Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Infinite balanced equations

In principle, an infinite number of balanced equations can be written for any reaction. The equations... [Pg.61]

The Flory principle allows a simple relationship between the rate constants of macromolecular reactions (whose number is infinite) with the corresponding rate constants of elementary reactions. According to this principle all chemically identical reactive centers are kinetically indistinguishable, so that the rate constant of the reaction between any two molecules is proportional to that of the elementary reaction between their reactive centers and to the numbers of these centers in reacting molecules. Therefore, the material balance equations will comprise as kinetic parameters the rate constants of only elementary reactions whose number is normally rather small. [Pg.170]

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. [Pg.173]

This leaves (n — 1) independent flux balance equations. The rate v can be found straightforwardly for a binary system as the composition of a and 7 are fixed at any temperature. In a ternary system there is a frirther degree of freedom as the number of thermodynamically possible tie-lines between the a and 7 phases is infinitely large. However, each tie-line may be specified uniquely by the chemical potential of one of the three components and thus there are only two unknowns and two equations to solve. The above approach can be generalised for multi-component systems and forms the platform for the DICTRA software package. [Pg.452]

Now we have to try to solve the mass-balance equations for each of these species to obtain [M](t), [AMj] r), and [P](r), remembering that j and n can go to infinity. We obviously have to solve an infinite number of equations sequentially in the general case. However, we can make the problem quite simple (but obtain a solution that is not entirely accurate) by setting all kpS and aU k S equal. Now we can play games with summations and make solution of this reaction set a rather straightforward problem. [Pg.456]

This quick test does not, however, tell us that there will be only one stable limit cycle, or give any information about how the oscillatory solutions are born and grow, nor whether there can be oscillations under conditions where the stationary state is stable. We must also be careful in applying this theorem. If we consider the simplified version of our model, with no uncatalysed step, then we know that there is a unique unstable stationary state for all reactant concentrations such that /i < 1. However, if we integrate the mass-balance equations with /i = 0.9, say, we do not find limit cycle behaviour. Instead the concentration of B tends to zero and that for A become infinitely large (growing linearly with time). In fact for all values of fi less than 0.90032, the concentration of A becomes unbounded and so the Poincare-Bendixson theorem does not apply. [Pg.77]

For a perfectly insulated reactor, with no heat loss through the walls, the Newtonian cooling time rN becomes infinite (because x - 0)- The mass- and heat-balance equations become... [Pg.188]

A classic chemical engineering problem of the form under consideration here is that of a non-isothermal reaction occurring in a catalytic particle or packed bed into which a single gaseous participant diffuses from a surrounding reservoir (Hatfield and Aris 1969 Luss and Lee 1970 Aris 1975 Burnell et al. 1983). This scenario is also appropriate to the technologically important problem of spontaneous combustion of stockpiled, often cellulosic, material in air (Bowes 1984). If we represent the concentration of the gaseous species as c, the mass- and heat-balance equations for reaction in an infinite slab are... [Pg.259]

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... [Pg.206]

The balance equations used to model polymer processes have, for the most part, first order derivatives in time, related with transient problems, and first and second order derivatives in space, related with convection and diffusive problems, respectively. Let us take the heat equation over an infinite domain as... [Pg.393]

It has been shown by Giddings [67], Van Deemter et al. [29] and Haarhoff and Van der Linde [68] (see Section 2.2.6) that when the mass transfer kinetics are fast but not infinitely fast, the system of mass balance equations (Eq. 2.2) and kinetic equations (Eq. 2.5) can be replaced by the following equation ... [Pg.47]

The ideal model of chromatography, which has great importance in nonlinear chromatography, has little interest in linear chromatography. Along an infinitely efficient column, with a linear isotherm, the injection profile travels unaltered and the elution profile is the same as the injection profile. We also note here that, because of the profound difference in the formulation of the two models, the solutions of the mass balance equation of chromatography for the ideal, nonlinear model and the nonideal, linear model rely on entirely different mathematical techniques. [Pg.290]

In the ideal model, we assume that the column efficiency is infinite, hence the rate of the mass transfer kinetics is infinite and the axial dispersion coefficient in the mass balance equation (Eq. 2.2) is zero. The differential mass balances for the two components are written ... [Pg.390]

Analogously, it can be shown for the third ideal geometry, an infinitely long cylinder, that the balance equation becomes... [Pg.344]

We have defined the reservoir conditions such that /ij, is infinite and is zero. If we insert these values in Eq. 8.19, we see that both sides are large without bound (i.e., infinite). The reservoir condition, which is the most convenient reference condition for temperature, pressure, and density, is therefore a very poor reference condition for the cross-sectional area we will choose a better one. In any such flow there is or could be a state at which the Mach number is exactly 1, Even if such a state does not exist for the flow in question, pretending that it exists will help us solve the problem. Let us refer to this state as the critical state and denote it by an asterisk. The mass balance equation between some arbitrary state and the critical state is... [Pg.297]

The characteristic scale of the polymerization wave (i.e., the spatial region over which the major variation of the temperature and the species concentrations occurs) is typically much smaller than the length of the tube. Thus, on the scale of the polymerization wave the tube can be considered infinite, — oo < X < 00. It is convenient to introduce a moving coordinate x = x — (fit, y), where (p is the position of a characteristic point of the wave. The specific choice of (p will be described later. Expressed in the moving coordinate system, the mass and energy balance equations become... [Pg.200]

Kinetic approaches represent realistic and comprehensive description of the mechanism of network formation. Under this approach, reaction rates are proportional to the concentration of unreacted functional groups involved in a specific reaction times an associated proportionality constant (the kinetic rate constant). This method can be applied to the examination of different reactor types. It is based on population balances derived from a reaction scheme. An infinite set of mass balance equations will result, one for each polymer chain length present in the reaction system. This leads to ordinary differential or algebraic equations, depending on the reactor type under consideration. This set of equations must be solved to obtain the desired information on polymer distribution, and thus instantaneous and accumulated chain polymer properties can be calculated. In the introductory paragraphs of Section... [Pg.198]

The actual amount of vapor required at individual points in the column can easily be calculated with balance equations by assuming an infinitely high number of plates [Kaibel 1989a]. For a two-component mixture the minimum vapor quyntity G at any position in a distillation column with a contration x of the light boiler in the liquid is given by Equation (2.3.2-38), where the quantities are in moles ... [Pg.124]

The last term of the equation involves the additional entropy caused by different structures of the chain at the same composition. Minimizing the equation is carried out for chains of infinite length taking material balance equations into consideration. In general, if there are q metal ions on the polymer chain containing m active centers, it is necessary to consider the combinatorial analysis of complexes formed having different structures, i.e. the number of combinations (/) of q molecules on m centers of the macroligand t =... [Pg.69]

At the same time, the transformation eliminates the reaction term in the balance equation. The operating line for the rectifying section of a reaction column is formally identical to the operating line of a non-reactive column. An infinite reflux ratio gives an expression that is formally identical to the one for calculating conventional distillation lines [5, 6]. Accordingly, we will refer to lines that have been calculated by this procedure as RD lines. These analogies are found for all the relationships that are important in distillation [7, 8]. [Pg.35]

The popularity of steady-state theories of explosion is easily understood. The neglect of reactant consumption dearly divides the solutions of the heat balance equation into two classes. Bdow a critical heat rdease rate, access temperatures tend to finite, steady values, vdmeas above this critical value temperatures become infinite in finite times. It is this topological distinction betwerai submtical and supncritical solutions whidi ensures the existenoe of critical conditions. [Pg.366]

We consider again the classical problem of mass or heat transfer in a semi-infinite domain. Let us consider a mass transfer problem where a gas is dissolved at the interface and then diffuses into liquid of unlimited extent. Ignoring the gas film mass transfer resistance, the mass balance equations in nondimensional form are... [Pg.558]

Adding the above two equations, we obtain the total mass balance equation (9.2-lb), which basically states that the total hold-up in both phases is governed by the net total flux contributed by the two phases. This is true irrespective of the rate of mass exchange between the two phases whether they are finite or infinite. Now back to our finite mass exchange conditions where the governing equations are (9.4-3a) and (9.4-3b). To solve these equations, we need to impose boundary conditions as well as define an initial state for the system. One boundary is at the center of the particle where we have the usual symmetry ... [Pg.582]


See other pages where Infinite balanced equations is mentioned: [Pg.172]    [Pg.155]    [Pg.282]    [Pg.77]    [Pg.179]    [Pg.181]    [Pg.167]    [Pg.683]    [Pg.128]    [Pg.564]    [Pg.77]    [Pg.97]    [Pg.49]    [Pg.652]    [Pg.45]    [Pg.169]    [Pg.466]    [Pg.45]    [Pg.23]    [Pg.300]    [Pg.13]    [Pg.110]    [Pg.629]    [Pg.653]   
See also in sourсe #XX -- [ Pg.154 ]




SEARCH



Balance equation

Mass Balance in an Infinitely Small Control Volume The Advection-Dispersion-Reaction Equation

© 2024 chempedia.info