Charl Gabriel Du Toit

• Doctor of Philosophy
• Chair at North-West University

A proper understanding of the porous structure of packed beds of spheres is imperative in the analysis and design of the processes involving fluid flow and heat and mass transfer. The radial variation in porosity is of specific interest. When the positions and sizes of the spheres are known, the radial variation in porosity can be determined using...

This paper proposes a vertical line-based methodology for the calculation of the porosity of packed beds of spheres. The results for the overall porosity, angular variation in porosity and tangentially averaged radial variation in porosity obtained through the use of this methodology are compared with the results of pre-existing methodologies and f...

The packing structure of pebble beds has a significant influence on their macroscopic thermal and mechanical performance. An advanced experimental X-ray tomography technique was applied to study the structural properties of binary pebble beds. Based on the reconstructed geometry, structural properties were analyzed employing different approaches. T...

1D/3D co-simulation is a promising approach for the modelling and simulation of complex systems. These systems are typically computationally intensive. By combining the desired features of a one-dimensional (1D) systems code and a three-dimensional (3D) Computational Fluid Dynamics (CFD) code, the required resources are optimized whilst still retai...

The paper deals with the evaluation of the distribution of the porosity of a model of the HTR-10 packed bed reactor which was generated using a Discrete Element Method (DEM) approach. The porosity of the packed bed was determined using semi-analytical methodologies. To evaluate the validity of the model, selected aspects of the DEM approach were fi...

The focus in the paper is on an evaluation of the friction factors as a function of the modified Reynolds number for the pressure drop through randomly and structured packed beds consisting only of uniform sized spheres. In the evaluation of the friction factor for large aspect ratio randomly packed beds it is confirmed that the use of the Ergun co...

In this study, steady-state single-phase natural circulation of water in square loops is simulated using a 1D network code. Fundamental explanations are given for the behaviour of the buoyancy force and mass flow rate. The methodology and model are validated by the very good agreement with the generalized correlation of the predicted steady-state R...

The paper discusses the numerical solution of the one-dimensional radially axi-symmetric non-linear second-order differential equation to model the conduction and radiation transfer through a spherical domain as a result of an exothermic heat source. The equation is transformed to a non-dimensional form. The dimensionless numbers emanating from the...

The influence of different packing structures of packed bed particles on the fluid flow through packed beds was investigated. The flow through cylindrical packed beds consisting of mono-sized spherical particles was simulated using an explicit approach. These beds had cylinder to particle diameter ratios between 1.6 and 6.33. The results for the fr...

The paper provides a comprehensive summary of the analysis of the porous structure of cylindrical packed beds with aspect ratios between 1 and 2. For the first time a proper geometrical explanation is given for the shape of the curve for the overall porosity as a function of the aspect ratio. Errors found in the published equations for the lateral...

This study investigates firstly, the effects of varying insulation thickness and flow path width between the core barrel and RCCS-concrete pressure vessel of an upward-flow helium-cooled pebble bed HTGR. The effects on component temperatures, parasitic heat loss as well as accident condition temperatures are determined. Secondly, the effect of refl...

Modelling prismatic block high temperature gas-cooled reactor units poses several challenges and typically requires significant computational resources and computational times. This is largely due to the complexity of prismatic block reactor models as these models must amongst others account for the fuel-compacts, graphite and coolant channels in f...

The VHTR (Very High Temperature Reactor) Prismatic block reactor is one of the reactor designs considered in the range of 4th Generation Reactors which are under development today. The heat generated by the gas-cooled reactor can be used to produce electricity or as process heat to produce amongst others hydrogen. For the operation of the reactor u...

The heat transfer and flow through the prismatic fuel blocks are very important for the safe operation of the VHTR core. Computational Fluid Dynamics (CFD) has been used in the industry to simulate different phenomena in the fuel blocks to a very accurate level. A convection heat transfer correlation was developed by Travis & El-Genk (2013b) to cal...

The effective thermal conductivity is an important parameter that represents the overall heat transfer in packed beds, including packed pebble beds. Experimental test facilities have been used to determine the effective thermal conductivity of packed pebble beds; however, the experimental results are often case specific and do not account for wall...

In this paper methodologies are described for the analysis of the porous structure of cylindrical and annular packed beds consisting of spheres when the coordinates and sizes of the spheres are known. Methodologies to determine the overlap between adjacent spheres and between the spheres and the walls of the container; axial variation in the porosi...

Part-load operation conditions of a centrifugal pump will impose potentially strenuous effects on the operation and reliability of the turbomachine in question due to the risk of suction recirculation. Suction recirculation in centrifugal pumps can be described as the flow reversal of the outer streamline in the impeller suction eye during operatio...

This paper presents a numerical investigation into thermal–fluid phenomena in a supercritical carbon dioxide-cooled pebble bed reactor (SCPBR) core under steady state using computational fluid dynamic. In this study, a three-dimensional model with the capability to simulate the fluid flow and heat transfer in the SCPBR core has been developed. The...

The effective thermal conductivity is an important parameter that is representative of the overall heat transfer in a packed bed of spheres. It includes the effects of conduction through the solid material and contact areas between spheres, conduction through the stagnant gas phase as well as thermal radiation between the surfaces of the spheres. A...

The effective thermal conductivity represents the overall heat transfer characteristics of a packed bed of spheres and must be considered in the analysis and design of pebble bed gas-cooled reactors. During depressurized loss of forced cooling conditions the dominant heat transfer mechanisms for the passive removal of decay heat are radiation and c...

The physical phenomena associated with the modes of heat transfer within a packed pebble bed can be modelled by explicitly solving the associated partial differential equations using a finite volume method approach. Convection from the spheres to coolant, conduction between spheres as well as radiation between various components are the main modes...

A Rotationally Symmetric Spherical Discretisation (RSSD) technique has recently been developed for the discretisation of scattering phase functions in the Discrete Ordinates Method (DOM) in a consistent and rational manner. RSSD has inherent energy conservation, is suitable for any quadrature scheme used, minimising run-time matrix calculation, and...

The GAMMA+ and Flownex codes are both based on a one-dimensional flow network modelling approach and both can account for any complex network of different heat transfer phenomena occurring simultaneously. However, there are notable differences in some of the detail modelling aspects, such as the way in which the convection in the reactor cavity is...

The GAMMA+ and Flownex codes are both based on a one-dimensional flow network modelling approach and both can account for any complex network of different heat transfer phenomena occurring simultaneously. However, there are notable differences in some of the detail modelling aspects, such as the way in which the convection in the reactor cavity is...

The hydrodynamic equations for a long-bowl gas centrifuge can be simplified to obtain a one-dimensional differential equation for the variation of the non-dimensional axial velocity in the radial direction. In this study the solution of the one-dimensional differential equation was revisited and an approximate finite element solution of the differe...

Purpose The flow characteristics in packed beds with small cylinder diameter to particle diameter ratios were investigated numerically using the commercial CFD package Star-CCM+®. Local rises in total pressure on the surfaces of the particles were observed. This paper reviews the relationship between the rise in total pressure on the surfaces of th...

The hydrodynamic equations for a long-bowl gas-centrifuge can be simplified to obtain a onedimensional differential equation for the variation of the non-dimensional axial velocity in the radial direction. This equation can be solved subject to the relevant boundary conditions and conservation of mass to provide the required velocity distribution....

Thermal-fluid simulations are used extensively to predict the maximum fuel temperatures, flows, pressure drops and thermal capacitance of pebble bed gas cooled reactors in support of the reactor safety case. The PBMR company developed the HTTF test facility in cooperation with M-Tech Industrial (Pty) Ltd. and the North-West University in South Afri...

Thermal-fluid simulations are used extensively to predict the maximum fuel temperatures, flows, pressure drops and thermal capacitance of pebble bed gas cooled reactors in support of the reactor safety case. The PBMR company developed the HTTU non-nuclear test facility in cooperation with M-Tech Industrial (Pty) Ltd. and the North-West University i...

This paper presents a new approach to the calculation of the effective thermal conductivity in packed pebble bed reactors, namely the Multi-sphere Unit Cell model. The model specifically accounts for the porous structure, which is characterised using the radial variation in porosity, coordination number and contact angles between adjacent spheres....

Engineers are faced with two major challenges when carrying out the thermal-fluid design of a complex system consisting of many interacting components. The first challenge is to predict the performance of all the individual thermal-fluid components. The second challenge is to predict the performance of the integrated plant consisting of all its sub...

This study deals with the determination of the positions of the spheres in packed beds using X-ray computed tomography. The quality of the data is assessed by considering the overlap between the spheres and the overall porosity obtained from the axial variation in the porosity. Good agreement is obtained with the theoretical predictions for the ove...

This paper presents a review of the literature describing the packing structure and effective thermal conductivity of randomly packed beds consisting of mono-sized particles. In this study particular attention was given to the packing structure (porosity, coordination number, and contact angles) and heat transfer by solid conduction, gas conduction...

Engineers are faced with two major challenges when carrying out the thermal-fluid design of a complex system consisting of many interacting components such as the PBMR power plant. The first challenge is to predict the performance of all the individual thermal-fluid components. The second challenge is to predict the performance of the integrated pl...

A finite element technique is described to determine the wall shear stresses and the axial temperature gradient for hydraulically and thermally fully developed laminar flow in concentric annuli with prescribed uniform heat fluxes at the walls as part of the primary solution. The technique was used to determine the Reynolds friction factors and the...

The paper focuses on the Global Galerkin finite element (GGFEM) solution of the fourthorder differential for an Euler-Bernoulli beam. In this approach the point loads and point moments are treated in a natural way as part of the system of external forces and are included in the differential equation during the definition of the problem. The Galerki...

This paper highlights the usefulness of applying analytical methods for uncertainty analysis to a least-squares polynomial fit. It also presents an equation to calculate the confidence interval in the slope of a regression polynomial. The specific area of application is the determination of the temperature gradient as a function of position from a...

Most numerical models simulate the superficial velocity distribution in a region inside a packed bed where it can be assumed that the fluid flow is fully developed. The in- and outlet regions of the packed bed are thus ignored since no correlation, describing the porosity distribution in these regions of the packed bed, is available. Special attent...

The question arose whether the correlation for the pressure drop prescribed for cylindrical pebble bed reactors by the Nuclear Safety Commission (KTA) of Germany could still be applied to the proposed annular configuration of the Pebble Bed Modular Reactor (PBMR) currently being developed in South Africa. An approach is described which uses the ext...

A proper understanding of the mechanisms of heat transfer, flow and pressure drop through a packed bed of spheres is of utmost importance in the design of a high temperature pebble bed reactor. A thorough knowledge of the porous structure within the packed bed is important to any rigorous analysis of the transport phenomena, as all the heat and flo...

This paper discusses the numerical analysis of the porous structure of an annular packed bed. Numerical procedures are described to determine the variation in the porosity, the variation in the coordination number (measure of the number of particles in contact with a specific particle) and the variation in the contact angle between adjacent particl...

This paper describes the results of a study of the variation in the radial direction in the porosity of annular packed beds. A comparison is made between the predictions by a number of empirical correlations found in literature, the porosities obtained from physical experiments and the porosities obtained from the analysis of numerically generated...

Complex thermal-fluid systems may consist of many interacting components such as pipes, heat exchangers, turbines and boilers. The first of two major design challenges is to predict the performance of all the thermal-fluid components on the system level and the second is to predict the performance of the integrated plant consisting of all its sub-s...

This paper provides an overview of the theoretical basis for a new thermal-fluid systems CFD simulation model for high temperature gas-cooled reactors, contained in the Flownex software code. Flownex provides for detailed steady-state and transient thermal-fluid simulations of the complete power plant, fully integrated with core neutronics and cont...

The theoretical basis and conceptual formulation of a comprehensive reactor model to simulate the thermal-fluid phenomena of the PBMR reactor core and core structures is given. Through a rigorous analysis the fundamental equations are recast in a form that is suitable for incorporation in a systems CFD code. The formulation of the equations results...

The paper focuses on the finite element solution of the fourth-order differential for an Euler-Bernoulli beam. In this approach the point loads and concentrated moments are treated in a natural way as part of the system of external forces and are included in the differential equation during the definition of the problem. The Galerkin formulation is...

The accuracy of two-phase flow simulations is mainly determined by the mathematical formulations and resolution of the computational mesh used in the numerical model. Selective mesh refinement improves the accuracy of results while maintaining low resource requirements. Object-oriented simulation software was developed that allows the mathematical...

A finite-element technique is described to determine the wall shear stresses and heat fluxes for fully developed laminar flow at the wall of a rectangular duct. It is shown that these values are accurate and that they can in some instances be independent of the grid resolution. These values are then used to obtain the friction factors and Nusselt n...

The pebble-bed reactor can be considered as a packed bed reactor. The mechanisms of fluid flow and heat transfer in such a bed are sensitive to the porosity distribution of the bed. In this paper numerical procedures are presented to evaluate the axial and radial variations of the porosity in a cylindrical packed bed. The application of the procedu...

For the efficient operation of electrostatic precipitators gas flow patterns should be optimised and particle collection must be effective. A numerical model is presented to investigate the electrohydrodynamic conditions inside a wire-plate precipitator. The finite-volume approach to the electrostatic and fluid dynamic models and Lagrangian approac...

System simulation is a valuable tool in the design of thermo-fluid systems like the Pebble Bed Modular Reactor nuclear power plant currently being developed for ESKOM in South Africa. A simplified and easy to use cycle simulation can be employed successfully in the conceptual design phase when parametric studies are being performed to determine the...

A segregated finite element algorithm for the solution of the SUPG formulation of the incompressible steady-state Navier–Stokes equations for non-isothermal flow is presented in this paper. The method features equal order interpolation for all the flow variables. The SIMPLER and SIMPLEST algorithms are employed and the sets of non-symmetric linear...

A segregated finite element algorithm for the solution of the SUPG formulation of the incompressible steady-state Navier-Stokes equations is investigated in this paper. The method features equal order interpolation for all the flow variables. The SIMPLEST algorithm is employed which results in symmetric coefficient matrices for the momentum equatio...

Making use of radiation and convective heat transfer theory the efficiency of a collector fitted with an absorber consisting of spaced parallel tubes is modelled on a personal computer. The absorptance of the tubes for sunlight and the convection coefficient between the back insulation and the tubes are adjusted until agreement between experimental...

The influence of the air flow pattern on the effectiveness of a mechanical–draught heat exchanger under the influence of a cross–wind is studied. A thorough understanding of the flow field and associated phenomena is imperative for the optimal design of such systems.

The results are reported of velocity measurements in oscillatory flow over rippled beds. Velocities were measured with a laserdoppler anemometer in both an oscillating tray rig and an oscillatory flow U-tube. Both self-formed and artificial ripples were examined. The results obtained in the two test rigs were compared and it was concluded that the...

The results are reported of velocity measurements in oscillatory flow over rippled beds. Velocities were measured with a laser-doppler anemometer in both an oscillating tray rig and an oscillatory flow water channel. Both self-formed and artificial ripples were examined. In addition, some measurements were made with an apparently plane bed with int...

. A segregated finite element algorithm for the solution of the SUPG formulation of the incompressible steady-state Navier-Stokes equations for non-isothermal flow is presented in this paper. The method features equal order interpolation for all the flow variables. The SIMPLER and SIMPLEST algorithms are employed and the sets of nonsymmetric linear...

Abstract – The theoretical basis and conceptual formulation of a comprehensive,reactor model to simulate the thermal-fluid phenomena ,of the PBMR reactor core and core structures is given. Through a rigorous ,analysis the fundamental,equations are recast in a form that is suitable for incorporation in a network code. This formulation of the equatio...

In the last few decades computational fluid dynamics has become a versatile tool for the analysis and solution of problems that are of considerable interest to the engineer. The purpose of this paper is to give a brief overview of the field of CFD and its application. This is done by first of all looking at the theoretical formulations which form t...

Thesis (doctoral)--University of Cambridge, 1980.