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.     

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.     

Mass and heat transfer by natural convection in a vertical cavity

This paper reports a fundamental study of laminar natural convection in a rectangular enclosure with heat and mass transfer from the side, when the bouyancy effect is due to density variations caused by either temperature or concentration variations. In the first part of the study scale analysis is used to determine the scales of the flow, temperature and concentration fields in boundary layer flow for all values of Prandtl and Lewis numbers. In particular, scale analysis shows that in the extreme case where the flow is driven by bouyancy due to temperature variations, the ratio of mass transfer rate divided by heat transfer rate scales as $\text{Le}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ only if (Pr > 1, Le < 1) or (Pr < 1, Sc < 1), and as $\text{Le}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ if (Pr > 1, Le > 1) or (Pr < 1, Sc > 1). In the second part of the study, the boundary layer scales derived in the first part are used to determine the heat and mass transport characteristics of a vertical slot filled with fluid. Criteria for the existence of distinct thermal and concentration boundary layers in the slot are determined. Numerical solutions for the flow and concentration fields in a slot without distinct thermal boundary layers are reported. These solutions support further the method of scale analysis employed in the first part of the study     

Optimum fin spacing of rectangular fins on a vertical base in free convection heat transfer

The steady-state natural convection heat transfer from aluminum vertical rectangular fins extending perpendicularly from vertical rectangular base was investigated experimentally. Thirty different fin configurations were tested. Experiments were performed for fin lengths of 250 and 340 mm. Fin thickness was kept fixed at 3 mm. Fin height and fin spacing were varied from 5 to 25 mm and 5.75 to 85.5 mm, respectively. Five heat inputs ranging from 25 to 125 W were supplied for all fin configurations, hence; the base-to-ambient temperature differences were measured in order to evaluate the heat transfer rates from fin arrays. The results of experiments have shown that the convective heat transfer rate from fin arrays depends on geometric parameters and base-to-ambient temperature difference. The separate roles of fin height, fin spacing and base-to-ambient temperature difference were investigated. It was found that, for a given base-to-ambient temperature difference, the convective heat transfer rate from fin arrays takes on a maximum value as a function of fin spacing and fin height and an optimum fin spacing value which maximizes the convective heat transfer rate from the fin array is available for every fin height. These measurements were to extend data obtained earlier from aluminum fin-arrays using the same experimental system and method (Yüncü and Güvenç in Heat Mass Transfer 37:409–416, 2001). Data collated from earlier and present work cover the range of fin spacing from 4.5 to 85.5 mm. The fin length range was from 100 to 340 mm, the fin height from 5 to 25 mm and the number of fins per array 3 to 34. The range of base-to-ambient temperature difference was quite extensive, from 30 to 150 K. These results indicate that the optimum fin spacing is between 6.1 and 11.9 mm, for the fin arrays employed in the earlier and present work. A scale analysis is performed in order to estimate the order-of-magnitude of optimum fin spacing at a given fin length and base-to-ambient temperature difference. From the scale analysis, correlations to evaluate the optimum fin spacing value and the corresponding maximum convective heat transfer rate at a given fin length and base-to-ambient temperature difference were obtained.     

Natural convection and conduction heat transfer in open shallow cavities with bounding walls

A numerical study has been carried out on inclined open shallow cavities, which are formed by a wall and horizontal fins. Constant heat flux is applied on the surface of the wall inside the cavity while its other surface was kept isothermal. The wall and the fins are conductive. Conjugate heat transfer by natural convection and conduction is studied by numerically solving equations of mass, momentum and energy. Streamlines and isotherms are produced, heat and mass transfer is calculated. A parametric study is carried out using following parameters: Rayleigh number from 10⁶ to 10¹², conductivity ratio from 1 to 60, open cavity aspect ratio from 1 to 0.125, dimensionless end wall thickness from 0.05 to 0.20, horizontal walls from 0.01 to 0.15 and inclination of the end wall from 90° to 45°. It is found that the volume flow rate and Nusselt number are a decreasing function of the cavity aspect ratio, horizontal fin thickness and conductivity ratio. They are an increasing function of end wall thickness and inclination angle, except in the latter case optima exist at high Rayleigh numbers.     

Experimental research on heat transfer of natural convection in vertical rectangular channels with large aspect ratio

This work presents the experimental research on the steady laminar natural convection heat transfer of air in three vertical thin rectangular channels with different gap clearance. The much higher ratio of width to gap clearance (60–24) and the ratio of length to gap clearance (800–320) make the rectangular channels similar with the coolant flow passage in plate type fuel reactors. The vertical rectangular channels were composed of two stainless steal plates and were heated by electrical heating rods. The wall temperatures were detected with the K-type thermocouples which were inserted into the blind holes drilled in the steal plates. Also the air temperatures at the inlet and outlet of the channel were detected. The wall heat fluxes added to the air flow were calculated by the Fourier heat conduction law. The heat transfer characteristics were analyzed, and the average Nusselt numbers in all the three channels could be well correlated with the Rayleigh number or the modified Rayleigh number in a uniform correlation. Furthermore, the maximum wall temperatures were investigated, which is a key parameter for the fuel’s integrity during some accidents. It was found that even the wall heat flux was up to 1500 W/m², the maximum wall temperature was lower than 350 °C. All this work is valuable for the plate type reactor’s design and safety analysis.     

Self-induced thermocapillary convection in film boiling heat transfer from a vertical surface in saturated conditions and viscous regimen

Self-induced thermocapillary convection and its significance with regard to film boiling heat transfer from a vertical saturated surface is discussed. Utilizing a simplified geometrical model, an analytical expression (multiplier factor) for the vapor film thickness and heat transfer coefficient corrected by thermocapillary phenomena was derived. The above equation is a new theoretical viewpoint for the enhancement in the heat transfer coefficient observed in the experimental data in the viscous regime and agree qualitatively with available experimental measurements made on R113 coolant.     

环形腔内双层薄液层热毛细对流的渐近解

为了解水平温度梯度作用下环形腔内双层薄液层热毛细对流的基本特性,采用渐近线方法获得了热毛细对流的近似解. 环形腔外壁被加热,内壁被冷却,上、下壁面绝热. 结果表明,当环形腔宽度与内半径比趋于零时,环形腔退化为矩形腔,所得到的主流区速度场和温度场的表达式演化为Nepomnyashchy 等得到的矩形腔内的结果;与数值模拟结果的比较发现,在主流区渐近解与数值解吻合较好.      

Natural convection in an inclined rectangular porous cavity with uniform heat flux from the side

The problem of natural convection in an inclined rectangular porous layer enclosure is studied numerically. The enclosure is heated from one side and cooled from the other by a constant heat flux while the two other walls are insulated. The effect of aspect ratio, inclination angle and Rayleigh number on heat transfer is studied. It is found that the enclosure orientation has a considerable effect on the heat transfer. The negative orientation sharply inhibits the convection and consequently the heat transfer and a positive orientation maximizes the energy transfer. The maximum temperature within the porous medium can be considerably higher than that induced by pure conduction when the cavity is negatively oriented. The peak of the average Nusselt number depends on the Rayleigh number and the aspect ratio. The heat transfer between the two thermally active boundaries is sensitive to the effect of aspect ratio. For an enclosure at high or low aspect ratio, the convection is considerably decreased and the heat transfer depends mainly on conduction.     

Effect of free convection on thermodiffusion in a liquid mixture filling an inclined rectangular cavity

The convective motion which develops in an inclined cavity upon heating from above determines to a significant degree the form of the concentration field produced by thermodiffusion. The interaction of convective and thermodiffusion fluxes at small thermal Grashof numbers Gr causes the appearance of longitudinal jumps in concentration. Increase in temperature difference intensifies convection and encourages reduction in concentration gradients. The dominant role of convection for fixed Gr is determined by the angle of inclination of the liquid layer [1, 2]. A significant feature of liquid solutions is their low diffusion coefficient and thus high Schmidt number. This fact does not permit use of results obtained for gas mixtures, and greatly complicates numerical simulations. In contrast to [2], the present study will investigate thermodiffusion separation in a cavity with impermeable boundaries.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 73–76, September–October, 1986.In conclusion, the authors thank G. Z. Gershun for evaluation of the results and helpful remarks.     

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.     

Effect of viscous dissipation on mixed convection heat transfer in a vertical tube with uniform wall heat flux

 The laminar and parallel flow of a Newtonian fluid in a vertical cylindrical duct with circular cross section has been analysed. Both the viscous dissipation effect and the buoyancy effect have been taken into account. The momentum balance equation and the energy balance equation have been solved by means of a perturbation method, in the case of a uniform heat flux prescribed at the wall of the duct. The velocity distribution, the temperature distribution, the Nusselt number and the Fanning friction factor have been evaluated analytically. Moreover, the velocity and temperature of the fluid have been compared with those obtained in two special cases: forced convection with viscous dissipation (i.e. negligible buoyancy effect); mixed convection with negligible effects of viscous dissipation. Received on 26 June 2000     

Analytical investigation of Bénard-Marangoni convection heat transfer in a shallow cavity filled with two immiscible fluids

The mechanism of natural and Marangoni convection in a system with two stratified fluid layers without mass transfer at the interface is investigated. The basis of the analytical solution is an assumption of parallel flow over a large portion of the system. The two cases of heat fluxes through horizontal or vertical opposite walls are considered. It is demonstrated that four different patterns of convection can be observed in the present system. The zone of occurrence of these flow patterns are specified in terms of non-dimensional parameters. Velocity and temperature distributions, stream function and Nusselt number are presented over a wide range of the governing parameters. The results obtained are explained in terms of the basic physical mechanisms that govern these flows showing many interesting aspects of the complex interaction between the buoyant and surface tension mechanisms.     

Simulation studies on multi-mode heat transfer from an open cavity with a flush-mounted discrete heat source

Prominent results of a simulation study on conjugate convection with surface radiation from an open cavity with a traversable flush mounted discrete heat source in the left wall are presented in this paper. The open cavity is considered to be of fixed height but with varying spacing between the legs. The position of the heat source is varied along the left leg of the cavity. The governing equations for temperature distribution along the cavity are obtained by making energy balance between heat generated, conducted, convected and radiated. Radiation terms are tackled using radiosity-irradiation formulation, while the view factors, therein, are evaluated using the crossed-string method of Hottel. The resulting non-linear partial differential equations are converted into algebraic form using finite difference formulation and are subsequently solved by Gauss–Seidel iterative technique. An optimum grid system comprising 111 grids along the legs of the cavity, with 30 grids in the heat source and 31 grids across the cavity has been used. The effects of various parameters, such as surface emissivity, convection heat transfer coefficient, aspect ratio and thermal conductivity on the important results, including local temperature distribution along the cavity, peak temperature in the left and right legs of the cavity and relative contributions of convection and radiation to heat dissipation in the cavity, are studied in great detail.     

Augmentation of heat transfer in a bubble agitated vertical rectangular channel

This paper presents the results of an experimental study of convective heat transfer between three parallel vertical plates symmetrically spaced with and without bubble agitation to ascertain the degree of augmentation of the heat transfer coefficients due to agitation. The centre plate was electrically heated, while the other side plates were water-cooled forming two successive parallel vertical rectangular channels of dimensions 20 cm × 3.5 cm × 35 cm (length W, gap L, height H) each. At the bottom of the hot and cold plates air spargers were fitted. Water/ethylene glycol (100%) was used to fill the channels. The superficial gas velocity ranged from 0.0016 to 0.01 m/s. Top, bottom and sides of the channels were open to the water/ethylene glycol in the chamber which is the novel aspect of this study. Experimental data have been correlated as under: Natural convective heat transfer: Nu = 0.60 Gr ^0.29, r = 0.96, σ = 0.186, 1.17 E6 < Gr < 1.48 E7; Bubble agitated heat transfer: St = 0.11(ReFrPr ²)^−0.23, r = 0.82, σ = 0.002, 1.20 E−2 < (ReFrPr ²) < 1.36 E2.     

Vibrational thermal convection in a rectangular cavity

Vibrational thermal convection in a rectangular cavity under conditions of weightlessness is studied. Some equilibrium configurations were obtained in earlier papers of two of the authors [1, 2] and their linear stability investigated. In the present paper, a numerical investigation is made of the developed vibrational convection which arises under conditions when equilibrium is impossible. The structure of the average vibrational-convective flows and the characteristics of the heat transfer are determined. The change of regimes and the connection with the stability problem are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–99, July–August, 1982.     

Effect of free surface heat transfer on thermocapillary flow in double-layer fluid structures

In this paper, we examine the effect of heat transfer at the free encapsulant-air surface on thermocapillary flow in a rectangular melt-encapsulant double-layer fluid structure. We show that increased heat transfer to the double-layer system through the free surface weakens thermocapillary convection in the encapsulant phase and enhances the convection in the melt phase.     

黏弹性纳米流体在垂直板上的自然对流与传热分析

基于分数阶Maxwell模型和分数阶Fourier定律构建黏弹性纳米流体在垂直板上的非定常二维边界层自然对流与传热控制方程,利用有限差分和L1算法获得数值稳定解,对不同物理参数下的速度、温度、平均表面摩擦系数和平均Nusselt数的变化趋势进行图形化分析。结果显示,速度和温度边界层均表现出短暂记忆和延迟特性;速度分数导数参数削弱了自然对流,而速度松弛时间的影响却相反;温度分数导数参数削弱了自然对流和热传导,而温度松弛时间的影响却相反。     

Bifurcation of thermocapillary convection in a shallow annular pool of silicon melt

In order to understand the nature of surface patterns on silicon melts in industrial Czochralski furnaces, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary convections in thin silicon melt pools in an annular container. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. The results show that the flow is steady and axisymmetric at small temperature difference in the radial direction. When the temperature difference exceeds a certain threshold value, hydrothermal waves appear and bifurcation occurs. In this case, the flow is unsteady and there are two possible groups of hydrothermal waves with different number of waves, which are characterized by spoke patterns traveling in the clockwise and counter-clockwise directions. Details of the flow and temperature disturbances are discussed and number of waves and traveling velocity of the hydrothermal wave are determined. The project supported by the National Natural Science Foundation of China (50476042) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education of China. The English text was polished by Yunming Chen.