Mathematical momentum balance

Full mathematical models also include momentum balance equations, which have been omitted here. 

Mathematical modeling is very much an art. It takes experience, practice, and brain power to be a good mathematical modeler. You will see a few models developed in these chapters. You should be able to apply the same approaches to your own process when the need arises. Just remember to always go back to basics mass, energy, and momentum balances applied in their time-varying form. 

Writing mass-, heat-, energy-, and/or momentum-balance equations to obtain the model equations that relate the system input and output to the state variables and the physico-chemical parameters. These mathematical equations describe the state variables with respect to time and/or space. 

Cyclic Steady State is the condition whereby die state at the end of each cycle is identical to that at its beginning. For a non-isotiiermal adsorption systan this may be represented by a mathematical model, comprising material, energy and momentum balances as well as adsorption equilibrium and kinetic models, the CSS can be repressed by ... 

Significant progress is being made in fundamental approaches. The current powerful computational fluid dynamics (CFD) tools (e.g., FLUENT and CFX software)—based on the solution of differential mass and momentum balances—have made it possible to allow simulations of the flow patterns within the crystallizer. Both physical and mathematical modeling add to our knowledge and understanding of the nature of high-concentration suspension flows. 

In this representation, particular emphasis has been placed on a uniform basis for the electron kinetics under different plasma conditions. The main points in this context concern the consistent treatment of the isotropic and anisotropic contributions to the velocity distribution, of the relations between these contributions and the various macroscopic properties of the electrons (such as transport properties, collisional energy- and momentum-transfer rates and rate coefficients), and of the macroscopic particle, power, and momentum balances. Fmthermore, speeial attention has been paid to presenting the basic equations for the kinetie treatment, briefly explaining their mathematical structure, giving some hints as to appropriate boundary and/or initial conditions, and indicating main aspects of a suitable solution approach. 

Basic physical phenomena occurring during a chromatographic separation are described in Chapter 2. A quantitative description is possible using suitable mathematical models, which are typically based on material, energy, and momentum balances, in addition to equations that quantify the thermodynamic equilibria of the distribution of the solutes between the different phases. A good model has to be as... 

Photochemical reactor design involves simultaneous solution of the mass, energy, and momentum balance equations (as in normal reactors) along with equations for the radiation field and energy source (which are specific to photochemical reactors). Two approaches are possible (1) the intensity of the incident light, irrespective of the source, is used as the inlet boundary condition incidence models)-, (2) the emission from the source itself is part of the mathematical description emission models). The first approach has been extensively used but suffers from the weakness that the incident light is a function of scale, and hence a priori design from laboratory scale data tends to be uncertain. The second approach is formally correct, and involves no such uncertainty. 

The momentum balance will not be discussed in more detail here, because the first simulation tests for periodic process control of trickle-bed reactors do not consider the momentum balance. A complete mathematical model for a three-phase reactor would thus be made up of the respective material, heat and momentum balances for the gas phase , for the liquid phase , and for the solid phase (catalyst) , but their complete solution currently encounters major difficulties. 

Practically, mathematical models are based on the conservation laws of mass, energy and momentum, which lead to mass, energy and momentum balances. The balances, together with transport and kinetics equations, form a set of equations (ODE or PDE) whose solution gives the component concentrations, temperature and pressure profiles inside the reactor. Mass and heat transport coefficients, reactants and products physical properties, catalyst efficiency factor and all parameters appearing in model equations have to be expressed. 

In contrast with the mathematical dividing surface, which has an isotropic interfacial tension (7, the forces acting on the material interface are not isotropic. They are characterized by the interfacial stress tensor, a, which is a two-dimensional counterpart of the bulk stress tensor, P. The two-dimensional analogue of the momentum balance equation (1) written in the bulk is called the interfacial momentum balance equation. Note, that the inter-phase exchanges momentum also with the contiguous bulk phases and the corresponding balance equation reads (Slattery 1990, Edwards et al. 1991) ... 

The RD model consists of sets of algebraic and differential equations, which are obtained from the mass, energy and momentum balances performed on each tray, reboiler, condenser, reflux drum and PI controller instances. Additionally, algebraic expressions are included to account for constitutive relations and to estimate physical properties of the components, plate hydraulics and column sizing. Moreover, initial values are included for each state variable. A detailed description of the mathematical model can be found in appendix A. The model is implemented in gPROMS /gOPT and solved using for the DAE a variable time step/variable order Backward Differentiation Formulae (BDF). 

Mathematical analysis of chemical reactors is based on mass, energy, and momentum balances. The main features to be considered in reactor analysis include stoichiometry, thermodynamics and kinetics, mass and heat transfer effects, and flow modeling. The different factors governing the analysis and design of chemical reactors are illustrated in Figure 1.3. In subsequent chapters, all these aspects will be covered. 

It may not be possible to theoretically design a H2 PSA process with such accuracy without using the actual experimental process performance data to fine tune the design model. The reasons are that (i) the practical PSA processes are fairly complex and (ii) the key input data (multi-component adsorption equilibria, kinetics and isosteric heats) for the mathematical design model (integration of coupled partial differential equations describing the mass, the heat, and the momentum balances inside the adsorber) may not be very accurate [27]. The PSA process models often act as amphfiers of errors in the input data. 

Free-radical high-pressure LDPE tubular reactors can be modelled in terms of a system of non-linear differential equations. Over the past twenty years, several modelling studies have been reported in the literature (Thies and Schoenemann I 1Agrawal and Han I 2j, Chen et al. 1 3H, Thies 1 4], Lee and Marano l5J, Donati et al. 1 6J, Goto et al. 7J, Yoon and Rhee [ S y Gupta et al. 9 ). In the present work, a detailed mathematical model is developed for an LDPE tubular reactor. The variation of the physical properties of the reaction mixture with position is accounted for. The elements of the model are the reaction mechanism, the mass, energy and momentum balances, and the moments of live and dead polymer distributions. Polyethylene is produced via the following set of elementary reactions k. 

The analysis given here is concerned with the movement of fluids through porous media the equation describing motion (momentum balance) is thus of central importance. Following the original work of Hemy Darcy, mathematical descriptions of liquid flow in porous media are based on Darcy s law. This law states that the volumetric flow rate Q of hquid through a specimen of porous material is proportional to the hydrostatic pressure difference Ap across the specimen, inversely proportional to the length of the specimen, and proportional to the cross-sectional area. Darcy s law is expressed simply as ... 

The problems experienced in drying process calculations can be divided into two categories the boundary layer factors outside the material and humidity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differences inside the material during the drying process. The mathematical discussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its dependency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

---