Effect of the use of the balance and gradient methods as a result of experimental investigations of natural convection action with regard to the conception and construction of measuring apparatus

Measurement apparatus designed and constructed according to conceptions of the authors, enabled a more precise calculation of the heat transfer coefficient with the balance and gradient methods. Construction and use of the apparatus and devices are described below, results of experimental investigations for horizontal and vertical, isothermal, flat plates obtained independently with the balance and gradient methods, are also presented. The following equations were found:Nu=0.612 · (Ra)^1/4 10⁴ ≦Ra ≦ 10⁸ for vertical platesNu=0.766 · (Ra)^1/5 10⁴ ≦Ra ≦ 10⁷ Nu=0.173 · (Ra)^1/3 10⁵ ≦Ra≦ 10⁸ for horizontal plates. On the basis of the results obtained from both these methods, differences of natural convection acting from vertical and horizontal plates are discussed. The usefulness of the balance and gradient methods have been considered for qualitative and quantitative investigations of heat transfer by natural convection.     

Experimental analysis of thermal fields in horizontally eccentric cylindrical annuli

This paper presents new experimental results on thermal field and heat transfer in a two-dimensional annulus between horizontally eccentric cylinders. The study is conducted by means of optical techniques, for 1.07×10⁴≤Ra _L≤8.27×10⁴ and a wide eccentricity range. The horizontal eccentricity of the inner cylinder substantially alters the thermal field and the geometry of the plume, but, in analogy to the behaviour for vertical eccentricity, the average Nu is slightly affected in the investigated range of eccentricity. The concentric geometry is also considered mainly to validate the experimental technique and evaluate the accuracy of the adopted methodology by comparison with available results. Both shearing interferometer and reference beam interferometer are obtained by means of Wollaston prisms with appropriate splitting angles, so that the temperature and local Nu distributions may be evaluated quantitatively from the original pictures via digital image processing.     

Nusselt-Rayleigh correlations for free convection in 2D air-filled parallelogrammic enclosures with isothermal active walls

In the present study Nu-Ra-α correlations are proposed to calculate the steady-state natural convection heat transfer taking place in 2D air-filled cavities of parallelogrammic section. The thermal conditions and the dimensions of the enclosures permit to cover a large range of Rayleigh numbers, 1.7 × 10³  ≤ Ra ≤ 3.0 × 10⁹, suitable for diverse engineering applications. The two active walls of the cavities are kept vertical and isothermal at hot and cold temperatures T _h and T _c respectively. Separated by a horizontal distance H, they have the same height H and are connected by a closed adiabatic channel whose upper and lower walls can be inclined at an angle α with respect to the horizontal, varying between −60° to +60°. That gives rise to a conducting or insulating cavity, in the convective sense of the term (diode cavity). A computational model based on the finite volume method is used to solve the governing equations. The large number of treated configurations led to propose Nu-Ra-α correlations for large ranges of Ra and α which can be applied to many engineering areas. The results of this numerical study have been successfully compared with calculated and measured available data.     

Combined convection and wall conduction effects in laminar pipe flow: numerical predictions and experimental validation under uniform wall heating

Laminar combined convection in horizontal circular ducts is investigated both numerically and experimentally, under uniform wall heating. A series of experiments for the heating of water in a long horizontal copper tube are simulated numerically in order to assess the reliability of the theoretical results. Peripheral and axial wall conduction effects, inherently present in the experiments, are accounted for in the numerical model. The cross validation of experimental and numerical data allows significant conclusions to be reached on conjugate conduction and convection with buoyancy effects in horizontal duct flows. Buoyancy is considered for values of the modified Rayleigh number,Ra _qo , up to 5·10⁶; the forced convection contribution is considered for two values of the entry Reynolds number,Re _o=500 and 1000.     

Subcooled forced convection film boiling over a vertical flat plate embedded in a fluid-saturated porous medium

The problem of subcooled forced convection film boiling on a vertical flat plate embedded in a porous medium was attacked exploiting similarity transformations on the governing equations and boundary conditions in both vapor and liquid layers. Similarity solutions were obtained to investigate the effects of the vapor super-heating and liquid subcooling. The heat transfer groupingNu _x /Ra _x ^1/2 is expressed in terms of a function of three parameters associated with the degree of liquid subcooling (Sub), the degree of vapor superheating (Sup) and the vapor buoyancy effect relative to the liquid forced convection effect (R). It is found that the level ofNu _x /Ra _x ^1/2 increases asSup orR decreases and asSub increases. Furthermore, asymptotic expressions were reduced considering the physical limiting conditions, namely, thin and thick vapor films.     

Heat transfer by laminar flow in a vertical and horizontal pipe with twisted-tape inserts

 This article presents the results of laboratory research on heat exchange while heating water in horizontal and vertical tubes with twisted-tape inserts. The scope of the research: 70 ≤ Re ≤ 4000 3.6 ≤ Pr ≤ 5.9 8.6 ≤ Gz ≤ 540 The research was held for three cases: – horizontal experimental tube – vertical experimental tube, the direction of flow according to the free convection vector – vertical experimental tube, the direction of flow not in accordance with the free convection vector For such cases the correlation equation was defined Nu_T=f(Gz; y), Nu = f(Gz) and the proportion Nu_T/Nu was analysed. Received on 30 March 2000     

Natural convection in inclined two dimensional rectangular cavities

Steady two-dimensional natural convection in fluid filled cavities is numerically investigated. The channel is heated from below and cooled from the top with insulated side walls and the inclination angle is varied. The field equations for a Newtonian Boussinesq fluid are solved numerically for three cavity height based Rayleigh numbers, Ra = 10⁴, 10⁵ and 10⁶, and several aspect ratios. The calculations are in excellent agreement with previously published benchmark results. The effect of the inclination of the cavity to the horizontal with the angle varying from 0° to 180° and the effect of the startup conditions on the flow pattern, temperature distribution and the heat transfer rates have been investigated. Flow admits different configurations at different angles as the angle of inclination is increased depending on the initial conditions. Regardless of the initial conditions Nusselt number Nu exhibits discontinuities triggered by gradual transition from multiple cell to a single cell configuration. The critical angle of inclination at which the discontinuity occurs is strongly influenced by the assumed startup field. The hysteresis effect previously reported is not always present when the calculations are reversed from 90° to 0°. A comprehensive study of the flow structure, the Nu variation with varying angle of inclination, the effect of the initial conditions and the hysteresis effect are presented.     

The Onset of Convection in a Heterogeneous Porous Medium with Vertical Throughflow

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from vertical throughflow, is studied analytically using linear stability theory. It is found that, to first order, a linear variation of the reciprocal of permeability with depth has no effect on the critical value of the Rayleigh number Ra _c based on the harmonic mean of the permeability, but a quadratic variation increasing in the upwards direction leads to a reduction in Ra _c.     

Heat transfer from the heated concave wall of a return bend with rectangular cross section

The characteristics of the turbulent heat transfer along the heated concave walls of return bends which have rectangular cross sections with large aspect ratio have been examined for various clearances of the ducts in detail. The experiments are carried out under the condition that the concave walls are heated at constant heat flux while the convex walls are insulated. Water as the working fluid is utilized. Using three kinds of clearance of 9, 34, and 55 mm, the Reynolds number in the turbulent range are varied from 5×10³ to 8×10⁴ with the Prandtl numbers ranging from 4 to 13. As a result it is elucidated that both the mean and the local Nusselt numbers are always greater than those for the straight parallel plates or for the straight duct, respectively. This is attributed to Görtier vortices, which are visualized here. It is also found that the more the clearance increases, the more both the local and the mean Nusselt numbers increase. Correlation equations for the mean and the local Nusselt numbers are determined in the range of parameters covered. Introducing the Richardson number, it appears that the local Nusselt number,Nu _x , may be described as the following equation:Nu _x =447.745 ·Re _x ^1.497 ·De _x ^?1.596 ·F ^0.960 ·Pr ^0.412     

Conjugate heat transfer in enclosures with openings for ventilation

Conjugate heat transfer by natural convection and conduction in enclosures with openings has been studied by a numerical method. The enclosure contained a chimney consisting of a vertical solid wall, which was insulated on one side and a constant heat flux applied on the other. Vertical boundaries with openings were isothermal and horizontal boundaries adiabatic. These problems are encountered in heat transfer in buildings and heat management in electronic equipment. Two dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation are solved using the Simpler method. Various geometrical parameters were: aspect ratio, A from 0.5 to 2.0, openings' heights, h ₁ and h ₂ from 0.10 to 0.30, orifice height, h ₃ from 0.05 to 0.15, insulation thickness, w ₁ from 0 to 0.10, wall thickness, w ₂ from 0.05 to 0.15 and chimney width, w ₃ from 0.05 to 0.15. Rayleigh number, Ra was varied from 10 ⁸ to 10 ¹² and the conductivity ratio, k _r was from 1 to 40. The results are reduced in terms of the normalized Nusselt number, Nu and volume flow rate, V? as a function of Ra number, and other non dimensional geometrical parameters. The isotherms and streamlines are produced for various Ra numbers and geometrical conditions. It is found that Nu and V? are both an increasing function of Ra, h ₁ at high Ra numbers, h ₃, and k _r. They are a decreasing function of h ₁ at low Ra numbers, h ₂, and w ₂. Nu and V? have optima with respect to w ₁, w ₃ and A.     

A transition phenomenon of ice formation in water flow between two horizontal parallel plates

Experiments have been performed to investigate the icetransition profiles and heat-transfer characteristics for water flows between two horizontal parallel plates. The experiments are carried out under the condition that upper plate is cooled at uniform temperature kept less than freezing temperature of water, while the lower plate is heated at uniform temperature kept higher than the temperature of water flow. The temperatures of the upper and lower plates range from ?8 to ?14°C and from 10 to 60 °C, respectively, with inlet-water temperature varied from 1.5 to 4.5 °C. The cooling and heating temperature ratios, θ_c and θ_h, are ranging from 1.78 to 9.33 and from 1.22 to 39, respectively. By using three kinds of heightH of 16, 30 and 40 mm between the horizontal parallel plates, the Reynolds and Grashof numbers are varied from 3.2 × 10² to 1.5 × 10⁴ and from 3.4 × 10³ to 8.97 × 10⁶, respectively. As a result of this investigation two ice-transition modes are observed. The first ice-transition mode is due to an interruption of upper and lower thermal boundary layers, while the second mode is due to an instability of laminar boundary layer formed on water-ice interface. In order to determine the kind of ice-transition mode, criterion correlation formulas including the Reynolds numberRe _H , Grashof numberGr _H , and heating temperature ratio θ_h are determined and may be written as follows: For thermal icetransition mode (th.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23<1.6×10^?3 and for hydrodynamical ice-transition mode (hy.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23>2.3×10^?3 By introducing the freezing parameterB _f , correlation equations for local and mean Nusselt numbers along the water-ice interface at steady-state condition are determined. From the current experimental results it is found that the local Nusselt number may be described as the following equation:Nu _x =0.835 Re _H ^0.278 · B _f ^0.834 ·x/H)^?0.139     

Nonlinear dynamics between two differentially heated vertical plates in the presence of stratification

We consider the numerical simulation of the flow between infinite, differentially heated vertical plates with positive stratification. We use a two-dimensional Boussinesq approximation, with periodic boundary conditions in the vertical direction. The relative stratification parameter ${\gamma=(\frac{1}{4}Ra S)^{1/4}}$ , where Ra is the Rayleigh number and S the adimensional stratification, is kept constant and equal to 8. The Prandtl number is 0.71. We derive a complex Ginzburg-Landau equation from the equations of motion. Coefficients are computed analytically, but we find that the domain of validity of these coefficients is small and rely on the numerical simulation to adjust the coefficients over a wider range of Rayleigh numbers. We show that the Ginzburg-Landau equation is able to accurately predict the characteristics of the periodic solution at moderate Rayleigh numbers. Above the primary bifurcation at Ra = 1.63 × 10⁵, the Ginzburg-Landau model is found to be Benjamin-Feir unstable and to be characterized by modulated traveling waves and phase-defect chaos, which is supported by evidence from the DNS. As the Rayleigh number is increased beyond Ra = 2.7 × 10⁵, nonlinearities become strong and the flow is characterized by cnoidal waves.     

Rayleigh–Bénard convection driven by a long wavelength heating

Natural convection in a two-dimensional horizontal layer has been investigated. The layer is confined between two parallel horizontal plates. The upper plate is kept isothermal, while the lower plate has an externally imposed, long wavelength, spatially sinusoidal heating with the amplitude expressed in terms of the Rayleigh number Ra and the wavelength characterized by the wave number α. Only steady-state flow structures and their bifurcations have been considered. The detailed analysis has been carried out for two Prandtl numbers, i.e. Pr = 0.7 and Pr = 7, and only small differences in the bifurcation diagrams have been observed. When Ra < Ra _cr = 427, convection has a simple topology consisting of one pair of counter-rotating rolls per heating period. Secondary motion in the form of rolls aligned in the direction of the primary rolls and concentrated around the hot spots occurs for Ra > 427. When 427 < Ra < ～470 and α < ～0.14, the secondary motion is described by the supercritical pitchfork bifurcation. One of the branches of this bifurcation is associated with an odd number of secondary rolls per half wavelength, with rolls above the hot spots rotating in the direction opposite to the primary rolls. The other branch is associated with an even number of secondary rolls per half wavelength, with the rolls above the hot spots co-rotating with the primary rolls. The new rolls are pinched off in pairs when α decreases. When Ra > ～470 and α > ~0.14, bifurcation assumes the form of “bifurcation from infinity”. The main branch is associated with one pair of rolls per heating period for α > 0.25. Decrease in α along this branch results in the formation of secondary rolls, with the rolls at the hot spot co-rotating with the primary rolls. The lower part of the other branch is associated with one pair of rolls per heating period in the limit α → 0. Increase in α results in pinching off a single roll which counter-rotates with respect to the primary roll at the hot spot.     

Computation of buoyancy-driven flow in an eccentric centrifugal annulus with a non-orthogonal collocated finite volume algorithm

A computational study is performed on two-dimensional mixed convection in an annulus between a horizontal outer cylinder and a heated, rotating, eccentric inner cylinder. The computation has been done using a non-orthogonal grid and a fully collocated finite volume procedure. Solutions are iterated to convergence through a pressure correction scheme and the convection is treated by Van Leer's MUSCL scheme. The numerical procedure adopted here can easily eliminate the ‘Numerical leakage’ phenomenon of the mixed convection problem whereby strong buoyancy and centrifugal effects are encountered in the case of a highly eccentric annulus. Numerical results have been obtained for Rayleigh number Ra ranging from 7×10³ to 10⁷, Reynolds number Re from 0 to 1200 and Prandtl number Pr from 0.01 to 7. The mixed rotation parameter σ (=Ra/PrRe²) varies from ∞ (pure natural convection) to 0.01 with various eccentricities ε. The computational results are in good agreement with previous works which show that the mixed convection heat transfer characteristics in the annulus are significantly affected by σ and ε. The results indicate that the mean Nusselt number Nu increases with increasing Ra or Pr but decreases with increasing Re. In the case of a highly eccentric annulus the conduction effect becomes predominant in the throat gap. Hence the crucial phenomenon on whereby Nu first decreases and then increases can be found with increasing eccentricity. © 1998 John Wiley & Sons, Ltd.     

Natural convection in a liquid metal heated from above and influenced by a magnetic field

Natural convection in a liquid metal heated locally at its upper surface and affected by a vertical magnetic field is investigated both experimentally and numerically. The experiments are conducted in a cylindrical test cell of large aspect ratio which is typical for application. The cell is filled with the liquid alloy GaInSn in eutectic composition. Temperature and velocity are measured using thermocouples and an electric potential probe, respectively. In the absence of the magnetic field the experimental results indicate a dependence of the Nusselt number on the Rayleigh number according to the law Nu∝Ra^0.191. The particular value of the scaling exponent is in excellent agreement with the prediction of a scaling analysis for laminar, boundary layer-type flow in a low-Prandtl number fluid. Furthermore the experiments demonstrate that the Nusselt number and therefore the convective heat losses can be decreased by about 20% when a magnetic field of moderate strength (B=0.1 T) is present. The numerical simulations solve the Boussinesq equations in an axisymmetric geometry using a finite element method. The results of the simulations are both quantitatively and qualitatively in good agreement with the experimental observations. Deviations are attributed to the three-dimensional characteristics of the flow.     

Mixed convection in the presence of horizontal impermeable surfaces in saturated porous media with variable permeability

Boundary layer approximation is applied for mixed convection about a horizontal flat plate in a saturated porous medium with aiding external flows. Similarity solutions are obtained, incorporating the variation of permeabilty, for 1) horizontal flat plate at zero angle of attack with constant heat flux; 2) stagnation point flows about horizontal flat plates with wall temperature varying asT _w ∝x ². The temperature and velocity profiles for different values of Ra/(RePr)^3/2 and the parameters governing the flow are obtained. The heat transfer rate is calculated and its implications in a geothermal application is discussed. Further, the criteria for pure mixed convection about horizontal flat plates in a porous media are established.     

Messung des gleichzeitigen Wärme- und Stoffübergangs am horizontalen Zylinder bei freier Konvektion

Simultaneous heat and mass transfer at horizontal cylinders under free convection conditions have been investigated by an electrochemical method. Over all and local measurements of the mass transfer coefficient have been executed. The over all results can be represented by two empirical equations: $$\begin{gathered} Nu = 0,858 \cdot (Gr \cdot Pr)^{0,22} \hfill \\ Sh = 0,23 \cdot [(Gr' \cdot Sc)^{1,07} + 1,49\sqrt {Sc/Pr} \cdot Gr \cdot Sc]^{0,28} \hfill \\ \end{gathered}$$ The ranges of the dimensionsless groups were as follows:Gr: 7,31 · 10² to 8,69 · 10⁵ Gr′: 1,21 · 10³ to 2,79 · 10⁵ Pr: 10 to 7,56Sc: 3440 to 1890     

Effect of flow diverters on free convection heat transfer from a pair of vertical arrays of isothermal cylinders

Laminar free convection heat transfer from two vertical arrays of five isothermal cylinders separated by flow diverters is studied experimentally using a Mach-Zehnder interferometer. The width of flow diverters is kept constant to two-cylinder diameters and the cylinders vertical center-to-center spacing is equal to three-cylinder diameter. Effect of the ratio of the horizontal spacing between two cylinder arrays to their diameter (S_h/D) on heat transfer from the cylinders is investigated for various Rayleigh numbers. The experiments are performed for S_h/D = 2-4, and the Rayleigh number based on the cylinder diameter ranging from 10³ to 3 × 10³. It is observed that for small S_h/D ratios, the flow diverters have a negative effect on the total rate of heat transfer from the arrays; while by increasing the horizontal center to center spacing, they tend to enhance the overall cooling rate of the array. Moreover, increasing Ra and S_h/D generally results in a higher average Nusselt number for each cylinder in the array.     

Large-scale tomographic PIV in forced and mixed convection using a parallel SMART version

Large-scale tomographic particle image velocimetry (tomographic PIV) was used to study large-scale flow structures of turbulent convective air flow in an elongated rectangular convection cell. Three flow cases have been investigated, that is, pure forced convection and mixed convection at two different Archimedes numbers. The Reynolds number was constant at Re?=?1.04?×?10⁴ for all cases, while the Archimedes numbers were Ar?=?2.1 and 3.6 for the mixed convection cases, corresponding to Rayleigh numbers of Ra?=?1.6?×?10⁸ and 2.8?×?10⁸, respectively. In these investigations, the size of the measurement volume was as large as 840?mm?×?500?mm?×?240?mm. To allow for statistical analysis of the measured instantaneous flow fields, a large number of samples needed to be evaluated. Therefore, an efficient parallel implementation of the tomographic PIV algorithm was developed, which is based on a version of the simultaneous multiplicative reconstruction technique (SMART). Our algorithm distinguishes itself amongst other features by the fact that it does not store any weighting coefficients. The measurement of forced convection reveals an almost two-dimensional roll structure, which is orientated in the longitudinal cell direction. Its mean velocity field exhibits a core line with a wavy shape and a wavelength, which corresponds to the height and depth of the cell. In the instantaneous fields, the core line oscillates around its mean position. Under the influence of thermal buoyancy forces, the global structure of the flow field changes significantly. At lower Archimedes numbers, the resulting roll-like structure is shifted and deformed as compared to pure forced convection. Additionally, the core line oscillates much more strongly around its mean position due to the interaction of the roll structure with the rising hot air. If the Archimedes number is further increased, the roll-like structure breaks up into four counter-rotating convection rolls as a result of the increased influence of buoyancy forces. Moreover, large-scale tomographic PIV reveals that the orientation of these rolls reflects a ??W??-like shape in the horizontal X?CZ-plane of the convection cell.     

Average and extremal properties of heat transfer and shear stress on a wall surface in Rayleigh–Bénard convection

In this study surface-averaged and extremal properties of heat transfer and shear stress on the upper wall surface of Rayleigh–Bénard convection are numerically examined. The Prandtl number was raised up to 10³, and the Rayleigh number was changed between 10⁴ and 10⁷. As a result, average Nusselt number Nu and shear rate τ/Pr depends on Pr, Ra, and the entire numerical results are distributed between two correlation equations corresponding to small and large Pr. The small and large Pr equations are closely related to steady and unsteady flow regimes, respectively. Nevertheless, a single relation τ/Pr ~ Nu ^3.0 exists to explain the entire results. Similarly the change of local maximal properties Nu _max and τ _max/Pr depends on Pr, Ra, and these values are also distributed between two correlation equations corresponding to small and large Pr cases. Despite such complicated dependence we can obtain a correlation equation as a form of τ _max/Pr ~ Nu _max^2.6, which has not been obtained theoretically.