Balancing equations using models

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... 

The model of the distillation column used throughout the paper is developed by [10], composed by the mass, component mass and enthalpy balance equations used as basis to implement a SIMULINK model (figure 1) which describes the nonlinear column dynamics as a 2 inputs Q, Lvo/) and 2 output (Ad, Ab ). Implementations details for the overall column dynamics are given in [11]. 

Marchisio, D. L. 2009 On the use of bi-variate population balance equations for modelling barium titanate nanoparticle precipitation. Chemical Engineering Science 64, 697-708. 

When we think we have finished the balancing, we should always do a complete check that the total number of atoms for each of the elements on the reactant side matches that found on the product side of the equation. A molecular view of this balanced equation using ball-and-stick models is shown next (element color coding black = C, red = O, white = H). 

A numerical heat transfer model of thin fibrous materials under high heat flux eonditions (bench-top burner) was developed by Torvi and Dale [37]. The model is applicable to two common, flame resistant fabrics, Nomex IIIA and Kevlar /PBI. A fabric-air gap-test sensor system (Figure 12.4) is used in which heat transfer is assmned to be one-dimensional. The fabric s thermal properties represent the average thermal property values of the fibrous stmcture. Mass transfer, hot gas flow and fabrie stmctural changes are not considered. The fabric s thermal properties are taken as fimetions of temperature only. The authors use energy balance equations and models of heat transfer modes to develop a differential equation (equation 12.26), and initial and boimdary conditions ... 

Herbst et al. [International J. Mineral Proce.ssing, 22, 273-296 (1988)] describe the software modules in an optimum controller for a grinding circuit. The process model can be an empirical model as some authors have used. A phenomenological model can give more accurate predictions, and can be extrapolated, for example from pilot-to full-scale apphcation, if scale-up rules are known. Normally the model is a variant of the popiilation balance equations given in the previous section. 

Consider the mass, thermal and momentum balance equations. The key assumption of the present analysis is that the Knudsen number of the flow in the capillary is sufficiently small. This allows one to use the continuum model for each phase. Due to the moderate flow velocity, the effects of compressibility of the phases, as well as mechanical energy, dissipation in the phases are negligible. Assuming that thermal conductivity and viscosity of vapor and liquid are independent of temperature and pressure, we arrive at the following equations ... 

Using models in learning about ehemieal equations has proved a successful tool, espeeially by students with diffieulties in eoneept understanding. By eounting the number of atoms in partiele representations they better understood the meaning of a balanced equation. Some students still have problems with balaneing chemical equations when models are not avail-... 

The basic scheme for the numerical solution is the same as that used for the 1 -D model, except that in this case the solid temperature field used to solve the DAE system for each monolith channel must be calculated from the three-dimensional solid-phase energy balance equation. The three-dimensional energy balance equation can be solved by a nonlinear finite element solver (such as ABAQUS) for the solid-phase temperature field while a nonlinear finite difference solver for the DAE system calculates the gas-phase temperature and... 

The specification of the initial state may be partial and lead to incomplete descriptions of various segments of the plant. The modeling facilities of MODEL.LA. contain a complete set of the balance equations, phase and chemical equilibrium, and rate relationships. These relationships are used to propagate the user-supplied specifications and thus complete the description of the initial state throughout the plant. 

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

A similar finite-differenced equivalent for the energy balance equation (including axial dispersion effects) may be derived. The simulation example DISRET involves the axial dispersion of both mass and energy and is based on the work of Ramirez (1976). A related model without reaction is used in the simulation example FILTWASH. 

As shown in this chapter, in the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaeed by their finite-differenced equivalents. These finite-differenced forms of the model equations are shown to evolve as a natural eonsequence of the balance equations, according to the manner of Franks (1967). The approximation of the gradients involved may be improved, if necessary, by using higher order approximations. Forward and end sections can... 

The derivation method of the model equations for extraction columns with backmixing are explained in Sec. 4.4.3. Here the balances are easier to formulate because concentrations are used. The form of the component balance equations in terms of concentrations are as follows. 

Although the Lewis cell was introduced over 50 years ago, and has several drawbacks, it is still used widely to study liquid-liquid interfacial kinetics, due to its simplicity and the adaptable nature of the experimental setup. For example, it was used recently to study the hydrolysis kinetics of -butyl acetate in the presence of a phase transfer catalyst [21]. Modeling of the system involved solving mass balance equations for coupled mass transfer and reactions for all of the species involved. Further recent applications of modified Lewis cells have focused on stripping-extraction kinetics [22-24], uncatalyzed hydrolysis [25,26], and partitioning kinetics [27]. 

Evolution of the component concentrations in the two phases of the bioreactor are modeled by using an iteractive program (Runge-Kutta). All the kinetic equations and the experimentally determined constants are introduced into the program. The mass balance equation is included. 

This section describes the continuous flux melting model used in Bourdon et al. (2003) and has many similarities with the model of Thomas et al. (2002). A significant difference is that the model described here keeps track of the composition of the slab as it dehydrates. This model is based on mass balance equations for both the mantle wedge and the slab. We assume secular equilibrium in the U-series decay chain initially ... 

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