Analytical and numerical study of natural convection heat transfer in an inclined composite enclosure

Laminar natural convection heat transfer in inclined fluid layers divided by a partition with finite thickness and conductivity is studied analytically and numerically. The governing equations for the fluid layers are solved analytically in the limit of a thin layered system with constant flux boundary conditions. The study covers of the range of Ra from 10³ to 10⁷, from 0° to 180° and the thermal conductivity ratio of partition to fluid ratioK from 10^–2 to 10⁶. The Prandtl number was 0.72 (for air). Results are obtained in terms of an overall Nusselt number as a function of Rayleigh number, angle of inclination of the system, mid layer thickness, and mid layer thermal conductivity. The critical Rayleigh number for the onset of convection in a bottom-heated horizontal system is predicted. The results are compared with the numerical results obtained by solving the complete system of governing equations, using SIMPLER method, as well as with the limiting cases in the literature.     

Buoyancy-induced flow and convective heat transfer in an inclined arc-shape enclosure

Numerical and experimental investigations have been conducted to study the flow and heat transfer characteristics for the buoyancy-induced flow inside an inclined arc-shape enclosure. Mathematical model in form of a stream function-vorticity formulation representing the laws of conservation in mass, momentum, and energy is expressed in a curvilinear coordinate frame and solved by a finite-volume discretization method. Heat transfer and flow pattern are predicted at various Grashof numbers and inclination angles. Meanwhile, an experimental system is developed and a flow-visualization technique using smoke is employed to observe the flow pattern. Results show that only when the Grashof number is higher than 10⁵, the increase in natural convection heat transfer becomes appreciable. Both the strength and the pattern of the buoyancy-induced vortex are found to be greatly dependent on the inclination angle. The range of the Grashof number considered in this study is up to 10⁷ and the inclination angle is varied from 0 to π.     

Transient heat flux measurement of natural convection in an inclined enclosure with time-periodically-varying wall temperature

The transient natural convection in an inclined enclosure filled with water is studied experimentally for the time-periodically-varying wall temperature on one side wall and constant average temperature on the opposing side wall. This system has no temperature difference between the opposing two side walls in time-averaged sense. The temperatures of two opposing walls and the heat flux across the enclosure are measured by a heat flux meter. Based on the experimental results, the effects of time-periodically-varying wall temperature and inclined angles of the enclosure on heat transfer characteristics are studied. The experimental results show that, with the upper wall temperature oscillating, the heat flux across the enclosure is also periodically varied with time, and the net heat flux is from the lower wall to the upper wall. Numerical computations are also conducted and numerical results are qualitatively assured by the experimental measurements.     

Combined natural convection heat and mass transfer in an enclosure having finite thickness walls

Unsteady three-dimensional conjugate heat and mass transfer in an enclosure having finite thickness heat-conducting walls has been analyzed numerically. The governing unsteady, three-dimensional flow, energy and contaminant transport equations for the gas cavity and unsteady heat conduction equation for solid walls, written in dimensionless terms of the vector potential functions, the vorticity vector, the temperature and the concentration, have been solved using an iterative implicit finite-difference method. Main attention was paid to the effects of the Rayleigh number, buoyancy ratio and the dimensionless time on the flow structure and heat and mass transfer regimes. It should be noted that the dominant cause of the oscillations in the dimensionless time dependences of the average Nusselt number on the heat source surface and the average Sherwood number on the contaminant source surface at Ra>5?10⁵ is the mutual influence of the analyzed object geometry and the thermo-diffusivity impact on the flow. The change in the buoyancy ratio can lead to the essential modifications of the flow, temperature and concentration fields owing to the significant influence of the concentration gradient.     

Thermal convection in a cylindrical enclosure with intense internal heat generation in the presence of a longitudinal magnetic field

Steady convection in a gaseous medium with intense heat generation is considered without making use of the Boussinesq approximation. The effect on convection of the thermal boundary conditions at the walls of the enclosure is investigated, together with the influence of a longitudinal magnetic field which is effective when the medium is strongly heated and becomes electrically conducting.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 11–18, May–June, 1992.The authors are grateful to the participants in the G. A. Lyubimov seminar for discussing their work.     

Natural convection due to heat and mass transfer in a composite system

A numerical study is performed to analyse heat and mass transfer phenomena due to natural convection in a composite cavity containing a fluid layer overlying a porous layer saturated with the same fluid. The flow in the porous region is modelled using Brinkman–Forchheimer-extended Darcy model that includes both the effect of macroscopic shear (Brinkman effect) and flow inertia (Forchheimer effect). The vertical walls of the two-dimensional enclosure are isothermal whilst the horizontal walls are adiabatic. The two regions are coupled by equating the velocity and stress components at the interface. The resulting coupled equations in non-dimensional form are solved by an alternating direction implicit method by transforming them into parabolic form by the addition of false transient terms. The numerical results show that the amount of fluid penetration into the porous layer depends strongly upon the Darcy, thermal and solutal Rayleigh numbers. Average Nusselt number decreases while average Sherwood number increases with an increase of the Lewis number. The transfer of heat and mass on the heated wall near the interface depends strongly on the Darcy number. Received on 11 May 1998     

Unsteady flow with heat and mass transfer due to impingement of slot jet on an inclined plate

The unsteady laminar incompressible boundary layer flow due to a two-dimensional slot jet on a flat plate at an angle of attack has been studied. The unsteadiness in the flow field is due to the free stream velocity distribution or wall temperature (concentration) which varies with time. The governing partial differential equations in primitive variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The effect of the variation of the free stream velocity distribution with time is found to be more pronounced on the skin friction than on the heat or mass transfer. The Prandtl number and the variation of the wall temperature with time strongly affect the heat transfer. Similarly, the Schmidt number and the variation of the concentration at the wall with time strongly affect the mass transfer. Beyond a certain critical value of the viscous dissipation parameter, the plate gets heated instead of being cooled.     

Experimental investigation of natural convection heat transfer in horizontal and inclined annular fluid layers

In the present study, an experimental investigation of heat transfer and fluid flow characteristics of buoyancy-driven flow in horizontal and inclined annuli bounded by concentric tubes has been carried out. The annulus inner surface is maintained at high temperature by applying heat flux to the inner tube while the annulus outer surface is maintained at low temperature by circulating cooling water at high mass flow rate around the outer tube. The experiments were carried out at a wide range of Rayleigh number (5 × 10⁴ < Ra < 5 × 10⁵) for different annulus gap widths (L/D _o = 0.23, 0.3, and 0.37) and different inclination of the annulus (α = 0°, 30° and 60°). The results showed that: (1) increasing the annulus gap width strongly increases the heat transfer rate, (2) the heat transfer rate slightly decreases with increasing the inclination of the annulus from the horizontal, and (3) increasing Ra increases the heat transfer rate for any L/D _o and at any inclination. Correlations of the heat transfer enhancement due to buoyancy driven flow in an annulus has been developed in terms of Ra, L/D _o and α. The prediction of the correlation has been compared with the present and previous data and fair agreement was found.     

Experimental investigation of mixed convection heat transfer in a horizontal and inclined rectangular channel

 Mixed convection heat transfer in rectangular channels has been investigated experimentally under various operating conditions. The lower surface of the channel is subjected to a uniform heat flux, sidewalls are insulated and adiabatic, and the upper surface is exposed to the surrounding fluid. Experiments were conducted for Pr=0.7, aspect ratios AR=5 and 10, inclination angles 0° ≤ θ ≤ 30°, Reynolds numbers 50 ≤ Re ≤ 1000, and modified Grashof numbers Gr*=7.0 × 10⁵ to 4.0 × 10⁷. From the parametric study, local Nusselt number distributions were obtained and effects of channel inclination, surface heat flux and Reynolds number on the onset of instability were investigated. Results related to the buoyancy affected secondary flow and the onset of instability have been discussed. Some of the results obtained from the experimental measurements are also compared with the literature, and a good agreement was observed. The onset of instability was found to move upstream for increasing Grashof number and increasing aspect ratio. On the other hand, onset of instability was delayed for increasing Reynolds number and increasing inclination angle. Received on 19 March 2001 / Published online: 29 November 2001     

A numerical study on heat transfer of a ferrofluid flow in a square cavity under simultaneous gravitational and magnetic convection

Theoretical and Computational Fluid Dynamics - Thermomagnetic convection is based on the use of external magnetic fields to better control heat transfer fluxes in ferrofluids, finding important...     

Similarity solutions for hydromagnetic simultaneous heat and mass transfer by natural convection from an inclined plate with internal heat generation or absorption

 The problem of coupled heat and mass transfer by natural convection from a semi-infinite inclined flat plate in the presence of an external magnetic field and internal heat generation or absorption effects is formulated. The plate surface has a power-law variation of both wall temperature and concentration and is permeable to allow for possible fluid wall suction or blowing. The resulting governing equations are transformed using a similarity transformation and then solved numerically by an implicit, iterative, finite-difference scheme. Comparisons with previously published work are performed and good agreement is obtained. A parametric study of all involved parameters is conducted and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter, average Nusselt number, and the average Sherwood number is illustrated graphically to show typical trends of the solutions. Received on 26 October 1998     

MHD Couette two-fluid flow and heat transfer in presence of uniform inclined magnetic field

The MHD Couette flow of two immiscible fluids in a parallel plate channel in the presence of an applied electric and inclined magnetic field is investigated in the paper. One of the fluids is assumed to be electrically conducting, while the other fluid and the channel plates are assumed to be electrically insulating. Separate solutions with appropriate boundary conditions for each fluid are obtained and these solutions are matched at the interface using suitable matching conditions. The partial differential equations governing the flow and heat transfer are transformed to ordinary differential equations and closed-form solutions are obtained in both fluid regions of the channel. The results for various values of the Hartmann number, the angle of magnetic field inclination, the loading parameter and the ratio of the heights of the fluids are presented graphically to show their effect on the flow and heat transfer characteristics.     

Natural convection heat transfer from a long heated vertical cylinder to an adjacent air gap of concentric and eccentric conditions

In this work, the natural convection heat transfer from a long vertical electrically heated cylinder to an adjacent air gap is experimentally studied. The aspect and diameter ratios of the cylinder are 55.56 and 6.33, respectively. The experimental measurements were obtained for a concentric condition and six eccentricities from 0.1 to 0.92 at five different heat fluxes. The surface temperature of the heated rod is measured at different heights, and the Nusselt number is calculated at the temperature measurement locations. A correlation is suggested to determine the Nusselt number based on the variation of the eccentric ratio values. The experimental results show a good agreement with other studies.     

Influence of lateral mass flux on mixed convection heat and mass transfer over inclined surfaces in porous media

 The effect of lateral mass flux on mixed convection heat and mass transfer in a saturated porous medium adjacent to an inclined permeable surface is analyzed. A similarity solution is obtained when surface temperature and concentration, free stream velocity and injection/suction velocity of fluid are prescribed as power functions of distance from the leading edge. The cases when the flow and buoyancy forces are in the same and opposite directions are discussed both for aiding and opposing buoyancy effects. The governing parameters are the mixed convection parameter Gr, the Lewis number Le, the buoyancy ratio N, the lateral mass flux parameter f _w, representing the effects of injection or withdrawal of fluid at the wall, and λ which specifies three cases of the inclined plate. The interactive effect of these parameters on heat and mass transfer rates are presented. It is observed that the diffusion ratio (Le) has a more pronounced effect on concentration field than on flow and temperature fields. It is found that the rates of heat and mass transfer increase with suction and decrease with injection of the fluid. Received on 31 August 2000 / Published online: 29 November 2001     

Onset of convection due to heating from above and heat release at an interface

A specific non-Rayleigh mechanism of convective instability in systems with an interface, developing in the presence of heating from above (anticonvection), was identified in [1, 2]. The onset of this type of instability requires a sharp difference in the physical parameters of the fluids. The effect of heat release and heat absorption on the onset of convection in systems for which this instability mechanism is possible is examined. In the presence of surface heat sources the directions of the temperature gradients in the two media may be different. The interaction of Rayleigh and non-Rayleigh types of instability is investigated. It is shown that for the water-mercury system on a certain interval of the parameters the oscillatory mode of instability is the most dangerous.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 16–20, May–June, 1990.     

Determination of surface heat transfer coefficients from measured temperature data for spherical and cylindrical bodies during cooling

In this paper which is a combination of the methodological and experimental aspects, models were developed for determining surface heat transfer coefficients for spherical and cylindrical bodies from their center temperature measurements during forced-cooling. Experiments involved the cooling of the individual spherical and cylindrical products as test samples in the air flow. The cooling parameters in terms of the cooling coefficients and lag factors were also determined to use in the present models. The results show that the surface heat transfer coefficients of the individual spherical and cylindrical products increased with an increase in the flow velocities from 1 to 2 m/s. It can be concluded that the present models have the capabilities of determining the surface heat transfer coefficients for spherical and cylindrical bodies with a single transient experiment.     

Experimental and numerical study on laminar natural convection in a cavity heated from bottom due to an inclined fin

Natural convection heat transfer in an inclined fin attached square enclosure is studied both experimentally and numerically. Bottom wall of enclosure has higher temperature than that of top wall while vertical walls are adiabatic. Inclined fin has also adiabatic boundary conditions. Numerical solutions have been done by writing a computer code in Fortran platform and results are compared with Fluent commercial code and experimental method. Governing parameters are Rayleigh numbers (8.10⁵ ≤ Ra ≤ 4 × 10⁶) and inclination angle (30° ≤ and ≤ 120°). The temperature measurements are done by using thermocouples distributed uniformly at the wall of the enclosure. Remarkably good agreement is obtained between the predicted results and experimental data. A correlation is also developed including all effective parameters on heat transfer and fluid flow. It was observed that heat transfer can be controlled by attaching an inclined fin onto wall.     

Experimental study of natural convection heat transfer between an outer horizontal cylindrical envelope and an inner concentric heated square cylinder with two slots

This paper presents the results of an experimental study of natural convection heat transfer between a horizontal cylindrical envelope and an internal concentric heated square cylinder with two slots. The internal cylinder was a hollow one with horizontal slots on its top and bottom surfaces. The ratio of slot widthS to the side heightH was 0.0612 and 0.3878. The ratio of the envelope inner diameterD _o to the side heightH was 2.653. Air was used as the working fluid. The range of Ray-leigh number was 1.77×10²8.72×10⁶ forS/H=0.0612 and 1.32×10²6.25×10⁶ forS/H=0.3878. The results show that there are three different heat transfer regimes in different Ray-leigh number regions, i.e. pure conduction regime, transition regime and convection regime. The average heat transfer results were correlated into two empirical equations. Comparison was made with the non-slotted case. It is found that slots of the internal cylinder can significantly enhance the heat transfer.

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Experimentelle Untersuchung des Wärmeübergangs bei natürlicher Konvektion zwischen einer horizontalen zylindrischen Außenhülle und einem konzentrischen, beheizten, quadratischen Prisma mit zwei Schlitzen

Zusammenfassung In der Arbeit werden die Ergebnisse einer experimentellen Untersuchung des Wärmeübergangs bei natürlicher Konvektion zwischen einer horizontalen zylindrischen Außenhülle und einem beheizten quadratischen Prisma mit zwei Schlitzen vorgestellt. Das Prisma selbst ist hohl und weist in der oberen und unteren Begrenzungsfläche je einen horizontalen Längsschlitz auf. Das Verhältnis von SchlitzweiteS zu SeitenhöheH beträgt 0,0612 und 0,3878, das des HülleninnendurchmessersD _o zur SeitenhöheH beträgt 2,653. Als Arbeitsmedium diente Luft. Die Rayleigh-Zahlen variierten zwischen 1,7·10² und 8,72·10⁶ fürS/H=0,0612 und zwischen 1,32·10² und 6,25·10⁶ fürS/H=0,3878. Die Ergebnisse belegen die Existenz dreier unterschiedlicher Wärmeübergangsregime in den verschiedenen Rayleigh-Zahl-Bereichen, und zwar reiner Leitungsbereich, Übergangsgebiet und Konvektionsbereich. Die Ergebnisse für den Wärmeübergang werden im Vergleich mit jenen für ein Prisma ohne Schlitze durch zwei Korrelationbeziehungen dargestellt. Es zeigt sich, daß durch Anbringung von Schlitzen am Innenprisma der Wärmeübergang wesentlich verstärkt werden kann.

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Nomenclature C _p specific heat at constant pressure, J/(kg·K) - D _i diameter of the related circular cylinder whose circumferential area is equal to that of the unslotted square cylinder, m - D _o internal diameter of the outer circular envelope, m - F _i surface area of the inner two slot cylinder, m - g gravitational acceleration, m/s² - H distance between the opposite sides of the square cylinder with two slots, m - K _eq dimensionless equivalent thermal conductivity - L axial length of the test section, m - m ratio of the area of the unslotted square cylinder surface to that of the slotted square cylinder - P pressure in the enclosure, Pa - Q total power input to the enclosure, W - Q _cond radial heat conduction, W - Q _conv convective heat transfer, W - Q _r radiation heat transfer, W - Q _los end heat dissipation, W - R air gas constant, J/(kg·K) - Ra Rayleigh number - S slot width, m - T _i wall temperature of the inner cylinder, K - T _o wall temperature of the outer envelope, K - T _m mean temperature, K - T temperature difference=T _i –T _o , K - W maximum gap width of the test annuli=(D _o –H)/2 for the square case, m Greek symbols ₀ black body radiation constant, W/(m²·K⁴) - _s equation system emissivity - air thermal conductivity, W/(m·K) - _eq equivalent thermal conductivity, W/(m·K) - air dynamic viscosity, kg/(m·s) This work was supported by the National Natural Science Foundation of China.     

The deviation of heat transfer calculation for laminar free convection of gas due to ignoring the variable thermophysical properties

According to the similarity transformation method described in [1–3], the ordinary differential governing equations of laminar free convection of gases with Boussinesq's approximation are set up. The equations are corresponding to the caseT _w /T 1 and _{c p} 0 as discussed in [2, 3]. The deviations of the predicted local heat transfer coefficients and local Nusselt numbers due to ignoring the variable thermophysical properties are calculated out theT _w /T T for several gases, and the necessity of the treatment for considering variable thermophysical properties is pointed out.

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Die Abweichungen der Berechnungsergebnisse für die laminare freie Konvektionsströmung von Gasen infolge Vernachlässigung der Veränderlichkeit ihrer Stoffwerte

Zusammenfassung Entsprechend der in [1–3] beschriebenen Methode der Ähnlichkeitstransformation werden die gewöhnlichen Differential — Grundgleichungen der laminaren freien Konvektion von Gasen unter Verwendung der Boussinesq — Approximation aufgestellt. Diese Gleichungen entsprechen, wie in [2, 3] beschrieben, dem FallT _w /T 1 undn _{c p} 0. Die Abweichungen der berechneten örtlichen Nusselt-Zahlen infolge Vernachlässigung der Veränderlichkeit der thermophysikalischen Stoffwerte werden in Abhängigkeit vonT _w /T für mehrere Gase ermittelt, wobei sich herausstellt, daß die Veränderlichkeit der Stoffwerte bei der Berechnung zu berücksichtigen ist.