One-dimensional Steady-state Model Research Articles

Modeling of temporal variations of vertical concentration profile of 7Be in the atmosphere

Study of the vertical concentration profile and of the deposition of cosmogenic radionuclides provides information on the vertical transport in the stratosphere and troposphere and the processes of scavenging of aerosol particles by precipitation. Information on the distribution of atmospheric aerosols is important for the understanding of the physical processes relating to the studies in weather climate, air pollution, and aerosol physics. In this work the one-dimensional steady-state model of vertical concentration profile was established and the values of turbulent diffusion coefficient and scavenging coefficient determined by model using experimental data of the 7Be monthly average atmospheric activity concentrations and monthly deposition fluxes in Bratislava are presented. The temporal variations of the vertical distribution profiles of 7Be for each month are also calculated.

Vertical velocity distributions through and above submerged, flexible vegetation

A one-dimensional steady-state model was constructed and used to study vertical profiles of longitudinalvelocities in open channel flows through, and above, submerged vegetation. The eddy viscosity was evaluated by using the analogue of the mixing length theory. The model of water velocity profiles takes into account the surface roughness of the channel bottom and the drag exerted by submerged flexible stems. The verification of the calculated velocity profiles was carried out based on data obtained in laboratory experiments. The proportionality coefficients for the analogue of the mixing length formulae in both layers—within homogenous flexible stems and above them—were determined.

Numerical Analysis of a Pilot-Scale Fixed-Bed Reactor for Dimethyl Ether (DME) Synthesis

Dimethyl ether (DME, CH3OCH3) is the simplest of ethers and is considered one of the leading candidates in the quest for a substitute for petroleum-based fuels. In this work, we analyzed the one-step synthesis of DME in a shell-and-tube type fixed-bed reactor. We have conducted simulations using a one-dimensional steady-state model of a heterogeneous catalyst bed. This model considered heat and mass transfer between the catalyst pellets and reactants and the effectiveness factor of the catalysts, together with the reactor cooling through the reactor tube wall. The reactor simulation was carried out under steady-state conditions. Thereafter, we compared the data of simulation results with the data obtained from the operation of a pilot-scale reactor and found acceptable agreement between the two data sets. Moreover, we analyzed effectiveness factors of the catalyst pellet and along the length of the reactor where we also analyzed temperature profiles and concentrations of the components. The analyses showed that complex reactions, when coupled with pore diffusion within the catalyst pellets, result in unusual values of the effectiveness factor along the length of the reactor. Given these results, the reactor demonstrated high performance for such variables as CO conversion and DME yield. On the contrary, operations over a high temperature range are unavoidable even though this type of reactor has been in general use. Eventually more effective cooling strategies in the operation of this type of reactor should be developed and studied.

Effect of capillary pressure on liquid water removal in the cathode gas diffusion layer of a polymer electrolyte fuel cell

In order to investigate the effect of capillary pressure on the transport of liquid water in the cathode gas diffusion layer (GDL) of a polymer electrolyte fuel cell, a one-dimensional steady-state mathematical model was developed, including the effect of temperature on the capillary pressure. Numerical results indicate that the liquid water saturation significantly increases with increases in the operating temperature of the fuel cell. An elevated operating temperature has an undesirable influence on the removal of liquid water inside the GDL. A reported peculiar phenomenon in which the flooding of the fuel cell under a high operating temperature and an over-saturated environment is more serious in a GDL combined with a micro-porous layer (MPL) than in a GDL without an MPL [Lim and Wang, Electrochim. Acta 49 (2004), 4149–4156] is explained based on the present analysis.

Effects of functional surface on performance of a micro heat pipe

The effect of a functional surface with the axial ladder contact angle distribution on the thermal performance of a triangular micro heat pipe has been analyzed based on a one-dimensional steady-state model. Compared with the traditional micro heat pipe (MHP) with a uniform contact angle distribution on its surface, the simulation results show that a MHP with a functional surface can remove a greater amount of heat under the same condition. The increase in thermal performance is more obvious with the increase in the ladder difference of the contact angles between the adjacent sections of the MHP. The increased thermal performance associated with the functional surface can be attributed to the increase of the liquid capillary force as well as the no obvious increase of the liquid shearing force provided by the functional surface, which also brings about the increase in condensate mass flow rate through the adiabatic section–evaporation section interface. It is also found that for the traditional MHP with uniform contact angle surface, there is an optimal contact angle leading to the maximum heat input. The deviation of the optimal value will decrease the capillary force and thermal performance of the MHP.

Modelling Gas Holdup in Flotation Column Froths

AbstractA one‐dimensional steady‐state model is developed for the prediction of axial variation of the gas holdup in flotation column froths. Froth is considered as an inverse fluidized bed of bubbles and hence the frictional pressure gradient is obtained based on the energy balance. Pressure gradient can also be obtained from the Ergun equation with adjustable constants. The model correctly captures the effect of superficial gas velocity, superficial liquid velocity and bubble diameter on the variation of the gas holdup along the froth height. The predictions of the model are in agreement with the experimental data from the literature.

An adaptive high-order discontinuous Galerkin method with error control for the Hamilton–Jacobi equations. Part I: The one-dimensional steady state case

We propose and study an adaptive version of the discontinuous Galerkin method for Hamilton–Jacobi equations. It works as follows. Given the tolerance and the degree of the polynomial of the approximate solution, the adaptive algorithm finds a mesh on which the approximate solution has an L ∞-distance to the viscosity solution no bigger than the prescribed tolerance. The algorithm uses three main tools. The first is an iterative solver combining the explicit Runge–Kutta discontinuous Galerkin method and the implicit Newton’s method that enables us to solve the Hamilton–Jacobi equations efficiently. The second is a new a posteriori error estimate based on the approximate resolution of an approximate problem for the actual error. The third is a method that allows us to find a new mesh as a function of the old mesh and the ratio of the a posteriori error estimate to the tolerance. We display extensive numerical evidence that indicates that, for any given polynomial degree, the method achieves its goal with optimal complexity independently of the tolerance. This is done in the framework of one-dimensional steady-state model problems with periodic boundary conditions.

Model development and validation: Co-combustion of residual char, gases and volatile fuels in the fast fluidized combustion chamber of a dual fluidized bed biomass gasifier

A one-dimensional steady state model has been developed for the combustion reactor of a dual fluidized bed biomass steam gasification system. The combustion reactor is operated as fast fluidized bed (riser) with staged air introduction (bottom, primary and secondary air). The main fuel i.e., residual biomass char (from the gasifier), is introduced together with the circulating bed material at the bottom of the riser. The riser is divided into two zones: bottom zone (modelled according to modified two phase theory) and upper zone (modelled with core-annulus approach). The model consists of sub-model for bed hydrodynamic, conversion and conservation. Biomass char is assumed to be a homogeneous matrix of C, H and O and is modelled as partially volatile fuel. The exit gas composition and the temperature profile predicted by the model are in good agreement with the measured value.

Development of Phase Change Heat Spreader for Treatment of Intractable Neocortical Epilepsy

A combined analytical and experimental investigation was conducted to develop a flat, phase change heat spreader to enable focal thermoelectric cooling as a treatment for intractable neocortical epilepsy. The design parameters required minimum transport capacity of 5 W with an associated temperature drop of less than 0.5°C. A one-dimensional steady-state model was developed and the predicted performance characteristics compared with the results obtained from the experimental evaluation of three conceptual designs of varying complexity. These designs varied in terms of the materials used, but all three used water as the working fluid. The experimental results indicate that the minimum transport limit of 5 W can be achieved by one of the three concepts evaluated with a maximum overall temperature drop of 0.36°C at 5 W, well below the 0.5°C limit and within the experimental uncertainty of the temperature measurement technique used. A simple model was used to aid in the selection of an appropriate heat pipe. Using the verified model, the initial design was optimized, and based upon this optimized design, a number of test articles were fabricated and evaluated experimentally.

On the effectiveness of Leverett approach for describing the water transport in fuel cell diffusion media

This paper investigates the effectiveness of employing the traditional Leverett approach for describing the capillary-induced flow in thin-film fuel cell diffusion media (DM) with mixed wettability. A one-dimensional steady-state analytical model is developed to analyze the capillary transport of liquid water through the thin-film DM. A capillary pressure–liquid saturation relation is derived based on the experimental data reported by Gostick et al. [J.T. Gostick, M.W. Fowler, M.A. Ioannidis, M.D. Pritzker, Y.M. Volfkovich, A. Sakars, J. Power Sources, 156 (2006) 375]. The empirical fit is applicable to the fuel cell DMs tested by reference [J.T. Gostick, M.W. Fowler, M.A. Ioannidis, M.D. Pritzker, Y.M. Volfkovich, A. Sakars, J. Power Sources, 156 (2006) 375] under a limited set of conditions, providing a means to evaluate the effectiveness of the traditional Leverett approach. Furthermore, a new characteristic relative permeability correlation appropriate for the tested DMs under the given conditions was obtained by fitting the experimental capillary pressure data [J.T. Gostick, M.W. Fowler, M.A. Ioannidis, M.D. Pritzker, Y.M. Volfkovich, A. Sakars, J. Power Sources, 156 (2006) 375] into four well-established empirical models. The empirically-derived relationships are then integrated into an analytical model framework in order to compare the liquid saturation profiles predicted by both approaches. The results show that use of the standard Leverett approach equipped with Leverett J-function can lead to significant errors; therefore, an extension of this approach appropriate for fuel cell DM with mixed wettability is needed for reliable model predictions. Finally, a sensitivity analysis is performed to assess the relative significance of the various input parameters on the predicted saturation profiles.

Study on water transport in PEM of a direct methanol fuel cell

Water transport phenomenon in PEM and the mechanism of occurrence and development of a two-phase countercurrent flow with corresponding transport phenomenon in the PEM are analyzed. A one-dimensional steady state model of heat and mass transfer in porous media system with internal volumetric ohmic heating is developed and simulated numerically. The results show that two dimensionless parameters D and N, which reflect the liquid water flow rate and inner heat source in the PEM, respectively, are the most important factors for the water fraction and thermal balance in the PEM. The saturation profiles within the two-phase region at various operating modes are obtained. Smaller mass flow rate of liquid water and high current density are the major contributions to the membrane dehydration.

Geophysical survey of the intra-caldera icefield of Mt Veniaminof, Alaska

AbstractMt Veniaminof is a large active stratovolcano located on the Alaska Peninsula (56.2° N, 159° W). We present results of the first geophysical survey of the icefield that fills much of the 10 km×8 km caldera that was most recently modified during the last major eruption roughly 3700 BP. The subglacial topography and ice volume are derived from an 8MHz radio-echo sounding survey conducted in July 2005. Prominent internal reflectors are assumed to be isochronal ash/acid deposits related to local eruptions. Accumulation rates and basal melt rates are calculated using a Nye one-dimensional steady-state accumulation model applied at a location that approximates an ice divide and calibrated by matching internal reflectors with published records of recent local volcanic eruptions. The model yields order of magnitude estimates of the accumulation rate of 4ma–1 water equivalent and 2 ma–1 of basal melt. The subsequent geothermal flux of ∽19Wm–2 is similar to active hydrothermal vents in volcanic lakes. We suggest that these values represent an upper limit for the geothermal flux within the ice-covered regions of the main caldera. We also analyze likely subglacial meltwater flow paths to examine the implications of recent eruption activity at an active intra-caldera cinder cone. Two lava-producing eruptions from the cinder cone in 1983–84 and 1993–94 melted roughly 0.17km3 of ice. The lack of significant deformation of the internal stratigraphy to the south and east of the melt hole suggests that any subglacial drainage in those directions was entirely within subglacial deposits. We suggest that the more likely drainage route was northwest into a large outlet glacier.

Energy analysis of silicon solar cell modules based on an optical model for arbitrary layers

An optical model for arbitrary layers is developed and a one-dimensional steady-state thermal model is applied to analyze the energy balance of silicon solar cell modules. Experimental measurements show that simulations are in good agreement, with a maximum relative error of 8.43%. The wind speed v wind, ambient temperature T amb and irradiance G are three main factors influencing the temperature of a photovoltaic panel. Over the course of a day the electrical output is reduced by the module temperature to only 32.5% of the rated value. Optical studies reveal that before 8:00 hours and after 16:00 hours, significant incident energy is lost by reflection because of the large angle of incidence θ in, while at other times of day optical losses are nearly the same due to only small changes of transmission for θ in < 45°. In addition, some optical losses result from the mismatched refractive indexes of encapsulating materials, especially at the ethylene-vinyl-acetate (EVA)/anti-reflection coating (ARC) and the ARC/Si interfaces. The uses of SiO 2 and TiO 2 as ARC materials for un-encapsulated and encapsulated Si solar cells are investigated by simulation. Comparing the results indicates that TiO 2 as ARC reduces the reflective optical loss within λ = 0.4–1.1 μm after encapsulation, while SiO 2 as ARC increases the loss by ∼5%. Energy allotment analysis shows that from 9:00 to 15:00, the reflective and transmissive optical losses are relatively steady at ∼26% and ∼13% of the incident energy, while the convective and radiative heat losses account for a further ∼30% and ∼24%, respectively. Thus, only ∼7% of incident energy is converted to electrical power.

Modeling and experimental investigation of looped separate heat pipe as waste heat recovery facility

A looped separate heat pipe as waste heat recovery facility for the air-conditioning exhaust system has been developed in this work. A one-dimensional steady-state model is presented for determining the upper and lower operating boundaries of the initial filling ratio of the working fluid, as a function of the separate heat pipe geometry, vapor temperature of working fluid and power throughput, combined with two-phase heat exchange characteristics and distribution of the liquid film velocity along with the liquid film thickness direction. A parametric analysis is performed to investigate the effects of the length of the evaporator, vapor temperature, and power throughput on the critical values of the upper and lower boundaries. Simulation results show that the length of the evaporator makes almost no influence on the upper boundary, but great effect on the lower boundary. An increase of the vapor temperature leads to the easier arriving of the lower boundary. Moreover, operation ranges of the separate heat pipe vary with the working fluids. Water and methanol were used separately. An experiment was implemented to validate the simulated results. The numerical predictions compare favorably with experimental results.

Asymptotic profile in a one-dimensional steady-state nonisentropic hydrodynamic model for semiconductors

We study the stationary flow for a one-dimensional nonisentropic hydrodynamic model for semiconductor devices. This model consists of the continuous equations for the electron density, the electron current density and electron temperature, coupled the Poisson equation of the electrostatic potential. In a bounded interval supplemented by the proper boundary conditions, we investigate the zero-electron-mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit, respectively. We show the strong convergence of the sequence of solutions and give the associated convergence rate.

Sensitivity of the dynamics of volcanic eruption columns to their shape

This paper presents a one-dimensional steady-state model to investigate the sensitivity of the dynamics of sustained eruption columns to radius variations with height due to thermal expansion of the entrained air, and decreases in atmospheric pressure with height. In contrast to a number of previous models using an equation known as the entrainment assumption, the new model is based on similarity arguments to derive an equation set equivalent to the model proposed by Woods [Bull Volcanol 50:169–193, 1988]. This approach allows investigation of the effect of gas compressibility on the entrainment rate of ambient air, which has been little examined for a system in which a decrease in pressure significantly affects the density stratification of a compressible fluid. The new model provides results that include two end members: one in which the volume change within the eruption columns affects only the radial expansion without changing the vertical motion, and the other is the converse. The Woods [Bull Volcanol 50:169–193, 1988] model can be regarded as being between those two end members. The range of uncertainty arises because the extremely high temperature of discharged materials from a volcanic vent, and the exceptional terminal height of the eruption columns, allow significant expansion of the gas component in the eruption columns, making them behave differently from common turbulent plumes. This study indicates that the maximum height of the eruption columns is affected considerably by this uncertainty, particularly when the eruption columns extend above a height of 10 km, at which the pressure is about one-fourth the pressure at the ground surface. Column collapse may also be suppressed in wider parameter ranges than previously estimated. However, the uncertainty can be reduced by measuring column radii through a vertical profile during actual volcanic eruptions. Accordingly, this paper suggests that appropriate observation of eruption column shapes is essential for improving our understanding of the dynamics of eruption columns.

Modelling potentials, concentrations and current densities in porous electrodes for metal recovery from dilute aqueous effluents

One-dimensional steady-state models have been developed for the recovery of Pb(II) ions from lead–acid battery recycling plant effluent by simultaneous lead and lead dioxide deposition, including oxygen evolution/reduction and hydrogen evolution as loss reactions. Both monopolar and bipolar reactor with porous graphite electrodes were modelled, as a design aid for predicting spatial distributions of potentials, concentrations, current densities and efficiencies, as well as specific electrical energy consumptions and by-pass currents. Since the industrial effluent contains a large excess of supporting electrolyte (Na2SO4), the electrical migrational contribution to reactant transport rates was neglected and the current density–potential relationship was described by the Butler–Volmer equation, allowing for both kinetic and mass transport control. The models were implemented and the governing equations solved using commercial finite element software (FEMLAB). The effects were investigated of electrolyte velocity, applied cathode potential, dissolved oxygen concentration and inlet Pb(II) ion concentration on single-pass conversion, current efficiency and specific electrical energy consumptions. According to model predictions, de-oxygenation of the inlet process stream was found to be crucial to achieving acceptable (i.e. >0.8) current efficiencies. Bipolar porous electrodes were also determined to be inappropriate for the recovery of Pb(II) from effluents, as the low concentration involved resulted in the predicted fraction of current lost as by-pass current, i.e. current not flowing in and out of the bipolar electrode, to be greater than 90% for the ranges of the variables studied.

Catalytic Combustion Systems for Microscale Gas Turbine Engines

As part of an ongoing effort to develop a microscale gas turbine engine for power generation and micropropulsion applications, this paper presents the design, modeling, and experimental assessment of a catalytic combustion system. Previous work has indicated that homogenous gas-phase microcombustors are severely limited by chemical reaction timescales. Storable hydrocarbon fuels, such as propane, have been shown to blow out well below the desired mass flow rate per unit volume. Heterogeneous catalytic combustion has been identified as a possible improvement. Surface catalysis can increase hydrocarbon-air reaction rates, improve ignition characteristics, and broaden stability limits. Several radial inflow combustors were micromachined from silicon wafers using deep reactive ion etching and aligned fusion wafer bonding. The 191mm3 combustion chambers were filled with platinum-coated foam materials of various porosity and surface area. For near stoichiometric propane-air mixtures, exit gas temperatures of 1100K were achieved at mass flow rates in excess of 0.35g∕s. This corresponds to a power density of ∼1200MW∕m3; an 8.5-fold increase over the maximum power density achieved for gas-phase propane-air combustion in a similar geometry. Low-order models, including time-scale analyses and a one-dimensional steady-state plug-flow reactor model, were developed to elucidate the underlying physics and to identify important design parameters. High power density catalytic microcombustors were found to be limited by the diffusion of fuel species to the active surface, while substrate porosity and surface area-to-volume ratio were the dominant design variables.

Model-based prediction of suitable operating range of a SOFC for an Auxiliary Power Unit

This paper presents a one-dimensional steady state model of a solid oxide fuel cell (SOFC) to be used in an Auxiliary Power Unit (APU). The fuel cell is fed a prereformed gas from an external autothermic reformer. In addition to the three electrochemical reactions (reduction of oxygen at the cathode, oxidation of hydrogen and carbon monoxide at the anode) the water–gas shift reaction and the methane steam reforming reaction are taken into account in the anode channel. The model predicts concentrations and temperatures and uses an equivalent circuit approach to describe the current–voltage characteristics of the cell. The model equations are presented and their implementation into the commercial mathematical software FEMLAB is discussed. An application of this model is used to determine suitable operating parameters with respect to optimum performance and allowable temperature.

A Model of Heat and Mass Transfer Beneath an Ablating Concrete Surface

This paper presents a one-dimensional steady-state model of heat and water vapor transport just beneath an ablating concrete surface. In the model an evaporation front separates a dry porous region through which water vapor flows to the ablation front from a semi-infinite region that is partially wet with evaporable water. The predicted water vapor pressures at the evaporation front are quite high and could conceivably cause the concrete to spall. The model is quantitatively compatible with spallation events observed during tests involving the pouring of molten steel onto concrete and is capable of explaining the disparate results obtained in two rather extensive test series on the penetration of induction heated metallic pools into concrete.