Thermodynamic control volume

Flow induced nucleation A Control volume thermodynamics approach... 

Here, we will formulate a novel model of flow induced nucleation by using control volume thermodynamics. The general idea is very simple and goes through the lines indicated along this chapter, that is, by assuming that nonequilibrium equations of state can be formulated in a way consistent with the laws of thermodynamics when a flow is imposed on the system. 

Using control volume thermodynamics, we showed that the nonequilibiium equations of state heuristically proposed by Frenkel and co-workers is in fact consistent with the general expression of the first law of thermodynamics, in which the heat and the work exchanged... 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

A control volume is a volume specified in transacting the solution to a problem typically involving the transfer of matter across the volume s surface. In the study of thermodynamics it is often referred to as an open system, and is essential to the solution of problems in fluid mechanics. Since the conservation laws of physics are defined for (fixed mass) systems, we need a way to transform these expressions to the domain of the control volume. A system has a fixed mass whereas the mass within a control volume can change with time. 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

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. 

For a steady-state situation, the first law of thermodynamics applied to the control volume states... 

Fig. 9.7 Thermodynamic analysis of an SOFC. CV represents the control volume for the analysis.

The scope definition is similar to the definition of the control volume in the thermodynamic analysis or the battery limits in process design, and for the LCA in terms of space and time (e.g., we follow the use of product X in the process from the raw materials to the time it is disposed by the consumer. Throughout the lifetime of the product, we analyze the environmental burden). The reasons for the study are also clearly defined (e.g., is the study necessary to make a decision about a process ), as well as an answer must be given as to who is performing the study and for whom. Consider the following hypothetical example ... 

Fig. 2.1 The control volume (broken curve) and the thermodynamic system (solid curve) at time t + At in a flowing medium. The control volume and the thermodynamic system coincide at time, t.

The law of conservation of mass for fluids in flow processes is most conveniently] written so as to apply to a control volume, which is equivalent to a thermodynamic] system as defined in Sec. 2.3. A control volume is an arbitrary volume enclosed] by a bounding control surface, which may or may not be identified with physical boundaries, but which in the general case is pervious to matter. The flow processes] of interest to chemical engineers usually permit identification of almost the entire control surface with actual material surfaces. Only at specifically provided entrances and exits is the control surface subject to arbitrary location, and heie it is universal practice to place the control surface perpendicular to the direction of flow, so as to allow direct imposition of idealizations 1 and 2. An example of a control volume with one entrance and one exit is shown in Fig. 7.1, The actual... 

In Chap. 2 the first law of thermodynamics was applied to closed systems (nonflo processes) and to single-stream, steady-state flow processes to provide specifi equations of energy conservation for these important applications. Our purpos here is to present a more general equation applicable to an open system or to control volume. 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

Summary of Equations of Balance for Open Systems Only the most general equations of mass, energy, and entropy balance appear in the preceding sections. In each case important applications require less general versions. The most common restrictedTcase is for steady flow processes, wherein the mass and thermodynamic properties of the fluid within the control volume are not time-dependent. A further simplification results when there is but one entrance and one exit to the control volume. In this event, m is the same for both streams, and the equations may be divided through by this rate to put them on the basis of a unit amount of fluid flowing through the control volume. Summarized in Table 4-3 are the basic equations of balance and their important restricted forms. 

Until this point we have limited our thermodynamic description to simple (closed) systems. We now extend our analysis considering an open system. In this case the material control volume framework might not be a convenient choice for the fluid dynamic model formulation because of the computational effort required to localize the control volume surface. The Eulerian control volume description is often a better choice for this purpose. 

As explaned in chap 1, for a fixed control volume the first law of thermodynamics can be written ... 

Figure 3.6 shows a non-adiabatic (heat is either lost to or gained from the environment surrounding the control volume) two-phase equilibrium-based process. There are two feed streams and two exit streams. The exit streams are in thermodynamic equiUbrium. 

Thus, if the inlet and outlet compositions are known, then VjO can be determined. For an extraction process, this corresponds to the required solvent/diluent flow ratio. The ratios O jF and V/F could be determined with the appropriate substitution in one of the component balances. IRemember these ratios are determined by mass balances around the conflol volume. Nothing has been stated about thermodynamic equilibrium within the control volume.]... 

We note that this conservation principle, for a closed system such as the material control volume, is precisely equivalent to the first law of thermodynamics, which we can obtain from it by integrating with respect to time over some finite time interval. 

We have seen that the energy conservation principle, applied to a material control volume of fluid, is equivalent to the first law of thermodynamics. A natural question, then, is whether... 

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