scholarly journals Computing Internodal Conductivities in Numerical Modeling of two Dimensional Unsaturated Flow on Rectangular Grid

2011 ◽  
Vol 57 (2) ◽  
pp. 215-225 ◽  
Author(s):  
A. Szymkiewicz ◽  
K. Burzynski
Keyword(s):  
Unsaturated Flow ◽  
Numerical Solutions ◽  
Geometric Mean ◽  
Flow Equation ◽  
Two Dimensional ◽  
Rectangular Grid ◽  
Recent Approach ◽  
State Approximation ◽  
Steady State Approximation ◽  
Upstream Weighting

Abstract This paper compares numerical solutions of transient two-dimensional unsaturated flow equation by using different averaging schemes for internodal conductivities. Averaging methods such as arithmetic mean, geometric mean, upstream weighting, and integrated mean are taken into account, as well as a recent approach based on steady-state approximation. The latter method proved the most flexible, producing relatively accurate solutions for both downward and upward flow cases.

Synthesis and Kinetics of Highly Substituted Piperidines in the Presence of Tartaric Acid as a Catalyst

2018 ◽  
Vol 21 (4) ◽  
pp. 302-311
Author(s):  
Younes Ghalandarzehi ◽  
Mehdi Shahraki ◽  
Sayyed M. Habibi-Khorassani
Keyword(s):  
Ambient Temperature ◽  
Tartaric Acid ◽  
Aromatic Aldehydes ◽  
One Pot ◽  
Rate Determining Step ◽  
State Approximation ◽  
Solvent Free Conditions ◽  
Steady State Approximation ◽  
Absorbance Changes ◽  
The One

Aim & Scope: The synthesis of highly substituted piperidine from the one-pot reaction between aromatic aldehydes, anilines and β-ketoesters in the presence of tartaric acid as a catalyst has been investigated in both methanol and ethanol media at ambient temperature. Different conditions of temperature and solvent were employed for calculating the thermodynamic parameters and obtaining an experimental approach to the kinetics and mechanism. Experiments were carried out under different temperature and solvent conditions. Material and Methods: Products were characterized by comparison of physical data with authentic samples and spectroscopic data (IR and NMR). Rate constants are presented as an average of several kinetic runs (at least 6-10) and are reproducible within ± 3%. The overall rate of reaction is followed by monitoring the absorbance changes of the products versus time on a Varian (Model Cary Bio- 300) UV-vis spectrophotometer with a 10 mm light-path cell. Results: The best result was achieved in the presence of 0.075 g (0.1 M) of catalyst and 5 mL methanol at ambient temperature. When the reaction was carried out under solvent-free conditions, the product was obtained in a moderate yield (25%). Methanol was optimized as a desirable solvent in the synthesis of piperidine, nevertheless, ethanol in a kinetic investigation had none effect on the enhancement of the reaction rate than methanol. Based on the spectral data, the overall order of the reaction followed the second order kinetics. The results showed that the first step of the reaction mechanism is a rate determining step. Conclusion: The use of tartaric acid has many advantages such as mild reaction conditions, simple and readily available precursors and inexpensive catalyst. The proposed mechanism was confirmed by experimental results and a steady state approximation.

scholarly journals Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

2021 ◽  
Vol 11 (8) ◽  
pp. 3421
Author(s):  
Cheng-Yu Ku ◽  
Li-Dan Hong ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Wei-Po Huang
Keyword(s):  
Porous Media ◽  
Time Domain ◽  
Numerical Solutions ◽  
Space Time ◽  
Two Dimensional ◽  
Transient Flows ◽  
Layered Porous Media ◽  
Time Marching ◽  
Time Basis ◽  
Marching Scheme

In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

scholarly journals Enzyme kinetics and metabolic control. A method to test and quantify the effect of enzymic properties on metabolic variables

1990 ◽  
Vol 269 (3) ◽  
pp. 697-707 ◽  
Author(s):  
L Acerenza ◽  
H Kacser
Keyword(s):  
Enzyme Concentration ◽  
Time Dependent ◽  
Experimental Systems ◽  
State Approximation ◽  
Metabolic Systems ◽  
Steady State Approximation ◽  
Before And After ◽  
Factor Alpha ◽  
Metabolic Variables ◽  
Time Invariant

It is usual to study the sensitivity of metabolic variables to small (infinitesimal) changes in the magnitudes of individual parameters such as an enzyme concentration. Here, the effect that a simultaneous change in all the enzyme concentrations by the same factor alpha (Co-ordinate-Control Operation, CCO) has on the variables of time-dependent metabolic systems is investigated. This factor alpha can have any arbitrary large value. First, we assume, for each enzyme measured in isolation, the validity of the steady-state approximation and the proportionality between reaction rate and enzyme concentration. Under these assumptions, any time-invariant variable may behave like a metabolite concentration, i.e. S alpha = Sr (S-type), or like a flux, i.e. J alpha = alpha Jr (J-type). The subscripts r and alpha correspond to the values of the variable before and after the CCO respectively. Similarly, time-dependent variables may behave according to S alpha (t/alpha) = Sr (t) (S-type) or to J alpha (t/alpha) = alpha J r (t) (J-type). A method is given to test these relationships in experimental systems, and to quantify deviations from the predicted behaviour. A positive test for deviations proves the violation of some of the assumptions made. However, the breakdown of the assumptions in an enzyme-catalysed reaction, studied in isolation, may or may not affect significantly the behaviour of the system when the component reaction is embedded in the metabolic network.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

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