Kinetic theory analysis of explosive boiling of a liquid droplet

The process of rapid phase transition from highly superheated liquid to vapor is frequently so fast and violent that it is called explosive boiling. The paper uses the kinetic theory of evaporation to study growth of an internal vapor bubble produced by homogeneous nucleation within a highly superheated liquid droplet boiling explosively in a hot medium. Evaporation/condensation coefficient is estimated by comparing the predictions of the theory with available experimental data. We show that the value of the evaporation coefficient can be very low for high reduced temperatures (0.06 for butane at 378 K), in agreement with recent molecular dynamic simulations.     

Linear stability analysis of convection in two-layer system with an evaporating vapor-liquid interface

Classical theories have successfully provided an explanation for convection in a liquid layer heated from below without evaporation. However, these theories are inadequate to account for the convective instabilities in an evaporating liquid layer, especially in the case when it is cooled from below. In the present paper, we study the onset of Marangoni convection in a liquid layer being overlain by a vapor layer. A new two-sided model is put forward instead of the one-sided model in previous studies. Marangoni-Bénard instabilities in evaporating liquid thin layers are investigated with a linear instability analysis. We define a new evaporation Biot number, which is different from that in previous studies and discuss the influences of reference evaporating velocity and evaporation Biot number on the vapor-liquid system. At the end, we explain why the instability occurs even when an evaporating liquid layer is cooled from below. The project supported by the National Natural Science Foundation of China (10372105) and the Knowledge Innovation Program of Chinese Academy of Sciences (KJCX2-SW-L05 and KGCX-SW-409) The English text was polished by Keren Wang.     

Evaporation of thin liquid droplets on heated surfaces

We carry out combined experimental and theoretical studies of liquid droplet evaporation on heated surfaces in a closed container filled with saturated vapor. The droplets are deposited on an electrically heated thin stainless steel foil. The evolution of droplet shapes is studied by optical methods simultaneously with high-resolution foil temperature measurements using thermochromic liquid crystals. A mathematical model is developed based on the assumptions that the droplet surface has uniform mean curvature and the contact line is pinned during evaporation. Both the dynamics of liquid–vapor interface and the temperature profiles at the foil are shown to be in good agreement with the experimental data.     

Grouping and trapping of evaporating droplets in an oscillating gas flow

A new approach to the analysis of droplet grouping in an oscillating gas flow is suggested. This is based on the investigation of droplet trajectories in the frame of reference moving with the phase velocity of the wave. Although the equations involved are relatively simple, the analysis shows distinctive characteristics of grouping and non-grouping cases. In the case of grouping, droplet trajectories converge to the points for which the ratio of flow velocity in this frame of reference and the amplitude of flow oscillations is less than 1, and the cosine of the arc sine of this ratio is positive. In the case of non-grouping, droplet trajectories in this frame of reference oscillate around the translational velocity close to the velocity of flow in the same frame of reference. The effect of droplet size on the grouping pattern is investigated. It has been pointed out that for the smaller droplets much more stable grouping is observed. The effect of droplet evaporation is studied in the limiting case when the contribution of the heat-up period can be ignored. It is shown that evaporation can lead to droplet grouping even in the case when the non-evaporating droplets are not grouped. This is related to the reduction in droplet diameter during the evaporation process. Coupling between gas and droplets is shown to decrease the grouping tendency. A qualitative agreement between predictions of the model and in-house experimental observations referring to Diesel engine sprays has been demonstrated.     

Transient analysis of sub-critical evaporation of fuel droplet in non-isothermal stagnant gaseous mixtures: effects of radiation and thermal expansion

Transient sub-critical droplet evaporation in non-isothermal stagnant gaseous mixtures taking into account the effects of radiation, liquid volumetric expansion and droplet heating is investigated numerically. We obtained equations for Stefan velocity and the rate of change of the droplet radius taking into account liquid volumetric expansion, and derived the boundary conditions taking into account the effect of liquid thermal expansion. It is shown that in the case of sub-critical evaporation neglecting the liquid volumetric expansion causes underestimation of the evaporation rate at the initial stage and overestimation of the evaporation rate at the final stage of droplet evaporation.

Enhancement of evaporation and combustion of liquid fuels through porous media

Existing designs of most conventional liquid fuel burners have relied solely on spray atomizers, with a large amount of very fine droplets forming in a relatively large combustion chamber, resulting in a relatively low combustion intensity. Against this background, a novel down-flow compact porous burner system was developed for burning kerosene without the need of using a spray atomizer. Successive development on this burner research is important in view of the need to create energy by an efficient device based on simple technology. The focus has been on the introduction of the packed bed emitter installed downstream of the porous burner. The evaporation process and combustion phenomena that occurred are described through the coupled interaction of the solid phase (porous burner), the liquid phase (kerosene) and the gas phase. Enhancement of evaporation and combustion are evaluated through the measured thermal structures in terms of temperature distribution along the burner length and emission characteristics at the burner exit. Stable combustion with low emission of pollutants was realized even though the combustion flame was confined in-between the porous burner and the packed bed emitter with an increase in the back-pressure. The effects of various parameters including heat input and equivalence ratio on the combustion characteristics were clarified to confirm improvement in mixing of the fuel vapor/air mixture and turn-down ratio of the burner. The effect of the introduced packed bed emitter with suitable bed length and its installation location is investigated as an efficient method for enhancement of evaporation and combustion of the liquid fuels without a spray atomizer. Future applications of this type of burner system are suggested.     

Infrared visualization of thermal motion inside a sessile drop deposited onto a heated surface

Drop evaporation is a basic phenomenon but the mechanisms of evaporation are still not entirely clear. A common agreement of the scientific community based on experimental and numerical work is that most of the evaporation occurs at the triple line. However, the rate of evaporation is still predicted empirically due to the lack of knowledge of the governing parameters on the heat transfer mechanisms which develop inside the drop under evaporation. The evaporation of a sessile drop on a heated substrate is a complicated problem due to the coupling by conduction with the heating substrate, the convection/conduction inside the drop and the convection/diffusion in the vapor phase. The coupling of heat transfer in the three phases induces complicated cases to solve even for numerical simulations. We present recent experimental results obtained using an infrared camera coupled with a microscopic lens giving a spatial resolution of 10 μm to observe the evaporation of sessile drops in infrared wavelengths. Three different fluids fully characterized, in the infrared wavelengths of the camera, were investigated: ethanol, methanol and FC-72. These liquids were chosen for their property of semi-transparency in infrared, notably in the range of the camera from 3 to 5 μm. Thus, it is possible to observe the thermal motion inside the drop. This visualization method allows us to underline the general existence of three steps during the evaporating process: first a warm-up phase, second (principal period) evaporation with thermal-convective instabilities, and finally evaporation without thermal patterns. The kind of instabilities observed can be different depending on the fluid. Finally, we focus on the evolution of these instabilities and the link with the temperature difference between the heating substrate and the room temperature.     

超空化条件下液体射流热稳定性研究

基于线性稳定性理论,建立了描述超空化条件下液体射流热稳定性的数学模型,并对数学模型及其求解方法进行了验证;在此基础上,对超空化条件下液体射流与周围气体间的温差对射流稳定性的影响进行了研究。研究结果表明,液体射流与周围气体间存在温差时,射流稳定性变差,扰动波波数范围拓宽,且拓宽的程度随温差的增加有明显加大的趋势;温度扰动对射流稳定性的影响与扰动模式关系不大;温度扰动会在一定程度上削弱超空化对射流稳定性的作用,并有可能完全抑制超空化对扰动波最大波数的作用,只有当超空化达到一定程度后,才能克服温度扰动的抑制作用,使扰动波最大波数变大。     

Experimental analysis of the local heat transfer coefficient of falling film evaporation with and without co-current air flow velocity

This paper presents the experimental results of the local heat transfer for falling film evaporation of water sheet by solving the inverse heat conduction problem. It is shown that the local heat transfer coefficients increase by increasing the air flow velocity, the film liquid flow rate or decreasing the inlet bulk film temperature. Correlations for the mean heat transfer coefficients in the absence of superimposed flow for the stagnation region, the thermally developed region and the bottom of the heated cylinder are proposed.     

Experimental investigation of evaporation-induced convection in water using laser based measurement techniques

Recent studies have shown that the evaporation of water can induce surface tension gradients along the water surface that ultimately lead to a surface driven flow, known as Marangoni convection. To visualize and characterize the Marangoni convection in water, this study generated evaporation driven convection in pure water with a vacuum pump to control and increase the evaporation rate of water within a rectangular cuvette that was placed within a vacuum chamber, and investigated the velocity and temperature distributions of the generated convection. The investigation was performed as the vacuum chamber pressure ranged from ∼250 Pa to ∼820 Pa. The temperature field obtained from thermocouple measurements and temperature planar laser induced fluorescence (temp-PLIF) measurements indicated that no buoyancy driven motion was generated during the investigation. Velocity vector fields captured with stereo particle image velocimetry (stereo-PIV) demonstrated a convection pattern that was strong and symmetric with the centerline of the cuvette. The strength of the convection was found to be correlated with the mean evaporation rate of water. The estimated Marangoni number exceeded the critical value typically used to characterize the onset of Marangoni convection. The convection had a similar pattern as Marangoni convection observed in volatile liquids evaporated from capillary tubes. In both cases, the convection scaled with the width of the liquid container even though the sizes of the containers differ by an order of magnitude. In addition, the size of the convection in this study was much larger than the Marangoni convection in water that was observed in previous studies.     

Local Minimizers and the Schmid Law in Corner-Shaped Domains

This paper focuses on the mathematical analysis of biaxial loading experiments in martensite, more particularly on how hysteresis relates to metastability. These experiments were carried out by Chu and James and their mathematical treatment was initiated by Ball, Chu and James. Experimentally it is observed that a homogeneous deformation y ₁ is the stable state for “small” loads while y ₂ is stable for “large” loads. A model was proposed by Ball, Chu and James which, for a certain intermediate range of loads, predicts crucially that y ₁ remains metastable (that is, a local—as opposed to global—minimiser of the energy). This result explains convincingly the hysteresis that is observed experimentally. It is easy to get an upper bound on the load at which metastability finishes. However, it was also noticed that this bound (the Schmid Law) may not be sharp, though this required some geometric conditions on the sample. In this research, we rigorously justify the Ball–Chu–James model by means of De Giorgi’s Γ-convergence, establish some properties of local minimisers of the (limiting) energy and prove the metastability result mentioned above. An important part of the paper is then devoted to establishing which geometric conditions are necessary and sufficient for the counter-example to the Schmid Law to apply, namely, the presence of sharp corners in the sample.     

Traveling Waves in Diffusive Random Media

The current paper is devoted to the study of traveling waves in diffusive random media, including time and/or space recurrent, almost periodic, quasiperiodic, periodic ones as special cases. It first introduces a notion of traveling waves in general random media, which is a natural extension of the classical notion of traveling waves. Roughly speaking, a solution to a diffusive random equation is a traveling wave solution if both its propagating profile and its propagating speed are random variables. Then by adopting such a point of view that traveling wave solutions are limits of certain wave-like solutions, a general existence theory of traveling waves is established. It shows that the existence of a wave-like solution implies the existence of a critical traveling wave solution, which is the traveling wave solution with minimal propagating speed in many cases. When the media is ergodic, some deterministic \hbox{properties} of average propagating profile and average propagating speed of a traveling wave solution are derived. When the media is compact, certain continuity of the propagating profile of a critical traveling wave solution is obtained. Moreover, if the media is almost periodic, then a critical traveling wave solution is almost automorphic and if the media is periodic, then so is a critical traveling wave solution. Applications of the general theory to a bistable media are discussed. The results obtained in the paper generalize many existing ones on traveling waves. AMS Subject Classification: 35K55, 35K57, 35B50     

An experimental investigation of roll waves in high pressure two-phase inclined pipe flows

This paper examines the effects of small upward inclinations on the formation of roll waves and the properties of fully developed roll waves at high pressure conditions. A total of 984 experiments were conducted at six positive pipe inclinations θ = 0.00°, 0.10°, 0.25°, 1.00°, 2.50° and 5.00° using a 25 m long 10 cm i.d. pipe. Sulfur hexafluoride (SF₆) was used at 8 bara giving a gas density of 50 kg/m³. Two independent mechanisms for the formation of roll waves were identified; (1) interaction between 2D shallow water waves and (2) a visible long wavelength instability of the stratified layer. Viscous long wavelength linear stability analysis predicted the critical liquid flow rate and liquid height for the initiation of roll waves when roll waves were formed due to the second mechanism. A simple equation from shallow water wave theory agreed with measurements for critical liquid flow rate when roll waves were formed due to the first mechanism. Shallow water wave speed agreed with critical wave speeds at transition and nonlinear wave speeds for fully developed roll waves in certain cases. The increase in interfacial friction due to the presence of large waves was compared with models from the literature.     

Heat transfer and two-phase flow characteristics of refrigerants flowing under varied heat flux in a double-pipe evaporator

A mathematical model based on the annular flow pattern is developed to simulate the evaporation of refrigerants flowing under varied heat flux in a double tube evaporator. The finite difference form of governing equations of this present model is derived from the conservation of mass, energy and momentum. The experimental set-up is designed and constructed to provide the experimental data for verifying the simulation results. The test section is a 2.5 m long counterflow double tube heat exchanger with a refrigerant flowing in the inner tube and heating water flowing in the annulus. The inner tube is made from smooth horizontal copper tubing of 9.53 mm outer diameter and 7.1 mm inner diameter. The agreement of the model with the experimental data is satisfactory. The present model can be used to investigate the axial distributions of the temperature, heat transfer coefficient and pressure drop of various refrigerants. Moreover, the evaporation rate or the other relevant parameters that is difficult to measure in the experiment are predicted and presented here. The results from the present mathematical model show that the saturation pressure and temperature of refrigerant decrease along the tube due to the tube wall friction and the flow acceleration of refrigerant. The liquid heat transfer coefficient increases with the axial length due to reducing the thickness of the liquid refrigerant film. Due to increase of the liquid heat transfer coefficient, increasing wall heat flux is obtained.Finally, the evaporation rate of refrigerant increases with increasing wall heat flux.     

Assessment of evaporation equilibrium and stability concerning an acoustically excited drop in combustion products

The evaporation of drops in a sound field has been the subject of numerous studies aimed at determining its role in combustion instability. The models generally assume local equilibrium evaporation at the interface. We determine here the conditions of validity of this assumption, without calling into question other a priori assumptions of the classical model, in particular spherically symmetric quasi-steady evolution in the gas phase and liquid phase thermal unsteadiness with pure heat conduction.     

Appearance of Nucleation Shocks in a Boiling Liquid

Depressurization of high-pressure vessels filled with a liquid coming to the boil with a decrease in pressure is investigated. After depressurization, which takes less than 1 ms, a rarefaction wave propagates into the vessel. Experiments [1–5] demonstrated that the pressure behind the wave goes over to a constant value, which is independent of the vessel diameter and lies between the saturation point and atmospheric pressure.The correspondence between the experiments and different bubble formation theories (bubble formation on the vessel walls, due to rupture of the bonds between the water molecules or boiling on foreign particles) is analyzed. A “ mechanical nucleation” theory is proposed, in which it is assumed that the liquid comes to the boil on a limited number of foreign particles, and the bubbles formed on the nucleation centers undergo multiple fragmentation due to the instability developing under the action of centrifugal accelerations of the bubble surface in the course of bubble growth.The calculations demonstrated that, after depressurization, bubble fragmentation occurs at the vessel outlet. Due to the growth of the phase interface in the course of fragmentation, the boiling intensity increases, the pressure grows, and a shock wave propagates into the vessel, following the rarefaction wave. Multiple fragmentation of the bubbles occurs on the shock front. This wave is followed by a series of waves with smaller amplitudes. The pressure in the vessel attains a stable level, without any shocks. This level is characterized by the metastability or superheating of the liquid, i.e. the difference between the liquid temperature and the saturation point. It is demonstrated that the resultant pressure in the vessel is independent of the number of initial boiling centers or the initial pressure in the vessel and is determined only by the initial temperature. For water, the dependence of the superheating of the liquid on the initial temperature is found and compared with experimental data.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 103–117.Original Russian Text Copyright © 2005 by Ivashnev and Smirnov.     

Collapsing and Explosion Waves in Phase Transitions with Metastability,Existence, Stability and Related Riemann Problems

Collapsing waves were observed numerically before and were used to explain the ring formations in dynamic flows involving phase transitions with metastability. In this paper, necessary and sufficient conditions for collapsing type of waves to exist are given. The conditions are that the wave speed of the collapsing wave is not less than a number and is supersonic on both sides of the wave. Existence and non-existence conditions for the explosion waves are also found. The stability of these waves are studied numerically. Although there are infinitely many collapsing (or explosion) waves for a fixed downstream state, the collapsing (or explosion) wave appeared in the solution of Riemann problem is numerically verified to be the one with the slowest speed. Although a Riemann problem in the zero viscosity limit may have two solutions, one with, the other without, a collapsing (or explosion) wave, from the vanishing viscosity point of view, the one with a collapsing (or explosion) wave is numerically verified to be admissible.     

Molecular gas dynamics approaches to interfacial phenomena accompanied with condensation

This paper deals with the condensation coefficient of methanol, which is evaluated from a condensation rate at the vapor–liquid interface. Film condensation is induced on the endwall of a vapor-filled shock tube, when a shock wave is reflected at the endwall and the vapor becomes supersaturated there. The liquid film grows with the lapse of time. The evolution in time of the liquid film thickness is measured by an optical interferometer with a high accuracy, and thereby the net condensation mass flux at the interface is obtained. The mass flux is incorporated into the kinetic boundary condition (KBC) at the interface for the Gaussian–BGK Boltzmann equation. Such a treatment of the boundary condition makes it possible to formally eliminate the evaporation and condensation coefficients in KBC and to obtain the unique numerical solution of the vapor–liquid system. In this way, the instantaneous condensation coefficient is accurately evaluated from the conformity with experiment and numerical solution. It is found that the values of condensation coefficient are, near vapor–liquid equilibrium states, close to those evaluated by molecular dynamics simulations.     

Heat transfer and evaporation/condensation problems based on the linearized Boltzmann equation

A polynomial expansion procedure and an analytical discrete-ordinates method are used to solve four basic problems, all based on the linearized Boltzmann equation for rigid-sphere interactions, that describe heat transfer and/or evaporation–condensation between two parallel surfaces or for the case of a semi-infinite half space. Relevant to the case of two surfaces, the basic problem of heat transfer driven by a temperature difference at two confining walls described by a general Maxwell gas–surface interaction law (a mixture of specular and diffuse reflection) is solved for the case where different accommodation coefficients can be used for each of the two bounding surfaces. In addition, the classical problem of “reverse temperature gradient” in the theory of evaporation and condensation is also solved for the case of two parallel liquid–vapor interfaces kept at different temperatures. In regard to half-space applications, an evaporation/condensation problem based on a presumed known interface condition and a heat-conduction problem (with no net flow) driven by energy flow from a bounding surface with know properties are each solved with what is considered a high degree of accuracy.