Study of the analytical solution to the heat transfer problem and surface temperature in a semi-infinite body with a constant heat flux at the surface and an initial temperature distribution

In many practical cases, one heats a semi-infinite solid with a constant heat flux source. For such an unsteady heat transfer problem, if the body has a uniform initial temperature, the analytical solution has been given by Carslaw and Jaeger. The surface temperature of the semi-infinite body follows the $\sqrt t $ -rule, that is, the surface temperature changes in proportion to square root of heating time. But if, instead of the uniform initial temperature, the body has a temperature distribution at the beginning of heating, the analytical solution has not yet been developed. Analytical solutions to the same problem with an exponential or a linear initial temperature distribution are obtained in this paper. It is shown, that in the case of a linear initial temperature distribution the surface temperature also changes according to $\sqrt t $ -rule Approximating the initial temperature distribution near the surface by its tangent at the surface, it is found that the surface temperature within a short time after the start of heating should also satisfy the $\sqrt t $ -rule, in spite of an arbitrary initial temperature distribution. The experimental data support this argument. Furthermore, the constant heat flux can be calculated after relationship between the surface temperature and heating time according to the equation derived in this paper, if the initial temperature distribution or its first-order derivative at the surface is known.     

A note on the free convection boundary layer on a horizontal circular cylinder with constant heat flux

The equations governing the free convection boundary-layer flow on a horizontal circular cylinder on which there is a prescribed surface heat flux are solved using a finite-difference scheme. This numerical solution is then used to compare the accuracy of two proposed series expansions, a Blasius expansion and a Görtler-type expansion. It is shown that the former method is better at estimating temperature profiles while the latter is better at estimating velocity profiles.

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Bemerkung über die freie Konvektionsgrenzschicht an einem horizontalen Kreiszylinder mit gleichförmigem Wärmestrom

Zusammenfassung Die Grenzschichtgleichungen für die freie Konvektion an einem horizontalen Kreiszylinder mit gleichförmigem Wärmestrom durch die Oberfläche wird mit Hilfe eines finiten Differenzverfahrens gelöst. Die numerisch ermittelten Ergebnisse werden nachher für den Vergleich der Genauigkeit von zwei Reihendarstellungen der Lösung der Grenzschichtsgleichungen benützt. Diese Reihen sind vom Blasiusbzw. Görtier-Typ. Es wird bemerkt, daß die Reihendarstellung von Blasius die Temperaturprofile besser beschreibt, während die Reihenentwicklung der Görtlerschen Art für die Geschwindigkeitsprofile eine gute Übereinstimmung mit der exakten Lösung zeigt.

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Nomenclature a radius of the cylinder - g acceleration of gravity - P _r Prandtl number - Q prescribed (constant) heat flux - T temperature of the fluid - t ₀ temperature of the ambient fluid - u velocity in thex-direction - v velocity in they-direction - x co-ordinate measuring distance round the cylinder - y co-ordinate measuring distance normal to the cylinder - G _r Grashof number=g Q a ⁴/v² - coefficient of thermal expansion - x thermal conductivity - v kinematic viscosity - _w skin friction     

Treatment of moisture condensation on fins

An analysis of natural convection from a vertical plate fin when the fin base temperature is below the dew point of the surrounding air is presented in this paper. The analytical solution derived is based upon a constant heat and mass transfer coefficient and is also valid for forced convection. The results of this simplified theory are compared with a numerical solution where the coupling of convection and conduction is taken into account. An experimental verification of the results is also shown.

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Aus Kondensation von Feuchtigkeit an Rippen

Zusammenfassung Es wird eine Analyse der freien Konvektion an einer vertikalen plattenförmigen Rippe dargestellt, bei der die Temperatur im Anfangsbereich der Rippe unterhalb des Taupunktes der umgebenden Luft liegt. Die abgeleitete analytische Lösung beruht auf einem konstanten Wärme- und Stoffübergangskoeffizienten und gilt auch für die erzwungene Konvektion. Die Resultate dieser vereinfachten Theorie werden mit einer numerischen Lösung verglichen, in der die Verbindung von Konvektion und Wärmeleitung in Betracht gezogen wird. Angeführt wird auch eine experimentelle Bestätigung der Resultate.

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Nomenclature a _f thermal diffusivity of air - A, B constants in Eq. (7) - c constant defined in Eq. (3) - D diffusion coefficient - f an arbitrary function ofT andx in Eq. (12) - F ₁,F ₂ coefficients in differential Eq. (13) - g gravitational acceleration - h heat transfer coefficient - h _m mass transfer coefficient - k thermal conductivity of fin - k _f thermal conductivity of air - l latent heat of moisture condensation - L total length of fin - L _w length of wet fin - m parameter, (h/kt)^1/2 - m _l dimensionless parameter, 1+ B/T _r - m _y parameter,m m _l ^1/2 - p pressure of surrounding air - p _ws saturation pressure of water vapor - p _w partial pressure of water vapor in air - Pr Prandtl number,/a _f - q total heat fluxl - q _c convective heat flux - q _m heat flux - q _r radiative heat flux - R parameter in Eq. (14) - R _w specific gas constant of water vapor - t half thickness of fin - T temperature - T _b base temperature of wet fin - T _c base temperature of dry fin=saturation temp. of vapor - T _r reference temperature defined in Eq. (15) - T temperature of surrounding air - T temp, difference between fin surface and surroundings - v initial temperature for quasilinearization - x vertical coordinate, see Fig. 1 - y horizontal coordinate, see Fig. 1 - coefficient of thermal expansion - emissivity - dimensionless parameter in Eq. (14) - ø _d heat flux of dry fin - ø _tot total heat flux of dry-wet fin - kinematic viscosity - Stefan-Boltzman coefficient - relative humidity of air     

Laminar boundary layer of infinitely long elliptic cylinders for an arbitrary yaw angle

Results are presented of the calculation of the laminar boundary layer on infinitely long elliptic cylinders in a supersonic perfect gas flow at an arbitrary angle of attack. It was assumed that the Prandtl number is constant and equal to 0.7, the dynamic viscosity coefficient follows a power-law variation ( T^0.76) with temperature, and there is high heat transfer at the body surface (H_1w=0.05).The calculations showed that a change of the body shape—the ellipticity coefficient =b/a—has a significant effect on the nature of the distribution and the magnitude of the local heat flux.In evaluating the thermal fluxes at the blunt leading edges, swept wings are usually considered as infinitely long yawed cylinders. In studying heat transfer at the surface of bodies of small aspect ratio at high angles of attack, wide use is made of the hypothesis of plane sections, when each section, orthogonal to the longitudinal axis of the body, is considered equivalent to a corresponding yawed infinite cylinder.By now quite detailed studies have been made of the behavior of the boundary layer on an infinitely long yawed circular cylinder with both the laminar and turbulent flow regimes for a compressible gas [1, 2]. However, there are no data on the heat transfer at the surface of a yawed infinite cylinder with arbitrary cross section, although the availability of such data is urgently needed, for example, for the proper selection of the form of the leading edges of the swept wing.This article presents the results of the calculation of the characteristics of the laminar boundary layer on the surface of infinite elliptic cylinders in a supersonic perfect gas flow. The calculations were made over a quite wide range of flight Mach number M, yaw angle , and ellipticity factor . The data presented on the distribution of the relative heat flux along the cylinder directrix may be used also for estimating the heat flux with account for the real properties of air if we know the corresponding value of the heat flux in the vicinity of the stagnation line.     

Prediction of roll temperature with a non-uniform heat flux at tool and workpiece interface

In a metal forming process, plastic deformation of the workpiece takes place at tool and workpiece interface region. Tool has been identified as one of the key parameters in controlling the productivity of any manufacturing industry. The deformation of metals and friction at the contact region produce large amount of heat, a part of that heat is conducted towards the tool where it is removed by forced convection. These cooling and heating cycles finally result in a substantial change in the temperature distribution in the roll. In this paper, an attempt is made to study the temperature and heat flux distribution in the roll by considering a non-uniform heat flux at the roll-workpiece interface for a cold rolling process. Adopting an elemental approach, a methodology has been proposed to model non-uniform heat flux at the interface. For this purpose both tool and workpiece has been considered together, thus a coupled approach is used to model both deformation and heat transfer phenomenon. It is demonstrated that the present approach of modeling is more general than that available in the literature. For example, a constant value of heat flux at the interface that is considered by several investigators is shown to be a special case of the present investigation, particularly when the deformation and relative velocity is very small. It is shown that the error in maximum temperature associated with constant heat flux assumption could be more than 5% in situations when reduction and relative velocity is high. The results are presented for temperature and heat flux distributions in the roll for different operating conditions.a thermal diffusivity, (m²/sec) - B pre-strain coefficient - C yield stress at unit strain, (N/m²) - e rate of deformation heat generation per unit volume, (W/m³) - f friction factor - h heat transfer coefficient, (W/m² °C) - k thermal conductivity, (W/m °C) - K yield stress at unit strain, (N/m²) - L bite length, (m) - n strain hardening exponent - P pressure between tool and workpiece, (N/m²) - q heat flux, (W/m²) - q_f friction heat flux, (W/m²) - heat flux entering towards the roll for any arbitrary element j (W/m²) - R roll radius, (m) - S_o yield stress in plane strain, (N/m²) - T temperature difference (T = T_r – T_o), (°C) - T surrounding temperature, (°C) - y strip thickness, (m) - V_rel relative slipping velocity, (m/sec) - V velocity, (m/sec) - Pe Peclet number - Bi Biot number - _T Total bite angle - mean effective strain - mean true stress, (N/m²) - mean strain rate - friction stress, (N/m²) - coefficient of friction - angle between heating and cooling regions - angle of cooling spray region - r, polar coordinates - x, y Cartesian coordinates - o initial value - f final value - r related to roll - s related to strip - a average value - j elemental region     

Transient temperatures from periodic surface heat flux for slabs,cylinders, and spheres

The transient temperatures resulting from a periodically varying surface heat flux boundary condition have numerous applications. In this work, explicit, analytic solutions are presented for the transient surface and medium temperatures due to periodically varying step changes in surface heat flux for geometries such as a slab, cylinder, and sphere. The nonlinear case allowing for the added effects of radiation from the surface into an external ambient are studied numerically.

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Temperaturschwankungen aus periodischen Änderungen des Wärmeflusses an der Oberfläche von Platten, Zylindern und Kugeln

Zusammenfassung Temperaturschwankungen herrührend von periodischen Änderungen des Wärmeflusses an der Oberfläche und der Grenzschichtbedingungen haben zahlreiche Anwendungen. In dieser Arbeit wird eine explizite analytische Lösung für die transienten Temperaturen an der Oberfläche und in der Mitte von Platten, Zylindern und Kugeln angegeben, die durch periodische stufenweise Änderungen des Wärmeflusses an der Oberfläche entstehen. Der nichtlineare Fall mit zusätzlichem Einfluß der Wärmestrahlung in die Umgebung wurde numerisch studiert.

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Nomenclature f ₀ reference heat flux - f() dimensionless applied surface heat flux=q(t)/f₀ - F _i dimensionless stepchange in surface heat flux for linear problem - J _i (z) Bessel function - k thermal conductivity - L half thickness of slab, half radius of cylinder and sphere - N conduction-to-radiation parameter= - P period of on-off surface heat flux - q (t) applied surface heat flux - t time - T(x, t) temperature - T _r reference temperature=(f ₀/)^1/4 - U(z) unit step function - x physical distance Greek symbols thermal diffusivity - _m eigenvalues - ₀ surface emissivity - dimensionless spacial distance=x/2L - (, ) dimensionless temperature=T/T _r - 0 ₀ dimensionless initial temperature - _i dimensionless times at which step changes in surface heat flux occur - dimensionless time=t/2 L² - Stefan-Boltzmann constant - fraction of periodP during which the surface heat flux is non-zero - (, ) dimensionless temperature     

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

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

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

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

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

Thermal wave propagation within a medium with the inert heat source

A heat conduction equation of a new type is derived which takes into account the finite velocity of heat flux propagation and the relaxation of heat source capacity. The equation is solved for a semi-infinite body and a step change in temperature at the surface. The analysis shows that as the time increases the obtained solution moves from the solution of the classical hyperbolic equation without energy generation towards the solution of the classical hyperbolic equation with energy generation.

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Ausbreitung thermischer Wellen in einem Medium mit träger Wärmequelle

Zusammenfassung Es wird eine neuartige Wärmeleitungsgleichung abgeleitet, welche die endliche Geschwindigkeit der Ausbreitung des Wärmestromes und die Relaxation der Kapazität der Wärmequelle berücksichtigt. Die Gleichung wird für einen halbunendlichen Körper und eine schrittweise Temperaturänderung an der Oberfläche gelöst. Die Analyse zeigt, daß mit zunehmender Zeit sich die Lösung der klassischen hyperbolischen Gleichung ohne Wärmeerzeugung in eine solche mit ebenfalls klassischer hyperbolischer Gleichung mit Wärmeerzeugung wandelt.

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Nomenclature a thermal diffusivity,k/( c _p - c _p specific heat at constant pressure - C speed of heat propagation - C ₁,C ₂ constants - k thermal conductivity - q _v steady capacity of internal heat source - q _vd transient capacity of internal heat source - r ₁,r ₂ roots of characterisitc equation - t time - t _k relaxation time of heat flux - t _q relaxation time of internal heat source capacity - T temperature - T ₀ surface temperature - u() unit step function - x, y, z Cartesian coordinates - X dimensionless coordinate - , constant coefficients - dimensionless temperature - density - dimensionless time - _r-t_qt_k ratio of relaxation times - dimensionless steady capacity of internal heat source - _d dimensionless transient capacity of internal heat source     

Unsteady free convection on a vertical cylinder with variable heat and mass flux

The unsteady natural convection boundary layer flow over a semi-infinite vertical cylinder is considered with combined buoyancy force effects, for the situation in which the surface temperature T ^′ _w(x) and C ^′ _w(x) are subjected to the power-law surface heat and mass flux as K(T ^′/r) = −ax ⁿ and D(C ^′/r) = −bx ^m . The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson method. Numerical results are obtained for different values of Prandtl number, Schmidt number ‘n’ and ‘m’. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt and Sherwood numbers are shown graphically. The local Nusselt and Sherwood number of the present study are compared with the available result and a good agreement is found to exist. Received on 7 July 1998     

Free convection above a horizontal circular disk in a saturated porous medium

The boundary-layer flow above a horizontal impermeable circular disk embedded in a saturated porous medium is considered in the cases both when the disk is held at a constant temperature above ambient and when heat is supplied to the convective fluid by the disk at a constant rate. Series solutions are obtained based on the flat plate solution, which holds at the edge of the disk, as the leading order terms. These series solutions can then be used to describe the flow nearly all the way across the disk. A simple approximate solution, based on an integrated form of the energy equation, is also obtained and is shown, for the constant wall temperature case, to be useful in indicating how the solution behaves near the centre of the disk. The solutions asr0, wherer measures distance from the centre is discussed in both cases, and it is shown that the solution develops a singularity with the boundary layer having a thickness of 0[(–logr)^1/2].

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Freie Konvektion über einer waagerechten Kreisscheibe in einem gesättigten porösen Medium

Zusammenfassung Es wird die Grenzschicht über einer waagerechten undurchlässigen Kreisscheibe, die in einem gesättigten porösen Medium eingebettet ist, untersucht. Zwei Möglichkeiten werden betrachtet: bei der einen ist die Oberflächentemperatur der Scheibe konstant, aber größer als die der Umgebung, bei der anderen ist die Scheibe gleichförmig beheizt. Es werden Lösungen in Form von Reihenentwicklungen erzielt, wobei die ersten Glieder den Lösungen einer ebenen Platte entsprechen, die an dem Rand der Scheibe gültig sind. Danach werden diese Lösungen zur Beschreibung der Bewegung der Flüssigkeit über einen großen Bereich der Scheibe benutzt. Eine einfache Lösung für den Fall der konstanten Wandtemperatur wird durch Integration der Energiegleichung erhalten und zur Beschreibung des Verhaltens der Lösung in der Nähe der Scheibenmitte verwendet. Man untersucht die Lösung fürr0 in beiden betrachteten Fällen, wobeir der Abstand von der Scheibenmitte ist. Bei einer Dicke von 0[(–logr)^1/2] der Grenzschicht weist die Lösung eine Singularität auf.

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Nomenclature a radius of disk - g acceleration due to gravity - K permeability of the porous medium - k thermal conductivity - q prescribed wall heat flux - Q (non-dimensional) heat transfer coefficient - r, coordinate measuring distance from the centre of the disk - R _a,R _a ^* Rayleigh number - T temperature of the convective fluid - T ₀ ambient temperature - T ₁ prescribed wall temperature - u Darcy's law velocity in ther-direction - V _w (non-dimensional) wall velocity - w Darcy's law velocity in thez-direction - x coordinate measuring distance from the edge of the disk - z, coordinate measuring distance normal to the disk - equivalent thermal diffusivity - coefficient of thermal expansion - non-dimensional temperature - _w (non-dimensional) wall temperature - viscosity of the convective fluid - stream-function - stream function at the edge of the boundary layer     

Development of mixed convection flow over a vertical plate due to an impulsive motion

The development of the mixed convection flow of an incompressible laminar viscous fluid over a semi-infinite vertical plate has been investigated when the fluid in the external stream is set into motion impulsively, and at the same the surface temperature is suddenly raised from its ambient temperature. The problem is formulated in such a way that at time t = 0, it reduces to Rayleigh type of equation and as time t , it tends to Blasius type of equation. The scale of time has been selected such that the traditional infinite region of integration becomes finite which significantly reduces the computational time. The nonlinear coupled singular parabolic partial differential equations governing the unsteady mixed convection flow have been solved numerically by using an implicit finite-difference scheme. The surface shear stress and the heat transfer increase or decrease with time when the buoyancy parameter is greater or less than a certain valve. There is a smooth transition from the initial steady state to the final steady state. The skin friction and heat transfer for the constant heat flux case are more than those of the constant wall temperature case. Also they increase with the buoyancy force.     

Effect of surface layers on the constriction resistance of an isothermal spot. Part I: Reduction to an integral equation and numerical results

The problem of the constriction resistance of a circular spot on a half-space covered with a uniform layer of different material is considered. For the general case of any specified axisymmetric distributions of temperature over the spot and heat flux over the rest of the surface, the mixed boundary value problem governing the heat flow from the spot to the underlying layer-substrate composite is converted to a non-homogeneous Fredholm integral equation of the second kind. For the particular case of isothermal spot on otherwise insulated surface, the evaluation of the constriction resistance is reduced to the reciprocal of a simple integral with the solution of the relevant integral equation as integrand. The integral equation is solved numerically and very accurate results are obtained for the constriction resistance over four orders of magnitude variation of the ratio, , of layer thickness to spot radius and the ratio, k_r, of layer to substrate conductivities for both conducting (k_r > 1) and insulating (k_r < 1) layers. An extensive discussion of the numerical results is presented with particular emphasis on their implications for the contact resistance of practical joints in the presence of interfacial layers. Further, in the light of the numerical results, two widely used analytical approximations for the constriction resistance – the first of which results from replacing the isothermal condition over the spot by a special flux (herein called the Equivalent Isothermal Heat Flux, EIHF) condition which is believed to render the spot nearly isothermal and the second is a consequence of the assumption (herein termed the Thin Insulating Layer Approximation, TILA) that, for thin insulating layers like oxide films, the heat flow in the layer region right beneath the spot is purely axial – are assessed as to their levels of accuracy and ranges of applicability with respect to both and k_r.     

The effects of temperature-dependent fluid properties on free convective flow along a semi-infinite vertical plate by the presence of radiation

The effects of temperature-dependent density, viscosity and thermal conductivity on the free convective steady laminar boundary layer flow by the presence of radiation for large temperature differences, are studied. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The usual Boussinesq approximation is neglected due to the large temperature difference between the plate and the fluid. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the shooting method. The effects of the density/temperature parameter n, the thermal conductivity parameter , the viscosity/temperature parameter _r and the radiation parameter F are examined on the velocity and temperature fields as well as the coefficient of heat flux and the shearing stress at the plate.     

Theoretical analysis of convective heat transfer enhancement of microencapsulated phase change material slurries

This paper analyzes the convective heat transfer enhancement mechanism of microencapsulated phase change material slurries based on the analogy between convective heat transfer and thermal conduction with thermal sources. The influence of each factor affecting the heat transfer enhancement for laminar flow in a circular tube with constant wall temperature is analyzed using an effective specific heat capacity model. The model is validated with results available in the literature. The analysis and the results clarify the heat transfer enhancement mechanism and the main factors influencing the heat transfer. In addition, the conventional Nusselt number definition of phase change slurries for internal flow is modified to describe the degree of heat transfer enhancement of microencapsulated phase change material slurries. The modification is also consistent evaluation of the convective heat transfer of internal and external flows.c volumetric concentration of microcapsules - c_m mass concentration of microcapsules - c_p specific heat, kJ kg^–1 K^–1 - h_fs phase change material heat of fusion, kJ kg^–1 - h_m* modified convective heat transfer coefficient, W m^–2 K^–1 - k thermal conductivity, W m^–1 K^–1 - k_e effective thermal conductivity of slurry, W m^–1 K^–1 - k_b slurry bulk thermal conductivity, W m^–1 K^–1 - ML dimensionless initial subcooling - Mr dimensionless phase change temperature range - Nu conventional Nusselt number - Nu* improved Nusselt number - q_wⁿ wall heat flux, Wm^–2 - Pe Peclet number - Pr Prandtl number - Re Reynolds number - r radial coordinate, m - r₀ duct radius, m - r₁ dimensionless radial coordinate - Ste Stefan number - T temperature, K - T₁ lower phase change temperature limit, K - T₂ upper phase change temperature limit, K - T_i slurry inlet temperature, K - u axial velocity, m/s - v radial velocity, m/s - x axial coordinate, m - x₁ dimensionless axial coordinate - thermal diffusivity, m²/s - dimensionless temperature - dynamic viscosity, N·s/m² - kinematic viscosity, m²/s - _t width of thermal boundary, m - degree of heat transfer enhancement, = h_m*/(h_m*)_single - b bulk fluid (slurry) - b0 slurry without phase change - l liquid - m mean - s solid - f suspending fluid - p microcapsule particles - w wall - single single-phase fluid     

Bubble growth predictions by the hyperbolic and parabolic heat conduction equations

A model for bubble growth in a uniformly superheated liquid is presented which is valid for both inertia and heat diffusion controlled growth. Two different heat transfer equations are considered: The Fourier (parabolic) equation and the hyperbolic heat conduction equation. It is shown that for short times, bubble growth prediction based on the Fourier equation, differs considerably from that based on the hyperbolic heat conduction equation. For long times, both predictions coincide. Using the hyperbolic heat conduction equation is important for bubble growth prediction in fluids like Helium II, in which thermal disturbances have a low speed of propagation. In such liquids the second sound effects must be considered long after the inertia and dynamic effects become unimportant.The validity of using a semi-infinite approximation to the heat conduction problem during the bubble growth period is investigated. An analytical solution in spherical coordinates reveals that the ratio between the spherical and semi-infinite solutions is a function of the Jakob number. Results of the present model, using the Fourier equation, are shown to be in better agreement with data for bubble growth in water, than other published solutions.

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Beschreibung des Blasenwachstums durch Wärmeleitungs-Gleichungen von hyperbolischer und parabolischer Form

Zusammenfassung Es wird ein Modell für Blasenwachstum in überhitzter Flüssigkeit vorgestellt, das sowohl bei durch Trägheit als auch bei durch Wärmediffusion kontrolliertem Blasenwachstum verwendbar ist. Zwei unterschiedliche Wärmeübertragungsbeziehungen werden in Betracht gezogen: Die Fourier-Gleichung (parabolisch) und eine Wärmeleitungs-Gleichung in hyperbolischer Form.Es wird gezeigt, daß die Modellergebnisse basierend auf der Fourier-Gleichung für schnelle Blasenwachstumszeiten signifikant von vergleichbaren Ergebnissen basierend auf der hyperbolischen Gleichung abweichen, während sie für langsames Wachstum mehr oder weniger identisch sind. Die Verwendung der hyperbolischen Wärmeleitungsgleichung in Blasenwachstumsmodellen ist vor allem in Fluiden wie Helium II von Bedeutung, wo thermische Störungen eine geringe Ausbreitungsgeschwindigkeit haben. Hier müssen die second sound-Effekte noch berücksichtigt werden, wenn die dynamischen und die Einflüsse der Trägheit schon vernachlässigbar sind.Es wurde untersucht, ob die Benutzung einer semi-unendlichen Approximation des Wärmeleitungsproblems während des Blasenwachstums zulässig ist. Eine analytische Lösung in Kugelkoordinaten zeigt, daß das Verhältnis zwischen letzteren und semi-unendlichen Ergebnissen eine Funktion der Jakob-Zahl ist.Schließlich wird gezeigt, daß die Resultate des vorgestellten Modells bei Benutzung der Fourier-Gleichung experimentelle Ergebnisse von Blasenwachstum in Wasser besser wiedergeben als andere bekannte Lösungen.

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Nomenclature a thermal diffusivity - B _s sphericity correction factor - b temperature decay coefficient - c propagation speed of thermal disturbances - E parameter, Eq. (37) - f function of the dimensionless time and bubble radius, Eq. (34) - h _v heat of evaporation - Ja Jakob number, Eq. (35) - k thermal conductivity - N /T - P pressure - P _i initial system pressure - P _v vapour pressure - Q* dimensionless heat flux (Stanton number) - q heat flux - transformed heat flux - q _wL heat flux into the liquid at the bubble boundary - R bubble radius - R* dimensionless bubble radius, Eq. (16) - R ₀ initial (critical) bubble radius - r radial coordinate - s the Laplace transform parameter - T temperature - T _i initial liquid temperature - T _s saturation temperature - T _v instantaneous bubble temperature - T ₀ initial saturation temperature,T _s (0) - T temperature difference,T _i–T _s (0) - t time - t* dimensionless time, Eq. (16) - y dimensionless distance from the bubble surface - Z constant of integration, Appendix A - a proportionality constant - temperature function, Eq. (8) - transformed temperature function - _v vapour density - _L liquid density - _vi initial vapour density - relaxation time,a/c ² - normalized temperature distribution, Eq. (15)     

Thermodynamic analysis of the one-dimensional heat conduction with inhomogeneities in the heat flux direction

Starting from the results of Li, Prigogine and others about the one-dimensional heat conduction with constant temperature boundary conditions, the aim of this paper is to study, according to the methods and the purposes of the generalized thermodynamics, the more general case of one-dimensional heat conduction in systems, whose conductivity is function of both temperature and the coordinate in the heat flux direction and presents a finite number of discontinuities.

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Thermodynamische Analyse für eindimensionale Wärmeströmung mit Ungleichartigkeiten in der Wärmeflußleitung

Zusammenfassung Ausgehend von den Ergebnissen von Li, Prigogine und anderen, für eindimensionale Wärmeströmung mit konstanten Temperaturen an den Grenzen, versucht diese Arbeit, gemäß den Methoden und den Zielen der verallgemeinerten Thermodynamik, den allgemeineren Fall der eindimensionalen Wärmeströmung mit Ungleichartigkeiten zu examinieren.

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Nomenclature c volumetric specific heat - G discriminating parameter [G (t) =k ₂ (t)t ² orG _i (t)=k_i(t) t², when the separation of the variables for thermal conductivity can be done, or in general: G(x, t)=k(x,t)t ²] - J heat transfer rate (generalized flux) [W] - J ₀ heat flux [W/m²] - k thermal conductivity - k ₁ component of thermal conductivity depending upon the coordinate in the heat flux direction - k ₂ component of thermal conductivity depending upon temperature - k _i component of the thermal conductivity of a homogeneous layer (i) depending upon temperature - L ₁,L ₂ extreme coordinates - Lip Lipschitz's function - P entropy production rate - (P(T))_min temperature distribution in a system corresponding to the minimum of entropy production rate - p ₁ thermokinetic potential - (P ₁ (T))_min temperature distribution in a system corresponding to the minimum of thermokinetic potential - P ₂ generalized force potential - Q differential form - dQ total differential form - ¯R set of all real numbers - S(x) area of the isothermal surface corresponding to the coordinatex - T system temperature distribution - t absolute temperature - ¯T set of all the possible system temperature distributions - x symbol of Cartesian product - x coordinate in the heat flux direction - X local generalized force - B,Y ₀,Y ₀ ^*,Z sets (defined in this paper) - function representing the time evolution of the temperature distribution in a system - time - ₀ reference time interval     

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     

The extended Graetz problem with arbitrary boundary conditions in an axially heat conducting tube

An analytical solution for the extended Graetz problem with prescribed wall temperature, heat flux or a linear combination of temperature and heat flux is presented. The present solution is derived from a superposition principle which is proposed here in order to reduce the problem to equivalent integral equations. The complex Fourier transform and the convolution theorem is used to obtain analytical expansions in terms of exponential functions for the temperature distribution and heat flux at the wall. The solutions obtained in this way are proved to be mathematically simple and easy to handle in small sized computers. A general solution for the particular geometry of a pipe is presented. This solution is extended to the case of an axially heat conducting pipe. The combined effect of both the Péclet number and the wall to the fluid conductivity ratio on the temperature and the heat flux distributions at the wall is analysed.Nomenclature e thickness of the tube wall - g complex function given by Eqs. (31) and (46) - h film coefficient - k thermal conductivity of the fluid - k _s thermal conductivity of the wall - q _i heat flux at the inner wall - q _o heat flux at the outer wall - r radial coordinate - v _x velocity component in the axial direction - x axial coordinate - B _i Biot number (=hR/k) - C _p fluid specific heat at constant pressure - K conductance ratio (=k _se/kR) - M(a, b, z) Kummer's function - O(x) function of order x - P _e Péclet number (=c _pVR/k) - R tube inner radius - T _b temperature of the environment outside the pipe - T _s wall temperature - T ₀ fluid temperature upstream at x=– - T _f fluid temperature downstream at x=+ - V characteristic velocity - _m ^± characteristic roots for >0 and <0 - dimensionless axial coordinate (=x/P _eR) - dimensionless radius (=r/R) - dimensionless temperature (=(T–T ₀)/(T _f–T ₀)) - _s dimensionless wall temperature (=(T _s-T₀)/(T _f-T₀)) - _n modified eigenvalues (= _n P _e) - dimensionless velocity (=v _x/V) - fluid density - dimensionless heat flux (=q _iR/k(T _f–T ₀)) - dimensionless heat flux (=q _o R/k(T _f–T ₀)) - complex number     

The temperature field in rotational rheometers and flow curve correction for viscous dissipation

A solution is presented for incompressible non-Newtonian liquids of the one-dimensional stationary temperature field which arises due to heat dissipation between two concentric cylinders, the outer fixed and thermostated, the inner rotating at a constant angular velocity. The object of the study is to outline a simple procedure for determining the temperature rise of the liquid and, primarily, to ascertain the corrections of the consistent variables and D which enable the experimenter to rectify the rheogram on the basis of measurement of the shear stress and the angular velocity . The results obtained are summarized in graphical form as diagrams of the temperature and velocity fields and, to facilitate practical application of the correction procedure, in a table relating the dimensionless temperature function (, n, ) to the geometry , the flow behaviour index n, and the coefficient of temperature rise and showing the function (1) as well.List of symbols a radius of the inner cylinder - b radius of the outer cylinder - constant angular velocity of the inner cylinder - r* dimensionless radial coordinate r/b - * dimensionless angular velocity of the liquid - K fluid consistency index - n flow behaviour index - dimensionless temperature rise (T–T ₀)/T ₀ - T temperature of measured liquid (K) - T ₀ temperature of the thermostated bath - Br Brinkman criterion - _f thermal conductivity of liquid - C constant of integration - coefficient of sensitivity in consistency-temperature law - coefficient of sensitivity divided by flow behaviour index: /n - (r*) dimensionless temperature function - coefficient of temperature rise; =Br· - ratio of the radii of inner and outer cylinder - T(1) temperature on the inner wall of the outer cylinder, i.e. for r*=1 - outer cylinder wall thickness - coefficient of heat transfer - q heat flux - k overall heat transfer coefficient - h height of measured liquid - _s thermal conductivity of the outer cylinder - (1) derivative of the dimensionless temperature function at point r*=1 - dimensionless heat transfer constant - _i (r*) dimensional temperature function calculated for isothermal wall; T(1)=T ₀ - dynamic viscosity - _i () maximum value of the dimensionless temperature function - dimensionless symbol — ratio of C/C ₀ - D rate of shear - shear stress - rate of shear (not considering dissipation) - shear stress (not considering dissipation) - D ⁺ rate of shear corrected for the inner cylinder temperature - ⁺ shear stress on the inner cylinder obtained by measurement on the rheometer used - _j rate of shear on the inner cylinder for j-th measurement referred to a single constant temperature     

Free-forced convection from a heated longitudinal horizontal cylinder embedded in a porous medium

The flow and heat transfer from a heated semi-infinite horizontal circular cylinder which is moving with a constant speed into a porous medium is considered. It is assumed that the Grashof and Reynolds numbers are large so that the governing equations are the three dimensional boundary-layer equations. A numerical procedure for solving these equations is described and the asymptotic solutions which are valid both near and distant from the leading edge of the cylinder are presented. The range of validity of these asymptotic solutions is discussed and the results are compared in detail with the full numerical solution. The problem is of practical importance, for example in the drilling of pipes into a geothermal reservoir.

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Freie erzwungene Konvektion von einem beheizten schlanken horizontalen Zylinder, eingebettet in ein poröses Medium

Zusammenfassung Es wird die Strömung und der Wärmeübergang an einem beheizten, halbunendlichen horizontalen Kreiszylinder betrachtet, der mit konstanter Geschwindigkeit sich in ein poröses Medium bewegt. Dabei wird angenommen, daß die Grashof- und Reynolds-Zahlen groß sind, so daß die Bestimmungsgleichungen von den dreidimensionalen Grenzschichtgleichungen gebildet werden. Es wird ein numerisches Verfahren zur Lösung dieser Gleichungen beschrieben und eine asymptotische Lösung präsentiert, die sowohl in der Nähe als auch in großem Abstand von dem vorderen Ende des Zylinders gültig ist. Der Gültigkeitsbereich dieser asymptotischen Lösungen wird diskutiert und die Ergebnisse werden im Detail mit vollständigen numerischen Lösungen verglichen. Das Problem ist z.B. beim Eindringen von Rohrleitungen in geothermische Reservoire von praktischer Wichtigkeit.

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Nomenclature a radius of cylinder - Gr Grashof number (=g(T_w-Ta/²) - g acceleration due to gravity - permeability in the porous medium - Nu local Nusselt number - total heat flux from cylinder - q _w heat flux from cylinder - r radial co ordinate - Ra Rayleigh number (=g (T_w - T_t8) a/ ) - Re Reynolds number (=U _t8 a/) - T temperature - u, v, w speeds inx, , r directions - x axial co ordinate - equivalent thermal diffusivity - thermal expansion coefficient - ratioGr/Re - similarity variable - dimensionless temperature (=(T- T)/(T _w- T) - kinematic viscosity - azimuthal co ordinate - w cylinder surface - free stream