Exponential Nonlinearity

In some models, particularly those from chemical engineering applications, the nonlinearity in the heat balance and mass balance equations contain exponential terms. Examples of exponential nonlinearity are, 

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The analysis of steady-state multiplicity of a nonisothermal chemical reactor is complicated due to the number of parameters involved and the exponential nonlinearity in the temperature dependence of the kinetic function. In this and the next subsection, the results of the analysis of steady-state multiplicity are presented. The derivations of these results are detailed in a review chapter by Morbidelli et al. (1986). 

For nonisothermal processes, exponential nonlinearity, expressed through dependence of rate and equihbrium constants on temperature, results in appearance of thermo-kinetic oscillations in temperature and concentrations. 

The applications of this simple measure of surface adsorbate coverage have been quite widespread and diverse. It has been possible, for example, to measure adsorption isothemis in many systems. From these measurements, one may obtain important infomiation such as the adsorption free energy, A G° = -RTln(K ) [21]. One can also monitor tire kinetics of adsorption and desorption to obtain rates. In conjunction with temperature-dependent data, one may frirther infer activation energies and pre-exponential factors [73, 74]. Knowledge of such kinetic parameters is useful for teclmological applications, such as semiconductor growth and synthesis of chemical compounds [75]. Second-order nonlinear optics may also play a role in the investigation of physical kinetics, such as the rates and mechanisms of transport processes across interfaces [76]. 

For those pesticides which are utilized as microbial growth substrates, sigmoidal rates of biodegradation are frequentiy observed (see Fig. 2). Sigmoidal data are more difficult to summarize than exponential (first-order) data because of their inherent nonlinearity. Sigmoidal rates of pesticide metabohsm can be described using microbial growth kinetics (Monod) however, four kinetics constants are required. Consequentiy, it is more difficult to predict the persistence of these pesticides in the environment. 

The relaxation time r of the mean length, = 2A Loo, gives a measure of the microscopic breaking rate k. In Fig. 16 the relaxation of the average length (L) with time after a quench from initial temperature Lq = 1.0 to a series of lower temperatures (those shown on the plot are = 0.35,0.37, and 0.40) is compared to the analytical result, Eq. (24). Despite some statistical fluctuations at late times after the quench it is evident from Fig. 16 that predictions (Eq. (24)) and measurements practically coincide. In the inset is also shown the reverse L-jump from Tq = 0.35 to = 1.00. Clearly, the relaxation in this case is much ( 20 times) faster and is also well reproduced by the non-exponential law, Eq. (24). In the absence of laboratory investigations so far, this appears the only unambiguous confirmation for the nonlinear relaxation of GM after a T-quench. 

A numerical study of the MMEP kinetics, as described by the system of nonlinear differential equations (26), subject to mass conservation (Eq. (27)), has been carried out [64] for a total number of 1000 monomers and different initial MWDs. As expected, and in contrast to the case of wormlike micelles, it has been found that during relaxation to a new equilibrium state the temporal MWD does not preserve its exponential form. 

It turns out that a rather simple description of this nonlinear relaxation in terms of a single relaxation time,, depending on the final average chain length Loo, is suggested by a scaling plot of L t) for different L o, as shown in Fig. 18 for an initial exponential MWD. It is evident from Fig. 18 that the response curves, L o — L t), for different L o may be collapsed onto a single master curve, 1 - L t)lLoo = /(V Loo) measured in units of a... 

This shows that p (energy) increases exponentially at a fixed value of the phase Mathieu equation (6-127). Omitting the intermediate calculations (6-128) and (6-129), and taking instead of (6-130) the series solution in the form... 

Hicks (H6) and Frazer and Hicks (F3) considered the ignition model in which exothermic, exponentially temperature-dependent reactions occur within the solid phase. Assuming a uniformly mixed solid phase, the one-dimensional unsteady heat-flow equation relates the propellant temperature, depth from the surface, and time by the nonlinear equation ... 

The principal difficulty with these equations arises from the nonlinear term cb. Because of the exponential dependence of cb on temperature, these equations can be solved only by numerical methods. Nachbar has circumvented this difficulty by assuming very fast gas-phase reactions, and has thus obtained preliminary solutions to the mathematical model. He has also examined the implications of the two-temperature approach. Upon careful examination of the equations, he has shown that the model predicts that the slabs having the slowest regression rate will protrude above the material having the faster decomposition rate. The resulting surface then becomes one of alternate hills and valleys. The depth of each valley is then determined by the rate of the fast pyrolysis reaction relative to the slower reaction. 

When combining Eqs. (A.l) and (A.4), we obtain a second-order nonlinear differential equation for /o(r) which is mathematically very difficult to solve. Therefore, in DH theory a simplified equation is used The exponential terms of Eq. (A.4) are expanded into series and only the first two terms of each series are retained [exp(y) 1 +y]. When we include the condition of electroneutrality and use the ionic strength we can write this equation as... 

Numerical techniques are iterative and require considerable computer processing power. With modern desktop computers, this is usually not an issue and solutions of root uptake over days or weeks typically take a few seconds to generate. However, for some strongly nonlinear problems, such as the development of rhizosphere microbial populations (Sect. Ill), where the increase in microbial biomass may be exponential over time, processing time may become important with solutions requiring >60 min to calculate on a modern PC. 

Watts (1994) dealt with the issue of confidence interval estimation when estimating parameters in nonlinear models. He proceeded with the reformulation of Equation 16.19 because the pre-exponential parameter estimates "behaved highly nonlinearly." The rate constants were formulated as follows... 

Because the term r(CA T) is exponentially dependent on T and can be nonlinear as well, a numerical solution or piecewise linearization must be used. To simplify the numerical manipulations, equations in Table IX are normalized by = z/L, r = ut/L, and jc = 1 - C,/(C,)0, where i is normally S02. y also is a normalized quantity. The Peclet numbers for mass and heat are written PeM = 2Rpu/D) and PeH = 2Rpcpul t for a spherical particle. They are also written in terms of bed length as Bodenstein numbers. It is... 

Since the reaction rate constant appearing in equations 12.3.100 and 12.3.104 depends exponentially on temperature, these equations are coupled in a nonlinear fashion and cannot be considered independently. 

This observation is expected from theory, as the observed thickness distributions are exactly the functions by which one-dimensional short-range order is theoretically described in early literature models (Zernike and Prins [116] J. J. Hermans [128]). From the transformed experimental data we can determine, whether the principal thickness distributions are symmetrical or asymmetrical, whether they should be modeled by Gaussians, gamma distributions, truncated exponentials, or other analytical functions. Finally only a model that describes the arrangement of domains is missing - i.e., how the higher thickness distributions are computed from two principal thickness distributions (cf. Sect. 8.7). Experimental data are fitted by means of such models. Unsuitable models are sorted out by insufficient quality of the fit. Fit quality is assessed by means of the tools of nonlinear regression (Chap. 11). 

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

As introduced in sections 3.1.3 and 4.2.3, the Arrhenius equation is the normal means of representing the effect of T on rate of reaction, through the dependence of the rate constant k on T. This equation contains two parameters, A and EA, which are usually stipulated to be independent of T. Values of A and EA can be established from a minimum of two measurements of A at two temperatures. However, more than two results are required to establish the validity of the equation, and the values of A and EA are then obtained by parameter estimation from several results. The linear form of equation 3.1-7 may be used for this purpose, either graphically or (better) by linear regression. Alternatively, the exponential form of equation 3.1-8 may be used in conjunction with nonlinear regression (Section 3.5). Seme values are given in Table 4.2. 

Equations 8.5-34 and -35 are nonlinearly coupled through T, since kA depends exponentially on T. The equations cannot therefore be treated independently, and there is no exact analytical solution for cA(r) and T(r). A numerical or approximate analytical solution results in tj expressed in terms of three dimensionless parameters ... 

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