Thermal Non-equilibrium Conditions Research Articles

Microchannels enhanced by porous materials: Heat transfer enhancement or pressure drop increment?

The analytical heat transfer performance study (i.e. HTP, introduced as the ratio of the heat transfer rate enhancement to the pressure drop increment) for microchannels enhanced by a porous material is performed. The Darcy–Brinkman equation of motion and the two-energy equation model for heat transfer within the porous material (i.e. the local thermal non-equilibrium condition, LTNE) are used to find the Poiseuille (Po) and Nusselt (Nu) numbers in the slip-flow regime (i.e. when the Knudsen number, Kn, <0.1). To cover a wide range of heat exchangers, effects of hydrodynamical characteristics of the porous material (i.e. porosity, ϕ, and Darcy number, Da), the effective thermal conductivity ratio (k), the interphase convective heat exchange between the two phases (Bi), the velocity-slip and temperature-jump coefficients (α and β), the dimensionless thermal resistance of the walls (Rw), and the internal heat generations within the solid and fluid phases of the porous material (ωf and ωs) are considered. HTP maps are presented for a wide variation range of all pertinent parameters to design or to select a micro-heat exchanger according to the heat transfer enhancement with respect to the pressure drop cost. Results reveal that the HTP may effectively be increased by inserting rarefied porous inserts. The present analysis suggests the implementation of porous inserts within flow passages especially when 0.001<Kn<0.1 (the slip-flow regime).

Numerical Investigation of Transient Phase Change in Horizontal Porous Channel with Localized Heating using Two-Equation Model

ransient phase change through a horizontal channel that subjected to discrete heat flux and filled with porous media of high-conductivity material, copper, is numerically investigated with n-pentane as the working fluid. The thermal non-equilibrium model is used in conjugate with the multiphase mixture model to analyze the transient behavior of fluid and solid phases. Three cases are studied regarding the position of the discrete heat flux: (1) discrete heat flux at the lower wall, (2) discrete heat flux at the upper wall and (3) discrete heat flux at both the lower and upper walls. Results show that the minimum liquid saturation and the maximum solid temperature (thermal non-equilibrium condition) are located above the heated suction. Results also show that the dryout zone is formed first at the upper wall due to the lower heat transfer from the upper heated wall. Temperature distribution for both solid and fluid phases, liquid saturation, vapor and liquid velocities are presented and analysed. KeywordsMedia; Transient; Phase Change; Thermal non- equilibrium model; Horizontal Channel.

Evolution of holographic dark energy with interaction term Q ∝ H ρ de $\boldsymbol {Q\propto H \rho _{\text {de}}}$ and generalized second law

A flat FLRW Universe with dark matter and dark energy, which are interacting with each other, is considered. The dark energy is represented by the holographic dark energy model and the interaction term is taken as proportional to the dark energy density. We have studied the cosmological evolution and analysed the validity of the generalized second law of thermodynamics (GSL) under thermal equilibrium conditions and non-equilibrium conditions. We have found that the GSL is completely valid at the apparent horizon but violated at the event horizon under thermal equilibrium condition. Under thermal non-equilibrium condition, for the GSL to be valid, we found out that the temperature of the dark energy must be greater than the temperature of the apparent horizon if the dark energy behaves as a quintessence fluid.

On the thermally developing forced convection through a porous material under the local thermal non-equilibrium condition: An analytical study

The aim of the present analytical study is to investigate a thermally developing forced convective heat transfer inside a channel filled with a porous medium whose walls are imposed to a constant heat flux (i.e. the iso-flux thermal boundary condition). The Darcy’s law of motion and the two-energy equation (i.e. local thermal non-equilibrium, LTNE) model are considered. A perturbation analysis is conducted to avoid using any model for the iso-flux thermal boundary condition. Thermally developing forced convection inside a porous-filled channel has previously not been analyzed without implementing a heuristic model for the iso-flux thermal boundary condition. The temperature difference between the fluid and solid phases (called the LTNE intensity) is analytically obtained. Results concerning the LTNE intensity is compared with the available models of the iso-flux thermal boundary condition. In addition, the Nusselt number of a thermally developing convection through a porous material is obtained for the LTNE condition. Also, an expression representing the developing length, i.e. the dimensionless axial location at which the flow becomes fully developed, for a porous-filled channel is proposed for the first time.

Two-temperature thermodynamic and transport properties of SF6–Cu plasmas

SF6 and Cu are widely adopted in electrical equipment as a dielectric medium and for conductive components, respectively. SF6–Cu plasmas are frequently formed, particularly in high-voltage circuit breaker arcs and fault current arcs, due to erosion of the Cu components. In this paper, calculated values of the thermodynamic and transport properties of plasmas in SF6–Cu mixtures are presented for both thermal equilibrium and non-equilibrium conditions. The composition is determined by the two-temperature Saha equation and Guldberg–Waage equation in the form derived by van de Sanden. The composition and the thermodynamic properties are evaluated through a classical statistical mechanics approach. For the transport coefficients, the simplified Chapman–Enskog method developed by Devoto, which decouples the electrons and heavy species, has been applied using the most recent collision integrals. The thermodynamic and transport properties are calculated for different electron temperatures (300–40 000 K), ratios of electron to heavy-species temperature (1–10), pressures (0.1–10 atm) and copper molar proportions (0–50%). It is found that deviations from thermal equilibrium strongly affect the thermodynamic and transport properties of the SF6–Cu plasmas. Further, the presence of copper has different effects on some of the properties for plasmas in and out of thermal equilibrium. The main reason for these changes is that dissociation reactions are delayed for non-thermal equilibrium plasmas, which in turn influences the ionization reactions that occur.

Finite Element Analysis of Heater Length in a Porous Annulus - Part A

The focus of present study is to investigate the influence of discrete heating by an isothermal heater placed at the inner radius of a vertical annular cylinder containing porous medium between its inner and the outer radius. Finite element method is used to solve the governing partial differential equations by employing 3 noded triangular elements. Darcy model is used to represent the flow behavior inside the porous medium. It is assumed that the thermal non-equilibrium condition exists between the fluid and solid phases of the porous medium. The study is conducted for different lengths of heater corresponding to 20%, 35% and 50% of the total height of the cylinder. It is found that the Nusselt number for fluid, solid phases as well as total Nusselt number initially decreases and the increases along the length of heater.

Stability of a porous Benard-Brinkman layer in local thermal non-equilibrium with Cattaneo effects in solid

The stability of a porous Benard layer of Brinkman fluid under local thermal non-equilibrium conditions and obeying the Cattaneo flux law in the solid is studied. The preference for stationary and overstable linear motions is examined in the whole parameter space of the problem. It is shown that overstability occupies a region within a branch of a rectangular hyperbola-like curve in the first quadrant of the (τˆ,H) plane, where τˆ,H are the Straughan and thermal inter-phase interaction parameters, respectively, so that overstability persists even when H→∞. The Brinkman effect tends to stabilize the layer but enhances the region of preference of overstability in the parameter space. The influence of the other parameters on the critical mode is also identified. The nonlinear development of the amplitude A of the linear wave motions of both stationary and overstable modes is found to be governed by a first order evolution equation. The analysis of the evolution equation shows that the layer can exhibit supercritical instability for both stationary and overstable modes and subcritical instability and stability for the stationary mode depending on the relative magnitudes of the parameters of the problem. The Brinkman effect is found to reduce the amplitude of the supercritical instability while the porosity modified ratio of thermal conductivities and the ratio of thermal diffusivities tend to promote subcritical instabilities. The implication of these results on the nonlinear global stability of the model is discussed.

Forced convection gaseous slip flow in a porous circular microtube: An exact solution

Rarefied phenomena can occur when a gas flows through a microchannel. However, most available convective solutions were obtained under the local thermal equilibrium condition. In this study, gaseous slip flow in a circular microtube filled with a porous medium is analytically investigated under the local thermal non-equilibrium condition. The first-order velocity slip and temperature jump conditions at the tube wall are invoked in order to account for the rarefaction effects. Rigorous analytical solutions are obtained for the velocity and temperature distributions as well as the average Nusselt number. Theoretical predictions are then compared to those of existing limiting cases in the literature. Results indicate that the degree of rarefaction, represented by the Knudsen number ranging from 10−3 to 10−1, has a significant effect on the velocity, temperature, pressure drop and heat transport within the microtube for various combinations of pertinent parameters such as the porosity, effective thermal conductivity ratio, Biot number and porous media shape factor.

Numerical Simulation of Natural Convection in a Porous Cavity with Linearly Temperature Distributions Under the Local Thermal Nonequilibrium Condition

A numerical investigation of the natural convection heat transfer in a rectangular cavity filled with a heat-generating porous medium by adopting the local thermal nonequilibrium (LTNE) model is reported in this paper. The top and bottom walls of the enclosure are adiabatic, the left wall is linearly cooled, and the right wall is cooled by a linear or uniform temperature profile. The results show that the isotherms for the fluid and solid phases become similar with the increase of the interphase heat transfer coefficient H; the increasing heat transfer between the two phases brings their temperatures closer to each other and thus the solid and fluid phases are in a state of the thermal equilibrium at higher values of H. For case A, the interphase heat transfer coefficient has little influence on the heat transfer rate of the solid phase of the porous cavity and the heat transfer profile of the solid phase (Nusy) is symmetrical with respect to the center point of line Y = 0.5. For case B, the interphase heat transfer coefficient H has a significant effect on that at the right wall, and the total heat transfer of the heat-generating porous cavity is implemented by the right-side wall. The total heat transfer rate Q of case B is higher than that of case A at a high thermal conductivity ratio γ (γ = 1, 10).

Analytical Study of Heat Flux Splitting in Micro-channels Filled with Porous Media

The conditions of occurring heat flux splitting (bifurcation) phenomenon in a micro-channel filled with a porous medium including internal heat generations within both the solid and fluid phases under the local thermal non-equilibrium (LTNE) condition is analytically studied in the slip regime. The channel walls are subjected to a constant heat flux. Exact solutions for both the dimensionless temperatures of the two phases and the Nusselt number are obtained. Effects of the pertinent parameters such as heat generation parameter (\(\omega \)), the interphase heat transfer parameter (Vadasz in J Porous Media 15:249–258, 2012) or Biot number (\(\textit{Bi}\)), the fluid-to-solid effective conductivity ratio (k), and the temperature jump coefficient (\(\beta \)) on the dimensionless temperature profile (\(\theta \)) of the two phases as well as the Nusselt number are investigated. Moreover, the validity of one-equation model (the local thermal equilibrium assumption) is analyzed by comparing the Nusselt number obtained by one-equation model (LTE) with that obtained by the two-equation model (LTNE). Results reveal that the conditions at which the heat flux bifurcation (splitting) occurs in the slip regime is the same as those of the no-slip regime. In addition, a kind of heat flux bifurcation in which the solid and fluid phases have the same dimensionless temperature sign is observed in the slip regime, while it was not previously observe in the no-slip regime. It is discussed that the Nusselt number can increase or decrease with respect to \(\omega \) and may have either positive or negative values in both the no-slip and slip regimes. The presence of internal heat generation intensifies the role of \(\beta \) in the Nusselt number reduction. In addition, the accuracy of LTE model increases with increased \(\textit{Bi}\) and with decreased \(\beta \), while it is not a monotonic function of k in the presence of internal heat generation.

Electronic doping and redox-potential tuning in colloidal semiconductor nanocrystals.

Electronic doping is one of the most important experimental capabilities in all of semiconductor research and technology. Through electronic doping, insulating materials can be made conductive, opening doors to the formation of p-n junctions and other workhorses of modern semiconductor electronics. Recent interest in exploiting the unique physical and photophysical properties of colloidal semiconductor nanocrystals for revolutionary new device technologies has stimulated efforts to prepare electronically doped colloidal semiconductor nanocrystals with the same control as available in the corresponding bulk materials. Despite the impact that success in this endeavor would have, the development of general and reliable methods for electronic doping of colloidal semiconductor nanocrystals remains a long-standing challenge. In this Account, we review recent progress in the development and characterization of electronically doped colloidal semiconductor nanocrystals. Several successful methods for introducing excess band-like charge carriers are illustrated and discussed, including photodoping, outer-sphere electron transfer, defect doping, and electrochemical oxidation or reduction. A distinction is made between methods that yield excess band-like carriers at thermal equilibrium and those that inject excess charge carriers under thermal nonequilibrium conditions (steady state). Spectroscopic signatures of such excess carriers, accessible by both equilibrium and nonequilibrium methods, are reviewed and illustrated. A distinction is also proposed between the phenomena of electronic doping and redox-potential shifting. Electronically doped semiconductor nanocrystals possess excess band-like charge carriers at thermal equilibrium, whereas redox-potential shifting affects the potentials at which charge carriers are injected under nonequilibrium conditions, without necessarily introducing band-like charge carriers at equilibrium. Detection of the key spectroscopic signatures of band-like carriers allows distinction between these two regimes. Both electronic doping and redox-potential shifting can be powerful tools for tuning the performance of nanocrystals in electronic devices. Finally, key chemical challenges associated with nanocrystal electronic doping are briefly discussed. These challenges are centered largely on the availability of charge-carrier reservoirs with suitable redox potentials and on the relatively poor control over nanocrystal surface traps. In most cases, the Fermi levels of colloidal nanocrystals are defined by the redox properties of their surface traps. Control over nanocrystal surface chemistries is therefore essential to the development of general and reliable strategies for electronically doping colloidal semiconductor nanocrystals. Overall, recent progress in this area portends exciting future advances in controlling nanocrystal compositions, surface chemistries, redox potentials, and charge states to yield new classes of electronic nanomaterials with attractive physical properties and the potential to stimulate unprecedented new semiconductor technologies.

On the effects of exothermicity and endothermicity upon the temperature fields in a partially-filled porous channel

Forced convection of heat in a two-dimensional channel, partially filled by a porous insert is considered. This system is assumed under fully developed conditions and constant wall heat flux. Further, the fluid and solid phases can feature internal heat generation (exothermicity) and consumption (endothermicity). Analytical solutions are developed for the solid and fluid temperature fields by applying local thermal non-equilibrium (LTNE) conditions and the Darcy-Brinkman model of momentum transport. Two existing interface models (Models A and B) are employed to describe the thermal boundary conditions at the porous-fluid interface. The developed solutions for the temperature fields are compared to those found by applying the local thermal equilibrium (LTE) assumption and, therefore, the validity of the LTE is examined. This is done for a wide range of pertinent parameters including Biot number, conductivity ratio, Darcy number and thickness of the porous insert. It is found that the thermal behaviour of the investigated partially filled system is influenced by the heat sources in both solid and fluid phase. It is further shown that the LTE approach remains an acceptable assumption only for some specific regions of the parametric space. Furthermore, the occurrence of temperature gradient bifurcation on the surface of the porous-fluid interface is examined. It is demonstrated that this effect is highly sensitive to the intensity of the energy sources.

Numerical investigation of mist flow regime in a vertical tube

In this research, the mist flow regime, consisting dispersed water droplets in vapor flow, is simulated numerically in a vertical tube; using a discrete phase model (DPM). In this method, in addition to solving transport equations for continuous phase, a discrete phase is simulated in a Lagrangian approach and the coupling between phases is modeled through interaction terms in the transport equations. The aims of this research are to investigate thermal equilibrium and evaluate the heat transfer coefficient of mist flow regime. The results show, when thermodynamic non-equilibrium takes happen, the rate of heat transfer from the vapor to the droplets is too slow that their presence is ignored, therefore by increasing the water mass flow rate in this case, due to reduction of vapor mass flow rate, the heat transfer coefficient decreases. But when complete thermodynamic equilibrium condition is established, the rate of heat transfer from the vapor to the droplets is too fast that the vapor temperature remains at the saturation temperature until all the droplets have been evaporated and by increasing the water mass flow rate, the heat transfer coefficient will increase. In order to simulate mist flow regime in thermal equilibrium and non-equilibrium conditions, water droplets with two different diameters are injected into steam flow. The numerical results of heat transfer coefficient and wall temperature in four different vapor qualities are investigated in each state, which show good agreement with experimental data and correlations.

Simulation of complete liquid–vapour phase change process inside porous evaporator using local thermal non-equilibrium model

The present article demonstrates the necessity and the usefulness of a newly proposed smoothing algorithm for the effective diffusion coefficient in order to avoid non-physical “jump” in the predicted properties during numerical simulations of the complete liquid–vapour phase change process within porous media. The formulation is based on the two-phase mixture model (TPMM) along with the assumption of local thermal non-equilibrium (LTNE) condition where the governing equations have been solved using the finite volume method. For the purpose of demonstration, one-dimensional phase change problem of water has been considered. Comparison between results obtained with the local thermal equilibrium (LTE) and LTNE models indicates that wherever applicable, particularly for lower inlet Reynolds number and higher heat input, the latter should be used. The effect of employing different models for the partitioning of wall heat flux using LTNE model has been found negligible owing to the very high convective heat transfer coefficient between the solid and the fluid phases. A representative parametric study suggests that the diffusive energy transport process in the upstream direction through the solid and the fluid phases has a strong influence on the initiation and the termination of the phase change process.

Thermodynamic and Transport Properties of Nitrogen Plasma Under Thermal Equilibrium and Non-equilibrium Conditions

Assuming a seven species model of nitrogen plasma, thermodynamic and transport properties are calculated under thermal equilibrium and non-equilibrium conditions. Species densities, mass densities, specific heat, enthalpy, viscosity, thermal conductivity, collision frequency, and electrical conductivity are studied as a function of temperature, pressure, and different degrees of thermal and chemical non-equilibrium. Effect of both electrons and ions are considered in the shielding for charged–charged interactions under shield coulomb potential and obtained results are found to be in excellent agreement with experimentally obtained results, especially at higher temperatures. Accounting up to third ionized states, results are presented with temperatures ranging from 300 to 50,000 K, the ratio of electron temperature (Te) to the heavy particle temperature (Th) ranging from 1 to 20 and the pressure ranging from 0.1 to 7 atmospheres. Under atmospheric pressure, results obtained for thermodynamic equilibrium (Te = Th) are compared with number of published results under similar conditions. Computed properties under different degrees of chemical non-equilibrium are also discussed.

Analysis and analytical characterization of bioheat transfer during radiofrequency ablation

Understanding thermal transport and temperature distribution within biological organs is important for therapeutic aspects related to hyperthermia treatments such as radiofrequency ablation (RFA). Unlike surface heating, the RFA treatment volumetrically heats up the biological media using a heating probe which provides the input energy. In this situation, the shape of the affected region is annular, which is described by an axisymmetric geometry. To better understand the temperature responses of the living tissues subject to RFA, comprehensive characteristics of bioheat transport through the annular biological medium is presented under local thermal non-equilibrium (LTNE) condition. Following the operational features of the RFA treatment, based on the porous media theory, analytical solutions have been derived for the blood and tissue temperature distributions as well as an overall heat exchange correlation in cylindrical coordinates. Our analytical results have been validated against three limiting cases which exist in the literature. The effects of various physiological parameters, such as metabolic heat generation, volume fraction of the vascular space, ratio of the effective blood to tissue conductivities, different biological media and the rate of heat exchange between the lumen and the tissue are investigated. Solutions developed in this study are valuable for thermal therapy planning of RFA. A criterion is also established to link deep heating protocol to surface heating.

Constant wall heat flux boundary condition in micro-channels filled with a porous medium with internal heat generation under local thermal non-equilibrium condition

Forced convection heat transfer in a micro-channel filled with a porous material saturated with rarefied gas with internal heat generation is studied analytically in this work. The study is performed by analysing the boundary conditions for constant wall heat flux under local thermal non-equilibrium (LTNE) conditions. Invoking the velocity slip and temperature jump, the thermal behaviour of the porous-fluid system is studied by considering thermally and hydrodynamically fully-developed conditions. The flow inside the porous material is modelled by the Darcy–Brinkman equation. Exact solutions are obtained for both the fluid and solid temperature distributions for two primary approaches models A and B using constant wall heat flux boundary conditions. The temperature distributions and Nusselt numbers for models A and B are compared, and the limiting cases resulting in the convergence or divergence of the two models are also discussed. The effects of pertinent parameters such as fluid to solid effective thermal conductivity ratio, Biot number, Darcy number, velocity slip and temperature jump coefficients, and fluid and solid internal heat generations are also discussed. The results indicate that the Nusselt number decreases with the increase of thermal conductivity ratio for both models. This contrasts results from previous studies which for model A reported that the Nusselt number increases with the increase of thermal conductivity ratio. The Biot number and thermal conductivity ratio are found to have substantial effects on the role of temperature jump coefficient in controlling the Nusselt number for models A and B. The Nusselt numbers calculated using model A change drastically with the variation of solid internal heat generation. In contrast, the Nusselt numbers obtained for model B show a weak dependency on the variation of internal heat generation. The velocity slip coefficient has no noticeable effect on the Nusselt numbers for both models. The difference between the Nusselt numbers calculated using the two models decreases with an increase of the temperature jump coefficient.

Entropy generation of viscous dissipative nanofluid flow in thermal non-equilibrium porous media embedded in microchannels

Based on the first-law and second-law of thermodynamics, we investigate thermal performance and entropy generation of water–alumina nanofluid flows in porous media embedded in a microchannel under local thermal non-equilibrium condition. Analytical closed-form solutions of two-dimensional temperature distributions are obtained for the cases with and without the viscous dissipation term in the energy equation. The thermal non-equilibrium entropy generation function is derived using the differential method. Due to the embedment of the porous medium in the microchannel and the suspension of the nanoparticle in the working fluid, the viscous dissipation effect is magnified significantly, altering thermal characteristics and entropy generation of the system. For the case where the viscous dissipation effect is neglected, total entropy generation and fluid friction irreversibility are overrated while heat transfer irreversibility is remarkably underestimated. In a low-aspect-ratio microchannel, the suspension of nanoparticles in the fluid decreases the thermodynamic efficiency from the second-law point of view. Utilization of nanofluids in a high-aspect-ratio microchannel enhances exergetic effectiveness in low-Reynolds-number flow regime. By reducing the nanoparticle size, entropy generation can be decreased by as much as 73%. The optimum Reynolds number associated with minimum entropy generation for nanofluid flow in a porous microchannel is identified. The optimum range of porous medium permeability is characterized by Da⩾10-1. It is observed that effectiveness of the interstitial heat transfer between the solid and fluid phases of the porous medium induces a pronounced effect on the entropy generation, signifying the importance to consider the thermal non-equilibrium condition in the second-law performance analysis of porous-medium flow.

New models for heat flux splitting at the boundary of a porous medium: three energy equations for nanofluid flow under local thermal nonequilibrium conditions

One of the challenging points in the simulation of a nanofluid flowing through a porous medium is modeling the surface heat flux in the presence of nanoparticles and internal solid matrix. The question is how much energy is absorbed by the solid phase, fluid phase, and particles at the surface of imposing heat flux? To reach a suitable answer, a local thermal nonequilibrium approach (including three energy equations) is presented in this paper and three heat flux models are proposed for the first time. The proposed models are compared and analyzed. The effects of interstitial heat transfer coefficients on the heat transfer in a porous channel are completely studied. The fluid temperature distributions and heat transfer rate obtained by homogenous and nonhomogenous approaches (for the proposed models) are completely studied and compared. The results show that the nonhomogeneous approach experiences larger Nusselt number than the homogeneous one for all the recommended heat flux models.

Comment on “Contributions of vacancies and self-interstitials to self-diffusion in silicon under thermal equilibrium and nonequilibrium conditions”

Comment on “Contributions of vacancies and self-interstitials to self-diffusion in silicon under thermal equilibrium and nonequilibrium conditions”