Second order polynomial equation

Many systems that cannot be represented by a first-order empirical model can be described by a full second-order polynomial equation, such as that for two factors. 

Development of the Nomograph. Tw o main sources of data were used to develop the nomograph McAuliffe and Price. The hydrocarbons were divided into 14 homologous series as listed in Table 1. Solubilities at 25°C were then regressed with the carbon numbers of the hydrocarbons in order to obtain the best fit for each homologous series. A second order polynomial equation fits the data very well ... 

The MO concentrations versus time profiles were fitted to second order polynomial equations and the parameters estimated by nonlinear regression analysis. The initial rates of reactions were obtained by taking the derivative at t=0. The reaction is first order with respect to hydrogen pressure changing to zero order dependence above about 3.45 MPa hydrogen pressure. This was attributed to saturation of the catalyst sites. Experiments were conducted in which HPLC grade MIBK was added to the initial reactant mixture, there was no evidence of product inhibition. 

Tensile Modulus. Tensile samples were cut from the 0.125 in. plates of the compositions according to Standard ASTM D638-68, into the dogbone shape. Samples were tested on an Instron table model TM-S 1130 with environmental chamber. Samples were tested at temperatures of -30°C, 0°C. 22°C, 50°C, 80°C, 100°C and 130°C. Samples were held at test temperature for 20 minutes, clamped into the Instron grips and tested at a strain rate of 0.02 in./min. until failure. The elastic modulus was determined by ASTM D638-68. Second order polynomial equations were fitted to the data to obtain the elastic modulus as a function of temperature for each of the compositions. 

In order to make the comparison between Ep and Ep/2 measurements summarized in Table 9, the two quantities were measured in separate experiments. A recent study by Eliason and Parker has shown that this is not necessary [57]. Analysis of theoretical LSV waves by second-order linear regression showed that data in the region of Ep are very nearly parabolic. The data in Fig. 9 are for the LSV wave for Nernstian charge transfer. The circles are theoretical data and the solid line is that described by a second-order polynomial equation. It was concluded that no detectable error will be invoked in the measurement of LSV Ep and Ip by the assumption that the data fit the equation for a parabola as long as the data is restricted to about 10 mV on either side of the maximum. This was verified by experimental measurements on both a Nernstian and a kinetic system. 

Fig. 9. Theoretical LSV data in the region of the current maximum along with the curve fit to a second-order polynomial equation.

The experimental data (Table 5) were analyzed by the response surface regression (Proc RSREG) procedure to fit the following second-order polynomial equation (SAS, 1990) ... 

The RSREG procedure for SAS was employed to fit the second-order polynomial equation 1 to the experimental data—percent weight conversions (Table 5). Among the various treatments, the greatest weight conversion (96.5%) was treatment 22 (12h, 45°C, 50% enzyme, substrate molar ratio 4 1, added water 10%), and the smallest conversion (only 22.4%) was treatment 12 (16h, 55°C, 20% enzyme, substrate molar ratio 5 1, added water 5%). From the SAS output of RSREG, the second-order polynomial equation is given below ... 

The effect of composition on each of the nine parameters describing the shapes of the DMS temperature curves was investigated enpirically by fitting these parameters into second order polynomial equations of the type ... 

Prediction of the log reduction of an inoculated organism as a function of acid concentration, time, and temperature can also be done by a mathematical model developed for this purpose, using the second-order polynomial equation to fit the data. The following tests justified the reliability of the model the analysis of variance for the response variable indicated that the model was significant (P < 0.05 and R2 = 0.9493) and had no significant lack of fit (P > 0.05). Assumptions underlying the ANOVA test were also investigated and it was demonstrated that with the normal probability plot of residuals, plot of residuals versus estimated values for the responses, and plot of residuals versus random order of runs, that the residuals satisfied the assumptions of normality, independence, and randomness (Jimenez et al., 2005). 

The results were modeled with a second-order polynomial equation, i.e. each response j was fitted by a quadratic model, given as follows for four factors. 

A 3 full factorial design provides sufficient data for the fitting of a second-degree expression. In this sense, the following second order polynomial equation explains the data obtained... 

Statistical optimisation was carried out by Mallick and co-workers [25] for the bacterium Nostoc muscorum to optimise the physical and chemical parameters. A five-level four-factorial central composite design was employed to determine the interactions between the variables for the production of PHA. A second-order polynomial equation was obtained using RSM, which resulted in an increase of product yield along with a decreased use of acetate and propionate. Yang and co-workers [27] reported the optimisation of Cupriavidus necator HI 6 for CDW, PITA content and 3HV monomer composition. A simplex lattice method was formulated using the Minitab V14 program. Optimisation in this study resulted in a 4-fold increase in cell growth and PHA production. 

A second order polynomial equation as shown in Eq.l was assumed in order to find out the regression equation. 

The values of ao to 39 and CoefBcient of determination (R ) were determined. values obtained were not very high. The particle release was correlated with load and time for particle of predefined size. A second order polynomial equation as shown in Eq. 2 is assumed in order to find out the regression equation. 

Typically, a second-order polynomial equation is sufficient to capture the nonlinearity. To do this, correlations are developed as shown in Figure 15.9 based on Table 15.2, which describes the relationship between valve position Vp and steam flow, M ... 

The Formal Graph in Graph 11.44 is merely an adaptation of the one drawn in Chapter 9 to the apparent variables used in this model. It results from the symmetrical splitting of Graph 11.43c made for solving the model (by reducing the degree of temporal operators on the template of a second-order polynomial equation). 

Fig. 6.4 Traids for the percentage particles exhaled and the time needed to fall a distance that equals the diameter of a peripheral airway (0.43 mm), both as function of the aerodynamic particle diametra-. The percentage exhaled for 1 pm particles is obtained from extrapolation (using a second order polynomial equation) exhalation data for 1.5 3 and 6 pm particles derived from Usmani et al. [11] and extrapolation of the correlationships towards particles of 1 pm...

The following method uses the method of least squares. In this case, all data points are used to generate a second-order polynomial equation. This equation is then differentiated and evaluated at the point where the value of the derivative is required. For example, Microsoft Excel can be employed to generate the regression equation. Once all the coefficients are known, the equation has only to be analytically differentiated ... 

The last method also uses five data points but only three coefficients are generated for a second-order polynomial equation of the form f=A- -BxA- Cx. Another set of equations are used to evaluate the derivative at each point using this method. The equations are provided below ... 

Subsequently, the difference Ad(no) (in angstrom) is calculated from the theoretical value and the measured d value of the (1 1 0) reflection. Using the empirical cahbration curve (a second-order polynomial. Equation 3.5), the pyrolusite concentration can be calculated or it may be taken from a diagram, as shown in Figure 3.5. 

From Figure 10.5, we found that this non-linear relation can be fit perfectly by a second order polynomial equation. In general, although the reaction rate of end chain scissions is faster than the rate of random chain scissions, oligomer production is, however, still dominated by random chain scissions for most cases. We apply a general polynomial form with flexible orders to express the relation between chain scission number and oligomer production number ... 

Response surface morphologies were utilized to analyze the experimental data (Table 15.5) and a second order polynomial equation was fitted by multiple regression analysis [44-45]. The quality of fit for the model was estimated by the coefficients of determination (R ) and the analysis of variances. After the coefficients were examined and the model was fine-tuned, all the insignificant coefficients were omitted. As a result, the quadratic response model was fitted to the following equation ... 

The data were fitted to a second-order polynomial (Equation 10.1) and the values of the second-, first- and zero-order coefficients were calculated for each solution as shown in Table 10.2. The second-order coefficient is reported as an indicator of the solute-solute interactions in the solution. The first-order coefficient is proportional to the variation in the solution density (molar volume) with the addition of vanadium sulphate and the last coefficient is the density of the acid-water mixture. The values of these coefficients at different acid concentrations do not show any trends. This might be due to the fact that the variations might be too small and lie within experimental error. 

Pure gas permeation data of membranes prepared using these additives were obtained from a constant pressure permeation system for CO2, CH4, O2 and N2. In order to know the effect of the structure of the nonsolvents, nonlinear regression analysis was attempted. Each additive was split into structural components groups. Several linear polynomial first and second order equations as well as nonlinear polynomial equations were attempted to derive an empirical correlation between the number of structural components and gas permeation data. A second order polynomial equation was derived to predict the pure CO2/CH4 permeance ratio from the structural components of the nonsolvents. From the structural studies the authors concluded that nonsolvent additives that possess a long straight hydrocarbon chain such as 2-ethyl-l-hexanol, 1-octanol and 2-decanol showed the highest pure gas permeance ratio. 

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