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

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

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

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

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

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

On the laminar forced convection with axial conduction in a circular tube with exponential wall heat flux

The values of the fully developed Nusselt number for laminar forced convection in a circular tube with axial conduction in the fluid and exponential wall heat flux are determined analytically. Moreover, the distinction between the concepts of bulk temperature and mixing-cup temperature, at low values of the Peclet number, is pointed out. Finally it is shown that, if the Nusselt number is defined with respect to the mixing-cup temperature, then the boundary condition of exponentially varying wall heat flux includes as particular cases the boundary conditions of uniform wall temperature and of convection with an external fluid.

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Über laminare Zwangskonvektion mit Längswärmeleitung in einem Kreisrohr mit exponentiell veränderlichem Wandwärmefluß

Zusammenfassung Es werden die Endwerte der Nusselt-Zahlen für vollausgebildete laminare Zwangskonvektion in einem Kreisrohr mit Längswärmeleitung und exponentiell veränderlichem Wandwärmefluß analytisch ermittelt. Besondere Betonung liegt auf dem Unterschied zwischen den Konzepten für die Mittel- und die Mischtemperatur bei niedrigen Peclet-Zahlen. Schließlich wird gezeigt, daß bei Definition der Nusselt-Zahl bezüglich der Mischtemperatur die Randbedingung exponentiell veränderlichen Randwärmeflusses die Spezialfälle konstanter Wandtemperatur und konvektiven Wärmeaustausches mit einem umgebenden Fluid einschließt.

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Nomenclature A _n dimensionless coefficients employed in the Appendix - Bi Biot numberBi=h _e r ₀/ - c _n dimensionless coefficients defined in Eq. (17) - c _p specific heat at constant pressure of the fluid within the tube, [J kg^–1 K^–1] - f solution of Eq. (15) - h ₁,h ₂ specific enthalpies employed in Eqs. (2) and (4), [J kg^–1] - h _e convection coefficient with a fluid outside the tube, [W m^–2 K^–1] - rate of mass flow, [kg s^–1] - Nu bulk Nusselt number,2r ₀ q _w /[(T _w –T _b )] - Nu _H fully developed value of the bulk Nusselt number for the boundary condition of uniform wall heat flux - Nu _T fully developed value of the bulk Nusselt number for the boundary condition of uniform wall temperature - Nu ^* mixing Nusselt number,2r ₀ q _w /[(T _w –T _m )] - Nu _C ^* fully developed value of the mixing Nusselt number for the boundary condition of convection with an external fluid - Nu _H ^* fully developed value of the mixing Nusselt number for the boundary condition of uniform wall heat flux - Nu _T ^* fully developed value of the mixing Nusselt number for the boundary condition of uniform wall temperature - Pe Peclet number, 2r ₀/ - q ₀ wall heat flux atx=0, [W m^–2] - q _w wall heat flux, [W m^–2] - r radial coordinate, [m] - r ₀ radius of the tube, [m] - s dimensionless radius,s=r/r ₀ - T temperature, [K] - T ₀ temperature constant employed in Eq. (14), [K] - T reference temperature of the fluid external to the tube, [K] - T _b bulk temperature, [K] - T _m mixing or mixing-cup temperature, [K] - T _w wall temperature, [K] - u velocity component in the axial direction, [m s^–1] - mean value ofu, [m s^–1] - x axial coordinate, [m] Greek symbols thermal diffusivity of the fluid within the tube, [m² s^–1] - exponent in wall heat flux variation, [m^–1] - dimensionless parameter - dimensionless temperature =(T _w –T)/(T _w –T _b ) - ^* dimensionless temperature ^*=(T _w –T)/(T _w –T _m ) - thermal conductivity of the fluid within the tube, [W m^–1 K^–1] - density of the fluid within the tube, [kg m^–3]     

Natural convection in inclined two dimensional rectangular cavities

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

Laminar heat transfer in the entrance region of ducts

A finite difference technique is used for the evaluation of the rate of heat transfer in the thermal entrance region of ducts with axial conduction. The velocity profile is fully developed and flow in a tube and between parallel plates is studied. Local and average Nusselt numbers and mixing temperatures are presented as a function of the Péclet number. A criterion is also established which proves useful for predicting the conditions under which axial conduction may be ignored.Nomenclature C transformation constant - c _v specific heat, constant volume - D _h hydraulic diameter - h local convective film coefficient, Eq. (15) - h* local convective film coefficient, Eq. (16) - h _m ^* mean convective film coefficient, Eq. (17) - k thermal conductivity - Nu local Nusselt number, hD _h/k - Nu* local Nusselt number, h*D _h/k - Nu _m ^* mean Nusselt number, hQD _h/k - Pe Péclet number, D _h v _m/ - q rate of heat transfer - r radial coordinate - r _o tube radius - R nondimensional radial coordinate, r/r _o - S transformed axial coordinate, Eq. (10) - T temperature - T _e entrance temperature - T _m mixing temperature, Eq. (18) - T _w wall temperature - v _z axial velocity - v _m mean axial velocity - V nondimensional axial velocity, v _z /v _m - y transverse coordinate in parallel plate flow - y _o half width of parallel plate duct - Y nondimensional transverse coordinate, y/y _o - z axial coordinate - Z nondimensional axial coordinate, z/r _o or z/y _o - Z ⁺ nondimensional axial coordinate divided by Peclet number, Z/Pe - thermal diffusivity - nondimensional temperature, (T–T _w)/(T _e–T _w) - mean nondimensional temperature, - _m nondimensional mixing temperature, Eq. (22) - density - i axial position index - j radial or transverse position index     

Heat and mass transfer characteristics of simulated high moisture flue gases

The heat transfer process occurring in a condensing heat exchanger where noncondensible gases are dominant in volume is different from the condensation heat transfer of the water vapor containing small amount of noncondensible gases. In the process the mass transfer due to the vapor condensation contributes an important part to the total heat transfer. In this paper, the Colburn-Hougen method is introduced to analyze the heat and mass transfer process when the water vapor entrained in a gas stream condenses into water on the tube wall. The major influential factors of the convective-condensation heat transfer coefficient are found as follows: the partial pressure of the vapor p _v , the temperature of the outer tube wall T _w , the mixture temperature T _g , Re and Pr. A new dimensionless number Ch, which is defined as condensation factor, has been proposed by dimensional analysis. In order to determine the relevant constants and investigate the convection-condensation heat and mass transfer characteristics of the condensing heat exchanger of a gas fired condensing boiler, a single row plain tube heat exchanger is designed, and experiments have been conducted with vapor-air mixture used to simulate flue gases. The experimental results show that the convection-condensation heat transfer coefficient is 1.52 times higher than that of the forced convection without condensation. Based on the experimental data, the normalized formula for convention-condensation heat transfer coefficient is obtained. A heat transfer area m² - Ch condensation factor - c _p specific heat at constant pressure, J/(kg·K) - G mass flux Kg/(m²·s) - h heat transfer coefficient W/(m²·K) - J J-factor - Nu Nusselt number - pa pressure - Pr Prandtl number - Q heat transfer rate - q heat flux W/m² - r latent heat, kJ/kg - Re Reynolds number - Sc Schmidt number - T temperature, C or K - heat conductivity m W/(m·K) - density, kg·m³ - g gas - h moistened hot air - i interface - v vapor - w water     

Free convection from a vertical plate with concentrated and distributed thermal sources

An analysis is developed for the laminar free convection from a vertical plate with uniformly distributed wall heat flux and a concentrated line thermal source embedded at the leading edge. We introduce a parameter=(1 +Q _L/Q_w)^–1=(1 + Ra_L/Ra_w)^–1 to describe the relative strength of the two thermal sources; and propose a unified buoyancy parameter=( Ra_L+ Ra_w)^1/5 with=1/(1 +Pr ^–1) to properly scale the dependent and independent variables. The variables are so defined that the resulting nonsimilar boundary-layer equations can describe exactly the buoyancy-induced flow from the dual sources with any relative strength to fluids of any Prandtl number from very small values to infinity. These nonsimilar equations are readily reducible to the self-similar equations of an adiabatic wall plume for=0, and to those of free convection from uniform flux plate for=1. Rigorous finite-difference solutions for fluids of Pr from 0.001 to are obtained over the entire range of from 0 to 1. The effects of both relative source strength and Prandtl number on the velocity profiles, temperature profiles, and the variations of wall temperature, are clearly illustrated.

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Freie Konvektion an einer vertikalen Platte mit einer konzentrierten und einer gleichmäßig verteilten Wärmequelle

Zusammenfassung Für die freie Konvektion an einer vertikalen Platte mit einer gleichmäßig verteilten Wandwärmestromdichte und einer in der Vorderkante eingebetteten linienförmigen Wärmequelle wird eine Berechnungsmethode entwickelt. Zur Beschreibung der relativen Stärke der beiden Wärmequellen führen wir einen Parameter=(1 + Q_L/Q_w)^–1=(1 + Ra_L/Ra_w)^–1 ein und schlagen einen vereinheitlichten Auftriebsparameter=( Ra _L+ Ra _w)^1/5 mit=1/(1 +Pr ^–1 für die Skalierung der abhängigen und unabhängigen Variablen vor. Die Variablen werden so definiert, daß mit den sich ergebenden unabhängigen Grenzschichtgleichungen die von den beiden Wärmequellen beliebiger Stärke verursachte Auftriebsströmung von Fluiden beliebiger Prandtl-Zahl genau beschrieben werden kann. Diese unabhängigen Gleichungen können ohne weiteres auf die selbstähnlichen Gleichungen für den Fall einer lokalen Wärmezufuhr an einer sonst adiabatischen Wand für=0 und jenen der freien konvektion an einer Platte mit einheitlichem Wärmestrom für=1 zurückgeführt werden. Für Fluide mit der Prandtl-Zahl zwischen 0,001 und Unendlich werden nach der strengen finite Differenzen-Methode Lösungen im Bereich von zwischen 0 und 1 erhalten. Der jeweilige Einfluß der relativen Quellenstärke und der Prandtl-Zahl auf die Geschwindigkeits- und Temperaturprofile sowie die Veränderung der Wandtemperatur werden deutlich dargestellt.

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Nomenclature C _f friction coefficient - C _p specific heat - f reduced stream function - g gravitational acceleration - k thermal conductivity - L width of the plate - Nu local Nusselt number - Pr Prandtl number - q _w wall heat flux - Q _L heat generated by the line source - Q _w heat released by the uniform-flux wall from 0 tox, q _w Lx - Ra _L local Rayleigh number, g T _L ^* x ³/( ) - Ra _w local Rayleigh number,g T _w ^* w ³/( ) - T fluid temperature - T temperature of ambient fluid - T _L ^* characteristic temperature of the line source,Q _{L/(C p} L) - T _w ^* characteristic temperature of the uniform flux wall, =q _w x/k=Q _w /(C _p L) - u velocity component in then-direction - U₀ dimensionless velocity,u/(/x) Ra _L ^2/5 - U ₁ dimensionless velocity,u/(/x) Ra _w ^2/5 - velocity component in they-direction - x coordinate parallel to the plate - y coordinate normal to the plate - thermal diffusivity - thermal expansion coefficient - pseudo-similarity variable,(y/x) - dimensionless temperature, (T–T )/(T _L ^* +T _w ^* ) - ₀ dimensionless temperature, (Ra_l)^1/5 (T–T )/T _L ^* - ₁ dimensionless temperature, (Ra_w)Ra_w)^1/5 (T–T )/T _w ^* - (Ra _L+Ra_w)^1/5 - kinematic viscosity - (1 +Ra _L/Ra_w)^–1=(1 +T _L ^* /T _w ^* )^–1=(1 + Q_L/Q_w)^–1 - density - Pr/(1 +Pr) - _w wall shear stress - stream function     

Fully developed laminar flow and heat transfer in an arbitrarily shaped triangular duct

Hydrodynamic and thermal characteristics of the fully developed laminar flow and heat transfer in an arbitrarily shaped triangular duct are evaluated using a finite difference technique. The hydrodynamic information encompasses the friction factor, the length from the tube entrance necessary for complete hydrodynamic development, and the incremental pressure drop due to flow development in the entrance section. The Nusselt numbers in the case of an allover isothermal ductNu _T , as well as for a duct heated by an axial uniform heat flux while its transverse local periphery is at a constant temperatureNu _H , are presented. Comparison of the isosceles results with those from the work of Shah [1], Sparrow and Haji-Sheikh [2], and Schmidt and Newell [3] revealed a maximum difference of about –0.2% in theNu _Hi data, less than ±0.5% in theNu _T ,about +0.3% in the friction factor, a –0.47% in the incremental pressure drop, and around –1% in the developing entrance length. The deviations from the results of other authors become smaller as the triangular geometry approaches the equilateral.

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Vollständig ausgebildete laminare Strömung und Wärmeübergang in einem willkürlich geformten dreieckigen Kanal

Zusammenfassung Mittels einer finiten Differenzenmethode wurde das hydrodynamische und thermische Verhalten einer vollständig ausgebildeten laminaren Strömung und der Wärmeübergang in einem willkürlich geformten dreieckigen Kanal untersucht. Hydrodynamische Erkenntnisse wurden über den Reibungskoeffizienten, die Einlauflänge bis zur vollständig ausgebildeten Strömung und den differentiellen Druckverlust im Einlaufgebiet gewonnen. Weiterhin wird sowohl die Nusselt-Zahl eines überall isothermen Kanals vorgestellt (Nu _T ), als auch die im Falle konstanter HeizflächenbelastungNu _Hi Ein Vergleich der eigenen Ergebnisse mit denen von Shah [1], Sparrow und Haji-Sheikh [2] sowie Schmidt und Newell [3] zeigt eine maximale Abweichung bei den Werten vonNu _Hi von ungefähr –0,2%, weniger als ±0,5% fürNu _T , ungefähr +0,3% beim Reibungskoeffizient, –0,47% beim differentiellen Druckverlust und etwa –1% bei der hydraulischen Einströmlänge. Die Abweichungen der Ergebnisse von anderen Autoren werden kleiner, wenn sich die Dreiecksgeometrie der Rechtecksgeometrie annähert.

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Nomenclature D _h Equivalent Diameter=4A/p - f Friction Factor - K Incremental Pressure Drop - L _e Entrance Length - Nu _Hi Nusselt Number — Constant Axial Heat Flux, Isothermal Local Periphery - Nu _T Nusselt Number — Isothermal Duct - Re Reynolds Number     

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.     

Particle statistics in x-ray diffractometry

Summary Expressions are derived for the relative r.m.s. error of diffractometer intensity measurements. The result for stationary specimens:=4R[sin/m w h N _eff]^1/2, withh=1/2(h _F+h_S) andN _eff=cAv/v², is identical with the result of Alexander c.s.¹, except for a slight difference in the numerical constant and in the definition ofw. The value of this parameter is found to lie betweenR+(w_F, w_S)_min (the last term indicating the smallest of the widthsw _F andw _S) andR+1/2(w _F+w_S); it reaches the latter limit in the case of integrated intensities being measured by totalizing counts while scanning through a line. For rotating specimens the particle statistics error turns out to be almost independent ofw. The following approximative formula is established:=6.5R sin/h(mN _eff)^1/2, showing that the factor of improvement resulting from specimen rotation is of the order of (h/w)^1/2.Part. II: Experiments, by P. M. de Wolff, Jeanne M. Taylor and W. Parrish, is in the course of preparation.Work done when on leave of absence (Nov. 1954–May 1955) from Technisch Physische Dienst T.N.O. and T.H., Delft, Netherlands.     

Natural convection in rectangular enclosures with adiabatic fins attached on the heated wall

The heat transfer by natural convection in vertical and inclined rectangular enclosures with fins attached to the heated wall is numerically studied using the energy and Navier-Stokes equations with the Boussinesq approximation. The range of study covers 10⁴Ra2×10⁵,A=H/L=2.5 to ,B=l/L=0 to 1,C=h/L=0.25 to 2 andPr=0.72. The inclination angle from the vertical was from 0 to 60 degree. The variation of the local Nusselt numberNu _loc along the enclosure height and the average Nusselt numberNu as a function ofRa are computed. Streamlines and isotherms in the enclosure are produced. The results show thatB is an important parameter affecting the heat transfer through the cold wall of the enclosure. The heat transfer is reduced for decreasingC and it passes from a maximum for an inclination angle. The results show that the heat transfer can generally be reduced using appropriate geometrical parameters in comparison with a similar enclosure without fins.Die Wärmeübertragung bei freier Konvektion in vertikalen und geneigten rechtwinkligen Behältern mit Rippen an den beheizten Wänden wird unter Verwendung der Energie- und Navier-Stokes-Gleichungen sowie der Boussinesq-Approximation numerisch untersucht. Der Bereich der Studie liegt bei 10⁴Ra2·10⁵,A=H/L=2,5 bis ,B=l/L=0 bis 1,C=h/L=0,25 bis 2 undPr=0.72. Der Neigungswinkel der Wand liegt zwischen 0 und 60 Grad. Die Veränderung der lokalen Nusselt-Zahl entlang der Höhe der Behälterwände und die mittlere Nusselt-Zahl in Abhängigkeit derRa-Zahl werden berechnet. Strömungslinien und Isothermen werden im Behälter erzeugt. Die Ergebnisse zeigen, daßB ein wichtiger Parameter für die Wärmeübertragung an der nicht beheizten Wand des Behälters ist. Die übertragene Wärmemenge verringert sich mit abnehmendemC und durchschreitet ein Maximum für eine bestimmte Wandneigung. Die Ergebnisse zeigen, daß im Vergleich zu einer Anordnung ohne Rippen, die Wärmeübertragung bei geeigneten geometrischen Parametern allgemein reduziert werden kann.     

Natural convection in shallow enclosures with multiple conducting partitions

Laminar natural convection and conduction in shallow enclosures having multiple partitions with finite thickness and conductivity have been studied. An approximate analytical solution is obtained by using the parallel flow approximation in horizontal shallow enclosures heated isothermally at two vertical ends while adiabatic on horizontal end walls. The same problem is solved also using a finite difference formulation and the control volume method. The study covers the range ofRa from 105 to 107,A=H/L0.2, C=1/L from 0 to 0.15, and the thermal conductivity ratio of partition to fluidk _r from 10^–4 to 10¹¹. The partition numberN was varied from 0 to 5. The Prandtl number was 0.72 (for air). The results are reduced in terms ofNu as a function ofRa, k, and various geometrical parameters (A, C). The streamlines and isotherms are produced to visualize the flow and temperature fields.Es wird der kombinierte Einfluß von laminarer Naturkonvektion und Leitung in flachen Behältern mit mehreren Trennwänden endlicher Dicke und Leitfähigkeit untersucht. Eine analytische Näherungslösung läßt sich über die Parallelstromapproximation bezüglich horizontaler flacher Behälter finden, deren zwei vertikale Begrenzungswände isotherm beheizt sind, während die Horizontalflächen adiabat sein sollen. Das selbe Problem wird unter Verwendung eines Differenzverfahrens und der Kontrollvolumen-Methode gelöst und zwar für die Parameterbereiche 105 Ra 10⁷;A=H/L<0.2;>C=1/L 0.15; 10^–4k_r 1011, wobei der letzte Parameter das Verhältnis der Leitfähigkeit von Trennwand und Fluid bezeichnet. Die Zahl der TrennwändeN variierte Zwischen 0 und 5, die Prandtl-Zahl betrug 0.72 (Luft). Die Ergebnisse werden in dimensionsloser Form gemäß der BeziehungNu =f (Ra, k _r ,A, C) mitgeteilt bzw. durch Diagrammdarstellungen der Stromlinien- und Isothermenfelder veranschaulicht.Financial support from the Natural Sciences and Engineering Council Canada is acknowledged. Financial support to A. Kangni from Canadian Fellowship Program For French Speaking Countries is also acknowledged.     

Conduction and convection heat transfer in composite solar collector systems with porous absorber

Steady natural convection and conduction heat transfer has been studied in composite solar collector systems. The system consists of a glazing, a porous layer and a massive wall installed in a room. The heat transfer in this system is studied by assuming the glazing and the vertical bounding wall isothermal at different temperatures, two horizontal bounding walls adiabatic and porous layer without vents. The aspect ratioA was from 0.1 to 1.0 but the detailed study was carried out withA=1. The thickness of the porous wallF _p varied from 1/3 to 1, while the solid wall thickness was kept constant. The conductivity ratio of porous layer was from 10^–2 to 10²,Ra from 10³ to 10⁷. The results are presented in terms of thermal parameters as function ofRa and non-dimensional geometrical parameters. The isotherms and stream lines within the system are produced.

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Wärmetransport infolge Leitung und Konvektion durch ein zusammengesetztes Solarkollektor-System mit porösem Absorber

Zusammenfassung Die Untersuchung bezieht sich auf den stationären Wärmetransport infolge natürlicher Konvektion und Leitung in zusammengesetzten Solarkollektor-Systemen. Ein solches System besteht aus einer Verglasung, einer porösen Schicht und der massiven Außenwand eines Raumes. Der Wärmetransport in diesem System wird unter der Annahme ermittelt, daß sich die Verglasung und die Hinterwand auf verschiedenen konstant Temperaturen befinden, Boden und Decke des Raumes adiabat sind und daß die poröse Schicht luftundurchlässig ist. Das VerhältnisA (Höhe zu Tiefe des Raumes) variierte von 0,1 bis 1,0, wobei die eingehenderen Untersuchungen fürA=1,0 erfolgten. Die bezogene DichteF _p der porösen Schicht bewegte sich von 1/3 bis 1, die Dicke der Außenwand blieb konstant. Das Wärmeleitfähigkeitsverhältnis bezüglich der porösen Schicht lag im Bereich 10^–2 bis 10² und die Rayleigh-ZahlRa reichte von 10³ bis 10⁷. In der Ergebnisdokumentation sind die thermischen Parameter als Funktionen vonRa und den Geometrieverhältnissen wiedergegeben. Stromlinien und Isothermen im Inneren des Systems wurden generiert und dargestellt.

     

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

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

Free convection from horizontal screened plates

Experimental investigations of laminar free convection from horizontal isothermal surfaces screened by cylindrical vertical walls are presented. Screen diameters (D) were equal to the diameter of a heating plate while their heights (H) were varied. Results obtained for several heating fluxes and two different liquids indicate that depending on the (H/D) ratio of cylindrical screens, the heat flux transferred from the heating plate to the liquid varies. Three subranges of screening effect have been distinguished: primary inhibition, intensification and secondary inhibition. For primary inhibition, a theoretical model has been proposed and critical ratio of (H/D) has been predicted. The results of flow visualization are also discussed.

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Freie Konvektion von horizontalen, seitlich abgeschirmten Platten

Zusammenfassung Es werden experimentelle Untersuchungen der laminaren freien Konvektion an horizontalen, isothermen Flächen mit vertikalen zylindrischen Abschirmflächen beschrieben. Die Schirmdurchmesser (D) waren gleich dem einer Heizplatte, während die Schirmhöhen (H) variiert wurden. Die Resultate, welche für mehrere Wärmeflüsse und mit zwei verschiedenen Flüssigkeiten gewonnen wurden, zeigen, daß — abhängig von VerhältnisH/D der zylindrischen Schirme — der von der Heizplatte an das Fluid übergehende Wärmestrom stark variiert. Drei Bereiche konnten bezüglich des Abschirmeffektes unterschieden werden: erste Hemmung, Verstärkung und zweite Hemmung. Für den Fall der ersten Hemmung wird ein theoretisches Modell vorgeschlagen und das kritischeH/D-Verhältnis vorausberechnet. Abschließend erfolgt die Diskussion von Ergebnissen, welche aus der sichtbaren Darstellung des Strömungsfeldes gewonnen wurde.

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Nomenclature a thermal diffusivity [m²/s] - D external diameter of the plate, internal diameter of screens [m] - g gravitational acceleration [m/s²] - H height of screen [m] - Nu= D/ Nusselt number - Nu _(H/D)=0 Nusselt number for plate without screens - Nu _H/D Nusselt number for plate with screen of heightH - Ra=g (T _w1–T _F)D ³/( a) Rayleigh number - Ra _H=g (T _w–T _w1)H ³/( a) Rayleigh number inside screen - q heat flux density [W/m²] - T _w surface temperature of the heating plate [°C] - T _w1 boundary temperature of the motionless liquid [°C] - T _F bulk temperature of the liquid [°C] - heat transfer coefficient [W/(m²·K)] - coefficient of volumetric expansion [1/K] - kinematic viscosity - thermal conductivity [W/(m·K)]     

Nonsimilarity solutions for non-Darcy mixed convection from horizontal surfaces in a porous medium

Non-Darcy mixed convection in a porous medium from horizontal surfaces with variable surface heat flux of the power-law distribution is analyzed. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ζ _f =Ra^* _x /Pe² _x is found to characterize the effect of buoyancy forces on the forced convection with K ^′ U _∞/ν characterizing the effect of inertia resistance. The second region covers the natural convection dominated regime where the dimensionless parameter ζ _n =Pe _x /Ra^*1/2 _x is found to characterize the effect of the forced flow on the natural convection, with (K ^′ U _∞/ν)Ra^*1/2 _x /Pe _x characterizing the effect of inertia resistance. To obtain the solution that covers the entire mixed convection regime the solution of the first regime is carried out for ζ _f =0, the pure forced convection limit, to ζ _f =1 and the solution of the second is carried out for ζ _n =0, the pure natural convection limit, to ζ _n =1. The two solutions meet and match at ζ _f =ζ _n =1, and R ^* _h =G ^* _h . Also a non-Darcy model was used to analyze mixed convection in a porous medium from horizontal surfaces with variable wall temperature of the power-law form. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ξ _f =Ra _x /Pe _x ^3/2 is found to measure the buoyancy effects on mixed convection with Da _x Pe _x /ɛ as the wall effects. The second region covers the natural convection dominated region where ξ _n =Pe _x /Ra _x ^2/3 is found to measure the force effects on mixed convection with Da _x Ra _x ^2/3/ɛ as the wall effects. Numerical results for different inertia, wall, variable surface heat flux and variable wall temperature exponents are presented. Received on 8 July 1996     

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

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

The effect of a nonlinear temperature dependent Prandtl-number on heat transfer of fully developed flow of liquids in a straight tube

If Nu_o is the Nusselt Number for a temperature-independent Prandtl number Pr, and Nu the Nusselt number for a temperature dependent Prandtl number, it is usual to define the correction factor Nu/Nu_o=C. A correction factor which has been defined in this form has, up to now, only been expressed as a function of two characteristic Pr numbers. Thus it was simply assumed that the Pr number was a linear function of the temperature. Fluids with very large Pr numbers show a (T;Pr) relationship which deviates considerably from a linear one. This leads to a very large difference (up to 70%) between the calculated and the measured values of the Nusselt number. In the following study we shall determine a so-called curvature parameter of the (T;Pr) curve and obtain a semi-empirical formula for C. This formula has a deviation from the actual relationship many times smaller than that of the formulae for a linear (T;Pr) relationship.

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Zusammenfassung Ist Nu_o die Nusseltzahl bei temperaturunabhangiger und Nu die Nusseltzahl bei temperaturabhangiger Prandtlzahl Pr, so ist es üblich, mit Nu/Nu_o=C den Korrekturfaktor zu bezeichnen. Ein in dieser Form definierter Faktor C ist bisher als Funktion nur zweier charakteristischer Pr-Zahlen ausgedrückt worden. Es wurde somit nur eine lineare Abhängigkeit von der Pr-Zahl von der Temperatur T vorausgesetzt. Flüssigkeiten mit großen Pr-Zahlen weisen eine (T;Pr)-Charakteristik auf, die sehr stark von der linearen abweicht. Sehr große Abweichungen (bis — 70%) der gerechneten von den gemessenen Nu-Zahlen sind eine Folge davon. In vorliegender Arbeit bilden wir mit einer dritten charakteristischen Pr-Zahl einen sogenannten Krümmungsparameter der Kurve (T;Pr) und leiten eine semiempirische Formel für C ab, die um ein großes Vielfaches kleinere Fehler aufweist, als die Formeln für den linearen (T;Pr)-Verlauf.

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Nomenclature

Material constants c_p specific heat at constant pressure [J/kgK] - k heat conductivity [W/mK] - density [kg/m³] - a temperature diffusivity, a=k/c_p [m²/s] - dynamic viscosity [Ns/m²] - kinematic viscosity [m²/s] Fluid dynamics D inner diameter of the tube [m] - L length of tube [m] - w mean speed of fluid [m/s] Heat transfer h coefficient of heat transfer [W/m²K] - T absolute temperature [K] - T_b bulk temperature (corresponding to the adiabatic mixing temperature) [K] - T_w tube wall temperature [K] - T_f=(T_b+T_w)/2 film temperature [K] - T=T_b-T_w temperature forcing difference of heat transfer [K] Characteristic quantities without dimensions Re=wD/ Reynolds number - Pr=/a Prandtl number - Nu=hD/k Nusselt number - related temperature - related Prandtl number - curvature parameter of the Prandtl number Various - C=Nu_b/Nu_o correction factor according to Eq.(5) - p exponent in Eq.(6), (a), (8) and (16) Indices o corresponding to the quasi-isothermal heat transfer - b,w,f with reference to quantities, including physical properties which correspond to the temperatures T_b, T_wor T_f - Pr,k,, for quantities calculated corresponding to the Prandtl number, the thermal conductivity coefficient, the density or the dynamic viscosity - H,C for heating or cooling exp for quantities calculated from experimental data