On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.      

On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.     

On heat exchange and heat transport in a geothermal plant

In the long term, the only way to address the challenging task of power supply, is to make renewable energy sources economically attractive and to use them efficiently. In particular, geothermal energy is promising to take over the base load of the power supply. Nevertheless, a lot of investigations needs to be made to use the almost inexhaustible source of thermal energy in the interior of the earth effectively. Starting from the initially isothermal state, a cold fluid is injected through a borehole into a rock. By the rising pressure gradient, the fluid flows through the porous rock and escapes through another borehole. While the fluid passes the micro cracks in the hot rock, the water is heated by the rock due to the heat exchange between the constituents. This process is simulated based on the Theory of Porous Media (TPM). The presented modelling approach of the heat transport and the flow processes in a fully saturated subsurface includes two non-isothermal constituents: an elastically deformable, materially incompressible solid skeleton where thermal expansion is neglected, and a viscous, materially incompressible fluid constituent. To solve the initial-boundary-value problem, the governing primary variables of the coupled model are spatially approximated by mixed finite elements, and the time-discretisation is carried out by an implicit Euler time-integration scheme. The aim of the presented numerical simulations is to study the heat transport and to evaluate the efficiency by varying flow rates. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nondestructive experimental approach for heat flux determination on direct‐chill cast aluminium alloy billet surface

In this paper a new nondestructive experimental method for the determination of the billet surface heat flux in direct‐chill casting of aluminium alloys is presented. The new measurement method is based on measurements of the water film temperatures along the billet surface and determination of the local heat flux from the energy balance. Experiments using the new technique are carried out in a real plant environment. The results are used for the validation of a computational model of the process.     

Well-posedness for an extended Penrose–Fife phase-field model with energy balance supplied by Dirichlet boundary conditions

In this study we consider the non-isothermal phase-field model proposed by Penrose and Fife [Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62]. The system consists of the energy balance law (a nonlinear heat equation) and an equation that describes space-time changes in the order parameter (the Ginzburg–Landau equation). For the energy balance law, we consider the general nonlinear heat flux arising in non-equilibrium thermodynamics and impose the Dirichlet boundary condition. For the order parameter, we impose a constraint and thus consider a parabolic variational inequality. We prove the well-posedness of the problem: the system yields a unique solution that depends continuously upon given data.     

A three-phase FE-model for the simulation of partially saturated soils

While for many numerical simulations in geotechnics use of a two-phase model is sufficient, separate consideration of all three phases is mandatory for numerical simulations of partially saturated soils subjected to compressed air. This is a common technique frequently applied for temporary ground support in tunnelling. For the numerical simulation of tunnelling using compressed air, a multiphase model for soft soils is developed, in which the individual constituents of the soil – the soil skeleton, the fluid and the gaseous phase – and their interactions are considered. The three phase model is formulated within the framework of the Theory of Porous Media (TPM), based upon balance equations and constitutive relations for the soil constituents and their mixture. Water is modelled as an incompressible and air as a compressible phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local sensitivity analysis for the heat flux-temperature integral relationship in the half-space

This paper presents insight into the heat flux-temperature (q″ ? T) integral relationship based on constant thermophysical properties. This relationship is often used in one-dimensional, transient heat transfer studies involving null-point calorimetry and heat flux investigations. This study focuses on a short transient studies where energy has not fully penetrated the body as the result of an imposed surface heating condition. A full nonlinear heat transfer model is developed involving a half-space planar region. Temperature results are then introduced into the constant property integral relationship and a newly derived Kirchoff integral relationship for retrieving the local heat flux. Good agreement is observed between the fully nonlinear results and locally linearized system. Additionally, a sensitivity study is presented which involves perturbing the average thermophysical properties of thermal conductivity and heat capacity.     

Influence of thermal dispersion on forced convection in a composite parallel-plate channel

This paper concentrates on the analytical study of the effect of thermal dispersion on fully developed forced convection in a parallel-plate channel partly filled with a fluid saturated porous medium. The walls of the channel are subject to a constant heat flux. The central part of the channel is occupied by a homogeneous fluid, while peripheral parts of the channel are occupied by a fluid saturated porous medium of uniform porosity. It is assumed that the momentum flow in the porous region is described by the Brinkman-Forchheimer-extended Darcy equation. Since thermal dispersion becomes appreciable in high speed flows, that is, for the same situation when accounting for the Forchheimer term in the momentum equation is essential, the effect of thermal dispersion should be taken into account simultaneously with accounting for the Forchheimer term in the momentum equation. The objective of the present research is to determine in which situations accounting for thermal dispersion can significantly influence the solution.     

Modeling the transient outlet pressure and mass flow during flashing of HCFC-22 in a small nonadiabatic vessel

A model was derived from basic thermodynamic principles to describe the rate of depressurization within a small vessel. Major assumptions for the model included negligible kinetic and potential energy terms, saturated conditions within the vessel, and equal phase velocities through the exiting orifice. The two-phase homogeneous equilibrium and homogeneous frozen models, along with the single-phase model were used to predict the mass flux. A verification of the model was provided against available experimental data. Model predictions showed that the orifice size, initial refrigerant amount, and wall heat transfer have a more significant impact on the depressurization process than initial pressure and vessel volume for the test conditions.     

Influence of radiation on MHD free convection from a vertical flat plate embedded in porous media with thermophoretic deposition of particles

Magneto-hydrodynamics and thermal radiation effects on heat and mass transfer in steady laminar boundary layer flow of a Newtonian, viscous fluid over a vertical flat plate embedded in a fluid saturated porous media in the presence of the thermophoresis particle deposition effect is studied in this paper. The governing equations are transformed by special transformations. Brownian motion of particles and thermophoretic transport are considered in the flow equations. The magnetic field is considered to be applied. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting similarity equations are solved numerically by the fourth-order Runge–Kutta method with shooting technique. Many results are obtained and representative set is displayed graphically to illustrate the influence of the various parameters on the wall thermophoretic deposition velocity, concentration, temperature and velocity profiles.     

A reduced model to locate low contrast inclusions in poroelastic media

This paper introduces a numerical method to localize inclusions having slightly different elastic coefficients than those of a fully saturated poroelastic matrix, whose detection is often difficult. This method can be used to find weakly stiffer or softer objects in saturated soils or diseased biological tissues at early stages. To this end, we propose a reduced model from the Biot’s equations by splitting the fluid pressure into two parts: one embedded into an elasticity model and the other one used as a corrector term. By applying the small amplitude homogenization method, we can successfully retrieve the position and extension of inclusions in poroelastic media employing this simplified model. Numerical results show a good agreement for the location of inclusions when the contrast is below 30% stiffer or softer than the matrix, and for a noise level up to 5% for frequencies below 50 Hz.     

Analysis of performance of an iron-bath reactor using computational fluid dynamics

The performance of an iron-bath reactor has been studied using a comprehensive numerical model that combines a computational fluid dynamics approach for the gas phase and a heat and mass balance model for the bath. The model calculates:

• •coal, ore, flux and oxygen consumption;
• •post-combustion ratio (PCR);
• •heat-transfer efficiency (HTE);
• •off-gas temperature and composition;
• •heat transfer and chemical reactions between gas and iron and slag droplets; and
• •heat transfer between gas and bath, refractories and lance.

The model was validated with data reported by the Nippon Steel Corporation for a 100 t pilot plant, and the calculated and measured data are in good agreement. Modelling results showed that the dominant mechanisms of heat transfer from the gas to the bath are radiation to the slag surface and convection heat transfer to droplets.     

Non-linear analysis of the consolidation of an elastic saturated soil with incompressible fluid and variable permeability by FEM

Research interest in the mechanical behaviour of soils is growing as a result of an increasing number of geomechanical problems involving consolidation effects. The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid and variable permeability. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the convergence of the method using a technique based on the proof of solution’s existence. Finally, we then solved this constitutive model by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data. Therefore, the model can be used for quantitative predictions of consolidation behaviour of soils with permeability dependent on the settlement.     

Ignition Properties of Thermally Thin Plastics: The Effectiveness of Non-Competitive Char Formation in Reducing Flammability

The retardancy effect of char formation upon the flammability of thermally thin products is investigated. The char is formed in a single-step non-competitive scheme and is assumed to be thermally stable. The criterion for ignition is that of a critical mass flux of volatiles from the solid into the as phase. Both steady-state and transient formulations of the model are considered. In the high activation energy limit the critical heat flux efficiency in the steady-state model is proportional to c/(1 _ c), where c is the fraction of char formed. In the transient model the efficiency in reducing the maximum heat release rate, average heat release rate, and total heat released is given by c and is independent of activation energy and heat flux. The specific application that we have in mind for our model is piloted ignition in the cone calorimeter.     

Adjoint based topology optimization of conjugate heat transfer systems

This paper presents a topology optimization method for coupled thermal problems. Heat transfer linked with the forced convection flow inside cooling channels is investigated using a conjugate model. This model includes both the full Navier-Stokes equations for the fluid medium and the energy equations for both fluid and solid. In this present work, the adjoint method is extended to such conjugate heat transfer (CHT) systems to optimize their performance by the use of gradient based methods. This performance is usually a compromise between an increase in heat flux or temperature distribution at a surface and maintaining a low pressure loss within the system. To exemplify the method a uniform temperature distribution is chosen and evaluated numerically. For implementation the open source CFD Software OpenFOAM is used. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Entropy equation and absolute temperature

In this paper we consider the equivalence between the heat and the entropy balance laws. These two equations are related by an integrating factor, which defines the absolute temperature. This result is obtained applying the thermodynamic laws to a perfect fluid. So that, by means of the entropy equation we introduce the Second Law of Thermodynamics. Two particular cases of constitutive equations, both for the internal energy and for the heat flux, are considered and their corresponding differential equations, useful to study the behaviour of these materials, are also given.     

Modelling and simulation of heat exchange and transport in a geothermal plant

The production of energy by use of the high temperature in the earth's mantle has played an increasingly important role in recent years. However, large uncertainties concerning the conditions in the subsurface make it difficult to use power plants efficiently. An appropriate modelling and simulation of the heat exchange and transport provides a promising tool for further investigations of the process and optimisation of the productivity. Starting from the isothermal state at high temperatures, a cold fluid is injected through a borehole into a porous rock by applying a pressure difference between at least two wells. Passing the fractured rock, the water is heated at the crack interfaces. In addition to the convection of the temperature due to the water flow, the conduction of heat in the rock and the water has to be considered. The modelling approach of this coupled process is based on the Theory of Porous Media (TPM). Both, the rock and the water, are assumed to be materially incompressible and the thermal expansion is solely considered for the fluid, since the expansion of the rock is negligible for the occurring temperature differences. Furthermore, it is assumed that the subsurface is saturated with water. To solve the generated initial-boundary-value problem, the governing primary variables of the coupled model are spatially approximated by mixed finite elements and the time discretisation is carried out by an implicit Euler time-integration scheme. Since in the considered problem the convective transport is dominant, a streamline upwinding scheme is used for the numerical stabilisation to obtain non-oscillatory solutions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a fully nonlinear degenerate parabolic system modeling immiscible gas–water displacement in porous media

In this paper, we are interested in the simultaneous flow of two immiscible fluid phases within a porous medium. We consider a two-phase flow model where the fluids are immiscible and there is no mass transfer between the phases. The medium is saturated by compressible/incompressible phase flows. We study the gas–water displacement without simplified assumptions on the state law of gas density. We establish an existence result for the nonlinear degenerate parabolic system based on new energy estimate on pressures.