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In this paper, the integration scheme is employed to solve a coupling problem of transient heat conduction and displacement due to thermal deformation. Based on the resulting displacement, gap or contact pressure on the interface of two thermoelastic bodies in imperfect contact is estimated. Such information is then used to determine the interfacial heat transfer between the two thermoelastic bodies. This again affects the temperature distribution and thus the thermal deformation. In the course of heat transfer and thermal deformation, effect of thermal rectification also is taken into account. Numerical solutions of transient and steady-state quantities including gap formation, normal stress, and temperature are demonstrated for a pair of stainless steel and aluminum with conforming shapes. Such an analysis of conduction-deformation interaction based on a finite-difference-like scheme has not been studied in the past.

The effect of anisotropy on transport phenomena in anisotropic porous media is studied in this paper. For convenience, a bank of circular cylinders which can be treated as an anisotropic porous medium is employed such that both Darcy-Forchheimer drag and effective thermal conductivity can be accurately determined. Two problems including a forced flow and a natural convection are illustrated to investigate the effect of anisotropy on fluid flow and heat transfer through a bank of circular cylinders. The solutions reveal that inclination of the cylinder bundle could give rise to an influence of more than 100% on the heat transfer rate for both forced convection and natural convection. Hence, the anisotropy of an anisotropic porous medium should not be ignored.

Based on the microscale temperature, the effective thermal conductivity in the principal direction of an anisotropic porous medium is evaluated. A correlation of the effective thermal conductivity then is employed to predict the volume-averaged temperature of a two-dimensional nonhomogeneous anisotropic porous medium. Through the example, the present model is found to possess an excellent performance even when the principal direction of the anisotropic medium is not parallel to the physical coordinates. The results also reveal that using the classic mixing rule could lead to a significant error when the thermal conductivity ratio of the solid and fluid is large. The present numerical procedure is believed to be able to determine the effective thermal conductivity for most porous media as long as their microstructure is regular and known.

Transient conjugate heat transfer relating the heat conduction inside a solid body of arbitrary shape and the natural convection around the solid is studied in the present investigation. A single set of governing equations covering both solid and fluid is solved by using the weighting function scheme along with the NAPPLE algorithm and the SIS solver. All of the computations are performed on a Cartesian grid system. Numerical results of velocities and isotherms are presented for Pr = 7 and Gr = 10 7 and 10 8 . An interesting observation on formation, growth, merging and decay of several plumes is discussed. The solution shows that there is a dead fluid inside the cavity under the solid body due to an upward temperature gradient. The maximum temperature thus occurs there in a late stage of the heat transfer process. An increase in the Grashof number is found to enhance the heat transfer. Through this study, the present numerical method is shown to have a good performance for conjugate heat transfer on solid body of arbitrary shape without the need of a body-fitted grid system.

A latent heat method is modified in the present study to properly handle the latent heat evolved in the solidification process of a binary alloy system. The eutectic state ( C E , T E ) is imposed on the eutectic front such that no species equation is needed in the pure solid region. Both momentum and species equations in the irregular domain consisting of the mushy zone and the liquid region are successfully solved by using the weighting function scheme along with the NAPPLE algorithm. To trace the interface of the mushy zone and the pure liquid region, an interpolation technique is proposed. Two examples were conducted to examine the performance of the present method. Satisfactory agreement with the existing experimental results is observed on the total volume of the mushy zone.

Forced convection on a rotating cylinder cooled with an air jet is investigated in the present study. To properly handle the jet flow before and after hitting the cylinder, a computational domain having an irregular shape is employed. Results of streamlines, isotherms and Nusselt numbers are presented for jet Reynolds numbers of Re j = 100 , 500 and 1000 under various rotation Reynolds numbers in the range of 0 ≤ Re ω /Re j ≤ 1. It is interesting to find two solution modes in the present problem. For small rotation speeds, the jet flow is separated into two branches upon hitting the cylinder. Each branch has a separation point. With the aid of the rotating cylinder, one of the separation points moves downstream, while the other receives little influence. Hence, the overall heat transfer is enhanced. For high rotation speeds, however, this heat transfer behavior is reversed due to a layer of dead air round the cylinder. Nevertheless, a more uniform heat transfer is achieved. For both cases of Re j = 500 and 1000, dual solutions according to each of the two solution modes exist in a range of rotation Reynolds numbers. Like the transition from laminar to turbulent flow, the heat transfer characteristics jump from one solution mode to the other depending on the flow instability. However, the transition between the two solution modes is not clear for a jet Reynolds number Re j as small as 100.

An enthalpy formulation is proposed in the present investigation for a phase change material (PCM) having a distinct freezing temperature. The latent heat is separated from the sensible heat such that there exists a dependent variable (the sensible heat) that is a continuous function over the entire physical domain. Inside each control volume, the latent heat is rigorously evaluated from the fraction of the liquid phase to achieve an even latent heat evolution. In the dimensionless transformation, the characteristic time is denned in terms of the Stefan number. The coefficients of the unsteady terms thus are always less than unity. This will achieve a good numerical stability for any Stefan number. In addition, this particular transformation makes the present enthalpy formulation applicable to single phase problems if an infinite Stefan number is assigned. To account for a thermal diffusivity jump at the liquid-solid interface, a modified weighting function scheme is developed. Through a few examples, the present enthalpy formulation is seen to produce an accurate and smooth liquid-solid interface for PCM having a distinct freezing point.

A numerical formulation based on the weighting function scheme is developed in the present investigation for convection-conduction phase change problems. The solid phase is regarded as a liquid having an infinite viscosity. Such a treatment allows the stream-vorticity formulation to apply over the entire physical domain including the liquid-solid interface. The weighting function scheme then is used to handle the sharp viscosity jump at the liquid-solid interface. A problem dealing with pure tin is employed to test the performance of the present method. The predicted interface profiles are found to agree with available experimental data qualitatively. The computation effort required by the present numerical technique, however, is less than 2% ofthat needed by a two-region method for this same problem. Unlike the existing single-region methods, the present numerical technique produces smooth streamlines and isotherms even in the vicinity of the solidification front.

The present investigation studies the cooling of a continuous moving sheet of finite thickness. The effect of the buoyancy force is also taken into account. The temperature distribution along the solid-fluid interface is determined by solving a conjugate heat transfer problem that consists of heat conduction inside the sheet and induced mixed convection adjacent to the sheet surface. For a better numerical stability, the weighting function scheme along with an axial coordinate transformation is employed to solve the transformed boundary layer equations. Three parameters are found to exist in the present investigation. They are the Prandtl number of the fluid Pr, the buoyancy parameter Ω and the heat capacity ratio C. Numerical results including the Biot number, the surface temperature and the overall heat transfer rate of the sheet are presented for 0.7 ⩽ Pr ⩽ 100, 0 ⩽ Ω ⩽ 10 and 0.1 ⩽ C ⩽ 1. The buoyancy force is seen to have a significant effect on the results. The heat capacity ratio, however, is the most important parameter. Based on the present results, it is concluded that using a liquid as the cooling medium could obtain a better cooling performance than using a gas. This is because the liquid has a larger heat capacity than a gas. The Prandtl number has only a minor effect.

This paper presents a finite element solution to the problem of low Peclet number fluid flow in the thermal entrance region of a round pipe. The velocity is assumed to be laminar and fully developed throughout the pipe and the fluid temperature is kept uniform atX = —∞. The pipe wall is adiabatic at X ≤ 0 and cooled convectively at X ≥ 0. The solutions include temperature distributions and Nusselt numbers for the parameters, Bi = 0.04, 0.4, 4, 20 and Pe = 1, 3, 5, 10, 20, 30, which are in excellent agreement with the existing analytic solution except in the region near the singular point Δ A temperature discrepancy in the analytic solution at this point is physically impossible. The finite element method overcomes this mathematical difficulty and shows a greater value in the Nusselt number due to a higher wall temperature at X ≥ 0. On presente, dans le travail, une solution utilisant une methode d'elements finis pour le probleme de l'ecoulement d'un fluide a faibles nombres de Peclet dans la region d'entree thermique d'un tube rond. On suppose que la vitesse est laminaire et etablie d'un bout a l'autre du tube et que la temperature du fluide est maintenue uniforme a X = — ∞. La paroi du tube est adiabatique a X ≤ 0 et elle est refroidie d'une maniere convective a X ≥ 0. Les solutions comprennent des distributions de temperatures et des nombres de Nusselt pour les parametres Bi = 0.04, 0.4, 4, 20 et Pe = 1, 3, 5, 10, 20, 30, elles concordent tres bien avec la solution analytique existante, sauf dans la region a proximite du point singulier X = 0, r = 1; il est physiquement impossible d'obtenir une difference de temperature dans la solution analytique en ce point. La methode des elements finis remedie a cette difficulte mathematique et fournit un nombre Nusselt de aleur plus grande, par suite d'une temperature plus elevee de la paroi a X ≥0.

The combined effects of axial conduction and solidification on heat transfer and pressure drop in pipe flows are investigated by the use of a modified Galerkin finite element method. To allow for the upstream heat conduction, the domain of study is extended from X = -∞ to X = -∞ as has been done in previous analyses. As a preliminary study on the effect of axial conduction, the present investigation assumes a superheat ratio that is sufficiently large (To > Tf or Tw ≈ Tf) such that solidification begins at a location near X = 0. For numerical convenience, the infinite domain -∞ ≤ X ≤ ∞ is transformed onto a finite domain -1 ≤ z ≤ 1. The energy equation for the liquid-phase is then solved by a modified Galerkin finite element method. For better numerical stability, a procedure is proposed for controlling the numerical error that might propagate from the singular point (X, R) = (O, Ro). The profile of the solid-liquid interface, the heat transfer rate and the pressure drop are presented for various values of Peclet number, Pe = 1, 3, 5, 10 and 30, and for the modified superheat ratio, c = 0.1, 0.5, 5.0, and ∞. A l'aide d'une methode d'elements finis de type Galerkin modifie, nous avons etudie les effets combines de la conduction axiale et de la solidification sur le transfert de chaleur et la perte de charge dans des ecoulements dans des tubes. Pour permettre la conduction de la chaleur en amont, le champ d'etude a ete etendu de X = ∞ a X = -∞ comme nous l'avons fait dans les analyses anterieures. Comme etude preliminaire sur l'effet de la conduction axiale, les presentes recherches supposent un rapport de surchauffe qui soit suffisamment grand (To ≤ Tf ou Tw ≈ Tf) pour permettre a la solidification de demarrer a un point pres de X = 0. Pour faciliter le calcul numerique, le domaine infini -∞ ≤ X ≤ ∞ est transforme en un domaine fini -1 ≤ z ≤ 1. L'equation d'energie pour la phase liquide est alors resolue par une methode d'elements finis de type Galerkin modifie. Pour obtenir une meilleure stabilite numerique, nous proposons une methode pour contrǒler l'erreur numerique qui peut se propager du point singulier (X, R) = (0, Ro). Le profil de l'interface solides-liquide, le taux de transfert de chaleur et la perte de charge sont presentes pour diverses valeurs du nombre de Peclet, Pe = 1, 3, 5, 10 et 30 et pour le rapport modifie de surchauffe, c = 0, 1, 0,5, 5,0 et ∞.

The purpose of the present paper is to propose a new hybrid method investigating the effect of the surface curvature of a solid body on hyperbolic heat conduction. The difficulty encountered in the numerical solutions of hyperbolic heat conduction problems is the numerical oscillation in vicinity of sharp discontinuities. In the present study, we have developed a new hybrid method combined the Laplace transform, the weighting function scheme [Shong-leih Lee, Weighting function scheme and its application on multidimensional conservation equations, Int. J. Heat Mass Transfer 32 (1989) 2065–2073], and the hyperbolic shape function for solving time dependent hyperbolic heat conduction equation with a conservation term. Four different examples have been analyzed by the present method. It is found from these examples that the present method is in good agreement in the analytical solutions [Tsai-tse Kao, Non-Fourier heat conduction in thin surface layers, J. Heat Transfer 99 (May) (1977) 343–345] and does not exhibit numerical oscillations at the wave front and the surface temperature is modified by the surface curvature during the short period when the non-Fourier effect is significant. The curvature will increase or decrease the temperature of the wave front, depending on whether the surface is concave or convex.