Quadratic Temperature Profile Research Articles

A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results

A one-dimensional two-phase Stefan problem for the melting of a semi-infinite material with a power-type temperature-dependent thermal conductivity was considered. The assumption of taking thermal parameters as functions of temperature found its basis in physical and industries applications, allowing for a more precise and realistic description of phase change processes. By imposing a Dirichlet condition at the fixed face, a theoretical and approximate study was developed. Through a similarity transformation, an equivalent ordinary differential problem was obtained from which an integral problem was deduced. The existence of at least one analytical solution was guaranteed by using the Banach fixed point theorem. Due the unavailability of an analytical solution, a heat balance integral method was applied, assuming a quadratic temperature profile in space, to simulate temperature variations and the location of the interface during the melting process. For constant thermal conductivity, results can be compared with the exact solution available in the literature to check the accuracy of the approximate method.

On the application of the double integral method with quadratic temperature profile for spherical solidification of lead and tin metals

• High Stefan and Biot solidification problem considered and modelled. • The proposed method achieved a reduction of 92.73% compared to previous approaches. • Gaussian Quadrature is 3 orders of magnitude less expensive compared to Runge-Kutta. In this paper we developed the double integral method to solve the solidification problem of liquid tin and lead metals in spherical geometry with a high Stefan and Biot numbers. The solidification time solutions were obtained by deriving the heat conduction equation using a quadratic temperature profile. The set of equations were solved using a Gaussian Quadrature approach; a novelty for this problem. Hence, we intend to improve the numerical accuracy against the traditional simple integral methods in the literature. Overall, the double integral method with the quadratic profile performed better than the single integral method with a quadratic profile; for Bi = 10 and Ste = 2, the reduction in solidification time relative error using the double integral method was 92,73% compared to the simple integral method. Compared to the experimental data of lead and tin, the double integral method performed better than the single integral method for all Biot and Stefan conditions; for tin cooled by water, the double integral method was 20,06% more accurate in predicting the solidification time compared to the single integral method. Furthermore, we evaluated that using a Gaussian Quadrature integrator led to a reduction in computational cost up to 3 orders of magnitude compared to traditional Runge-Kutta methods. Therefore, our studies attested to the correctness and suitability of the developed double integral method for applications in the solidification of metals.

Multi-dimensional Simulation of Phase Change by a 0D-2D Model Coupling via Stefan Condition

Considering phase changes associated with a high-temperature molten material cooled down from the outside, this work presents an improvement of the modelling and the numerical simulation of such processes for an application pertaining to the safety of light water nuclear reactors. Postulating a core meltdown accident, the behaviour of the core melt (aka corium) into a steel vessel is of tremendous importance when evaluating the vessel integrity. Evaluating correctly the heat fluxes requires the numerical simulation of the interaction between the liquid material and its solid counterpart which forms during the solidification process, but also may melt back. To simulate this configuration, encountered in various industrial applications, one considers a bi-phase model constituted by a liquid phase in contact and interaction with its solid phase. The liquid phase may solidify in presence of low energetic source, while the solid phase may melt due to an intense heat flux from the high-temperature liquid. In the frame of the in-house legacy code, several simplifying assumptions (0D multi-layer discretization, instantaneous heat transfer via a quadratic temperature profile in solids) are made for the modelling of such phase changes. In the present work, these shortcomings are illustrated and further overcome by solving a 2D heat conduction model in the solid by a mixed Raviart-Thomas finite element method coupled to the liquid phase due to heat and mass exchanges through Stefan condition. The liquid phase is modeled with a 0D multi-layer approach. The 0D-liquid and 2D-solid models are coupled by a Stefan like phase change interface model. Several sanity checks are performed to assess the validity of the approach on 1D and 2D academical configurations for which exact or reference solutions are available. Then more advanced situations (genuine multi-dimensional phase changes and an “industrial-like scenario”) are simulated to verify the appropriate behavior of the obtained coupled simulation scheme.

Thermo-Mechanical Analysis for an Axisymmetric Functionally Graded Rotating Disc under Linear and Quadratic Thermal Loading

The study presents thermo-mechanical analysis of functionally graded (FG) rotating disc whose material properties, namely, Young’s modulus, density and coefficient of thermal expansion in radial direction are tailored from inner to outer radius using power law form. The disc is considered to be under the influence of internal pressure, centrifugal body force and thermal loading of the form linear as well as quadratic. Response of FG disc under linear and quadratic temperature profile subjected to internal pressure as well as centrifugal body force is analysed. An exact solution for stress in radial and tangential directions, under mechanical and thermal loading is presented. Numerical solutions for stresses under internal pressure with uniform thermal loading are obtained using finite element method and its comparison with analytical results is presented graphically. Results for radial displacement, radial stress and tangential stress are depicted graphically and their interpretation has been discussed.

Integral balance methods applied to non-classical Stefan problems

We consider two different Stefan problems for a semi-infinite material for the non-classical heat equation with a source that depends on the heat flux at the fixed face. One of them, with constant temperature at the fixed face, was already studied in literature and the other, with a convective boundary condition at the fixed face, is presented in this work. Due to the complexity of the exact solution it is of interest to compare with approximate solutions obtained by applying heat balance integral methods, assuming a quadratic temperature profile in space. A dimensionless analysis is carried out by using the parameters: Stefan number and the generalized Biot number. In addition it is studied the case when Biot number goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to verify the accuracy of the approximate methods.

Heat balance integral methods applied to the one-phase Stefan problem with a convective boundary condition at the fixed face

In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles.

Turbulent flux modelling with a simple 2-layer soil model and extrapolated surface temperature applied at Nam Co Lake basin on the Tibetan Plateau

Abstract. This paper introduces a surface model with two soil-layers for use in a high-resolution circulation model that has been modified with an extrapolated surface temperature, to be used for the calculation of turbulent fluxes. A quadratic temperature profile based on the layer mean and base temperature is assumed in each layer and extended to the surface. The model is tested at two sites on the Tibetan Plateau near Nam Co Lake during four days during the 2009 Monsoon season. In comparison to a two-layer model without explicit surface temperature estimate, there is a greatly reduced delay in diurnal flux cycles and the modelled surface temperature is much closer to observations. Comparison with a SVAT model and eddy covariance measurements shows an overall reasonable model performance based on RMSD and cross correlation comparisons between the modified and original model. A potential limitation of the model is the need for careful initialisation of the initial soil temperature profile, that requires field measurements. We show that the modified model is capable of reproducing fluxes of similar magnitudes and dynamics when compared to more complex methods chosen as a reference.

A uniform source-and-sink scheme for calculating thermal conductivity by nonequilibrium molecular dynamics

A uniform source-and-sink (USS) scheme, which combines features of the reverse [F. Müller-Plathe, J. Chem. Phys. 106, 6082 (1997)] and improved relaxation [B. Y. Cao, J. Chem. Phys. 129, 074106 (2008)] methods, is developed to calculate the thermal conductivity by nonequilibrium molecular dynamics (NEMD). The uniform internal heat source and sink are realized by exchanging the velocity vectors of individual atoms in the right half and left half systems, and produce a periodically quadratic temperature profile throughout the system. The thermal conductivity can be easily extracted from the mean temperatures of the right and left half systems rather than by fitting the temperature profiles. In particular, this scheme greatly increases the relaxation of the exited localized phonon modes which often worsen the calculation accuracy and efficiency in most other NEMD methods. The calculation of the thermal conductivities of solid argon shows that the simple USS scheme gives accurate results with fast convergence.

Natural convection along a heated vertical plate immersed in a nonlinearly stratified medium: application to liquefied gas storage

We consider free convection driven by a heated vertical plate immersed in a nonlinearly stratified medium. The plate supplies a uniform horizontal heat flux to a fluid, the bulk of which has a stable stratification, characterized by a non-uniform vertical temperature gradient. This gradient is assumed to have a typical length scale of variation, denoted Z0, while Ψ0 is the order of magnitude of the related heat flux that crosses the medium vertically.We derive an analytic solution to the Boussinesq equations that extends the classical solution of Prandtl to the case of nonlinearly stratified media. This novel solution is asymptotically valid in the regime RaS ≫ 1, where RaS denotes the Rayleigh number of nonlinear stratification, based on Z0, Ψ0, and the physical properties of the medium.We then apply the new theory to the natural convection affecting the vapour phase in a liquefied pure gas tank (e.g. the cryogenic storage of hydrogen). It is assumed that the cylindrical storage tank is subject to a constant uniform heat flux on its lateral and top walls. We are interested in the vapour motion above a residual layer of liquid in equilibrium with the vapour. High-precision axisymmetric numerical computations show that the flow remains steady for a large range of parameters, and that a bulk stratification characterized by a quadratic temperature profile is undoubtedly present. The application of the theory permits a comparison of the numerical and analytic results, showing that the theory satisfactorily predicts the primary dynamical and thermal properties of the storage tank.

Effects of approximations regarding stress and temperature profiles on the solution in thermal elastohydrodynamically lubricated line contacts

The assumptions of both a Couette flow dominance and a quadratic temperature profile across the film are commonly used in the analysis of thermal non-Newtonian elastohydrodynamic lubrication. In such calculations, the conduction across the film and the energy dissipation are assumed to be the main phenomena in the prediction of temperature rise. Consequently the gradients of the physical properties across the film as well as the reverse flow and the heat convection in all directions are assumed to have no or little impact on the general characteristics of the temperature profiles and therefore can be neglected. The objective of this paper is to assess the quality of these approximations. A comparison between both the simplified and the complete non-Newtonian thermal approach is made for the line contact problem. The complete solution is based upon the resolution of a two-dimensional energy equation, which is fully coupled with the non-Newtonian Reynolds equation. The analysis reveals some local discrepancies between both the simplified and the complete solutions. These differences are clearly visible at the inlet of the contact where a bump in the temperature profiles disappears in the simplified approach. The results also show a significant difference in the levels of temperature predicted within the high-pressure region between the simplified and complete approach. In regard to the friction coefficient, the simplified calculation forecasts a lower coefficient than that obtained with the complete formulation.

On the effects of the temperature profile approximation in thermal Newtonian solutions of elastohydrodynamic lubrication line contacts

The assumptions of a quadratic temperature profile and mean viscosity across the film, which are frequently used in the analysis of thermal elastohydrodynamic lubrication (EHL), are examined and discussed. Two different approaches for solving the thermal EHL problem are compared for line contact conditions, namely (a) a one-dimensional model based upon both the assumption of a quadratic temperature profile and the adoption of the mean physical properties across the film and (b) a two-dimensional model which includes changes in temperature and physical properties of a lubricant across the film and which takes into account the conditions of reverse flow and heat convection across the film occurring at the inlet region. A multi-grid algorithm is implemented to solve these two conditions. The temperature profile and the general solutions in the conjunction obtained in both approaches are compared. For the one-dimensional model, results reveal that temperature peaks just prior to the inlet of the conjunction. This feature is not apparent in the two-dimensional model and results in lower values in film thickness and larger frictional coefficients than in the two-dimensional model. In the high-pressure region, both equations yield almost the same mean temperature.

Penetrative convection in an internally heated layer of water near the maximum density point

A model for penetrative convection in which a stably stratified layer of fluid is bounded by two unstable layers is considered. This configuration is obtained in a horizontal water layer around its density extremum when a quadratic temperature profile is maintained by internal heating. A linear stability analysis shows that either stationary or oscillatory modes set in at the onset of instability depending on the values of the control parameter. Moreover, two types of stationary modes have been identified that differ by the value of their critical wave number and the number of vertical cells. These results are discussed in the light of recent studies on related topics.

Approximate solutions of heat conduction in snow with linear variation of thermal conductivity

The approximate heat balance integral method (HBIM) * is extended to the case of variable properties media such as snow. The case of linear variation of thermal conductivity was investigated. An alternative heat balance integral method (AHBIM) was developed. Both constant surface temperature and surface heat flux were considered. Comparison of temperature distribution from HBIM, AHBIM and the extension of the analytical solution of Jaeger ** was given for the case of constant surface temperature. In general, for small value of time (τ), results agree quite well with the analytical solution but as τ increases, the difference becomes more pronounced. It was found that AHBIM with a quadratic temperature profile gave a somewhat better result especially when the value of η is small. For specific property function of F(η)=e η, closed form solutions were obtained. The results were compared with those from HBIM, AHBIM and the analytical method and was found to agree exceptionally well with the analytical especially for large values of τ. * Goodman (1958). ** Jaeger (1956).

Buoyancy-Surface Tension Instability Due to an Internal Heat Source

The stability of a horizontal fluid layer with an internal heat source is considered using linear stability theory for the case of buoyancy and surface tension driven flow. Under the assumption that the neutral state is a stationary one, the method of a power series expansion is used to obtain the eigenvalue equation which is then computed numerically. As compared with the corresponding linear temperature case, it is found that the quadratic temperature profile makes the layer less stable and the cells appearing at marginal stability narrower.