Constant Flux Approximation Research Articles

Imperfect narrow escape problem.

We consider the kinetics of the imperfect narrow escape problem, i.e., the time it takes for a particle diffusing in a confined medium of generic shape to reach and to be adsorbed by a small, imperfectly reactive patch embedded in the boundary of the domain, in two or three dimensions. Imperfect reactivity is modeled by an intrinsic surface reactivity κ of the patch, giving rise to Robin boundary conditions. We present a formalism to calculate the exact asymptotics of the mean reaction time in the limit of large volume of the confining domain. We obtain exact explicit results in the two limits of large and small reactivities of the reactive patch, and a semianalytical expression in the general case. Our approach reveals an anomalous scaling of the mean reaction time as the inverse square root of the reactivity in the large-reactivity limit, valid for an initial position near the extremity of the reactive patch. We compare our exact results with those obtained within the "constant flux approximation"; we show that this approximation turns out to give exactly the next-to-leading-order term of the small-reactivity limit, and provides a good approximation of the reaction time far from the reactive patch for all reactivities, but not in the vicinity of the boundary of the reactive patch due to the above-mentioned anomalous scaling. These results thus provide a general framework to quantify the mean reaction times for the imperfect narrow escape problem.

Numerical Computation of Fission-Product Poisoning Build-up and Burn-up Rate in a Finite Cylindrical Nuclear Reactor Core

All fission products are classified as reactor poisons because they absorb neutrons to some extent, most of which buildup slowly as the fuel burns up and eventually constitutes a long term reactivity effect in the core. Amidst the numerous fission fragments produced per fission, the presence of Xenon-135 and Samarium-149 has the greatest effect on a reactor core multiplication factor because of their large absorption cross-sections. In this study, we present a modified one-group time independent neutron diffusion equation using the method of Eigen functions and also provided an algorithm to calculate the temperature variations of the neutron fluxes. The solution obtained from the diffusion equation was used to determine the initial thermal neutron flux needed for the reactor startup. The four basic fission-product poisoning buildup and burn-up rate equations were solved using direct integration method and constant flux approximation over a particular time interval. Furthermore, a computer algorithm called Java code for Fission-Product Poisioning Build-up and Burn-up (Jac-FPPB) code was designed to calculate the temperature variations of the neutron fluxes, fission -isotopes cross sections and the atom concentrations of the fission products over a given time interval. The result from Jac-FPPB code showed that the neutron fluxes and neutron energies increase as the temperature of the fuel increases. In addition, the computed atom concentrations of each fission isotopes at any given time interval showed that the isotopes increasingly build up steadily at the initial time interval and rises to a constant level where the buildup rate of the isotopes approximately equals its burn up rate. This study concluded that the designed algorithm (JaC-FPBB code) proved efficient as it could compute the build-up and burn-up rates for the two important fission fragments in a nuclear reactor core. The code is easily accessible and could serve as a tool for the development of nuclear energy in developing countries, especially Nigeria.

Iso-flux tension propagation theory of driven polymer translocation: the role of initial configurations.

We investigate the dynamics of pore-driven polymer translocation by theoretical analysis and molecular dynamics (MD) simulations. Using the tension propagation theory within the constant flux approximation we derive an explicit equation of motion for the tension front. From this we derive a scaling relation for the average translocation time τ, which captures the asymptotic result τ∝N0(1+ν), where N0 is the chain length and ν is the Flory exponent. In addition, we derive the leading correction-to-scaling term to τ and show that all terms of order N0(2ν) exactly cancel out, leaving only a finite-chain length correction term due to the effective pore friction, which is linearly proportional to N0. We use the model to numerically include fluctuations in the initial configuration of the polymer chain in addition to thermal noise. We show that when the cis side fluctuations are properly accounted for, the model not only reproduces previously known results but also considerably improves the estimates of the monomer waiting time distribution and the time evolution of the translocation coordinate s(t), showing excellent agreement with MD simulations.

Laminar flow with injection through a long dead‐end cylindrical porous tube: Application to a hollow fiber membrane

AbstractEffects of membrane length and hydraulic resistance on the steady‐state laminar flow of a fluid with injection in a dead‐end cylindrical porous tube have been investigated using the perturbation approach of the Navier‐Stokes and continuity equations. Analytic solutions of the dynamic equations, reduced to non linear differential equations, were obtained and applied to submerged, dead‐end hollow fiber membranes of large resistances and small wall Reynolds numbers. The velocity and pressure profiles were solved using the method of separation of variables as with physical properties of the membrane and fluid and the wall velocity at the dead end. The constant flux approximation was found to be valid only for a short membrane with a large hydraulic resistance. The constant permeability approximation used in this study is universal for a membrane of an arbitrary length, through which the fluid velocities and pressure increase exponentially from the dead end to the open end. © 2010 American Institute of Chemical Engineers AIChE J, 2011

Applicability of Constant Flux Approximation in Method of Characteristics with Filtering to Tiny Regions

For the method of characteristic (MOC), a system with large-gradient neutron flux caused by a strong absorber or neutron leakage is reported to entail large errors in spatial mesh discretization and require very fine mesh spacing. To apply the MOC to such fine models, the ray-trace path width has to be fine in order to make many paths cross each region. Our new method intends to obtain good accuracy with a coarse path width. With a coarse path width on the MOC, some tiny regions have less ray-tracing paths. The reliability of flux calculation for the regions can be evaluated with the calculation volumes that are estimated in ray tracing. If the discrepancy between the calculation and true volumes becomes large, the accuracy cannot be expected. In this study, the discrepancies were numerically evaluated, and it is found that the discrepancies occur on a very tiny region. To make the flux calculation of such tiny regions more reliable, an approximation, in which the outgoing flux is equal to the incoming neutron, is applied instead of the usual MOC equation. The criteria switching on the approximation, which is called filtering, was numerically evaluated for PWR assemblies. The method is validated with numerical benchmark calculations.

Applicability of Constant Flux Approximation in Method of Characteristics with Filtering to Tiny Regions

For the method of characteristic (MOC), a system with large-gradient neutron flux caused by a strong absorber or neutron leakage is reported to entail large errors in spatial mesh discretization and require very fine mesh spacing. To apply the MOC to such fine models, the ray-trace path width has to be fine in order to make many paths cross each region. Our new method intends to obtain good accuracy with a coarse path width. With a coarse path width on the MOC, some tiny regions have less ray-tracing paths. The reliability of flux calculation for the regions can be evaluated with the calculation volumes that are estimated in ray tracing. If the discrepancy between the calculation and true volumes becomes large, the accuracy cannot be expected. In this study, the discrepancies were numerically evaluated, and it is found that the discrepancies occur on a very tiny region. To make the flux calculation of such tiny regions more reliable, an approximation, in which the outgoing flux is equal to the incoming neutron, is applied instead of the usual MOC equation. The criteria switching on the approximation, which is called filtering, was numerically evaluated for PWR assemblies. The method is validated with numerical benchmark calculations.

Estimation of Higher Sobolev Norm from Lower Order Approximation

Given an accurate piecewise constant flux approximation $q$ of a smooth (but unknown) exact flux $p$, how can one compute a Sobolev norm of $p$ over a small domain $\omega$? This question arises in a duality argument of goal-oriented finite element a posteriori error analysis. Two equivalent practical estimators $\eta_{M}$ and $\eta_{\mathcal{E}}$ are presented for an approximation $\!q\!$ of $|p|_{H^1(\omega)}$ and model situations are addressed where $\!q\!$ is asymptotically an upper and +lower bound of $|p|_{H^1(\Omega)}$.

An Enthalpy Control Volume Method for Transient Mass and Heat Transport with Solidification

A single domain enthalpy control volume method is developed for solving the coupled fluid flow and heat transfer with solidification problem arising from the continuous casting process. The governing equations consist of the continuity equation, the Navier–Stokes equations and the convection–diffusion equation. The formulation of the method is cast into the framework of the Petrov–Galerkin finite element method with a step test function across the control volume and locally constant approximation to the fluxes of heat and fluid. The use of the step test function and the constant flux approximation leads to the derivation of the exponential interpolating functions for the velocity and temperature fields within each control volume. The exponential fitting makes it possible to capture the sharp boundary layers around the solidification front. The method is then applied to investigate the effect of various casting parameters on the solidification profile and flow pattern of fluids in the casting process.

Approximating the Becker–Döring cluster equations

The Becker–Döring equations model the dynamics of coagulation and fragmentation of clusters of identical particles. The model is an infinite system of ordinary differential equations (ODEs) which specify the rates of change of the concentrations of r-particle clusters. For numerical computation the system is truncated at clusters of a finite size, but this might have to be prohibitively large to capture the metastable behaviour of the system. In this work we derive and investigate the properties of approximations of the Becker–Döring equations which aim to capture the metastable behaviour of the problem with a much reduced system of equations. The schemes are based on a piecewise constant flux approximation, a Galerkin method using a discrete inner product, a discretization of a PDE that in turn approximates the Becker–Döring equations and a lumped coefficients model. We establish a posteriori error estimates for three of these schemes and report on the results of a set of numerical experiments involving metastable behaviour in the solution. Three of the schemes give accurate results on a reduced set of equations, and the PDE scheme appears to be more efficient than the others for this problem.

The Correlation of Simultaneous Heat and Mass Transfer Experimental Data for Aqueous Lithium Bromide Vertical Falling Film Absorption

A study of simultaneous heat and mass transfer was conducted on a vertical falling film absorber to better understand the mechanisms driving the heat and mass transfer processes. Thermographic phosphors were successfully used to measure the temperature profile along the length of the absorber test tube. These measures of the local variations in temperature enabled calculation of the bulk concentration along the length of the absorber. The bulk concentration varied linearly, which infers that the concentration gradient in the direction of flow is approximately constant. The implication is that the mass flux and therefore the absorber load can be solved for using a constant flux approximation. Design data and correlations are sparse in the open literature. Some experimental data are available; however, all literature data to date have been derived at mass fractions of lithium bromide ranging from 0.30 to 0.60. Experiments were therefore conducted with no heat and mass transfer additive on an internally cooled smooth tube of 0.01905-m outside diameter and of 1.53-m length. The data, for testing at 0.62 and 0.64 mass fraction, were scaled and correlated into both Nu and Sh formulations. The average absolute error in the Nu correlation is about ±3.5% of the Nu number reduced from the experimental data. The Sh correlation is about ±5% of the reduced Sh data. Data from the open literature were reduced to the authors Nu and Sh formulations, and were within 5% of the correlations developed in the present study. The study therefore provides test data with no heat and mass transfer additive and correlations for the coupled heat- and mass-transfer process that are validated against the extensive experimental data.

Theory of the diffusion-influenced substrate binding rate to a buried and gated active site

The effects of stochastic gating on the diffusion-influenced substrate binding rate to a buried active site are studied. An approximation introduced by Samson and Deutch [J. Chem. Phys. 68, 285 (1978)] is shown to be equivalent to making the constant-flux approximation on the entrance to the active site. The constant-flux approximation is then extended to the case where the entrance to the active site is stochastically gated because of conformational fluctuations of the enzyme. The stochastically gated rate constant, ksg, is found to be given by the relation 1/ksg=1/k+wo/wc(wo+wc)ĥ(wo+wc), where k is the rate constant in the absence of gating, ĥ(s) is the Laplace transform of the total flux across the entrance after the substrate is started from an equilibrium distribution outside the entrance, and wo and wc are the transition rates between the open and closed gating states. This relation reduces to an approximate relation derived earlier for a more restrictive situation, where the reactivity within the active site is gated. The leading term in the expansion of sĥ(s) for large s is DA〈exp(−βU)〉(s/D)1/2/2, where D is the diffusion coefficient of the substrate, A is the total area of the entrance, and 〈exp(−βU)〉 is the average Boltzmann factor on the entrance. The time scale of conformational fluctuations, ∼ a few picoseconds, is much shorter than the time scale of diffusion, so this leading term is useful for estimating (wo+wc)ĥ(wo+wc). A further consequence of the disparity in time scales is that the value of (wo+wc)ĥ(wo+wc) is much larger than k. As a result the decrease of the rate constant due to gating is relatively small (unless the entrance to the active site is closed nearly all the time). This suggests that a buried and gated active site may play the important role of controlling enzyme specificity without sacrificing efficiency.

Measurements of concentration gradients of HNO 2 and HNO 3 over a semi-natural ecosystem

Gradients of HNO 2 and HNO 3 have been measured over a semi-natural grassland in Central Switzerland. The site was characterised by a high excess of NH3 over HNO 3. In such a situation the measured product of NH3 and HNO 3 often exceeds the thermodynamic equilibrium product over solid NH 4NO 3. As a consequence, transient formation and re-evaporation of NH 4NO 3 can occur leading to a height-dependent flux of HNO 3. Consequently, the constant flux approximation, which is a pre-requisite for the determination of HNO 3 deposition by the gradient method, is violated. Measured HNO 2 concentrations are higher as can be explained by production of homogeneous gas-phase reaction alone. But no evidence for a ground-based source from the vegetation-covered soil has been found.

A SVAT scheme for NO, NO2, and O3 ? Model description and test results

A soil/vegetation/atmosphere transfer (SVAT) scheme for determining the dry deposition and/or emission fluxes of NO, NO2, and O3 in the atmospheric surface layer over horizontally uniform terrain covered with fibrous canopy elements is presented and discussed. This transfer scheme is based on the micrometeorological ideas of the transfer of momentum, heat and matter near the Earth's surface, where chemical reactions between these trace gases are included. The fluxes are parameterized by first-order closure principles. The uptake processes by vegetation and soil are described in accord with Deardorff (1978). The SVAT scheme requires only routine data of wind speed, dry- and wet-bulb temperatures, short wave and long wave radiation, and the concentrations of O3 and nitrogen species provided by stations of monitoring networks. First model results indicate that the dry deposition fluxes of NO, NO2, and O3 are not only influenced by meteorological and plant-physiological parameters, but also by chemical reactions between these trace species and by NO emission from the soil. Furthermore, a small displacement in the concentrations of NO, NO2, and O3 within in the range of the detection limits of the chemical sensors can produce large discrepancies in the flux estimates, which are manifested here by the shift from height-invariant fluxes substantiated by the photostationary state to strongly height-dependent fluxes caused by the departure from that state. Especially in the case of these nitrogen species the widely used ‘big leaf’ multiple resistance approach, which is based on the constant flux approximation seems to be inappropriate for computing dry deposition fluxes and deposition velocities.

On the dry deposition of ozone and reactive nitrogen species

Dry deposition of ozone and reactive nitrogen species, such as NO, NO2, NO3, N2O5, HNO3, NH3 and NH4NO3, is examined in the context of numerical methods. These methods are based on the generally accepted micrometeorological ideas of the transfer of momentum, sensible heat, and matter near the Earth's surface, where chemical reactions among these trace species are considered. The fluxes in the turbulent region of the atmospheric surface layer are parameterized by first-order closure principles. The uptake processes by vegetation and the soil are described by a Deardorfl=type soil-vegetation-atmosphere transfer scheme. As in the case of HNO3 and NH3 the resistance of the system vegetation—soil against uptake of matter seems to be of minor importance, parameterization approaches for the more important transfer resistances of the interfacial sublayer adjacent to the surface are evaluated.The model results show that especially the dry deposition fluxes of reactive nitrogen species are not only influenced by micrometeorological and plant-physiological parameters, but also strongly affected by chemical reactions. In most cases, the fluxes of these trace constituents vary strongly with height and often show a change of direction. These flux results differ considerably from those derived with the constant flux approximation, sometimes up to several hundred percent. Thus, in such cases the most widely used “big leaf” multiple resistance approach which is based on the constant flux approximation seems to be inappropriate for deriving dry deposition fluxes and deposition velocities of reactive nitrogen species.

Diffusion-controlled reaction rate to asymmetric reactants under Coulomb interaction

The rate constant for diffusion-controlled reactions between asymmetric reactants described by the simple model of Solc and Stockmayer under the influence of Coulomb-type interaction is considered. Using the method of dual series relations, we calculate the rate constant with a high accuracy and obtain some approximate analytical formulas. We compare our results with an approximate analytical formula derived before by the constant-flux approximation and with numerical calculations based on the Brownian dynamics simulation. It is shown that in the case of strong attractive potential the rate constant only slightly depends on the size of the active site and tends to the classical Debye result for isotropically reactive particles with a further increase in the Onsager length. Moreover, it is shown that for small-sized active sites the effect of the interaction potential is to scale the rate constant for neutral reactants by a Boltzmann factor, which was first conjectured by Zhou [Biophys. J. 64, 1711 (1993)].