Mixed convection and conduction heat transfer in open cavities

A numerical study has been carried out on mixed convection and conduction in open cavities. The study covers the Rayleigh number from 10 to 10⁶, the Reynolds number from 1 to 10³ andk _r =k _w /k _a from 1 to 100 forPr=0.72 (air) and cavity aspect ratioB from 0.8 to 1.3. The flow is assumed to be laminar and two-dimensional. The density variation is taken into account by the Boussinesq approximation. The controlvolume approach is used for solving the governing equations of conjugate heat transfer involving conduction in the walls. Streamlines and isotherms in the system are produced and the results are represented in terms of the Nusselt number as function of other parameters. Correlations are derived to calculate heat transfer through the cavity openings.     

Natural convection heat transfer in enclosures with conducting multiple partitions and side walls

The flow and heat transfer in enclosures with conducting multiple partitions and side walls were numerically analyzed. Side walls were kept at isothermal conditions, while top and bottom walls were insulated. Employing control volume approach, a computer program based on SIMPLE algorithm was developed. Computations were carried out to investigate the effects of Rayleigh number, number of partitions and cavity aspect ratios on the heat transfer rate. The mean Nusselt numbers were calculated from computed temperature fields. It was observed that, the mean Nusselt number decreases with increasing partition number. It is inversely proportional to (1+N) for N≤4. For all partition numbers, the mean Nusselt number increases with increasing Rayleigh number. On the other hand, the cavity aspect ratio does not affect the mean Nusselt number to a considerable extent for considered aspect ratios in this study.     

Natural convection in vertical cavities with internal heat generating porous medium

Steady natural convection heat transfer in a two dimensional cavity filled with a uniform heat generating, saturated porous medium has been studied. The boundary conditions were: Two isothermal walls at different temperatures, two horizontal adiabatic walls. The aspect ratio was varied from 0.1 to 10 and the Rayleigh number from 10⁰ to 10⁸. The results are presented in terms of the isotherms and stream functions, the temperature variation and maximum temperature in the cavity and heat transfer from the vertical walls. The study indicates that asymmetric vertical boundary conditions with _h >0 has an important effect on the temperature and flow fields as well as on the heat transfer characteristics of the cavity with highly asymmetric results. Various heat transfer modes are identified dependent on the Rayleigh number and the aspect ratio.Es wurde stetiger Wärmeübergang durch freie Konvektion in einen zweidimensionalen Hohlraum, gefüllt mit gleichmäßig wärmeerzeugendem porösen Medium, untersucht. Dabei wurden folgende Randbedingungen festgelegt: zwei isotherme Wände unterschiedlicher Temperatur, zwei horizontale adiabate Wände. Das Verhältnis vonH/L wurde zwischen 0,1 und 10, die Rayleigh-Zahl zwischen 10⁰ und 10⁸ variiert. Die Ergebnisse werden durch Terme der Temperaturverläufe, der maximalen Temperatur im Hohlraum, der Isothermen und der Strömungsfunktion, sowie der Wärmeübertragung der vertikalen Wände dargestellt. Die Studie zeigt, daß asymmetrische Randbedingungen mit _h >0 einen großen Einfluß auf Temperatur- und Strömungsfelder, als auch auf die Wärmeübertragungskennzahlen des Hohlraumes mit stark asymmetrischen Ergebnissen haben. Die verschiedenen Arten der Wärmeübertragung werden in Abhängigkeit von der Rayleigh-Zahl und dem Verhältnis vonH/L beschrieben.     

Effect of vertical heat transfer on thermocapillary convection in an open shallow rectangular cavity

In order to understand the effect of the vertical heat transfer on thermocapillary convection characteristics in a differentially heated open shallow rectangular cavity, a series of two- and three-dimensional numerical simulations were carried out by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and the aspect ratio ranged from 12 to 30. Results show that thermocapillary convection is stable at a small Marangoni number. With the increase of the heat flux on the bottom surface, thermocapillary convection transits to the asymmetrical bi-cellular pattern with the opposite rotation direction. The roll near the hot wall shrinks as the Marangoni number increases. At a large Marangoni number, numerical simulations predict two types of the oscillatory thermocapillary flow. One is the hydrothermal wave, which is dominant only in a thin cavity. The other appears in a deeper cavity and is characterized by oscillating multi-cellular flow. The critical Marangoni number for the onset of the oscillatory flow increases first and then decreases with the increase of the vertical heat flux. The three-dimensional numerical simulation can predict the propagating direction of the hydrothermal wave. The velocity and temperature fields obtained by three-dimensional simulation in the meridian plane are very close to those obtained by two-dimensional simulation.     

Forced convection — axial conduction between parallel walls with unequal heat fluxes

Laminar flow heat transfer is computed for a situation in which a fluid moves along a parallel plate channel with unequal wall heat fluxes (one wall is insulated). The fluid enters the heating section through an upstream region which is perfectly insulated. This situation serves to describe an upper bound for the commonly encountered case of double pipe heat exchangers with identical thermal conditions. A control volume approach has been employed for the numerical work enabling a fast calculation for the thermally developing regime in the parallel plate channel. The merits of the adopted procedure are assessed by comparison with other results available in the literature for the one-region and for the two-region problem.     

A steady conjugate heat transfer problem with conduction and free convection

Summary A steady conjugate heat transfer problem dealing with conduction in a heat-generating slab and free convection in the surrounding fluid is studied analytically. Free convection is analyzed by a Görtler-type series solution to the boundary-layer equations for non-uniform surface-temperature variations, while conduction is treated by the standard technique of Fourier transforms. Interfacial temperature and heat flux variations from both solutions in series forms are then formally matched to yield algebraic relations for the coefficients in the series. These coefficients can then be simply evaluated in a given problem in terms of three physical parameters. A numerical example is shown.     

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     

Entropy generation and natural convection in square cavities with wavy walls

The aim of the present work is to study the entropy generation in the natural convection process in square cavities with hot wavy walls through numerical simulations for different undulations and Rayleigh numbers, while keeping the Prandtl number constant. The results show that the hot wall geometry affects notably the heat transfer rate in the cavity. It has been found in the present numerical study that the mean Nusselt number in the case of heat transfer in a cavity with wavy walls is lower, as compared to heat transfer in a cavity without undulations. Based on the obtained dimensionless velocity and temperature values, the distributions of the local entropy generation due to heat transfer and fluid friction, the local Bejan number, and the local entropy generation are determined and plotted for different undulations and Rayleigh numbers. The study is performed for Rayleigh numbers 10³ < Ra < 10⁵, irreversibility coefficients 10^?4 < φ < 10^?2, and Prandtl numbers Pr = 0.71. The total entropy generation is found to increase with increasing undulation number.     

Natural convection heat transfer along vertical rectangular ducts

Experimental investigations have been reported on steady state natural convection from the outer surface of vertical rectangular and square ducts in air. Seven ducts have been used; three of them have a rectangular cross section and the rest have square cross section. The ducts are heated using internal constant heat flux heating elements. The temperatures along the vertical surface and the peripheral directions of the duct wall are measured. Axial (perimeter averaged) heat transfer coefficients along the side of each duct are obtained for laminar and transition to turbulent regimes of natural convection heat transfer. Axial (perimeter averaged) Nusselt numbers are evaluated and correlated using the modified Rayleigh numbers for laminar and transition regime using the vertical axial distance as a characteristic length. Critical values of the modified Rayleigh numbers are obtained for transition to turbulent. Furthermore, total overall averaged Nusselt numbers are correlated with the modified Rayleigh numbers and the area ratio for the laminar regimes. The local axial (perimeter averaged) heat transfer coefficients are observed to decrease in the laminar region and increase in the transition region. Laminar regimes are obtained at the lower half of the ducts and its chance to appear decreases as the heat flux increases.     

Natural convection heat transfer from sinusoidal wavy surfaces

This paper presents the results of an experimental study of the natural convection heat transfer characteristics of sinusoidal wavy surfaces on vertical plates maintained at a constant temperature. Local heat transfer coefficients were obtained with a Mach-Zehnder interferometer. The heat transfer from the wavy surfaces, compared to a plane plate of equal projected area, increased with increasing amplitude-to-wavelength ratio. The heat transfer was increased by about 15 percent at an amplitude-to-wavelength ratio of 0.3; for this case a flow instability was detected. A quantitative comparison with a previously published numerical investigation is also presented. In general, there is agreement between the two studies.     

Natural convection heat transfer from round horizontal plate

A new approach to the model of natural convection from a horizontal, isothermal round plate and a simplified analytical solution of this model have been presented. In this model two separate regions with different fluid motions have been distinguished. In the first region, inside the boundary layer, the fluid flows concentrically towards the centre of the plate, while in the second one (stagnation region) the fluid is motionless. The presented theory has been verified experimentally.Ein neuer Lösungsweg für das Modell der freien Konvektion an einer isothermen, kreisförmigen, horizontalen Platte und eine vereinfachte analytische Lösung für dieses Modell werden hier vorgestellt. An diesem Modell wird zwischen zwei Bereichen mit verschiedenen Fluidbewegungen unterschieden. Im ersten Bereich, innerhalb der Grenzschicht, strömt das Fluid konzentrisch in Richtung Plattenmitte, während im zweiten Bereich (Stau-Bereich) die Flüssigkeit in Ruhe ist. Diese Theorie wurde experimentell überprüft.     

Natural convection heat transfer in enclosures with microemulsion phase change material slurry

This paper has dealt with the natural convection heat transfer characteristics of microemulsion slurry composed of water, fine particles of phase change material (PCM) in rectangular enclosures. The microemulsion slurry exhibited non-Newtonian pseudoplastic fluid behavior, and the phase changing process can show dramatically variations in both thermophysical and rheological properties with temperature. The experiments have been carried out separately in three subdivided regions in which the state of PCM in microemulsion is in only solid phase, two phases (coexistence of solid and liquid phases) or only liquid phase. The complicated heat transfer characteristics of natural convection have appeared in the phase changing region. The phase change phenomenon of the PCM enhanced the heat transfer in natural convection, and the Nusselt number was generalized by introducing a modified Stefan number. However, the Nusselt number did not show a linear output with the height of the enclosure, since a top conduction lid or stagnant layer was induced over a certain height of the enclosure. The Nusselt number increased with a decrease in aspect ratio (width/height of the rectangular enclosure) even including the side-wall effect. However, the microemulsion was more viscous while the PCM was in the solid phase, the side-wall effect on heat transfer was greater for the PCM in the solid region than that for the PCM in the liquid region. The correlation generalized for the PCM in a single phase is $ Nu = 1/3(1 - C_1 )Ra^{{1 \over {3.5n + 1}}} , $ where C ₁ = e ^–0.09AR for the PCM in solid phase and C ₁ = e ^–0.33AR for the PCM in liquid phase. For the PCM in the phase changing region, the correlation can be expressed as $ Nu = CRa^{{1 \over {7n + 2}}} Ste^{ - (1.9 - 1.65n)} , $ where C = 1.22 – 0.035AR for AR > 10 and C = 0.55 – 16.4e ^–1.1AR for AR < 10. The enclosure height used in the present experiments was varied from H = 5.5 [mm] to 30.4 [mm] at the fixed width W = 120 [mm] and depth D = 120 [mm]. The experiments were done in the range of modified Rayleigh number 7.0 × 10² ≤ Ra ≤ 3.0 × 10⁶, while the enclosure aspect ratio AR varied from 3.9 to 21.8.     

Natural convection in partially cooled tilted cavities

Two-dimensional numerical simulations of laminar natural convection in a partially cooled, differentially heated inclined cavities are performed. One of the cavity walls is entirely heated to a uniformly high temperature (heat source) while the opposite wall is partially cooled to a lower temperature (heat sink). The remaining walls are adiabatic. The tilt angle of the cavity is varied from 0° (heated from left) to −90° (heated from top). The fast false implicit transient scheme (FITS) algorithm, developed earlier by the same authors, is modified to solve the derived variables vorticity-streamfunction formulation. The effects of aspect ratio (AR), sink–source ratio and tilt angle on the average Nusselt number are examined through a parametric study; solutions are obtained for two Grashof numbers, 10⁵ and 10⁷. Flow patterns and isotherms are used to investigate the heat transfer and fluid flow mechanisms inside the cavity. © 1998 John Wiley & Sons, Ltd.     

Natural convection in cavities containing internal sources

 Numerical predictions are reported for buoyancy-induced circulations in a 2D closed cavity with internal heat sources. Two cases are considered: (A) two vertical plates with uniform heat generation, forming a short vertical channel within the enclosure; and, (B) a rectangular heating block with uniform wall temperature, placed central in the enclosure. Air, with a Prandtl number 0.71, is considered as the working fluid. The vertical enclosure walls are isothermal, while the horizontal enclosure walls are adiabatic. Results are presented for two values of the Grashof number, one below the stability limit for laminar flow, and one well above it. In those latter cases, the long-term behaviour of the numerical solution is time-dependent, i.e. no steady-state can be reached. Heat transfer results are compared with predictions from standard correlations for isolated surfaces. Received on 17 January 2000     

Natural convection heat transfer in a uniformly heated horizontal pipe

Natural convection heat transfers inside horizontal pipes were measured. The Rayleigh numbers were varied from 6.8 × 10⁸ to 1.5 × 10¹², while the Prandtl number was fixed at 2,094. Based on the analogy concept, a copper sulfate electroplating system was adopted to measure mass transfer rates in place of heat transfer rates. Test results using single-piece electrodes were in good agreement with the work of Sarac and Korkut. The angle-dependent mass transfer rates, measured using piecewise electrodes, were compared with the results of studies on natural convection in concentric annuli, and showed similar trends. The experiments were expanded to the turbulent region, and a transition criterion was proposed. Angle-dependent natural convection heat transfer correlations for the laminar and turbulent regions were derived.     

Natural convection heat transfer on surfaces of copper micro-wires

The natural convection heat transfer characteristics and mechanism for copper micro-wires in water and air were investigated experimentally and numerically. The wires with diameters of 39.9, 65.8 and 119.1 μm were placed horizontally in water inside of a sealed tube and in air of a large room, respectively. Using Joule heating, the heat transfer coefficients and Nusselt numbers of natural convection for micro-wires in ultra pure water and air were obtained. A three dimensional incompressible numerical model was used to investigate the natural convection, and the prediction with this model was in reasonable accordance with the experimental results. With the decrease of micro-wire diameter, the heat transfer coefficient of natural convection on the surface of micro-wire becomes larger, while the Nu number of natural convection decreases in water and air. Besides, the change rate of Nu number in water decreases apparently with the increase of heat flux and the decrease of wire diameter, which is larger than that in air. The thickness of boundary layer on the wall of micro-wire becomes thinner with the decrease of diameter in both water and air, but the ratio of boundary layer thickness in water to the diameter increases. However, there is almost no change of this ratio for natural convection in air. As a result, the proportion of conduction in total heat transfer of natural convection in water increases, while the convective heat transfer decreases. The velocity distribution, temperature field and the boundary layer in the natural convection were compared with those of tube with conventional dimension. It was found that the boundary layer around the micro-wire is an oval-shaped film on the surface, which was different from that around the conventional tube. This apparently reduces the convection strength in the natural convection, thus the heat transfer presents a conduction characteristic.     

Natural convection heat transfer from a horizontal ice cylinder

Heat transfer characteristics passing through the maximum density point around a horizontal ice cylinder immersed in water was studied both theoretically and experimentally. For the sake of a precise comparison, the stagnation point Nusselt number was measured and results then compared with those of the numerical computations that were obtained by solving the full Navier-Stokes equations. A fairly good agreement was seen between the theory and the experiment.At about 6°C of water temperature where the stagnation Nusselt number takes its minimum value, the instability of the flow was observed. It was found that two different computer solutions exist, which shows unstable aspects corresponding to the experimental result.     

Free convection heat transfer in an open system of vertical rods

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 88–95, September–October, 1990.