Numerical methods reaction with temperature

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. 

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

Assessing the dependence of rate on concentration from the point of view of the rate law involves determining values, from experimental data, of the concentration parameters in equation 4.1-3 the order of reaction with respect to each reactant and the rate constant at a particular temperature. Some experimental methods have been described in Chapter 3, along with some consequences for various orders. In this section, we consider these determinations further, treating different orders in turn to obtain numerical values, as illustrated by examples. 

ROP of p-lactones is highly prone to numerous side reactions, such as transester-fication, chain-transfer or multiple hydrogen transfer reactions (proton or hydride). Specifically, the latter often causes unwanted functionalities such as crotonate and results in loss over molecular weight control. Above all, backbiting decreases chain length, yielding macrocyclic structures. All these undesired influences are dependent on the reaction conditions such as applied initiator or catalyst, temperature, solvent, or concentration. The easiest way to suppress these side reactions is the coordination of the reactive group to a Lewis acid in conjunction with mild conditions [71]. p-BL can be polymerized cationically and enzymatically but, due to the mentioned facts, the coordinative insertion mechanism is the most favorable. Whereas cationic and enzymatic mechanisms share common mechanistic characteristics, the latter method offers not only the possibility to influence... 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

Unfortunately, the exponential temperature term exp(- E/RT) is rather troublesome to handle mathematically, both by analytical methods and numerical techniques. In reactor design this means that calculations for reactors which are not operated isothermally tend to become complicated. In a few cases, useful results can be obtained by abandoning the exponential term altogether and substituting a linear variation of reaction rate with temperature, but this approach is quite inadequate unless the temperature range is very small. 

If the reaction rate is a function of pressure, then the momentum balance is considered along with the mass and energy balance equations. Both Equations 6-105 and 6-106 are coupled and highly nonlinear because of the effect of temperature on the reaction rate. Numerical methods of solution involving the use of finite difference are generally adopted. A review of the partial differential equation employing the finite difference method is illustrated in Appendix D. Figures 6-16 and 6-17, respectively, show typical profiles of an exothermic catalytic reaction. 

A combined analytical and numerical method is employed to optimize process conditions for composites fiber coating by chemical vapor infiltration (CVI). For a first-order deposition reaction, the optimum pressure yielding the maximum deposition rate at a preform center is obtained in closed form and is found to depend only on the activation energy of the deposition reaction, the characteristic pore size, and properties of the reactant and product gases. It does not depend on the preform specific surface area, effective diffusivity or preform thickness, nor on the gas-phase yield of the deposition reaction. Further, this optimum pressure is unaltered by the additional constraint of prescribed deposition uniformity. Optimum temperatures are obtained using an analytical expression for the optimum value along with numerical... 

To calculate the amount of catalyst for a particular case, mass and heat balance have to be considered they can be described by two differential equations one gives the differential CO conversion for a differential mass of catalyst, and the other the associated differential temperature increase. As analytical integration is not possible, numerical methods have to be used for which today a number of computer programs are available with which the calculations can be performed on a powerful PC in the case of shift conversion. Thus the elaborate stepwise and graphical evaluation by hand [592], [609] is history. For the reaction rate r in these equations one of the kinetic expressions discussed above (for example, Eq. 83) together with the function of the temperature dependence of the rate constant has to be used. 

Numerous methods have been described for the preparation of niobium(V) chloride, among them the reaction of niobium(V) oxide with thionyl chloride in a sealed system. In such a procedure some niobium(V) oxide trichloride, NbOCls, is almost always formed, and it is difficult to obtain the pentachloride completely free from this impurity, even by repeated sublimation. The simple, efficient method described here consists in allowing hydrous niobium(V) oxide to react with thionyl chloride at room temperature. Almost quantitative conversion is observed, the pentachloride dissolving in the thionyl chloride, from which it may be recovered, free of oxide trichloride, by vacuum evaporation... 

In order to find the evolution of species concentration or temperature with time, the above equations must be integrated. For complex reaction mechanisms this usually means integration by numerical methods. There are a large number of schemes for the numerical integration of coupled sets of differential equations, but not all will be suitable for the types of mechanisms we are discussing. Chemical systems form a difficult problem because of the differences in reaction time-scale between each of the... 

Historically, phospholanium salts have been prepared by quaternization in high yield of a selected phospholane with an alkyl halide. Although numerous methods exist for the preparation of the desired phospholanes (recently reviewed ), most of these have major drawbacks, including (1) very critical conditions, such as reaction time, dilution effects, and temperature (2) expensive and/or difficult to manipulate reagents and (3) the necessity of a multistep synthetic sequence giving overall low yields of phospholane. A general example of the latter would be the reaction of a substituted dihalophosphine with a 1,3-diene and hydrolysis to the phospholene oxide, which then can be catalytically reduced to the phospholane oxide and subsequently converted to the phospholane. ... 

The purpose o this work is to show that the values for temperatures o mold and storage bulb are important parameters in injection molding process o rubber. The study was performed by using two different thicknesses for rubber sheets and two values for the cure enthalpy. The problem was solved by applying an explicit numerical method with finite differences (6). Although rubber vulcanization consists of a complex series of reactions (8), the overall cure heat could be described by a first-order reaction with a single activation energy, as shown previously (6). 

Accordingly, the Arrhenius equation should yield a straight line of slope —EJR and intercept A if n k is plotted against 1/71 Implicit in this statement is the assumption that E is constant over the temperature range in question. Despite the fact that E generally varies significantly with temperature, the Arrhenius equation has wide applicability in industry. This method of analysis can be used to test the rate law, describe the variation of k with T, and/or evaluate E. The numerical value of E will depend on the choice and units of the reaction velocity constant. 

This method has been used for the preparation of numerous amides. However, with many esters it is necessary to heat the reaction mixture to 200-250° for a few hours. Ethyl mandelate is like ethyl lactate in that it gives a good yield (75-80 per cent of the theoretical amount) of mandelamide at room temperatures. 

The temperature, T, is a function of radial position, r, /Cg is an effective thermal conductivity, and -AHrxn is the heat of reaction. Again, there is no flux at the center, and the temperature at the outer boundary is set at Tr. Now the reaction rate depends upon both concentration and temperature [and Eq. (9.3) has to be written with R(c, T)]. Such problems can be very difficult owing to the rapid change in the kinetic constants with temperature, and they provide a severe test of any numerical method. 

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