Partial differential equations computer software

The material and energy balances of a tubular vessel are based on the conservation law, Eq 2.42, applied to a differential ring between r and r+dr and z and z+dz. A material balance is derived, for example, in problem P5.08.01, and is quoted in Table 2.6 along with the heat balance. The result is a pair of second order partial differential equations, usually nonlinear, that must be solved numerically. Table 2.6 indicates one possible procedure, but computer software is plentiful. 

Many advances in digital simulation have taken place since the publication of the first edition. Some of these advances have occurred in hardware through the development of the personal computer. Others have taken place by the development of commercial software that will perform specific simulations or will create a computer environment (e.g., a spreadsheet) that will allow one to do simulations without having to write a computer program. Finally, there have been theoretical advances where newer implicit algorithms are used to solve the necessary partial differential equations more efficiently than is possible using the more intuitive explicit methods described herein. 

J. Adams, MUDPACK Multigrid FORTRAN Software for the Efficient Solution of Linear Elliptic Partial Differential Equations, Appl. Math. Comput, 34 (1989). 

A dynamic tubular reactor model, comprising a set of partial differential equations, has been used to test the computational efficiency and the data handling capabilities of the various software packages. Experimental data of three time-varying model inputs, i.e. the reactor temperature, the fluid velocity and the reactant inlet concentration, are used to estimate the model parameters fix)m experimental data of the reactor temperature measured at several fixed reactor locations as a function of time. This problem was originally published in 1992 [3]. 

The numerical solution is performed by the method of lines. Spatial discretization of the partial differential-equation system using finite differences on statically adapted grids leads to large systems of ordinary differential and algebraic equations. This system of coupled equations is solved by an implicit extrapolation method using the software package LIMEX [14]. The code computes species mass-fraction and temperature profiles in the gas phase, fluxes at the gas-surface interface, and surface temperature and coverage as function of time. 

Complex coupled flow and heat transfer problems can be solved using numerical techniques in which the partial differential equations are converted to a large set of coupled algebraic equations, and the algebraic equations are then solved using conventional methods developed specificaUy to be efficient on digital computers. The concept by which the numerical solution of the partial differential equations is obtained is rather straightforward, and we wiU describe it here. Actual implementation into an efficient, user-friendly computer code is difficult and tedious, however, and most users employ commercial software. 

To analyze the airflow pattern, simulation of airflow was carried out using a fluid flow analysis package. Fluent 6.1 [1,6-12]. To solve the three-dimensional airflow field inside the nozzles, a CFD model was developed using the above software. Fluid flow and related phenomena can be described by partial differentiation equations, which caimot be solved analytically except in over-simplified cases. To obtain an approximate solution numerically, a discretization method to approximate the differential equations by a system of algebraic equations, which can be then numerically solved on a computer. The approximations were applied to small domains in space and/or time so the numerical solution provides results at discrete locations in space and time. Much of the accuracy depends on the quality of the methodology used, for which CFD is a powerful tool to predict the flow behavior of fluid inside any object. It provides various parameters such as air velocity profiles (axial, tangential, resultant etc.) and path lines trajectory, which are important for subsequent analysis. It was for those reasons that a CFD package. Fluent 6.1, which uses a Finite Volume (FV) method, was employed for airflow simulation. 

---