Section 2 Fluid Mechanics

The chapters of this section deal with those areas of fluid mechanics that are important to unit operations. The choice of subject matter is but a sampling of the huge field of fluid mechanics generally. Chapter 2 treats fluid statics and some of its important applications. Chapter 3 discusses the important phenomena 

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Separate chapters are devoted to each of the principal operations, which are grouped in four main sections fluid mechanics, heat transfer, equilibrium stages and mass transfer, and operations involving particulate solids. One-semester or one-quarter courses may be based on any of these sections or combinations of thenu... 

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter AIChE J., 7, 445 48 [1961]), Wheeler and Wissler (AJChE J., 11, 207-212 [1965]), Bird, Armstrong, and Hassager Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977), and Skelland Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). 

In one of the most successful and typical popular texts of natural philosophy, which went through twenty editions from 1766 to 1827, Adam Walker arranged the contents in sections of mechanics, astronomy, light (including optics), fluids (including air), chemistry (including heat), magnetism, and electricity.25... 

Traditionally the fluid mechanics of the extrusion process are summarized by the simple plate model illustrated in Fig. A7.1 and as described in Section 7.4. The motion of the screw is unchanged, but the reference frame has been moved to transform the problem to a fixed boundary problem for the observer. The flow in the rectangular channel is reduced into the x-direction flow across the channel and the z-direction flow down the channel. 

In the first section, the mechanisms involved in size exclusion chromatography are discussed this is an area where additional understanding and clarification still are needed. Data treatment with respect to statistical reliability of the data along with corrections for instrumental broadening is still a valid concern. Instrumental advances in the automation of multiple detectors and the developm.ent of a pressure-programmed, controlled-flow supercritical fluid chromatograph are presented. 

Since we do not want to consider the fluid mechanics of this reactor in detah, we will assume that the hquid falls with a constant average velocity and forms a hquid film of thickness R — R, with / j the radius in the tube at the surface of the hquid film. If these assumphons hold, then and R — i j are independent of the position z in the tube of length L (actually the height, since the tube must be vertical). Similarly, if the hquid film thickness is constant, then the cross section occupied by the gas is constant, and the velocity of the gas is also independent of z if the density of the gas is constant. These distances are sketched in Figure 12-8. 

It is impossible to read much of the literature on viscosity without coming across some reference to the equation of motion. In the area of fluid mechanics, this equation occupies a place like that of the Schrodinger equation in quantum mechanics. Like its counterpart, the equation of motion is a complicated partial differential equation, the analysis of which is a matter for fluid dynamicists. Our purpose in this section is not to solve the equation of motion for any problem, but merely to introduce the physics of the relationship. Actually, both the concentric-cylinder and the capillary viscometers that we have already discussed are analyzed by the equation of motion, so we have already worked with this result without explicitly recognizing it. The equation of motion does in a general way what we did in a concrete way in the discussions above, namely, describe the velocity of a fluid element within a flowing fluid as a function of location in the fluid. The equation of motion allows this to be considered as a function of both location and time and is thus useful in nonstationary-state problems as well. 

The objective of this section is to establish a relationship between the time rate of change of an extensive property of a system and the behavior of the associated intensive property within a control volume that surrounds the system at an instant in time. This kinematic relationship, described in terms of the substantial derivative, is central to the derivation of conservation equations that describe fluid mechanics. 

Fig. P3.16 Bipolar coordinate system. The shaded area denotes the cross section of the fluid, and the constant a, the distance of the pole from the origin. [Reprinted by permission from R. Bird, R. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Volume 1, Fluid Mechanics, Second edition, Wiley, New York, 1987.]...

The FEM, which was originally developed for structural analysis of solids, has been very successfully applied in the past decades to viscous fluid flow as well. In fact, with the exponentially growing computer power, it has become a practical and indispensable tool for solving complex viscous and viscoelastic flows in polymer processing (20) and it is the core of the quickly developing discipline of computational fluid mechanics (cf. Section 7.5). 

Fluid Mechanics (Determination of Bed Cross-sectional Area and Flow Regime... 

At least in the traditional domains of chemical engineering and in the traditional core of instructions that chemical engineers receive during their education, fluid mechanics (transport phenomena) has played a key role. Also one of the principal motivations for creating nonequilibrium thermodynamics was an attempt to make fluid mechanics manifestly compatible with equilibrium thermodynamics. Even the noncanonical Hamiltonian structures that play such an important role in the multiscale nonequilibrium thermodyna mics presented in Section 3 have been first discovered... 

Complex fluids are the fluids for which the classical fluid mechanics discussed in Section 3.1.4 is found to be inadequate. This is because the internal structure in them evolves on the same time scale as the hydro-dynamic fields (85). The role of state variables in the extended fluid mechanics that is suitable for complex fluids play the hydrodynamic fields supplemented with additional fields or distribution functions that are chosen to characterize the internal structure. In general, a different internal structure requires a different choice of the additional fields. The necessity to deal with the time evolution of complex fluids was the main motivation for developing the framework of dynamics and thermodynamics discussed in this review. There is now a large amount of papers in which the framework is used to investigate complex fluids. In this review we shall list only a few among them. The list below is limited to recent papers and to the papers in which I was involved. 

We proceed now to the problems (Problem 2) and (Problem 3). At least two levels of description are involved in direct molecular simulations. The first one is the level of the np-particle kinetic theory and the second is the level of fluid mechanics on which the external forces and the final results that we seek are formulated. We shall use the multiscale formulation developed above and combine the two levels. The two levels that we consider in this section are... 

Experimental observations of the time evolution of externally unforced macroscopic systems on the level meSo l show that the level eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if Cmeso is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here Ath) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for Ath (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description (i) Mslow/ (ii) Aneso 2 -> Ath, (ui) Aneso l -> Aneso 2, and (iv) Aneso i —> Aneso 2 —> Ath- The first two are the same as (111). We now turn our attention to the third one, that is,... 

Using the procedure outlined in this chapter for using the boundars laser equations to find the-forced convective heat transfer rate from a circular cylinder buried in a saturated porous medium, investigate the heat transfer rate from cylinders with an elliptical cross-section with their major axes aligned with the forced flow. The surface velocity distribution should be obtained from a suitable book on fluid mechanics. 

Unlike the other examples in this section, the equation governing the electrostatics here [i.e., Eq. (53)] is not the linearized Poisson-Boltzmann equation. However, considering interactions outside of thin double layers does have the effect of linearizing the problem. In Eq. (54), n is the fluid viscosity, K is the conductivity, is the zeta potential of the z th surface, and is a bipolar coordinate that is constant on the sphere and wall surfaces. It is this last condition (54), derived by Bike and Prieve [36] as a requirement to satisfy charge conservation, that couples the fluid mechanics with the electrostatics. 

Of course, no subject with the long-term vitality and impact of fluid mechanics and transport phenomena will remain for long without the appearance of major new directions, tools, and challenges. These developments will be the focus of the last section of this paper. However, before leaving this brief synopsis of the past, a few clarifications may be worthwhile. 

In the remainder of this section, I discuss the significance of these future trends in fluid mechanics and transport phenomena, beginning with the evolution of new theoretical tools and concluding with a more detailed description of future objectives, challenges, and opportunities for fluid mechanics research at the microscale. Some of the latter discussion is drawn from a more comprehensive report, The Mechanics of Fluids with Micro-structure written in 1986 by Professor R. A. Brown (MIT) and myself (with substantial input from the chemical engineering research community) as part of a general NSF-sponsored study of The Future of Fluid Mechanics Research. 

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