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In thermodynamics, it is essential to distinguish between state functions and process functions. The reason is that the simple compressible thermodynamic system is a bivariate-process system, and the change of internal energy, a state function, corresponds to two process functions, heat and work. Among the state functions in thermodynamics, entropy is a special one because it has to be defined through a process function, exchanged heat δ Q , and a unique factor of integration, 1/T. In heat transfer, it is shown that Fourier's law and the differential equation of heat conduction are both relations of state quantities alone, and process quantities appear when an integration with respect to time is applied. Moreover, an incompressible heat conduction medium element without conversion between heat and work is a univariate-process system governed by a single variable, temperature. In this case, the change of the thermal energy (“heat content”) stored in the system, a state quantity as a function of T alone, corresponds to only one process quantity, the transferred heat. Therefore, on the one hand, it is unnecessary to strictly distinguish between state quantities and process quantities in heat transfer, and on the other hand, there is no need to use a factor of integration to prove entransy a state quantity in heat transfer. Thermodynamics and heat transfer are two parallel sub-disciplines in thermal science. It is incorrect to deny entransy as a state quantity in heat transfer by the uniqueness of the factor of integration for entropy in thermodynamics, and entransy has significant physical meaning in the analysis and optimization of heat transfer processes.

A newly developed frosting simulation method, pseudo-VOF (p-VOF) method, has applied to simulate the dynamic frost formation on the NACA0012 airfoil under strong convection. The p-VOF method is a pseudo volume of fraction simulation method of the multiphase flow with phase change. By solving a simplified mass conservation equation explicitly instead of the original volume fraction equations in CFD software, the efficiency and robustness of calculation are greatly improved. This progress makes it possible to predict a long-time frost formation. The p-VOF method was successfully applied to the simulation of dynamic frosting on the two-dimensional NACA0012 airfoil under strong convection conditions with constant frost physical properties. The simulation result shows that the average thickness of the frost layer increases, and the frost bulges and flow separation appear earlier, when the airfoil surface temperature decreases or the air humidity increases. The frost bulges and flow separation appear earlier, when the air velocity is faster, the growth rate of the frost layer at the early stage is greater, but the final frost layer is thinner.

Because of its high specific surface area, metal foam is expected to further enhance the flow and thermal performance of a minichannel heatsink. In this study, fully developed flow in minichannels with porous fins was studied. The porous geometry was constructed based on Laguerre-Voronoi tessellation and simulated in a pore scale, which eliminated the influence of parameter selection in a macroscale method. The influence of porosity, filling ratio, and Reynolds number on velocity field and pressure drop was analyzed. The general relationship between the Poiseuille number and filling ratio was determined, and a correlation was obtained using the study results. The results of this study were compared with those of the macroscale method, and finally, the influence of effective viscosity was discussed.

Recently, the analogy between the processes of heating a solid body and charging an electric capacitor was deemed as “heat-work analogy” and argued that it was false and violated the second law of thermodynamics. It was claimed that the two processes were not analogous because the process of heating a solid body was classified as a purely thermal system, while that of charging an electric capacitor was classified as a purely mechanical system. In response, this note discusses the analogy between heat and electric conductions and that between heating a solid body and charging an electric capacitor. It is shown that the analogies have nothing to do with heat-work conversion, and never violate the second law of thermodynamics. The classification of charging an electric capacitor as a purely mechanical system is inappropriate. It is also shown that what we concern in the analogy analysis are the commonalities between the processes, rather than the differences.

The entropy generation minimization and the entransy theory are widely used in the optimization of heat transfer. In this paper, the two theories are applied to coupled steady heat transfer systems. The two different definitions of radiative entransy flux are discussed. It is found that the extremum principle of entransy dissipation for coupled heat transfer systems can be obtained only when temperature is treated as the driving force of radiative heat transfer and the definition of entransy flux for radiative heat transfer is the same as that in conductive and convective heat transfer. Taking temperature as the driving force of radiative heat transfer, we have analyzed three coupled heat transfer examples, which are the one-dimensional coupled conductive and radiative heat transfer, the coupled convective and radiative heat transfer and the coupled conductive, convective and radiative heat transfer. The results show that the extremum principle of entransy dissipation always leads to the best system performance, while the entropy generation minimization does not always. The definition of radiative entransy flux in which the blackbody emissive power is treated as the driving force is also used for the three examples. However, the results show that neither the extremum radiative entransy dissipation rate nor the extremum conductive/convective entransy dissipation rate results in the best system performance. Therefore, this definition is not suitable for coupled heat transfer systems.

Let ( N , A 1 , A 2 , … ) be a sequence of random variables with N ∈ N ∪ { ∞ } and A i ∈ R + . We are interested in asymptotic properties of non-negative solutions of the distributional equation Z = (d) ∑ i = 1 N A i Z i , where Z i are non-negative random variables independent of each other and independent of ( N , A 1 , A 2 , … ) , each having the same distribution as Z which is unknown. For a solution Z with finite mean, we prove that for a given α > 1 , P ( Z > x ) is a function regularly varying at ∞ of index − α if and only if the same is true for P ( Y 1 > x ) , where Y 1 = ∑ i = 1 N A i . The result completes the sufficient condition obtained by Iksanov & Polotskiy (2006) on the branching random walk. A similar result on sufficient condition is also established for the case where α = 1 .

Open-cell metal foam mini-channel evaporator (OMFME) is proposed to enhance heat transfer in electronics cooling. In present study, flow boiling in an OMFME is investigated using R134a. The OMFME has nine channels, which are 507 μm in width and 1097 μm in depth. Comparing with a mini-channel evaporator (ME) with the same footprint area, the heat transfer coefficient (HTC) is enhanced up to 1.5 times and the dry out is delayed. Flow patterns are presented and compared with an existed flow patterns map. HTCs generally increases with the quality, the heat flux and it is slightly influenced by the mass flow rate. Nucleate boiling and evaporation in the porous fins are the two heat transfer mechanisms in different quality range. A correlation is presented to calculate frictional pressure drop.

The transient heat transfer performance of spray cooling with R21 is studied experimentally. Three variation patterns of surface temperature depending on the initial surface temperature are observed. There is only one rapid temperature drop at lower initial surface temperature and two drops with a shoulder transition at higher initial surface temperature. Further increasing the initial surface temperature will deteriorate heat transfer. Based on the pool boiling heat transfer curve and one-dimensional transient heat conduction model, the temperature variation patterns and processes are discussed which relate to nucleate boiling, transition boiling and film boiling, respectively. The experimental results reveal that the deterioration temperature decreases with increase of the heating flux and nozzle-to-heater distance, while increases with the spray flow rate. Based on the experimental results a region map of temperature variation patterns in transient spray cooling is given.

Experiments are carried out on the frost and defrost characteristics of fine tube bundles under coolant temperature between −20 and −58 °C. The diameter of the tube is 2 mm. The frost and defrost processes are observed by taking photos and measuring pressure drops. The results indicate that the frost layer thickness on the tube surface increases with decreasing coolant temperature, but decreases with increasing air humidity. A model is given to explain the variation of frost thickness. The experiments verified that spraying ethanol is effective on inhibiting the formation of the frost.

We prove necessary and sufficient conditions for the L p -convergence, p > 1 , of the Biggins martingale with complex parameter in the supercritical branching random walk. The results and their proofs are much more involved (especially in the case p ∈ ( 1 , 2 ) ) than those for the Biggins martingale with real parameter. Our conditions are ultimate in the case p ≥ 2 only.

Droplet impact onto thin film is a ubiquitous phenomenon in many industrial applications. In the present study, the VOF method is adopted to simulate the crown evolution during the impact process with different surface tension coefficient, viscosity and gravity. Especially, the crown behavior is investigated in the energy aspect. The kinetic energy, gravitational potential energy, surface energy and viscous dissipation are taken into consideration. The results reveal that higher surface tension coefficient leads to larger crown diameter. The evolution of gravitational potential energy can be divided into three phases: decrease phase, increase phase and collapse phase. Similarly, the evolution of surface energy can be divided into four phases: increase phase, drop phase, flat phase and collapse phase. The drop phase is caused by the film rupture in the center zone. In the case with zero gravity the flat phase is replaced by a second increase phase. In particular, it is found that larger viscosity will postpone the film rupture and the film even doesn't break at high viscosity.

Entransy method is developed for the analyses of thermal processes with variable thermophysical properties. The entransy and enthalpy entransy balance equations for thermal processes with variable thermophysical properties are derived based on the definitions of differential entransy and enthalpy entransy. The definitions of the entransy and enthalpy entransy for an object are presented. The optimization principles for heat transfer and heat-work conversion processes with variable thermophysical properties are discussed with the concept of entransy. Entransy analyses for a two-stream heat exchanger and a closed Brayton cycle with variable thermophysical properties are conducted and compared with the entropy generation analyses. For the heat exchanger, the results show that both the minimum entransy-dissipation-based thermal resistance and the minimum revised entropy generation number correspond to the maximum heat exchanger effectiveness when the inlet condition of the heat exchanger is prescribed, but the minimum entropy generation rate and the minimum entropy generation number do not. For the closed Brayton cycle, the results show that both the maximum entransy loss rate and the minimum entropy generation rate correspond to the maximum output power and both the maximum entransy loss coefficient and the minimum entropy generation number correspond to the maximum heat-work conversion efficiency when the inlet conditions of the streams and the environmental temperature are prescribed. When the inlet temperature of the hot stream increases, larger entransy loss rate and larger entransy loss coefficient still correspond to larger output power and larger heat-work conversion efficiency respectively.

Entransy balance equations and the corresponding optimization principles for heat conduction, heat convection and thermal radiation have been established. This paper develops the entransy balance equation for a low speed steady flow system with vaporization processes. The definition of enthalpy entransy for a substance from solid state to vapor state is proposed. The expressions of entransy and entransy dissipation are derived with differential volume element integration and the concept of enthalpy entransy. The heat transfer between a vaporization stream and a hot stream in a heat exchanger is also analyzed with the entransy approach. The theoretical results show that the minimum entransy-dissipation-based thermal resistance always leads to the maximum heat exchanger effectiveness. A numerical example compared with the entropy generation approach verifies the theoretical analyses. For a heat exchanger network undergoing vaporization processes, the minimum entransy-dissipation-based thermal resistance is proved to be applicable as the optimization criteria for the maximum heat exchange rate. The minimum entransy-dissipation-based thermal resistance principle is applied to two numerical cases for the thermal conductance distribution optimization to obtain the maximum heat exchange rate.

The analyses of thermal systems are necessary because of the energy situation. In this paper, two kinds of thermal systems with work, a common heat–work conversion system and a common heat pump system, are analyzed with the entransy theory. The entransy terms and variations for thermal system with heat–work interaction are discussed. The concept of entransy variation associated with work is proposed. The results show that the entransy variation associated with work is directly related to output work and has the same change tendency as that of the work.

The heat transfer during spray cooling was studied experimentally using deionized water to investigate the spray characteristics and the differences between spray cooling on a smooth silicon surface and micro-, nano- and hybrid micro/nano-structured surfaces. The spray cooling experiments show that the heat transfer rates were better for the nano-structured surface, followed by the smooth surface coated with the SiO2 film and the pure silicon surface since the contact angle was smallest on the nano-structured surface and increased on the latter two. The droplet parameter results show that most droplets were 40–60 μm in size. The heat transfer coefficient increased and the wall temperature decreased on the 25G × 25S surface coated with the SiO2 film compared with the 50G × 50S surface coated with the SiO2 film as the heat transfer moved into the partial dryout region due to the SiO2 film’s stronger hydrophilicity so the heated area was more fully utilized, while the CHF was larger for the 50G × 50S surface. Coating the micro-structured surfaces with carbon nano-tube (CNT) films having characteristic sizes smaller than the droplet size was more effective than on the surfaces with larger characteristic sizes. The CHF was largest on the 25G × 25S surface coated with 4 carbon nano-tube films with a 75.3% increase over the smooth surface. The wall temperature increase and the temperature fluctuations were small in the boiling regime as the power increases for the enhanced surfaces.

This paper reports the comparison on the applicability between the concepts of entropy generation in terms of thermal conductance and generalized thermal resistance to analyzing heat exchangers. Six typical two-stream heat exchangers and one three-stream heat exchanger are analyzed. When the heat capacity flow rates of the discussed heat exchangers are fixed, the minimum generalized thermal resistance always corresponds to the maximum heat exchanger effectiveness, while the minimum entropy generation in terms of thermal conductance does not always. When the heat capacity flow rates of the discussed heat exchangers are not fixed, smaller generalized thermal resistance always leads to larger heat transfer rate, while smaller entropy generation in terms of thermal conductance does not always. Therefore, the concept of generalized thermal resistance is more appropriate than the concept of entropy generation in terms of thermal conductance for describing the performance of the discussed heat exchangers.

The performance analysis of heat transfer and heat–work conversion is very important in energy utilization. T – q diagram for heat transfer is discussed and extended to heat–work conversion processes. The results show that T – q diagram is convenient for expressing entransy dissipation and analyzing heat transfer performance and can express work entransy and entransy loss intuitively. It could have an application in analyzing the performance of heat–work conversion processes.

This paper presents the optimization of an air conditioning system, in which the heat from the hot stream is transported to do work first, and then the work is used to pump heat from the environment into the room. It is shown that the system can deliver more heat into the room than the direct heating by the hot stream. A functional is set up to optimize the distribution of the total thermal conductance for the system, and the numerical results show that the functional is effective. The system is also discussed with the concepts of entropy and entransy. It is found that the minimum values of the entropy generation rate, entropy generation number, revised entropy generation number, entransy loss and entransy loss coefficient all correspond to the maximum heat flow rate into the room with prescribed heat flow rate from the hot stream, but the values of the parameters all increase with increasing heat flow rate from the hot stream and that into the room.

In this study, degradation of ethylene diamine tetraacetic acid (EDTA) solution by the UV/Cu(II)/H2O2 process under various conditions was evaluated via chemical oxygen demand (COD) removal. The reactions were performed in a lab-scale batch photoreactor. The effects of different parameters such as H2O2 dosage, copper catalyst dosage, initial pH, and initial EDTA concentration on the oxidation efficiency of the process were investigated. The results indicated that the UV/Cu(II)/H2O2 process could effectively remove COD in both synthesized EDTA wastewater and real copper-plating wastewater. According to the experimental results, a reasonable mechanism was proposed to interpret the oxidation process. When treating real copper-plating wastewater (COD = 3,660 mg/L) catalyzed by self-contained 1.4 g/L copper, 98.6% of COD removal was obtained using 0.6 M H2O2 in 180 min. In addition to the removal of organic matter, copper complexes were decomposed to form easily removed free copper ions. This study sho...

In this paper, heat pump systems are analyzed with entransy increase and entropy generation. The extremum entransy increase principle is developed. When the equivalent temperatures of the high and low temperature heat sources are fixed, the theoretical analyses and numerical results both show that the maximum COP leads to the maximum entransy increase rate for fixed input power, while it leads to the minimum entransy increase rate for fixed heat flow absorbed from the low temperature heat source. The minimum entropy generation principle shows that the minimum entropy generation rate always leads to the maximum COP for fixed input power or fixed heat flow absorbed from the low temperature heat source when the equivalent thermodynamic forces of the high and low temperature heat sources are given. Further discussions show that only the entransy increase rate always increases with increasing heat flow rate into the high temperature heat source for the discussed cases.

Let (Z n ) be a supercritical branching process in an independent and identically distributed random environment ζ = (ζ 0, ζ 1,…), and let W be the limit of the normalized population size Z n / $\mathbb{E}$ (Z n |ζ). We show a necessary and sufficient condition for the existence of weighted moments of W of the form $\mathbb{E}$ , where α ≥ 1 and l is a positive function slowly varying at ∞.

Heat entransy loss is defined and both the concepts of entropy generation and heat entransy loss are applied to the analyses of heat-work conversion and heat transfer processes in this paper. In heat-work conversion, it is found that the minimum entropy generation rate relates to the maximum output power under the conditions of the prescribed heat absorption and the equivalent thermodynamic forces corresponding to the heat absorption and release of the system, while the maximum heat entransy loss relates to the maximum output power under the conditions of the prescribed heat absorption and the equivalent temperatures corresponding to the heat absorption and release of the system. In heat transfer, the maximum entropy generation rate is consistent with the maximum heat transfer rate with prescribed equivalent thermodynamic force difference, while the minimum entropy generation rate corresponds to the minimum equivalent thermodynamic force difference with prescribed heat transfer rate. Furthermore, when the concept of heat entransy loss is used, the maximum heat entransy loss rate corresponds to the maximum heat transfer rate with prescribed equivalent temperature difference, while the minimum heat entransy loss rate corresponds to the minimum equivalent temperature difference with prescribed heat transfer rate. The numerical results of some examples verify the theoretical analyses.

Heat transfer optimization methods to effectively improve heat transfer performance is of great importance for energy conservation and pollution reduction. A recently developed heat transfer optimization method based on entransy theory and related peer-reviewed papers published between 2003 and 2010 are reviewed and updated in this paper to describe entransy, entransy dissipation, optimization criteria and optimization principles and their applications to different heat transfer modes (thermal conduction, convection and radiation) and to different levels (heat transfer element, heat exchanger, and heat exchanger network). Entransy theory is then compared with entropy theory in several aspects, including the heat transfer purpose, irreversibility and optimization principle for energy savings or weight reductions of thermal facilities. Finally, entransy theory is also compared with constructal theory in terms of optimization objective, optimization method and optimized results.

The concepts of entransy and thermomass have been developed for heat transfer analyses and optimizations. Thermomass is the equivalent mass of heat based on the Einstein’s mass–energy relation. The concept of entransy reflects the energy of thermomass. The entransy balance equations of heat conduction and heat convection, which are the basis of the heat transfer optimization principles, are derived from the energy equation of thermomass in this paper. The relationship between the entransy and the thermomass is discussed.

Entransy is a concept developed in recent years to measure the heat transfer ability of an object or a system at a temperature and has been used in heat transfer optimization. In this contribution, the closed system undergoing thermodynamic processes is analyzed from the viewpoint of entransy. Heat entransy flow and work entransy flow are defined with the consideration of heat interaction and work interaction respectively. The entransy balance equation is developed for the thermodynamic cycle and the processes of the Carnot cycle are discussed. In the cycle, the net heat entransy flow is showed to be equal to the net work entransy flow. The difference between heat entransy flow and work entransy flow is discussed. Work entransy flow can be input from low temperature to high temperature and vice versa. Heat entransy flow can only be transferred from high temperature to low temperature spontaneously, leading to entransy dissipation. The entransy balance equation for the closed system with irreversible processes and heat engines is further introduced by defining the irreversible entransy loss. The entransy balance equations are applied to analyzing the heat-work conversion in a system with given heat input. It is shown that reducing the irreversible entransy loss or the entropy generation can improve the heat-work conversion efficiency. For the reversible heating and air conditioning example in which heat from the high temperature heat source is first used to do work and then the work is used to pump more heat from environment into the target room, the change in work entransy flow can be used to measure the cost for the additional heat pumped from environment, but entropy generation is zero and could not be related to the benefit.

At present, there have been few direct molecular dynamics simulations on the thermal conductivity of polycrystalline nanofilms. In this paper, we generate polycrystalline argon nanofilms with random grain shape using the three-dimensional Voronoi tessellation method. We calculate the out-of-plane thermal conductivity of a polycrystalline argon nanofilm at different temperatures and film thicknesses by the Muller-Plathe method. The results indicate that the polycrystalline thermal conductivity is lower than that of the bulk single crystal and the single-crystal nanofilm of argon. This can be attributed to the phonon mean-free-path limit imposed by the average grain size as well as the grain boundary thermal resistance due to the existence many grain boundaries in polycrystalline materials. Also, the out-of-plane thermal conductivity of the polycrystalline argon nanofilm is insensitive to temperature and film thickness, and is mainly dominated by the grain size, which is quite different from the case of single-crystal nanofilms.

Let ( N , A 1 , A 2 , … ) be a sequence of random variables with N ∈ N ∪ { ∞ } and A i ∈ R + . We are interested in asymptotic properties of solutions of the distributional equation Z = ∑ i = 1 N A i Z i , where Z i are nonnegative random variables independent of each other and independent of ( N , A 1 , A 2 , … ) , each has the same distribution as Z which is unknown. For a solution Z ⩾ 0 with finite mean, we show that under a natural moment condition, the regular variation of P ( Z > x ) ( x → ∞ ) is equivalent to that of P ( Y 1 > x ) , where Y 1 = ∑ i = 1 N A i . The results generalize the corresponding theorems of Bingham and Doney (1974, 1975) [1] , [2] and de Meyer (1982) [6] on Galton–Watson processes and Crump–Mode–Jirina processes, and improve those of Iksanov and Polotskiy (2006) [7] on branching random walks.

Entransy is a new concept developed in recent years to measure the transport ability of heat at a temperature in conduction and convection. This paper develops the concept of entransy flux for thermal radiation in enclosures with opaque surfaces. The entransy balance equation and entransy dissipation function are derived. The minimum principle of radiative entransy loss is developed. The potentials and the heat fluxes distribution which meet the Stefan–Boltzmann’s law and the energy balance equation would make the radiative entransy loss minimum if the net heat flux of each surface or the thermal potentials of the surfaces are given. The extremum entransy dissipation principles (EEDP) for thermal radiation are developed. The minimum radiative entransy dissipation leads to the minimum average radiative thermal potential difference for prescribed total heat exchange and the maximum radiative entransy dissipation leads to the maximum heat exchange for prescribed average radiative thermal potential difference. The minimum and maximum principle can be concluded into the minimum thermal resistance principle (MTRP) for thermal radiation by defining the thermal resistance with the entransy dissipation. The EEDP or MTRP is proved to be reliable when they are used to optimize some radiative heat transfer problems, and a comparison is made between the minimum principle of entropy generation and the EEDP.

The experiments are carried out to study the transient heat transfer characteristics of a minichannel heat sink under high heat flux density that varies from 150 W/cm 2 to 200 W/cm 2 , in an enclosed loop. The heat flux is supplied by the ethyne flame. The average rate of the cooling water is from 1.00 m/s to 1.86 m/s. The temperature rise of the heated surface and the temperature rise of the cooling water are measured. The results show that the temperature rise of the heated top surface rises quickly at the startup period of heating, and then increases slowly. The variation of the top surface temperature is similar to the voltage in the RC circuit. The surface temperature is a function of thermal resistance and thermal capacity of the heat sink. The temperature of the cooling water at outlet also rises quickly at the startup period of heating, and then becomes stable. The temperature increases are related to the heat flux density and the rate of the cooling water. There is a time delay between the temperature rise of the inlet cooling water and the startup of heating. This delay is related to the length of the enclosed loop and the rate of the cooling water. The pressure drop across the heat sink decreases slightly with time at high heat flux since the water viscosity reduces with the increasing temperature.

A new physical quantity, E h = 1 2 Q vh T , has been identified as a basis for optimizing heat transfer processes in terms of the analogy between heat and electrical conduction. This quantity, which will be referred to as entransy, corresponds to the electric energy stored in a capacitor. Heat transfer analyses show that the entransy of an object describes its heat transfer ability, as the electrical energy in a capacitor describes its charge transfer ability. Entransy dissipation occurs during heat transfer processes as a measure of the heat transfer irreversibility. The concepts of entransy and entransy dissipation were used to develop the extremum principle of entransy dissipation for heat transfer optimization. For a fixed boundary heat flux, the conduction process is optimized when the entransy dissipation is minimized, while for a fixed boundary temperature the conduction is optimized when the entransy dissipation is maximized. An equivalent thermal resistance for multi-dimensional conduction problems is defined based on the entransy dissipation, so that the extremum principle of entransy dissipation can be related to the minimum thermal resistance principle to optimize conduction. For examples, the optimum thermal conductivity distribution was obtained based on the extremum principle of entransy dissipation for the volume-to-point conduction problem. The domain temperature is substantially reduced relative to the uniform conductivity case. Finally, a brief introduction on the application of the extremum principle of entransy dissipation to heat convection is also provided.

The Volume-to-Point (VP) problem is a base problem of heat conduction optimization. The nonlinear two-dimensional optimization problem of VP is discretized and transformed to a combinatorial optimization problem, which can be solved by some modern optimization algorithms. Algorithms for VP problem using simulated annealing and genetic algorithm are developed. Results for different cases are obtained using these algorithms. Analyses of the results and algorithms are also presented, that shows these algorithms are better than bionic optimization algorithm and constructal theory for VP problem, and can be generalized to complex conditions.

Let $(Y_n)_{n\ge0}$ be a Mandelbrot's martingale defined as sums of products of random weights indexed by nodes of a Galton-Watson tree, and let $Y$ be its limit. We show a necessary and sufficient condition for the existence of weighted moments of $Y$ of the forms $\mathbb{E}Y^{\alpha}\ell(Y)$, where $\alpha>1$ and $\ell$ is a positive function slowly varying at $\infty$. We also show a sufficient condition in the case of $\alpha=1$. Our results complete those of Alsmeyer and Kuhlbusch (2010) for weighted branching processes by removing their extra conditions on $\ell$.

The field synergy equation for steady laminar convection heat transfer was derived by conditional variation calculus based on the least dissipation of heat transport potential capacity. The optimum velocity field with the best heat transfer performance and least flow resistance increase can be obtained by solving the synergy equation. The numerical simulation of laminar convection heat transfer in a straight circular tube shows that the multi-longitudinal vortex flow in the tube is the flow pattern that enhances the heat transfer enormously. Based on this result, a novel enhanced heat transfer tube, the discrete double-inclined ribs tube (DDIR-tube), is developed. The flow field of the DDIR-tube is similar to the optimal velocity field. The experimental results show that the DDIR-tube has better comprehensive heat transfer performance than the current heat transfer enhancement tubes. The present work indicates that new heat transfer enhancement techniques could be developed according to the optimum velocity field.

Experiments are carried out on the convection inside a novel alternating elliptical axis tube. The unified correlations of heat transfer and flow resistance for 500

This work studied the thermal conductance of thin layers with randomly oriented composites by the percolation theory, and developed an effective-medium approximation (EMA) model for triple bond percolation systems. The results showed that the thin layer is anisotropic in conductivities when its thickness is lower than the correlation length. The conductivity in normal direction increases with decreasing thickness while the in-plane conductivity declines. The significance of this thickness effect is a function of the concentration of good conductor and the thermal conductivity ratio of the phases in the composites. A threshold concentration exists for the conductivity of bulk composites, beyond which the effective thermal conductivity increases significantly from that of the poor conductor with increasing concentration. The developed EMA model agrees quite well with the numerical simulation and is expected being applicable to predict thermal conductivities of composites using the coordination number as a fitting parameter

This study uses the molecular dynamics computational technique to investigate the thermal conductivity of solid thin films in the direction perpendicular to the film plane. In order to establish a benchmark reference, the computations are based on the widely used Lennard-Jones argon model due to its agreement with experimental liquid-phase data, its physically meaningful parameters, and its simple two-body form. Thermal conductivity increases with film thickness, as expected from thin-film experimental data and theoretical predictions. The calculated values are roughly 30 percent higher than anticipated. Varying the boundary conditions, heat flux, and lateral dimensions of the films causes no observable change in the thermal conductivity values. The present study also delineates the conditions necessary for meaningful thermal conductivity calculations and offers recommendations for efficient simulations. This work shows that molecular dynamics, applied under the correct conditions, is a viable tool for calculating the thermal conductivity of solid thin films. More generally, it demonstrates the potential of molecular dynamics for ascertaining microscale thermophysical properties in complex structures. [S0022-1481(00)02303-3]

At high temperature the effective thermal conductivity of a bulk material with a large temperature difference may differ from the local equivalent thermal conductivity of a unit cell at the mean temperature due to radiation. This paper analyzed the local equivalent thermal conductivity and the effective thermal conductivity for porous materials with cylindrical and spherical air cavities. Prediction for a spherical case agrees well with experimental data. Investigation shows that the equivalent thermal conductivity at mean temperature can well represent the effective thermal conductivity for a cavity diameter of

Let ( Z n ) be a supercritical Galton–Watson process, and let W be the limit of the normalized population size Z n / m n , where m = E Z 1 > 1 is the mean of the offspring distribution. Let l be a positive function slowly varying at ∞. Bingham and Doney (1974) [4] showed that for α > 1 not an integer, E W α l ( W ) ∞ if and only if E Z 1 α l ( Z 1 ) ∞ ; Alsmeyer and Rosler (2004) [2] proved the equivalence for α > 1 not a dyadic power. Here we prove it for all α > 1 .

A systematic study has been carried out numerically and compared with some experimental data to examine the heating effect on the corner recirculation zone (CRZ) in sudden-expansion gas flows. The heat addition to such flows will lead to the reduction of the CRZ length, and the CRZ can even disappear if the heating intensity is sufficiently large. The concept of thermal drag (heating induced pressure drop in duct flows) has been used to clarify the underlying mechanism of the heating effect on CRZ. The heating induced reduction of the adverse pressure gradient in sudden-expansion flows should be largely responsible for the shrinkage of the CRZ due to heating. Computational results also show that heating the CRZ and the upstream part of expansion flow are more efficient for changes of the CRZ.

Using the atomistic Green's functions method in combination with the Landauer formula, we show that the phonon transmission and thermal conductance of periodic rectangular-shaped and triangular-shaped Si/Ge interfaces are both enhanced and tunable by roughness. For triangular-shaped interface, there is maximum phonon transmission and conductance with increasing roughness height, and the conductance can be tuned maximally by 22.3% compared with the flat interface. The maximum conductance of rectangular-shaped interface is enhanced by about 11.1%. The competing mechanisms between the broadening frequency transport window of rough interface and the increasing diffusing phonon scattering at the interface with higher roughness introduce the maximum transmission and conductance. Similar result is also obtained in non-periodic interfaces. The presented results provide insights into the thermal design of interfaces in nanoscale devices.

It has been estimated that, of all the worldwide energy utilization, more than 80% involves the heat transfer process, and the thermal engineering has for a long time recognized the huge potential for conserving energy and decreasing CO2 release so as to reduce the global warming effect through heat transfer efficiency techniques (Bergles, 1988, 1997; Webb, 1994; Zimparov, 2002). In addition, since the birth of electronic technology, electricity-generated heat in electronic devices has frequently posed as a serious problem (Arden, 2002; Chein & Huang, 2004), and effective cooling techniques are hence needed for reliable electronic device operation and an increased device lifespan. In general, approaches for heat transfer enhancement have been explored and employed over the full scope of energy generation, conversion, consumption and conservation. Design considerations to optimize heat transfer have often been taken as the key for better energy utilization and have been evolving into a well-developed knowledge branch in both physics and engineering. During the last several decades and promoted by the worldwide energy shortage, a large number of heat transfer enhancement technologies have been developed, and they have successfully cut down not only the energy consumption, but also the cost of equipment itself. However, comparing with other scientific issues, engineering heat transfer is still considered to be an experimental problem and most approaches developed are empirical or semi-empirical with no adequate theoretical base (Gu et al., 1990). For instance, for a given set of constraints, it is nearly impossible to design a heat-exchanger rig with the optimal heat transfer performance so as to minimize the energy consumption. Therefore, scientists developed several different theories and methods to optimize heat transfer, such as the constructal theory (Bejan, 1997) and the minimum entropy generation (Bejan, 1982). Then heat transfer processes were optimized with the objective of minimum entropy generation. Based on this method, several researchers (Nag & Mukherjee, 1987; Sahin, 1996; Sekulic et al., 1997; Demirel, 2000; Sara et al., 2001; Ko, 2006) analyzed the influences of geometrical, thermal and flow boundary conditions on the entropy generation in various convective heat transfer processes, and then optimized them based on the premise that the minimum entropy generation will lead to the most efficient heat transfer performance. However, there are some scholars (Hesselgreaves, 2000; Shah & Skiepko, 2004; Bertola & Cafaro, 2008) who questioned whether the entropy generation is the universal irreversibility measurement for heat transfer, or the minimum entropy generation is the general optimization criterion for all heat transfer processes, regardless of the nature of the

Based on the relationship between entransy and microstate number, we discuss the variations of the available transport entransy, the unavailable transport entransy, the available conversion entransy and the unavailable conversion entransy with the microstate number. We focus on physical processes in which heat is used for heating/cooling or doing work. When heat is transported for heating or cooling, the available transport entransy increases if the increase in microstate number is due to the increase in internal energy of the system, and decreases if the increase in microstate number is due to spontaneous heat transfer. When heat is used to do work, both the available conversion entransy and the unavailable conversion entransy increase if the increase in microstate number relates to the growth in internal energy of the system. The available conversion entransy decreases and the unavailable conversion entransy increases if the increase in microstate number results from spontaneous heat transfer.

The entropy increase principle for an isolated system and the criteria of thermal equilibrium for an isolated system and systems with prescribed temperature and volume can be derived on the basis of the concept of entropy and the first and second laws of thermodynamics. In this paper, the entransy decrease principle for an isolated system is introduced on the basis of the concept of entransy. It is found that the entransy of an isolated system always decreases during heat transfer. This principle can be taken as an expression of the second law of thermodynamics for heat transfer. The thermal equilibrium criteria for an isolated system and a closed system are also introduced. It is found that when an isolated system reaches thermal equilibrium, its entransy is a minimum value. This criterion is referred to as the minimum entransy principle. When a closed system reaches thermal equilibrium, its free entransy is also a minimum value. This criterion is referred to as the minimum free entransy principle. Therefore, like entropy, entransy can be considered an arrow of time in heat transfer and used to describe the thermal equilibrium state.

The concept of entransy developed in recent years can describe the heat transport ability. This paper extends this concept to the open thermodynamic system and defines the concept of enthalpy entransy. The entransy balance equation of steady open thermodynamic systems, as well as the concept of entransy loss, is developed. The entransy balance equation is applied to analyzing and discussing the air standard cycle. It is found that the entransy loss rate can describe the change in net power output from the cycle but the entropy generation rate cannot when the heat absorbed by the working medium is from the combustion reaction of the gas fuel. When the working medium is heated by a high temperature stream, both the maximum entransy loss rate and the minimum entropy generation rate correspond to the maximum net power output from the cycle. Hence, the concept of entransy loss is an appropriate figure of merit that describes the cycle performance.

The relationship between entransy dissipation and the irreversibility of some thermodynamic processes, such as heat transfer, work-heat conversion, free expansion, isothermal diffusion etc., are analyzed in this paper. The results show that there is entropy generation but no entransy dissipation in irreversible work-heat conversion, free expansion and isothermal diffusion. Therefore, entransy dissipation cannot be used to describe the irreversibility of these processes. Both entropy generation and entransy dissipation exist in heat transfer process, which indicates that the entransy dissipation can be used to describe the irreversibility of heat transfer processes. Furthermore, the irreversibility of endoreversible cycles is analyzed. As all the irreversibility in endoreversible cycles is attributed to heat transfer between the heat sources and the working medium, entransy dissipation can be used to describe the irreversibility of this kind of cycles. To verify this conclusion, numerical examples of the endoreversible Carnot cycle are discussed.