Development of a mathematical model for a curved slat venetian blind with thickness

This article is about the development of a mathematical model for a venetian blind. The model is used for determining the performance of the glass window installed with a venetian blind in term of heat transmission. The blind, whose optical properties are considered nonspecular, is modeled as an effective layer. The optical properties of the effective layer are mainly dependent on the slat angle, slat properties and solar profile angle. The effect of slat curvature and the effect of slat thickness are included in the developed model. The shortwave optical properties of the effective layer, transmittance, reflectance and absorptance, are classified as the optical properties for direct radiation and the optical properties for diffuse radiation. The analysis for optical properties due to the interreflection of the direct radiation between the adjacent slat surfaces is done by using radiosity method on a 6 surface closed enclosure. The effect of the slat curvature and thickness causes the shaded area blocked by the blind itself increased in certain cases. The optical properties for diffuse radiation of the effective layer are determined considering the incident diffuse radiation from the sky and from the ground. The optical properties calculated from the developed model are also compared to the results obtained from the three previous models. The validation of the results predicted by the developed mathematical model is performed by comparing the predicted results with the experimental results. The results that used to validate in this study is the ratio of the transmitted radiation through the glass window installed with venetian blind to the incident radiation on the glass window. It is shown from the comparison that the developed mathematical model for the venetian blind included the curvature and thickness effect yields quite accurate predicted results.     

Comparison of single-cell testing,short-stack testing and mathematical modeling methods for a direct methanol fuel cell

In this paper, a comparison between direct methanol fuel cell (DMFC) measurements performed on a single cell and a short-stack, and the results of a mathematical model for a DMFC, is presented. The testing of a short-stack, which consists of 5 cells with an active area of 315 cm², was performed at various current densities, permeation current densities, and cathode flow rates (CFR) in order to determine the voltage outputs of each cell. Methanol concentration and stack temperature results obtained from short-stack testing were then integrated into the single cell test and single cell mathematical model as the input parameters. For the mathematical modelling, transport equations originating from methanol, water, and oxygen were coupled with the electrochemical relations. Therefore, a comparison between these three methods is made in order to gain a deeper understanding of the effects of the operating parameters on DMFC performance. This study showed that the model could describe experimental results well when lower methanol concentrations (under 1.2 M) and temperature (under 60 °C) values are used as input parameters. The results also show very good agreement at lower methanol permeation rates and therefore lower temperatures. It is found that the voltage output for a given current density is higher for the theoretical model than that of the experimental studies; and the differences in the results can be up to 0.04 V for a cell.     

A mathematical model for refinery hydrogen network synthesis integrating multi-stage compressors

The synthesis of refinery hydrogen network (HN) is generally performed under the assumption that compressors are single-stage ones. This assumption might overestimate the compression power, leading to deviation from the optimum designs. To tackle this problem, we integrate multi-stage compressors into hydrogen networks synthesis in the present work. The number of compression stages inside each compressor is defined as a variable which is simultaneously optimized with the network configurations. Taking advantage of the discrete nature of the defined stage variable, we linearize the constraint of compression power without losing generality. The proposed mathematical model for HN synthesis is formulated as a mixed-integer nonlinear programming problem. Three instances are conducted and the solution results show that the proposed model could be solved to global optimality. The total annualized costs of globally optimal solutions are 1.4–5.1% lower than those of the base solutions.     

A mathematical model of thermal performance of a solar air heater with slats

A mathematical model for computing the thermal performance of a single pass flat-plate solar air collector is presented. Air channels were formed by providing metal slats running along the circulated air passage linking the absorber plate by the bottom one in an endeavor to enhance the thermal efficiency of the solar air collector. A mathematical model, therefore, is developed by which the influence of the addition of the metal slats on the efficiency of the solar collector is studied. A computer code that employs an iterative solution procedure is constructed to solve for the governing energy equations to estimate the mean temperatures of the collector. The effect of volume airflow rate, collector length, and spacing between the absorber and bottom plates on the thermal performance of the present solar air heater was investigated. Furthermore, a numerical comparison of the present design with the most common type of solar air heaters is conducted. The results of the comparison have indicated that better thermal performance was obtained by the modified system.     

Numerical simulation of mass and charge transfer for a PEM fuel cell

A three-dimensional, steady state, single phase model is developed to study the mass and charge transfer within a proton exchange membrane (PEM) fuel cell. A single set of conservation equations is used for all PEM fuel cell layers and the governing equations are solved numerically using a finite-volume-based computational fluid dynamics technique. The numerical results for the flow field, species transport and phase potential are presented for two designs, namely a PEM fuel cell with conventional and interdigitated flow fields for the reactant supply.     

Advanced mathematical model for the passive direct borohydride/peroxide fuel cell

In the literature a mathematical model has been developed for the direct borohydride fuel cells by Verma et al. [1]. This model simply simulates the fuel cell system via kinetic mechanisms of the borohydride and oxygen. Their mathematical expression contains the activation losses caused by the oxidation of the borohydride and the concentration overpotential increased by the reduction of oxygen. In this study a direct borohydride/peroxide fuel cell has been constructed using hydrogen peroxide (H₂O₂) as oxidant instead of the oxygen. Therefore we created an advanced model for peroxide fuel cells, including the activation overpotential of the peroxide. The goal of our model is to provide the information about the peroxide reduction effect on the cell performance. Our comprehensive mathematical model has been developed by taking Verma’s model into account. KH₂O₂ used in the advanced model was calculated as 6.72 × 10⁻⁴ mol cm⁻² s⁻¹ by the cyclic voltammogram of Pt electrode in the acidic peroxide solution.     

A mathematical model of variable displacement wobble plate compressor for automotive air conditioning system

Based on the force balance equations, mass and energy conservation equations, a mathematical model of control valve used in the variable displacement wobble plate compressor (VDC) is developed firstly. The dynamic model of the moving components is developed then by analyzing the forces and force moments acting on the piston, piston rod, wobble plate, rotating journal and shaft sleeve. The compression process model is obtained by fitting the data from our experiments. And finally the steady-state mathematical model of VDC is developed by combining the three models above. In order to verify the mathematical model of compressor, a test bench for the control valve and the test system for the VDC have been established, and the simulated results agree well with the experimental data. The simulation results show that there are four operation modes for the VDC, i.e. constant rotary speed and constant piston stroke length (PSL), variable rotary speed and constant PSL, constant rotary speed and variable PSL, variable rotary speed and variable PSL, which have included almost all operation modes of the refrigeration compressor in common use.     

Two-dimensional two-phase mass transport model for methanol and water crossover in air-breathing direct methanol fuel cells

A two-dimensional two-phase mass transport model has been developed to predict methanol and water crossover in a semi-passive direct methanol fuel cell with an air-breathing cathode. The mass transport in the catalyst layer and the discontinuity in liquid saturation at the interface between the diffusion layer and catalyst layer are particularly considered. The modeling results agree well with the experimental data of a home-assembled cell. Further studies on the typical two-phase flow and mass transport distributions including species, pressure and liquid saturation in the membrane electrode assembly are investigated. Finally, the methanol crossover flux, the net water transport coefficient, the water crossover flux, and the total water flux at the cathode as well as their contributors are predicted with the present model. The numerical results indicate that diffusion predominates the methanol crossover at low current densities, while electro-osmosis is the dominator at high current densities. The total water flux at the cathode is originated primarily from the water generated by the oxidation reaction of the permeated methanol at low current densities, while the water crossover flux is the main source of the total water flux at high current densities.     

Influence of water vapor on performance of co-planar single chamber solid oxide fuel cells

Influences of feeding gas compositions on the performance of co-planar, single chamber solid oxide fuel cells (SC-SOFCs) are investigated with emphasis on the role of water vapor. The maximum open circuit voltage (OCV) and peak power density are obtained at a methane-to-oxygen ratio of 3.5 under the wet gas condition, and a stoichiometric ratio of 2.0 for methane partial oxidation under the dry gas condition. In addition to the partial oxidation of methane on the anode and electrocatalytic reactions, both steam reforming and methane combustion occur on the anode and cathode, respectively, in the presence of water vapor. Local volume expansion and a rise in temperature associated with these parasitic reactions intensify inter-mixing of the reactant and product gases by which the OCV and power density drastically deteriorate with decreasing anode-to-cathode gap distance, as confirmed by impedance analysis for the LSM-YSZ|YSZ|LSM-YSZ symmetrical cell.     

Application of mathematical model for the process of coal tar pitch modified petroleum asphalt

This paper firstly discusses the properties of modified asphalt by blending petroleum asphalt and coal tar pitch. Then, a new mathematical model which was called ZQL was constructed. Based on three variables, property, granularity size, and proportion, the three-dimension function was fit, where the property was set as a dependent variable and the rest of the variables were set as independent variables. The different functions were finally obtained. Finally, based on the British standard of road asphalt, we solve the ZQL model and obtain a suitable result which matches the experimental results very well.     

Comprehensive one-dimensional,semi-analytical,mathematical model for liquid-feed polymer electrolyte membrane direct methanol fuel cells

Polymer electrolyte membrane direct methanol fuel cells (PEM-DMFCs) have several advantages over hydrogen-fuelled PEM fuel cells; but sluggish methanol electrochemical oxidation and methanol crossover from the anode to the cathode through the PEM are two major problems with these cells. In the present work, a comprehensive one-dimensional, single phase, isothermal mathematical model is developed for a liquid-feed PEM-DMFC, taking into account all the necessary mass transport and electrochemical phenomena. Diffusion and convective effects are considered for methanol transport on the anode side and in the PEM, whereas only diffusional transport of species is considered on the cathode side. A multi-step reaction mechanism is used to describe the electrochemical oxidation of methanol at the anode. Stefan–Maxwell equations are used to describe multi-component diffusion on the cathode side and Tafel type of kinetics is used to describe the simultaneous methanol oxidation and oxygen reduction reactions at the cathode. The model fully accounts for the mixed potential effect caused by methanol crossover at the cathode. It shows excellent agreement with literature data of the limiting current density for different low methanol feed concentrations at different operating temperatures. At high methanol feed concentrations, oxygen depletion on the cathode side, due to excessive methanol crossover, results in mass-transport limitations. The model can be used to optimize the geometric and physical parameters with a view to extracting the highest current density while still keeping a tolerably low methanol crossover.     

Application of a detailed mathematical model to the gasifier unit of the dual fluidized bed gasification plant

A one-dimensional steady state model for biomass-steam gasification has been developed. The reactor is a bubbling fluidized bed. With respect to hydrodynamics the model distinguishes two zones namely: dense zone and freeboard zone. The gasification process is modelled in three steps: drying, devolatilization and gasification of biomass char. The model assumes that solids are well mixed while the gases are in plug flow regime. Mass and energy balance is solved globally across the entire gasifier. The gas composition and temperatures predicted by the model for wood chips as fuel agree well with values measured at an 8 MW (fuel power) commercial plant.     

A model for hydrogen sulfide poisoning in proton exchange membrane fuel cells

A polymer-electrolyte fuel cell model that incorporates the effects of hydrogen sulfide contaminant on performance is developed. The model is transient, fully two-phase and non-isothermal and includes a complex kinetic mechanism to describe the electrode reactions. Comparisons between the simulation results and data in the literature demonstrate that known trends are well captured. The effects of temperature and relative humidity variations in the anode stream are investigated, with further comparisons to experimental data and a proposed explanation for the nonlinear behaviour observed in the experiments of Mohtadi et al. [R. Mohatadi, W.-K. Lee, J. van Zee, Appl. Catal. B 56 (2005) 37–42)]. Extensions to the model and future work are discussed.     

A transient analysis of three-dimensional heat and mass transfer in a molten carbonate fuel cell at start-up

In this paper, we investigate the time-dependent heat and mass transfer in a molten carbonate fuel cell at start-up. Thus, a three-dimensional, transient mathematical model is presented through a comprehensive inclusion of various physical, chemical and electrochemical processes that occur within the different components of molten carbonate fuel cells. The model is proposed as a predictive tool to provide a three-dimensional demonstration of variable variations at system start-up. The local distribution of field variables and quantities are showcased. It reveals that the electrochemical reaction rate is dominated by the over-potential, not by the reactants' molar fraction. Reversible heat generation and consumption mechanisms of the cathode and anode are dominant in the first 10 s while the heat conduction from the solid materials to the gas phase is negligible. Meanwhile, activation and ohmic heating have nearly the same impact within the anode and cathode. Based on these findings, the importance of heat conduction and its main features are finally assessed.     

A simplified mathematical model for predicting the nocturnal output of a solar still

The production capacity of a solar still which converts saline water to fresh water can be increased by introducing hot feed water into the unit at night. A waste heat source, such as cooling water from a power plant, can be used to preheat the feed. The nocturnal production, i.e. the distilled water produced at night, seems to be influenced by several parameters. However, a simplified mathematical model suggests that the distillate depends only on the initial brine temperature, the drop in brine temperature and the brine depth. This was experimentally verified for different brine depths and for initial brine temperatures up to 150°F.     

单颗粒燃料滴在高温氧化环境中着火规律的研究

本文在前人对单个油滴着火规律研究的基础上，通过分析着火过程中油滴周围气相温度场、浓度场的特点，提出了油滴着火的分区反应模型，得到了直观的着火条件解析表达式。在此基础上，预报的临界着火温度与环境氧浓度的关系，与实验结果符合得很好。另外，用这个着火条件结合油滴加热过程的分析计算，得到了着火延迟时间与各因素的关系，与实验结果对比，证明了这种分区近似法是一种预报、分析油滴着火过程的较简单、准确的方法。     

A reactor model for hydrogen generation from sodium borohydride and water vapor

This paper reports new data on the production of hydrogen from water vapor plus NaBH₄, or NaBH₄ + 10% CoCl₂. Data were collected with the aid of an isothermal semi-batch reactor with in-situ H₂ rate measurement. The reaction of NaBH₄ to generate H₂ proceeds via three steps: deliquescence, dissolution and reaction. The deliquescence regime of NaBH₄ in the presence of 10 weight percent CoCl₂ is defined. The H₂ yield is quantified at various reaction conditions (reaction temperature 70–120 °C, relative humidity 31–69%). CoCl₂ significantly accelerates the rate of H₂ production compared to deliquescence + reaction of pure NaBH₄. It is also found that a combination of high temperature and high relative humidity contributes to high H₂ rate and yield, and either of the two factors dominates the reaction at different conditions. A two-part reactor model accounting for the mechanism of the steam hydrolysis by NaBH₄ is developed. The model captures the dissolution + reaction step as well as reaction-only step and was validated by experimental data.     

Developing a mathematical tool for hydrogen production,compression and storage

Among the few lessons learned presented in the literature, authors put in evidence the on-going need to investigate on station components and their integration. The specific power consumption of station units with on-site hydrogen generation is often subject to uncertainty, and it would have been desirable to find more details about the energy contribution of each component. To address this gap, this paper focuses on the development of a mathematical modeling as a dynamic and multi-physical design tool to predict the energy performance of hydrogen production systems. Particularly, the model aims to describe and analyze the energy performance of two different electrolyzer technologies (PEM and Alkaline), integrated with a compressor system and gaseous buffer storage. Multiple tank options and a switching strategy are investigated, as well as a control system to simulate a real infrastructure operation. Auxiliaries and components related to the thermal management system have been also included. A carbon-footprint analysis follows the energy one, focusing on the CO₂ emission reduction. Comparisons between literature data and model show that the hydrogen system proposed model is suitable to evaluate systems with respect to energy efficiency and system performance. The model could be a powerful tool for exploring control strategies and understanding the contributions to the overall energy consumption from the various internal components as a guide to researchers aiming for improved performance.     

Bi-modal water transport behavior across a simple Nafion membrane

The development of predictive mathematical models for water management in polymer electrolyte membrane fuel cells requires detailed understanding of water distribution and water transport across the Nafion layer. The anisotropic microstructure of Nafion suggests the measurement of water content and mass transport should be along the fuel cell functional direction, i.e. across the membrane. Non-invasive, high resolution, microscopy measurements of this type are very challenging.We report here the calibration of a minimal mathematical model for diffusive water transport in Nafion against data from high-resolution water content maps determined with a new magnetic resonance imaging methodology developed for this purpose. A mock fuel cell was designed to permit well-controlled wetting and drying boundary conditions. With no chemical potential driving force involved, we assume the water transport behavior will be dominated by diffusion. Moreover we show that, in this context, our model is mathematically equivalent to the traditional permeation models based upon saturation dependent pressure gradients via a capillary pressure ansatz.The non-linear equilibrium water distribution across the Nafion membrane measured in this work suggests a bi-modal diffusivity. The model constructed associates distinct transport behaviors to water contents above and below a critical threshold, consistent with a rearrangement of a micro-structural pore network. The experimental observation and the model prediction agree with the primary features of Weber's model of Nafion, which predicts distinct modes of transport for hydration fronts traversing the through-plane direction of the membrane.     

A two-dimensional two-phase mass transport model for direct methanol fuel cells adopting a modified agglomerate approach

A two-dimensional two-phase mass transport model for liquid-feed direct methanol fuel cells (DMFCs) is presented in this paper. The fluid flow and mass transport across the membrane electrode assembly (MEA) is formulated based on the classical multiphase flow theory in the porous media. The modeling of mass transport in the catalyst layers (CLs) and membrane is given more attentions. The effect of the two-dimensional migration of protons in the electrolyte phase on the liquid flow behavior is considered. Water and methanol crossovers through the membrane are implicitly calculated in the governing equations of momentum and methanol concentration. A modified agglomerate model is developed to characterize the microstructure of the CLs. A self-written computer code is used to solve the inherently coupled differential governing equations. Then this model is applied to investigate the mechanisms of species transport and the distributions of the species concentrations, overpotential and the electrochemical reaction rates in CLs. The effects of radius and overlapping angle of agglomerates on cell performance are also explored in this work.