Explicit Finite Difference Formulation Research Articles

Development of a fast thermal-hydraulic model to simulate heat and fluid flow in MNSR

Extending a reliable and fast thermal–hydraulic model is the first stage in developing a micro-simulator for a miniature neutron research reactor (MNSR) with training and educational objectives. The framework of the TH model, which comprises an explicit finite difference formulation for solving coupled hydraulic and thermal equations with variable water properties, is explored in this study. The hydraulic section uses the integral form of the momentum equation to relate the mass flow rate variation to buoyancy and friction forces. The thermal part of the model relies on the integral form of the energy equation for different elements of the reactor including fuel, core, reflector, tank, chimney, and pool. The model's results are compared to reactor operation data and available correlations for various reactor powers. According to the comparison, the model's output matches the trend of experimental data for all power in both short and long time periods and predicts the eventual temperature rise of the core with an error of less than 5%. The performance of the model in other operational conditions is also evaluated. The model's performance in different operational circumstances is also assessed, and the results suggest that reducing the reactor's initial temperature would enhance the core's temperature rise.

Interview

Dr. Patrick Goh of Universiti Sains Malaysia, talks to us about the work behind the paper ‘A voltage-in-current formulation for the latency insertion method for improved stability’, page 1904 Dr. Patrick Goh LIM is a fast transient circuit simulation technique for high-speed interconnects. It was first developed by Professor Jose E. Schutt-Aine at the University of Illinois at Urbana-Champaign and I had the opportunity to work with him for a couple of years to advance the method. With the continuous device scaling towards faster and more compact designs, the integrity of the signals on printed circuit boards and other interconnects often becomes an important factor in the design process to ensure proper operation. In some cases, it might even be the bottleneck that prevents more aggressive revisions of the product specifications. Thus, designers need better computer-aided design (CAD) tools that can accurately simulate their designs and do so in a fraction of the time taken by conventional simulators and this is where LIM comes in. Most circuit simulators rely on the solution of a large matrix that is formed by the elements of the circuit to solve for the voltages and currents in the circuit. When the number of elements increases, the size and solution time of this matrix can quickly get out of hand. LIM, instead, takes advantage of the latencies present (the capacitance and inductance) in circuits at high frequencies to perform a leapfrog solution of the voltages and currents, thus achieving a better scaling with respect to the circuit size. Published results thus far show that the solution time of LIM scales linearly with the number of nodes and branches in the circuit. In our paper, my student Tan Kin Hang and I have developed a reformulation of LIM that does not suffer from the stability limitation of the original LIM. In addition, the reformulation is presented in a closed-form expression which will minimise any increase in the computational burden when adopting the new formulation. The original LIM was well suited to handle circuits with inherent latencies such as in transmission lines. However, because of its explicit finite-difference formulation, it suffers from a stability limitation that is exacerbated when there are small latency elements in the circuit. Our reformulation alleviates this problem by going towards an implicit formulation, while at the same time minimising the computational burden that normally comes along with an implicit method through a forward branch marching scheme. We believe that this will open up a whole new possibility for LIM. Without the limitation on the time step size, we can now use LIM to solve problems that were previously notoriously ill-conditioned for LIM. We can even consider using LIM for any type of circuits, not just high-speed and high-frequency circuits. Of course, there will be challenges. All finite difference methods suffer from a numerical integration error, which is a function of the time step that accumulates over the solution period. When using the original LIM, this error is rarely an issue as the time step is always small enough due to the limitation by the stability condition, but with the option of using larger time steps in the new formulation, this error might become significant and will have to be addressed.

Finite difference thermal model of a latent heat storage system coupled with a photovoltaic device: Description and experimental validation

The use of photovoltaic (PV) systems has been showing a significant growth trend but for a more effective development of this technology it is essential to have higher energy conversion performances. Producers of PV often declare an higher efficiency respect to real conditions and this deviation is mainly due to the difference between nominal and real temperature conditions of the PV. To improve the solar cell energy conversion efficiency many authors have proposed a methodology to keep lower the temperature of a PV system: a modified PV system built with a normal PV panel coupled with a Phase Change Material (PCM) heat storage device. In this paper is described a thermal model analysis of the PV–PCM system based on a theoretical study using finite difference approach. The authors developed an algorithm based on an explicit finite difference formulation of energy balance of the PV–PCM system. To this aim, a forward difference at time t and a first-order central difference for the space derivative at position x was used. Two sets of recursive equations were developed for two types of spatial domains: a boundary domain and an internal domain .The reliability of the developed model is tested by a comparison with data coming from a test facility. Results of this experience confirm the performed numerical simulations and show that the proposed model is valid and can be used to determine the thermal behaviour of a solar cell coupled with a PCM heat storage device.

On the Efficacy of PCM to Shave Peak Temperature of Crystalline Photovoltaic Panels: An FDM Model and Field Validation

The exploitation of renewable energy sources and specifically photovoltaic (PV) devices have been showing significant growth; however, for a more effective development of this technology it is essential to have higher energy conversion performances. PV producers often declare a higher efficiency respect to real conditions and this deviation is mainly due to the difference between nominal and real temperature conditions of the PV. In order to improve the solar cell energy conversion efficiency many authors have proposed a methodology to keep the temperature of a PV system lower: a modified crystalline PV system built with a normal PV panel coupled with a Phase Change Material (PCM) heat storage device. In this paper a thermal model analysis of the crystalline PV-PCM system based on a theoretical study using finite difference approach is described. The authors developed an algorithm based on an explicit finite difference formulation of energy balance of the crystalline PV-PCM system. Two sets of recursive equations were developed for two types of spatial domains: a boundary domain and an internal domain. The reliability of the developed model is tested by a comparison with data coming from a test facility. The results of numerical simulations are in good agreement with experimental data.

Stability of soil nailed slope using strength reduction method

This paper describes the influence of soil nail reinforcement on the stability of slopes through a numerical analysis that uses the strength reduction method. The stability of nailed slopes is evaluated in terms of factor of safety and potential slip surface. In this paper, the dual-spring structure element in the commercial package (FLAC3D) based on an explicit finite difference formulation is adopted to simulate the characteristics of the nails. A 3D slope model is built to simulate the excavations of the slope and the reinforcement of soil nails. The effects of nail length, orientation, installation location and horizontal spacing on the factor of safety and potential slip surface of the slope, as well as on the nails’ mechanical responses, are numerically investigated.

Influence of stress path on tunnel excavation response – Numerical tool selection and modeling strategy

The actual stress path in a rock mass during tunnel excavation is complex. To capture the correct tunnel excavation response, it is important to correctly resemble the stress path in situ in the numerical tools. FLAC and Phase 2 are two powerful two-dimensional continuum codes for modeling soil, rock, and structural behavior, in the fields of geotechnical, geomechanics and in civil and mining engineering. FLAC is based on explicit finite difference formulation while Phase 2 is based on implicit finite element formulation. When the two codes are applied to the analysis of tunnel excavation problems, difference in results might occur simply due to the different formulation methodologies used in these codes. It is shown that for linear elastic tunnel excavation problems, both codes provide the same result because stress path is unimportant. For tunnel excavation in elasto-plastic materials using long-round drill and blast method, there is significant difference in terms of yielding zone distribution by the two codes if conventional modeling approach is used, especially when the rock strength is low relative to the in situ stress magnitude. The mechanism of the difference is investigated and recommendation provided for choosing appropriate tools and modeling strategies for tunnel excavation problems. The importance of honoring the true stress path in tunnel excavation response simulation is illustrated using a few examples.

Inverse Heat Transfer Solution of the Heat Flux Due to Induction Heating

The explicit finite difference formulation of an inverse heat transfer model to calculate the heat flux generated by induction is developed. The experimentally measured temperature data are used as the input for the inverse heat transfer model. This model is particularly suitable for a workpiece with low cross section Biot number. Induction heating experiments are carried out using a carbon steel rod. The finite difference method and thermocouple temperature measurements are applied to estimate the induction heat flux and workpiece temperature. Compared to measured temperatures, the accuracy and limitation of proposed method is demonstrated. The effect of nonuniform temperature distribution, particularly in the heating region during the induction heating, is studied. Analysis results validate the assumption to use the uniform temperature in a cross section for the inverse heat transfer solution of induction heat flux. Sensitivity to the grid spacing, thermocouple location, and thermophysical properties are also studied.

Discrete Particle Distribution Model for Advection-Diffusion Transport

A new methodology, named DisPar, based on a discrete probability distribution for a particle displacement, was developed to solve 1D advection-diffusion transport problems in water bodies. The discrete probability distribution for the particle displacement was developed as an average and variance function. These probabilities were used to predict the deterministic mass transfer between cells in one time step, and therefore the particle concentration in each cell was considered the state variable. The state equation was found to be similar to an explicit finite-difference formulation with a Eulerian grid. The model stability, positivity, and mass conservation are guaranteed by the probability distribution concept. DisPar produces solutions without numerical dispersion for constant velocity, diffusion coefficient, and cross-sectional area. In these conditions, DisPar was also developed as a function of space and time for an instantaneous mass spill. When the stability and positivity restrictions were respected, the model produced excellent results when compared to analytical solutions and other methods. The discrete particle displacement distribution concept differs from other numerical formulations, and therefore it represents a new modeling technique.

Heat Flow Modelling of the Laser Drilling Process as Applied to Unidirectional Carbon Fibre Composites

AbstractLaser drilling has been carried out with a pulsed Nd:YAG laser. The laser has been applied to the drilling of APC-2 (PEEK-61%C fibre), with specimens up to 1 mm in thickness and using both moving and stationary beams. Hole shapes and heat-affected zone (HAZ) microstructures were studied using SEM, optical microscopy of polished sections and optical interferometric surface profilometry. Two prominent microstructural features were noted in the HAZ of the composite. These were (a) swelling of the ends of the carbon fibres (from about 8 microns to about 12 microns in diameter) and (b) an extensive zone around the hole in which the matrix had clearly undergone melting and some flow. The fibre swelling was such that neighbouring fibres were brought into close contact. The degree of swelling was found to remain constant as the laser pulse energy was increased, although the morphology of the fibre ends became more wrinkled and porous. The phenomenon is thought to be caused by internal gas pressure generation and distortion of the turbostratic structure within the fibre.A heat flow model has been developed to simulate the drilling process, based on an explicit finite difference formulation and a Cartesian (3-D) mesh. The post-iterative heat accumulation method was used to simulate the effect of latent heat absorption. The explicit approach was found superior to implicit schemes, which tend to exhibit poor stability when handling the phase change algorithm in regions of extremely high thermal gradient. The total solution time was kept within reasonable limits by using a graded mesh, which was fine only in the regions close to the beam axis. Comparisons between predicted and observed hole dimensions showed good agreement over a range of conditions, including variations in beam energy and beam motion. This suggests that effects omitted from the model, such as melt ejection, were not playing a significant role.

Analysis of impulsively loaded reinforced concrete structural elements—I. Theory

A computational approach is presented for the analysis of reinforced concrete beams and slabs under the effects of transient uniformly distributed loads. This approach is based on the Timoshenko beam formulation with non-ideal support conditions. In addition, non-linear material properties, strain are effects, load-reversals, and rational structural behavior models have been included. The solution is achieved by implementing an explicit finite difference formulation. The numerical aspects of the problem are treated and the issue of stability and convergence are addressed.

A comparison of finite difference and reflectivity seismograms for marine models

Summary Finite difference and reflectivity synthetic seismograms are compared for laterally homogeneous seafloor models with step and ramp discontinuities in elastic properties. Given suitable numerical parameters in each case excellent agreement can be obtained. For step discontinuities between a liquid and a solid the explicit finite difference formulation of the elastic wave equation for heterogeneous media is unstable. It is necessary to introduce the boundary conditions specifically at the interface and suitable formulations correct to first and second order in the space increment are given. For ramp discontinuities between a liquid and a solid the finite difference formulation for heterogeneous media is stable and agrees with the reflectivity method when, in the latter case, the gradient is approximated by homogeneous layers thinner than one-fifth of the minimum compressional wavelength at the upper half-power frequency. For a step discontinuity between two solids the explicit finite difference formulation of the elastic wave equation for heterogeneous media, although stable, is inaccurate and boundary conditions must again be specifically introduced.

On a finite-difference solution for the constant-property turbulent boundary layer.

A finite-differe nce method has been developed for the solutions to the governing partial differential equations for the constant-property turbulent boundary layer. Prandtl's mixinglength concept was used to express the apparent turbulent shearing stress according to a hypothesized mixing-length distribution through the boundary layer. The model for the mixing-length distribution is in general agreement with experimental data for a variety of flow conditions. The method, an always stable explicit finite-difference formulation that requires no iterative procedures, is numerically more direct than other methods recently proposed and is unique in its evaluation of the apparent shearing stress by using an assumed mixing-length distribution as the only empirical input. The predicted skin-friction coefficients and velocity profiles agree well with experimental data for the several comparisons made, which included flows in both favorable and adverse pressure gradients as well as flat plate cases with and without blowing. The numerical method presented is not restricted to use with the particular mixing-length model tentatively proposed in this work, but can be used to compare various models and help establish their properties and range of applicability, thereby serving to further understanding of the most fundamental aspects of turbulent flow. a c Cf gc I

A Digital Computer Solution of the Laplace Equation Using the Dynamic Relaxation Method

SummaryThe Dynamic Relaxation solution of the Laplace equation introduces dynamic terms into the basic equation. When this is written as an explicit finite difference formulation it can be solved by an iterative process which only requires a simple substitution routine. The method is easy to programme and requires small storage in the computer. By studying problems involving wind tunnel interference in steady flow, the potentialities of the method are demonstrated.