12 Computational Flow Modeling

As mentioned in the previous section, the equations of fluid mechanics are analytically solvable for only a limited number of flows. Though known solutions are extremely useful in providing an understanding of the fluid dynamics, these rarely can be used for engineering analysis and design. Although many key ideas for the numerical solution of partial differential equations were established more than a century ago, these were of little use before the advent of digital computers. The revolution in the ability of computers to store the data and to perform algebraic operations has greatly accelerated the development of numerical techniques for the solution of equations of fluid mechanics. This has led to the birth of a specialized discipline called computational fluid dynamics (CFD). It takes little imagination to realize that rapid advances in computing power and CFD can make significant contributions to various engineering fields. Before we discuss applications to reactor engineering, various aspects of CFD and computational flow modeling (CFM) are introduced in this section.

CFD deals with the solution of fluid dynamic equations on digital computers and the related use of digital computers in fluid dynamic research. CFD requires relatively few restrictive assumptions and gives a complete description of the flow field for all variables. Quite complex configurations can be treated and the methods are relatively easy to apply. It can incorporate a variety of processes simultaneously. CFD simulations serve as a bridge between theory and reality. Simulations have the added advantage that diagnostic 'probing' of a computer simulation does not disturb the flow and normal operation! The detailed predicted flow field gives an accurate insight to the fluid behavior and can sometimes give information which cannot be obtained from experiments. CFD simulations may allow one to switch on and off various interactions included in the model to understand the relative contributions of each individual process, which is extremely difficult - if not impossible - to achieve in experiments. These simulations allow detailed analysis at an earlier stage in the design cycle, with less cost, with lower risk and in less time than experimental testing. It sounds almost too good to be true. Indeed, these advantages of CFD are conditional and may be realized only when the fluid dynamic equations are solved accurately, which is extremely difficult for most engineering flows of interest. It must be remembered that numerical simulations will always be approximate. There can be various reasons for differences between computed results and 'reality'. Errors are likely to arise from each part of the process used to generate numerical simulations:

• fluid dynamic equations;

• input data and boundary conditions;

• numerical methods and convergence;

• computational constraints;

• interpretation of results, and so on.

It is necessary to develop an appropriate methodology to harness the potential of CFD tools for engineering analysis and design despite some of the limitations. Computational flow modeling (CFM) includes such overall methodology and all the other activities required to use CFD to achieve the engineering objectives.

Computational flow modeling for reactor engineering requires broad-based expertise in process and reactor engineering and an in-depth understanding of various aspects of CFD, along with a generous dose of creativity. Activities involved in a typical computational flow-modeling project are shown in Fig. 1.11. The identification of objectives for flow modeling and the application of a validated CFD model to achieve the set objectives are discussed in the next section with specific reference to chemical reactor engineering. Other aspects of computational flow modeling are introduced in this section and are discussed in detail in Part II (Chapters 2 to 8) of this book.

After establishing the modeling goals, the starting point of any computational flow-modeling project is to develop a mathematical model (equations and boundary conditions) to describe the relevant flow phenomena. This involves a rigorous analysis of fluids and type of flow under consideration. The first step is the rheological characterization of fluids under consideration. For Newtonian fluids, the knowledge of fluid viscosity is sufficient to develop the governing fluid dynamic equations. For non-Newtonian and rheologically complex fluids, it may be necessary to characterize the behavior of the fluid by more than one parameter. In this book, scope is restricted to the analysis of Newtonian fluids. Once the viscosity is known, it is necessary to distinguish the type of flow to select or develop an appropriate flow model. Some of the types are as follows:

• compressible/incompressible;

• laminar/turbulent;

• steady/unsteady;

• isothermal/non-isothermal;

• passive/reactive;

• single phase/multiphase.

Each of these types will have special features. For example, unsteady flows may be either forced unsteady, like a flow generated by a rotating impeller, or inherently

Problem Definition: Identify key performance governing processes & relate these with fluid dynamics

Problem Definition: Identify key performance governing processes & relate these with fluid dynamics

Evolve Objectives for Flow Modeling Analysis of underlying physics, possible simplifications, desired accuracy, available

Development of Generic Flow Models

Select suitable framework, Develop model equations (turbulence, multiphase & reactive flows)

resources

Development of Specific Sub-models Reaction sources, Physical properties Interphase transport, Boundary condition^

Development of Specific Sub-models Reaction sources, Physical properties Interphase transport, Boundary condition^

Mapping these Models onto CFD Solver Select numerical method and solver, Geometry mapping & grid generation, Numerical simulations, Grid dependence & convergence analysis, Post-processing

Mapping these Models onto CFD Solver Select numerical method and solver, Geometry mapping & grid generation, Numerical simulations, Grid dependence & convergence analysis, Post-processing

Application for Design and

Optimization

Validation of CFD Models Error analysis, Qualitative checks to ensure correct flow features, Quantitative validation with experimental & plant data

Validation of CFD Models Error analysis, Qualitative checks to ensure correct flow features, Quantitative validation with experimental & plant data

FIGURE 1.11 Typical flow modeling project.

resources unsteady, like vortex shedding behind a bluff body. Multiphase flows cover a very wide range of flows and have several sub-types depending on the nature of the phases (see for example, Fig. 1.9). For more details on multiphase reactors and commonly encountered flow regimes, refer to the discussion in Section 1.1.

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 overall simulation costs is to isolate the expensive processes and to replace them with simplified phenomenological models. These phenomenologies are physically reasonable, approximate models. The parameters of such models are guesses, fits to experiments or calibrated from a more detailed but more restricted numerical model. Several phenomenological turbulence models are widely used in engineering analysis. It is essential to assess the underlying assumptions and limitations of such phenomeno-logical models before they are used for a specific application. Similar or even more caution is necessary for multiphase flows and reactive turbulent flows. In many of these, even the underlying physics is not adequately understood and an engineer has to negotiate the challenges with inadequately validated phenomenological models. It must be remembered that if a phenomenological model is used to treat one of the controlling physical or fluid-dynamic processes in a simulation, the overall simulation is no more accurate than the phenomenology. It is necessary to evolve an appropriate methodology to use and to interpret the results obtained by such simulations.

Reactive flows have special problems over and above the complexities of underlying fluid dynamics. A reactor engineer wishes to know the history (path followed through the reactor and concentrations along the path) of all the fluid elements flowing through the reactor. This poses fundamental difficulties in modeling turbulent reactive flows. In a Eulerian approach (in which fluid motion and mixing is modeled using a stationary frame of reference), the location of a fluid element may be known exactly but its state (concentration) is not known accurately. In a Lagrangian approach (in which fluid motions and mixing is modeled using a frame of reference moving with the fluid particles), the state of the fluid element may be known accurately, however, its location is not known exactly. Several hybrid approaches have been used to find a way around this difficulty. Here again, similar to any phenomenological model, appropriate care needs to be taken when developing the model and interpreting its results. More detailed discussion of the modeling of fluid dynamics with special emphasis on turbulent flows, dispersed multiphase flows and reactive flows is given in Part II. Suggested references for developing model equations for other types of flows are also provided. Apart from the basic governing equations, it is necessary to develop specific sub-models for the system under consideration, such as models for variations of physical properties, interphase transport terms (momentum, heat and mass) and reaction sources.

After developing suitable governing equations, it is necessary to set the required boundary (and initial) conditions to solve these equations. This includes decisions about the extent of the solution domain. The process of isolating the system under consideration from its surrounding environment and specifying the outside influences in terms of boundary conditions may not be as straightforward as it sounds. In many cases, inappropriate decisions about the extent of the solution domain and the boundary conditions may give misleading results. Some examples of this are discussed in Chapter 2.

After finalizing the model equations and boundary conditions, the next task is to choose a suitable method to approximate the differential equations by a system of algebraic equations in terms of the variables at some discrete locations in space and time (called a discretization method). There are many such methods; the most important are finite difference (FD), finite volume (FV) and finite element (FE) methods. Other methods, such as spectral methods, boundary element methods or cellular automata are used, but these are generally restricted to special classes of problems. All methods yield the same solution if the grid (number of discrete locations used to represent the differential equations) is adequately fine. However, some methods are more suitable to particular classes of problems than others and the preference is often determined by ease of application, required computational resources and familiarity of the user.

Finite difference (FD) is probably the oldest method for the numerical solution of partial differential equations (PDEs). In this method, the solution domain is covered by a computational grid. At each grid point, the terms containing partial derivatives in the differential equation are approximated by expressions in terms of the variable values at grid nodes. This results in one algebraic equation per grid node, in which the variable value at that node and a certain number of neighboring nodes appear as unknowns. Taylor series expansions or polynomial fitting is used to obtain approximations for the first- and second-order derivatives. For simple geometry and structured grids (grid types are discussed later in this chapter and then in Part II), the FD method is very simple and effective. However, in finite difference methods, conservation is not enforced unless special care is taken. This is one of the major limitations of FD methods from the reactor engineering point of view. The restriction to simple geometry is also a significant disadvantage, since most industrial reactors have complex geometrical constructions.

In the finite element (FE) method, the solution domain is broken into discrete volumes or finite elements (generally unstructured; in 2D they are triangles or quadrilaterals and in 3D they are tetrahedra or hexahedra). The distinguishing feature of FE methods is that the equations are multiplied by a weight function before they are integrated over the entire domain. This approximation is then substituted into the weighted integral of the conservation law. By minimizing the residual, a set of non-linear algebraic equations is obtained. An important advantage of the FE method is its superior ability to deal with a solution domain having complex geometry. It is, however, difficult to develop computationally efficient solution methods for strongly coupled and non-linear equations using FE.

The finite volume (FV) method uses the integral form of the conservation equations as its starting point to ensure global conservation. The solution domain is again divided into number of computational cells (similar to FE). The differential equation is integrated over the volume of each computational cell to obtain the algebraic equations. Variable values are stored at the cell centers and interpolation is used to express variable values at cell faces in terms of the cell center values. Surface and volume integrals are approximated using suitable quadrature formulae. As a result, one obtains an algebraic equation per computational cell, in which a number of neighboring cell center values appear as unknowns. The FV methods can accommodate any type of grid and is, therefore, suitable for handling complex geometry. All terms that need to be approximated have physical meaning in the FV approach. Finite volume methods are, therefore, quite popular with engineers. The disadvantage, however, of FV methods is that higher than second-order approximations of gradient terms are difficult to implement, especially in 3D. Despite this, FV is the method of choice of many engineers and so it is, for this book. Detailed descriptions of various aspects of the FV method will be given in Part II (Chapters 6 and 7).

Having selected the numerical method, it is necessary to generate an appropriate grid, i.e. discrete representation of the solution domain and discrete locations at which variables are to be calculated. Two types of grids, namely structured and unstructured grids, are briefly discussed here. In a structured grid, there are families of grid lines

following the constraint that grid lines of the same family do not cross each other and cross each member of the other families only once. The position of a grid point within the solution domain is, therefore, uniquely identified by a set of two (in 2D) or three (in 3D) indices. It is thus logically equivalent to a Cartesian grid. The properties of a structured grid can be exploited to develop very efficient solution techniques. One major disadvantage is the difficulty in controlling the grid distribution. In a structured grid, concentration of grid points in one region for more accuracy may unnecessarily lead to small spacing in other parts of the solution domain. A block-structured grid is used to eliminate or reduce this disadvantage. In a block-structured grid, the solution domain is divided into a number of blocks which may or may not overlap. Within each block, a structured grid is defined. This kind of grid is more flexible as it allows local (block-wise) grid refinement. For very complex geometry, unstructured grids, which can fit an arbitrary solution domain boundary, are used. In this case, there is no restriction on the shape of the control volume and the number of neighboring nodes. Triangles and quadrilaterals in 2D and tetrahedra or hexahedra in 3D are the most widely used grid shapes in practice. Examples of structured and unstructured grids are shown in Fig. 1.12. Unstructured grids can be refined locally and allow more control of the variation of aspect ratio, etc. The advantage of flexibility is often offset by the disadvantage of the irregularity of the data structure. The solvers for algebraic equation systems of unstructured grids are generally slower than those for structured grids. Methods of grid generation will not be covered in this book. Several excellent texts on grid generation are available (see for example, Thompson et al., 1985; Arcilla et al., 1991). Some discussion on assessing the quality of the generated grid and tips for rectifying observed deficiencies are given in Part II.

Following the choice of grid type, one has to select the approximations to be used in the discretization process. For the finite volume method, one has to select the methods of approximating surface and volume integrals. The choice of method of approximation influences the accuracy and computational costs. The number of nodes involved in approximation controls the memory requirements, speed of the code and difficulty in implementing the method in the computer program. More accurate approximation involves a larger computational molecule (more nodes) and gives fuller coefficient matrices. A judicious compromise between simplicity, ease of implementation, accuracy and computational efficiency has to be made. More discussion and details of choices with such a compromise in mind are given in Part II.

Discretization yields a large system of algebraic equations. The choice of solver depends on the type of flow, grid type and the size of the computational molecule (number of nodes appearing in each algebraic equation). A wide variety of solvers and accelerators are available; some of which are discussed in Part II. The set of algebraic equation is usually solved iteratively (so-called 'inner iterations'). The details of solvers of algebraic equations will not be discussed in this book. Interested readers may refer to specialized texts on these topics (for example, Press et al., 1992; Fergizer and Peric, 1995). The 'outer iterations' involve repeating this process many times over to deal with the non-linearity and coupling among the model equations. Deciding when to stop the iterative process (convergence criterion) at each level is important, from both accuracy and efficiency points of view. These issues are also briefly discussed in Part II. Some tips for assessing the degree of overall convergence and tuning of solver parameters are also given.

Of course, before tuning the solver parameters, it is necessary to develop a computer program to implement the numerical techniques selected to solve the mathematical model. It takes an organized and dedicated effort to design an efficient and error-free computer program. After adequate testing, the program can be a valuable tool for engineers trying to understand and manipulate the complex fluid dynamics in industrial equipment. The steps in developing and testing the program are, therefore, of the utmost importance. It is difficult to discuss guidelines for the development of computer programs since this encompasses widely different issues. Patankar (1980) has listed some such suggestions in his book. In this book, it will be assumed that an error-free computer program, which takes care of grid generation and the solver part, is available to the reactor engineer. This assumption is not as unrealistic as it sounds, since there is an increasing tendency, especially in chemical process industries, to use commercially available CFD codes. This brief introduction and the discussion in Part II will allow chemical engineers to use such programs efficiently and effectively. The emphasis in this book is on discussions concerning the modeling of fluid flows relevant to chemical reactor engineers and on the usage of such tools to solve the model equations and reactor engineering problems.

The path of chemical reactor engineers, hoping to use CFD tools and programs to enhance the productivity of a reactor, is crowded by complex challenges offered by multiphase flows, turbulent flows, reactive flows, non-Newtonian flows and so on. Understanding CFD and having access to a CFD program is only part of the solution. It is necessary to appreciate various issues and make decisions about modeling strategies and the interpretation of results. It is necessary to devise an appropriate methodology to achieve reactor engineering objectives by creatively employing the best available knowledge and tools.

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