4 TD T nD2 n2 D2 nrw D3 ; ; ; ; ; V A rw nD g rw D mn pD V A g m r ! n2 rw D3 m V 2A V 2A rw D m2=3 g rw D2 mnD ; ; D ¼ 0, ; ; ; ; ; 2=3 p p rw V A g p p g 1=3 rw 1=4 1=2 w n ; T ; T 2 ; (3) where, from the alternative choice of 16 parameters, any five can be selected, provided T, n, VA, g, m and r are each included at least once. 2.3. Choose the right coefficients One possible conventional solution leads to Eq. (4). ! T 1=4 nD2 m V 2A V 2A rw D ; ; ; ; ; D ¼ 0. 1=2 V A rw V A g p r1=4 w n (4) A further choice is now available in forming the non-dimensional equation. The one that leads to a meaningful functional relationship is ! T nD m V 2A V 2A rw ; ; ; ; ¼ 0. (5) rw n2 D4 V A Drw V A gD p Another method such as Buckingham’s P theorem could have been used to obtain the dimensionless terms in Eq. (5). 2.4. Rearrangement of the dimensionless parameters Rearranging the above dimensionless parameters and adding the linear terms that were excluded in non-dimensional form results in the dimensionless Eq. (6) for the thrust of a propeller operating in open water. ! T V A rw DV A V 2A p P0:7R c Dh ; ; ; ; ys ; yr ; ; a; N . ¼f ; ; (6) nD m gD rw V 2A D D D rw n2 D4 In the above equation there are several standard numbers and commonly used pffiffiffiffiffiffipropeller related terms. These are Froude number, F n ¼ V A = gD; Reynolds number, Rn ¼ rwDVA/m or DVA/n, where n is the kinematic viscosity of the fluid; propeller advance coefficient, J ¼ VA/nD; pitch ratio ¼ P0.7R/D; skew ¼ ys; rake ¼ yr; and thrust coefficient, KT ¼ T/rwn2D4. 2.5. Further study of the dimensionless parameters Variable of control m rw method, linear proportionalities are formed by combining the first six variables in pairs. Introducing rw as required to eliminate mass dimensions and using the reference tables provided in Barr (1969), Eq. (3) was obtained. Eq. (6) states in effect that if all the parameters on the righthand side have the same values for two geometrically similar but different sized propellers, the flow patterns will be similar and the value of T/rwn2D4 will be the same for each. When one is testing model propellers to predict the performance of a prototype propeller it is necessary to fulfill a number of conditions (laws of similitude) to ensure similarity between the full-scale and the model scale results. Here, it should be noted that the response now depends on 11 dimensionless parameters instead of 14 that existed before applying partial analysis to the system. In general, if there are n quantities and m fundamental dimensions (e.g., M, L, and T), there will be (n–m) dimensionless terms. However, to develop a functional relationship with 11 independent dimensionless parameters would require an extensive systematic experimentation program to generate the data for statistical analysis and modelling. As such, it may be prudent to further reduce the dimensionality of the problem. That is, use only those ARTICLE IN PRESS 239 M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 parameters that will significantly contribute to explaining the variation in the dependent parameter. An efficient technique for screening a large number of parameters based on modern experimental design methodologies will be discussed next. 3. Statistical design of experiments Before describing the experimental design and the subsequent analysis to screen out the statistically insignificant parameters for the problem at hand, a brief note about the statistical design of experiments or DOE is warranted. Engineers in general carry out a fair amount of physical experimentation in the laboratory and on the computer using a variety of numerical models. Experiments are carried out to (1) evaluate and compare basic design configurations, (2) evaluate material alternatives, (3) select design parameters so that the design will work well under a wide variety of field conditions (robust design), and (4) determine the key design parameters that impact performance. As with most engineering problems, time and budget are often limited. Hence it is necessary to gain as much information as possible and do so as efficiently as possible from an experimental program. In engineering, one often-used approach is the best-guess (with engineering judgment) approach. Another strategy of experimentation that is prevalent in practice is the one-factorat-a-time or OFAT approach. The OFAT method was considered the standard, systematic, and accepted method of scientific experimentation. Both of these methods have been shown to be inefficient and in fact can be disastrous (Montgomery, 2005). These methods of experimentation became outdated in the early 1920s when Ronald A. Fisher discovered much more efficient methods of experimentation based on factorial designs. This class of experimental designs includes the general factorial, two-level factorial, fractional factorial, and response surface designs among others. These statistically based experimental design methods are now simply called design of experiment methods or DOE methods. A recent application of DOE methods in ocean engineering can be found in Hawkins and Lye (2006), among others. Basically, DOE is a methodology for systematically applying statistics to experimentation. DOE lets experimenters develop a mathematical model that predicts how input variables interact to create output variables or responses in a process or system. DOE can be used for a wide range of experiments for various purposes including nearly all fields of engineering and science and even in marketing studies. The use of statistics is important in DOE but not absolutely necessary. In general, by using DOE, one can      learn about the process being investigated; screen important factors; determine whether factors interact; build a mathematical model for prediction; and optimize the response(s), if required. DOE methods are also useful as a strategy for building mechanistic models, and they have the additional advantage that no complicated calculations are needed to analyze the data produced from the designed experiment. It has now been recognized that the factorial-based DOE is the correct and the most efficient method of conducting multi-factored experiments; they allow a large number of factors to be investigated in few experimental runs. The efficiency stems from using settings of the independent factors that are completely uncorrelated with each other. That is, the experimental designs are orthogonal. The consequence of the orthogonal design is that the main effect of each experiment factor, and also the interactions between factors, can be estimated independent of the other effects. As stated earlier, many industries have recognized this fact and design of experiment methodologies is a key component of the Six-Sigma quality program used by many major corporations. Yet it is surprising that after about 90 years since the invention of modern experimental design it is still not widely taught in schools of engineering or science in our universities (Box et al., 2006). The wide variety of experimental designs and their statistical details can be found in many excellent texts including Montgomery (2005), Myers and Montgomery (2002), Box et al. (2005), Ryan (2007), Antony (2006), Box et al. (2006) and Berger and Maurer (2002), among others. In the next section, a two-level quarter-fractional factorial or in short, a 2k2 factorial design will be used to study the effect of the 11 factors (dimensionless parameters) on the thrust coefficient, KT of a propeller. 4. Application of fractional factorial design The following steps were followed to study the thrust coefficient of the propeller using the factorial design method. 4.1. Statement of the problem Evaluate how the geometric and motion parameters and their interactions affect the thrust coefficient of a propeller in open water conditions. 4.2. Choice of factors, levels, ranges and response variables From Eq. (6), the terms rwDVA/m, VA2/gD, and p/rwVA2 can be removed since they are extraneous in the sense that for constant g, p and rw, these terms mainly depends on speed of advance VA. Also it is difficult to take into account the viscosity m because it is hard to control. The modified equations take the form of Eq. (7):   T V A P0:7R c ys yr Dh ; ; ; ; ; ; ¼ f a ; N . (7) nD D D D D D rw n2 D4 In this reduced and simplified form there are still eight independent factors to deal with (Table 1). Before proceeding with the factorial design it is important that the response and the factors’ ranges are properly defined. The factors (variables of control) are given in Table 2 with the corresponding low and high values to be considered in the two-level fractional factorial design. The low and high values of the factors are chosen from a practical viewpoint. It is to be noted that the resulting model for the response is only valid within the ranges of the factors. Factors involving angles are modelled as actual angles rather than ratios. In traditional propeller experimental program design, variables are usually divided into two separate groups: geometry and motion parameters. Geometry parameters determine the number of propeller models that need to be built (materials and Table 2 The variables of control with their ranges (low and high values). A B C D E F G H Factors Low (1) High (+1) Number of blades, N Chord-diameter ratio, c/D Pitch-diameter ratio, p0.7R/D Rake angle, yr Skew angle, ys Hub taper angle (HTA), a Hub-diameter ratio, Dh/D Propeller advance coeff., J 3 0.33 0.80 151 01 151 0.20 0.2 5 0.52 1.20 151 901 151 0.30 1.0 ARTICLE IN PRESS 240 M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 fabrication cost) and motion parameters determine the required number of runs (facility time and staff labor cost). It is possible to deal with the parameters in the two groups separately using the proposed method and then integrate these two to give an integrated solution. However, in the current study, the two groups of parameters were studied together. Furthermore, the advance coefficient, J is the ‘‘primary’’ independent variable for the dependent variable, thrust coefficient, because propulsion performance charts are plotted as curves of thrust coefficient versus advance coefficient. One can argue that the advance coefficient should be definitely included in the final analysis without needing any DOE analysis. However, in the present study the advance coefficient was included in the analysis to illustrate the method for any hydrodynamic problem for completeness. The significance of the advance coefficient will automatically be confirmed from the subsequent analysis if it is indeed the most important parameter. More importantly, how it will interact with other parameters will also be quantified which cannot be done if it is analyzed on its own. 4.3. Choice of experimental design A fractional factorial design (FFD) was used to design the experiments to minimize the runs. With eight factors, the quarterfractional two-level factorial design (282) requires a combination of experimental 64 runs or calculation points. The 64 run combinations for the 282 design and the responses are shown in Table 3. The design is a Resolution V design, which means that all main effects and two-factor interactions can be estimated without ambiguity (Montgomery, 2005). 4.4. Perform the experiments The experimental design requires that the responses at high and low values of the control factors (independent variables) be obtained at different combinations. This would require a large number of model propellers and test runs to obtain the responses for the numerous combinations for all the control factors. In the current study, a computer simulation software package, PROPELLA, developed by Liu (2003–2008) was used to obtain the responses for the run combinations. PROPELLA is a time domain panel method computer code, which can calculate the propeller thrust, KT for all the combinations of experimental runs. The geometry part of the code can model the propeller with any practical geometry and the code has been validated against actual measurements with good agreement. All the calculations were done with a propeller rpm of 600, zero shaft angles and at a shaft depth of 2 diameters. Details of the geometry of the propellers are presented by Liu (2006). The predictions given by PROPELLA might or might not be as accurate as real physical measurements, but are sufficient to establish a standard procedure of implementing design of experiments techniques in a quantitative analysis of propulsive performance. Fig. 1 shows a few model propellers simulated using PROPELLA. Table 3 FFD data sheet. Std. order N c/D P0.7R/D yr ys a Dh/D J KT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.3337 0.3337 0.5157 0.5157 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 0.3 0.2 0.2 0.3 0.2 0.3 0.3 0.2 0.2 0.3 0.3 0.2 0.3 0.2 0.2 0.3 0.3 0.2 0.2 0.3 0.2 0.3 0.3 0.2 0.2 0.3 0.3 0.2 0.3 0.2 0.2 0.3 0.3 0.2 0.2 0.3 0.2 0.3 0.3 0.2 0.2 0.3 0.3 0.2 0.3 0.2 0.2 0.3 0.3 0.2 0.2 0.3 0.2 0.3 0.3 0.2 0.2 0.3 0.3 0.2 0.3 0.2 0.2 0.3 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.0115 0.3118 0.2597 0.0029 0.1685 0.4616 0.3718 0.2233 0.0195 0.2832 0.2382 0.0131 0.1469 0.4529 0.3737 0.1974 0.1193 0.0595 0.0637 0.3571 0.1704 0.1665 0.1381 0.3264 0.1308 0.059 0.0497 0.3144 0.1558 0.1733 0.157 0.3634 0.2251 0.0298 0.0466 0.2622 0.3362 0.2052 0.1821 0.4064 0.226 0.041 0.0605 0.2715 0.3296 0.2064 0.1774 0.4266 0.0334 0.227 0.1815 0.0663 0.122 0.3099 0.2732 0.2452 0.0371 0.2738 0.2119 0.1039 0.1331 0.305 0.3074 0.2618 4.5. Statistical analysis of the two-level fractional factorial design The main focus of this part is to determine the most significant factors among the eight factors in Table 2 for the propulsive performance of a propeller. A reduced number of factors will then be used to obtain a response surface design for fitting a secondorder polynomial model to the thrust coefficient. Design Experts 7.03 from Statease, a stand-alone software for design of experiments was used to design the experiment and analyze the results. The Pareto chart for the response thrust coefficient is shown in Fig. 2. This is a plot of the ordered absolute value of the effects estimates. The important main effects in descending order that emerge from this analysis are advance coefficient, J, pitch ratio, P0.7R/D, number of blades (N), skew angle (ys), and chord-diameter ratio (c/D). The two-factor interactions that stood out included, interaction of skew angle and advance coefficient, interaction of ARTICLE IN PRESS M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 241 together with the other parameters, how it will interact with the other parameters could not be determined and quantified. 4.6. Further study of the significant parameters Having now identified the most significant dimensionless parameters that explained the variation of the dependent dimensionless parameter of interest, follow up experiments can now be carried out to refine the models so that quadratic terms can be added to the model presented by Eq. (8). Adding quadratic terms would allow nonlinear effects to be modelled. Experimental designs for fitting second-order models are known as response surface methodology (RSM) in DOE terminology. This will be discussed in the next section. 5. Response surface modelling Fig. 1. Propeller models simulated by PROPELLA. From left to right (row-wise), Standard run # 1, 22, 35, 46, 55 and 64, as shown in Table 3. Fig. 2. Pareto chart for thrust coefficient in the FFD analysis. pitch ratio and skew angle, interaction of number of blades and advance coefficient, interaction of chord-diameter ratio and skew angle, interaction of pitch ratio and advance coefficient, and interaction of number of blades and skew. The model for KT obtained from the fractional factorial design in coded units is given in Eq. (8). c P0:7R þ 0:058  0:017ys  0:088J D D c P0:7R þ0:011N ys  0:015NJ þ 0:014 ys  0:016 ys D D P0:7R J þ 0:021ys J. þ0:011 (8) D K T ¼ 0:200 þ 0:030N þ 0:016 This simple linear model gave a R2 value of about 96% and predicted R2 of about 94%. It should be noted that the five main factors were identified to be: number of blades (N), chord-diameter ratio (c/D), pitch ratio (P0.7R/D), skew (ys), and the advance coefficient (J) with the advance coefficient being the most significant factor. From Eq. (8) it is clear that the advance coefficient, skew, and pitch ratio and their interactions play a dominant role in determining the thrust coefficient, KT, with the other terms playing a minor role. It should be pointed out that if the advanced coefficient was not analyzed Response surface methodology is a strategy to achieve a goal that involves experimentation, modelling, data analysis, and optimization. Usually a sequential experimentation strategy is considered. This facilitates an efficient search of the input factors to optimize the response by using a first order experiment, as was done above using a fractional factorial design, followed by a second-order experiment. The second-order design allows one to approximate the response surface relationship with a fitted second-order regression model to include nonlinearities in the response surface. There are several second-order designs available. Among the most popular designs are the central-composite designs or CCD and the Box-Behnken design or BBD. There are also special space-filling designs that are meant for computer experiments. Some of these newer designs are discussed in Ryan (2007). Details of the RSM design phases and theoretical background on the CCD and BBD are presented in Myers and Montgomery (2002) among others. In this paper, three second-order response surface designs were considered with five factors. They are the conventional Face-Centered Central-Composite design and the BoxBehnken design; and a space-filling design called Uniform Design (Fang et al., 2006). The response functions to be considered are given in Eq. (9) after screening out the insignificant factors, which are, rake angle (yr), hub taper angle (a), and the hub-diameter ratio (Dh/D).   T V A c P 0:7R ; ; ; ¼ f y ; N , (9) s nD D D rw n2 D4 where all terms are as defined earlier and in Table 2. The final five factors used to develop the second-order response surface and their ranges considered are given in Table 4. Other dimensions in defining propeller geometry and the operating condition are held constant. The thrust coefficient, KT of the propeller was calculated using the numerical simulation program PROPELLA described earlier for all the test points. 5.1. Central-composite design (CCD) The central-composite design is a quadratic design that contains an embedded factorial or fractional factorial design. In this design the treatment combinations are at the factorial or fractional factorial points plus axial points and at the center. These designs are rotatable or near rotatable and can give up to 5 levels of each factor. The axial points can be located at the face of the design space or located at a distance designed to give a rotatable design depending on the number of factors. Future details can be obtained from Myers and Montgomery (2002). The CCD is one of the most popular experimental designs for fitting a second-order ARTICLE IN PRESS 242 M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 Table 4 Control factors and their ranges for the response surface designs. A B C D E Parameter name Symbol Low value (1) Mid value (0) High value (+1) Number of blades Chord-diameter ratio Pitch-diameter ratio Skew Propeller advance coeff. N c/D P0.7R/D 3 0.33 0.80 01 0.20 4 0.43 1.00 451 0.50 5 0.52 1.20 901 0.80 ys J ¼ VA/nD Table 5 CCD data sheet. Std. order N c/D P0.7R/D ys J KT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 4 4 4 4 4 4 4 4 4 0.33 0.33 0.52 0.52 0.33 0.33 0.52 0.52 0.33 0.33 0.52 0.52 0.33 0.33 0.52 0.52 0.425 0.425 0.33 0.52 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 1 1 1 1 0.8 1.2 1 1 1 1 1 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 45 45 45 45 45 45 0 90 45 45 45 0.8 0.2 0.2 0.8 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.2 0.8 0.2 0.2 0.8 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2 0.8 0.5 0.024 0.2926 0.4719 0.0459 0.3421 0.2117 0.1744 0.46 0.1368 0.0577 0.081 0.2235 0.1127 0.2625 0.1952 0.2329 0.1976 0.2798 0.2211 0.264 0.1746 0.2988 0.2349 0.1721 0.3205 0.1434 0.2434 Fig. 3. Actual versus predicted values for the response, thrust coefficient in the CCD model. model. For this experiment, a half fractional factorial with five factors, 10 axial points at the face, and a center point were used giving a total of 27 design points. When the axial points are placed at the low and high values, the design is called a face-centered central-composite design. The design and corresponding responses are given in Table 5. The analysis part of the RSM consists of different distinct steps to ensure the predicted model acts well and the statistical assumptions are valid. Using a step-wise regression approach and using a significance level of 5%, the final quadratic regression equation in terms of the actual factors for the thrust coefficient, KT, is shown in Eq. (10). pffiffiffiffiffiffi c P0:7R K T ¼ 0:3623  0:0292N þ 1:8144  0:1602  0:0012ys D D c P0:7R 0:9981J  0:1917N: þ 0:1192N: þ 0:0003N:ys D D c P P P 0:7416 : 0:7R  0:0010 0:7R :ys þ 0:6826 0:7R :J þ 0:0037ys J D D D D 2 0:00002ys  0:1983J 2 . (10) The R2 is about 0.995 and the prediction R2 is about 0.953. The actual versus predicted values for the model is shown in Fig. 3. The addition of the quadratic terms is clearly reflected in the curved interaction plots between the skew and advance coefficient, and between the skew and the advance coefficient as shown in Figs. 4 and 5, respectively. Fig. 4. Interaction plot between number of blades and pitch-diameter ratio for the response, thrust coefficient in the CCD analysis. 5.2. Box-Behnken design (BBD) The Box-Behnken design is an independent quadratic design in that it does not contain an embedded factorial or fractional factorial design. In this design the treatment combinations are at the midpoints of edges of the process space and at the center. These designs are rotatable (or near rotatable) and require 3 levels of each factor. However, the use of the BBD should be confined to situations in which one is not interested in predicting response at ARTICLE IN PRESS 243 M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 Table 6 BBD data sheet. Fig. 5. Interaction plot between skew and advance coefficient for the response, thrust coefficient in the CCD analysis. the extremes, that is, at the corners of the cube. Futher details can be obtained from Myers and Montgomery (2002). For this design, a total of 46 experimental runs were required and the run combinations and responses are shown in Table 6. For the BBD, the final quadratic regression equation in terms of the actual factors for the thrust coefficient, KT, is shown in Eq. (11). pffiffiffiffiffiffi c P 0:7R þ 0:0006ys K T ¼ 0:0157 þ 0:0406N þ 0:0998 þ 0:6844 D D c P 0:7R P 0:7R  0:7259J þ 0:0021 :ys  0:0018 :ys þ 0:5044 :J D D D 2 P 2 þ 0:0026ys J  0:2553 0:7R  0:00002ys  0:2158J2 (11) D The R2 is about 0.992 and the prediction R2 is about 0.981. The actual versus predicted values for the model is shown in Fig. 6. 5.3. Uniform design (UD) The uniform design is an efficient, near orthogonal, and robust fractional factorial design for experiments where there are a large number of factors (four or more) and levels (three or more). It was first proposed by Kai-Tai Fang and Wang Yuan in 1980 and is described in great detail in Fang and Lin (2003) and in Fang et al. (2006). The uniform design is an important class of space-filling designs, which is useful for computer and industrial experiments when there is little or no information about the effect of factors on the response but the true model is suspected to be highly nonlinear. Design points are essentially regularly spaced over the design region. The construction of uniform designs for more than two factors is not as straightforward as the classical response surface designs. The construction depends on the measure of uniformity used. Fortunately, uniform designs developed by Fang et al. have been tabulated and are available at http://www.math.hkbu.edu.hk/ UniformDesign. Also, unlike classical designs, there is a wide choice of sample sizes for a given measure of uniformity, number of levels, and number of factors. For example, for a Un(q)s uniform design based on the centered L2-discrepancy measure, sample size n can range from 9 to 51 when the number of levels q is 3 and number of factors s is 5. The choice of the sample size hence depends on balancing the need for minimizing the number of experimental runs with the complexity of the model to be fitted and the type of modeling approach used. To implement the Std. order N c/D P0.7R/D ys J KT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 3 5 3 5 4 4 4 4 4 4 4 4 3 5 3 5 4 4 4 4 4 4 4 4 3 5 3 5 4 4 4 4 3 5 3 5 4 4 4 4 4 4 4 4 4 4 0.33 0.33 0.52 0.52 0.425 0.425 0.425 0.425 0.33 0.52 0.33 0.52 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.33 0.52 0.33 0.52 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.425 0.33 0.52 0.33 0.52 0.425 0.425 0.425 0.425 0.425 0.425 1 1 1 1 0.8 1.2 0.8 1.2 1 1 1 1 0.8 0.8 1.2 1.2 1 1 1 1 0.8 0.8 1.2 1.2 1 1 1 1 0.8 1.2 0.8 1.2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45 45 45 45 0 0 90 90 45 45 45 45 45 45 45 45 0 90 0 90 45 45 45 45 0 0 90 90 45 45 45 45 45 45 45 45 0 0 90 90 45 45 45 45 45 45 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2 0.2 0.8 0.8 0.5 0.5 0.5 0.5 0.2 0.2 0.8 0.8 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2 0.2 0.8 0.8 0.2 0.2 0.8 0.8 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.18 0.2562 0.2109 0.307 0.1546 0.3095 0.129 0.209 0.3017 0.3441 0.1224 0.1594 0.1402 0.1987 0.2435 0.3514 0.3509 0.2049 0.1165 0.1169 0.1538 0.1903 0.2773 0.3239 0.2067 0.2577 0.138 0.2075 0.2677 0.3731 0.0621 0.215 0.2646 0.3707 0.1151 0.1682 0.2367 0.238 0.1596 0.1906 0.2434 0.2434 0.2434 0.2434 0.2434 0.2434 Fig. 6. Actual versus predicted values for the response, thrust coefficient in the BBD model. ARTICLE IN PRESS 244 M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247   pffiffiffiffiffiffi hci P 0:7R K T ¼ 0:5060 þ 0:0476½N þ 0:0869  0:00052½ys  þ 0:3840 Table 7 UD data sheet. Std. order N c/D P0.7R/D ys J KT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 4 4 3 5 4 3 3 4 4 5 3 3 5 5 3 3 5 5 4 4 4 4 5 3 3 5 5 0.43 0.43 0.52 0.43 0.52 0.52 0.43 0.33 0.43 0.43 0.33 0.33 0.52 0.52 0.33 0.33 0.43 0.52 0.33 0.52 0.52 0.33 0.33 0.52 0.43 0.33 0.33 1.2 1 1 1.2 0.8 1.2 1 1 0.8 0.8 0.8 1.2 0.8 1.2 1 1 1 1 1.2 1.2 1 0.8 0.8 0.8 0.8 1.2 1 45 45 45 0 0 0 45 0 90 90 45 0 45 90 90 90 0 0 90 45 90 0 45 90 0 45 90 0.2 0.5 0.2 0.5 0.2 0.5 0.5 0.8 0.2 0.8 0.8 0.2 0.5 0.2 0.2 0.2 0.2 0.8 0.5 0.8 0.8 0.5 0.2 0.5 0.8 0.8 0.5 0.3731 0.2434 0.2809 0.3407 0.1694 0.2728 0.1976 0.111 0.1733 0.0898 0.0371 0.3421 0.2268 0.302 0.1591 0.1591 0.3799 0.1305 0.1923 0.2364 0.1372 0.1544 0.2817 0.1165 0.0271 0.2489 0.2075 uniform design in industrial experiments the following steps are necessary (Fang and Lin, 2003): 1. Choose factors and experimental domain as well as determine suitable number of levels for each factor. 2. Choose a suitable UD to accommodate the number of factors and levels. 3. From the UD table, randomize the run order if necessary and conduct the experiments. 4. Find a suitable model to fit the data. E.g., regression analysis, neural networks, wavelets, multivariate splines, etc., and 5. Knowledge discovery from the built model. Optimize response if required. As can be seen, the use of uniform designs is similar to the use of the classical fractional factorial designs or response surface designs except that there is a wider choice of levels, number of runs, and model fitting methods. This means that results obtained from a uniform design may be quite different depending on the various choices made at each step. For the problem at hand, a uniform design based on the centered L2-discrepancy uniformity measure that uses the same number of factors and levels as the CCD and BBD was used. The sample size chosen was 27. This was the same number as the half fractional face-centered central-composite design used earlier. The experimental run combinations and responses obtained are given in Table 7. A step-wise second-order centered quadratic regression at the 5% significance level was used to obtain the relationship between the factors and the response. The final quadratic regression equation in terms of the actual factors for the thrust coefficient, KT, is shown in Eq. (12). In this equation, the terms with square brackets are centered values. That is, ½N ¼ N  4:000; hci D ¼   c P 0:7R P 0:7R  0:9926, ¼  0:4230; D D D ½ys  ¼ ys  45:00; ½J ¼ J  0:4778 D D     hci P 0:7R P 0:7R :½ys   0:0022 :½ys  þ 0:1996 :½J D D D hci hci þ 0:0014½ys ½J  0:1140½N: þ 0:4540 ½J D D   P 0:7R 2  0:3710  0:000025½ys 2  0:3340½J2 . (12) D  0:2870½J þ 0:0051 The R2 is about 0.992 and the prediction R2 is about 0.942. The actual versus predicted values for the model is shown in Fig. 7. Square-root transformation is again necessary for the response to give a better fit to the assumptions of regression analysis. Centering the explanatory variables reduces the multicollinearity problem considerably. 5.4. Comparison of results From the above sections, one can see that all second-order models gave reasonably high R2 and predicted R2 values, with the BBD giving the highest values and the UD giving the lowest values. However, it is important to note that these R2 values are not the true measure on how well the models actually perform when tested against data that have not been used in developing the models. To compare the results among the three second-order experimental designs, 20 sets of the five factors (N, c/D, P0.7R/D, ys and J) within their respective ranges were randomly generated as shown in Table 8. As shown in Table 8, the propeller pitch ratios might be unusual numbers and might not normally be chosen for a propeller model. The random values of the parameters were merely used to test the accuracy of the models, not to provide practical propeller design parameter values. The responses of the propellers using the random parameter values were then calculated using PROPELLA. The responses were then compared to the predictions obtained from each of the regression models. The criterion used for the comparison is the mean absolute percentage error (MAPE). The most accurate model would be the one with the smallest MAPE. The results of the comparison are shown in Table 8 and the plots of actual versus predicted values for the thrust coefficient based on each model are shown in Figs. 8–10, respectively. The results show that the most accurate predictions for KT are those based on the BBD, followed by UD. The models based on the CCD although gave higher R2 and prediction R2 values than the models for the UD, did not perform as well for the 20 unseen data. The MAPE for the KT predictions were 11.96%, 5.35%, and 3.11% for the CCD, UD, and BBD, respectively. In terms of maximum percentage error, they were 33.61%, 16.01%, and 9.90% for the CCD, UD, and BBD, respectively. If we choose the best model, the MAPE is about 3% with a maximum percentage error of less than 10%, which would be quite acceptable for hydrodynamic experiments. It should also be noted that while the regression equations for all three experimental designs looked similar at first glance, they however do not possess the same terms. It is likely that if another uniform design was used, say one with a larger sample size, a different regression equation would result. Also, if a different fitting method was used, say multivariate splines or neural network, again a better goodness of fit could be obtained. Another point to make is that with five independent parameters, the CCD or UD with 27 experimental runs, one can study the linear and quadratic effects of each of the five parameters and their interactions. If the traditional one-factor-at-a time or OFAT approach has been used, 48 runs would be required just to study the effects of each of the five parameters with the same reliability as the fractional factorial design. Furthermore, the OFAT design would not inform us about the presence of interaction effects; this ARTICLE IN PRESS 245 M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 Fig. 7. Actual versus predicted values for the response, thrust coefficient in the UD model. Table 8 Comparison of results. Run N c/D P0.7R/D ys J KT KT_CCD KT_BBD KT_UD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 5 3 3 3 4 5 5 5 3 3 4 3 4 3 3 3 3 4 5 5 0.3687 0.3402 0.5194 0.4485 0.4554 0.3409 0.4598 0.4869 0.3919 0.4416 0.3794 0.3495 0.4777 0.4775 0.4783 0.3346 0.4820 0.4341 0.4609 0.4422 1.1346 1.0068 0.9772 0.8830 1.0259 1.1796 0.9575 0.9044 1.0783 0.8445 1.1565 1.1926 0.9060 1.0619 1.1149 1.0160 0.8834 1.1879 0.8388 1.0007 80.7882 87.3684 18.7508 73.8626 19.9483 16.7385 60.5916 64.7757 59.1554 37.0964 61.9974 1.4986 80.8365 18.2799 42.4605 69.8868 73.1526 10.5408 83.3857 48.0798 0.7084 0.5169 0.3502 0.6057 0.7281 0.6936 0.5412 0.7282 0.5516 0.6955 0.7350 0.3373 0.5762 0.7010 0.6728 0.7514 0.5758 0.3714 0.7711 0.7358 0.1984 0.1293 0.2608 0.1196 0.1795 0.2574 0.2488 0.1749 0.1813 0.0991 0.1931 0.3054 0.1624 0.1719 0.1933 0.1027 0.1333 0.3573 0.1238 0.1995 0.2336 0.1139 0.3484 0.1411 0.1655 0.2703 0.2235 0.1430 0.1857 0.1240 0.2034 0.3019 0.1644 0.1840 0.2090 0.0954 0.1660 0.3582 0.0949 0.1773 0.2060 0.1273 0.2651 0.1189 0.1719 0.2715 0.2450 0.1655 0.1912 0.0927 0.1983 0.3124 0.1610 0.1643 0.1978 0.1056 0.1330 0.3582 0.1115 0.1988 0.2109 0.1228 0.2356 0.1250 0.1763 0.2986 0.2612 0.1784 0.1969 0.0967 0.1962 0.3109 0.1722 0.1651 0.2046 0.0931 0.1426 0.3554 0.1206 0.2135 Run %KT_CCD %KT_BBD %KT_UD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17.72 11.95 33.61 17.99 7.80 5.03 10.18 18.22 2.41 25.15 5.33 1.16 1.25 7.05 8.10 7.10 24.51 0.24 23.34 11.12 3.81 1.56 1.63 0.57 4.22 5.47 1.54 5.38 5.46 6.41 2.69 2.30 0.85 4.40 2.30 2.78 0.21 0.25 9.90 0.37 6.32 5.01 9.68 4.49 1.78 16.01 4.97 1.98 8.63 2.41 1.61 1.80 6.02 3.95 5.86 9.34 6.96 0.54 2.56 7.03 ARTICLE IN PRESS 246 M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 Fig. 8. Actual versus predicted values of thrust coefficient in the CCD model. Fig. 9. Actual versus predicted values of thrust coefficient in the BBD model. is a serious defect. Even if we were to say that there might not be interaction, there is no decline in the merit of the factorial design. Indeed, in the OFAT design, there would be no way to determine that there is no interaction (Berger and Maurer, 2002). Hence, not only there would be cost savings in using factorial designs, it is the more correct and complete method of experimental design. 6. Conclusions This paper has presented a combined dimensional analysis and statistical design of experiments approach to reduce the high number of variables to a reduced set of significant variables. The system under consideration was a propeller operating in open water whose performance depends on many variables. A total number of fourteen variables were considered to control the thrust coefficient. The number of variables were first reduced by three using dimensional analysis, which also changes the variables to a more useful format in terms of dimensionless Fig. 10. Actual versus predicted values of thrust coefficient in the UDD model. parameters. A two-level fractional factorial design was then used to screen out any statistically insignificant parameters. A secondorder experimental design was then used to obtain information to define a functional relationship between the thrust coefficient and the five parameters. In developing the functional relationships, three response surface experimental designs were considered. Two of them, namely the CCD and BBD are well-known classical designs, whereas the UD is a rather new type of space-filling design that have been found to be useful in both industrial and computer experiments. The UD provides for a greater choice of designs and modeling approaches, which also means that the user would need a much greater range of statistical and mathematical expertise to use it. It was also seen in the comparison of the results that a different experimental design might give rise to different significant terms in the regression. For the problem at hand, the BBD gave the best prediction model. The final models as presented may not be as accurate as classical regression methods for interpolating and extrapolating the thrust coefficient. The purpose of this paper however is not to find the best models for the thrust coefficient but to illustrate how a problem with a large number of variables can be systematically handled to obtain functional relationships using a variety of modern statistical design of experiment tools combined with the more well-known dimensional analysis. Acknowledgments The authors would like to thank Pengfei Liu of Institute for Ocean Technology of National Research Council for giving the permission to use his panel method code PROPELLA, which was used to carry out all the numerical predictions on the performance characteristics of the propellers at various geometric configurations required by the DOE analysis process. Thanks are also extended to Brian Veitch of Memorial University of Newfoundland for all his support and suggestions in carrying out the work. References Antony, J., 2006. Design of Experiments for Engineers and Scientists. Elsevier, Butterworth-Heinemann. Barr, D.I.H., 1969. Method of synthesis: basic procedures for the new approach to similitude. Water Power, 148–153 pp. 183–8. ARTICLE IN PRESS M.F. Islam, L.M. Lye / Ocean Engineering 36 (2009) 237–247 Berger, P.D., Maurer, R.E., 2002. Experimental Design with Applications in Management, Engineering, and the Sciences. Duxbury Press. Box, G., et al., 2006. 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