Thermo-diffusive Model Research Articles

Phonon thermophoresis of crystalline nanoparticles in liquids.

Our nonequilibrium thermodynamic model of thermodiffusion in molecular liquid systems is used to examine the role of thermal phonons in the thermophoresis of liquid suspensions of crystalline nanoparticles, which tend to have high thermal conductivity. The Soret coefficient used to characterize stationary thermodiffusion is related to differences in entropy between a particle and the body of liquid that it displaces. Calculated phonon Soret coefficients for graphite and diamond nanoparticles in three polar solvents are used to establish parameters where the phonon mechanism is expected to dominate particle thermophoresis compared to slip-flow caused by forces induced in the surface layer by the temperature gradient. Because the active mode of thermal conductivity in crystals varies with particle size, phonon thermophoresis is expected to dominate within a specific range of particle size, which varies with the properties of the particle and suspending liquid. For graphite and diamond particles in polar solvents the model estimates a size range of around 10-100 nm. Finally, thermophoretic particle accumulation is ultimately limited by the increasing concentration of particles having high thermal conductivity because the zone of particle concentration decreases the local temperature gradient that drives thermophoresis. The respective nonperturbing concentration is evaluated as the function of the size of a given material.

A novel mathematical model on generalized thermoelastic diffusion theory

In this work, a new linear model of thermo-diffusion in elastic solids has been proposed in the context of generalized thermo-elasticity. The theories of coupled thermoelastic diffusion and of generalized thermoelastic diffusion problem with one relaxation time and also of generalized thermoelastic diffusion in GN models as well as Lord-Shulman, Green-Nagdhi, two-phase-lag and three-phase-lag models on thermo-elasticity follow as limit cases. A uniqueness theorem of the proposed model followed by the reciprocity theorem for diffusion is derived. The variational theorem for the governing equations has also been derived. Also, some applications of the model in dynamic and quasi-static problems are considered.

Nonequilibrium thermodynamic model of thermoelectricity and thermodiffusion in semiconductors.

We present a self-consistent model rooted in nonequilibrium thermodynamics for defining concentration gradients in the electron/hole pairs and electric-field gradients in an intrinsic semiconductor created upon exposure to a temperature gradient. The model relies on the equation for entropy production expressed through phenomenological equations for the electron/hole flux, with the imposed condition that the resulting concentration profiles of the electrons and holes are identical. The chemical potentials of electrons, holes, and parent atoms of the lattice, which are contained in the flux equations, are calculated on the basis of the temperature-dependent equilibrium dissociation reaction: lattice atom ↔ electron + hole. Electron/hole concentration profiles resulting from the temperature gradient, along with the associated gradient in the electric field, are expressed through equilibrium microscopic parameters of the semiconductor, which include the effective masses of electrons and holes, the energy gap width, and the Debye temperature. The resulting expressions contain neither kinetic nor fitting parameters, and predict values in reasonable (order-of-magnitude) agreement with empirical data. Finally, the model predicts a measurable additional thermodiffusion-based Seebeck effect when the temperature difference is on the order of several tens of degrees across a nonisothermal semiconductor working as a power supply under conditions of optimal power transport.

Transient responses to an infinite solid with a spherical cavity according to the MGT thermo-diffusion model with fractional derivatives without nonsingular kernels

The main purpose of this paper is to describe the coupling mechanism between heat transfer and diffusion in thermoelastic materials. This will be investigated by deriving a new thermal diffusion model that allows waves to move at limited speeds, unlike the traditional thermodiffusion models. Based on the thermal relaxation concept, the fractional Moore-Gibson-Thompson equation can be used to describe the equations for heat conduction and mass diffusion. The system of equations is developed using the Green-Naghdi Type III theories and the Atangana–Baleanu-type (AB) fractional differential operator instead of the integer derivatives of time. Under the proposed concept of the fractional thermo-diffusion model, a problem of an unbounded elastic structure with a spherical hole is investigated. The surface within the bore is free of traction and exposed to thermal shock and chemical potential. The system of governing equations was combined into a sixth-order differential equation to determine the analytical results. For this purpose, the Laplace transform was used, and a numerical algorithm to calculate the inversions was applied. The numerical results demonstrate that the improved fractional operator considerably impacts the dynamics of all research fields.

On the velocity, size, and temperature of gaseous dendritic flames

Dendritic combustion in Hele–Shaw cells is investigated qualitatively using a simplified one-dimensional thermo-diffusive model. Formulas for the velocity, size, and temperature of the flamelets are derived. The temperature and velocity of the flames increase for small radii to allow for their survival regardless of the activation energy. In addition, the results obtained with very large activation energy were compared with experimental results, finding that additional tests are required due to the strong influence of gravity on the velocity and size estimations. Conditions for the existence of this anomalous propagation are investigated, confirming analytically that it can only happen for low Lewis numbers.

Effects of curvature on triple flame propagation in fuel-oxidizer counterflow

In this study, we analyze the propagation of an expanding triple flame in an axisymmetric counterflow of a fuel against an oxidizer. The problem is formulated as a thermo-diffusive model with one-step global reaction. An asymptotic analysis in the limit of large activation energy and weak strain rate is conducted. The study is supported and complemented by numerical simulations carried out for arbitrary values of the strain rate. In addition to the flame front curvature 1/Rt associated with the variation in the reactants concentrations transverse to the mixing layer, the propagation of the expanding triple flame also depends on the azimuthal curvature 1/Rf where Rf is the front leading edge radial distance. As the triple flame expands to large radial distances, its propagation becomes quasi-steady. Under a quasi-steady state assumption, an explicit expression is derived for the displacement speed of the triple flame, which is found to be linearly proportional to the total curvature 1/Rf+1/Rt. Two-dimensional axisymmetric simulations are conducted to validate in particular the quasi-steady assumption. These include transient simulations of the expanding triple flame which are compared to the numerical solution of the steady eigenvalue problem obtained in a frame attached to the propagating front under a quasi-steady assumption. Following a transient ignition phase, the triple flame is found to propagate in a quasi-steady manner when Rf (measured with the stoichiometric planar flame thickness) exceeds 5, approximately. Although the theoretical analysis is performed in the weak strain limit, the linear dependence of the triple flame speed on the curvature 1/Rfis found to be applicable over a wide range of strain rates. Besides, the analysis is extended for inwardly propagating triple flames (flame holes) and similar expressions describing the relationship between displacement speed of the triple flame and curvatures are obtained.

Effects of thermal expansion on Taylor dispersion-controlled diffusion flames

A theoretical analysis is developed to investigate the effects of gas expansion due to heat release on unsteady diffusion flames evolving in a pipe flow in which the mixing of reactants is controlled by Taylor's dispersion processes thereby extending a previously developed theory based on the thermo-diffusive model. It is first shown that at times larger than radial diffusion times, the pressure gradient induced by the gas expansion is, in the first approximation, small in comparison with the prevailing pressure gradient driving the flow, indicating that corrections to the background velocity profile are small. The corrections to the velocity components along with the leading-order mixing variables such as the concentrations, temperature and density are solved for a Burke–Schumann flame. Due to the dependence of the effective Taylor diffusion coefficients on the gas density, quantitative and sometimes qualitative departures in predictions from the thermo-diffusive model are observed.

The combined effects of chemical reaction order and stoichiometry on nonpremixed edge flames

We examined in this work the combined effects of chemical reaction order n and the stoichiometric mixture fraction on the propagation of nonpremixed edge flames in a counterflow configuration. We began by formulating the problem mathematically using the constant density thermo-diffusive model. The problem was then solved numerically using the finite element method. It has been shown that the reaction order and stoichiometry have an interchangeable role in modifying flame propagation. The study has shown that triple flames exist only for a limited range of values of both the chemical reaction order n and the stoichiometric mixture fraction . Over such values, modifying n and were found to have a great impact on many characteristics related to the flame structure, such as the flame location, the extent of both the premixed flame wings and the trilling diffusion flame, and the flame curvature. In addition, the flame location was found to vary subjected to the value of the reaction order in a manner similar to that observed by varying the mixture fraction. The study also identified the influence of the fuel Lewis number on the flame propagation and was found that the effect of is more pronounced when the reaction order differs from unity. Finally, the study has examined the role of the strain rate on the flame propagation and concluded that the effect of the chemical order n is negligible for stoichiometric mixtures but tends to be significant for off-stoichiometric mixtures.

Solution of three-dimensional generalized thermodiffusive elastic half-space with phase lag in the presence of chemical potential shock and thermal pulse

An isotropic homogeneous thermoelastic half-space has been considered with permeating gas, thermal pulse, and chemical potential on the bounding plane. A three-dimensional generalized thermo-diffusion model with three-phase lag has been constructed in the context of Memory-dependent derivative (MDD). Laplace transform (time domain) and Double Fourier transform (space domain) have been used in the formulation of the problem and thus the solutions for temperature increment, invariant stress, chemical potential and diffusion concentration are obtained using eigenvalue approach. Finally the effect of kernel function has been discussed and also the results are depicted in several dimensions.

Nonlinear resultant theory of shells accounting for thermodiffusion

The complete nonlinear resultant 2D model of shell thermodiffusion is developed. All 2D balance laws and the entropy imbalance are formulated by direct through-the-thickness integration of respective 3D laws of continuum thermodiffusion. This leads to a more rich thermodynamic structure of our 2D model with several additional 2D fields not present in the 3D parent model. Constitutive equations of elastic thermodiffusive shells are discussed in more detail. They are formulated from restrictions imposed by the resultant 2D entropy imbalance according to Coleman–Noll procedure extended by a set of 2D constitutive equations based on heuristic assumptions.

Optical pumping through a black-As absorbing-cooling layer in graphene-based heterostructure: thermo-diffusion model

We study the optical near- and mid-infrared pumping of the heterostructure based on graphene with a black-As layer. This layer serves for the optical generation and cooling of the electron-hole pairs to be injected into the graphene layer. Due to the cooling of the electron-hole pairs, their energy in the case of the absorbing-cooling layer with the optimized thickness can be close to the energy gap of the black-As layer. Owing to a relatively narrow energy gap of the black-As layer Δ G, the energy of the injected electron-hole pairs can be smaller than the energy of optical phonons in in graphene (ℏ ω 0≃0.2 eV. This can provide the formation of the cold electron-hole plasma in the graphene-layer that is beneficial for achieving of the interband population inversion and the interband terahertz lasing. The obtained results can be used for the optimization of the terahertz lasers with the optical pumping.

A Numerical Simulation to the Influence of Unsteady Strain Flow on Twin Premixed Flames

The aim of this paper was to examine the response of twin premixed flames formed in a counterflow configuration to the presence of an unsteady straining flow. We began by describing the problem mathematically using the thermodiffusive model with constant density and then adopted a finite elements approach to solve the problem numerically. The study has shown that the role of flow on flame propagation is determined by three main parameters, namely, flow amplitude A, strain rate ε, and fuel Lewis number LeF. For LeF ≥ 1, the flow is seen to promote flame extinction, while LeF < 1 the flow clearly enhances the flame reactivity. Qualitatively, it has been shown that for LeF = 1, there exists a critical value of A (that varies with ε) below which the reactivity decreases monotonically with A. For small LeF < 1, on the other hand, the reactivity was seen to increase with A. For LeF > 1, however, a nonmonotonic dependence, especially for small ε, is predicted.

The Combined Effects of Heat Loss and Reversibility on the Propagation of Planar Premixed Counterflow Flames

Abstract We study in this paper the combined effect of heat loss and reversibility on the propagation of planar flames formed within the counterflow configuration. The problem has been formulated first using the thermodiffusive model with constant density and then solved numerically using finite elements. The impact of four main parameters, namely the reversibility r, the heat loss κ, the strain rate ε, and the activation energy β, on the propagation of planar flames has been discussed in details. The study has shown that planar flames under reversible conditions behave qualitatively similar to those observed for irreversible reactions, which agree with the asymptotic findings. In the presence of heat loss, the problem exhibits multiplicity of solutions whose number and stability were found to vary according to the strain rate ε. In addition, the study has predicted the existence of a certain value of the reversibility parameter r beyond which the impact of reversibility becomes negligible. Finally, we have examined the stability of the solutions and determined the domain of stability of solutions and their multiplicity for this problem.

Diffusion mechanism of fingering in smolder combustion of thin solid fuel

ABSTRACTThe frontal propagation of smolder combustion of thin solid fuel in airflow in a narrow channel is considered. It is shown that the governing process that determines the propagation of the smolder combustion front is the diffusion of oxygen in flow and its absorption by the front. A two-dimensional thermodiffusion model of propagation of the smolder combustion front is considered and the equations describing the front propagation have been obtained. The solution of the equations was obtained in general form that allows constructing an algorithm for the numerical simulation. The stability of the planar smolder combustion front is investigated. It is shown that diffusion of oxygen and thermal conductivity in airflow are competing processes and can play either a stabilizing or destabilizing role depending on the mutual direction of airflow and the propagation of the smoldering front. So, if airflow occurs against the propagating smoldering front, the diffusion of oxygen in airflow plays a destabilizing role while the thermal conductivity is a stabilizing factor. If, however, airflow and the propagation of the smoldering front occur in the same direction, the phenomenon will be reversed. The diffusion of oxygen in airflow is a stabilizing factor while the thermal conductivity plays a destabilizing role.

Asymptotic analysis to the effect of temperature gradient on the propagation of triple flames

We study asymptotically in this paper the influence of the temperature gradient across the mixing layer on the propagation triple flames formed inside a porous wall channel. The study begins by formulating the problem mathematically using the thermo-diffusive model and then presents a thorough asymptotic analysis of the problem in the limit of large activation energy and thin flames. Analytical formulae for the local burning speed, the flame shape and the propagation speed in terms of the temperature gradient parameter have been derived. It was shown that varying the feed temperatures can significantly enhance the burning of the reactants up to a critical threshold, beyond which no solutions can be obtained. In addition, the study showed that increasing the temperature at the boundaries will modify the usual triple structure of the flame by inverting the upper premixed branch and extending it to the boundary, which may have great implications on the safety of the adopted combustion chambers.

Influence of Fuel Premixing on Partially Premixed Flames in Porous-Wall Channels

Abstract This paper examines the influence of mixing a fuel with an oxidizer-rich reactant on the existence and propagation of partially premixed flames formed in a porous-wall channel. The amount of the injected fuel has been taken as a parameter to generate different states of the combustion mixtures. The problem has been mathematically formulated in the framework of the thermodiffusive model under a single-step chemical reaction. Numerical simulations based on finite elements were then employed to solve the governing partial differential equations. The study has shown that the degree of fuel premixing in the combustion mixtures as well as the value of the Lewis number of fuel and oxidizer plays a great role on the existence and structure of triple flames. It is found also that flame extinguishes for larger values of the premixing parameter. Flame structures were also found to exhibit several changes such as the shift of the leading edge toward the fuel supply side and the formation of twin flames. It has been predicted that the premixing can substantially enhance the reactivity of the combustion mixture, but on the other hand it tends to shift the flame to the boundary which may destroy the underlying combustion chamber. For nonunit Lewis numbers, fuel premixing is found to shift triple flames from the stoichiometric surface and attach the flame to the channel walls. It is predicted also that the maximum reaction rate reduces with increasing the Lewis numbers.

Dynamical thermodiffusion model of graphene synthesis on polymer films by laser irradiation and application to strain sensors

A method of graphene synthesis at a desired position on polymer substrates by laser irradiation has been studied. Polyethylene naphthalate films were used as substrates, on which a thin Ni film was deposited as a catalyst layer. The irradiation of a focused laser made a hole in the Ni film and the graphene was synthesized in the hole by the surface decomposition of the polymer. The laser power dependence of the hole radius was successfully explained by a dynamical model of thermodiffusion. The quality of the laser-synthesized graphene was studied by micro-Raman scattering. Typical ambipolar characteristics of the synthesized graphene were observed in field-effect transistors. We studied the application of the laser-synthesized graphene to strain sensors using the sensitivity of the electric conductance to the strain induced by the bending deformation of substrates.

On the Stability of Planar Premixed Flames Under Nonadiabatic Conditions and Preferential Diffusion

In this paper, we provide a numerical study of the stability analysis of a planar premixed flame. The interaction of preferential diffusion and heat loss for a planar premixed flame is investigated using a thermodiffusive (constant density) model. The flame is studied as a function of three nondimensional parameters, namely, Damköhler number (ratio of diffusion time to chemical time), Lewis number (ratio of thermal to species diffusivity), and heat loss. A maximum of four solutions are identified in some cases, two of which are stable. The behavior of the eigenvalues of the linearized system of stabilty is also discussed. For low Lewis number, the heat loss plays a major role in stabilizing the flame for some moderately high values of Damköhler number. The results show the effect of increasing or decreasing Lewis number on adiabatic and nonadiabatic flames temperature and reaction rate as well as the range of heat loss at which flames can survive.

Transient Heat and Solute Transfers in Liquid-Saturated Porous Media

We investigate transient heat and solute transfers in liquid-saturated porous media. The macroscopic equivalent models are obtained by a homogenization process from the pore-scale description. The large value of the Lewis number in liquid mixtures introduces a possible separation of scales between the heat and solute diffusion wavelengths \(L^{\mathrm{heat}}\) and \(L^{\mathrm{solute}}\) at a given excitation time. Thus, two separations of scales could be present between the pore scale of characteristic length l and \(L^{\mathrm{heat}}\) and \(L^{\mathrm{solute}}\). Depending on the ratio of these two separations of scales results in different equivalent macroscopic models or in non-homogenizable situations. Among numerous possibilities, we present two different homogenizable problems and a non-homogenizable situation. As the characteristic excitation time is decreased, the solute pseudo-wavelength and the heat pseudo-wavelength increase; a unique separation of scales between both solute and heat transfers and the pore scale appears, and we recover the classical macroscopic thermodiffusion model.

Propagation of symmetric and non-symmetric lean hydrogen–air flames in narrow channels: Influence of heat losses

In this paper we present results of direct numerical simulations of lean hydrogen–air flames freely propagating in a planar narrow channel with varying flow rate, using detailed chemistry and transport and including heat losses through the channel walls. Our simulations show that double solutions, symmetric and non-symmetric, can coexist for a given set of parameters. The symmetric solutions are calculated imposing symmetric boundary conditions in the channel mid-plane and when this restriction is relaxed non-symmetric solutions can develop. This indicates that the symmetric solutions are unstable to non-symmetric perturbations, as predicted before within the context of a thermo-diffusive model and simplified chemistry. It is also found that for lean hydrogen–air mixtures an increase in heat losses leads to a discontinuity of the steady state response curve, with flames extinguishing inside a finite interval of the flow rate. Non-symmetric flames burn more intensely and in consequence are much more robust to flame quenching by heat losses to the walls. The inclusion of the non-symmetric solutions extends therefore the parametric range for which flames can propagate in the channel. This analysis seems to have received no attention in the literature, even if it can have important safety implications in micro-scale combustion devices burning hydrogen in a lean premixed way.