COMPREHENSIVE,THEORETICALLY BASED,CORRELATING EQUATIONS FOR FREE CONVECTION FROM ISOTHERMAL SPHERES

Correlating equations are developed for the local and mean Nusselt number for free convection from an isothermal sphere as a function of the Rayleigh and Prandtl numbers. These expressions are based primarily on theoretical solutions for limiting cases, and hence are presumed to be more reliable than purely empirical correlations. The predictions of the proposed expressions are, however, validated by comparisons with prior experimental data. The expressions for the mean Nusselt number are shown to be applicable for all Ra and Pr. The expressions for the local Nusselt number are limited in applicability to the laminar boundary layer regime. The same equations are applicable to mass transfer and to combined heat and mass transfer in terms of the Sherwood, Schmidt and appropriately modified Rayleigh numbers.     

RADIATION EFFECT ON FREE CONVECTION LAMINAR FLOW FROM AN ISOTHERMAL SPHERE

Natural convection laminar flow from an isothermal sphere immersed in a viscous incompressible optically dense fluid in the presence of radiation effects has been investigated. The governing boundary layer equations are transformed into a non dimensional form and the resulting nonlinear systems of partial differential equations are reduced to local non-similarity boundary layer equations, which are solved numerically by two distinct, efficient methods, namely, (i) implicit finite difference method together with the Keller box scheme and (ii) local non-similarity method. Numerical results of the velocity and temperature profiles of the fluid are presented. The results of the shearing stress and the heat transfer rate in terms of skin-friction coefficient and Nusselt number respectively are also presented for a wide range of the radiation-conduction parameter or Planck number R _d (=0.0, 1.0, 2.0, 3.0), the surface heating parameter θ _w (=1.1, 1.2, 1.4), and Prandtl number Pr (=7.0, 10.0, 15.0, 20.0).     

RESIDENCE TIME DISTRIBUTIONS FOR FREE CONVECTION IN PACKED BEDS

Experimental tracer pulse responses were recorded for a laboratory simulation of a fixed-bed reactor. The simulation consisted of a packed bed fitted with an electrical heater and supplied with a gas throughflow. Radically different responses were observed depending on whether the bed was undergoing free convection or not as inferred from the calculated Rayleigh number. For the free-convection cases, the response was characterized by a large early peak followed by a second smaller peak in contrast to the single broad peak characteristic of dispersed plug flow. Interpretation of the experimental results was obtained from numerical calculations of the Residence Time Distribution (RTD) and resulting plots of steady state stream function and concentration contours for different times after injection of the pulse. The successive peaks were seen to arise from the rotation of a high-concentration zone around a convection cell. The time between peaks is thus the mean circulation time of the convection cell.

The method is rapid to use and allows immediate determination of the convecting state of a packed bed. It should be possible to use it for the verification of theoretical results, particularly for stability limits.     

FREE AND FORCED CONVECTION HEAT TRANSFER FROM SHORT INCLINED CIRCULAR CYLINDERS

Free and forced convective heat transfer rates from circular cylinders whose length to diameter ratio is small have been experimentally measured. The cylinders were arranged with their axes in a vertical plane and tests were conducted with the axes set at angles of between 0° and 90° to the horizontal. In the forced convective tests the cylinders were exposed to a horizontal air stream. The majority of the tests were undertaken with models having diameters of 19 mm and 25 mm with length to diameter ratios of between 1.5 and 16. The heat transfer rates were determined by the transient method i.e. by using solid aluminum models, heating them and measuring their temperature-time variation while they cooled. Significant differences between the present results and those observed in previous studies with models having large length to diameter ratio have been found. Methods of correlating both the free convective results and the forced convective results have been derived.     

FREE AND FORCED CONVECTION HEAT TRANSFER FROM SHORT INCLINED CIRCULAR CYLINDERS

Free and forced convective heat transfer rates from circular cylinders whose length to diameter ratio is small have been experimentally measured. The cylinders were arranged with their axes in a vertical plane and tests were conducted with the axes set at angles of between 0° and 90° to the horizontal. In the forced convective tests the cylinders were exposed to a horizontal air stream. The majority of the tests were undertaken with models having diameters of 19 mm and 25 mm with length to diameter ratios of between 1.5 and 16. The heat transfer rates were determined by the transient method i.e. by using solid aluminum models, heating them and measuring their temperature-time variation while they cooled. Significant differences between the present results and those observed in previous studies with models having large length to diameter ratio have been found. Methods of correlating both the free convective results and the forced convective results have been derived.     

CORRELATING EQUATIONS FOR THE RECOVERY FACTOR AND THE BLASIUS FUNCTIONS FOR FLOW ALONG A FLAT PLATE

This paper describes the development of a complete set of continuous, comprehensive correlating equations for the recovery factor, and for the various functions of the velocity field which are required for calculation of heat transfer in the laminar boundary layer regime. These expressions are based on theoretical asymptotes and intermediate numerical values. Asymptotes were derived in the few instances for which they were not already known. Additional numerical values were computed to support the construction of the correlating equation for the recovery factor, but prior computed values were sufficient for the velocity functions.     

CORRELATING EQUATIONS FOR THE RECOVERY FACTOR AND THE BLASIUS FUNCTIONS FOR FLOW ALONG A FLAT PLATE

This paper describes the development of a complete set of continuous, comprehensive correlating equations for the recovery factor, and for the various functions of the velocity field which are required for calculation of heat transfer in the laminar boundary layer regime. These expressions are based on theoretical asymptotes and intermediate numerical values. Asymptotes were derived in the few instances for which they were not already known. Additional numerical values were computed to support the construction of the correlating equation for the recovery factor, but prior computed values were sufficient for the velocity functions.     

THE LOCAL VOLUME-AVERAGED EQUATIONS OF MOTION FOR A SUSPENSION OF NON-NEUTRALLY BUOYANT SPHERES

The local volume averages of the equations of motion as well as the appropriate boundary conditions are developed for a flowing suspension of non-neutrally buoyant, uniform spheres in an incompressible Newtonian fluid under conditions such that inertial effects can be neglected. These equations do not represent an asymptotic theory with respect to the volume fraction of solids. Higher order terms have been retained everywhere, except where it has been necessary to estimate the velocity distribution within the immediate neighborhood of each sphere by neglecting the effects of the other spheres present. The resulting local volume-averaged equations of motion and boundary conditions involve no free or undetermined parameters.

For the special case of a very dilute suspension of neutrally buoyant spheres, the total local volume average of Cauchy's first law reduces to the form of the Navier-Stokes equation with the effective viscosity computed by Einstein (1906, 1956).

In two succeeding papers, we demonstrate for several flows [in vertical tubes, in a cone-plate viscometer, between rotating concentric cylinders (Couette flow), and between rotating parallel plates] that our general theory describes more concentrated neutrally buoyant suspensions than does its limiting case of very dilute suspensions.     

STUDY OF THE LAMINAR NATURAL CONVECTION PROBLEM ALONG AN ISOTHERMAL VERTICAL PLATE BASED ON DISCRIMINATED DIMENSIONAL ANALYSIS

The results derived from the application of “classical” dimensional analysis to the natural convection heat transfer problem are poor, particularly in the case of two- or three-dimensional problems. “Discriminated” dimensional analysis provides more precise solutions, which frequently lead to only one dimensionless monomial. The main feature of the discrimination, on one hand, is that the dimensional basis is extended, taking into account that each of the spatial coordinates of a problem is dimensionally independent. On the other hand, discrimination provides different dimensional equations (derived from the basic laws) for quantities of vectorial character according to its inherent spatial direction. As a result, the quantities that take part in the solution are related each other in a more restricted way than in the case of classical dimensional analysis, leading, in general, to a minor number of dimensionless groups. Many of the known dimensionless numbers, such as the Rayleigh, Grashof, Boussinesq, and Nusselt numbers, have dimensions in a discriminated basis and, per se, do not play an independent role in the solution of the problem. Discriminated dimensional analysis groups these numbers, both among themselves and with other quantities, forming new (discriminated) dimensionless monomials with a precise physical significance. Here, we study natural convection on an isothermal vertical plate using this technique and compare the results with those obtained by classical dimensional analysis.     

COMBINED LAMINAR FREE AND FORCED CONVECTION FROM A VERTICAL PLATE TO POWER LAW FLUIDS

The boundary layer heat transfer for the steady two-dimensional incompressible combined laminar free and forced convection from a vertically static or moving plate to a power-law non-Newtonian fluid is analyzed by local similarity method. The effects of the flow index, buoyancy parameter, modified Prandtl number and streamwise distance on the heat transfer rate are discussed for the prescribed surface temperature and prescribed wall heat flux cases.     

COMBINED LAMINAR FREE AND FORCED CONVECTION FROM A VERTICAL PLATE TO POWER LAW FLUIDS

The boundary layer heat transfer for the steady two-dimensional incompressible combined laminar free and forced convection from a vertically static or moving plate to a power-law non-Newtonian fluid is analyzed by local similarity method. The effects of the flow index, buoyancy parameter, modified Prandtl number and streamwise distance on the heat transfer rate are discussed for the prescribed surface temperature and prescribed wall heat flux cases.     

DERIVATION OF CONSTITUTIVE EQUATIONS FOR INTERFACIAL FORCE AND REYNOLDS STRESS FOR A SUSPENSION OF SPHERES USING ENSEMBLE CELL AVERAGING

A novel new ensemble cell averaging technique has been used to derive various interfacial forces and Reynolds stresses for inviscid two-fluid models of two-phase. New terms, not seen in the spatially averaged conservation equations, have been found. Moreover, deficiencies inherent in spatial averaging techniques have been demonstrated.     

EQUATIONS FOR CALCULATION OF THE TERMINAL VELOCITY AND DRAG COEFFICIENT OF SOLID SPHERES AND GAS BUBBLES

An analysis of the correlations proposed in the literature for calculation of the drag coefficient (C_D) and the terminal velocity of a falling rigid sphere has been made. Among the correlations describing C_D vs. Re, that of Turton and Levenspiel fits the experimental data almost perfectly. However, it is not explicit in the terminal velocity. The available explicit correlations do not fit the experimental data well. The present paper shows that a simple and precise explicit correlation can be developed if C_D is related to the Archimedes instead of the Reynolds number. The precision of the correlation proposed is similar to that of the Turton and Levenspiel (1986), while it is explicit in the terminal velocity. On the basis of this correlation, a model is proposed to calculate the drag coefficients and the terminal velocities of free falling or rising spherical particles in an infinite fluid as well as gas bubbles with any volume and shape rising in a contaminated liquid.     

基于模糊综合评判理论的陶瓷窑炉先进性评价模型

运用二级模糊综合评判评价陶瓷窑炉先进性的方法和模型，为用量化的办法客观、公正、科学、规范地做好陶瓷窑炉先进性评价工作提供了一种可供借鉴的模式．并通过实例验算，汪明结果可靠，操作简便，有广泛的应用前景。     

A GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR COMBINED FREE AND FORCED CONVECTION FLOW ABOUT INCLINED SURFACES IN POROUS MEDIA

In this article, we apply the differential transform method (DTM) to obtain approximate analytical solutions of combined free and forced (mixed) convection about inclined surfaces (or wedges) in a saturated porous medium. Both aiding and opposing flows are considered. It is found that the parameter mixed convection from inclined surfaces in porous media is Gr/Re, where Gr is the local Grashof number and Re is the local Reynolds number. DTM solutions are obtained for mixed convection from an isothermal vertical flat plate as well as an inclined plate with constant heat flux having an inclination of 45°. Temperature and velocity profiles for these two cases at different values of Gr/Re are presented. The similarity transformations are applied to reduce the governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. DTM is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Thereafter a Padé approximant is applied to the solutions to increase the convergence of the given series. Excellent correlation between DTM-Padé and numerical quadrature (shooting) solutions is achieved. The DTM-Padé simulation is shown to be a robust benchmarking tool providing an excellent means of validation of numerical methods. The study has applications in geothermal energy systems, chemical engineering filtration systems, and packed beds.     

FREE CONVECTIVE DRAFT INDUCED BY THERMAL AND CONCENTRATION GRADIENTS INSIDE AN ISOTHERMAL, VERTICAL CYLINDER†

Laminar, free convective flow through a vertical cylinder induced by the thermal and concentration buoyancy forces is investigated. The numerical studies involve development of a steady-state, two-dimensional heat and mass transfer model for the moist air core of the vertical tube. The stream function-vorticity method is employed to simplify the governing, coupled conservation equations which were then numerically solved by the successive over-relaxation (SOR) and alternating direction implicit ((ADI) methods.

A graphical correlation was found between dimensionless flow rate and dimensionless tube length as a function of the buoyancy force ratio N = Gr/Grc. Excellent agreement was obtained for the dimensionless flow rale results with those of Davis and Perona¹⁶ and Kageyama and Izumi¹³ for the case when only the-thermal buoyancy force is considered.

The combined buoyancy force from thermal and species diffusion provides larger local Nusselt Nu and local Sherwood Sh numbers relative to the case when just one buoyancy force is accounted for. Both local Nu and local Sh are seen to asymptotically approach a constant value as flow develops.     

二元低压汽液平衡p、y、χ关系及其在加压中的近似应用

把汽液平衡的真实状态看成是3个虚拟状态的组合,结合Gibbs-Dubem方程,推得了等温低压汽液平衡时p-y-x之间的一个近似关系式,并把它推广应用到加压及含超临界组分体系的汽液平衡计算.     

二元低压汽液平衡p、y、χ关系及其在加压中的近似应用

把汽液平衡的真实状态看成是3个虚拟状态的组合,结合Gibbs-Dubem方程,推得了等温低压汽液平衡时p-y-x之间的一个近似关系式,并把它推广应用到加压及含超临界组分体系的汽液平衡计算.