Linearisation

The linearised Poisson-Boltzmann equation is obtained by taking only the first term in the expansion, giving ... 

In some cases, however, it is possible, by analysing the equations of motion, to determine the criteria by which one flow pattern becomes unstable in favor of another. The mathematical technique used most often is linearised stabiHty analysis, which starts from a known solution to the equations and then determines whether a small perturbation superimposed on this solution grows or decays as time passes. 

Guha [5] pointed out some limitations in the linearised analyses developed by Horlock and Woods to determine the changes in optimum conditions with the three parameters n (and n ),/ and Not only is the accurate determination of (Cpg)i3 (and hence n ) important but also the fuel-air ratio although small, it cannot be assumed to be a constant as r is varied. Guha presented more accurate analyses of how the optimum conditions are changed with the introduction of specific heat variations with temperature and with the fuel-air ratio. 

An example of a modem instrument of this type is the Coming Model 410 flame photometer. This model can incorporate a lineariser module which provides a direct concentration read-out for a range of clinical specimens. Flame photometers are still widely used especially for the determination of alkali metals in body fluids, but are now being replaced in clinical laboratories by ion-selective electrode procedures (see Section 15.7). 

At a high substrate concentration, die rate can be simplified and a linearised model is obtained ... 

In evaluation of kinetic parameters, the double reciprocal method is used for linearisation of the Michaelis-Menten equation (5.7.3). 

A linearisation model is used to explain the equation of a simple straight line 9... 

A.C. McIntosh. The linearised response of the mass burning rate of a premixed flame to rapid pressure changes. Combustion Science and Technology, 91 329-346, 1993. 

Branched arabinans with a backbone of (l->5)-linked a-L-arabinofuranosyl residues are present in pectic fractions of apples. During ripening a slow linearisation of the arabinans occurs. 

System stability ean also be analysed in terms of the linearised differential model equations. In this, new perturbation variables for concentration C and temperature T are defined. These are defined in terms of small deviations in... 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

The two linearised model equations have the general solution of the form... 

The types of system behaviour predicted, by the above analysis are depicted in Figs. 3.16 and 3.17. The phase-plane plots of Fig. 3.17 give the relation of the dependant variables C and T. Detained explanation of phase-plane plots is given in control textbooks (e.g., Stephanopoulos, 1984). Linearisation of the reactor model equations is used in the simulation example, HOMPOLY. 

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples, THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG and STABIL. 

Based on a linearisation approach as applied to the model equations, Clough and Ramirez (1971) predict multiple, steady-state solutions for the reactor. Is this confirmed and if not, why not ... 

Likelihood or probability density 114 Limit cycles 155 Linearisation 154 Liquid... 

The linearisation method of Naphtali and Sandholm has been used by Fredenslund et al. (1977) for the multicomponent distillation program given in their book. Included in then-book, and coupled to the distillation program, are methods for estimation of the liquid-vapour relationships (activity coefficients) using the UNIFAC method (see Chapter 8, Section 16.3). This makes the program particularly useful for the design of columns for... 

Due to the fact that the outlet concentration of each contaminant is not at its maximum, the models derived each take the form of an MINLP. The nonlinearities are linearised using the relaxation-linearisation technique proposed by Quesada and Grossman (1995). The linearised model that results is used to generate an initial solution for the exact non-linear model. 

One would notice that there are a number of nonlinear terms in the above constraints, specifically in the contaminant balance constraints. The linearisation technique used to remove these nonlinearities is that proposed by Quesada and Grossman (1995), the general form of this linearization technique can be found in Appendix A. During the application of the model to the illustrative examples,... 

The methodology takes the form of an MINLP, which must be linearised to find a solution. The linearization method used was the relaxation-linearization technique proposed by Quesada and Grossman (1995). During the application of the formulation to the illustrative examples it was found that only one term required linearization for a solution to be found. 

The first minor change to the mass balance constraints from the scheduling formulation is found in constraint (8.2), which defines the size of a batch. In the synthesis formulation, the batch size is determined by the optimal size of the processing unit. Due to this being a variable, constraint (8.2) is reformulated to reflect this and is given in constraint (8.59). The nonlinearity present in constraint (8.59) is linearised exactly using Glover transformation (1975) as presented in Chapter 4. 

The application of the maximum outlet concentration condition (Savelski and Bagajewicz, 2000) allows for the simplification of constraint (9.5) and subsequent linearisation of two nonlinear terms present in the resulting constraint. This is done as follows. 

There are two forms of nonlinearities in constraint (9.77). The first comprises of a continuous variable and a binary variable and the second comprises of two continuous variables. The first nonlinearity can be linearised exactly using a Glover transformation (1975) and the second can be linearised using a relaxation-linearization technique proposed by Quesada and Grossman (1995), where necessary. 

A note must be made on the solution time of the second case in the second illustrative example. The solution time is excessive, approximately 900 CPU seconds. The long solution time is due to the solution procedure used. The solution time for the MILP accounted for more than 99% of the total solution time. Shorter solution times might have been achieved if a different solution procedure had been followed, e.g. only partially linearising the MINLP. However, the final solution found is globally optimal, which justifies the usage of the solution procedure. 

Sharp [48] has described a dry combustion-direct injection system built for oceanographic analyses. This unit used 100 xl samples, injected into a 900 °C oven in an atmosphere of oxygen. The output from a non-dispersive infrared carbon dioxide analyser was linearised and integrated. 

This is a non-linear relationship. Linearising using Taylor s series, as given in equation 7.24, Volume 3 ... 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

A further simplification (known as the Leveque approximation [57]), linearises the profile in the vicinity of the surface (where a local coordinate x, perpendicular to the surface, is taken) ... 

Obviously, this approximate treatment fails when <5, > R, as then the Leveque linearisation is clearly unapplicable. For an approximation to account for lateral effects of nonactive walls in box-like channels, see ref. [46]. 

Figure 5b shows the resulting steady-state flux. / s (obtained as the positive solution for c f from equation (23)) for a range of c"M values. At low c"M values (usually associated with low values), there is a linear dependence between and r M, as expected from the linearisation of the Langmuir isotherms (see equation (31), below). At large c M values, the usual Michaelis-Menten saturating effect of is also seen. 

Fe in marine water), known as the excess of ligand case, which allows linearisation of the problem via ... 

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