## 24 Discussion

The basic governing equations (2.1 to 2.10) along with appropriate constitutive equations and boundary conditions govern the flow of fluids, provided the continuum assumption is valid. To obtain analytical solutions, the governing equations are often simplified by assuming constant physical properties and by discarding unimportant

= "fe except pressure

Four tangential inlets

Four tangential inlets

terms (refer to Section 2.1.4). For example, many analytical solutions of the Euler equations are available (Lamb, 1932; Milne-Thompson, 1955; Frisch, 1995). Solutions of low Reynolds number hydrodynamics (creeping flows) are discussed by Happel and Brenner (1983) and Kim and Karrila (1991). For some laminar flow problems, analytical solutions can be obtained by asymptotic methods or by using scaling arguments (Leal, 1992). Despite these extensive efforts, for most practical flow problems, it is not possible to obtain analytical solutions.

Recent advances in numerical methods and algorithms allow solution of these governing equations using digital computers. The modeling of flow processes occurring in industrial equipment may, however, require additional modeling steps beyond the formulation of basic governing equations. Any engineer familiar with fabrication drawings of an industrial reactor will know the complexity and co-existence of a wide range of geometric and velocity scales within the reactor. An industrial reactor may contain several internals such as cooling coils, feed pipes, distributors and so on. Developing a single model to simulate all the flow processes occurring in such reactors is almost always impractical (if not impossible). More often than not the reactor engineer has to divide the overall problem into different components. Each component can then be modeled separately to learn about its fluid dynamics. At each stage, the reactor engineer has to recombine the understanding and knowledge gained during the modeling of these components to build the overall model. Of course, at each stage, the implications of considering the separate components and extrapolation of these results to formulate the overall model need to be constantly evaluated. The approach to modeling flow processes in complex industrial reactors is discussed further with the help of case studies in Parts III and IV.

Recent progress in computing technologies has resulted in an order of magnitude increase in our capacity to use and solve these basic equations to simulate complex flow processes. This may lead to thinking that given sufficient computing power, these equations can be solved numerically to make a priori predictions of any complex flow process. Unfortunately, this is NOT true, especially for those flow processes in which a reactor engineer will be interested.

A close examination of the basic equations will reveal that non-linearity is at the core of these equations and there are no general ways of solving non-linear problems. The non-linearity in the governing equations manifests in the form of turbulence under certain conditions (high Reynolds number or Grashoff number). Turbulence is the most complex fluid motion, making even its precise definition difficult. Despite tremendous progress in the last few decades, it is not yet possible to compute, from first principles, how much power one would need to pump a given volumetric flow of liquid through a pipe if the flow rate (Reynolds number) is high enough! In order to describe real-life, complex flow processes, it is, therefore, necessary to develop and use additional models to complement the basic governing equations discussed above. For reactor engineering applications, the problem of turbulence is further complicated by the presence of chemical reactions and by multiple phases. The modeling of turbulent flows, multiphase flows, and reactive flows, is discussed in detail in the following chapters.

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