Thermodynamics and the conservation equations

Every process is subject to the laws of thermodynamics and to the conservation laws for mass and momentum, and we can expect every dynamic simulation of an industrial process to need to invoke one or more of these laws. The interpretation of these laws as they apply to different types of processes leads to different forms for the describing equations. This chapter will begin by reviewing the thermodynamic relations needed for process simulation, and it will go on to derive the conservation equations necessary for modelling the major components found in industrial processes. Finally, the different equations arising from lumped-parameter and distributed-parameter systems containing fluids will be brought out. 

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Atmospheric models for research and forecasting of weather, climate, and air quality are all based on numerical integration of the basic equations governing atmospheric behavior. These equations are the gas law, the equation of continuity (mass), the first law of thermodynamics (heat), the conservation equations for momentum (Navier-Stokes equations), and usually equations expressing the conservation of moisture and air pollutants. At one extreme, atmospheric models deal with the world s climate and climate change at the other extreme, they may account for the behavior of local flows at coasts, in mountain-valley areas, or even deal with individual clouds. This all depends on the selected horizontal scale and the available computing resources ... 

In order to calculate the density of reactant B about A, it is necessary to know by what means the reactants migrate in solution. Under most circumstances, diffusion is a very adequate description (the limitations of and complications to diffusion are discussed in Sect. 6, Chap. 8 Sect. 2 and Chap. 11). In this simple analysis of diffusion, Fick s laws will be used with little further justification, save to note that Fick s second law is identical to the equation satisfied by a random walk function. Hardly a surprising result, because diffusion is a random walk with no retention of information about where the diffusing species was before its current location. In Chap. 3 Sect. 1, the diffusion equation is derived from thermodynamic considerations and the continuity equation (law of conservation of mass). 

A means to find or estimate required constitutive properties that appear in the conservation equations. These can include equations of state, thermodynamic and transport properties, and chemical reaction rates. 

Overall our objective is to cast the conservation equations in the form of partial differential equations in an Eulerian framework with the spatial coordinates and time as the independent variables. The approach combines the notions of conservation laws on systems with the behavior of control volumes fixed in space, through which fluid flows. For a system, meaning an identified mass of fluid, one can apply well-known conservation laws. Examples are conservation of mass, momentum (F = ma), and energy (first law of thermodynamics). As a practical matter, however, it is impossible to keep track of all the systems that represent the flow and interaction of countless packets of fluid. Fortunately, as discussed in Section 2.3, it is possible to use a construct called the substantial derivative that quantitatively relates conservation laws on systems to fixed control volumes. 

The constitutive relations along with the conservation equations give the basic equations of fluid mechanics, which are a set of five nonlinear partial differential equations involving the seven variables, p, g,e, P, and T. Because five equations [Eqs. (1), (2), (3), (5), and (6)] cannot determine seven quantities, the equations are closed by expressing any two variables of the set (p,e,P,T) in terms of the other two remaining variables. This is done by using the assumption of local equilibrium and thermodynamic equations of state. 

According to the classical ZND model, the thermodynamic quantities of a material in the unshocked state and the final shocked state are related by the conservation equations of mass, momentum and energy across the shock front according to ... 

For a ternary mixture, equations above can describe thermodynamically and mathematically coupled mass and energy conservation equations without chemical reaction, and electrical, magnetic and viscous effects. To solve these equations, we need the data on heats of transport, thermal diffusion coefficient, diffusion coefficients and thermal conductivity, and the accuracy of solutions depend on the accuracy of the data. 

One way to recognize the significance of this equation is to remember that the ultimate objective of chemical thermodynamics is to calculate the equilibrium composition of a system of reactions. A chemical reaction system has R independent equilibrium constant expressions and C conservation equations, and this is just enough information to calculate the equilibrium concentrations of N species. Equation 7.1-9 is useful because it makes it possible to calculate a conservation matrix from a stoichiometric number matrix. In doing this with the operation NullSpace we will see again that it yields a basis for the conservation matrix. 

The local, phase-volume-averaged treatment of flow and heat and mass transfer has been addressed by Voller et al. [154], Prakash [155], Beckermann and Ni [156], Prescott et al. [157], and Wang and Beckermann [158], and a review is given by Beckermann and Viskanta [143]. Here we will not review the conservation equations and thermodynamic reactions. They can be found, along with the models for the growth of bubble-nucleated crystals, in Ref 158. 

Hydrodynamic models of the atmosphere on a grid or spectral resolution that determine the surface pressure and the vertical distributions of velocity, temperature, density, and water vapor as functions of time from the mass conservation and hydrostatic laws, the first law of thermodynamics, Newton s second law of motion, the equation of state, and the conservation law for water vapor. Abbreviated as GCM. Atmospheric general circulation models are abbreviated AGCM, while oceanic general circulation models are abbreviated OGCM. geomorphology... 

The liquid phase model proposed below considers the mass transfer inside the droplet and the changes in liquid phase properties due to the temperature and composition changes. In derivation of the following equations, it has been assumed that liquid circulation is absent, the droplet surface is at local thermodynamic equilibrium state, momentum, energy and mass transfer are spherically symmetric within the droplet, and the two liquids (fuel and water) are immiscible. With these assumptions, the conservation equations for the total mass, mass of water, and the energy equation are written as follows [14] ... 

Similar to the mass conservation laws, the macroscopic formulation of the First Law of Thermodynamics—that is, energy is conserved—is applicable to a wide range of chemical processes and process elements. The energy conservation equation for a multicomponent mixture can be written in a bewildering array of equivalent forms. For the volume of qpace shown in Fig. 2.2-1 containing i species, one form of the conservation equation for the energy (internal + kinetic -I- potential) content of the system is... 

As illustrated in the following example, it is preferable to start with the conservation equations, rate expression, and thermodynamic relationships, rather than immediately applying Pick s Second Law to a diffusion problem. 

In this section a number of methods ate described for the experinrental determination of molecular diffitsioa coefficients. The purpose is not ottty to acquaint the reader with some of these tedmiques but also to illustrate, by means of the associated analyses, the proper fomwlation and solution of the appropriate mathematical model for the particular experirtrental dif km situation. In all cases, the governing equations follow ftom simplifying the conservation equations for total mass and particular qiecies, using the flux expression and necessary thermodynamic relationships, artd qiplying appropriate boundary conditions established from the physical situation. Further descriptions of experirtrental methods tttay be found in the books by Jost and Cussler. Many mathematical solutions of the diffusion equation ate found in Ctank. ... 

According to the theory of relativity, E = ntc, nuclear fission decreases mass and converts it to energy. However, because the mass reduction is insignificant for engineering purposes (one atomic mass imit for 931.4 MeV), energy and mass conservation equations are normally decoupled in thermodynamics. Instead, mass is considered to be conserved and fission energy is treated as an energy source term. 

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