Hertz-Knudsen Equation Research Articles

Experimental Vapor Pressure Determination for C2H4, C2H6, CH3OH, CH4, CO, CO2, H2O, and N2 Molecules for Astrophysical Relevant Temperatures. Implications for the Presence of Volatiles in Kuiper Belt Objects and Trans-Neptunian Objects

Vapor pressure is a relevant quantity that is necessary in order to improve the study of the atmosphere dynamics that take place within astrophysical scenarios. The aim of this study was to obtain the vapor pressure values of the following molecules: C2H4, C2H6, CH3OH, CH4, CO, CO2, H2O, and N2 through experimentation, as well as to determine their empirical relationship with the temperature, applying the results to the persistence of volatiles in trans-Neptunian objects (TNOs) and Kuiper Belt objects (KBOs). The experimental determination was performed by measuring the sublimation rate for each molecule at different temperatures. The Hertz–Knudsen equation was used to obtain the vapor pressures for the aforementioned molecules, taking the necessary considerations into account, and the sublimation rate was measured using a quartz crystal microbalance. In order to check the validity of the methods used, the results obtained for water ice were compared with those of previous studies from the literature. The values obtained for CO, N2, and CH4 are of particular interest in the study of the TNOs' and KBOs' atmosphere composition. The results of this study improve the understanding of the surface and atmospheric composition of objects in the cold scenarios of the solar system, in particular, in KBOs and TNOs.

Well-posedness of an evaporation model for a spherical droplet exposed to an air flow

In this paper, we address the well-posedness of an evaporation model for a spherical liquid droplet taking into account the convective impact of an air flow in the ambient gas phase. From a mathematical perspective, we are dealing with a coupled ODE–PDE system for the droplet radius, the temperature distribution, and the vapor concentration. The nonlinear coupling arises from the evaporation rate modeled by the Hertz–Knudsen equation. Under physically meaningful assumptions, we prove existence and uniqueness of a weak solution until the droplet has evaporated completely. Numerical simulations are performed to illustrate how different air flows affect the evaporation process.

Application of pulsed laser ablation thermal model in nanosecond pulsed laser removal of the epoxy resin paint film

The effect of laser surface paint removal depends on the mechanism of paint removal. The mechanism of nanosecond laser removal of epoxy resin paint film on aluminum alloy surfaces was studied theoretically and experimentally. Firstly, a laser thermal paint removal model was established based on the heat conduction equation, Hertz-Knudsen equation, and Hooke's law. The adsorption relationship between the paint film and the aluminum alloy substrate molecules and the adsorption force between the paint film and the substrate were analyzed and calculated. A laser thermal paint removal model was solved using COMSOL finite element software. By analyzing the changes in temperature and thermal stress with laser fluence, we obtained the theoretical threshold for laser paint removal. Subsequently, experimental research was conducted on laser paint removal. Through theoretical and experimental studies, it was found that the paint removal threshold of nanosecond pulse laser is 0.47 J/cm2. The primary mechanism of nanosecond laser paint removal is thermal stress stripping. When the laser fluence is low, the nanosecond pulse laser can only ablate the paint on the surface, and the thermal stress between the paint film and the substrate makes it difficult to remove the paint film. As the laser fluence increases, the nanosecond laser will ablate the surface paint film, and the thermal stress between the paint film and the substrate will cause the stripping of the underlying paint film and the substrate. The primary mechanism of laser paint removal is stripping.

Modelling of liquid oxygen and nitrogen injection under flashing conditions

The present numerical investigation of two-phase flashing flows examines the injection of liquid oxygen and liquid nitrogen into near-vacuum conditions prevailing in the upper-stage boosters of rocket engines. The predictive capability of a pressure-based solver and a density-based solver, each employing distinct approaches related to the imposed phase-change rate and thermodynamics closure, has been comparatively evaluated. Regarding the pressure-based solver, the departure from thermodynamic equilibrium during phase-change has been taken into account via the implementation of a bubble-dynamics model employing the Hertz-Knudsen equation. In contrast, the density-based solver relies on the adoption of thermodynamic equilibrium while real-fluid thermodynamic properties are assumed by loading tabulated values to the solver. Each thermodynamic property value was calculated in advance by solving the Helmholtz Equation of State (EoS) for a wide range of density and internal energy conditions. Numerical findings have been compared against experimental data available in the literature. The comparison demonstrates the capability of both methodologies in capturing the evolution of cryogenic flashing flow expansion, phase-change, and spray formation. The salient features identified in the numerical results, i.e., the expansion sphere immediately downstream of the injector exit, the bell-shaped topology of the spray, as well as the dependency of the spray cone angle on superheat, are in agreement with experimental measurements. Especially the density-based approach has been proven highly accurate with respect to the steady expanding flow described by a level of superheat in the range of 3 to 245, while also being independent of any parameter tuning.

Nonuniform heating of a substrate in evaporative lithography

This work is devoted to a method to generate particle cluster assemblies and connected to evaporative lithography. Experiments are carried out using nonuniform evaporation of an isopropanol film containing polystyrene microspheres in a cylindrical cell. The local inhomogeneity of the vapor flux density is achieved by exploiting the temperature gradient. A copper rod is mounted in the central part of the bottom of the cell for further heating. The thermocapillary flow resulting from the surface tension gradient, due in turn to the temperature drop, transfers the particles that were originally at rest at the bottom of the cell. The effect of the initial thickness of the liquid layer on the height and base area of the cluster formed in the central region of the cell is studied. The velocity is measured using particle image velocimetry. A model describing the initial stage of the process is developed. The equations of heat transfer and thermal conductivity are used to define the temperature distribution in the liquid and in the cell. The fluid flow is simulated using the lubrication approximation. The particle distribution is modeled using the convection–diffusion equation. The evaporation flux density is calculated using the Hertz–Knudsen equation. The dependence of the liquid viscosity on the particle concentration is described by Mooney's formula. Numerical results show that the liquid film gradually becomes thinner in the central region, as the surface tension decreases with the increasing temperature. The liquid flow is directed to the heater near the substrate, and it transfers the particles to the center of the cell. The volume fraction of the particles increases over time in this region. The heat flow from the heater affects the geometry of the cluster for two reasons: First, the Marangoni flow velocity depends on the temperature gradient, and second, the decrease in film thickness near the heater depends on the temperature. The results of the simulation are in general agreement with the experimental data.

Modelling of liquid oxygen nozzle flows under subcritical and supercritical pressure conditions

The two-phase flow of liquid oxygen in a converging-diverging nozzle has been numerically predicted at conditions resembling those that prevail in the lower-stage boosters of rocket engines realising lift off, as well as in the respective upper stages operating in sub-atmospheric pressures. A comparative evaluation of the predictive capability of a pressure and a density-based solver with various approaches regarding the imposed phase-change rate and thermodynamics closure have been performed. The departure from thermodynamic equilibrium during phase-change has been taken into account via implementation of a bubble-dynamics model employing the Hertz-Knudsen equation in the pressure based solver, whereas thermodynamic equilibrium is adopted in the density-based solver. Tabulated data for the variation of the fluid thermodynamic properties have been derived by the Helmholtz Equation of State (EoS) in a modelling approach universal for both the sub- and supercritical states. This approach has been comparatively assessed in the sub-critical regime against the bubble-dynamics-based model including different EoS for the liquid/vapour phases and against a different tabulated approach based on the NIST dataset for supercritical injection. In terms of flow physics, more severe flow expansion in the diverging part of the nozzle has been detected for subcritical pressures, leading to supersonic flow velocities and significant cooling of the fluid mixture. Complementary Detached Eddy Simulations (DES) have provided detailed insight on the complex expansion phenomena and flow instabilities manifesting on the divergent part of the nozzle for subcritical-injection conditions. The comparison of the numerical predictions against available experimental data and analytical solutions demonstrates the suitability of the employed methodologies in describing the evolution of the cryogenic oxygen flow expansion and phase-change.

Evaporation and propagation of liquid drop streams at vacuum pressures: Experiments and modeling.

Evaporation of streams of liquid droplets in environments at vacuum pressures below the vapor pressure has not been widely studied. Here, experiments and simulations are reported that quantify the change in droplet diameter when a steady stream of ≈100 μm diameter drops is injected into a chamber initially evacuated to <10^{-8}bar. In experiments, droplets fall through the center of a 0.8 m long liquid nitrogen cooled shroud, simulating infinity radiation and vapor mass flux boundary conditions. Experimentally measured changes in drop diameters vary from ≈0 to 6 μm when the initial vapor pressure is increased from 10^{-6} to 10^{-3} bar by heating the liquid. Measured diameter changes are predicted by a model based on the Hertz-Knudsen equation. One uncertainty in the calculation is the "sticking coefficient" β. Assuming a constant β for all conditions studied here, predicted diameter changes best match measurements with β≈0.3. This value falls within the range of β reported in the literature for organic liquids. Finally, at the higher vapor pressure conditions considered here, the drop stream disperses transverse to the main flow direction. This spread is attributed to forces imparted by an absolute pressure gradient produced by the evaporating stream.

Dynamics of hygroscopic aqueous solution droplets undergoing evaporation or vapour absorption

Abstract

Numerical modeling of the liquid film flow with evaporation on the basis of the generalized interface conditions

A thin layer of a viscous incompressible liquid flowing down an inclined, non-uniformly heated substrate under conditions of a concomitant gas flow is studied. The flow is accompanied by evaporation at the thermocapillary interface. Dynamic processes in a gas are not taken into account (one-side model). The mathematical model of a thin liquid layer flow is based on the Navier-Stokes equation and the heat transfer equation, as well as generalized kinematic, dynamic and energetic conditions at the thermocapillary boundary. The local mass flux at the interface is determined using the Hertz-Knudsen equation. Analytical solutions of the problem are constructed for the zeroth-order terms of the expansions of the desired functions in powers of a small parameter of the problem. Parametric analysis of the problem is performed. An evolutionary equation for determining the thickness of the liquid layer is obtained. An algorithm for numerical solution of the evolutionary equation is constructed. The results of numerical studies of flows of a thin layer of ethanol and HFE-7100 with evaporation at the interface are obtained.

Numerical modeling of a thin liquid layer flow on an inclined substrate based on classical convection equations and generalized conditions at the interface

The flow of a thin layer of a viscous incompressible liquid along an inclined non-uniformly heated substrate is investigated in the two-dimensional case. Evaporation is taken into account at the thermocapillary interface. The system of Oberbeck-Boussinesq equations is used as a mathematical model. The kinematic, dynamic and energy conditions generalized for the case of a nonzero mass flux are assumed to be fulfilled at the interface. The value of the local mass flux is determined using the Hertz-Knudsen equation. Solutions to problems for the main terms of power decomposition of a small parameter of the problem are obtained. The evolution equation to determine the position of the interface is obtained. A numerical algorithm for solving the problem is constructed. The problem of periodic runoff of an incompressible liquid layer is considered. The effect of additional terms in the energy condition on the dynamics of the thickness of the liquid layer is studied.

Coating density as the key factor behind hydrogen embrittlement of cadmium-plated 4340 steel

The aerospace industry uses 200-h sustained-load test (SLT), described in ASTM F-519, to evaluate hydrogen embrittlement (HE) of their 4340 steel electroplated parts. The cadmium plating used in ASTM F-519 suggest that the coating density may affect HE in an important manner. Therefore, in this work, 4340 steel notched bars were plated with five different cadmium coating densities. Then, the SLT was combined with thermal desorption spectroscopy (TDS) to evaluate: (i) the degree of HE, and (ii) the hydrogen content of the cadmium-plated 4340 steel samples. The results show strong evidences that the major factor causing HE of cadmium-plated 4340 steel is the coating density. Moreover, an advanced analysis of TDS curves using the Hertz–Knudsen equation allows distinguishing hydrogen in steel from hydrogen in the cadmium coating. The advanced analysis also gives an estimation of the coating density, and therefore indicates if the baking will be effective at relieving embrittlement. Finally, TDS could be a good complementary tool to the SLT in evaluating HE.

Evaporation-driven liquid flow through nanochannels

Evaporation-driven liquid flow through nanochannels has attracted extensive attention over recent years due to its applications in mass and heat transfer as well as energy harvesting. A more comprehensive understanding is still expected to reveal the underlying mechanisms and quantitatively elucidate the transport characteristics of this phenomenon. In this study, we investigated evaporation-driven liquid flow through nanochannels using molecular dynamics simulations. The evaporation flux from the solid-liquid interface was higher than that from the middle region of the channel or the liquid-vapor interface. This finding may explain why experimental observations of evaporation flux through nanochannels exceed the limits predicted by the classical Hertz–Knudsen equation. Upon increasing the interaction strength between liquid atoms, the liquid exhibited enhanced solid-liquid interfacial evaporation and higher surface tension, albeit with reduced total flux. We also found that lyophilic channels exhibited higher evaporation fluxes than lyophobic channels, which can be interpreted by a Gibbs free energy analysis. The energy conversion analysis indicated that the effective pressure gradient exerted on a liquid flow by evaporation depends on the channel length. This was consistent with our simulations. Evaporation-driven liquid flow through nanochannels could be modeled quantitatively using this knowledge.

Theoretical analyses of chemical and magnesium and silica isotope fractionation during vaporization of ca-al-inclusion of chondrite

Experimental study of changes in the composition of Ca-Al-inclusions of chondrites during evaporation indicates a close relationship of chemical and isotopic fractionation of this substance. The theoretical description of the coupling is carried out using the equations of the evaporation rate of the melt component (the Hertz-Knudsen equation) and isotope Rayleigh fractionation. The form of Rayleigh equation taken in foreign literature, derived from the Hertz-Knudsen equation, faces difficulties in interpreting experimental data. The discrepancy between the experimental and model data is explained by the fact that the «ideal» isotope fractionation factor used in the Rayleigh equation does not take into account its dependence on the temperature and composition of the evaporating melt. The article presents an alternative expression of the Rayleigh equation and a new expression of the rate of evaporation of Hertz-Knudsen, taking into account the activity of the melt component. The activity of the component is determined by the acidity-basicity index of the melt Ca-Al-inclusion, which, in turn, affects the evaporative fractionation of magnesium and silicon isotopes.

Theoretical Analysis of Mg and Si Chemical and Isotopic Fractionation at Vaporization of Ca–Al Inclusions of Chondrites

Experimental study of changes in the composition of Ca–Al inclusions of chondrites during evaporation indicates the chemical and isotopic fractionation of this material are closely interrelated. The coupling is theoretically described using an equations for the evaporation rate of a component of melt (the Hertz–Knudsen equation) and Rayleigh isotope fractionation. The form of the Rayleigh equation (which was derived from the Hertz–Knudsen equation) conventionally used in the foreign literature faces difficulties in interpreting experimental data. The discrepancy between the experimental and model data is explained by the fact that the “ideal” isotope fractionation factor used in the Rayleigh equation does not take into account its dependence on the temperature and composition of the evaporating melt. The paper presents an alternative expression for the Rayleigh equation and a new expression of the Hertz–Knudsen evaporation rate with regard for the activity of melt components. The activity of the components is determined by the acidity–basicity index of the melt Ca–Al inclusion, which, in turn, affects the evaporative fractionation of Mg and Si isotopes.

Simulation of mass and heat transfer in liquid hydrogen tanks during pressurizing

In order to accurately predict the heat and mass transfer behaviors and analyze key factors affecting pressurization process in the hydrogen tank, a comprehensive 2D axial symmetry Volume-Of-Fluid (VOF) model is established by Computational Fluid Dynamics (CFD) method. The effects of phase change, turbulence and mass diffusion are included in the model and relationships between physical properties and temperature are also comprehensively considered. The phase change model is based on Hertz-Knudsen equation and the mass transfer time relaxation factor is determined by the NASA’s experimental data. The mass diffusion model is included in gaseous helium pressurizing. The key factors including the inlet temperature, inlet mass flow rate, injector types and pressurizing gas kinds are quantitatively analyzed. Compared with the experiment, the simulation results show that the deviation of pressurizing gas mass consumption, condensing mass and ullage temperature are 3.0%, 7.5% and 4.0% respectively. The temperature stratification is existed along the axial direction in the surface liquid region and the ullage region, and the bulk liquid is in subcooled state during pressurizing. The location of phase change mainly appears near the vapor–liquid interface, and the mass transfer expressing as condensation or vaporization is mainly determined by the heat convection and molecular concentration near the vapor–liquid interface. The key factors show that increasing the inlet temperature and inlet mass flow rate could shorten the pressurizing time interval and save the pressurizing gas mass. The proportion of the total energy addition of the tank absorbed by the ullage region, the liquid region and the tank wall respectively is greatly influenced by the injector types and more heat transferred into the ullage would result in a faster pressure rising rate. Gaseous hydrogen pressurization has a higher efficiency than gaseous helium pressurization. The simulation results presented in this paper can be used as a reference for design optimization of the pressurization systems of cryogenic liquid launch vehicles so as to save the mass of pressurizing gases and shorten the pressurizing time interval.

Sublimation of buried cometary ice

The sublimation of ice covered by dust is a common way of outgassing of comets. The rate of this process depends on the thermal conductivity and permeability of dust, as well as the properties of the ice itself. It can be calculated using the Hertz-Knudsen equation, corrected for temperature dependent sublimation coefficient αs(T) (Gundlach et al., 2011; Kossacki et al., 1999; Kossacki & Leliwa-Kopystynski, 2014; Kossacki et al., 2017) and the Knudsen equation for the permeability of dust. In this work we present direct experimental verification of this set of equations using sand with different granulation: coarse (grain radii rg = 0.5 − 1 mm), medium (rg = 0.25 − 0.5 mm), fine (rg = 0.125 = 0.25 mm) and unsorted (rg = 0 − 1 mm, rg, av = 0.5 mm). We have found that at T < 220 K the measured rate of recession of the surface is correctly reproduced if αs(T) is the same as determined for the uncovered ice (Kossacki & Leliwa-Kopystynski, 2014). The characteristic radius of pores in the Knudsen equation is approximately equal to, or greater than the arithmetic average of the largest and the smallest radii of grains.

Numerical examination of the evaporation process within a vacuum induction furnace with a comparison to experimental results

This paper discusses a mathematical model for and presents the experimental results of the metal evaporation process in a vacuum induction furnace. An in-house-developed coupling procedure was utilized to predict the electromagnetic, flow and temperature fields in a simplified axisymmetric domain. Evaporation kinetics were simulated by means of a Hertz-Knudsen equation and implemented as source terms in transport equations. The metal vapour above the molten metal bath was described by an additional conservation equation, with gradients of diffusion flux and evaporated metal source terms included. The diffusion flux was the solution of Fick’s law. To fully analyse the evaporation process, several numerical computations were performed to examine the influence of the input power of the inductor, the crucible position inside the copper coil and the amount of charge. The validation of the mathematical description was performed according to aluminium mass loss measurements. A comparison with the experimental results confirmed that the proposed mathematical model of evaporation kinetics can be applied to the evaporation process modelled within a vacuum induction furnace. A numerical case study allowed for the proper identification of operating conditions to intensify the evaporation process within the induction furnace. Moreover, the obtained results confirmed that even small changes in the charge temperature during the evaporation process might have a crucial influence on the evaporation rate.

Studying heating of magnetron target based on measurement of substrate temperature

The present work aims to develop the method for indirect determination of the magnetron target temperature. This method is based on the measurements of heating kinetic curves of the thermocouple sensor's sensing element. The measurements were performed during operation of the magnetrons with cold and hot titanium targets. In order to calculate the hot target temperature, a transcendental equation was derived using the Stefan-Boltzmann law and the Hertz-Knudsen equation. In the general case, the equation uses the difference of the fluxes transferred to the sensing element. The study revealed that in the target temperature range of 1000–1500 °C, the influence of the heat flux transferred by the evaporated atoms of the titanium target on the substrate temperature can be neglected. And the heat fluxes caused by the gas discharge with a current density in the range of 50–100 mA/cm2 have a significant influence on the substrate temperature.

Enhancement of pulsed laser ablation assisted with continuous wave laser irradiation

Continuous wave (CW) laser irradiation is employed to enhance the pulsed laser ablation of silicon and stainless steel (316L) samples. Different surface temperatures generated by the CW laser irradiation are set as the initial working circumstance for the pulsed laser ablation. The diameter and depth of laser-ablated craters are measured to study threshold fluence, pulse incubation coefficient and ablation rate under different surface temperatures. Numerical simulation employing Heat Transfer in Solid and Deformed Geometry Interface modules in COMSOL is performed to estimate ablation rate theoretically based on Hertz-Knudsen equation. The realized crater-related data are analyzed to further obtain their dependences on surface temperature. The parametric and morphological studies indicate that the weakened plasma shielding effect and thermal diffusion in the ablated region induced by the CW laser irradiation lead to the enhanced pulsed laser ablation significantly.

A model framework to retrieve thermodynamic and kinetic properties of organic aerosol from composition-resolved thermal desorption measurements

Abstract. Chemical ionization mass spectrometer (CIMS) techniques have been developed that allow for quantitative and composition-resolved measurements of organic compounds as they desorb from secondary organic aerosol (SOA) particles, in particular during their heat-induced evaporation. One such technique employs the Filter Inlet for Gases and AEROsol (FIGAERO). Here, we present a newly developed model framework with the main aim of reproducing FIGAERO-CIMS thermograms: signal vs. ramped desorption temperature. The model simulates the desorption of organic compounds during controlled heating of filter-sampled SOA particles, plus the subsequent transport of these compounds through the FIGAERO manifold into an iodide-CIMS. Desorption is described by a modified Hertz–Knudsen equation and controlled chiefly by the temperature-dependent saturation concentration C*, mass accommodation (evaporation) coefficient, and particle surface area. Subsequent transport is governed by interactions with filter and manifold surfaces. Reversible accretion reactions (oligomer formation and decomposition) and thermal decomposition are formally described following the Arrhenius relation. We use calibration experiments to tune instrument-specific parameters and then apply the model to a test case: measurements of SOA generated from dark ozonolysis of α-pinene. We then discuss the ability of the model to describe thermograms from simple calibration experiments and from complex SOA, and the associated implications for the chemical and physical properties of the SOA. For major individual compositions observed in our SOA test case (#C=8 to 10), the thermogram peaks can typically be described by assigning C25∘C* values in the range 0.05 to 5 µg m−3, leaving the larger, high-temperature fractions (&gt;50 %) of the thermograms to be described by thermal decomposition, with dissociation rates on the order of ∼1 h−1 at 25 ∘C. We conclude with specific experimental designs to better constrain instrumental model parameters and to aid in resolving remaining ambiguities in the interpretation of more complex SOA thermogram behaviors. The model allows retrieval of quantitative volatility and mass transport information from FIGAERO thermograms, and for examining the effects of various environmental or chemical conditions on such properties.