Modélisation en régime transitoire des mouvements convectifs dans une cellule d'habitation

This study aims thermal phenomena modelling inside a dwelling cell in order to determine temperature distribution there. A coupling of the zonal method and integral analysis in walls vicinity is carried out. The zonal method lies on breaking up the cell into isothermal zones. Mass and energy balances are established, in transient flow, for each zone. The various convective transfer types are listed and studied: transfer between air and walls, transfer between air layers, transfer between air and cold air jet, transfer between air and transmitter. Conductive transfer through walls and radiative transfer inside the cell are also taken into account. An experimental validation campaign is also achieved in a testing room, two heat transmitter types are tested: distributed and located. Validation results are satisfying and air diffusion importance is emphasised. The influence of air nodes number is also studied.     

Modélisation par éléments finis du chauffage infrarouge des membranes thermoplastiques semi-transparentes

This paper reports on a simplified approach for analysing the temperature evolution in a semi-transparent thin membrane of the amorphous polyethylene terephtalate type (PET) exposed to a radiative source. It is based on a 3D finite elements method. The thermophysical properties of the PET are assumed independent of temperature while the internal radiative intensity absorption is taken as one-dimensional and is governed by the Beer–Lambert law. To avoid the difficult problem of computing the shape factors, a semi-analytical approach is used. Finally, the numerical simulations have allowed to validate the analytical and experimental results.In the first place, we have written the energy conservation equation in absence of convection (based on the first law of thermodynamics) (1.a), the radiative source term (1.b) and the boundary conditions (2). As for the finite elements method, the Galerkin approach is used for the formulation of the 3D heat transfer equation (3) and the diagonalisation of the heat capacity matrix, “lumped matrix”, is adopted [M.A. Dokainish, K. Subbaraj, A survey of direct time-integration methods in computational structural dynamics, Comput. Struct. 32 (6) (1989) 1371–1386]. A single step implicit time integration scheme is used for the computation [M.A. Dokainish, K. Subbaraj, A survey of direct time-integration methods in computational structural dynamics, Comput. Struct. 32 (6) (1989) 1371–1386]. After recalling the classical expressions for the radiative flux divergence (5) and the flux itself (6), we have rewritten the radiative source term for an homogeneous medium as a function of spectral intensity (7) and given the integro-differential equation of radiative transfer [R. Siegel, J.R. Howel, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 1992], Eq. (8). The spectral relations linking coefficients of extinction, absorption, scattering and scattering albedo are given by Eqs. (9) and (10), while Eq. (11) expresses the boundary conditions.In the second place, on the basis of the non-scattering behaviour of amorphous PET [K. Esser, E. Haberstroh, U. Hüsgen, D. Weinand, Infrared radiation in the processing of plastics: Precise adjustment-the key to productivity, Adv. Polymer Technol. 7 (2) (1987) 89–128; M.D. Shelby, Effects of infrared lamp temperature and other variables on the reheat rate of PET, in: Proceedings of ANTEC'91 Conference, 1991, pp. 1420–1424; G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249], we have assumed that extinction and absorption coefficients are equal, the corresponding albedo being zero. The radiative transfer equation can thus be rewritten in a reduced form (12). Moreover, for the temperature range prevailing in the processes of PET thermoforming and preforms blowing, temperatures of radiation sources are generally much higher than those used for the forming of these thermoplastic media. Under these conditions, the cold medium hypothesis is used [Y. Le Maoult, F.M. Schmidt, V. Laborde, M. El Hafi, P. Lebaudy, Measurement and calculation of perform infrared heating: a first approach, in: Proceeedings of the Fourth International Workshop on Advanced Infrared, 1997, pp. 321–331], which allows to express transmission of spectral intensity across the material as a function of position and direction (13), and to write the radiative flux divergence in a simplified form as well (14). For one-dimensional radiation, solution of the radiative transfer equation [R. Siegel, J.R. Howel, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 1992] is given by Eq. (15), which combines with Eq. (14) to yield the spectral flux generated inside the semi-transparent medium, Eq. (16). In case of radiation propagation in the same direction as the normal to the polymeric membrane surface, radiation intensity can then be expressed by the Beer–Lambert law (17). This permits to deduce the expression of spectral flux transmitted across the PET membrane thickness (18).Infrared radiation intercepted by the membraneFor determining the infrared radiation intercepted by the membrane surface, the medium between the radiation source and the thermoplastic material is assumed to be completely transparent to radiation. It is also assumed that the surface of the source is diffuse. This allows to infer, from spectral expressions of the energy emitted by the source (Eqs. (19) and (20)) and of the energy received by the semi-transparent membrane (Eq. (21)), a relation for radiation intercepted by the surface at all wavelengths (Eqs. (22) and (23)). For an isotropically radiating source (Lambertian source) when considering an average emissivity of the heating source, Eq. (24), total radiation intercepted by the surface is given by Eq. (25).Infrared radiation absorbed by the membraneFor the determination of infrared radiation absorbed by the polymeric membrane, first, based on the principle of energy conservation, spectral relation (27) which links reflectivity, absorptivity and transmissivity is recalled, as well as relation (28) associating spectral absorption and transmission coefficients following the Beer–Lambert law. Consequently, this allows to find the spectral energy absorbed by the semi-transparent medium (29) with the aid of Eq. (25). Concentrating on wavelength bands that constitute the transmission spectrum of the medium, Eq. (29) is then replaced by Eq. (30) where average values of reflectivity (31) and transmissivity (32) are considered. As a result, the volumetric total energy absorbed by the medium, the flux divergence, is then given by Eqs. (33) and (34). Furthermore, considering that reflectivity is low for polymeric materials (generally lower than 5%), the latter is neglected leading to Eq. (35) instead of Eq. (34). Fig. 1 illustrates a typical transmissivity curve for PET [G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249].Shape factorIn order to take into account arbitrary shapes of sources and preforms, a semi-analytical approach is used for the computation of shape factors [G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249]. The definition of shape factor is recalled in Eq. (36) and Fig. 2. An equivalent form based on the contour principle [R. Rammohan, Efficient evaluation of diffuse view factors for radiation, Int. J. Heat Mass Transfer. 39 (1996) 1281–1286] is given by Eq. (37), while the semi-analytical formula [F. Erchiqui, N.G. Dituba, Analyse comparative des méthodes de calcul des facteurs de formes pour des surfaces à contours rectilignes, Internat. J. Thermal Sci. 46 (2007) 284–293] is expressed by Eq. (39). Validation of this semi-analytical approach is obtained by comparison with the analytical solution [H.C. Hottel, Radiant heat transmission between surfaces separated by non-absorbing media, Trans. ASME 53 (1931) 265–273, FSP-53-196; A. Feingold, Radiant-interchange configuration factors between various selected plane surfaces, Proc. Roy. Soc. London Ser. A 292 (1996) 51–60; J.R. Ehlert, T.F. Smith, View factors for perpendicular and parallel rectangular plates, J. Thermophys. Heat Trans. 7 (1993) 173–174] as well as with results yielded by three different techniques, i.e. the area-integration method, the Gauss quadratic technique and the contour method. Table 1 summarizes the numerical results obtained in each case and gives the relative error.Analytical validation of the reheatingRegarding the numerical validation, we have considered a PET semi infinite medium subjected to a uniform incident radiative flux density. The boundaries of the semi transparent medium are taken as adiabatic. Thermophysical properties and geometrics are given in Table 2. The one-dimensional Laplace equation which governs the radiation heat transfer is solved analytically with the aid of Laplace transforms. The temperature evolution within the depth of the semi transparent medium is obtained by analytical solution [A.B. De Vriendt, in: G. Morin, (Ed.), La transmission de la chaleur, vol. 1, Chicoutimi, Québec, 1984], Eq. (40). Fig. 3 shows a comparison between the numerical solution obtained by the 3D finite elements method and the analytical one. It is seen that, for the three cases, the relative error between numerical and analytical results stays lower than 0.1%.Numerical modeling of infrared heatingAn amorphous PET sheet is considered. The face of the membrane is a square of side 20 cm with a thickness of 1.5 mm. The lateral walls are assumed adiabatic. For modeling with the 3D finite elements method, the sheet is meshed with identical hexahedra comprising eight nodes. The thermophysical properties used for this study are given in [S. Monteix, F. Schmidt, Y. Le Maoult, R Ben Yedder, R.W. Diraddo, D. Laroche, Experimental study and numerical simulation of perform or sheet exposed to infrared radiative heating, Journal of Materials Processing Technology 119 (2001) 90–97] and the reheating time is 35 seconds. The heat transfer coefficient h on front and rear faces is 10 W m^?2 K^?1.Heat flux received by the polymer during heatingTo estimate the incident heat flux distribution on the surface of the polymer, we first calculate the shape factors between each pair of emitting/receiving surfaces. Then the flux distribution is obtained with Eq. (25). Fig. 4 illustrates this distribution.Validation against experimental resultsFigs. 5 and 6 compare the numerical results obtained by the simulation (via MEF 3D) with those obtained experimentally. Fig. 5 displays the temperature distribution on the front and rear surfaces.ResultsFigs. 7–10 illustrate the numerical results obtained for the temporal temperature evolution, during the first five seconds of heating, of the upper receiving face (surface directly exposed to the infrared source), the central plane and the lower face of the PET membrane, respectively.Figs. 11 and 12 give the temperature evolution at four positions on upper and lower faces of the membrane: 0 cm (edge), 2.5 cm, 5 cm and 10 cm (middle). It is observed that the trend is nearly linear during the period of infrared heating.On Figs. 13 and 14 is displayed the temperature evolution in the center of the membrane along its thickness during periods 0–4 and 5–35 seconds.     

Estimation des propriétés thermiques à l'échelle millimétrique par méthodes périodiques: Résolution du problème direct et du problème inverse

Estimation of thermal properties by periodic methods: direct problem and inverse problem solving. This article presents two processes of thermal diffusivity measurement for homogeneous material. The experimental bench uses a periodic method dedicated to millimeter scale study. Simulation of this experimental method is studied in detail. The case of a homogeneous material of unknown diffusivity to be measured is studied. Sensitivity coefficient and calculus of Cramer–Rao bounds (BCR) prove the good conditioning of diffusivity estimation and illustrate the action of thermal losses. Simulations of Monte Carlo compare performances of least squares estimator to optimal performances defined by the BCR. Two experimental processes are validated by a study of the iron ARMCO chosen as material of reference.     

Transferts couples de chaleur et de masse dans des matériaux consolides utilises en génie civil

The method of measuring the humidity in the materials based on the determination of the thermal conductivity is presented and the principal characteristics are analysed. The water diffusion coefficients due to the water content and temperature gradients are measured and utilized in the numerical calculation of a drying process. These results are compared to an experiment and discussed. We note the difficulty of describing the superficial mass transfer with a single coefficient function of the heat transfer coefficient. We present some enthalpic balance which proves of interest in taking into account the presence of humidity in walls of buildings.     

Modélisation d'un panache d'émetteur de chaleur pour le logiciel de simulation énergétique des bâtiments SPARK

Plume's convector modeling for object-oriented thermal building simulation software SPARK. In this paper, a zonal model used to predict air movement and temperature distribution in a room is presented. This model is based on a rough partitioning of the room: it is an intermediate approach between one-node models (that consider a homogeneous temperature in each room) and CFD models. Flow rates are calculated in respect to the pressure distribution in low velocity domains and specific laws describe plumes and jets. The airflow model is coupled with a building envelope model. They are implemented in an object-oriented environment called SPARK. The modularity of SPARK allows the creation of very flexible tools, and its strict syntax permits having the simulations automatically generated.     

Contact périodique en regime etabliresistances thermiques apparentes limites pour des durees de contact nulles et unfinies

The knowledge of heat transfer between periodically contacting metallic surfaces is necessary in numerous engineering applications.The analytical solution for quasi-steady state heat transfer in periodically contacting finite regions was first given by NIKHAILOV[1], and then by VICK and OZISIK [2]. These solutions are revisited. The limit cases at infinite or nul contact duration are studied. It is shown that the apparent limit resistances of the system are: These expressions are convenient physical tools easier to implement than the exact analytical solution.     

Simulation numérique des jets turbulents subsoniques à masse volumique variable par le modèle k–ε

A numerical investigation is reported for round free turbulent non-isothermal binary mixing incompressible jets discharging into a quiescent atmosphere. The standard k–ε model is used. The standard closure schemes in Favre averaged variables are first introduced. The parabolic numerical simulation method of Patankar and Spalding [Heat and Mass Transfer in Boundary Layer, Intertext Books, London, 1970] is followed. The numerical simulations show a satisfactory agreement with the experimental results of Chassaing [Mélange turbulent de gaz inertes dans un jet de tube libre, Thèse d'état, INPT, 1979], Birch et al. [J. Fluid Mech. 88 (3) (1978) 431–449] and Panchapakesan et Lumley [J. Fluid Mech. 246 (1993) 197–223, 225–247]. The developed numerical code is used to study the sensitivity of turbulent characteristics to the density ratio between the jet and the ambient air. The decay rate of the mean axial velocity, temperature and mass fraction are shown to increase with decreasing density ratio. This confirms a higher mixing efficiency (parameter which determines the quantity of mass or heat injected at the jet exit and found further from the axis) when the density ratio between the jet and the quiescent air decreases. Finally, it is shown that the density effects are affected by the buoyancy terms in the similarity region of the jet.     

Représentation non locale d'un milieu hétérogène en diffusion pure

For a one-dimensional diffusive transfer through a spatially periodic heterogeneous medium, the quadrupoles method allows to obtain the exact non-local (in time) equation describing the transfer. The case of reversible (that is to say when input and output can be permuted) and non-reversible media is examined separately. Some examples are treated for illustration: degenerated media (made of pure resistances or capacitances) or real media. The obtained equation cannot be reduced to the hyperbolic Cattaneo–Vernotte equation, which is unable to describe diffusive transfer through heterogeneous media.     

Les aspects théoriques régissant l'instrumentation d'un capteur thermique pariétal à faible inertie

The theoretical aspects of the instrumentation of a weak inertia parietal thermal sensor. From a simple semi-infinite model, we present theoretical developments allowing the optimisation of the instrumentation of parietal thermal sensors. Generally equipped with two thin thermocouples, these sensors must present a weak thermal inertia in order to characterise thermal boundary conditions in fast unsteady regime. Performances of such sensors are described by three characteristics: response time, measurement sensitivity and accuracy. For a given material, the constraint on the response time determines diameter and position of the second thermocouple. In a second time, the position of the first thermocouple results from a good compromise between measurement sensitivity and accuracy. This compromise depends itself on the sensor response time, particularly at the beginning of the studied thermal phenomena.     

Prevision numérique de la convection forcée turbulente dans une cavité bidimensionnelle entrainée

Turbulent flow and associated heat transfer in confined geometry (driven closed cavity flow in two dimensions) has been studied using a finite-difference numerical method in primitive variables. Turbulence modelling is based on one point closures derived from the classicalk-ε model. Calculated mean velocity and turbulent kinetic energy are compared with available experimental data. In spite of its limitations, the k-ε model proved to be a useful tool for prediction of global quantities. The case of forced heat convection with fixed wall temperature is considered. Mean temperature field and overall thermal properties of the cavity flow are studied. Correlations giving Nusselt numbers at each face of the cavity versus Reynolds number are deduced from numerical results, they sum up mean transfer properties of such a flow configuration.     

Étude numérique de l'évaporation dans un courant d'air humide laminaire d'un film d'eau ruisselant sur une plaque inclinée

Numerical study of the evaporation in laminar humid air flow of a liquid film flowing over an inclined plate. By using an implicit centered finite differences method with a non-uniform grid, the authors study numerically the evaporation of a thin liquid film flowing over an inclined plate in a forced humid-air flow. They consider the existence of two-dimensional laminar boundary-layers with variable physical properties and show that the term of enthalpy diffusion is always negligible, whether the plate is adiabatic, isothermal or heated by a constant heat flux density. By using in the liquid film transfer equations which are one-dimensional, partially two-dimensional and two-dimensional, the authors additionally show the following features. If the plate is adiabatic, the liquid mass flow rate is without influence on the transfers and the gas–liquid interface behaves like an isotherm surface at rest. In this case, one may use a one-dimensional model in the film whatever liquid mass flow rate is. If the wall is isotherm or heated by a constant heat flux and when the liquid mass flow rate is less than 10⁻³ kg·m⁻¹·s⁻¹, the one-dimensional model is sufficient; if it is included in the interval [10⁻³ kg·m⁻¹·s⁻¹, 10⁻² kg·m⁻¹·s⁻¹[, the partially two-dimensional model is useful; if it is superior to 10⁻² kg·m⁻¹·s⁻¹, it is necessary to use the two-dimensional model. Generally, whatever the thermal conditions on the plate are, heat transfer is dominated by the liquid-vapor transition.     

Étude numérique de l'ébullition en convection mixte dans une couche poreuse verticale

By using an enthalpic method (two-phase mixture model), we have studied numerically boiling with mixed convection in a vertical porous layer with a discrete heating. Liquid is injected at the top face. Finite volume method is used for numerical resolution of equations of volumetric enthalpy and pressure. Results giving time–space evolution of temperature, pressure, velocities of the fluids and evaporated volume are presented and analyzed. Parametric studies to assess the effects of inlet velocity, imposed heat flux and permeability, were performed. Results show that boiling is important if the effects of both, natural and forced convections are similar. The evaporated volume will decrease at high values of intrinsic permeability of porous medium or at high values of inlet velocity.     

Séchage superficiel d'un matériau poreux humide par convection forcée d'air chaud: couplage entre les équations de transfert dans le matériau et celles de la couche limite

The authors considered drying of a thick slab of humid porous material immersed in a laminar steady flow of hot air parallel to its surface. They wrote the boundary layer equations in air (continuity, momentum, energy and mass), those describing humidity and heat transfer in the porous medium deduced from Luikov's theory. Then, they coupled them at the air-product interface by expressing the continuity of the thermal and mass fluxes taking into account the evaporation. They solved numerically the resulting system of differential equations using an implicit finite-difference method. They determined the instantaneous evaluation of the spatial distributions of heat and humidity, the local values of the Nusselt and Sherwood numbers. They also studied the influences of principal parameters of the system.     

Evolution thermique d'une sonde électrostatique dans un gaz raréfié et partiellement ionisé

Based on the classical Langmuir theory, an electrostatic probe can be used in an ionized gas to measure electron temperature, electron density and electron distribution function. In many experimental cases the characteristic curve (applied potential vs total current collected) presents a complex and reproducible hysteresis effect. Consequently, electron measurements are impossible. The hysteresis effect decreases when the probe potential frequency increases and is related to the thermal probe evolution. The thermal probe evolution is determined by a numerical method taking into account plasma energy flux, radiation and probe conduction. The numerical results show that the hysteresis effect appears simultaneously with a thermal probe hysteresis.     

Évaporation en convection forcée d'un film liquide mince ostwaldien ruisselant en régime laminaire permanent sur une surface plane isotherme et inclinée

The authors study, in forced convection, the evaporation of an Ostwaldian film flowing over an isothermal inclined plane surface to determine the influence of the behaviour index of the liquid on the dynamic and thermal characteristics of liquid-air system. The liquid flow is considered partially two-dimensional whereas for the air it is two-dimensional. The coupled equations with the interfacial conditions are solved using a fully implicit finite differences method. From the study, it appears that the behaviour index influences considerably the transfers which are more important for pseudoplastic liquids than for dilatant ones.     

Un modèle simple de la pénétration couplée de chaleur et de matière dans l'absorption gaz-liquide en film ruisselant laminaire

In this paper we present a simple model based on the coupled penetration of heat and mass fronts for the study of the absorption kinetics of vapor in a concentrated salt solution falling in laminar flow down an adiabatic wall. Though this model is very simple, it permits a concrete and intuitive comprehension of the numerical integration of differential equations. The forecasts of this model, in the case of the absorption of water vapor in the lithium bromide solution, are presented and compared to the results of the numerical solution.     

Convection thermique pour l'écoulement de couette avec débit axial; cas d'un fluide pseudo-plastique

The different patterns of psuedo-plastic fluid flow between two coaxial cylinders, with an axial component of velocity, have been determined. A new mode on stability was set obviousness. The incidence of the flow's structure and heat flux on the Nusselt number has been examined. The variation of rheological properties has been accounted for in the correlation between the mean Nusselt number and the usual dimensionless parameters.     

Planform structure of turbulent Rayleigh-Bénard convection

The planform structure of turbulent Rayleigh-Bénard convection is obtained from visualising a liquid crystal sheet stuck to the bottom hot surface. The bottom plate of the convection cell is Plexiglas and the top plate is glass. Water is the 3est liquid and the Rayleigh number is 4 × 10⁷. The planform pattern reveals randomly moving hot streaks surrounded by cold regions suggesting that turbulent Rayleigh-Bénard convection is dominated by quasi-two-dimensional randomly moving plumes. Simultaneous temperature traces from two vertically separated thermocouples indicate that these plumes may be inclined forward in the direction of horizontal motion. The periodic eruption of thermals observed by Sparrow et al [3] and which forms the basis of Howard's model is not observed.     

Conductivité thermique de matériaux poreux humides: évaluation théorique et possibilité de mesure

We develop here a mathematical model of simultaneous heat and mass transfer in unsaturated porous media. We put forward the influence of the total pressure gradient in the gaseous phase on the mass transfer contrary to the classical models where only two driving forces are taken into account: a temperature and a moisture gradient. We use this model on the one hand to investigate the theoretical notion of the apparent thermal conductivity of a wet porous medium and on the other hand to simulate a hot plate measurement and an unsteady state one (“flash” method). We show the great importance of both the total pressure gradient and the boundary conditions, that is to say of the measurement technique on the results.