Differential balance chemical species

Reactive Organic Chemical Mass Balance (Friedlander). In the original formulation of the CMB receptor model (1) it was recognized that the fractional amounts of various chemical species emitted by a source are not necessarily conserved during the transport of the species to the receptor site. This could occur through both physical (differential dispersion or deposition) or chemical (removal due to atmospheric reactions) processes. This possibility was acknowledged by writing the CMB equations in the form... 

Chromatographic separation is achieved by selective adsorption of chemical species in a packed column. Consider a one-dimensional, isothermal chromatographic column, which is fed with a mixture of m-species at axial velocity v, uniform across the cross-section. Let C be the moles per unit volume concentration of species i in liquid phase, and be the concentration of the species in the moles per unit volume. The local mass balances for each of the species result in the following set of first-order partial differential equations (Rhee et al., 1986) ... 

The conservation of species equations, whether expressed in terms of mass Eq, (2.2-2)) or moles [Eq. 2.2-4)], are entirely equivalent) the choice of which to use is made on the basis of convenience. The steady-state forms of these material balances are universally employnd in the desiga and analysis of chemical processes including separation devices.J 4 Many examples will be encountered in later chaprem of this book. The primary purpose here is to set the stage for the differential balance formulation to follow shortly. 

The derivation of differential mass balances or continuity equations for the components of an element of fluid flowing in a reactor is considered in detail in texts on transport processes (e.g.. Bird et al. [1]). These authors showed that a fairly general form of the continuity equation for a chemical species j reacting in a flowing fluid with varying density, temperature, and composition is ... 

Expressions of the conservation of mass, a particular chemical species, momentum, and energy are fundamental principles which are used in the analysis and design of any separation device. It is appropriate to formulate these laws first without specific rate expressions so that a clear distinction between conservation laws and rate expressions is made. Some of these laws contain a source or generation term, for example, for a particular chemical species, so that the particular quantity is not actually conserved. A conservation law for entropy can also be formulated which contributes to a useful framework for a generalized transport theory. Such a discussion is beyond the scope of this chapter. The conservation expressions are first presented in their macroscopic forms, which are applicable to overall balances on energy, mass, and so on, within a system. However, such macroscopic formulations do not provide the information required to size equiprrwnt. Such analyses usually depend on a differential formulation of the conservation laws which permits consideration of spatial variations of composition, temperature, and so on within a system. 

Under such simplified description of the streamer development, we could model the subsequent evolution of the gas phase in a standard way, using the continuity equations for each chemical species and solving a system of mono-dimensional first-order differential equations easily and quickly tackled by numerical integration (Riccardi, 2000). From a chemical engineering point of view, indeed it means that the model can be formulated as a well-mixed reactor (Benson, 1982). The gas-phase composition in the reactor is determined by the chemical reactions among the reactive species and the transport processes. The time evolution of the concentration of the different N sp>ecies in the gas phase is determined by integrating each balance equation for the density nk of the Id spiecies ... 

Here i = 1, 2,. .., n, if the total number of species in the system is n. This is the general differential equation of balance for species i in molar units in a system in the presence of any chemical reactions involving species i. An alternative form in vector notation is... 

Differential balance equations for some quantity (/ , which could be the concentration of some chemical species, or the x-, y- or -momentum, are given in Appendix 7. A. We wish to give an idea of how finite difference equations are formulated by presenting a finite difference equivalent of the one-dimensional balance equation, Eq. (7.A.2), in its steady state form ... 

A transient chemical species Lavoisier mass balance is performed, which incorporates Equations 19.2 and 19.3. A simple, linear, ordinary differential equation results ... 

The fundamental equations of a pit model are the differential mole balances for chemical species dissolving in the pit. The mole balance for species i is... 

The mathematical description of simultaneous heat and mass transfer and chemical reaction is based on the general conservation laws valid for the mass of each species involved in the reacting system and the enthalpy effects related to the chemical transformation. The basic equations may be derived by balancing the amount of mass or heat transported per unit of time into and out of a given differential volume element (the control volume) together with the generation or consumption of the respective quantity within the control volume over the same period of time. The sum of these terms is equivalent to the rate of accumulation within the control volume ... 

The concentration of any of these species depends on the total concentration of dissolved aluminum and on the pH, and this makes the system complex from the mathematical point of view and consequently, difficult to solve. To simplify the calculations, mass balances were applied only to a unique aluminum species (the total dissolved aluminum, TDA, instead of the several species considered) and to hydroxyl and protons. For each time step (of the differential equations-solving method), the different aluminum species and the resulting proton and hydroxyl concentration in each zone were recalculated using a pseudoequilibrium approach. To do this, the equilibrium equations (4.64)-(4.71), and the charge (4.72), the aluminum (4.73), and inorganic carbon (IC) balances (4.74) were considered in each zone (anodic, cathodic, and chemical), and a nonlinear iterative procedure (based on an optimization method) was applied to satisfy simultaneously all the equilibrium constants. In these equations (4.64)-(4.74), subindex z stands for the three zones in which the electrochemical reactor is divided (anodic, cathodic, and chemical). 

Employing a high recirculating flow rate in this small laboratory reactor, the following assumptions can be used (i) there is a differential conversion per pass in the reactor, (ii) the system is perfectly stirred, (iii) there are no mass transport limitations. Also, it can be assumed that (iv) the chemical reaction occurs only at the solid-liquid interface (Minero et al., 1992) and (v) direct photolysis is neglected (Satuf et al., 2007a). As a result, the mass balance for the species i in the system takes the following form (Cassano and Alfano, 2000) ... 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

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