Incompressible fluids, availability

With incompressible fluids the energy F is either lost to the surroundings or causes a very small rise in temperature. If the fluid is compressible, however, the rise in temperature may result in an increase in the pressure energy and part of it may be available for doing useful work. 

The equation of motion for the turbulent flow of an incompressible fluid is obtained from the Navier-Stokes equations by replacing the instantaneous values of each point quantity by the sum of the average and its fluctuating component, and time averaging. This results in the Reynolds equations for incompressible turbulent motion in which there are more unknowns than available equations. Therefore additional relations are needed to solve the equations. 

The last term on the RHS represents the loss of useful or available energy that occurs in an incompressible fluid because of friction [185, 114]. 

We see that application of the angular acceleration principle does reduce, somewhat, the imbalance between the number of unknowns and equations that derive from the basic principles of mass and momentum conservation. In particular, we have shown that the stress tensor must be symmetric. Complete specification of a symmetric tensor requires only six independent components rather than the full nine that would be required in general for a second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently independent unknowns and only four independent relationships between them. It is clear that the equations derived up to now - namely, the equation of continuity and Cauchy s equation of motion do not provide enough information to uniquely describe a flow system. Additional relations need to be derived or otherwise obtained. These are the so-called constitutive equations. We shall return to the problem of specifying constitutive equations shortly. First, however, we wish to consider the last available conservation principle, namely, conservation of energy. 

Some partial contributions to the total availability are then considered. Thus, a simple expression for the pressure availability of an incompressible fluid is developed. Formulae for the chemical availability of hydrocarbon fuels obtained by Szargut and Styrylska are then discussed and summarized in a separate table. Equations for the average value of the specific heat of various solid fuels between some fixed temperature and some other variable one are also given, as is a technique to estimate the lower heating value of a fuel of known atomic composition. Finally, a simplified approach used in approximating the thermal availability of tars is described. 

The new integral term corresponds to the pressure availability, ap, of equation [3]. In the incompressible fluid model this term is readily evaluated as... 

The pressure availability can be obtained by using the formula developed earlier in this paper for the incompressible fluid model, namely... 

Figure 2.27. Graph of the potential wells of hydrogen atoms filled with valence electrons considered as an incompressible fluid. Bonding between two atoms is the result of delocalization of the electron density fluid the vessel for the fluid becomes bigger and the level drops. Similarly, when an electron with spin a (right-slanted) has more space available its average kinetic energy drops. Similarly for spin b (left-slanted). The two fluids with different spins can occupy the same space and ignore each other as a first approximation. Reprinted with permission from the Journal of Chemical Education, Vol 65, 1988, p. 581 copyright 1988, Division of Chemical Education, Inc.

As hydrocarbons become more valuable and also less available in the United States and Canada, it is clear that the increased use of inert gases such as carbon dioxide, nitrogen and waste flue gases will be emphasized. Laboratory research will continue to try to understand the mechanisms and researchers will try to devise better techniques for using these inexpensive low-viscosity fluids more effectively. Research on efforts to use that almost incompressible fluid, water, to drive the special gases through the reservoir will undoubtedly receive more emphasis. 

The last term on the RHS represents the loss of useful or available energy that occurs in an incompressible fluid because of friction [114, 184]. With the given restrictions this equation is identical to the ID mechanical energy equation for the flow between the two points (1) and (2), as can be derived directly from (1.133) [11, 103] ... 

As already explained the necessity to satisfy the BB stability condition restricts the types of available elements in the modelling of incompressible flow problems by the U-V P method. To eliminate this restriction the continuity equation representing the incompressible flow is replaced by an equation corresponding to slightly compressible fluids, given as... 

Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations or continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conseiwation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. 

One usually distinguishes two types of lattice models. The first type may be called lattice-gas models. In this case, the number of molecules in the system is less than the number of available sites. In other words, there are vacant sites. The second type of lattice models may be called lattice fluids. In this case, all lattice sites are filled exactly by the molecular components in the system the system is considered to be incompressible. It is easily shown that a two-component incompressible lattice fluid model can be mapped on a one-component lattice gas one. In other words, it is possible to interpret vacant sites to be occupied by a ghost ... 

Marshall s extensive review (16) concentrates mainly on conductance and solubility studies of simple (non-transition metal) electrolytes and the application of extended Debye-Huckel equations in describing the ionic strength dependence of equilibrium constants. The conductance studies covered conditions to 4 kbar and 800 C while the solubility studies were mostly at SVP up to 350 C. In the latter studies above 300°C deviations from Debye-Huckel behaviour were found. This is not surprising since the Debye-Huckel theory treats the solvent as incompressible and, as seen in Fig. 3, water rapidly becomes more compressible above 300 C. Until a theory which accounts for electrostriction in a compressible fluid becomes available, extrapolation to infinite dilution at temperatures much above 300 C must be considered untrustworthy. Since water becomes infinitely compressible at the critical point, the standard entropy of an ion becomes infinitely negative, so that the concept of a standard ionic free energy becomes meaningless. 

Fluent is a commercially available CFD code which utilises the finite volume formulation to carry out coupled or segregated calculations (with reference to the conservation of mass, momentum and energy equations). It is ideally suited for incompressible to mildly compressible flows. The conservation of mass, momentum and energy in fluid flows are expressed in terms of non-linear partial differential equations which defy solution by analytical means. The solution of these equations has been made possible by the advent of powerful workstations, opening avenues towards the calculation of complicated flow fields with relative ease. 

Several texts are available for further reading on turbulent flow, including Pope (Turbulent Flows, Cambridge University Press, Cambridge, U.K., 2000), Tennekus and Lumley (ibid.), Hinze (Turbulence, McGraw-Hill, New York, 1975), Landau and Lifshitz (Fluid Mechanics, 2d ed., Chap. 3, Pergamon, Oxford, 1987) and Panton (Incompressible Flow, Wiley, New York, 1984). 

Each of these different types of flows is governed by a set of equations having special features. It is essential to understand these features to select an appropriate numerical method for each of these types of equations. It must be remembered that the results of the CFD simulations can only be as good as the underlying mathematical model. Navier-Stokes equations rigorously represent the behavior of an incompressible Newtonian fluid as long as the continuum assumption is valid. As the complexity increases (such as turbulence or the existence of additional phases), the number of phenomena in a flow problem and the possible number of interactions between them increases at least quadratically. Each of these interactions needs to be represented and resolved numerically, which may put strain on (or may exceed) the available computational resources. One way to deal with the resolution limits and... 

Thus the objective here is a generally applicable simulation of steady, two-dimensional, incompressible flow between rigid rolls with film splitting. The results reported are solutions of the full Navier-Stokes system including the physically required boundary conditions. The analysis is also extended to a shearthinning fluid. The solutions consist of velocity and pressure fields, free surface position and shape, and the sensitivities of these variables to parameter variations, valuable information not readily available from the conventional approach (10). The rate-of-strain, vorticity, and stress fields are also available from the solutions reported here although they are not portrayed. Moreover, the stability of the flow states represented by the solutions can also be found by additional finite element techniques (11), and the results of doing so will be reported in the future. 

To use the energy balances, we will need to relate the energy to more easily measurable properties, such as temperature and pressure (and in later chapters, when we consider mixtures, to composition as well). The interrelationships between energy, temperature, pressure, and composition can be complicated, and we will develop this in stages. In this chapter and in Chapters 4,5, and 6 we will consider only pure fluids, so composition is not a variable. Then, in Chapters 8 to 15, mixtures will be considered. Also, here and in Chapters 4 and 5 we will consider only the simple ideal gas and incompressible liquids and solids for which the equations relating the energy, temperature, and pressure are simple, or fluids for which charts and tables interrelating these properties are available. Then, in Chapter 6, we will discuss how such tables and charts are prepared. 

The starting point is a Mathematical Model, i.e. the set of equations and boundary conditions, which covers the physics of the flow most suitable. For some problems the governing equations are known accurately (e.g. the Navier-Stokes equations for incompressible Newtonian fluids). However for many phenomena (e.g. turbulence or multiphase flow) and especially for the description of ceramic materials or wall slip phenomena the exact equations are either not available or a numerical solution is not feasible. 

It is commonly accepted that the finite element methods offer the most rigorous numerical schemes for the simulation of fluid flow phenomena. The inherent flexibility of these schemes and their ability to cope with complicated geometries and boundary conditions can be used very effectively to solve the governing equations of complex flow regimes. In particular, the finite element simulation of steady, incompressible laminar flow is very well-established, and an extensive literature in this area is available. Galerkin finite element schemes based on different types of Lagrange elements are the most frequently used techniques in these simulations [8]. In flow domains with porous walls, however, more recent work... 

Ever since the pioneering work of Stokes in 1851, considerable research effort has been expended in studying the hydrodynamic aspects of sphere motion in Newtonian fluids consequently, a voluminous body of information has accrued over the last 150 years or so, dealing with different aspects of sphere motion in incompressible Newtonian fluids. Excellent treatises summarizing the rich literature on this subject are available [e.g., see Happel and Brenner (1965), Clift et al. (1978), O Neill (1981), and Kim and Karrila (1991)]. 

A spherical particle is unique in that it presents the same projected area to the oncoming fluid irrespective of its orientation. For nonspherical particles, on the other hand, the orientation must be known before their terminal velocity or the drag force can be calculated. Conversely, nonspherical particles tend to attain a preferred or most stable orientation, irrespective of their initial state, in the free-settling process. Vast literature, although not as extensive as that for spherical particles, is also available on the hydrodynamic behavior of nonspherical—regular as well as irregular type— particles in incompressible Newtonian fluids and these studies have been summarized in the aforementioned references, whereas the corresponding aerodynamic literature has been comprehensively reviewed by Hoerner (1965). 

The deformation of a material is governed not oidy by a constitutive relation between deformation and stress, like the neo-Hookean equation discussed above, it also must obey the principles of conservation of mass and conservation of momentum. We have already used die mass conservation principle (conservation of volume for an incompressible material) in solving the uniaxial extension example, eq. 1.4.1. We have not yet needed the momentum balance because the balance was satisfied automatically for the simple deformations we chose that is, they involved no gravity, no flow, nor any inhomogeneous stress fields. However, these balances are needed to solve more complex deformations. They are presented for a flowing system because we will use these results in the following chapters. Here we see how they simplify for a solid. Detailed derivations of these equations are available in nearly every text on fluid or solid mechanics. 

---