Semi Implicit Research Articles

Numerical Computation of Heat and Mass Transfer in Fluidized Beds with Liquid Injection

AbstractThe goal of this work is the simulation of concentration and temperature distributions insides fluidized bed with a uniform liquid distribution. Further, a physically based 2D model is developed for the heat and mass transfer processes in fluidized beds with a spray nozzle. The model is a coupled and semi‐linear system of convection‐reaction‐diffusion equations. We considered the numerical solution of these semi‐linear partial differential equations with discrete boundary conditions using linear finite elements on an adaptive triangular grid in space and implicit methods in time. We present calculations using, semi implicit and implicit methods in time, and different solvers for solving the linear systems. The complex correlations of mass and liquid flow rates, mass transfer, heat transfer, drying, and transient two dimensional air humidity, air temperature, degree of wetness, liquid film temperature and particle temperature were simulated. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multi-scale finite element analyses of sheet metals by using SEM-EBSD measured crystallographic RVE models

In this study, two multi-scale analyses codes are newly developed by combining a homogenization algorithm and an elastic/crystalline viscoplastic finite element (FE) method (Nakamachi, E., 1988. A finite element simulation of the sheet metal forming process. Int. J. Numer. Meth. Eng. 25, 283–292; Nakamachi, E., Dong, X., 1996. Elastic/crystalline viscoplastic finite element analysis of dynamic deformation of sheet metal. Int. J. Computer-Aided Eng. Software 13, 308–326; Nakamachi, E., Dong, X., 1997. Study of texture effect on sheet failure in a limit dome height test by using elastic/crystalline viscoplastic finite element analysis. J. Appl. Mech. Trans. ASME(E) 64, 519–524; Nakamachi, E., 1998. Elastic/crystalline viscoplastic finite element modeling based on hardening–softening evaluation equation. In: Proc. of the 6th NUMIFORM, pp. 315–321; Nakamachi, E., Hiraiwa, K., Morimoto, H., Harimoto, M., 2000a. Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification. Int. J. Plasticity 16, 1419–1441; Nakamachi, E., Xie, C.L., Harimoto, M., 2000b. Drawability assessment of BCC steel sheet by using elastic/crystalline viscoplastic finite element analyses. Int. J. Mech. Sci. 43, 631–652); (1) a “semi-implicit” finite element (FE) code and (2) a “dynamic explicit” FE code. These were applied to predict the plastic strain induced yield loci and the formability of sheet metal in the macro scale, and simultaneously the crystal texture and hardening evolutions in the micro scale. The isotropic and kinematical hardening laws are employed in the crystalline plasticity constitutive equation. For the multi-scale structure, two-scales are considered. One is a microscopic polycrystal structure and the other a macroscopic elastic plastic continuum. We measure crystal morphologies by using the SEM-EBSD apparatus with a unit of about 3.8 μm voxel, and define a three dimensional (3D) representative volume element (RVE) for the micro polycrystal structure, which satisfy the periodicity condition of crystal orientation distribution. A “micro” finite element modeling technique is newly established to minimize the total number of finite elements in the micro scale. Next, the “semi-implicit” crystallographic homogenization FE code, which employs the SEM-EBSD measured RVE, is applied to the 99.9% pure-iron uni-axial tensile problem to predict the texture evolution and the subsequent yield loci in the various strain paths. These “semi implicit” results reveal that the plastic strain induced anisotropy in the micro and macro levels can be predicted by our FE analyses. The kinematical hardening law leads a distinct plastic strain induced anisotropy. Our “dynamic-explicit” FE code is applied to simulate the limit dome height (LDH) test problem of the mild steel DQSK, the high strength steel HSLA and the aluminum alloy AL6022 sheet metals, which were adopted as the NUMISHEET2005 Benchmark sheet metals (Smith, L.M., Pourboghrat, F., Yoon, J.-W., Stoughton, T.B., 2005. NUMISHEET2005. In: Proc. of 6th Int. Conf. Numerical Simulation of 3D Sheet Metal Forming Processes, PART A and B(Benchmark), pp. 409–451) to estimate formability. The “dynamic explicit” results reveal that the initial crystal orientation distribution has a large affects to a plastic strain induced texture and anisotropic hardening evolutions and sheet formability.

A microwave plasma model for a PCVD setup

A numerical model is constructed using the PLASIMO toolkit to simulate a microwave configuration which is similar to that which is in use for optical glass fibre production. The simulations offer the flow patterns and the electromagnetic (EM) energy incoupling of a two-temperature argon plasma. The Yee algorithm was used for the EM module, whereas the Semi implicit method for pressure linked equations algorithm was used to calculate the pressure and velocity field. It is found that at 400 W coupling of 2.46 GHz EM radiation in 1000 Pa argon results in a plasma that is not in local thermodynamic equilibrium, in the sense that Te/Th ≥ 4 and the ionization degree is smaller than that predicted by the Saha equations. The model results are subjected to various sanity checks.

Numerical simulation of liquid metals in differentially heated enclosure undergoing orthogonal rotation

In this work a numerical investigation has been performed to examine the characteristics of mixed convective heat transfer in square enclosures undergoing orthogonal rotation i.e. rotation axis and gravity axis are orthogonal to each other. A semi implicit finite difference code on a collocated grid is used to solve the momentum and energy equations subject to Boussinesq approximation. The study is carried out for a wide range of operating parameters such as Rayleigh number ( Ra), Taylor number ( Ta), Rotational Rayleigh number ( Ra w) for a fixed Prandtl number ( Pr). The numerical experiments have been carried out for a fixed Pr = 0.01, Ra varies from 10 5 to 10 7 while Ta and Ra w vary from almost 0 to 10 9. Results reveal that significant increase or decrease in heat transfer rates can be achieved by the rotational effects, mainly influenced by centrifugal force.

Isothermal flow in an optical fiber coating applicator and die system

The isothermal flow generated by a fast-moving optical fiber in a coating applicator and die was studied numerically. Although it is expected that thermal effects can be important in typical coating processes, the isothermal flow behavior should be thoroughly investigated before including the heat transfer considerations. Thus, isothermal flow is studied in this paper, and key parameters that play a major role in the optical fiber coating process are investigated in detail. The coupled partial differential equations that govern the fluid flow are solved on a transformed nonuniform staggered grid. A finite volume method is employed with a semi implicit method for pressure linked equations (SIMPLE)-based algorithm for the pressure calculations. The fiber speed is varied from 0.1 up to 22 m/s, and various applicator sizes are considered. The diameters at the die entrance and at the die exit are varied separately, since the geometry of the applicator and of the die can play an important role. Both free and solid wall surfaces are taken for the boundary condition at the top surface in the applicator. The shear-free condition can be regarded as direct contact of the coating fluid with the cooling gas at the top surface. Three different viscosities were chosen, based on reference temperatures, taken as the fluid inlet temperature, the fiber inlet temperature, and the average temperature. Relatively low pressure differences, compared to those in the applicator and the die, were found at the dynamic contact point where the fiber first enters the coating applicator. This suggests that air or gases used for cooling the fiber may be able to enter through this location. It was also found that the shear rates in this region are comparatively higher. The temperature-dependent viscosities were found to be of critical importance in the flow. The change in size of the applicator did not significantly affect the mass flow rates at the die exit, whereas the pressure gradients and shear rates near the dynamic contact point were strongly affected. The flow field in the die changed substantially with a change in the exit size. Maximum pressures and shear rates in the die increased drastically when the die exit became smaller. This study also suggests that a more accurate representation of the upper meniscus is needed to simulate flow near the fiber entry into the applicator.

Jacobian–Free Newton–Krylov Methods for the Accurate Time Integration of Stiff Wave Systems

Stiff wave systems are systems which exhibit a slow dynamical time scale while possessing fast wave phenomena. The physical effects of this fast wave may be important to the system, but resolving the fast time scale may not be required. When simulating such phenomena one would like to use time steps on the order of the dynamical scale for time integration. Historically, Semi-Implicit (SI) methods have been developed to step over the stiff wave time scale in a stable fashion. However, SI methods require some linearization and time splitting, and both of these can produce additional time integration errors. In this paper, the concept of using SI methods as preconditioners to Jacobian---Free Newton---Krylov (JFNK) methods is developed. This algorithmic approach results in an implicitly balanced method (no linearization or time splitting). In this paper, we provide an overview of SI methods in a variety of applications, and a brief background on JFNK methods. We will present details of our new algorithmic approach. Finally, we provide an overview of results coming from problems in geophysical fluid dynamics (GFD) and magnetohydrodynamics (MHD).

An algorithm for the elastic flow of surfaces

A semi implicit fully discrete finite element scheme for the computation of the parametric elastic flow of two dimensional surfaces in $\\r^3$ based on a variational form is presented. Linear finite elements are used for the space discretization of a mixed formulation. Time discretization is carried out by a semi implicit method to linearize the problem.

A 2D high- β Hall MHD implicit nonlinear solver

A nonlinear, fully implicit solver for a 2D high- β (incompressible) Hall magnetohydrodynamics (HMHD) model is proposed. The task in non-trivial because HMHD supports the whistler wave. This wave is dispersive ( ω∼ k 2) and therefore results in diffusion-like numerical stability limits for explicit time integration methods. For HMHD, implicit approaches using time steps above the explicit numerical stability limits result in diagonally submissive Jacobian systems. Such systems are difficult to invert with iterative techniques. In this study, Jacobian-free Newton–Krylov iterative methods are employed for a fully implicit, nonlinear integration, and a semi-implicit (SI) preconditioner strategy, developed on the basis of a Schur complement analysis, is proposed. The SI preconditioner transforms the coupled hyperbolic whistler system into a fourth-order, parabolic, diagonally dominant PDE, amenable to iterative techniques. Efficiency and accuracy results are presented demonstrating that an efficient fully implicit implementation (i.e., faster than explicit methods) is indeed possible without sacrificing numerical accuracy.

An efficient numerical method for distributed-loop models of the urine concentrating mechanism

In this study we describe an efficient numerical method, based on the semi-Lagrangian (SL) semi-implicit (SI) method and Newton’s method, for obtaining steady-state (SS) solutions of equations arising in distributed-loop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newton-type solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newton-type solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newton-type solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.

Physical and mathematical modelling of flow and residence time distributions in different tundish designs

A steady state, three‐dimensional, turbulent flow model has been developed in‐house for analysis of melt flow and residence time distribution phenomena in steelmaking tundish system. The governing equations of flow, turbulence and tracer dispersion were derived in terms of the Cartesian co‐ordinate systems and solved numerically with their associated boundary conditions adapting a control volume based finite difference procedure. In the numerical solution scheme, the pressure‐velocity coupling was treated via the popular Simple (semi implicit method for pressure linked equations) algorithm.Prior to carrying out elaborate numerical predictions for tundish geometry, the model was applied to several standard test problems and evaluated against corresponding bench mark results. Thus, several typical test problems such as, flows in a cubic cavity, flows in ducts of rectangular cross‐section, flow over flat plate and so on were simulated numerically to assess the adequacy and appropriateness of the computational procedure developed. Results thus obtained together with the bench mark solutions indicated that the mathematical model is internally consistent and sufficiently robust. Accordingly, the turbulent flow model was applied to simulate flow and Residence Time Distributions (RTD) in four different tundish designs . These included, a single strand and a two strand slab casting tundish systems, a six strand rectangular shaped tundish and a six strand delta shaped billet casting tundish. Various RTD parameters (e.g., minimum break through time, tmin, time at which peak concentration occurs, tpeak and average residence time, tav) were computed numerically in the four tundish systems and these were subsequently compared with corresponding experimental measurements derived from equivalent water model tundish systems. Except for the single strand tundish system, large differences between measurements and prediction (particularly on tmin and tpeak) were noted for the other three tundish geometries. Furthermore, the extent of such discrepancy was found to be relatively more pronounced for the multi‐strand tundish system. The possible reasons for such discrepancy is discussed in the text and it was shown computationally that relatively better agreement between theory and measurement can be achieved if, instead of the high Reynolds number k‐ε turbulence model, a low Reynolds number turbulence model is applied in the computational procedure.

CFD DEVELOPMENT AND SIMULATION OF FLOW FIELDS IN VENTILATED SPACES WITH MULTIPLE OCCUPANTS

Simulation of flow fields in ventilated spaces with multiple occupants (cows) was conducted using a computational fluid dynamics (CFD) technique. Airflow in a ventilated space is affected by the locations of the inlet and outlet, inlet or room shape, and positions of the occupants. Three simulation test cases considered are: (1) four cows standing in a row with one end wall simulating as an inlet, and the opposite end wall simulating as an outlet; (2) four cows standing in a row with the two end walls simulating as inlets, and an outlet located at the center of the ceiling; and (3) ten cows placed at random with inlet and outlet arrangements the same as in Case (2). The body–fitted coordinate (BFC) system is adopted to characterize the shape of a cow, and the governing equations in the Cartesian coordinate system are transformed into the BFC system. To generate the grid under the BFC system, partial differential equations and algebraic interpolation methods were used. The finite volume method is applied to transform the partial differential equations into algebraic ones, and the SIMPLE (Semi–Implicit Method for Pressure–Linked Equation) algorithm is extended to the BFC system to solve the flow field. The effects of obstructions (cows), and the positions of cows, inlets, and outlet on flow distribution were clearly demonstrated by velocity vectors, velocity profiles, and streamline plots generated. The flow field was more evenly distributed when cows were positioned at random than when in a row.

Numerical simulation of two-dimensional room air flow with and without buoyancy

This investigation deals with the velocity and temperature distribution in a room heated by a warm air stream introduced at various levels. Numerical solutions of Navier Stokes equations and energy equation have been obtained by SIMPLE and Semi Implicit Method for Pressure Linked Equation-Consistent (SIMPLEC) algorithms. Steady, laminar, and incompressible flow under Boussinesq's approximation has been considered. Solutions are presented for various locations of inlets and outlets, and for different values of Grashof number and Reynolds number. If the outlet is located at a lower level than the inlet, an increase in Gr makes the warm jet almost horizontal, which finally bends downwards towards the exit. An increase in Gr increases the intensity of recirculation and yields uniform temperature distribution. Location of outlet at higher level than the inlet leads to better temperature distribution.

A projection method for shallow water equations

The dynamics of shallow water has been studied and an algorithm for this dynamics has been developed. Results have been obtained with data of the Venice lagoon using a model made expressively by a semi‒implicit method based on a finite element method in space. Comparison has been made between field data and the results of the simulation. Very good agreement is shown over a long period of simulation. © 1998 John Wiley & Sons, Ltd.

Multidomain implicit numerical scheme

A multidomain method for the solution of elliptic CFD problems with an ADI scheme is described. Two methods of treatment of internal boundary conditions for ADI functions are discussed, namely an explicit and a semi‒implicit method. Stability conditions for the proposed methods are derived theoretically. The semi‒implicit scheme is more stable than the explicit scheme, leading to improved numerical efficiency for multidomain computations. Numerical computations for a linear convection-diffusion equation, for buoyancy‒driven recirculating flow in a square cavity and for turbulent flow in a square duct confirmed the theoretical results. Computer runs of the multidomain code in a distributed memory multiprocessor system were successful and efficient and produced reliable results. © 1997 John Wiley & Sons, Ltd.

The droplet diffusion model—An empirical model for micromixing in reacting gas phase systems

An empirical mixing model for mixing of reacting species in a fluid is proposed where the mixing process is divided into macromixing and micromixing. Macromixing is assumed to be instantaneous and it is modelled by dividing the injected fluid into a number N d of identical droplets, each surrounded by the same amount of bulk fluid. Micromixing is modelled as a molecular diffusion process in each droplet. The model can be used for any fluid residence time distribution E( t). The model is solved by means of collocation polynomials and numerical integration of the set of coupled differential equations using a third order semi implicit Runge-Kutta method. The proposed model is used to simulate pilot plant experiments of the selective non-catalytic reduction of NO by NH 3. The experiments have previously been simulated using an ideal plug flow reactor model without success. Combining the empirical kinetic model for the reactions with this mixing model gives satisfying results when choosing the mixing parameter corresponding to a mixing time of 20–30 ms.

Modelling of slurry reactors. A fundamental approach

Multiphase reactors in the form of bubble columns and stirred tank reactors are being used in large numbers throughout the petrochemical and biochemical industry today, and steadily for more complicated reaction systems. It has become increasingly more important to develop good, and theoretically sound models for scaling up and design of the reactors. The most common models used today rely heavily upon empirical information obtainable only from pilot size experiments and this makes “a priori” design from small scale data difficult. The model presented is based on a “first principles” approach where the parameters are fundamental in nature and as far as possible, obtainable from general fluid dynamics literature. It is a dynamic two phase model incorporating a k-ε turbulence model, compressible gas phase and axisymmetry. The equations are discretisized using a finite volume technique with an upwind or hybrid differencing scheme and solved using a variation of the SIMPLE (Semi Implicit Method for Pressure Linked Equations) method. The results from simulations of a two-phase bubble column are compared with measured radial distributions of gas hold-up, liquid velocity and turbulent kinetic energy using air and desalinated water as media. As chemical reaction, the production of methanol from synthesis gas is implemented. Simulation results using a pseudohomogenous 3-phase slurry model are compared with data from literature.

Numerical Study of a Dump Diffuser Flow Field in a Short Annular Combustor

A computer program for solving the two-dimensional steady flow field of the dump diffuser is made based on the streamtube method and SIMPLE (Semi–Implicit Method for Pressure linked Equations) algorithm. Some concrete measures or improvements are developed to treat the irregularly shaped calculation domain, the inlet distribution of k and e to use the variable relaxation factors. The flow field in a short dump diffuser model with pre–diffuser and blind flame tube is calculated with the program. The calculated results agree well with the experimental data.Copyright © 1985 by ASME

A FINITE ELEMENT METHOD FOR STORM SURGE AND TIDAL COMPUTATION

The paper reports the current progress in developing a finite element method for the shallow water equations. Some recent developments as the implementation of a semi implicit scheme or the use of an incident wave condition are described. Different realistic applications are presented concerning tidal and storm surge simulations.