Effect Of Axial Diffusion Research Articles

The effect of a turbulent jet on gas transport during oscillatory flow.

Axial mass transport due to the combined effects of flow oscillation and a turbulent jet was studied both experimentally and with a simple theoretical model. The experiments show that the distance over which turbulence enhances transport is greatly increased by flow oscillation, and is particularly sensitive to tidal volume. The jet flow rate and jet configuration are relatively less important. To analyze the results, the region influenced by the jet is divided into two zones: a near field in which the time-mean flow velocities are larger than the turbulent fluctuations, and a far field where the time-mean flow is essentially zero. In the far field, axial mass transport is increased due to the turbulence which decays in strength away from the jet. When oscillatory flow is superimposed upon the steady jet flow, the turbulence in the far field interacts with the flow oscillations to augment the transport of turbulence energy and of mass. This transport enhancement is modeled by introducing an effective axial diffusivity analogous to that used in laminar oscillatory flow.

LAMINAR MIXED CONVECTION IN A SYMMETRICALLY OR ASYMMETRICALLY HEATED VERTICAL CHANNEL

A numerical investigation is made of laminar mixed convection of air in a vertical channel. The thermal boundary conditions considered are symmetric heating, where both plates are heated, and asymmetric heating, where one plate is heated and the other is adiabatic. Results are obtained for Rayleigh numbers of 10 3 W. T. Lawrence and J. C. Chato , Heat Transfer Effects on the Developing Laminar Flow inside Vertical Tube , J. Heat Transfer , vol. 88 , no. 2 , pp. 214 – 222 , 1966 .[Crossref], [Web of Science ®] , [Google Scholar], 10 5 W. L. Wang and P. A. Longwell Laminar Flow in the Inlet Section of Parallel Plates , AIChEJ. , vol. 10 , no. 3 , pp. 323 – 329 , 1964 .[Crossref], [Web of Science ®] , [Google Scholar], and 10 6 J. S. Vrentas , J. L. Duda , and K. G. Bargeron , Effect of Axial Diffusion of Vorticity on Flow Development in Circular Conduits Part 1, Numerical Solutions , AIChE J. , vol. 12 , no. 5 , pp. 837 – 844 , 1966 .[Crossref], [Web of Science ®] , [Google Scholar] and Gr/Re 2 L. N. Tao On Combined Free and Forced Convection in Channels , J. Heat Transfer , vol. 82 , no. 3 , pp. 233 – 238 , 1960 .[Crossref] , [Google Scholar] values of 0.1, 1, 3, and 5. The temperatures are observed to increase with increasing Gr/Re 2 L. N. Tao On Combined Free and Forced Convection in Channels , J. Heat Transfer , vol. 82 , no. 3 , pp. 233 – 238 , 1960 .[Crossref] , [Google Scholar] and decreasing Rayleigh number. The velocity profile peaks near the hot plates and exhibits a concavity, which, in the symmetric heating case, occurs around the centerline. With increasing Gr/Re 2 L. N. Tao On Combined Free and Forced Convection in Channels , J. Heat Transfer , vol. 82 , no. 3 , pp. 233 – 238 , 1960 .[Crossref] , [Google Scholar] the velocity near the hot wall increases while the velocity near the centerline decreases. The Nusselt number attains its maximum value near the inlet of the channel and increases with decreasing Gr/Re 2 L. N. Tao On Combined Free and Forced Convection in Channels , J. Heat Transfer , vol. 82 , no. 3 , pp. 233 – 238 , 1960 .[Crossref] , [Google Scholar] values.

Parallel‐Plate Electrochemical Reactor Model: A Method for Determining the Time‐Dependent Behavior and the Effects of Axial Diffusion and Axial Migration

A method is presented for determining the effects of time dependence, axial diffusion, and axial migration in a parallel‐plate electrochemical reactor (PPER). The method consists of formulating the governing equations and applying a numerical integration technique to solve a set of time‐dependent, nonlinear, coupled, multidimensional equations. This formulation reveals that the steady‐state performance of the PPER depends on the cell potential and three dimensionless groups. Predictions of the concentration, potential, and local current distributions in a PPER are presented for the electrowinning of copper from an aqueous, hydrochloric acid solution. These predictions show that axial diffusion and axial migration are significant when the aspect ratio (i.e., the ratio of electrode separation to electrode length) is greater than 0.5.

Convective diffusion to a slowly rotating spherical electrode; The effect of axial diffusion in the bulk liquid for Re 10

A complete mathematical model has been solved of the steady axially symmetric convective diffusion toward the surface of a spherical electrode of radius R rotating at an angular velocity Ω under the creeping flow conditions Re ≡ ΩR2ρ/η < 10 and Pe ≡ Ω2R4ρ/(12Dη) > 10 by the method of singular perturbations. For Pe > 300 the effect of axial diffusion has been found entirely negligible; for 10 < Pe < 300 it causes an increase of local transfer coefficients by 1-10%. For Pe < 10 the applied asymptotic method of solution, assuming Pe >> 1 is no longer applicable.

The effects of axial diffusion and permeability barriers on the transient response of tissue cylinders. II. Solution in time domain

A mathematical description of transient mass transfer in a Krogh tissue cylinder, for which a solution in transform space was presented previously, is solved in the time domain. The solution is found in the form of an expansion in terms of the eigenfunctions of a non-self-adjoint differential operator, with the eigenvalues being found by way of a computational scheme which makes use of the known characteristics of the constitutive compartments of the system. The solution is compared with previous solutions of both a complete and an approximate nature, and two modifications of the single-phase axial dispersion model are found to be especially useful: the previously-used flow-limited approximation is satisfactory for highly permeable solutes, while the apparently novel barrier-limited approximation is accurate for poorly permeable solutes at the early times of most experimental interest. Although the neglect of axial diffusion does not affect the qualitative nature of the solution, e.g. in predicting a bimodal response curve, significant discrepancies shed doubt on this practice when truly impulsive inputs are used. The results obtained raise several questions regarding existing approaches to interpretation of indicator dilution experiments. These include the use of extraction ratios and of exponential extrapolation of the tails of response curves.

An analytic benchmark solution to the linear, time-dependent isothermal laminar flow fission-product plateout problem

The time-dependent convective-diffusion equation with radioactive decay is solved analytically in axisymmetric cylindrical geometry for laminar and slug velocity profiles under isothermal conditions. Concentration-dependent diffusion is neglected. The laminar flow solution is derived using the method of separation of variables and Frobenius' technique for constructing a series expansion about a regular singular point. These solutions, which describe the transport of fission products in a flowing stream, are then used to determine the pointwise and integrated concentrations of radioactive material deposited on a conduit wall using a standard mass-transfer model. Extensive parametric investigations have been conducted by varying the wall mass-transfer coefficient, diffusion coefficient, flow velocity, pipe radius and species half-life in the deposition models. The computational results indicate that the plateout estimates for the slug flow model are typically 5–100% greater than for the laminar model. The effect of axial diffusion is necessarily negligible for Péclet numbers greater than 100. Little increased plateout is observed for Péclet numbers less than 100; an additional 8% is predicted for a Péclet number of 20 if axial diffusion is included. Characteristic stream, wall and integrated deposition profiles are shown. Representative results from the analysis of fission-product deposition measurements for diffusion tubes in the Fort St Vrain high-temperature gas-cooled reactor plateout probe are presented. Using single-region slug and laminar models, the wall mass-transfer coefficients, diffusion coefficients and inlet concentrations are determined using least-squares analysis. The diffusion coefficients and inlet concentrations are consistent between tubes. The computed diffusion coefficients and wall mass-transfer coefficients are in agreement with known literature values. The analysis indicates that little difference can be discerned between computed laminar and slub flow-model parameters unless accurate data is obtained under strictly regulated conditions.

Laser modification of thermophoretic deposition

The modified chemical vapor deposition process (MCVD) used in the manufacture of optical fiber preforms depends on the thermophoretic deposition of an aerosol of silicon and germanium dioxide particles borne in a background carrier gas. In order to increase the efficiency of this process by increasing the temperature gradient that drives the motion, we consider the absorption of an incident beam of laser radiation by the aerosol particles themselves, and the subsequent conduction of this energy into the background gas. The full set of equations governing the laminar tube flow of an aerosol-gas mixture is formulated. The effect of laser energy deposition is modeled as a source term in the energy equation, and is proportional to the concentration-dependent absorption coefficient. The particle concentration equation contains the effects of axial convection, diffusion, and thermophoretic motion, and is therefore coupled to the energy equation. Scattering and spontaneous emission have been neglected. Analytic results are presented to special cases of the full equations in order to estimate the magnitude of the temperature increase, as well as the change in aerosol concentration, to be expected from laser irradiation. These cases are limited either to circumstances in which the aerosol concentration is decoupled from the temperature (to obtain an upper bound on the temperature increase) or to one-dimensional model problems showing the effect of thermophoretic displacement of the aerosol on modulating the laser heating that can occur. In addition, it is shown that laser heating can cause 100% deposition efficiencies to occur in sufficiently long tubes and some qualitative experiments that indicate the effectiveness of this interaction are described.

On the computation of substrate levels in perfused tissues

The recent development of experimental techniques to study the isolated perfused heart and brain have renewed interest in the mathematical modeling of capillary-tissue structures. A new analytical representation is developed for the Krogh cylinder model for blood-tissue structures, and a scheme is presented for determining an asymptotic series approximation for this solution. This solution is explored for model parameters of experimental interest, and the contribution and effect of axial diffusion is demonstrated.

The effects of axial diffusion and permeability barriers on the transient response of tissue cylinders. I. Solution in transform space

Transient mass transfer in a Krogh tissue cylinder is described by a model taking into account axial diffusion in both blood and tissue, a localized permeability barrier at the capillary membrane and a diffusion barrier on the outer surface and at the ends of the cylinder. Radial diffusion in both blood and tissue is assumed to be infinitely fast. In contrast to previous work, which has usually relied on numerical methods for solving the equations, an exact solution is presented here in Laplace transform space. This allows calculation of the moments of the concentration at any point in the cylinder. Numerical results indicate that the moments of the residence time distribution are affected by the boundary conditions used, and that the discrepancies between the predictions using different conditions may be large in some physiological situations. Order-of-magnitude calculations are used to estimate when the use of simpler models may be feasible. The transform space solution may also be useful for parameter estimation, but it seems preferable to extend the present results to a time-domain solution for this purpose.

Role of longitudinal diffusion in the extravascular pulmonary space on parameter estimates derived from data of multiple indicator dilution

A mathematical model of transcapillary exchange has been developed that considers in detail the role of axial diffusion in the extravascular tissue region on estimates of such physiological parameters as lung water (VE) and pulmonary capillary permeability-surface area products (PS), obtained from multiple indicator dilution studies. The experimental cases considered correspond to two animal models of pulmonary oedema in which the integrity of the pulmonary capillary membrane is disrupted and the effects of extravascular axial diffusion may be important. A novel feature of the computational scheme is the use of an Array Processor in the solution of the governing equations, initial and boundary conditions. Computer time is reduced to 2-3 min for parameter identification, thereby allowing a wide range of values for extravascular axial diffusion coefficients (D'/L2) to be studied at little computational expense. The results indicate that diffusion in the extravascular region does not influence parameter estimates for PS to urea. A statistical correlation is suggested between values for VE, PS to water, and D'/L2.

Effects of gravity and branching on longitudinal gas mixing in glass tube models.

We investigated the effects of gravity and branching on gas mixing in glass tube models. The mixed gas (5% He and 5% SF6) was made to flow through a straight tube and tubes with bifurcated branches, and the effective axial diffusion coefficient (Deff) was measured. The direction of the branches was varied from upward to downward by 90% turns, and the flow distribution to each branch was separately controlled. In the case of a straight horiontal tube, the flow velocity dependency of the Deff of He and SF6 was observed as we expected from Aris' equation. In the case of glass tubes with branches, the Deff of SF6 was greater in the downward-directed branches than it was in the upward-directed branches when gas mixing occurred at the front of the gas mixture; however, when gas mixing occurred at the tail, the opposite effects were observed. These characteristics were more marked at slower flow rates and when the flow to each branch was different. Our findings suggest that gravity has an effect on gas mixing in glass tube airway models, especially at a low flow rate and uneven flow distribution.

Mass transfer with a chemical reaction of the first order. Effect of axial diffusion

The role of diffusion in the direction of convection transport, which is usually neglected in mass (heat) transfer problems, is systematically investigated. Analytical solutions and asymptotic expressions are derived for cases of mass transfer coupled with a homogeneous and heterogeneous irreversible chemical reaction of the first order in plug flow, in Couette flow and in Couette flow with a moving interface. Irrespective of the flow situation, the effect of axial diffusion is of importance for small transformed distances in the direction of the convective transport. Then, the local mass transfer co-efficient is a square of the coefficient which is evaluated without the axial diffusion term. As expected, with an increase in the transformed distance both mass transfer coefficients tend to the same value. In the case of heterogeneous (surface) chemical reactions of the first order, the diffusion in the direction of convective transport is of minor importance. The same is observed when the case of simultaneous homogeneous and heterogeneous chemical reactions of the first order in uniform flow is considered.

The effects of chemical reaction on diffusive ion loss processes in a flowing afterglow

Numerical calculations are described that model the flow, diffusive and reactive processes that occur in a typical flowing afterglow apparatus. These calculations differ from previous solutions to the problem in that the changes in the ambipolar diffusion coefficients that occur with changes in ion composition are considered explicitly. It is shown that these changes can cause distinct curvature in plots of log(ion concentration) versus neutral reactant flow. The effects of axial diffusion and reactant inlet geometry are also considered explicitly. The results are compared with recent experimental data on ion/molecule reactions between UF 6 and various negative ions.

Mathematical analysis of oxygen transport to tissue

The mathematical analysis of steady state oxygen distribution in a Krogh cylinder is presented in this paper. Perturbation techniques are used to determine the effect of axial diffusion when the diameter of the Krogh cylinder is small compared to its length. Separate asymptotic expansions are developed for the arterial end, the central region, and the venous end of the Krogh cylinder. Solutions are obtained for each of these regions, and they are combined, using the methods of matched asymptotic expansions, to obtain a solution that is uniformly valid throughout the entire Krogh cylinder. The accuracy of the perturbation analysis is examined by comparing the approximate solution with the exact solution for the case of a linear oxyhemoglobin dissociation relationship. A simple analytic criterion is derived for determining when axial diffusion is important and when it can be neglected.

Convective diffusion with homogeneous and heterogeneous reactions in a tube

A new complete solution for laminar flow with axial diffusion and first-order homogeneous and wall reactions in a tube is presented. The solution is in terms of an eigenfunction series expansion. Effects of axial diffusion, homogeneous reaction, and wall reaction on the concentration field are discussed. Comparison of the simplified results derived from the present analysis by neglecting axial diffusion and wall reaction with those of previous workers shows good agreement. Conditions are established for the necessity of applying the present results to describe the system accurately. 11 figures.

The simulation of binary adsorption in continuous countercurrent operation and a comparison with other operating modes

AbstractSimulation studies are presented of three different modes of operating a two‐component adsorption system. The efficiency with which an activated carbon adsorbent is utilized is compared for a single fixed bed, a periodic countercurrent system in which the adsorbent is equally distributed over two, four, and six beds and a continuous countercurrent system. The carbon utilization in the single fixed bed was shown to be one third to one half of that in a continuous countercurrent adsorber of the same length, whereas the periodic countercurrent system achieved a utilization of 79 to 98% of that in the continuous system.The mathematical models used to describe the three different modes of column operation include the effects of axial diffusion in the columns, mass transfer resistance in the boundary layer surrounding each particle, and fluid diffusion within the porous particles.

Mass transfer with axial diffusion and chemical reaction

Mass transfer with axial diffusion and chemical reaction is studied for three different cases. The first one is plug flow mass transfer in semi-infinite region. The second one is laminar flow mass transfer in a semi-infinite region. The third one is laminar flow convective diffusion in the region of x ≤ 0 while mass transfer with chemical reaction in the region of x ≥ 0. The first problem is solved in terms of Bessel functions while the second in terms of confluent hypergeometric functions. In order to solve the third problem, orthonormal functions are constructed by linear combination of the eigenfunctions of the two semi-infinite regions respectively. The series expansion coefficients of the solutions are determined by solving a set of forty simultaneous equations obtained through matching boundary conditions at x = 0.Solutions are generally obtained for low Peclet number (Pe = 1, 5, 10) and high reaction rate parameters. Effects of axial diffusion and chemical reaction on the concentration field are discussed.

Tubular Flow Reactors With First-Order Kinetics.

A method is presented for automatically calculating true first order rate constants for gas phase and wall reactions from experimentally observed decay parameters in tubular flow reactors. It includes the effects of axial and radial diffusion and Poiseuille flow.

A Flow‐Through Porous Electrode Model: Application to Metal‐Ion Removal from Dilute Streams

A one‐dimensional model for flow‐through porous electrodes operating above and below the limiting current of a metal deposition reaction has been developed. The model assumes that there is one primary reactant species in an excess of supporting electrolyte, and that a simultaneous side reaction may occur. The model predicts nonuniform reaction rates due to ohmic, mass‐transfer, and heterogeneous kinetic limitations; the effects of axial diffusion and dispersion are included. Results are compared with the experimental data observed by various authors for the deposition of copper from sulfate solutions with the simultaneous generation of dissolved hydrogen. Satisfactory agreement between model predictions and experimental data on over‐all reactor performance and deposit distributions has been accomplished. For an upstream counterelectrode, distributions of reaction rate (for both single and multiple reactions), concentration, and potential describe the detailed system behavior.

ChemInform Abstract: STEADY‐STATE POLARIZATION AT POROUS, FLOW‐THROUGH ELECTRODES WITH SMALL PORE DIAMETER. I. REVERSIBLE KINETICS

AbstractEs wird ein Modell für reversible Einelektro‐ nenübertragungsreaktionen an porösen Durchflußelektroden entwickelt und einige Voraussagen experi‐mentell bestätigt.