Natural Convection Induced by a Centrifugal Force Field in a Horizontal Annular Porous Layer Saturated with a Binary Fluid

The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, j{\varphi } , the radius ratio of the cavity, R, the normalized porosity, e{\varepsilon } , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations.     

Analytical and numerical study of double diffusive convection in a vertical enclosure

This paper presents an analytical and numerical study of natural convection of a double-diffusive fluid contained in a rectangular slot subject to uniform heat and mass fluxes along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number, Ra _T ; buoyancy ratio, N; Lewis number, Le; Prandtl number, Pr and aspect ratio of the cavity, A. In the first part of the analytical study a scale analysis is applied to the two extreme cases of heat-transfer and mass-transfer-driven flows. In the second part, an analytical solution, based on the parallel flow approximation, is reported for tall enclosures (A?1). Solutions for the flow fields, temperature and concentration distributions and Nusselt and Sherwood numbers are obtained in terms of the governing parameters of the problem. In the limits of heat-driven and solute-driven flows a good agreement is obtained between the prediction of the scale analysis and those of the analytical solution. The numerical solutions are based on the complete governing equations for two-dimensional flows, and cover the range 1≤Ra _T ≤10⁷, 0≤N≤10⁵, 10^-3≤Le≤10³, 1≤A≤20 and Pr=7. A good agreement is found between the analytical predictions and the numerical simulation.     

Natural convection in shallow enclosures with multiple conducting partitions

Laminar natural convection and conduction in shallow enclosures having multiple partitions with finite thickness and conductivity have been studied. An approximate analytical solution is obtained by using the parallel flow approximation in horizontal shallow enclosures heated isothermally at two vertical ends while adiabatic on horizontal end walls. The same problem is solved also using a finite difference formulation and the control volume method. The study covers the range ofRa from 105 to 107,A=H/L0.2, C=1/L from 0 to 0.15, and the thermal conductivity ratio of partition to fluidk _r from 10^–4 to 10¹¹. The partition numberN was varied from 0 to 5. The Prandtl number was 0.72 (for air). The results are reduced in terms ofNu as a function ofRa, k, and various geometrical parameters (A, C). The streamlines and isotherms are produced to visualize the flow and temperature fields.Es wird der kombinierte Einfluß von laminarer Naturkonvektion und Leitung in flachen Behältern mit mehreren Trennwänden endlicher Dicke und Leitfähigkeit untersucht. Eine analytische Näherungslösung läßt sich über die Parallelstromapproximation bezüglich horizontaler flacher Behälter finden, deren zwei vertikale Begrenzungswände isotherm beheizt sind, während die Horizontalflächen adiabat sein sollen. Das selbe Problem wird unter Verwendung eines Differenzverfahrens und der Kontrollvolumen-Methode gelöst und zwar für die Parameterbereiche 105 Ra 10⁷;A=H/L<0.2;>C=1/L 0.15; 10^–4k_r 1011, wobei der letzte Parameter das Verhältnis der Leitfähigkeit von Trennwand und Fluid bezeichnet. Die Zahl der TrennwändeN variierte Zwischen 0 und 5, die Prandtl-Zahl betrug 0.72 (Luft). Die Ergebnisse werden in dimensionsloser Form gemäß der BeziehungNu =f (Ra, k _r ,A, C) mitgeteilt bzw. durch Diagrammdarstellungen der Stromlinien- und Isothermenfelder veranschaulicht.Financial support from the Natural Sciences and Engineering Council Canada is acknowledged. Financial support to A. Kangni from Canadian Fellowship Program For French Speaking Countries is also acknowledged.     

Natural convection in rectangular enclosures with adiabatic fins attached on the heated wall

The heat transfer by natural convection in vertical and inclined rectangular enclosures with fins attached to the heated wall is numerically studied using the energy and Navier-Stokes equations with the Boussinesq approximation. The range of study covers 10⁴Ra2×10⁵,A=H/L=2.5 to ,B=l/L=0 to 1,C=h/L=0.25 to 2 andPr=0.72. The inclination angle from the vertical was from 0 to 60 degree. The variation of the local Nusselt numberNu _loc along the enclosure height and the average Nusselt numberNu as a function ofRa are computed. Streamlines and isotherms in the enclosure are produced. The results show thatB is an important parameter affecting the heat transfer through the cold wall of the enclosure. The heat transfer is reduced for decreasingC and it passes from a maximum for an inclination angle. The results show that the heat transfer can generally be reduced using appropriate geometrical parameters in comparison with a similar enclosure without fins.Die Wärmeübertragung bei freier Konvektion in vertikalen und geneigten rechtwinkligen Behältern mit Rippen an den beheizten Wänden wird unter Verwendung der Energie- und Navier-Stokes-Gleichungen sowie der Boussinesq-Approximation numerisch untersucht. Der Bereich der Studie liegt bei 10⁴Ra2·10⁵,A=H/L=2,5 bis ,B=l/L=0 bis 1,C=h/L=0,25 bis 2 undPr=0.72. Der Neigungswinkel der Wand liegt zwischen 0 und 60 Grad. Die Veränderung der lokalen Nusselt-Zahl entlang der Höhe der Behälterwände und die mittlere Nusselt-Zahl in Abhängigkeit derRa-Zahl werden berechnet. Strömungslinien und Isothermen werden im Behälter erzeugt. Die Ergebnisse zeigen, daßB ein wichtiger Parameter für die Wärmeübertragung an der nicht beheizten Wand des Behälters ist. Die übertragene Wärmemenge verringert sich mit abnehmendemC und durchschreitet ein Maximum für eine bestimmte Wandneigung. Die Ergebnisse zeigen, daß im Vergleich zu einer Anordnung ohne Rippen, die Wärmeübertragung bei geeigneten geometrischen Parametern allgemein reduziert werden kann.     

Mixed convection heat transfer in open ended channels with protruding heaters

Mixed convection heat transfer has been studied in vertical channels, open at the bottom and top, with protruding discrete heaters installed on one side. The flow is assumed to be steady, laminar and two-dimensional. The Boussinesq approximation is used to account for the density variation. Non-dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation are solved using the SIMPLER method. Heat transfer through the top and the right are calculated as functions of the Rayleigh number (0≤Ra≤10⁷), the Reynolds number (0≤Re≤200), various aspect ratios (1≤A≤6). The effect of the entrance and exit lengths and that of the position of the electronic components in the channel are also examined. Flow and temperature fields for various cases are produced, and the temperature variations in the electronic components are calculated. Received on 2 March 1998     

General methodology for the chemoselective N-alkylation of (2,2,6,6)-tetramethylpiperidin-4-ol: Contribution of microwave irradiation

A convenient method to access a broad variety of N-alkyl-(2,2,6,6)-tetramethylpiperidin-4-ol compounds is reported. The thermal treatment of a mixture of (2,2,6,6)-tetramethylpiperidin-4-ol and allyl or benzyl bromide derivatives gave the corresponding N–alkylated compounds in good yields while leaving the hydroxyl functional group intact. Whereas 40?h were needed to reach complete conversion, microwave irradiation allowed the reaction time to be reduced (20?min) and improved the yields in most cases.     

Natural convection in an inclined rectangular porous cavity with uniform heat flux from the side

The problem of natural convection in an inclined rectangular porous layer enclosure is studied numerically. The enclosure is heated from one side and cooled from the other by a constant heat flux while the two other walls are insulated. The effect of aspect ratio, inclination angle and Rayleigh number on heat transfer is studied. It is found that the enclosure orientation has a considerable effect on the heat transfer. The negative orientation sharply inhibits the convection and consequently the heat transfer and a positive orientation maximizes the energy transfer. The maximum temperature within the porous medium can be considerably higher than that induced by pure conduction when the cavity is negatively oriented. The peak of the average Nusselt number depends on the Rayleigh number and the aspect ratio. The heat transfer between the two thermally active boundaries is sensitive to the effect of aspect ratio. For an enclosure at high or low aspect ratio, the convection is considerably decreased and the heat transfer depends mainly on conduction.     

FEATURES AND APPLICATIONS OF BERTALANFFY-RICHARDS‘ AND SCHNUTE'S GROWTH EQUATIONS

ABSTRACT. Based on various ranges of the parameter m (or b), the paper analyzes the features and the integral forms of the Schnute and the Bertalanffy-Richards growth equations as well as the two aspects of their special cases (such as Gompertz, logistic and monomolecular models). It is a first attempt to investigate all the corresponding relationships among parameters of the models derived from the Schnute and the Bertalanffy-Richards growth equations. All the models from the two are empirically fitted by different data sets for eucalypt plantations. Unlike earlier papers, the results of this paper show that either of the two growth equations can be considered as a model for estimating forest growth given a parameter range, and both can produce similar growth performances. Some other aspects of the two growth equations are discussed so that the two can be used correctly.