Variable volume systems, mathematics

In variable volume systems the dV/dt term is significant. Although equation 3.0.9 is a valid one arrived at by legitimate mathematical operations, its use in the analysis of rate data is extremely limited because of the awkward nature of the equations to which it leads. Equation 3.0.1 is preferred. 

Mathematical Characterization of Simple Variable Volume Reaction Systems... 

In the study of thermodynamics we can distinguish between variables that are independent of the quantity of matter in a system, the intensive variables, and variables that depend on the quantity of matter. Of the latter group, those variables whose values are directly proportional to the quantity of matter are of particular interest and are simple to deal with mathematically. They are called extensive variables. Volume and heat capacity are typical examples of extensive variables, whereas temperature, pressure, viscosity, concentration, and molar heat capacity are examples of intensive variables. 

Observe that the units of a are no longer dimensionally consistent with the units of time. Instead, the units of a are given by [volume of reactor x time/total mass]. Even though the units of a are different to t, the underlying mathematical and geometric behavior of a is equivalent to residence time. Hence, a is still a useful measure of the volume of a reactor network, and thus it is an important variable when determining reactor structures with minimum total volume in variable density systems. 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

Let us first introduce some important definitions with the help of some simple mathematical concepts. Critical aspects of the evolution of a geological system, e.g., the mantle, the ocean, the Phanerozoic clastic sediments,..., can often be adequately described with a limited set of geochemical variables. These variables, which are typically concentrations, concentration ratios and isotope compositions, evolve in response to change in some parameters, such as the volume of continental crust or the release of carbon dioxide in the atmosphere. We assume that one such variable, which we label/ is a function of time and other geochemical parameters. The rate of change in / per unit time can be written... 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

The model is seen to be a series sequence of N equal sized CSTRs which have a total volume V and through which there is a constant flowrate Q. From the physical standpoint, it is natural to restrict N, the number of tanks, to integer values but, mathematically, this need not be the case. When N is considered as a continuous variable which lies between one and infinity, a model results which can be used to interpolate continuously between the bounds of mixing associated with the CSTR and PFR. For N less than unity, the model represents systems with partial bypassing [41]. For integral values of N eqn. (43) may be inverted directly (see Table 9, Appendix 1) to give... 

Although these systems involve two variables, their steady-state solutions can be calculated in general and a more complete mathematical analysis of dissipative structures is possible. From a practical point of view it is interesting to note that systems obeying equations of the form (2) may be found in artificial membrane reactors.22 Examples are presented by D. Thomas in this volume. 

We have implied in Section 1.1 that certain properties of a thermodynamic system can be used as mathematical variables. Several independent and different classifications of these variables may be made. In the first place there are many variables that can be evaluated by experimental measurement. Such quantities are the temperature, pressure, volume, the amount of substance of the components (i.e., the mole numbers), and the position of the system in some potential field. There are other properties or variables of a thermodynamic system that can be evaluated only by means of mathematical calculations in terms of the measurable variables. Such quantities may be called derived quantities. Of the many variables, those that can be measured experimentally as well as those that must be calculated, some will be considered as independent and the others are dependent. The choice of which variables are independent for a given thermodynamic problem is rather arbitrary and a matter of convenience, dictated somewhat by the system itself. 

The idea of hidden variables is fairly common in chemical models such as the kinetic gas model. This theory is formulated in terms of molecular momenta that remain hidden, and evaluated against measurements of macroscopic properties such as pressure, temperature and volume. Electronic motion is the hidden variable in the analysis of electrical conduction. The firm belief that hidden variables were mathematically forbidden in quantum systems was used for a long time to discredit Bohm s ideas. Without joining the debate it can be stated that this proof has finally been falsified. 

The seemingly obvious statement of transitive properties of the Zeroth Law has important ramifications at the outset consider only the case where the properties of a system can be specified in terms of a prevailing pressure P and volume V. We follow the procedure advocated by Buchdahl. Consider then two systems 1 and 2 that are initially isolated we use pressures Pi and P2 (forces per unit area) to deform their volumes V and V2. We may have to make thermal or other adjustments that will permit physically possible pairs of pressure-volume variables (Pi, Fi) and (P2, V2) to be independently established in the two systems. Let these two units now be joined and equilibrated it is an experience of mankind that under these conditions only three of the four variables can be independently altered. This restriction is expressed by a mathematical relation f3(P, Vi, P2, V2) = 0, where fs is an appropriate mathematical function that provides the interrelation between the indicated variables its detailed form is not of interest at this point. 

We now repeat the process for joining system 1 to a new system 3 characterized by the pressure-volume variables P3, V3. By the same line of argument, after setting up the compound system one encounters a second interrelation of the form i62(Pi, V l, P3, V3) = 0. Lastly, on joining systems 2 and 3 one must set up a third mathematical restriction of the form (P2, V2, P2,1 3) = 0. If equilibrium prevails after each combination, we require for consistency with the Zeroth Law that system 3 remain unaltered in its union with either system 1 or 2 this allows us to solve for P3 in the functions 2 and f to write P3 = [Pg.5]

The selected mathematical model is represented by a discretization method for approximating the differential equations by a system of algebraic equations for the variables at some set of discrete locations in space and time. Many different approaches are used in reactor engineering , but the most important of them are the simple finite difference methods (FDMs), the flrrx conservative finite volume methods (FVMs), and the accurate high order weighted residual methods (MWRs). 

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