Differential equations from mass conservation

If the system is closed, then the transport fluxes T and P, are zero and rf([A [B]) = 0, and the total mass of reactants in state A plus reactants in state B in the system remains constant. 

Here the symbol T has been used to denote mass flux in units of mass per unit time. Throughout this book we use the symbol J to denote a chemical concentration flux for constant volume systems. Chemical flux J has units of concentration per unit time. Writing Equations (3.2) through (3.4) in terms of chemical fluxes, we have 

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A major criticism of the stochastic probability approach is that relatively slow secondary reactions, for which the near-equilibrium assumption does not apply, cannot be accommodated. In this situation, it is necessary to derive and solve simultaneous partial differential equations for mass conservation and obtain expressions for the first and second moments of the elution profile and the concomitant plate height arising from slow kinetics of secondary equilibrium. If, once again, the process can be represented as involving the reversible binding of two forms, the resolution of the interconverting species can be given by [59]... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Theories are not used directly, as in the discussion presented in Sect. 3.1, but allow building a mathematical model that describes an experiment in the unambiguous language of mathematics, in terms of variables, constants, and parameters. As an example, when considering the identification of kinetic parameters of chemical reactions from isothermal experiments performed in batch reactors, the relevant equations of mass conservation (presented in Sect. 2.3.1) give a set of ordinary differential equations in the general form... 

The specific heat at constant volume and at constant pressure and Cp, the universal gas constant, R, and the ratio of specific heats, y, are related hy y = CpfCy, and R = Cp — c. For ambient air, Cp = kJ/kg K and a = 1.4. Four additional equations obtained from conservation of mass and energy for each layer are required to complete the equation set. The differential equations for mass in each layer are... 

The formulation step may result in algebraic equations, difference equations, differential equations, integr equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form. 

Because we need to know how long the refined section of the bar is, it is important to describe the ramping up of the compositions in a quantitative way. We can do this by writing a differential equation which describes what happens as the zone moves from some general position x to a new position x + 8x (Fig. 4.4g). For a bar of unit cross-section we can write the mass conservation equation... 

Semibatch or semiflow processes are among the most difficult to analyze from the viewpoint of reactor design because one must deal with an open system under nonsteady-state conditions. Hence the differential equations governing energy and mass conservation are more complex than they would be for the same reaction carried out batchwise or in a continuous flow reactor operating at steady state. 

For steady flow of a gas (at a constant mass flow rate) in a uniform pipe, the pressure, temperature, velocity, density, etc. all vary from point to point along the pipe. The governing equations are the conservation of mass (continuity), conservation of energy, and conservation of momentum, all applied to a differential length of the pipe, as follows. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

We will begin with domain discretization into control volumes. Consider our box used in Chapter 2 to derive the mass transport equation. Now, assume that this box does not become infinitely small and is a control volume of dimensions Ax, Ay, and Az. A similar operation on the entire domain, shown in Figure 7.1, will discretize the domain into control volumes of boxes. Each box is identified by an integer i, j, k), corresponding to the box number in the x-, y-, and z-coordinate system. Our differential domain has become a discrete domain, with each box acting as a complete mixed tank. Then, we will apply our general mass conservation equation from Chapter 2 ... 

A system of conservation equations, whose solution describes the velocity, temperature, and composition fields. These equations usually take the form of partial differential equations that are derived from physical laws governing the conservation of mass, momentum, and energy. 

Differential equations governing the kinetics of chemical reaction systems may be thought of as arising from statements of mass conservation. For example, consider the well mixed system illustrated in Figure 3.1, containing reactants A and B in a dilute system of constant volume, V. 

Takeuchi et al. 7 reported a membrane reactor as a reaction system that provides higher productivity and lower separation cost in chemical reaction processes. In this paper, packed bed catalytic membrane reactor with palladium membrane for SMR reaction has been discussed. The numerical model consists of a full set of partial differential equations derived from conservation of mass, momentum, heat, and chemical species, respectively, with chemical kinetics and appropriate boundary conditions for the problem. The solution of this system was obtained by computational fluid dynamics (CFD). To perform CFD calculations, a commercial solver FLUENT has been used, and the selective permeation through the membrane has been modeled by user-defined functions. The CFD simulation results exhibited the flow distribution in the reactor by inserting a membrane protection tube, in addition to the temperature and concentration distribution in the axial and radial directions in the reactor, as reported in the membrane reactor numerical simulation. On the basis of the simulation results, effects of the flow distribution, concentration polarization, and mass transfer in the packed bed have been evaluated to design a membrane reactor system. 

We have shown how a pointwise DE can be derived by application of the macroscopic principle of mass conservation to a material (control) volume of fluid. In this section, we consider the derivation of differential equations of motion by application of Newton s second law of motion, and its generalization from linear to angular momentum, to the same material control volume. It may be noted that introductory chemical engineering courses in transport phenomena often approach the derivation of these same equations of motion as an application of the conservation of linear and angular momentum applied to a fixed control volume. In my view, this obscures the fact that the equations of motion in fluid mechanics are nothing more than the familiar laws of Newtonian mechanics that are generally introduced in freshman physics. 

To derive the macroscopic transport equations, the conservation Relation [10] and [11] must be converted to differential equations. The main assumption needed is that the mean density and the mean velocity vary slowly in space and in time. Starting from Eq. [10] and [11], the macro dynamic equations describing the large-scale behavior of the lattice gas are obtained by multiple-scale perturbation expansion technique (Frisch et al., 1987). We shall not derive this formalism here. In the continuous limit, Eq. [10] leads to the macro dynamical conservation of mass or Euler equation... 

This line is termed a characteristic, which is a line in the x-t plane on which an ordinary differential equation may be written, in this case dp = 0. From Eqs. (5.4.19), (5.4.20), and the mass conservation relation dpidt = -djidx, it follows that the slope of the upward-propagating characteristic curve, or characteristic speed, is... 

It contains two p E, i) producing terms (either from different energy levels of A, term 1, or from p E, t), term 4) and two consuming terms, in which p E, t) is lost to other energy levels of A (term 2) or to B (term 3). For the population density of B an analogous ME exists. Both populations are coupled by the mass conservation requirement and therefore the set of coupled differential equations contains both species. We can define a new vector p E, t) which contains the populations of both isomers. This leads to the same eigenvalue equation as discussed earlier. 

For incompressible flows, p is constant along the flow, and Dp/Dt = 0. Equation (3.4) can also be obtained from the application of the Law of Mass Conservation on a differential element, in Cartesian coordinates for convenience, as shown in Figure 3.1. 

The DAE system (14.2 through 14.7) represents the system model. Differential equations (14.2) are derived from the application of conservation principles to fundamental quantities the rate of accumulation of a quantity within the boundaries of a system is the difference between the rate at which this quantity enters the system and the rate at which it comes out, plus the rate of its net internal production. In chemical process systems, the fundamental quantities that are being conserved are mass, energy, and momentum, and conservation laws are expressed as balances on these quantities. 

The governing equations for the continuous phase of multiphase flows can be derived from the Navier Stokes equations for single-phase flows. Considering the existence of dispersed particles, a volume-averaging technique is used to develop a set of partial differential equations to describe the mass and momentum conservation of the liquid phase. The continuity equation for the liquid phase can be given as... 

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