Second-order Response Surface Model Research Articles

The Research of the Assembly Precision of CNC Machine Tools Considering the Joint Surface Characteristics

In order to evaluate the assembly precision of the CNC machine tools influenced by the joint surface characteristics effectively, a method is proposed in this paper. Finite element analysis is used to obtain joint surface deformation data, and then a second-order response surface model is established as a surrogate model to express the function of joint surface static characteristics and deformation. The error propagation in assembly sequence is obtained, so that the assembly precision of CNC machine tools can be evaluated effectively.

Vanadium oxide–carbon nanotube composite electrodes for energy storage by supercritical fluid deposition: experiment design and device performance

Vanadium pentoxide (V2O5) deposited on porous multiwalled carbon nanotube (MWCNT) buckypaper using supercritical fluid CO2(scCO2) deposition shows excellent performance for electrochemical capacitors. However, the low weight loading of V2O5 is one of the main problems. In this paper, design of experiments and response surface methods were employed to explore strategies for improving the active material loading by increasing the organo-vanadium precursor adsorption. A second-order response surface model was fitted to the designed experiments to predict the loading of the vanadium precursors onto carbon nanotube buckypaper as a function of time, temperature and pressure of CO2, buckypaper functionalization, precursor type, initial precursor mass and stir speed. Operation conditions were identified by employing a model that led to a precursor loading of 19.33%, an increase of 72.28% over the initial screening design. CNTs–V2O5 composite electrodes fabricated from deposited samples using the optimized conditions demonstrated outstanding electrochemical performance (947.1 F g−1 of V2O5 at a high scan rate 100 mV s−1). The model also predicted operation conditions under which light precursor aggregation took place. The V2O5 from aggregated precursor still possessed considerable specific capacitance (311 F g−1 of V2O5 at a scan rate 100 mV s−1), and the significantly higher V2O5 loading (∼81%) contributed to an increase in overall electrode capacitance.

Grinding Machine Column Structure Dynamic Optimization

Static and dynamic characteristics of grinding machine column structure experientially designed are studied based on finite element models. Then propose structure modifications, analyze characteristics, and select the better one. Study the natural frequency changes in the sensitivity of the wall thickness and rib thickness, and determine the parameters to be optimized. Expert Design software is used to finish experimental design of the parameters, and establish the second-order response surface model. So far a multi-objective optimization model is set up easily. Here using the corresponding ideal point method with relative squared weighting method transforms it into a single - objective one. Some optimal design schemes are received by changing the weight coefficients. ANSYS simulation demonstrates that the result of optimized design is satisfied and the design method is suitable.

Dephosphorization of chromium-rich ashes from thermally-treated leather waste for utilization in pyrometallurgical process

Acid leaching was investigated to remove phosphorus from the ashes of thermally-treated (gasification and combustion process) footwear leather waste in order to obtain a good quality raw material for the production of a ferrochromium alloy. The ash contained high amounts of chromium (55wt.% Cr2O3) and phosphorus (1.42wt.% P). The following key factors were selected for study: temperature, H2SO4 concentration in the leaching solution, acid solution/ash ratio and the particle size of the ash. The response was the phosphorus removal percentage and, under some experimental conditions, chromium loss. The strategy was based on a 24 full factorial design with a central point followed by the addition of star points in a face-centered central composite design. The maximum chromium loss was 1.1wt.%. The significant effects (according to variance analysis) were used for a proposed second-order response surface model. The model was useful for finding the optimal operating conditions in the experimental region by the maximization of phosphorus leaching. 99.2±2.5% phosphorus removal from the ash was achieved under the following conditions: 60°C, 13.9wt.% of H2SO4 in the leaching solution, 4.80 (ml of acid solution)(g of ash)−1, 0.0370±0.0370mm particle size. Mass balance was used as a tool to indicate that the remaining phosphorus content was acceptable for producing ferrochromium alloys.

Predicting electroless Ni–Co–P coating using response surface method

Electroless Ni–Co–P coating has been studied considering pure copper (99.99 %) as a substrate material. The deposited mass per unit area of the substrate has been considered as the response variable. The individual as well as combined effects of the process parameters on the response variable has been studied. The bath temperature and the interaction of cobalt source concentration and reducing agent concentration have been found to be significant through regression analysis and Student's t test. Mathematical modelling has been carried out using a second-order response surface model with central composite design to take into account the effect of curvature in the predicted response. The equation for response surface has been determined using MATLAB software package. The response surface plots show that the rate of deposition increases significantly with the rise in temperature while the cobalt source concentration and reducing agent concentration have an interactive effect which influences the rate of deposition. The test of reliability for the predicting response surface equation shows that the equation gives an excellent fitting to the observed values.

Response Surface Optimization for Activation of Bentonite Using Microwave Irradiation

AbstractMicrowave irradiation as a means for heating bentonites during acid activation has been investigated in the past but it has never been optimized for industrial applications. The purpose of this study was to apply a factorial 23 experimental design to a Serbian bentonite in order to determine the influence of microwave heating on the acid-activation process. The effect of acid activation under microwave irradiation on the textural and structural properties of bentonite was studied as a model reaction. A mathematical, second-order response surface model (RSM) was developed with a central composite design that incorporated the relationships among various process parameters (time, acid concentration, and microwave heating power) and the selected process response of specific surface area of the bentonite. The ranges of values for the process parameters chosen were: time, 5–21 min; acid concentration, 2–7 M; and microwave heating power, 63–172 W. The effect of individual variables and their interaction effects on the textural and structural properties of the bentonite were determined. Statistical analysis showed that the duration of microwave irradiation was less significant than the other two factors. The model showed that increasing the time and acid concentration improved the textural properties of bentonites, resulting in increased specific surface area. This model is useful for setting an optimum value of the activation parameters for achieving the maximum specific surface area. An optimum specific surface area of 142 m2g−1 was achieved with an acid concentration of 5.2 M, activation time of 7.4 min, and microwave power of 117 W.

Parametric optimization of ultrasonic metal welding using response surface methodology and genetic algorithm

This paper focuses on the development of an effective methodology to determine the optimum welding conditions that maximize the strength of joints produced by ultrasonic welding using response surface methodology (RSM) coupled with genetic algorithm (GA). RSM is utilized to create an efficient analytical model for welding strength in terms of welding parameters namely pressure, weld time, and amplitude. Experiments were conducted as per central composite design of experiments for spot and seam welding of 0.3- and 0.4-mm-thick Al specimens. An effective second-order response surface model is developed utilizing experimental measurements. Response surface model is further interfaced with GA to optimize the welding conditions for desired weld strength. Optimum welding conditions produced from GA are verified with experimental results and are found to be in good agreement.

Comparative study of response surface designs with errors-in-variables model

This paper investigates the scaled prediction variances in the errors-in-variables model and compares the performance with those in classic model of response surface designs for three factors. The ordinary least squares estimators of regression coefficients are derived from a second-order response surface model with errors in variables. Three performance criteria are proposed. The first is the difference between the empirical mean of maximum value of scaled prediction variance with errors and the maximum value of scaled prediction variance without errors. The second is the mean squared deviation from the mean of simulated maximum scaled prediction variance with errors. The last performance measure is the mean squared scaled prediction variance change with and without errors. In the simulations, 1 000 random samples were performed following three factors with 20 experimental runs for central composite designs and 15 for Box-Behnken design. The independent variables are coded variables in these designs. Comparative results show that for the low level errors in variables, central composite face-centered design is optimal; otherwise, Box-Behnken design has a relatively better performance.

Small Box–Behnken design

Box-Behnken design has been popularly used for the second-order response surface model. It is formed by combining two-level factorial designs with incomplete block designs in a special manner—the treatments in each block are replaced by an identical design. In this paper, we construct small Box–Behnken design. These designs can fit the second-order response surface model with reasonably high efficiencies but with only a much smaller run size. The newly constructed designs make use of balanced incomplete block design (BIBD) or partial BIBD, and replace treatments partly by 2 III 3−1 designs and partly by full factorial designs. It is shown that the orthogonality properties in the original Box and Behnken designs will be kept in the new designs. Furthermore, we classify the parameters into groups and introduce Group Moment Matrix (GMM) to estimate all the parameters in each group. This allows us to significantly reduce the amount of computational costs in the construction of the designs.

Parameters optimization of a vertical ground heat exchanger based on response surface methodology

Vertical ground heat exchangers are one of the most important parts of geothermal utilization systems. In order to understand the effect of simultaneous variation of design parameters; a three dimensional computational fluid dynamics simulation was carried out. Based on the effective parameters on the heat exchanger efficiency and the total heat transfer efficiency in cooling mode, and with the aid of the second-order Response Surface Model, two functions for the total heat transfer efficiency and the heat exchanger efficiency were derived. The effects of different design parameters on the response variables were examined. Based on the Response Surface Model, it was found that the dimensionless inlet fluid temperature and the dimensionless pipe diameter significantly affect the response variables, while the response variables are weakly affected by dimensionless depth. Finally, an optimization was performed and the optimum heat exchanger is defined using the model equations.

D-Optimal and D-Efficient Equivalent-Estimation Second-Order Split-Plot Designs

Industrial experiments often involve factors that are hard to change or costly to manipulate and thus make it undesirable to use a complete randomization. In such cases, the split-plot design structure is a cost-efficient alternative that reduces the number of independent settings of the hard-to-change factors. In general, model estimation for split-plot designs requires the use of generalized least squares (GLS). However, for some split-plot designs (including not only classical agricultural split-plot designs, but also some second-order split-plot response surface designs), ordinary least squares (OLS) estimates are equivalent to GLS estimates. These designs are called equivalent-estimation designs and offer the advantage that estimation of the factor effects does not require estimation of the variance components in the split-plot model. As an alternative to these equivalent-estimation designs, one can use D-optimal designs that guarantee efficient estimation of the fixed effects of the statistical model that is appropriate given the split-plot structure. We explore the relationship between equivalent-estimation and D-optimal split-plot designs for a second-order response surface model and propose an algorithm for generating D-efficient equivalent-estimation split-plot designs. This approach allows for a flexible choice of the number of hard-to-change factors, the number of easy-to-change factors, the number of whole plots, and the total sample size.

Extension of Central Composite Design for Second-Order Response Surface Model Building

The central composite design (CCD) is perhaps the most popular class of second-order response surface designs. Even though the CCDs are popular for response surface designs, this class of design has some limitations such as it does not sometimes possess good statistical properties, and it does not fit complicated models well. In this article, we propose extended central composite designs (ECCDs) to overcome these limitations. We compare ECCDs with CCDs in terms of average prediction variance, and find that ECCDs are better than CCDs.

Robust Second-Order Slope-Rotatable Designs with Maximum Directional Variance

In response surface methodology, rotatability and slope-rotatability are natural and highly desirable properties for second-order regression models. In this article, we introduce the concept of robust slope-rotatable designs with equal maximum directional variance for second-order response surface models with correlated observations. This requires that the maximum variance of the estimated slope over all possible directions to be only a function of the distance of the point from the design origin, and independent of correlation parameter or parameters involved in the variance–covariance matrix of errors. It is derived that robust second-order rotatable designs of two factors are also robust slope-rotatable designs with equal maximum directional variance. It is also established that within the robust second-order symmetric balanced designs, robust rotatable designs are also robust slope-rotatable with equal maximum directional variance for more than two factors. We also investigate a class of robust second-order slope-rotatable designs with equal maximum directional variance for special correlation structures of errors.

On D-optimal robust second order slope-rotatable designs

Das and Park (2006) introduced slope-rotatable designs overall directions for correlated observations which is known as A-optimal robust slope-rotatable designs. This article focuses D-optimal slope-rotatable designs for second-order response surface model with correlated observations. It has been established that robust second-order rotatable designs are also D-optimal robust slope-rotatable designs. A class of D-optimal robust second-order slope-rotatable designs has been derived for special correlation structures of errors.

Exact [formula omitted]-optimal designs for a second-order response surface model on a circle with qualitative factors

In this article, exact D -optimal designs for a second-order response surface model on a circular design region with qualitative factors are investigated. Based on this design region, an exact D -optimal design with regular polygon structure is made up according to the remainder terms of the numbers of experimental trials at each qualitative levels divided by 6. The complete proofs of exact D -optimality for models including two quantitative factors and one 2-level qualitative factor are presented as well as those for a model with only quantitative factors. When the qualitative factor has more than 2 levels, a method is proposed for constructing exact designs with high efficiency. Exact D -optimal designs with minimal supports are also proposed for practical consideration.

Second-Order Response Surface Model with Neighbor Effects

This article considers the second-order response surface model in which the experimental units, i.e., plots experience the neighbor effects from immediate left and right neighboring plots assuming the plots to be placed adjacent linearly with no gaps. Conditions have been derived for the estimation of coefficients of second-order response surface model. A method of constructing designs for fitting second-order response surface in the presence of neighbor effects has been developed. The designs so obtained are found to be rotatable.

Removal of cobalt ions from aqueous solutions by polymer assisted ultrafiltration using experimental design approach. part 1: Optimization of complexation conditions

The polymer assisted ultrafiltration process combines the selectivity of the chelating agent with the filtration ability of the membrane acting in synergy. Such hybrid process (complexation–ultrafiltration) is influenced by several factors and therefore the application of experimental design for process optimization using a reduced number of experiments is of great importance. The present work deals with the investigation and optimization of cobalt ions removal from aqueous solutions by polymer enhanced ultrafiltration using experimental design and response surface methodological approach. Polyethyleneimine has been used as chelating agent for cobalt complexation and the ultrafiltration experiments were carried out in dead-end operating mode using a flat-sheet membrane made from regenerated cellulose. The aim of this part of experiments was to find optimal conditions for cobalt complexation, i.e. the influence of initial concentration of cobalt in feed solution, polymer/metal ratio and pH of feed solution, on the rejection efficiency and binding capacity of the polymer. In this respect, the central compositional design has been used for planning the experiments and for construction of second-order response surface models applicable for predictions. The analysis of variance has been employed for statistical validation of regression models. The optimum conditions for maximum rejection efficiency of 96.65% has been figured out experimentally by gradient method and was found to be as follows: [Co 2+] 0 = 65 mg/L, polymer/metal ratio = 5.88 and pH 6.84.

Combined effects of organic matter and pH on acute toxicity of cadmium to Paronychiurus kimi (Collembola): Development of response surface model

A second-order central composite design was employed to investigate the effect of organic matter (OM) content and soil acidity (pH) on the cadmium toxicity to Paronychiurus kimi (Lee), a collembolan species native to Korea. Two independent variables, OM and pH, were adjusted from 0 to 10% (in total dry wt) and 4.5 to 7.0, respectively. Cadmium concentrations tested were 10, 20, 40, 80, and 160 mg/kg dry soil. The toxic effects that cadmium exerted on P. kimi (28-day median lethal concentration (LC50s)) varied significantly with different combinations of OM content and pH. The second-order response surface model was appropriate for describing combined effects of OM and pH on the cadmium toxicity to P. kimi. Linear effects (OM and pH) were predominant contributors accounting for ≈70% of the total variance of cadmium toxicity to P. kimi. No significant quadratic (pH × pH) and interaction (OM × pH) effects were observed; therefore, the final second-order response surface model could be reduced as follows: 28-day LC50 of cadmium (mg/kg) = –94.87 + 26.69 × OM + 16.64 × pH – 1.82 × OM2. Validation of the developed response surface model using 4 soil media with different OM content and pH resulted in a significant correlation coefficient (R2 = 0.97) between predicted and observed 28-day LC50. Based on physico-chemical properties of the test media, developed models can be used to explain the variation of cadmium toxicities at various environmental conditions and may improve current environmental risk assessment procedures of metals.

도시화에 의한 야간 대기 냉각율 변화에 관한 연구

This work investigated the methodology to simultaneously optimize the ozone and relative humidity composition for the NO x degradation using soil biofilter. Experiments were made as a function of inlet ozone concentration (0~1,770 ppb) and relative humidity (38~81%). Factorial design (2 2 +3) and response surface methodology by central composite designs were used to examine the role of two factors and optimal response condition on NO x degradation. It was found that a second-order response surface model can properly interpret the experimental data with an R 2 -value of 0.9730 and F-value of 71.83, based on which the maximum NO x degradation was predicted up to 92.8% within our experimental conditions.

Mechanical properties and grinding performance on aluminum-based metal matrix composites

The metal matrix composites (MMCs) are attractive materials in the various areas of the industrial application because of their superior properties. This study presented the fabrication by the pressureless infiltration process under the nitrogen gas atmosphere, and the grinding of the aluminum-based MMCs reinforced with SiC particles. The working conditions such as the contents of the SiC and the Mg were examined how to influence the fabricating process of the MMCs. The effect of the grinding parameters on the surface roughness and the grinding forces was evaluated, and then the grinding parameters were optimized using the S/N ratio. The second-order response surface models were developed and the usefulness of the developed models was verified.