### Computational transport phenomena for engineering analyses

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Treatments of numerical, analytical, and computational solutions are presented side by side, often with sample code in MATLAB , to aid students' understanding and develop their confidence in using computational skills to solve real-world problems. Learning objectives and mathematical prerequisites at the beginning of each chapter orients students to what is required; summaries and over end-of-chapter problems help them retain the key points and check their understanding.

Online supplementary material including solutions to problems for instructors, supplementary reading material, sample computer codes, and case studies complete the package. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:.

## Transport phenomena in the human nasal cavity: a computational model.

Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location. Toggle Main Navigation. Books Books MathWorks. E-mail addresses: farmer sierraengineering. Farmer , pike she. Pike , gcheng uab. Farmer et al. The conservation equations of mass, momentum, and energy, when simplified to describe such flows, are solved with a variety of numerical techniques. The complexities arising from including turbulence, fluid property variation and reactions, and multi-phase flows in the simulations have not been addressed in a systematic fashion.

However, enough work has been accomplished, and some of it made available as leased computational software that basic computational methodology can be identified. A critique of these models is shown in the following. These models may aptly be called computational transport phenomena CTP , due to their applicability to multi-component and multi-phase flows. The models are meant to be illustrative only. A definitive critique is not possible due to commercial analytical codes and company codes being proprietary and to the current position of the government being to suppress federally funded research activities.

The scope of computational transport phenomena Two basic approaches are used to formulate the conservation equations for modeling complex chemical processes. Such a model is integrated over time, space and the internal coordinates to define the system. B and D are birth and death functions of the particles, i. These approaches are defined as deterministic systems by Himmelblau and Bischoff, Process Analysis and Simulation. These partial differential equations describe the process at a point and must be integrated over a volume to represent the entire system.

Historically, these equations have been simplified by assuming geometrically simple, usually steady-state flow conditions. CFD allows us to represent the process without making these simplifying assumptions. CFD simulations are numerical solutions obtained for a discrete number of grid points.

Before a discussion of grids is presented, variations of the basic model equations will be shown in order to demonstrate the generality and options available for modeling complex processes. Laminar flows To initiate our discussion, laminar flow of a multi-component, single-phase fluid will be described in a fixed coordinate system by the following. Cross-diffusion effects and non-Newtonian fluids are not considered at this point. The g is gravity. The subscripts i and k signify a species value. Turbulent flows Numerous methods of turbulence modeling are continuously under investigation.

The most fruitful is to average the laminar conservation equations over time. If the fluid density is constant, the time-average method of Reynolds is appropriate. For variable fluid density, the mass average method of Favre is useful. Physically, turbulence causes the fluid to act as though it has a very high viscosity and conductivity and diffusivity except near solid surfaces, where these transport mechanisms are reduced to laminar levels in very short distances.

Conservation equations in this section are shown in terms of rectangular Cartesian coordinates. These models are discussed in Kuo, Principles of Combustion. The subscript m denotes mass average, otherwise the term is time-averaged. Since this equation is in Cartesian tensor nomenclature, the repeated index k indicates a summation over the Xk coordinates.

Dakhoul, Improved averaging method for turbulent flow simulation dissertation. Filter functions are used to introduce an eddy size into turbulence models, thereby producing large eddy simulation LES methodology. LES models are not now computationally practical. Multi-phase flows For a fluid continuum with a sparse distribution of particulates solids, bubbles or droplets , Euler—Lagrange methods may be used.

The fluid flow is calculated then the particles are tracked through the flowfield. However, if the distribution of particulates is dense, both phases are more strongly coupled and the flow is more efficiently simulated with strictly Eulerian methods. These methods are described as follows. Multi-phases as coexisting continua Just as multi-component flows are treated as being continua in each component, multi-phase flows may be treated as being coexisting continua in each phase.

This allows us to write the Eulerian transport equations for each phase, including interaction terms between the phases, and solve the entire set to represent the process. A recent evaluation of this methodology is reported by van Wachem et al. The derived conservation equations may be solved directly or modeled transport equations may be solved for the probability density functions.

Transport equations from a PDF The Boltzmann equation for the velocity distribution function for monatomic, non-reacting mixtures of low-density gases. The concept of phase space, i. When the fluid is in motion, the distribution function must be corrected. When this is done, expressions for the laminar transport coefficients can be obtained. Similar expressions can be obtained for polyatomic gases, monatomic liquids, and polymeric liquids. However, no new information is generated which is not already available in the phenomenological statement of the transport equations.

Application of PDF methods to particulate flows Even if the particulates are not molecular in scale, the same statistical methods may be applied to their analysis. The internal coordinates are used to simulate the physical processes, which the particles are undergoing. Such methods are still under investi- gation; Verkoeijen et al. As with most chemical process analyses, such research has been primarily directed at investigating the physical processes the par- ticulates are encountering while over simplifying their motion, i.

Application of PDF methodology to turbulent flows With a goal of getting more detail into the analysis of turbulent flows, investigators have stopped short of averaging the laminar flow conservation equations to initiate their analyses. To date, such modeling approaches have not produced practical process simulation models, although they have contributed to a more basic understanding of turbulent flows.

These methods are well presented by Pope, Turbulent Flows. Extension to the description of reacting flows has been accomplished by Fox, Computational Models for Turbulent Reacting Flows. Realistic variation of density and complex chemical reactions has not yet been addressed with these methods. In the opinion of these investigators, the inclusion of density variations in the description of turbulent reacting flows is more important than the description of local unmixedness, which is obtained from the use of PDF methodology.

## Mechanophysics

Admittedly, adequate experimental justification of the relative importance of these effects is not available. However, chemical kinetics are determined by experiment, not theory. If unmixedness is important, in all likelihood, it also affected the reported experimental reaction rates.

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In any event, the computational efficiency of using PDF methodology precludes it from being a practical process simulation tool at this time. Coordinates and grids To solve the conservation equations numerically for a complex flow, a grid system is needed. In practice, general curvilinear grids are used; such grids are discussed in Fletcher, Computational Techniques for Fluid Dynamics, vol.

## Computational Transport Phenomena for Engineering Analysis (HB)

The extensive research on grids performed at Mississippi State University is contained in the tome Handbook of Grid Generation edited by Thompson, Soni, and Weatherill. Almost never will the very general grids developed to describe flow about aircraft be needed for process analyses. However, if environmental phenomena related to toxic releases or explosive accidents are simulated, the general grid methods should be considered. Moving grids are of great benefit for such simulations. What is needed is an elementary grid generator for curvilinear coordinates, which can be used to understand CFD analyses of transport phenomena.

Ultimately, a grid generator for physical curvilinear moving coordinates should be developed as a public-domain tool. Elaborate grid generators are not needed except to match the geometry of the process system. However, if an elementary grid code is available, its use is transparent to the practitioner. An additional benefit for using such coordinates is that tensor quantities are elegantly defined. Much of the transport literature is confusing on the definition and use of tensor analysis. Non-Newtonian flows Many process flows are non-Newtonian, such as polymeric flow and flows of suspensions.

However, simulation of the shear rate—shear stress can be only described with simple models, such as power law models. Laminar flows to interpret rheological experiments have been extensively modeled for example, see BSL, Transport phenomena, 2nd ed.

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The co- rotating and codeforming derivatives used in such models cannot be conveniently utilized to model process flows. In the first place, only differential and partial differential equations have established solution methodology. In the second place, only basic non-Newtonian flows have been investigated in the turbulent flow regime. Examples These investigators endorse the use of computational transport phenomena models for the practical analysis complex chemical processes.

For multi-component reacting flows, modest sized kinetics mechanisms can be analyzed with current methodology. Since kinetics models are experimentally determined, large kinetics models cannot be sufficiently validated to be justified, nor are they computationally efficient to analyze. Variable fluid properties can be easily included in the analysis. For multi-phase flows, additional interaction terms can be easily included in the computational analysis. The problem is one of validation. Using more elaborate models than can be validated is misleading and often renders the simulation impractical.

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Two-equation turbulence models have been shown to be very useful in simulations. Two examples are presented in the following to justify this position. Stirred-tank reactors Agitated vessels are widely used in the chemical, pharmaceutical, and petroleum refinery industries for mixing and chemical processing. An impeller is used to generate a turbulent flowfield which minimizes temperature and concentration gradients, suspends solid particles, maintains emulsions, disperses a gas into a liquid, and combines chemical reactants.

Two complementary simulation and validation investigations were conducted by the Chemical Engineering Department at Louisiana State University to model these stirred-tank systems. To model the flow in a baffled, tank stirred by a Rushton turbine impeller in which no reactions occurred, a CFD analysis was made and flowfield validation experiments were conducted.

The geometry of this tank is shown in Fig. The fluid studied Fig. Standard tank configuration with a Rushton turbine. A non-uniform, three-dimensional, cylindrical grid was used to discretize a quadrant of the tank. Periodic boundary conditions were used to match the flows at the sides of the quadrants. No-slip conditions were applied at the tank and baffle surfaces. The fluid surface was assumed to be flat. Flow into and out of the turbine blade was represented as specified velocity profiles into the turbine hub region and out of the tip region.

Notice, that the flow in the vicinity of the turbine blade and the fluid surface were simplified in order to keep the simulation practical. The mean velocity field through out the tank was measured with a hot wire anemometer. The measurements indicated that the turbulent flowfield was non-isotropic. The CFD simulation, with the improvement obtained by this non-isotropic model, agreed very well with the experimental test data. The second investigation involved reacting flow in the same tank.

Hydrochloric acid and sodium hydroxide were reacted. The hydroxide was injected near the turbine tip. Tomography was used to track the local concentrations and reacting flow front. This flow was simulated with the leased Fluent CFD code. The grid and boundary conditions were the same as for the non- reacting case. The non-isentropic turbulence model could not be used because only the source code was available. The instantaneous point measurements made with tomography are shown in Fig.

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The time-averages of these measurements are compared to the time-average simulation in Fig. Radial concentration predictions and measurements are shown in Fig. The radial velocity profile from the turbine blade tip to the wall is shown in Fig. The transport model agreed with the measured data very well and was, therefore, considered to be validated.