Simultaneous method

In this method, the equations describing the system are all solved simultaneously using, for instance, the Newton-Raphson technique. Applied to phase boundary conditions, the basic relationships in Equations 2.7, 2.8, and 2.12 take a special form. Since / is either zero at the bubble point or one at the dew point. Equation 

7 reduces to the trivial relationship Z = X or Z = T . Equation 2.8 need not be involved in the simultaneous solution of the system equations. Once the phase boundary conditions are determined, this equation may be solved independently to calculate the heat duty required to bring the feed to its bubble point or dew point. Thus, the only equation to be solved for bubble or dew point calculations is Equation 2.12. 

At the bubble point, the X S are known (X = Z and the temperature or pressure is given. The unknowns are C - 1 vapor compositions and the pressure or temperature. At the dew point, the T s are known (T = Z as well as the temperature or pressure, and the unknowns are C - 1 liquid compositions and the pressure or temperature. In either case, C equations (/ = 1 to C) must be solved for C unknowns. 

Applying the method to bubble point pressure prediction, for example, where the liquid composition and temperature are known. Equation 2.12 may be written in the form (Peng et al., 1977) 

The iterations are repeated until the Equation 2.22 is satisfied within a specified tolerance, or until the corrections are within the tolerance. 

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Various procedures for solving Eqs. (13-149) to (13-161), ranging from a complete tearing method to solve the equations one at a time, as shown by Distefano, to a complete simultaneous method, have been studied. Regardless of the method used, the following considerations generally apply ... 

Crescenzi et al. developed a multi-residue method for pesticides including propanil in drinking water, river water and groundwater based on SPE and LC/MS detection. The recoveries of the pesticides by this method were >80%. Santos etal. developed an on-line SPE method followed by LC/PAD and LC/MS detection in a simultaneous method for anilides and two degradation products (4-chloro-2-methylphenol and 2,4-dichlorophenol) of acidic herbicides in estuarine water samples. To determine the major degradation product of propanil, 3,4-dichloroaniline, the positive ion mode is needed for atmospheric pressure chemical ionization mass spectrometry (APCI/MS) detection. The LOD of 3,4-dichloroaniline by APCI/MS was 0.1-0.02 ng mL for 50-mL water samples. 

In vivo evaluation of the semi-simultaneous method for bioavailability estimation using controlled intravenous infusion as an... 

An alternative means of calculating reaction pathways is employed in so-called global methods. These methods treat the entire path as a succession of points [71] which are found simultaneously. Methods of this type (for example the conjugate peak refinement algorithm [72], available in the TRAVEL module of CHARMM, which has the advantage of requiring only first derivatives of the energy) have been used to determine reaction paths in a number of proteins [4, 73]. 

Here we briefly describe the first two approaches with the aim of contrasting them to simultaneous methods presented in Section V. In that section, the advantages of simultaneous methods also lead us to consider some open questions and concentrate on current work in this area. 

However, all of these studies determine only approximate or parameterized optimal control profiles. Also, they do not consider the effect of approximation error in discretizing the ODEs to algebraic equations. In this section we therefore explore the potential of simultaneous methods for larger and more complex process optimization problems with ODE models. Given the characteristics of the simultaneous approach, it becomes important to consider the following topics ... 

Instead, the simultaneous method can be extended to select adaptively a sufficient number of finite elements. Here, we note that even if we set any element length to zero, the collocation equations and the continuity equations are still satisfied. Thus, any number of zero length (or dummy) elements can be added without changing the control or state profiles, or the solution to the NLP. Vasantharajan and Biegler (1990) take advantage of this important property and propose an adaptive element addition approach embedded within the simultaneous solution strategy. 

The three methods described in the preceding paragraphs each offer distinct advantages and disadvantages. The first and most obvious difference between the methods is the distinction between the sequential methods (sequential simplex and prisma method) and the simultaneous method (mixture design). With the sequential method some experiments are performed, these are evaluated, and on the basis of this evaluation new design points are selected, these are evaluated etc. With the simultaneous... 

The choice for a sequential or a simultaneous method depends on the expected difficulty to obtain the optimum and the effort needed to perform one experiment. In this paper the problem is further complicated by the requirement of robustness. The optimization method will be determined after a discussion of robustness and the Taguchi method. 

Regression methods (simultaneous methods) are applied to model criteria as a function of factor adjustments. (Fractional) Factorial designs are often used to optimise process variables, e.g. pH or ionic strength. The levels of the factors can be chosen at will. Examples of factorial designs are given in [9,10,11]. A review on the subject has been given by Deming et al. [12]. 

With all of the point-successive methods, only one array in the computer is needed to store both o k) and 6 k+1) because the element o k) is never again needed once o k+1) is computed. This contrasts with the point-simultaneous methods, where, for each n, the elements of 6 k) must be available corresponding to the full range over which [s] m is finite. 

We have also learned that the point-successive methods need not demand more computer memory than point-simultaneous methods, and that the seemingly inherent asymmetry of the point-successive methods can be overcome. Furthermore, the linear methods described in this section are... 

Recent work on a production basis involving 5-fim spectra of 12CD3F has verified line-center frequency stability under deconvolution. Large numbers of experimental records of 12CD3F, simultaneously recorded with CO in the sample, were calibrated before and after deconvolution (point-simultaneous methods, Jansson algorithm). No systematic differences were detected in comparisons of the before and after frequencies of nonblended absorption lines. That is, the variance was consistent with the optomechanical precision of the spectrometer and the mean deviation summed to approximately zero, validating the frequency calibration of the deconvolved data. 

In the simultaneous method, which is the one most commonly used, the substrate is irradiated while in direct contact with the monomer. The monomer can be present as a vapor, liquid, or solution. This grafting process can occur via free radial or ionic mechanism. With the simultaneous method, the formation of homopol5mier is unavoidable, but there are several systems to minimize it. The advantage of this method is that both monomer and substrate are exposed to the radiation source and both form reactive sites. The other two techniques rely upon rupture of the bond to form reactive sites, and therefore require higher radiation doses. Thus, the simultaneous method is more suitable for substrates sensitive to radiation. The simultaneous method can utilize UV radiation besides EB source. Logically, the UV irradiation requires a photoinitiator or sensitizer to achieve an acceptable level of grafting. 

FTIR is a simultaneous method in that signal processing is global. Each spectral point is calculated from all the given data... 

A subset of simultaneous methods which overcomes the difficulty of mapping complex response surfaces by an exhaustive series of experiments are the interpretive methods, in which retention surfaces are modeled using a minimum number of experimental data points. Retention surfaces thus obtained for the individual solutes are then used to calculate (via computer) the total response surface according to some predetermined criterion. The total response surface is then searched for the optimum. 

Meuzelaar and co-workers [90] have described an on-line high pressure TG-GC-MS system which requires only very small amounts (10-100 mg) of sample and can be operated at high pressure under different atmospheres (N2, He, H2, etc.). The system has been used in recycling lower grade post-consumer polymers, as coloured polyethylene and polystyrene or used rubber tyres, by co-processing with coal. The main characteristics of non-simultaneous methods, TG-GC and techniques requiring a cold trap, are given in Table 1.2. 

Section 5.2 describes simultaneous methods of optimization. In these methods all experiments are performed according to a pre-planned schedule. After all the experiments have been performed, the optimum is located. 

The simplest form of a simultaneous method follows a path in figure 5.4 that can be characterized by 1011. The term grid search forms an illustrative description of such a... 

In the following two sections we will describe two kinds of interpretive methods. In section 5.5.1 we will discuss simultaneous methods, which involve a fixed experimental design. In the iterative procedures of section 5.5.2, an initial design that consists of a minimum number of experiments is used and the location of the next data point is calculated during the optimization process. 

Method Simultaneous methods without solute recognition Includes Grid search ... 

Simultaneous methods (without solute recognition) may be used for selectivity optimization in chromatography, but require large numbers of experiments. 

Figure 5 shows the influence of IPN component ratio on Tg. Each sample was of IPN 1 type, prepared by the simultaneous method. Increasing the amount of 23G lowered the Tg of the IPN. This result also suggests that these IPNs show a good ability to mix. The Tg of IPN containing 70% of 23G was shown to be about 0 C, just like IPN 2 synthesized by the sequential method. The structure of 70%-23G-IPN can be considered to be cured phenolic resol in a continuous poly 23G phase. So, the loss tangent peak of IPN 2 at 0°C was attributed to the structure of 23G rich IPN such as 70%-23G-IPN. 

Sprayer interface (hence not compatible with ultrafast chromatography). The MUX technique was validated for rabbit, rat, mouse, and dog plasma, and the authors concluded that the technique is well-suited for simultaneous method validations and early discovery studies. 

FIGURE 13.5 Two preparation routes used in making ceramic-metal nanocomposites. Method (a) solution mixing of all components simultaneously. Method (b) uses a premade sol to which a further solution is added before gelation. (From Roy, R.A., and Roy, R., Mater. Res. Bull, 19, 169, 1984. With permission.)... 

I should add, however, that the simultaneous method is equivalent to a procedure that regards current value as the sum of an infinite series of labour inputs in the past. Toavoid complex matrix algebra, 1 shall show this fora simple one-sector model in which corn and labourare used to produce com. Specifically, we assumed thataunitsofseedcomanddunitsof labour are needed to produce one unit of com. Setting x for the unknown labour value of one unit of com, we immediately have... 

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