An Approximate Analysis of Laminar Dispersion in Circular Tubes

1976 ◽  
Vol 43 (4) ◽  
pp. 537-542 ◽  
Author(s):  
J. S. Yu
Keyword(s):  
Order Approximation ◽  
Average Concentration ◽  
Small Time ◽  
Dimensionless Time ◽  
Circular Tubes ◽  
Basic Differential Equation ◽  
Flow Through ◽  
Time Limitations ◽  
And Diffusion ◽  
Small Time Approximation

The combined effects of convection and diffusion for dispersion on mass transport in fully developed laminar flow through circular tubes are investigated. The present method, which in general may be used to yield solutions at any arbitrary dimensionless time, in its zeroth-order approximation is identical to Taylor’s analysis for the average concentration. Solutions to the basic differential equation for an initial input of solute either concentrated at a section of the tube or uniformly distributed in the form of a slug of finite axial extension are developed. Numerical results for the former input are presented over a large range of dimensionless time and Peclet numbers. The time limitations of Taylor’s solution and Lighthill’s small time approximation [15] are placed on more reliable quantitative bases by comparison with the present calculations.

Velocity Profiles of Liquid Flow Through Circular Tubes and How They Affect Flow Measurement

1991 ◽  
Vol 113 (3) ◽  
pp. 206-210 ◽  
Author(s):  
D. Yogi Goswami
Keyword(s):  
Transition Region ◽  
Laser Doppler Velocimetry ◽  
Flow Measurement ◽  
Velocity Profiles ◽  
Turbulent Regime ◽  
Flow Meters ◽  
Circular Tubes ◽  
Flow Through ◽  
Hydrogen Bubble Technique

This paper analyzes velocity profiles for flow through circular tubes in laminar, turbulent, and transition region flows and how they affect measurement by flow-meters. Experimental measurements of velocity profiles across the cross-section of straight circular tubes were made using laser doppler velocimetry. In addition, flow visualization was done using the hydrogen bubble technique. Velocity profiles in the laminar and the turbulent flow are quite predictable which allow the determination of meter factors for accurate flow measurement. However, the profiles can not be predicted at all in the transition region. Therefore, for the accuracy of the flowmeter, it must be ensured that the flow is completely in the laminar regime or completely in the turbulent regime. In the laminar flow a bend, even at a large distance, affects the meter factor. The paper also discusses some strategies to restructure the flow to avoid the transition region.

Numerical analysis of the convection and diffusion of solutes to roots

Soil Research ◽  
1967 ◽  
Vol 5 (2) ◽  
pp. 149 ◽  
Author(s):  
JB Passioura ◽  
MH Frere
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Porous Medium ◽  
Diffusion Coefficient ◽  
Numerical Analysis ◽  
Average Concentration ◽  
Soil System ◽  
Dry Soil ◽  
The One ◽  
And Diffusion

A numerical method is given for solving a partial differential equation describing the radial movement of solutes through a porous medium to a root. Computer programmes based on the method were prepared and used to obtain solutions of the equation for an idealized root-soil system in which a solute is transported to the root by convection but is not taken up by the root. Various patterns of water uptake were considered, the most complex being a diurnally varying uptake from soil in which the water content is decreasing. The solutions suggest that the maximum build-up of solute at the surface of a root is trivial if the root is growing in a medium such as agar, in which the diffusion coefficient of the solute is high, but may be considerable, with a concentration up to 10 times higher than the average concentration in the soil solution, when the root is growing in a fairly dry soil. The application of the method to systems other than the one considered in detail is discussed.

Compact Algebraic Asymptotes for Small and Large Time Surface Temperatures for Regular Solid Bodies Heated With Uniform Heat Flux: Scrutiny of New Critical Fourier Numbers

2020 ◽  
Vol 13 (3) ◽  
Author(s):  
Agustín M. Delgado-Torres ◽  
Antonio Campo ◽  
Yunesky Masip-Macia
Keyword(s):  
Heat Flux ◽  
Large Time ◽  
Small Time ◽  
Uniform Heat Flux ◽  
Dimensionless Time ◽  
Time Approximation ◽  
Surface Temperatures ◽  
Uniform Heat ◽  
Relative Errors ◽  
Solid Bodies

Abstract The alternate infinite series at “small time” have been used to analyze the time variation of surface temperatures (ϕs) in regular solid bodies heated with uniform heat flux. In this way, compact algebraic asymptotes are successfully retrieved for ϕs in a plate, cylinder, and sphere in the “small time” sub-domain extending from 0 to the critical dimensionless time or critical Fourier number. For the “large time” sub-domain, the exact solution is approximated in two ways: with the “one-term” series and with the simple asymptotes corresponding to extreme “large time” conditions. Maximum relative errors of 1.23%, 6.24%, and 0.96% in ϕs for the plate, cylinder, and sphere are τcr obtained, respectively, with the “small time”—“large time” approximation using a traditional approach to fix the τcr value. An alternative approach to set the τcr is proposed to minimize the maximum relative error of the approximated solutions so that values of 1.19%, 3.93%, and 0.16% are then obtained for the plate, cylinder, and sphere, respectively, with the “small time”—“large time” approximation. For the “small time”—“one-term” approximation maximum relative errors of 0.024%, 1.33%, and 0.004% for the plate, cylinder, and sphere are obtained, respectively, with this approach.

Adsorption and Diffusion in HZSM-5 Zeolite Studied by an Oscillating Microbalance

1997 ◽  
Vol 62 (12) ◽  
pp. 1832-1842 ◽  
Author(s):  
Hans P. Rebo ◽  
De Chen ◽  
Marit S. A. Brownrigg ◽  
Kjell Moljord ◽  
Anders Holmen
Keyword(s):  
Gas Phase ◽  
High Sensitivity ◽  
Probe Molecules ◽  
Additional Mass ◽  
Transfer Resistance ◽  
Partial Pressures ◽  
Uptake Rates ◽  
Flow Through ◽  
Tapered Element Oscillating Microbalance ◽  
And Diffusion

A novel microbalance technique has been used to study diffusion and adsorption in a commercial HZSM-5 zeolite. This new technique uses an inertial microbalance TEOM (Tapered Element Oscillating Microbalance) to measure mass changes in the zeolite bed. Time resolution as short as 0.1 s, a flow-through design where all the probe molecules see the zeolite bed and high sensitivity allowing zeolite loadings down to a few milligrams are the three most important properties of the TEOM. The probe molecules studied were o-xylene, p-xylene and toluene which were introduced at 303, 373 or 473 K and at partial pressures in the range of 0.2-10 kPa. The inverse characteristic uptakes (D/L2), corrected (D0/L2) and steady-state (Dss/L2) diffusion times are reported. The thermodynamic correction used for D0/L2 calculations almost eliminated the concentration dependence of the diffusivities. The Dss/L2 values were found to be rather unaffected by both temperature (373-473 K) and concentration, suggesting a certain degree of unification for diffusivities. o-Xylene uptake rates in the TEOM were found to be significantly higher than in a gravimetric microbalance under identical conditions, probably as a result of additional mass transfer resistance other than intracrystalline diffusion caused by poor contact between the gas phase and the zeolite in a conventional gravimetric microbalance.

scholarly journals Dufour Effect with Ramped Wall Temperature and Specie Concentration on Natural Convection Flow Through A Channel

Physics ◽  
2019 ◽  
Vol 1 (1) ◽  
pp. 111-130 ◽  
Author(s):  
Jha ◽  
Gambo
Keyword(s):  
Wall Temperature ◽  
Numerical Solutions ◽  
Vertical Channel ◽  
Convection Flow ◽  
Physical Situation ◽  
Dimensionless Time ◽  
Dufour Effect ◽  
Laplace Transform Technique ◽  
Flow Through ◽  
The Mathematical Model

In this paper, we have obtained an analytical solution to the problem of unsteady free convection and mass transfer flow of an incompressible fluid through a vertical channel in the presence of Dufour effect (or diffusion thermo). The bounding plates are assumed to have ramped wall temperature as well as specie concentration. The mathematical model responsible for the physical situation is presented in dimensionless form and solved analytically using the powerful Laplace Transform Technique (LTT) under relevant initial and boundary conditions. In order to cross check the accuracy of the analytical results, numerical solutions are obtained using PDEPE solver in MATLAB. The expressions for temperature, concentration, and velocity are obtained. The effects of Dufour parameter, Prandtl number (Pr), Schmidt number (Sc), and dimensionless time are described during the course of these discussions. The temperature, concentration, and velocity profiles are graphically presented for some realistic values of Pr=0.025, 0.71, 7.0, 11.62, 100.0 and Sc=0.22, 0.60, 1.00, 2.62, while the values of all other parameters are arbitrarily taken.

scholarly journals Homogenization of Coupled Fast-Slow Systems via Intermediate Stochastic Regularization

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Maximilian Engel ◽  
Marios Antonios Gkogkas ◽  
Christian Kuehn
Keyword(s):  
Diffusion Coefficients ◽  
Sufficient Conditions ◽  
Analytical Techniques ◽  
Slow Variable ◽  
Small Time ◽  
Separation Parameter ◽  
Time Scale Separation ◽  
Weakly Coupled ◽  
Noise Limit ◽  
And Diffusion

AbstractIn this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter $$\varepsilon $$ ε such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit $$\varepsilon $$ ε to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering $$\varepsilon $$ ε to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study weakly-coupled systems, where the coupling only occurs in lower time scales.

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