Mass Transfer Balance Research Articles

Multi-Physics Modeling and the Sensitivity Analysis for the Critical Factors in Solid Oxide CO2-Steam Co-Electrolysis System Performances

A multi-physics modeling framework, which includes electrochemical and chemical reactions, mass transfer, and energy balance, has been developed and validated against experiment measurements to investigate the performance of solid oxide CO2-steam co-electrolysis (SOEC) under various operating conditions and cell designs. However, multi-physics modeling for complex SOEC systems can be computationally expensive and not tractable for general system design and optimization. Reduced-order models (ROMs) have been proven as a powerful tool for reducing computational costs, closely mimic the high-fidelity SOEC stack models and identify the operational condition parameters to which the SOEC stack is sensitive and needs appropriate controls for stable operation. In this study, the deep neural networks (DNN) algorithm is employed to construct ROMs according to multi-physics simulations for SOECs to systematically investigate the SOECs’ electrochemical performance and rank the contributions of operational condition parameters for both 2 cm2 button cell and 300 cm2 planar cells. Sensitivity analysis reveals that cell voltage, fuel composition (denoted by CO2/H2O ratio), and the operating temperature impact the cell performance most. The product ratio ΔCO/ΔH2 has a stronger dependence on the CO2/H2O ratio in the fuel compositions than on other operating condition parameters. The large 300 cm2 planar cells were modeled under both adiabatic and furnace environment. The simulation and sensitivity analysis indicate that the large cell operating in adiabatic environment usually provides relatively lower performances than the one in furnace environment. Additionally, the adiabatic environment also causes the cell’s performances are more sensitive to the operation conditions, such as external voltage, fuel flow rate, and fuel compositions. The large cell in adiabatic environment also leads to higher internal temperature variation, which may impact the structure reliability.

Performance study of organic photovoltaic/thermal system with synergistic effect of photocatalytic and thermal catalytic technology

Organic photovoltaic cells are considered to be an effective solution for maintaining excellent electrical energy output in high temperature environment. However, the application of OPV alone has a low utilization rate of the solar spectrum. The combined application of OPV and other technologies is explored to improve the utilization efficiency of the solar spectrum. Based on the above gaps, this study proposes an organic photovoltaic/thermal system with synergistic effect of photocatalytic and thermal catalytic (PC/TC-OPV/T). By integrating organic photovoltaic cells, photocatalytic technology, thermal catalysis technology and photothermal components, the utilization efficiency of the solar spectrum is improved, and the triple functions of formaldehyde removal, electrical energy and hot water are realized. Therefore, this study proposes a new air purification-thermal-electrical model, establishes the heat transfer balance equation inside the system and the air purification and mass transfer balance equation in the air duct, proposes the performance evaluation index of the system, and uses Ansys Fluent software to predict the working characteristics of the system under different operating conditions, and draw the following conclusions: (1) Under the synergistic action of photocatalytic and thermal catalytic technologies, the formaldehyde removal efficiency is 84.4%, which is 30% higher than that of photocatalytic alone. (2) The electrical efficiency of the new system benefits from high solar irradiance and environment temperature, enabling it to achieve the efficiency of 17.84% within the research scope. (3) Under the research conditions, the instantaneous water flow heating efficiency of the system is up to 69.6%.

A Preliminary Approach towards Rotor Icing Modeling Using the Unsteady Vortex Lattice Method

UAV rotors are at a high risk of ice accumulation during their operations in icing conditions. Thermal ice protection systems (IPSs) are being employed as a means of protecting rotor blades from ice, yet designing the appropriate IPS with the required heating density remains a challenge. In this work, a reduced-order modeling technique based on the Unsteady Vortex Lattice Method (UVLM) is proposed as a way to predicting rotor icing and to calculate the required anti-icing heat loads. The UVLM is gaining recent popularity for aircraft and rotor modeling. This method is flexible enough to model difficult aerodynamic problems, computationally efficient compared to higher-order CFD methods and accurate enough for conceptual design problems. A previously developed implementation of the UVLM for 3D rotor aerodynamic modeling is extended to incorporate a simplified steady-state icing thermodynamic model on the stagnation line of the blade. A viscous coupling algorithm based on a modified α-method incorporates viscous data into the originally inviscid calculations of the UVLM. The algorithm also predicts the effective angle of attack at each blade radial station (r/R), which is, in turn, used to calculate the convective heat transfer for each r/R using a CFD-based correlation for airfoils. The droplet collection efficiency at the stagnation line is calculated using a popular correlation from the literature. The icing mass and heat transfer balance includes terms for evaporation, sublimation, radiation, convection, water impingement, kinetic heating, and aerodynamic heating, as well as an anti-icing heat flux. The proposed UVLM-icing coupling technique is tested by replicating the experimental results for ice accretion and anti-icing of the 4-blade rotor of the APT70 drone. Aerodynamic predictions of the UVLM for the Figure of Merit, thrust, and torque coefficients agree within 10% of the experimental measurements. For icing conditions at −5 °C, the proposed approach overestimates the required anti-icing flux by around 50%, although it sufficiently predicts the effect of aerodynamic heating on the lack of ice formation near the blade tips. At −12 °C, visualizations of ice formation at different anti-icing heating powers agree well with UVLM predictions. However, a large discrepancy was found when predicting the required anti-icing heat load. Discrepancies between the numerical and experimental data are largely owed to the unaccounted transient and 3D effects related to the icing process on the rotating blades, which have been planned for in future work.

Novel algorithms for high-resolution prediction of canopy evapotranspiration in grapevine

Developing low-cost technology for custom water delivery to individual or small groups of plants is a critical next step to advance precision irrigation. Current systems for estimating evapotranspiration (ET), or plant water use, work on the scale of a full vineyard (e.g., 3–5 acres) or the scale of a single vine, but at a cost that prohibits monitoring past a small number of representative vines. To develop and evaluate low-cost ET sensors for individual grapevines, we used three head-pruned Zinfandel vines in pots and placed them on load cells to collect continuous weights indicative of actual ET. We mounted research-grade sensors for humidity, temperature, and wind speed on each vine and saved data at 2-minute intervals during three growing seasons. We developed three models based on first principles (Convective Mass Transfer or Mass Balance approaches) or simple correlations to predict actual single-plant ET from these data. We present here the results of a multi-year trial at the UC-Davis RMI vineyard to illustrate the performance of each of the models for ET estimation. Relative model performance was assessed by comparing model predictions to ground truth data provided by measurements from load cells–including assessments of estimated instantaneous ET rate, estimated cumulative water use over a one-hour window surrounding solar noon, and estimated cumulative water use over a full 24-hour period. The three algorithms developed consistently performed well, with single vine ET rate predictions showing a strong linear relationship with ground truth (range in r2 over three seasons CMT r2 = 0.61–0.86; MB r2 = 0.07–0.91; EM r2 = 0.57–0.92). The MB approach, which includes two measurements of relative humidity and temperature, was the most variable, likely due to the impact of sensor placement. In all seasons, we also examined the trend in the plant scaling factor found in each model, deemed As, which, based on model theory, is a function of vine size. Taken together, these results suggest that high-resolution irrigation (HRI) models are a promising new method for ET estimation at the single plant level.

An experimental study on the frost formation over a flat plate: Effect of frosting on heat transfer

Frost accumulation on cooled surfaces generates thermal barrier between the surface and air. Apart from the frost layer thickness, the thermal properties of the frost also affect its heat transfer characteristics. In this experimental study, the effect of the frosting on heat transfer is investigated at different cold surface temperatures and at different air velocities in a closed loop wind tunnel. A large number of experiments is performed by altering the parameters of free stream air velocity, relative humidity, and cold plate temperature. Frost layer impact on heat transfer rate is measured through a high precision thin film heat flux sensor for 60 min. Accordingly, heat flow was significantly obstructed due to frosting. The rate of diminishment in the heat flux was not found to be constant over time but shows a decreasing trend. Furthermore, an enhancement on the heat transfer was reported during the early stage of frost formation under certain environmental conditions, majorly due to the increase in effective heat transfer area induced by ice crystals and corresponding increase in latent heat generation. The densification of the frost layer and its effect on the total thermal resistance was discussed. Based on the convective heat and mass transfer balance over the frost layer, frost surface temperature was predicted. Lastly, a correlation for estimating the heat transfer decrease during frosting is proposed via non dimensional numbers.

Long-term storage of microalgae: determination of optimum cryopreservation conditions

AbstractMaintenance of eukaryotic microalgae strains for the long term is generally carried out using serial subculture techniques which require labour, time and cost. Cryopreservation techniques provide long-term storage of up to years for numerous microorganism strains and cell cultures. Ssu930ijn vbvbhnn8;l,n is related to a successfully designed mass and heat transfer balance throughout the cell. In this study, optimization of the cryopreservation process was carried out for two commercially used microalgal strains. The parameters to be optimized were DMSO percentage (0–25%), incubation time (1–15 min) and cryopreservation term (7–180 days) using a central composite design (CCD). Long-term storage up to 123.17 and 111.44 days corresponding to high cell viabilities was achieved for Chlorella vulgaris and Neochloris texensis, respectively. Generated models were found to be in good agreement with experimental results. The study also revealed holistic results for storage of microalgal strains in a stable state for industrial applications.

Recovery and Purification of Silicon From Diamond Wire Saw Wasted Silicon Powder by a Technique of Induction Smelting Followed by Directional Solidification

A technique of induction smelting followed by directional solidification to recover and purify silicon from diamond wire saw waste silicon powder without the use of additives is presented herein. The silicon reaction and the effect of smelting time on the removal of impurities from silicon powder waste were investigated using a simple induction smelting and directional solidification method. Through the detection of byproducts and condensed volatiles, it is inferred that Si is volatilized and removed as silicon monoxide during smelting. Byproducts containing Si, SiC, or SiO2 phases can be generated by the reaction between silicon monoxide and graphite crucible after volatilization. Because the crucible is sealed with byproducts, the flow of volatile materials in the crucible disrupts the mass transfer balance in the silicon melt. The amount of residual impurities in silicon ingot trends downwards with the extension of smelting time. A 99 pct oxygen removal rate was obtained after the material was smelted three times. The silicon ingot achieved the lowest impurity content after smelting for 4 hours. The silicon powder waste can be potentially used as a metallurgical-grade silicon product after a simple one-step induction smelting process. Further purification of the silicon ingot by directional solidification can reduce residual impurities to 74 ppmw.

Different configurations of humidification-dehumidification desalination systems: Thermal and economic assessment

Drought has hit several water-stressed regions, with serious consequences for water and food security. In this regard, the low-cost and low-grade thermally driven humidification-dehumidification desalination cycle has good potential to provide freshwater in arid regions. From energy and economic point of view, different configurations of the humidification-dehumidification cycle have been proposed in the literature without a comprehensive comparison to identify the most suitable one. Thus, this paper investigates the thermal and economic performance of several multi-humidifier water- and air-heated humidification-dehumidification cycles. Thermodynamic models are structured to study the behavior of each cycle. The models rely on heat and mass transfer balances for each component. The effectiveness method is implemented for humidifiers and dehumidifiers to evaluate heat and mass exchangers. The models are validated using previously published data. The series seawater flow, triple humidifier, water-heated humidification-dehumidification cycle could deliver 151.2 kg/h of fresh water at a maximum temperature of 70 °C, with a gain output ratio of 1.995 and a specific cost of 2.73 $/m3. This cost rises by around 16% when a maximum temperature of 60 °C is employed. Results reveal that using a double-humidifier in an air-heated humidification-dehumidification cycle produces freshwater of 102.2 kg/h at the lowest price of $2.06/m3. It is concluded that the multi-humidifier water-heated humidification-dehumidification cycle produces more freshwater and less gain output ratio than the multi-humidifier air-heated humidification-dehumidification cycle.

Production of Urea/Acetylated-lignin Sulfonate Matrix as SRFs and an Investigation on the Effect of Hydrodynamic Conditions on Release Rate Using the Biot Number

Background: This study aimed to investigate the effect of hydrodynamic conditions on the release rate of urea/acetylated lignin sulfonate (Ac-LS) matrix as slow-release fertilizers (SRFs). Therefore, two models were developed using the mass transfer balance for the determination of finite/infinite volume of fluids and solving finite integral transformation/separation of a variable. The Biot number, validating the hydrodynamic condition, was found in these models. Methods: In this study, the urea/Ac-LS matrix fertilizer was prepared. The morphological, thermal, chemical, and mechanical properties of the LS, Ac-LS, urea, and urea/Ac-LS matrix were analyzed using Fe-SEM, TGA, XRD, and SANTAM. Finally, the nitrogen release of the matrix fertilizer was investigated at 25°C for different impeller speeds. The models were also validated using the experimental data. Results: The results showed that the thermal and mechanical resistance of urea/Ac-LS due to strong interaction increased compared to pure urea or Ac-LS. The results further showed that the external resistance on the mass transfer decreased as the impeller speed increased, and the nitrogen release rate increased as the Biot number increased in both the states, i.e., finite and infinite. Conclusion: It was also observed that the release rate in the finite environment was less than that of the infinite one in the given hydrodynamic condition initially; however, the type of environment did not affect the release rate after a while.

Measurements of surface temperature distributions on coal dust particles

The rate of thermochemical conversion of solid fuel depends strongly on temperature. However, temperature can vary significantly across a single particle's surface and can locally change reactivity for combustion. To account for this phenomenon in numerical models, we carried out an experimental investigation of the temperature profiles for particles obtained from two chars prepared from hard coal and lignite. We used a custom-built, two-color pyrometer to measure surface temperature distributions for particles with an approximate diameter of 100 μm. We used CCD cameras as detectors what allowed for measure temperature with high resolution. The applied research method allowed for the analysis of thousands of individual particles in different conditions. We carried out experiments under O2/N2 and O2/CO2 atmospheres with the oxidizer concentration between 10 and 30% (vol%). We observed highly non-uniform temperature distributions on the surface of the reacting particles with differences greater than 100 K. The standard deviation of the surface temperatures increased with the mean temperature of the particles. The observed variations could be due to inherent particle heterogeneity for ash content, porosity, and shape. Non-uniform temperature profiles and reactivities may alter the Stefan flow at the outer surface of the particles, affecting the mass and heat transfer balance and changing the trajectory of the particles as they move through the combustion chamber. Therefore, the correction factors determined in this study may be applied to existing numerical models of solid fuel combustion.

Studies on the CO2 Capture by Coal Fly Ash Zeolites: Process Design and Simulation

At present, mitigating carbon emissions from energy production and industrial processes is more relevant than ever to limit climate change. The widespread implementation of carbon capture technologies requires the development of cost-effective and selective adsorbents with high CO2 capture capacity and low thermal recovery. Coal fly ash has been extensively studied as a raw material for the synthesis of low-cost zeolite-like adsorbents for CO2 capture. Laboratory tests for CO2 adsorption onto coal fly ash zeolites (CFAZ) reveal promising results, but detailed computational studies are required to clarify the applicability of these materials as CO2 adsorbents on a pilot and industrial scale. The present study provides results for the validation of a simulation model for the design of adsorption columns for CO2 capture on CFAZ based on the experimental equilibrium and dynamic adsorption on a laboratory scale. The simulations were performed using ProSim DAC dynamic adsorption software to study mass transfer and energy balance in the thermal swing adsorption mode and in the most widely operated adsorption unit configuration.

Long-term estimation of the canopy photosynthesis of a leafy vegetable based on greenhouse climate conditions and nadir photographs

In horticultural crop production, accurately estimating the canopy photosynthetic rate (Ac) based on greenhouse climate conditions is of practical value for predicting and controlling crop growth and yield. We developed a method for continuously estimating Ac in a greenhouse from readily obtainable information (i.e., climatic data and canopy photographs) by combining a canopy photosynthesis model with image analysis. The canopy photosynthesis model was based on a sun/shade representation of the crop canopy combined with models of single-leaf photosynthesis, stomatal conductance, mass transfer and leaf energy balance. This combination allowed 1) the incorporation of all major climatic variables (i.e., radiation, CO2 concentration, air temperature, humidity, and wind velocity) into the Ac estimation and 2) the simultaneous estimation of Ac, canopy transpiration rate (Ec), and leaf temperature (TL). The leaf area index (Lc), which changes considerably throughout the growth period of a crop canopy, was evaluated nondestructively from the gap fractions of nadir digital photographs (i.e., the fractions of nonleaf area). The canopy photosynthesis model and image analysis results were combined and used to estimate the Ac, Ec, and TL of spinach canopies grown at ambient and elevated CO2 concentrations (c. 400 and 800 µmol mol−1, respectively). Regardless of the CO2 concentration, the estimates of Ac, Ec, and TL were in good agreement with measurements obtained with the open chamber method throughout the growth period. Analyzing the sensitivity of Ac to input variables revealed that the effects of climatic variables on Ac can vary considerably depending on the Lc and incoming radiation. The proposed method enables not only the estimation of Ac from readily obtainable information but also the quantitative evaluation of the effect of climate control on Ac in horticultural greenhouses.

Recycling of municipal solid waste using solar energy

The article proposes a solar installation for thermal processing of municipal solid waste and presents a method of technical and economic analysis of systems for processing municipal solid waste based on solar energy. The object of the research is a solar installation for thermal processing of municipal solid waste. The analysis of the heat balance of the developed solar installation was compiled and carried out, and the energy requirement for thermal processing of municipal solid waste was determined. The work uses the methods of the theory of heat and mass transfer and body balances of heat power and solar installations. The created solar installation for anaerobic fermentation of solid waste to obtain landfill gas and bio fertilizers has been investigated. Balance methods of numerical and experimental studies of anaerobic fermentation of solid waste were used to analyze the results. A calculation has been made showing the energy efficiency of a solar installation for the processing of municipal solid waste using solar energy. Based on the research carried out, it was found that at an average heating temperature of 50-55 °C, 200-250 m3 of landfill gas was obtained from one ton of household waste.

Integration of multi effect evaporation and membrane distillation desalination processes for enhanced performance and recovery ratios

This work focuses on the integration of multiple effect evaporation (MEE) and membrane distillation (MD) processes. The MEE brine is treated by the MD allowing at the same time a preheating of the MEE feed water. Different structures of integration are assessed theoretically. Simulation models based on the heat and mass transfer balances are developed for the parallel feed (PF), parallel cross (PCF), forward feed (FF)-MEE, and direct contact MD (DCMD) configurations. The main performance parameters such as the performance ratio, recovery ratio, and specific energy consumption are investigated via simulation of various MEE-MD configurations and operating parameters. The results show that hybrid MEE and MD structures outperform the standalone MEE systems. The performance ratio increased by more than 25% when hybrid structure is used as compared to standalone MEE. The results revealed also that the hybrid parallel PF-MEE-MD configuration yields the best performance compared to other hybrid configurations. It also outperforms the standalone PCF-MEE which is widely adopted and used in practice. Besides, the performance of the hybrid configurations is especially sensitive to the feed temperature's variation in both MEE and MD units. Increasing MD feed temperatures has a positive impact on the recovery ratios of the combined systems.

The mathematical modeling of changes in grain moisture and heat loss on adsorption drying from parameters of grain dryer

The literary review has established that to reduce the heat loss of a dryer, the heat insulation of the dryer design, heat recovery and suction air flow are used. However, the use of prospective suction airflow is not fully disclosed. In the article, based on the equations of mass transfer, thermal and material balance, the dependences of grain moisture and heat consumption on drying from the regime parameters of the dryer, grain properties, and air flow used during drying are found. Analytical dependencies are illustrated by the corresponding schedules. Analysis of the analytical and graphical dependences of grain moisture and heat consumption on the corn grain sample has established that reducing the heat consumption when drying the grain, possibly reducing the pressure in the dryer, while simultaneously draining the air stream before it is fed to the drying chamber.

Irreversibility Analysis for Double Diffusive Convection Flow of a Gas Mixture in a Chamfered Corners Square Enclosure Filled with a Porous Medium

An analysis of thermodynamic irreversible principle (TPI), through determination of entropy generation rate, during double-diffusive convection in an octagonal shape, filled with saturated fluid in a porous enclosure, is numerically investigated in this work, by studying the influence of thermodiffusion effect on entropy generation variations. According to the fluid flow evolution under consideration, the influence of mass flux due to temperature gradient is incorporated into the governing equations of the problem, which are solved numerically, using the COMSOL software. Appropriated thermal and solutal Dirichlet boundary conditions are used under the Boussinesq approximation. For the porous medium, the Darcy–Brinkman model is assumed in coupling with energy and mass transfer balance equations. The flow model is described in terms of mass flux due to temperature gradient, buoyancy ratio, thermal Rayleigh number and porosity of the medium. Results of the variations of heat transfer and entropy generation in the studied enclosure are graphically illustrated and are basically discussed, through various physical aspects of the problem. The numerical computations are presented for various values of thermal Rayleigh number (RaT), thermal diffusion parameter (Kt), Darcy number (Da), buoyancy ratio (N) and porosity of the medium (e). In addition, the total entropy generation due to thermal, species and mixed diffusion gradients; Darcy–Brinkman dissipation; and fluid friction are studied and discussed. It is found that the entropy generation increases strongly when passing from the cooperative case of buoyancy forces to the opposite case. A similar behaviour is obtained with increasing the thermal diffusion ratio, at a constant buoyancy ratio. Also, an accentuated increase in entropy generation is observed when the Darcy number exceeds the value 10−6. Moreover, the results show that the augmentation of the thermodiffusion effect induces an increase in total entropy generation, at fixed value of the porosity.

Numerical simulation of oscillatory flow and heat transfer in pulsating heat pipes with multi-turns using OpenFOAM

The oscillatory flow and the heat transfer in two-dimensional pulsating heat pipes (PHPs) with multi-turns were simulated using OpenFOAM. The volume of fluid method was used for the phase change, and the behavior of the working fluid was achieved by considering the mass transfer balance between the evaporation and the condensation. Ethanol was used as the working fluid, and the liquid phase and the vapor phase were assumed to be incompressible. The result revealed that the temperature variation curves did not converge to one pattern according to the number of grids in the symmetric shape of the PHP because the starting time of the working fluid circulation was different. In the PHP with the asymmetric shape, the circulation started earlier than in the PHP with symmetric shape. When the bond number was 0, which means being in zero gravity, the working fluid dried out in the evaporator section of the PHPs with 5 and 10 turns. However, the working fluid still remained in the PHPs with 15 and 20 turns. The numerical analysis performed in this article is expected to help to simulate the flow phenomenon in PHP.

Preparation of a Simple Simulation Material for Learning Plate Theory by Visualizing the Mass Transfer and Mass Valance

A simple simulation sheet drawn a chromatographic column divided into five plates and fifty number cards indicating the amount of a solute possible to exist in the mobile or stationary phase of each plate were prepared. Two rectangular blanks each for indicating the mobile and stationary phases were also drawn on each plate. A simulation for plate theory was performed in the following three steps: (1) only the cards placed on the blanks of mobile phase are moved to that of the mobile phase in the next plate, (2) the two cards, indicating the numbers divided by the sum of numbers on the card(s) placed on each plate according to the distribution coefficient, are selected and replaced on the two blanks of each plate, (3) these steps continue until the fifth plate. This simulation helped students to progress in their learning the mass transfer and mass balance on plate theory in chromatography.

Numerical modeling of heat transfer at cooling a high-temperature steel cylinder by the flow of a water-air medium with account taken of the gamma - alpha transformation

Based on the mathematical model of the conjugate heat transfer from a steel cylinder to a water-air medium, the results of the calculation of the cooling rate and the mass fraction of the α-phase in supercooled austenite during the γ → α transformation are obtained. To find the mass fraction of the α-phase in austenite during supercooling, a modified Kolmogorov – Johnson – Mel – Avrami equation is used. The calculation results have been obtained during the cooling of the high-temperature metal cylinder by the laminar gas-liquid flow in an annular channel with account taken of vaporization in the liquid. The regulation of the heat and mass transfer balance of the system during the vaporization is based on the energy model of the heat balance and is performed using the effective thermophysical parameters of the cooling medium. The mathematical model is presented in a two-dimensional non-stationary formulation taking into account the symmetry of the cooling medium flow relative to the longitudinal vertical axis of the cylinder. To solve the system of differential equations, the control volume method is used. The parameters of the cooling medium flow are calculated using an algorithm SIMPLE. For the iterative solution of the system of linear algebraic equations, the methods of Gauss-Seidel and under-relaxation are used. The calculations, are carried out using a grid with a condensing profile at the «metal cylinder – liquid» and «liquid - outer metal ring» boundaries on the liquid side and the metal side.

Mathematical Modeling of Suspended Microbial Fuel Cells and Electron Transfer Mediator using Homotopy Perturbation Method

The theoretical model of microbial fuel cells with suspended cells and an additional electron transfer mediator is analyzed. Many biological, chemical and electrochemical reactions occur in the bulk liquid and on the surface of the electrode with the substrate (glucose), oxidized mediators and reduced mediators. The homotopy perturbation method (HPM) is used to solve the nonlinear diffusion equations in microbial fuel cells. Direct and approximate polynomial expression of a substrate (glucose), oxidized mediator and reduced mediator concentration are obtained at the mass transfer balance layer. The results of the experiment are compared with the results of the analytical and simulation and satisfactory agreement is noted.