Second-order model

Curved one-factor response surface showing (a) the limitation of a 2 factorial design for modeling second-order effects and (b) the application of a 3 factorial design for modeling second-order effects. 

FIG. 18 (a) Plot of the critical points Fc versus Fa for the ZGB-DD model ( ) second-order IPTs (o) first-order IPTs. (b) Detailed view of (a) in the limit Fc 0. 

FIG. 20 Plot of the critical points Yq versus Fa for the DMM model. Second order IPTs (v) and first-order IPTs ( and ). ( ) shows the location of the IPT of the MM model. 

To obtain the second-order regression model, second-order CCRD has been used. The number of design points (trials) for k=5 was 32. The design core has corresponded to half-replica 2s 1 with this generating ratio Xs=XrX2X3X4. The value and number of design points in the experimental center n0=6 are determined from Table 2.137. The design of experiments with outcomes is shown in Table 2.145. 

The design used is a function of the model proposed. Thus, if it is expected that the important responses vary relatively little over the domain, a first-order polynomial will be selected. This will also be the case if the experimenter wishes to perform rather a few experiments at first to check initial assumptions. He may then change to a second-order (quadratic) polynomial model. Second-order polynomials are those most commonly used for response surface modeling and process optimization for up to five variables. 

As in Part (b) segregation modeU second order except adiabatic As in Part (c) maxinumt mixe ss model, second order except adiabatic... 

Using the tank in series model second order reaction... 

Coeff. First-order model Second-order model Reduced cubic model ... 

For the Kelvin model, second-order and first-order equations for radius and thickness, respectively, are obtained. Rewriting the second-order equation as two first-order equations the system is transformed into the following equations [13] ... 

X3 First-Order Model Second-Order Model First-Order Model ... 

In contrast to the eddy viscosity approach, the Reynolds stress tensor, R, can be modeled directly by introducing one transport equation for each term. Such models are called Reynolds stress models, second-order closure models or second-moment closure models and have been pioneered by Rotta [47]. Since R is a symmetric second-order tensor, this requires six transport equations, and an additional three transport equations are needed to model T. Obviously, the predictive capabilities of the Reynolds stress models over the two-equation models are improved, but these models are significantly more complex and there is a considerable additional computational cost. A Reynolds stress model for spray applicatiOTis has been developed and tested by Yang et al. [63]. 

In this kind of models second-order differential terms appear in mass, energy, and momentum balances and two boundary conditions on the axial coordinate have to be imposed. The set of differential equations together with the boundary conditions represents a boundary value problem (BVP), since the constraints are imposed at the inlet and at the outlet of the tubular reactor. 

Eight cracking reactions were included in the model second-order gas oil cracking to gasoline fraction paraffins, olefins, naphthenes and aromatics and to LPG, first-order gas oil cracking to dry gas and to coke and first-order cracking of gasoline olefins to LPG. 

Second-order fluid Walters fluid model Second-order fluid... 

Ncop mol. % First-order Markov model Second-order Markov model Finemann-Ross References... 

Vydrov and van Voorhis proposed considerable modification of the original vdW-DFl [55] called the vdW-DF-09 [67]. A new dispersion model, second-order gradient-coefficient of and a new local functional part were introduced. Specifically, the LC-mPBE exchange-correlation functional was adopted for the nonlocal part of vdW-DF-09. Thus, vdW-DF-09 is defined as... 

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