Computational fluid dynamics conservation equations

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

Much effort has been expended in the last 5 years upon development of numerical models with increasingly less restrictive assumptions and more physical complexities. Current development in PEFC modeling is in the direction of applying computational fluid dynamics (CFD) to solve the complete set of transport equations governing mass, momentum, species, energy, and charge conservation. 

Here, the thermo-fluid analyses are performed using the computational fluid dynamics code STAR-CD (Computational Dynamics Ltd.) [9], In STAR-CD, the algebraic finite-volume equations are solved. The solid and fluid parts are divided into small discrete meshes, and in each mesh, the following differential equations governing the conservation of mass, momentum, and energy are solved. 

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 of continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conservation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

Takeuchi et al. 7 reported a membrane reactor as a reaction system that provides higher productivity and lower separation cost in chemical reaction processes. In this paper, packed bed catalytic membrane reactor with palladium membrane for SMR reaction has been discussed. The numerical model consists of a full set of partial differential equations derived from conservation of mass, momentum, heat, and chemical species, respectively, with chemical kinetics and appropriate boundary conditions for the problem. The solution of this system was obtained by computational fluid dynamics (CFD). To perform CFD calculations, a commercial solver FLUENT has been used, and the selective permeation through the membrane has been modeled by user-defined functions. The CFD simulation results exhibited the flow distribution in the reactor by inserting a membrane protection tube, in addition to the temperature and concentration distribution in the axial and radial directions in the reactor, as reported in the membrane reactor numerical simulation. On the basis of the simulation results, effects of the flow distribution, concentration polarization, and mass transfer in the packed bed have been evaluated to design a membrane reactor system. 

Governing Conservation Partial Equations Used in Computational Fluid Dynamic Approach... 

Computational fluid dynamics methods, which typically calculate flow field variables at himdreds of thousands of points inside the reactor to come up with overall reaction rates, are far better suited for the analysis of such systems. Another difference between CFD and traditional design methods is the minimal reliance of CFD on experimental data and extrapolation of that data to different scales, a process known as scale-up. Computational fluid dynamics relies on solving the fundamental equations of motion and conservation. These equations are scale independent and can be solved directly for the full-scale equipment. 

Computational fluid dynamics is based on the principle of solving conservation equations for all relevant variables. The conservation equations include the transport of the variable throughout the domain, as well as either its creation or its destruction. Conserved variables include ... 

Omran et al. have proposed a 3D, single phase steady-state model of a liquid feed DMFC [181]. Their model is implemented into the commercial computational fluid dynamics (CFD) software package FLUENT . The continuity, momentum, and species conservation equations are coupled with mathematical descriptions of the electrochemical kinetics in the anode and cathode channel and MEA. For electrochemical kinetics, the Tafel equation is used at both the anode and cathode sides. Results are validated against DMFC experimental data with reasonable agreement and used to explore the effects of cell temperature, channel depth, and channel width on polarization curve, power density and crossover rate. The results show that the power density peak and crossover increase as the operational temperature increases. It is also shown that the increasing of the channel width improves the cell performance at a methanol concentration below 1 M. 

The lattice Boltzmann method (LBM) is a relatively new simulation technique for complex fluid systems and has attracted great interests from researchers in computational physics and engineering. Unlike traditional computation fluid dynamics (CFD) methods to numerically solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy), LBM models the fluid as fictitious particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. Due to its particulate nature and local dynamics, LBM has several advantages over conventional CFD methods, especially in dealing with complex boundaries, incorporation of microscopic interactions, and parallel computation [1, 2]. 

Computational fluid dynamics, known as CFD, is the numerical method of solving mass, momentum, energy, and species conservation equations and related phenomena on computers by using programming languages. 

The lattice Boltzmann method (LBM) is a relatively new simulation technique for complex fluid systems which has attracted a great deal of interest from researchers in computational physics. Unlike the traditional computation fluid dynamics (CFD), which numerically solves the conservation equations of macroscopic properties (i. e., mass. 

The other form of mathematical model is the more rigorous computational fluid dynamics (CFD) approach that solves the complete three-dimensional conservation equations. These methods have been applied with encouraging results (Britter, 1995 Lee et al. 1995). CFD solves approximations to the fundamental equations, with the approximations being principally contained within the turbulence models—the usual approach is to use the K-e theory. The CFD model is typically used to predict the wind velocity fields, with the results coupled to a more traditional dense gas model to obtain the concentration profiles (Lee et al., 1995). The problem with this approach is that substantial definition of the problem is required in order to start the CFD computation. This includes detailed initial wind speeds, terrain heights, structures, temperatures, etc. in 3-D space. The method requires moderate computer resources. 

A single set of conservation eqnations valid for both porous electrodes and the free electrolyte region is derived and nnmerically solved using a computational fluid dynamics technique. This numerical methodology is capable of simulating a two-dimensional cell with the fluid flow taken into consideration. The motion of the liquid electrolyte is governed by the Navier-Stokes equation with the Boussinesq approximation and the continuity equation as follows ... 

Consequently, numerical solution of the equations of change has been an important research topic for many decades, both in solid mechanics and in fluid mechanics. Solid mechanics is significantly simpler than fluid mechanics because of the absence of the nonlinear convection term, and the finite element method has become the standard method. In fluid mechanics, however, the finite element method is primarily used for laminar flows, and other methods, such as the finite difference and finite volume methods, are used for both laminar and turbulent flows. The recently developed lattice-Boltzmann method is also being used, primarily in academic circles. All of these methods involve the approximation of the field equations defined over a continuous domain by discrete eqnalions associated with a finite set of discrete points within the domain and specified by the user, directly or through an antomated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD). 

---