Wide Range Of Rayleigh Numbers Research Articles

Natural convection from pin fin arrays

Pin fins are used to increase heat transfer from heated surfaces to air. Industrial experience has shown that for the same surface area, pin fins can transfer considerably more energy than straight fins. The analysis of a single pin fin is well known. However, when fins are placed in an array, the convective patterns become interrelated, and the resulting heat transfer coefficient has not been predicted. An experimental investigation was carried out during which three pin fin arrays were tested inline and staggered under a variety of heat inputs and inclinations. This investigation suggests that the most important geometric parameter influencing the heat transfer from pin fin arrays is the ratio of the fin diameter to the center-to-center spacing. From our measurements and other available experimental data, we developed an empirical model that predicts the performance of pin fin arrays for a wide range of Rayleigh numbers and geometries. The experimental results indicate that a pin fin array performs better than a plate fin array under the same conditions, and the best performance occurs when the ratio of fin diameter to center-to-center spacing is about 1 3 .

Influence of Variable Fluid Density on Free Convection in Rectangular Porous Media

Free convection in rectangular porous media has been numerically investigated incorporating the dependence of fluid density on temperature in all the conservation equations. The fluid density dependence on temperature is described through a parametric equation which can be used for different fluids. The method of successive accelerated replacement scheme has been employed to obtain numerical results for a wide range of Rayleigh numbers, 50 to 1000, aspect ratio, 0.1 to 10 and density difference ratio of 0 to 0.75. The increase in the average Nusselt number at a Rayleigh number of 100 and an aspect ratio of unity is 9.57 percent when density difference ratio is 0.25, whereas it is as high as 19.88 percent when the density difference ratio is 0.5.

An analysis of heat transfer using equivalent thermal conductivity of liquid phase during melting inside an isothermally heated horizontal cylinder

A method of analysis using the equivalent thermal conductivity of the liquid phase is presented for the melting process of a phase change material (PCM) inside an isothermally heated horizontal cylinder. The equivalent thermal conductivity of the liquid phase during the melting process is estimated from the results for natural convection heat transfer between concentric horizontal circular cylinders. The melting process is, thus, analyzed as a pure heat conduction problem. Experiments are also performed using noctadecane as a PCM. The present analytical results for melting rates show good agreement with the experimental data for a wide range of Rayleigh numbers. It is also shown that the effect of natural convection on the melting process can be negligible for Ra D < 10 5.

Natural convection in horizontal enclosures with multiple partitions

Natural convection in horizontal enclosures with multiple partitions is experimentally studied for a wide range of Rayleigh numbers. A thin partition with a high conductivity is employed. The heat transfer rates, temperature distributions and flow visualizations reveal that natural convection in each cell constructed by the partitions is identical to the ordinary Benard problem, i.e. thermal coupling by conduction through partitions is minute. It is also found from an engineering standpoint that the horizontal and vertical enclosures are equivalent in the thermal insulation capability of partitions under the same conditions, in spite of different flow patterns.

An experimental study on natural convection heat transfer in a hemisphere.

An experimental study was performed for natural convection in a hemisphere heated from below. The purpose of the present study was to obtain heat transfer characteristics for a wide range of Rayleigh numbers and to clarify the behavior of thermal plumes in connection with the laminar-turbulent transition of such a flow. Heat-transfer and flow-visualization experiments were carried out using various liquids. The ranges of Rayleigh and Prandtl numbers for the experiments were 1×106<Ra<5×1010 and 6<Pr<13 000 for the heat-transfer experiment and 5×105<Ra<2×109 and 6<Pr<13 000 for the flow-visualization experiment. Empirically obtained heat-transfer correlations and the behavior of the thermal plumes generated on the hot surface indicated that the laminar-turbulence transition occurs at Ra 109.

Study of natural convection in horizontal annuli

Natural convection of gases in a horizontal annulus, where the inner cylinder is heated by the application of a constant heat flux and the outer cylinder is isothermally cooled, is studied numerically. Detailed results of temperature, velocity and heat transfer are presented for a wide range of Rayleigh numbers extending from conduction to the convection-dominated steady flow regime, and diameter ratios of 1.2–10. A crescent-shaped eddy dominates for the small diameter ratio and a kidney-shaped flow pattern appears for the large diameter ratio as observed by previous investigators in their flow visualization studies. The inner wall temperature is a function of diameter ratio and Rayleigh number. An increase in Rayleigh number based on the same temperature difference for the inner wall boundary condition of constant heat flux or constant temperature increases the heat transfer rate; however, the increase is larger for the constant heat flux case. At large diameter ratios ( κ ≥ 10), the heat transfer rates are the same for both types of heating, and a single cylinder in an infinite atmosphere gives nearly the same results.

Natural convective heat transfer in a rectangular porous cavity with variable fluid properties—validity of the Boussinesq approximation

In this work, the applicability of the Boussinesq approximation is investigated for natural convection in a fluid-saturated porous cavity with vertical walls maintained at two different temperatures and horizontal walls completely insulated. Numerical calculations are performed for two different fluids of practical interest, water and air, in a wide range of Rayleigh numbers and aspect ratios. Flow and temperature fields and heat transfer rates obtained through the evaluation of a model that includes fluid temperature dependent properties, are presented. One of the most important conclusions is that the Nusselt number evaluated through the Boussinesq approximation can be substantially different from the Nusselt number obtained with this model.

Thermogravitational and thermocapillary convection heat transfer in concentric and eccentric horizontal, cylindrical annuli filled with two immiscible fluids

Laminar thermogravitational convection in concentric and eccentric horizontal, cylindrical annuli, filled with two immiscible fluids (water/air, water/silicon oil 10, water/silicon oil 100, Freon 113/ water) is studied numerically. Streamline and temperature distributions, local and average equivalent thermal conductivities are obtained over a wide range of Rayleigh number. The influence of thermocapillary convection (Marangoni convection) is similarly demonstrated for the water/air system in an annular enclosure.

Prandtl number effects on natural convection heat transfer in concentric and eccentric horizontal cylindrical annuli

The influence of various Prandtl numbers on the laminar convection flow between concentric and vertically eccentric cylinders is studied numerically. To overcome the difficulties associated with the complex physical domains a numerical transformation method is used to map this region on a rectangle. Although two independent computer programs which are based on different formulations of the governing equations were used, nearly identical results were obtained. Local heat transfer results are presented for a wide range of Rayleigh numbers for the first time. Local heat transfer rates are found to depend on the Prandtl number in addition to the Rayleigh number dependence.

The destruction of geochemical heterogeneities by differential fluid motions during mantle convection

Summary. A two-dimensional numerical method is presented which permits the observation of deformation within convecting fluids at high Prandtl number. This method is applied to steady state convection, from which it is concluded that deformation is achieved by simple shear, when considered over long time-scales. In numerical examples over a wide range of Rayleigh numbers and heat sources this shear is shown usually to be prograde, with the periphery of the convection cell rotating slower than the core. Occasionally retrograde shear occurs, in which the reverse is true. When the method is applied to time-dependent convection, the deformation of discrete bodies occurs by lateral eddy diffusion of mass, and exponential increase of surface area. The rates of these two processes are coupled and vary as Ra0.5 over a wide range of Rayleigh numbers. It is shown that an eddy diffusive approximation is only appropriate for horizontal scales greater than about 3 times the depth of a convecting layer and for time-scales of at least 300 Myr in the upper mantle. The effects of variable viscosity and three-dimensionality are discussed, and the results of the experiments applied to the Earth's mantle. It is concluded that any convecting layer within the mantle must be well mixed on a lateral scale of at least 2000 km after a time of 0.5–1 Gyr, depending on the processes of magma extraction from a streaky parental material. The deformation rate depends only upon the layer depth and the fluid viscosity. Hence whole mantle convection is 5 times more effective in dispersing geochemical heterogeneities than is convection confined to the upper mantle. Various models for the spatial distribution of geochemical reservoirs are discussed and the only viable models are those in which geochemical reservoirs are identified with separate fluid layers. Whole mantle convection is not favoured by these conclusions. On other grounds, some layered models are discarded and we support a conventional model with separate convection in the upper and lower mantle with a boundary at about 700 km depth. Isotopic anomalies erupted at the surface must be relatively recent additions to the otherwise homogeneous upper mantle.

Experimental Investigation of Free Convection in a Vertical Rod Bundle—A General Correlation for Nusselt Numbers

Free convection in two vertical, enclosed rod bundles has been experimentally investigated for a wide range of Rayleigh numbers. A uniform power dissipation per unit length is supplied to each rod, and the enclosing outer cylinder is maintained at constant temperature. Nusselt numbers for each rod, as well as an overall value for each bundle, have been obtained as a function of Rayleigh number. Comparison of the results for air and water as the working fluid indicate that, for a fixed Rayleigh number, an increase in the Prandtl number produces a reduction in the Nusselt number. This is contrary to what has been reported for vertical cavities and is attributed to curvature effects. Furthermore, the data reveal the interesting fact that it is quite possible for the individual rods in the bundle to exchange energy with the working fluid via different but coexisting regimes at a given power dissipation. Also, as the Rayleigh number is increased, the rods each tend to assume nearly the same heat transfer coefficient. Finally, a correlation for the overall convective Nusselt number is developed in terms of Rayleigh number and geometric parameters.

A second‐order approximation to natural convection for large Rayleigh numbers and small Prandtl numbers

AbstractThe problem under investigation is that of fluid flow within an enclosed rectangular cavity. It is assumed that one wall is maintained at a constant temperature T1 (hot wall) and the other wall is maintained at a constant temperature T0 (cold wall). At the remaining walls, two separate cases are studied. In the first, an adiabatic boundary condition is assumed. That is, the normal derivative of the temperature function is assumed to be 0. In the second, it is assumed the temperature varies linearly from T0 to T1.The purpose of this paper is the application of a second order numerical technique to the problem of fluid flow within a heated closed cavity. The method is a modification of a method developed by Shay1 and applied to the driven cavity problem. In order to test the viability of this technique, it was decided to extend the technique to the problem of natural convection in a square. Jones2 proposed that this problem is suitable for testing techniques that may be applied to a wide range of practical problems such as reactor insulation, cooling of radioactive waste containers, solar energy collection and others.3The technique makes use of second‐order finite difference approximations to all derivatives in the governing equations. Furthermore, second‐order approximations are also used to determine boundary vorticities and, when the adiabatic boundary condition is used, for the boundary temperatures as well. In some works, where second‐order approximations are used at interior points, second‐order boundary approximations have been sacrificed in favour of a more stable, but first‐order boundary approximation.The current approximations are generated by writing the unknown value of a function at a given interior node as a linear combination of unknown function values at all of the neighbouring nodes. Then the function values at these neighbouring nodes are expanded in a Taylor series about the given node. Through appropriate regrouping of terms and the use of the equations to the solved, constraints are imposed on the coefficients of the linear combination to yield a second‐order approximation. As it turns out, there are more unknowns than constraints and, as a result, we are left with some freedom in choosing coefficients. In this work this freedom was used to choose coefficients in such a way as to maximize stability of the resulting system of equations. In other words, the approximations to the governing partial differential equation are individually determined at each point dependent on the direction of flow in order to generate the best possible stability. This idea is analogous to that used in the derivation of the upwind method. However, the current method is second‐order accurate where the upwind method is only first‐order accurate. Thus, what is generated is an easily implemented second‐order method that yields a system of equations that has proved easy to solve.The system of equations is solved via the method of successive overrelaxation. The stability of the method is shown in the convergence for a wide range of Rayleigh numbers, Prandtl numbers and mesh sizes. Level curves of the stream, vorticity and temperature functions are provided for Rayleigh numbers (Ra) as large as 100,000, Prandtl numbers (Pr) as small as 0.0001, and mesh sizes as small as 0.0125. Values of the Nusselt number have also been calculated through the use of Simpson's rule, and a second order approximation to the normal derivative of the temperature along the cold wall. Comparisons are made with other current works to aid in the verification of this methods' accuracy and also with the first‐order upwind method to demonstrate superiority over the first‐order method.

Free convection of gases in a horizontal cylinder with differentially heated end walls

The free convective flow of gases in a horizontal cylinder (radius to length, r 0/ L = 0.1) with different end temperatures and a linear temperature gradient at the wall was characterized by laser Doppler anemometry. Experiments were conducted over a wide range of Rayleigh numbers (74 < Ra < 1.3 × 10 6, with Ra ∝ r 0 3), which were varied via gas species and gas pressure. In the vertical mid-plane the velocity profiles in the cylinder core region at low Ra and in the end regions at high Ra were as expected from 2-D models. For the end regions, however, pronounced 3-D flow behavior was observed. It was found that only part of the fluid flows along the cold (hot) thermode into the lower (upper) half of the cylinder. The remaining part changes the flow direction by dropping (rising) at the side wall over a considerable distance from the respective thermode.

The influence of side walls on finite-amplitude convection in a layer heated from below

Finite-amplitude convection rolls of an infinite-Prandtl-number fluid in a long channel heated from below are investigated. Because of the side walls, the convection rolls depend on all three spatial co-ordinates, although only two velocity components are of importance for a wide range of Rayleigh numbers and aspect ratios. Accurately converged solutions are presented for a range of aspect ratios between 0 (Bénard convection) to 100 (Hele Shaw convection) and for Rayleigh numbers up to about 50 times the critical linear stability value. The influence of rigid versus slip boundaries as well as the wavelength of the convection rolls on the heat transport is investigated in detail. Comparisons with existing results for the analogous problem of convection in a porous medium indicates that the similarity tends to disappear at Rayleigh numbers less than a few times the critical value. Whenever possible, the theoretical findings are compared with experimental results. In all cases close agreement is found.

HEAT TRANSFER BY NATURAL CONVECTION ACROSS VERTICAL AIR LAYERS

Finite-difference predictions of natural convection in vertical air layers are reported for a wide range of Rayleigh numbers and aspect ratios. Calculations were carried out for both perfectly conducting and adiabatic boundaries at the top and bottom ends of the layer. Comparisons between the predictions and measurements show that the calculated heat transfer rates are valid only over a limited range of parameters, and that the points of departure correlate closely with predicted points of instability.

Mean square velocities of b�nard convection and heat transfer

The approximate formula K ∩ a−2R(N−1), where a is a constant near 9 and R and N are the Rayleigh and Nusselt numbers, was proposed in [1] for the dimensionless kinetic energy K of convection in a horizontal layer of liquid. It is shown in the present paper that this expression is exact in linear and weakly nonlinear convection theory when the velocity and temperature fields are represented analytically [2–4]. The valuea is found to be 8.76 when the upper and lower boundaries of the layer are solid walls. The results are given of numerical calculations of the kinetic energy of the convection and the heat transfer in a wide range of Rayleigh numbers (up to 44 000) and Prandtl numbers (0.025 ⩽ P ⩽ 15). Analysis of the results shows that a is in fact a weak function of both R and P. If this is also the case at large R, it indicates a certain breaking of scaling of the mean convection characteristics at sufficiently large values of the Rayleigh number. It also indicates why laboratory experiments give values of n in the dependence N ∼ Rn which are generally slightly less than the theoretical value n = 1/3.

Free convective heat transfer to fluids in the near-critical region from vertical surfaces with uniform heat flux

Numerical predictions are made for laminar free convective heat transfer to fluids in the nearcritical region from a vertical flat plate with uniform surface heat flux. The variation of all the thermophysical properties have been taken into consideration. The governing equations are integrated using the Patankar-Spalding implicit finite difference scheme. Computations are made for carbon dioxide at pressures of 75 ( P P cr = 1.015 ), 80 ( P P cr = 1.083 ) and 100 ( P P cr = 1.354 ) bar and for water at pressures of 225 ( P P cr = 1.018 ) and 245 ( P P cr = 1.108 ) bar, for various values of wall-heat flux ranging from 1000 W/M 2 to 50 000W/M 2. Based on the results obtained, a correlation has been proposed to evaluate the local heat-transfer coefficient for a wide range of Rayleigh numbers ( Ra ∞ = 5 × 10 6−5 × 10 10).

Studies of finite amplitude non-Newtonian thermal convection with application to convection in the Earth's mantle

Studies of non-Newtonian thermal convection have been made to determine the importance of non-Newtonian viscosity on flow in the earth's mantle. Finite difference solutions have been obtained with a viscosity law representing the sum of deformation rates due to diffusion and dislocation creep. Non-Newtonian flow structures differ only slightly from corresponding Newtonian flows. The principal effect of the non-Newtonian viscosity is to increase the effective Rayleigh number. An effective Rayleigh number based on a strain rate squared averaged viscosity provides a good correlation between Newtonian and non-Newtonian flows over a wide range of Rayleigh numbers.

Natural Convection Adjacent to Horizontal Surface of Various Planforms

Natural convection adjacent to horizontal surfaces of circular, square, rectangular, and right triangular planforms has been studied experimentally. Electrochemical techniques were employed involving a fluid with a Schmidt number of about 2200. The results encompass a wide range of Rayleigh numbers thus providing information on both the laminar and the turbulent regimes. The data for all planforms are reduced to a single correlation in the laminar and turbulent regimes using the characteristic length, as recommended by Goldstein, Sparrow, and Jones. L* = A/p, where A is the surface area and p is the surface perimeter. The laminar data for all planforms are correlated by the expression Sh=0.54Ra1/4(2.2×104≤Ra≤8×106) and the data for the turbulent regime are correlated by the expression Sh=0.15Ra1/3(8×106≤1.6×109) Transition is found to occur at about Ra = 8 × 106. The present work thus significantly extends the Rayleigh number range of validity for the use of L* through the 1/4 power laminar regime into the turbulent 1/3 power regime. It also demonstrates the validity of the use of L* to correlate natural convection transfer coefficients for highly unsymmetrical planforms, which heretofore had not been demonstrated. Comparisons to analytical solutions and other experimental heat and mass transfer data are presented.

Crystal Growth from the Melt under Destabilizing Thermal Gradients

Direct time correspondence between crystal growth behavior and the thermal characteristics of the melt under destabilizing gradients was established by means of “time markers” which were introduced into the growing crystals and simultaneously registered on the continuous recording of the thermal behavior of the melt. Employing Te‐doped , it was found that as solidification progressed, the melt exhibited successively turbulent convection, oscillatory thermal instabilities, and, finally, thermal stability. During turbulent convection, the crystals underwent pronounced transient back‐melting and the average microscopic growth rate was found to be independent of, and about 20 times greater than, the average macroscopic growth rate; this microscopic growth rate was controlled by the convection characteristics of the melt and the thermal gradients in the solid. With the onset of oscillatory thermal instabilities, back‐melting became less pronounced and then ceased. In this region the average microscopic and macroscopic growth rates were equal and the crystals exhibited fluctuations in dopant concentration with a periodicity identical to that of the thermal oscillations in the melt. Finally, under stabilized thermal conditions, the microscopic and macroscopic growth rates were identical and no localized fluctuations in dopant concentration could be detected. The Rayleigh numbers of the melt during a growth experiment ranged from . In view of the fact that the vertical thermal gradient changed only by a factor of about two during the entire growth, the wide range of Rayleigh numbers was the result of the changes in melt height. In the turbulent convection region, the Rayleigh numbers ranged from about , in the oscillatory region from , and in the region of thermal stability from 103 to 0.