Eduardo Fontes

Electrochemical treatment of tumours: a simplified mathematical model

Electrochemical treatment of tumours implies that tumour tissue is treated with a direct current. During electrolysis, electrical energy is converted to chemical energy through electrochemical reactions at the electrodes. The anode is preferably placed in the tumour and the cathode in a blood vessel or in fresh surrounding tissue. The main electrochemical reactions are chlorine and oxygen evolution, at the anode, if platinum is used. Hydrogen evolution takes place at the cathode. The aim of this paper is to show how mathematical modelling can be used as a tool for defining and optimising the operating conditions of electrochemical treatment (ET) of tumours. A simplified mathematical model is presented for direct current treatment of tumours, focusing on tissue surrounding a spherical platinum anode. The tissue is treated as an aqueous solution of sodium chloride and only the major electrochemical reactions are considered. The model is based on transport equations of ionic species in dilute solutions. Kinetic expressions for the electrochemical reactions, at the anode surface, are introduced. Inputs to the model are the applied current density, and sizes of the anode and electrolyte domain. Concentration profiles of the ionic species and potential distribution, as a function of time, are calculated. In addition, current yields of the anode reactions are obtained from the model.

Mathematical modelling of physicochemical reactions and transport processes occurring around a platinum cathode during the electrochemical treatment of tumours.

The electrochemical treatment (EChT) of tumours is an anti-tumour therapy in which a continuous direct current is applied to electrodes, placed in or near a tumour. Promising results have been reported from clinical trials in China, where more than 10,000 patients have been treated with EChT during the past 10 years. Before clinical trials can be conducted outside of China, a reliable dose-planning strategy has to be developed. One approach in achieving this is the use of physicochemical simulation models. A simplified mathematical model of the physicochemical processes, occurring around a spherical platinum cathode during EChT, is developed and visualized in three steps in this paper. In the final step, tissue is treated as an aqueous solution of sodium chloride, containing a bicarbonate buffer system and organic constituents susceptible to reactions with hydroxyl ions. This model is shown to give a good description of the pH profile obtained around the cathode after EChT. The simulation results reveal a strong correlation between the pH profiles and size of experimentally measured lesions, thus indicating that it is the spreading of hydroxyl ions that determines the extent of tissue destruction around the cathode. In addition, the simulations indicate that the model could be of use in predicting the size of a lesion produced by EChT.

Influence of gas phase mass transfer limitations on molten carbonate fuel cell cathodes

The purpose of this paper is to elucidate to what extent mass transfer limitations in the gas phase affect the performance of porous molten carbonate fuel cell cathodes. Experimental data from porous nickel oxide cathodes and calculated data are presented. One and two-dimensional models for the current collector and electrode region have been used. Shielding effects of the current collector are taken into account. The mass balance in the gas phase is taken into account by using the Stefan–Maxwell equation. For standard gas composition and normal operating current density, the effect of gas-phase diffusion is small. The diffusion in the gaseous phase must be considered at operation at higher current densities. For low oxygen partial pressures, the influence of mass transfer limitations is large, even at low current densities. To eliminate the influence of conversion on polarization curves recorded on laboratory cell units, measurements should always be performed with a five to tenfold stoichiometric excess of oxygen. Two-dimensional calculations show rather large concentration gradients in directions parallel to the current collector. However, the influence on electrode performance is still small, which is explained by the fact that most of the current is produced close to the electrolyte matrix.

Mathematical modelling of the MCFC cathode

Abstract Different models for the NiO cathode have been compared with respect to their abilities to predict polarization curves and the influence of the amount of electrolyte on the electrode performance. It has been shown that the agglomerate model for the MCFC cathode gives more reasonable results when the exterior agglomerate surface area is specifically taken into account. In the cathode only the outermost layer of nickel oxide particles in the agglomerate is utilized for the electrochemical reaction. The pseudohomogeneous approach is questionable for these agglomerates since the individual particles constituting the agglomerate are of the same size as the reaction zone thickness. A thin-film model with a roughness factor for the electrode surface appears to be as good a model as the agglomerate model. A model based on a chain of spherical agglomerates and the partially drowned agglomerate model are physically more realistic models than the homogeneous agglomerate model for the prediction of the influence of electrolyte fill on the electrochemical performance.

A Simulation of the Tertiary Current Density Distribution from a Chlorate Cell: I. Mathematical Model

Numerical modeling is becoming an integral part of all research and development within the field of electrolytic systems. A numerical model that calculates the current density distribution and concentration profiles of a chlorate cell is presented here, The results are shown as functions of electrolyte velocity and exchange current density. The model takes into account the three transport mechanisms; diffusion, migration, and convection by considering the development of the flow velocity vector through the channel. It was seen that the developing velocity profile influences the concentration overpotentials, which in turn influences current density distributions. Results from the model show that the total current density decreased along the length of the anode, and that this distribution varied more at lower velocities. In addition, it was seen that migration contributes significantly to species transport, even within the diffusion layer. Finally, the model indicates that the hypochlorite ion is the main participant in the principal side reaction producing oxygen, and not the hypochlorous acid molecule. The results are useful as they increase knowledge of the chlorate process, and can be used to simulate future systems with a wide range of varying parameters such as cell geometry, flow, electrolyte composition, and electrode materials. The aim of the model is to use it as a tool for identifying the sources that contribute to the overpotential in the cell. This article concentrates on the concentration overpotential, which is one of the phenomena that can actually be influenced,

Mathematical modelling of the MCFC cathode : On the linear polarisation of the NiO cathode

Experimental polarisation curves for the porous lithiated NiO cathode used in molten carbonate fuel cells very often exhibit a linear shape over a wide potential range. It is shown by means of mathematical modelling that this linear behaviour can be explained by the interplay of intrinsic electrode kinetics, diffusion of electroactive species through an electrolyte film and the effective ohmic resistance of the pore electrolyte, providing that the cathodic transfer coefficient has a value of about 1.5. In contrast, with the generally assumed value of 0.5 of this transfer coefficient and with reasonable values of the effective electrolyte conductivity, predicted polarisation curves will always bend downwards over the overvoltage region of interest. The evolution of the polarisation curves with increasing electrolyte fill can be simulated by a model according to which the electroactive surface area becomes gradually blocked with the increasing amount of electrolyte.

A heterogeneous model for the MCFC cathode

A steady state agglomerate model for the MCFC cathode which takes into account the heterogeneous structure of this porous electrode is presented. The resulting model equations are solved by means of the finite element method. Calculations have been performed on two different test structures and include the influence of kinetics, porosity, outer surface area and distribution of electrolyte film. Comparisons with the filmed agglomerate model show that it is possible to obtain excellent agreement between the polarisation curves predicted by the two different models if a uniform film is used in the simulations. The tortuosity in the filmed agglomerate model is used as a fitting parameter. Other effects that evolve in the heterogeneous model due to variations in the local structure are not revealed in the pseudohomogeneous model. The effect of a non-uniform electrolyte film thickness was investigated by solving the problem for a structure with pore mouths filled by an electrolyte meniscus. Also the effect of an electrolyte meniscus between spherical agglomerates was investigated.

Effects of different design parameters on the performance of MCFC cathodes

The effects of electrode thickness, electrolyte filling and current collector geometry on the performance of MCFC cathodes are investigated by using a steady state mathematical model. A two-dimensional pseudo-homogeneous model for the three-phase system in the cathode is used, which includes the polarisation curves from the heterogeneous agglomerate model[1] as local source functions. The model takes into account the potential distribution in the electrolyte and catalyst phase but neglects mass transport limitations in the gas phase. The simulations show that, for cathodes with a finite electronic conductivity, there is a substantial potential distribution perpendicular to the depth of the electrode depending on the size of the gas holes in the current collector.

Impact of chlorine and acidification in the electrochemical treatment of tumours

Electrochemical treatment (EChT) of tumours offers considerable promise as a safe, simple and relatively noninvasive antitumour therapy. When platinum is used as electrode material, the major electrochemical reactions at the anode are chlorine and oxygen evolution. Conflicting opinions can be found in the literature over the role of chlorine in the underlying destruction mechanism behind EChT. In this present study, the impact of chlorine in EChT treatments is investigated by means of mathematical modelling. The analysis focuses on the tissue surrounding a spherical platinum anode when applying a constant current density. Tissue is modelled as an aqueous solution of sodium chloride containing a bicarbonate buffer system and organic constituents susceptible to reactions with chlorine. Except for the case of very low anode current densities, the simulations clearly show that it is the spreading of hydrogen ions – and not chlorine molecules – that determines the extent of tissue destruction around the anode. Moreover, it is found that the reactions of chlorine with tissue play important roles as generators of hydrogen ions. The contribution of these reactions to the acidification of tissue, surrounding the anode, is strongly dependent on the applied current density and increases with decreasing current density.

The Nordic fuel cell effort

Abstract Fuel cell technologies are playing an increasing part in Scandinavias efforts to minimize the production of greenhouse gases. In this article Ed Fontes, Eva Nilsson and Patrik Bosander provide an overview of the part that Scandinavian countries have played in the development of fuel cell systems.