Mass balance-based differential equations

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

To transfer these equations into mass balances based on concentrations, it is necessary to introduce different volumes. The overall differential volume dVc is the sum of the mobile dVint and the stationary phase d Vads- By means of the void fraction (Eq. 2.6) and the cross section A, of the column those are calculated as ... 

To transfer these equations into mass balances based on concentrations, it is necessary to introduce characteristic volumes. The overall differential volume dV/ is... 

Thus, the initial value of the initiator concentrations, [Il]° and [I2]°, are calculated with Equation 15, for given values of the initial loading, feed rates, temperature, and time for the main semi-batch step, and [M]° is fixed according to experimental data from the base case semi-batch step. The nonlinear differential equation for [M] in terms of [II] and [I2] is given by Equation 16. Equation 10, with a redefinition of terms, is the differential equation mass balance for [II] and [12]. In the finishing step, only one of the initiators would be added for residual monomer reduction. Thus, Qm = 0,... 

Based on this configuration, the reformer and combustor are modeled with partial differential equations. Since the thickness of the plates is relatively small, only the flow direction is considered. Using the equation of continuity, the component mass balances are constructed and the energy balance considering with heat loss and momentum balance are established as follows. 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

Although simple, the conceptual base of the method should be discussed. Consider a stage P somewhere in a column of stages. In general, for a vapor-liquid process, a vapor stream would enter the stage from stage P — 1 and a liquid would enter from stage P -t- 1. The differential equation for the mass balance on component i would be... 

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

The model-based controller-observer scheme requires to solve online the system of differential equations of the observer. The phenol-formaldehyde reaction model is characterized by 15 differential equations, and it is, thus, unsuitable for online computations. To overcome this problem, one of the reduced models developed in Sect. 3.8.1 can be adopted. In order to be consistent with the general form of nonchain reactions (2.27) adopted to develop the controller-observer scheme, the reduced model (3.57) with first-order kinetics has been used to design the observer. The mass balances of the reduced model are given by... 

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). 

The strategy for determining the differential equations for biochemical reactants, pH, and binding ions is to express the equations for reactants based on the stoichiometry of the reference reactions and to determine the kinetics of pH and binding ions based on mass balance. 

In 1987 Valdes104 developed a model for composite deposition at a RDE taking into account the various ways in which a particle is transported to the cathode surface. As starting point an equation of continuity for the particle number concentration, C p, based on a differential mass balance was chosen, that is ... 

Clearly, at one extreme—when q(a) is zero throughout the reactor and we have a general j a)—we have the equations for a segregated-flow model. On the other extreme—when (a) is a Dirac delta exactly at one point and we have a general nonzero q oi)—this model reduces to the Zwietering (1959) model of maximum mixedness. Also, we define Q(a) as the flow of molecules at point a. Based on this nomenclature, a differential mass balance on an element Aa leads to... 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

The transport approach has been used very early, and most extensively, to calculate the chromatographic response to a given input function (injection condition). This approach is based on the use of an equation of motion. In this method, we search for the mathematical solution of the set of partial differential equations describing the chromatographic process, or rather the differential mass balance of the solute in a slice of column and its kinetics of mass transfer in the column. Various mathematical models have been developed to describe the chromatographic process. The most important of these models are the equilibrium-dispersive (ED) model, the lumped kinetic model, and the general rate model (GRM) of chromatography. We discuss these three models successively. 

When the temperature is not constant, the bulk heat transfer equation complements the system and involves Equations 5.240, 5.241, and 5.276. The heat transfer equation is a special case of the energy balance equation. It should be noted that more than 20 various forms of the overall differential energy balance for multicomponent systems are available in the literature." " The corresponding boundary condition can be obtained as an interfacial energy balance." - Based on the derivation of the buUc and interfaciaT entropy inequalities (using the Onsager theory), various constitutive equations for the thermodynamic mass, heat, and stress fluxes have been obtained. 

Based on mass balances, a reaction network can be described in terms of linear equations, with parameters representing the stoichiometric coefficients and variables, the metabolic fluxes. The result is a system of ordinary differential equations,... 

The mathematical model for the mass transfer of an adsorbate in the LC column packed with the silicalite crystal particles is based on the assumptions of (1) axial—dispersed plug—flow for the mobile phase with a constant interstitial flow velocity (2) Fickian diffusion in the silicalite crystal pore with an intracrystalline diffus— ivity independent of concentration and pressure and (3) spherical silicalite crystal particles with a uniform particle size distribution. A detailed discussion of these assumptions can be found in (13). The differential mass balances over an element of the LC column and silicalite crystal result in the following two partial differential equations ... 

Various other ways of characterizing the role of mass transfer in PTC systems have been reported. For example, based on the two-film theory, Chen et al.(1991) derived algebraic expressions for the interphase flux of QY and QX. Nonlinear differential equations described the slow reaction in the organic phase, and coupled algebraic equations described the dissociation equilibria in the aqueous phase and the species mass balance. Model parameters were estimated... 

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