Laminar natural convection between partially heated vertical parallel plates

A numerical analysis has been performed to examine the characteristics of laminar natural convection in vertical channel with unheated entry and unheated exit. The heated section is subjected to uniform wall temperature (UWT) or uniform heat flux (UHF). Theoretical results for average Nusselt number and induced volume flow rateQ were derived under fully developed condition. Particular attention is paid to investigating the effects of the partially heated section on the induced volume flow rate and Nusselt number for various conditions. Results show that for UWT the induced volume flow rateQ increases with decreasing unheated entry length or increasing total length of heated section and unheated exit. For a fixed unheated entry length, the channel with a longer heated section length causes a greaterQ. Additionally, for both UWT and UHF, the average Nusselt number under fully-developed condition increases with increasing value ofE ₁/E ₂.

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Laminare Naturkonvektion zwischen teilbeheizten senkrechten Parallelplatten

Zusammenfassung Die numerische Untersuchung soll das Verhalten einer laminaren Strömung bei natürlicher Konvektion in einem senkrechten Kanal mit unbeheiztem Ein- und Austritt klären. Der beheizte Abschnitt wird entweder mit gleichförmiger Wandtemperatur (UWT) oder gleichförmigem Wärmefluß (UHF) beaufschlagt. Bezüglich voll ausgebildeter Strömung ließen sich theoretische Ergebnisse für die mittlere Nußelt-Zahl und den induzierten VolumenstromQ gewinnen. Besonderes Interesse galt der Untersuchung des Einflusses des teilbeaufschlagten Abschnittes auf Volumenstrom und Nußelt-Zahl unter verschiedenen Nebenbedingungen. Die Ergebnisse zeigen, daß im UWT-Fall der induzierte VolumenstromQ mit abnehmender unbeheizter Einlauflänge oder zunehmender Gesamtlänge des Heizabschnittes anwächst. Bei fester unbeheizter Einlauflänge erzeugt der Kanal mit längerem Heizabschnitt einen höheren StromQ.

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Nomenclature b half channel width - E ₁ ratio of unheated exit length to channel length,l ₁/l - E ₂ ratio of heated section length to channel length,l ₂/l - E ₃ ratio of unheated entry length to channel length,l ₃/l - Gr _L Grashof number, Eq. (4) - g gravitational acceleration - k thermal conductivity - l channel length - l ₁ unheated exit length - l ₂ heated section length - l ₃ unheated entry length - L dimensionless channel length, Eq. (4) - L ₁ dimensionless unheated exit length - L ₂ dimensionless heated section length - L ₃ dimensionless unheated entry length - average Nusselt number - Nu _x local Nusselt number - p pressure defect - P dimensionless pressure defect, Eq. (4) - Pr Prandtl number,/ - q _w wall heat flux of heated section - Q dimensionless induced volume flow rate, Eq. (8) - Ra _E Rayleigh number based on the heated section length,Ra _L/E₂ - Ra _L Rayleigh number based on the full channel length,Gr _LPr - T temperature - T ₀ inlet temperature - T _w wall temperature - u, v velocity components in thex andy directions, respectively - U, V dimensionless velocity components in thex andy directions, respectively, Eq. (4) - u ₀,U ₀ dimensional and dimensionless inlet velocity, respectively - x, y coordinates in thex andy directions, respectively - X, Y dimensionless coordinates in thex andy directions, respectively, Eq. (4) - X ratio of longitudinal distance from the entrance of heated section to the heated section length,X=[X–(L–L ₁–L ₂)]/L ₂ Greek symbols thermal diffusivity - thermal expansion coefficient - kinematic viscosity - dimensionless temperature, Eq. (4) - ₀ fluid density at ambient temperature     

Natural convection in inclined rectangular enclosures with perfectly conducting fins attached on the heated wall

The natural convection heat transfer in inclined rectangular enclosures with perfectly conducting fins attached to the heated wall is numerically studied. The parameters governing this problem are the Rayleigh number (10²≤Ra≤2×10⁵), the aspect ratio of the enclosures (2.5≤A=H′/L′≤∞), the dimensionless lengths of the partitions (0≤B=?′/L′≤1), the aspect ratio of micro-cavities (A≤C=h′/L′≤0.33), the inclination angle (0≤φ≤60^°) and the Prandtl number (Pr=0.72). The results indicate that the heat transfer through the cover is considerably affected by the presence of the fins. At low Rayleigh numbers, the heat transfer regime is dominated by conduction. When B≈0.75 and C≈0.33, the heat transfer through the cold wall decreases considerably. This trend is enhanced when the enclosure is inclined. Useful engineering correlations are derived for practical applications.     

Prediction of CHF in concentric-tube open thermosiphon using artificial neural network and genetic algorithm

In this paper, an artificial neural network (ANN) for predicting critical heat flux (CHF) of concentric-tube open thermosiphon has been trained successfully based on the experimental data from the literature. The dimensionless input parameters of the ANN are density ratio, ρ _l/ρ _v; the ratio of the heated tube length to the inner diameter of the outer tube, L/D _i; the ratio of frictional area, d _i/(D _i + d _o); and the ratio of equivalent heated diameter to characteristic bubble size, D _he/[σ/g(ρ _l−ρ _v)]^0.5, the output is Kutateladze number, Ku. The predicted values of ANN are found to be in reasonable agreement with the actual values from the experiments with a mean relative error (MRE) of 8.46%. New correlations for predicting CHF were also proposed by using genetic algorithm (GA) and succeeded to correlate the existing CHF data with better accuracy than the existing empirical correlations.     

Natural convection in a partitioned enclosure heated from below

Cubic spline collection numerical method has been developed to determine two dimensional natural convection in a partitioned enclosure heated from below. The both sides of impermeable partition are considered to have continuity in heat flux and temperatures. The governing equations are solved with aid of the SADI procedure. Parametric studies of the effects of the partition and Rayleigh number on the fluid flow and temperature fields have been performed. Results show that the location of the partition and Rayleigh number have a significant influence on the flow and heat transfer characteristics.

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Freie Konvektion in einem von unten beheizten, unterteiltem Hohlraum

Zusammenfassung Eine numerische dreidimensionale SplineMethode zur Berechnung der zweidimensionalen Naturkonvektion in einem von unten beheizten, unterteiltem Hohlraum wird vorgestellt. Der Wärmestrom und die Temperatur auf beiden Seiten der undurchlässigen Trennwand werden als konstant betrachtet. Mit Hilfe der SADI-Prozedur werden die beschreibenden Gleichungen gelöst. Über den Einfluß der Unterteilung und der Rayleigh-Zahl auf die Strömung des Fluids und das Temperaturfeld wird eine Parameter-Studie durchgeführt. Die Ergebnisse zeigen, daß die Anordnung der Unterteilung und die Rayleigh-Zahl einen entscheidenden Einfluß auf das Wärmeübertragungsverhalten haben.

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Nomenclature A aspect ratio=L/H - g gravitational acceleration - H enclosure height - H₁ distance between the top wall of enclosure and the partition - H₂ distance between the bottom wall of enclosure and the partition - k thermal conductivity of fluid - L enclosure length - m number of vertical grid lines - n number of horizontal grid lines - Nu Nusselt number - P pressure - Pr Prandtl number - Q heat transfer across enclosure - Ra Rayleigh number based onH - t time - T dimensional temperature - T _H temperature of warm horizontal wall - T _L temperature of cold horizontal wall - T ₀ average temperature=T(_H+T_L)/2 - T temperature difference between the hot and cold wall =T _H–T_L - u, U dimensional and dimensionless horizontal velocity - , V dimensional and dimensionless vertical velocity - x, X dimensional and dimensionless horizontal coordinate - y, Y dimensional and dimensionless vertical coordinate - fluid thermal diffusivity - coefficient of thermal expansion - viscosity - kinematic viscosity=/g9 - density - , dimensional and dimensionless stream function - dimensionless temperature - , dimensional and dimensionless vorticity - dimensionless time     

Conjugate natural convection from an array of discrete heat sources: part 2 — a numerical parametric study

Coupled conduction and natural convection transport within a discretely heated cavity have been investigated numerically. One vertical wall of the cavity is composed of discrete, isoflux heat sources mounted in a substrate of finite thermal conductivity. The opposite vertical wall and the horizontal walls are assumed to be isothermal and adiabatic, respectively. The governing steady-state partial differential equations for the fluid and solid region are solved simultaneously using a control volume formulation, coupled with an additive correction multigrid procedure that increases the convergence rate of the solution. The fluid Prandtl number and heater/fluid thermal conductivity ratio are fixed at 25 and 2350, respectively, corresponding to a dielectric fluid (FC-77) and heaters manufactured from silicon. With increasing modified Rayleigh number (10⁴ < Ra_Lz^* < 109), the cavity flow becomes more boundary layer-like along the vertical walls, and multiple fluid cells develop in the central region. Thermal spreading in the substrate increases with decreasing modified Rayleigh number and with increasing values of the substrate/fluid thermal conductivity ratio (10⁻¹ <- R_s ≤ 10³). For large R_s, the discrete heat sources lose their thermal identity, and the streamlines and isotherms resemble those associated with a differentially heated cavity. Thermal spreading in the substrate also has a significant effect on circulation in the cavity and on maximum surface temperatures.     

Laminar forced convection in a channel with arrays of thermal sources

The problem of laminar forced convection in a two-dimensional channel with arrays of thermal sources is studied numerically. The surfaces of the thermal sources are assumed to be isothermal. In order to promote the accuracy of the result, numerical computation is performed with alternating direction implicit (A.D.I.) method and SIMPLER algorithm. The primary purpose of this paper is to study the fundamental heat transfer phenomena in L.S.I. packages, which are widely used in microelectric equipments recently. The contribution of this study is not only on academic purpose but also on industrial application. Influence of Reynolds number on Nusselt number of thermal sources is discussed. Calculations are made under the conditions of different arrangements of the thermal sources. Flow patterns, pressure and temperature distributions are demonstrated.

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Laminare Zwangskonvektion in einem Kanal mit einem Wärmequellen-Feld

Zusammenfassung Hier wurde das Problem der laminaren Zwangskonvektion in einem zweidimensionalen Kanal mit einem Wärmequellen-Feld numerisch untersucht. Die Oberflächen der Wärmequellen wurden als isotherm vorausgesetzt. Um die Genauigkeit des Ergebnisses zu verbessern, wurden die numerischen Berechnungen mit dem A.D.I.-Verfahren und dem SIMPLER-Algorithmus durchgeführt. Das grundlegende Ziel dieser Arbeit sind die wesentlichen Wärmeübertragungsphänomene in L.S.I.-Paketen zu untersuchen, welche vorwiegend in Geräten der Mikroelektronik verwendet werden. Diese Untersuchung liefert nicht nur einen Beitrag für akademische Zwecke, sondern ist auch für die industrielle Anwendung gedacht. Der Einfluß der Reynolds-Zahl auf die Nusselt-Zahl der Wärmequellen wurde besprochen. Die Berechnungen wurden für verschiedene Zusammenstellungen der Wärmequellen gemacht. Strömungsbilder, Druck- und Temperaturverteilungen sind dargestellt worden.

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Nomenclature B ₁ channel width - B ₂ horizontal distance between heating elements - g magnitude of gravitational acceleration - i numbers of the horizontal grid points - j numbers of the vertical grid points - k thermal conductivity of fluid - l ₁ distance between two thermal sources - L channel length - L ₁ distance between the entrance and the first column of heating element - N number of thermal sources in a horizontal column - Nu Nusselt number - Nu average Nusselt number - p, P dimension and dimensionless pressure - Pr Prandtl number - Re Reynolds number - T local temperature - T _H temperature of the top surface of the thermal source - T _L inlet-flow temperature - U _O average inlet-flow velocity - u, U dimensional and dimensionless horizontal velocities - v, V dimensional and dimensionless vertical velocities - x, X dimensional and dimensionless horizontal coordinates - y, Y dimensional and dimensionless vertical coordinates - W width of the thermal source - fluid thermal diffusivity - coefficient of thermal expansion - lateral pitch of thermal sources - viscosity - kinematic viscosity=/ - density - dimensionless temperature=(T–T _O)/(T _H–T_O)     

Mixed convection in vertical channels with a discrete heat source

Two dimensional laminar mixed convection flow in vertical channels with a discrete heat source was numerically analyzed. An isoflux discrete heating element was located on the left wall, while the isothermal conditions were imposed on the other wall. The governing equations were solved using a finite difference method based on the control volume approach. The mean Nusselt number was calculated and the maximum component temperature was determined. The computations were carried out for different Grashof number, Reynolds number, heater locations and the channel width. It was observed that the location of the heating element does not play a considerable role on the flow. At low Reynolds numbers (Re<50), the mean Nusselt number and the maximum temperature are mainly controlled by the Grashof number. However, at higher Reynolds numbers, the Reynold number plays an important role on the flow. It was also found that at low Reynolds numbers, cooling is more effective when the channel width is large (W/H>1). However, at high Reynolds numbers more effective cooling is obtained in narrow channels.

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Mischkonvektion in vertikalen Kanälen mit einer lokalen Wärmequelle

Zusammenfassung Die zweidimensionale laminare Mischkonvektion in vertikalen Kanälen mit einer lokalen Wärmequelle wird numerisch untersucht. Ein Heizelement konstanter Wärmeleistung befindet sich auf der linken Kanalwand, die rechte hat konstante Temperatur. Die Lösung der Grundgleichung erfolgte mit Hilfe der auf dem Kontrollvolumenprinzip basierenden Finitdifferenzenmethode. Die mittlere Nusselt-Zahl sowie die Maximaltemperatur des Heizelementes wurden berechnet, und zwar unter Variation der Grashof-Zahl, der Reynolds-Zahl, der Lage des Heizelements und der Kanalbreite. Letztere hatte nur geringen Einfluß auf den Strömungsverlauf. Bei kleinen Reynolds-Zahlen (Re<50) werden Nusselt-Zahl und Maximaltemperatur vorrangig durch die Grashof-Zahl bestimmt, während bei hohen Reynolds-Zahlen letztere den Strömungsvorgang beherrscht. Ferner zeigte sich, daß bei niedrigen Reynolds-Zahlen die Kühlung für große Kanalbreite (W/H>1) effektiver wird und bei hohen Reynolds-Zahlen die Verhältnisse gerade umgekehrt liegen.

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Nomenclature g gravitational acceleration - Gr Grashof number (Gr=gqH ⁴/v²k) - H heater hight - k thermal conductivity of fluid - L height of the channel - Nu Nusselt number - P pressure - Pr Prandtl number - Re Reynolds number (Re=V ₀H/v) - S position of heater center - T temperature - T _c cold wall temperature - T ₀ inlet temperature - u velocity component inx-direction - U dimensionless velocity component inx-direction (U=u/V ₀) - x horizontal axis - X dimensionless horizontal axis (x/H) - v velocity component iny-direction - V dimensionless velocity component iny-direction (V=v/V ₀) - V ₀ inlet velocity - W width of the channel - y vertical axis - Y dimensionless vertical axis (y/H) Greek symbols a thermal diffusivity - thermal expansion coefficient - density of fluid - kinematic viscosity - dimensionless temperature (=(T–T _c)/[qH/k])     

Effects of viscous dissipation and boundary conditions on forced convection in a channel occupied by a saturated porous medium

Forced convection with viscous dissipation in a parallel plate channel filled by a saturated porous medium is investigated numerically. Three different viscous dissipation models are examined. Two different sets of wall conditions are considered: isothermal and isoflux. Analytical expressions are also presented for the asymptotic temperature profile and the asymptotic Nusselt number. With isothermal walls, the Brinkman number significantly influences the developing Nusselt number but not the asymptotic one. At constant wall heat flux, both the developing and the asymptotic Nusselt numbers are affected by the value of the Brinkman number. The Nusselt number is sensitive to the porous medium shape factor under all conditions considered.     

Natural convection in partitioned enclosed spaces of solar collector

The two-dimensional, steady, laminar natural convection phenomena in partitioned enclosure of solar collector has been studied numerically. Heat conduction along the partition is considered. An iterative finite-difference scheme is employed to solve the governing equations in the flow field. The effects of Rayleigh number, thermal conductivity ratio, partition angle, tilt angle, and aspect ratio on both the local and average heat transfer coefficients of the solar collector have been discussed. The range of Rayleigh number tested was up to 5 × 10⁴, the thermal conductivity ratio of 4.5 and 30, partition angle from 10 deg to 170 deg, tilt angle from 10 deg to 90 deg, and aspect ratio varied between 0.2 and 10. The results indicate that the convective heat transfer is strongly affected with the aspect ratio of the enclosures.

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Freie Konvektion in unterteilten Kammern von Solarkollektoren

Zusammenfassung Die zweidimensionale, stetige, laminare freie Konvektion in unterteilten Kammern von Solarkollektoren wurde numerisch untersucht. Die Wärmeübertragung entlang dieser Kammern wurde betrachtet. Ein iteratives Finite-Differenzen-Schema wurde angewandt um die Gleichungen, welche das Strömungsfeld beschreiben, zu lösen. Der Einfluß der Rayleigh-Zahl, der thermische Leitfähigkeit, des Kammerwinkels, des Neigungswinkels und der Längenverhältnisse auf die örtlichen und durchschnittlichen Wärmeübertragungskoeffizienten des Solarkollektors wurde diskutiert. Der Bereich der Rayleigh-Zahl erstreckte sich bis zu 5 × 10⁴, das Verhältnis der thermischen Leitfähigkeit betrug 4.5 und 30, der Kammerwinkel lag zwischen 10° und 170°, der Neigungswinkel zwischen 10° und 90° und das Längenverhältnis variierte zwischen 0.2 und 10. Die Ergebnisse beinhalten, daß die konvektive Wärmeübertragung sehr stark durch das Längenverhältnis der Kammern beeinflußt wird.

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Nomenclature a slope of the partition with respect to the horizontal - A H/L=cell aspect ratio - A _w t/L=wall aspect ratio - g acceleration due to gravity - h local heat transfer coefficient - average heat transfer coefficient - H cell length - k thermal conductivity of fluid within the cell - k _w thermal conductivity of the cell wall - L plate spacing - Nu _f h L/k=local cell Nusselt number - L/k=average cell Nusselt number - overall average Nusselt number - qL/k _w t(T _h–T _c)=average wall Nusselt number - Pr /=Prandtl number - q heat transfer in the cell wall from the hot to cold plate per unit depth - Ra g L ³ T/=Rayleigh number - R _k k _w/k=ratio of wall thermal conductivity to that of the fluid - t thickness of cell wall - T _c cold plate temperature - T _f temperature in cell - T _h hot plate temperature - T _w temperature in cell wall - u, U dimension and dimensionless velocities inx-direction - v, V dimension and dimensionless velocities iny-direction - x distance measured from the bottom of the cell (Fig. 1) - X x/L=normalized distance ofx - y distance measured from hot plate (Fig. 1) - Y y/L=normalized distance ofy - x ₁ distance measured in wall (Fig. 1) - X ₁ x/L=normalized distance ofx ₁ Greek symbols thermal diffusivity of fluid - coefficient of volumetric expansion of fluid - partition angle with respect to the hot plate - _f T _f–T _c/T _h–T _c=dimensionless temperature in cell - _w T _w–T _c/T _h–T _c=dimensionless temperature in cell wall - kinematic viscosity of fluid - enclosure tilt angle from horizontal - dimensional vorticity - L ²/=dimensionless vorticity - dimensionless streamline     

Control of Flow and Heat Transfer in a Porous Enclosure due to an Adiabatic Thin fin on the Hot Wall

Natural convection flow in a differentially heated square enclosure filled with porous matrix with a solid adiabatic thin fin attached at the hot left wall is studied numerically. The Brinkman–Forchheimer-extended Darcy model is used to solve the momentum equations, in the porous medium. The numerical investigation is done through streamlines, isotherms, and heat transfer rates. A parametric study is carried out using the following parameters: Darcy number (Da) from 10⁻⁴ to 10⁻², dimensionless thin fin lengths (L _p) 0.3, 0.5, and 0.7, dimensionless positions (S _p) 0.25, 0.5, and 0.75 with Prandtl numbers (Pr) 0.7 and 100 for Ra = 10⁶. For Da = 10⁻³ and Pr = 0.7, it is observed that there is a counter clock-wise secondary flow formation around the tip of the fin for S _p = 0.5 for all lengths of L _p. Moreover when Da = 10⁻² the secondary circulation behavior has been observed for S _p = 0.25 and 0.75 and there is another circulation between the top wall and the fin that is separated from the primary circulation. However, these secondary circulations features are not observed for Pr = 100. It is also found that the average Nusselt number decreases as the length of the fin increases for all locations. However, the rate of decrease of average Nusselt number becomes slower as the location of fin moves from the bottom wall to the top wall. The overall heat transfer rate can be controlled with a suitable selection of the fin location and length.     

Computation of conjugate heat transfer in the turbulent mixed convection regime in a vertical channel with multiple heat sources

This paper presents the results of a comprehensive numerical study to analyze conjugate, turbulent mixed convection heat transfer from a vertical channel with four heat sources, uniformly flush-mounted to one of the channel walls. The results are presented to study the effect of various parameters like thermal conductivity of wall material (k _s), thermal conductivity of flush-mounted discrete heat source (k _c), Reynolds number of fluid flow (Re _s), modified Richardson number (Ri ⁺) and aspect ratio (AR) of the channel. The standard k-ε turbulence model, modified by including buoyancy effects with physical boundary conditions, i.e. without wall functions, has been used for the analysis. Semi-staggered, non-uniform grids are used to discretise the two dimensional governing equations, using finite volume method. A correlation, encompassing a wide range of parameters, is developed for the non-dimensional maximum temperature (T ^*) using the asymptotic computational fluid dynamics (ACFD) technique.     

Free convective heat-transfer performance of a two-dimensional open thermosyphon with heat sources of cavity dotted along vertical wall

This paper is concerned with the heat-transfer characteristics in a vertical two-dimensional open thermosyphon whose heat sources are the heated cavities dotted along the vertical wall. Air is utilized for the measurement of heat transfer, while transformer oil for the observation of the flow patterns. Attention is particularly focussed on the effects of the depth of cavity and the clearance for main fluid-flow on the behavior of free convective heat transfer in the present open thermosyphon. Environmental temperature is maintained at 10°C, while temperature of the bottom-surface of cavity and the clearance of main fluid-flow are parametrically varied, as Rayleigh number ranging from 1.2×10¹ to 3.8×10⁶.It is found that the effect of the clearance on the heat-transfer characteristics in the two-dimensional open thermosyphon is unexpectedly large. Experimental results are finally given as plots of Nusselt number versus Rayleigh number. An experimental correlation is given for the Nusselt number as a function of the Rayleigh number and the clearance/length ratio of the open thermosyphon.

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Wärmeübertragungsverhalten bei freier Konvektion eines zweidimensionalen, offenen Thermosyphons mit längs der vertikalen Wand verteilten Hohlräume als Wärmequellen

Zusammenfassung Der Bericht befaßt sich mit dem Wärmeübertragungsverhalten in einem vertikalen, zweidimensionalen offenen Thermosyphon mit längs der vertikalen Wand verteilten, beheizten Hohlräumen als Wärmequellen. Zur Messung des Wärmeüberganges wird Luft, zur Strömungsbeobachtung Transformatorenöl verwendet. Besonderes Interesse gilt den Einflüssen der Hohlraumtiefe und der lichten Weite für den Hauptstrom auf das Verhalten des Wärmeüberganges bei freier Konvektion. Die Umgebungstemperatur wird auf 10°C gehalten, während die Hohlraumbodentemperatur und die lichte Weite für den Hauptstrom variiert werden mit Rayleigh-Zahlen zwischen 1.2×10¹ und 3.8×10⁶.Es wird festgestellt, daß der Einfluß der lichten Weite auf das Wärmeübertragungsverhalten unerwartet groß ist. Die experimentellen Ergebnisse werden in Diagrammen der Nusselt-Zahl über der Rayleigh-Zahl dargestellt. Ein Zusammenhang für die Nusselt-Zahl als Funktion von der Rayleigh-Zahl und dem Verhältnis von lichte Weite zu Länge wird gegeben.

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Nomenclature B distance between heated wall and opposing insulation wall,W+D - d _i diameter of inner tube - d ₀ diameter of outer heated tube - D depth of cavity along vertical wall, 0, 25, and 50 mm - g gravitational acceleration - H length of heated or un-heated wall, 100 mm - L length of thermosyphon, 500 mm for two-, 700 mm for three-, and 1100 mm for five-dotted heat sources - Nu _B Nusselt number based on B as reference length - Nu _x Nusselt number, defined in Eq. (1) - Pr Prandtl number, defined in Eq. (3) - q heat flux from heated wall - r equivalent heat-transfer radius - Ra _B Rayleigh number based on B as reference length - Ra _x Rayleigh number, defined in Eq. (2) - T _e temperature of entrance-fluid - T _w temperature of heated wall - T temperature difference between heated wall and entrance-fluid,T _w -T _e - W clearance for main fluid-flow - x reference length - X distance from bottom of thermosyphon Greek symbols coefficient of volumetric expansion - thermal diffusivity - thermal conductivity - kinematic viscosity     

Buoyant MHD flows in a vertical channel: the levitation regime

Buoyant magnetohydrodynamic (MHD) flows with Joulean and viscous heating effects are considered in a vertical parallel plate channel. The applied magnetic field is uniform and perpendicular to the plates which are subject to adiabatic and isothermal boundary conditions, respectively. The main issue of the paper is the levitation regime, i.e., the fully developed flow regime for large values of the Hartmann number M, when the hydrodynamic pressure gradient evaluated at the temperature of the adiabatic wall is vanishing. The problem is solved analytically by Taylor series method and the solution is validated numerically. It is found that the fluid velocity points everywhere and for all values of M downward. For small M’s, the velocity field extends nearly symmetrically (with respect to the mid-plane) over the whole section of the channel between the adiabatic and the isothermal walls. For large values of M, by contrast, the fluid levitates over a broad transversal range of the channel, while the motion becomes concentrated in a narrow boundary layer in the neighborhood of the isothermal wall. Accordingly, the fluid temperature is nearly uniform in the levitation range and decreases rapidly within the boundary layer in front of the isothermal wall. It also turns out that not only the volumetric heat generation by the Joule effect, but also that by viscous friction increases rapidly with increasing values of M, the latter effect being even larger than the former one for all M.     

Natural convection in partially heated vertical channels

A numerical analysis has been performed on laminar natural convection of air in open vertical channels partially heated at uniform wall temperature (UWT) or at uniform heat flux (UHF). The governing equations have been solved by means of a finite difference technique. Results showing axial velocity and temperature developments as well as heat transfer performances and correlations between non-dimensional groups, are presented.

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Natürliche Konvektion in teilweise erwärmten vertikalen Kanälen

Zusammenfassung Eine numerische Analyse wurde über die natürliche Luftkonvektion in vertikalen, mit gleichmäßiger Wandtemperatur (UWT) oder mit gleichmäßigem Wärmestrom (UHF) teilweise erwärmten Kanälen durchgeführt. Die analytischen Gleichungen des Problems wurden mit der Finit-Differenzen-Technik gelöst, und es werden Ergebnisse hinsichtlich der Geschwindigkeits- und Temperaturverteilungen im Inneren des Kanals sowie der thermischen Leistung des Systems aufgeführt.

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Nomenclature a thermal diffusivity of the fluid - c _p specific heat (at constant pressure) of the fluid - g acceleration due to gravity - Gr =[·g·S ³ ·(T₁-T₀)]/v²,Grashof number (UWT case) - Gr =[-g-S ⁴ -q ₁]/(v ²·k), Grashof number (UHF case) - Gr ^* =(S/H) Gr, modified Grashof number - H overall channel height - I, J X andY coordinate indexes - k thermal conductivity of the fluid - Nu mean Nusselt number of the channel - p difference between pressure inside the channel and pressureoutside, at the same heightx - P dimensionless difference pressure - Pr Prandtl number - q specific heat flux - q ₁ specific heat flux from heated plates (UHF case) - Q heat flux (per unit length in thez-direction) from walls - S channel width - T temperature - T _w reference wall temperature - T _o fluid temperature at the inlet section - T ₁ heated plates temperature (UWT case) - u, axial and transverse velocity of the fluid - u _o axial velocity of the fluid at the inlet section - U, V dimensionless axial and transverse velocity - U _o dimensionless axial velocity at the inlet section - x, y axial and transverse coordinate - X, Y dimensionless axial and transverse coordinate - X =H/(S·Gr), dimensionless overall channel height - thermal expansion coefficient of the fluid - dimensionless temperature - v kinematic viscosity of the fluid - density of the fluid     

Experimental investigation of mixed convection from an array of discrete heat sources at the bottom of a horizontal channel

Mixed convection heat transfer from an array of discrete heat sources inside a rectangular channel has been investigated experimentally under various operating conditions for air. The lower surface of the channel was equipped with 8 × 4 flush-mounted heat sources subjected to uniform heat flux, sidewalls and the upper wall are insulated and adiabatic. The experimental parametric study was made for an aspect ratio of AR = 10, Reynolds numbers 241 Re_Dh 980, and modified Grashof numbers Gr^* = 9.53 × 10⁵ to 1.53 × 10⁷ . From the experimental measurements, surface temperature distributions of the discrete heat sources were obtained and effects of Reynolds and Grashof numbers on these temperatures were investigated. Furthermore, Nusselt number distributions were calculated for different Reynolds and Grashof numbers, with emphasis on changes obtained for different discrete heat source locations. From these results, the buoyancy affected secondary flow and the onset of instability have been discussed. Results show that surface temperatures increase with increasing Grashof number and decrease with increasing Reynolds number. However, with the increase in the buoyancy affected secondary flow and the onset of instability, temperatures level off and even drop as a result of heat transfer enhancement. This outcome can also be observed from the variation of the row-averaged Nusselt number showing an increase towards the exit, especially for low Reynolds numbers.     

Non-similar Solution for Natural Convective Boundary Layer Flow Over a Sphere Embedded in a Porous Medium Saturated with a Nanofluid

A boundary layer analysis is presented for the natural convection past an isothermal sphere in a Darcy porous medium saturated with a nanofluid. Numerical results for friction factor, surface heat transfer rate, and mass transfer rate have been presented for parametric variations of the buoyancy ratio parameter N _r, Brownian motion parameter N _b, thermophoresis parameter N _t, and Lewis number L _e. The dependency of the friction factor, surface heat transfer rate (Nusselt number), and mass transfer rate (Sherwood number) on these parameters has been discussed.     

Momentum and heat transfer in the entrance region of a parallel plate channel: developing laminar flow with constant wall temperature

A new integral method of solution is presented for developing laminar flow and heat transfer in the entrance region of a parallel plate channel with uniform surface temperature. Unlike earlier Karman-Pohlhausen analyses, the new analysis provides solutions which are free from jump discontinuities in the gradients of the velocity and temperature distributions throughout and at the end of the entrance region. The hydrodynamic and thermal results from the present analysis therefore join smoothly and asymptotically to their fully-developed values. The heat transfer results obtained are further found to agree well with previously published numerical solutions.Nomenclature a half width of the channel, m - D _h hydraulic diameter (=4a), m - h local heat transfer coefficient,W/(m²·K) - h _m mean heat transfer coefficient defined by Eq- (9),W/(m²·K) - k thermal conductivity, W/(m·K) - L _H axial length of the hydrodynamic entrance region, m - L _T axial length of the thermal entrance region, m - L _in,H axial length of the hydrodynamic inlet region, m - L _in,T axial length of the thermal inlet region, m - Nu _x local Nusselt number,hD _h /k, dimensionless - Nu _m mean Nusselt number defined by Eq. (9),h _mD_h/k, dimensionless - P pressure, N/m² - P _O pressure at the entrance, N/m² - Pr Prandtl number,c _p /k, dimensionless - Re Reynolds number, 4aU _o /v, dimensionless - T absolute temperature, K - T _b fluid bulk temperature, K - T _c centerline temperature, K - T _w wall temperature, K - U _c centerline velocity, m/s - U ₀ velocity of the fluid at entrance, m/s - U core velocity, m/s - u velocity component inx direction, m/s - v velocity component iny direction, m/s - x spatial coordinate, axial distance, m - y spacial coordinate measured from channel wall, m Greek Letters molecular thermal diffusivity, m²/s - hydrodynamic shape factor, dimensionless - _T thermal shape factor, dimensionless - hydrodynamic boundary layer thickness, m - * /a, dimensionless - _T thermal boundary layer thickness, m - * _{T T} /a, dimensionless - dimensionless distance,y/ ory/a - Pohlhausen's shape factor, dimensionless - dynamic viscosity coefficient, kg/(m·s) - v kinematic viscosity,/, m²/s - dimensionless axial distance,x/(a·Re) - _H dimensionless axial length of the hydrodynamic entrance region (=L _H /(a·Re)) - _T dimensionless axial length of the thermal entrance region (=L _T /(a·Re)) - _in,H dimensionless axial length of the hydrodynamic inlet region (=L _in,H/(a·Re)) - _in,T dimensionless axial length of the thermal inlet region (L _in,T /(a·Re)) - fluid density, kg/m³     

Turbulent natural convection in vertical parallel-plate channels

The problem of buoyancy driven turbulent flow in parallel-plate channels is investigated. The investigation is limited to vertical channels of uniform cross-section with different modes of heating. The details of the flow and thermal fields are obtained from the solution of the conservation equations of mass, momentum, and energy in addition to equations of the low Reynolds number turbulence model. The study covers Rayleigh number ranging from 10⁵ to 10⁷ and focuses on the effect of channel geometry on the characteristic of the flow and thermal fields as well as the local and average Nusselt number variation. A Nusselt number correlation has been developed in terms of a modified Rayleigh number and channel aspect ratio for the cases of symmetrically heated isothermal and isoflux conditions.     

Effects of solid and slotted baffles on the convection characteristics of backward-facing step flow in a channel

This study investigates the enhancement of the laminar forced convection characteristics of backward-facing step flow in a two-dimensional channel through the installation of solid and slotted baffles onto the channel wall. The effects of the height of baffle H _b, inclination of baffle installation ϕ_b, height of slot in baffle H _t, inclination of slot in baffle ϕ_t, and distance between the backward-facing step and baffle D on the flow structure, temperature distribution and Nusselt number variation for the system at various Re are numerically explored. Results show that a slotted baffle can enhance the average Nusselt number for the heating section of channel plate by the maximum 190% when Pr=0.7, H _s=0.5, L=5, H _b ≤ 0.3, W _b ≤ 0.2, 0.1 ≤ D ≤ 0.5, 0° ≤ ϕ_b ≤ 45°, H _t ≤ 0.1, 0° ≤ ϕ_t ≤ 45° and 50 ≤ Re ≤ 400. As for the solid baffle, the enhancement may be up by 230%. The solid baffle might cause the re-separation of main stream, and consequently result in poor local heat transfer coefficient in the end region of heating section. This disadvantage can be obviously improved as the baffle is slotted. Besides the penalty of increase in pressure drop due to the baffle installation is much higher for the situation with solid baffle.