Nonlinear Surface Heating Research Articles

Modeling of the heterogeneous hypergolic ignition with particle packing model

The ignition process of a hypergolic hybrid rocket fuel presents a significant challenge due to its complex and interconnected nature. However, understanding its underlying physics is crucial for accurately estimating the delay time and ignition performance. In this study, we employed an approximate analytical method to directly address the transient conduction problem with nonlinear surface heat generation. By using the Arrhenius equation and considering dual heat flux directions in both the liquid and solid domains, we propose a model for determining the delay time of heterogeneous hypergolic ignition. Our solution is straightforward and computationally efficient. Notably, when subjected to identical initial conditions and property settings, our results exhibit similarity with the solutions derived from other models. Moreover, we advance our approach by developing a particle packing model through the utilization of a well-established algorithm. By simulating the particle arrangement within the pellet, the surface layer properties and the thermal time constant of certain fuel designs can be modeled. This simulation serves the purpose of incorporating the particle size effect of the fuel pellet into our ignition model. The results generated by our heterogeneous hypergolic ignition model, in conjunction with the particle packing model, demonstrated impressive agreement with the experimental data derived from droplet tests. Novelty and significance statementThis study utilized a novel solving method to address a transient conduction problem with non-linear heat generation on the surface during a heterogeneous hypergolic ignition process. The study successfully derived an analytical solution, providing a simpler method for calculating ignition delay times. A scenario involving different initial temperatures was also examined. Additionally, the research integrated particle packing model data and thermal time constants to assess the effect of particle size. The model results were in agreement with the experimental data, confirming its accuracy and offering a reliable method for estimating the heterogeneous hypergolic ignition delay time.

Geometrically nonlinear rapid surface heating in FGM hermetic capsule

Geometrically nonlinear rapid surface heating in FGM hermetic capsule

Isogeometric and NURBS-enhanced boundary element formulations for steady-state heat conduction with volumetric heat source and nonlinear boundary conditions

Boundary Element Method (BEM) is a numerical tool that is applied to many different types of engineering problems. BEM possesses several advantages over other numerical methods such as boundary-only discretization and its semi-analytical nature. Application of Isogeometric Analysis (IGA) to BEM is a recently proposed computational method where the main purpose is to use geometrical basis functions as the shape functions of field variables. Key features of combining these two techniques are the exact representation of the geometry, elimination or suppression of the meshing procedure, reduction of the problem dimension and obtaining highly accurate results especially for the flux values on the boundaries compared to conventional BEM. In this study, steady-state heat conduction problems with nonlinear boundary condition and surface heat source are analyzed where geometries are formed by NURBS, and field variables are described with Lagrangian (constant and quadratic) basis functions and NURBS basis functions. Convergence rates and CPU time of different types of element representations are discussed which eventually reveals the performance and capabilities of IGABEM.

Numerical Analysis of quenching and cooling process of Cr12MoV Die Steel under Thermo-mechanical Coupling condition

Cr12MoV die steel has the characteristics of good wear resistance, small quenching deformation, high hardenability and hardness, so it requires high heat treatment process. With the emergence of computer simulation technology, it is convenient for the evolution and calculation of quenching process. In this paper, by using the technical route of combining nonlinear surface heat transfer coefficient with finite element simulation, the thermo-mechanical coupling theory is applied to the quenching process of Cr12MoV steel, and the temperature field and stress field within 300s before quenching are studied with the help of ABAQUS finite element software. The results show that the node temperature decreases with the increase of quenching time, the lowest temperature at edges and corners is about 21°C, the heat flux vector increases with the increase of quenching time, the yield stress increases with the increase of quenching time, and the maximum corner stress is about 751MPa. The experimental results of this paper are beneficial to the evolution calculation of material quenching process and the optimization of heat treatment method.

Geometrically nonlinear rapid surface heating of temperature-dependent FGM arches

Based on the nonlinear dynamic analysis, thermally induced vibrations of the FGM shallow arches subjected to different sudden thermal loads are studied. Temperature and position dependence of the material properties are taken into account. Based on the uncoupled thermoelasticity assumptions, The non-linear one-dimensional transient heat conduction equation is solved numerically by a hybrid iterative GDQ method and Crank-Nicolson time marching scheme. A first order shear deformation arch theory (FSDT) is also combined with the von Kármán type of geometrical non-linearity and the Donnell kinematic assumption to obtain the equations of motion employing the Hamilton principle. Discretization of the highly coupled non-linear equations of motion is done by using the GDQ method in the arch domain. The solution of the system of the ordinary differential equations is established by means of a hybrid iterative Picard-Newmark scheme. Comparison is also made with the existing results for the case of isotropic homogeneous shallow arches, where good agreement is obtained. Also, parametric studies are proposed to show the effects of temperature dependency, geometrical non-linearity, arch thickness, power law index, and the type of thermal-mechanical boundary conditions upon the arch deflection.

Computing anode heating voltage in high-pressure arc discharges and modelling rod electrodes in dc and ac regimes

Numerical modelling of near-anode layers in arc discharges in several gases (Ar, Xe and Hg) is performed in a wide range of current densities, anode surface temperatures, and plasma pressures. It is shown that the density of energy flux to the anode is only weakly affected by the anode surface temperature and varies linearly with the current density. This allows one to interpret the results in terms of anode heating voltage (volt equivalent of the heat flux to the anode). The computed data may be useful in different ways. An example considered in this work concerns the evaluation of thermal regime of anodes in the shape of a thin rod operating in the diffuse mode. Invoking the model of nonlinear surface heating for cathodes, one obtains a simple and free of empirical parameters model of thin rod electrodes applicable to dc and ac high-pressure arcs provided that no anode spots are present. The model is applied to a variety of experiments reported in the literature and a good agreement with the experimental data found.

Observation of Thermal Cathodic Hot Spots in a Magnetically Rotating Arc Plasma Generator

At atmospheric pressure, cathodic arc root tends to shrink to a luminous hot spot, which limits arc column expanding and accelerates cathode ablative rates. To obtain a nonconstricted cathodic arc root, we built a magnetically rotating arc plasma generator that mainly consists of a cylindrical graphite anode chamber, a concentric lanthanum tungsten cathode, and a solenoid coil for producing an axial magnetic field (AMF). Evolution of self-organized multihot spots on cathode end is observed in this study. Results show that with the AMF, arc currents, or/and cathode temperature increasing, the spot’s quantity gradually increases, and at last multispots evolve into a diffuse annular one. When the arc column is constricted, the arc moves on the spots periodically. Thus, a single constricted cathodic root is formed. By controlling the axial gas flow, the constricted arc converts into a diffusive one which covers all spots simultaneously, and then multiroots or diffuse annular root are developed. The cathodic spots and roots formation mechanism are proposed, and experimental results support the prediction of nonlinear surface heating model .

Numerical Simulation of Cr12MoV Steel during Quenching Process

The excessive residual stress induced by quenching in steels will easily result in distortion and failure of parts. In order to obtain the more suitable quenchant, quenching process of Cr12MoV steel with different mediums involving water and oil are simulated, respectively. In present paper, the influence of nonlinear surface heat transfer coefficient, thermodynamic parameters and latent heat are considered comprehensively. The distribution of temperature, microstructure, hardness and residual stress after quenching for Cr12MoV steel are simulated by DEFORM finite element software. According to the results mentioned above, the variations of each field of the steel are analyzed.

A new nonlinear surface heat flux calibration method based on Kirchhoff transformation and rescaling principles

A novel surface heat flux calibration method is presented applicable to nonlinear inverse heat conduction problems. Quasi-linearization of the nonlinear heat equation is achieved by combining the Kirchhoff transform with time-domain rescaling based on the local temperature measurement. At each time step, the thermophysical properties are held constant throughout the spatial domain though allowed to vary with advancing time. The rescaled forms are then resolved through a calibration framework. The proposed calibration formulation is expressed in terms of Volterra integral equation of the first kind. This functional equation relates the rescaled net unknown surface heat flux to the rescaled net calibration surface heat flux and their corresponding rescaled Kirchhoff transformed variables for the in-depth temperature measurements during the calibration test and unknown runs. Tikhonov regularization is introduced for generating a family of predictions based on the Tikhonov parameter. The L-curve strategy is used for selecting the proper regularization parameter. In this paper, favourable numerical results are demonstrated verifying both accuracy and robustness of the rescaling calibration approach in the presence of significant experiment noise. The methodology works well for a variety of practical isotropic materials. This approach does not require knowledge of the probe position but presently requires knowledge of the host material’s thermal diffusivity.

Numerical Simulation of the Transient Temperature Field during Gas Quenching

Based on the theory of heat transfer, phase transformation and thermal non-elasticity, a nonlinear coupling heat-conduction equation considering phase transformation, nonlinear surface heat transfer coefficient and variable physical properties during gas quenching is proposed and solved by means of Finite Element Method (F.E.M). The transient temperature field is obtained and the influencing factors are analyzed and discussed. It might be valuable for some practical applications and for the development of theory.

Joule heat generation in thermionic cathodes of high-pressure arc discharges

The nonlinear surface heating model of plasma-cathode interaction in high-pressure arcs is extended to take into account the Joule effect inside the cathode body. Calculation results are given for different modes of current transfer to tungsten cathodes of different configurations in argon plasmas of atmospheric or higher pressures. Special attention is paid to analysis of energy balances of the cathode and the near-cathode plasma layer. In all the cases, the variation of potential inside the cathode is much smaller than the near-cathode voltage drop. However, this variation can be comparable to the volt equivalent of the energy flux from the plasma to the cathode and then the Joule effect is essential. Such is the case of the diffuse and mixed modes on rod cathodes at high currents, where the Joule heating causes a dramatic change of thermal and electrical regimes of the cathode. The Joule heating has virtually no effect over characteristics of spots on rod and infinite planar cathodes.

MJO change with A1B global warming estimated by the 40-km ECHAM5

This study estimates MJO change under the A1B greenhouse gas emission scenario using the ECHAM5 AGCM whose coupled version (ECHAM5/MPI-OM) has simulated best MJO variance among fourteen CGCMs. The model has a horizontal resolution at T319 (about 40 km) and is forced by the monthly evolving SST derived from the ECHAM5/MPI-OM at a lower resolution of T63 (about 200 km). Two runs are carried out covering the last 21 years of the twentieth and twenty-first centuries. The NCEP/NCAR Reanalysis products and observed precipitation are used to validate the simulated MJO during the twentieth century, based on which the twenty-first century MJO change is compared and predicted. The validation indicates that the previously reported MJO variances in the T63 coupled version are reproduced by the 40-km ECHAM5. More aspects of MJO, such as the eastward propagation, structure, and dominant frequency and zonal wavenumber in power spectrum, are simulated reasonably well. The magnitude in power, however, is still low so that the signal is marginally detectable and embedded in the over-reddened background. Under the A1B scenario, the T63 ECHAM5/MPI-OM projected an over 3 K warmer tropical sea surface that forces the 40-km ECHAM to produce wetter tropics. The enhanced precipitation variance shows more spectral enhancement in background than in most wavebands. The zonal winds associated with MJO, however, are strengthened in the lower troposphere but weakened in the upper. On the one hand, the 850-hPa zonal wind has power nearly doubled in 30–60-days bands, demonstrating relatively clearer enhancement than the precipitation in MJO with the warming. A 1-tailed Student’s t test suggests that most of the MJO changes in variance and power spectra are statically significant. Subject to a 20–100-days band-pass filtering of that wind, an EOF analysis indicates that the two leading components in the twentieth-century run have a close structure to but smaller percentage of explained-to-total variance than those in observations; the A1B warming slightly increases the explained percentage and alters the structure. An MJO index formed by the two leading principal components discloses nearly doubling in the number of prominent MJO events with a peak phase occurring in February and March. A composite MJO life cycle of these events favors the frictional moisture convergence mechanism in maintaining the MJO and the nonlinear wind-induced surface heat exchange (WISHE) mechanism also appears in the A1B warming case. On the other hand, the Slingo index based on the 300-hPa zonal wind discloses that the upper-level MJO tends to be suppressed by the A1B warming, although the loose relationship with ENSO remains unchanged. Possible cause for the different change of MJO in the lower and upper troposphere is discussed.

Excitation of Intraseasonal Variability in the Equatorial Atmosphere by Yanai Wave Groups via WISHE-Induced Convection

Abstract A mechanism is presented, based on multiscale interactions via nonlinear wind-induced surface heat exchange (WISHE), that produces eastward-propagating, intraseasonal convective anomalies in the tropical atmosphere. Simulations of convectively coupled disturbances are presented in two intermediate-complexity atmospheric models. One is a shallow water model with a simple WISHE-motivated heating term. The other model is also based on a first baroclinic mode but has an additional prognostic equation for humidity and a simple representation of moist convection based on a quasi-equilibrium approximation. In spite of many differences between the models, they robustly produce a coherent signal in westerly winds and convection that travels eastward at 4–10 m s−1. It is shown here that this slow signal is a forced response to an eastward-propagating Yanai (mixed Rossby–gravity) wave group. The response takes the form of a forced Kelvin wave that is driven nonlinearly, via WISHE, by meridional wind anomalies of the Yanai wave group and that travels considerably more slowly than the free convectively coupled Kelvin waves in these models. The Yanai waves are destabilized in the models used here by WISHE in the presence of mean easterlies, but more generally they could also be excited by stratiform instability in the absence of mean easterlies so that the mechanism described here could also operate without mean easterlies. Similarities to and differences from the Madden–Julian oscillation (MJO) and convectively coupled tropical waves are discussed.

The Moisture Mode in the Quasi-Equilibrium Tropical Circulation Model. Part II: Nonlinear Behavior on an Equatorial β Plane

Abstract Numerical calculations of a simplified quasi-equilibrium tropical circulation model (QTCM) on the equatorial β plane have been performed to explore the nonlinear regime of the moisture mode. Sensitivity tests have examined the effects of nonlinear advection and nonlinear wind-induced surface heat exchange (WISHE). Starting from a spatially homogeneous radiative–convective equilibrium with some background gustiness, the model develops quasi-stationary moisture modes, as expected from linear analysis. Upon nonlinear saturation due to thermodynamic limiting processes, a different regime emerges. A classical Gill model augmented with a prognostic humidity variable is found to be able to capture the nonlinear dynamics of the moisture mode; the time evolution solely resides in the humidity variable, and it is possible to understand its dynamics by examining the humidity budget. A scaling analysis shows that the approximation with a Gill model is valid for disturbances whose time scale is much longer than the damping time scale of equatorial waves. When nonlinear WISHE is included, a large-scale disturbance of wavenumber 1 grows and moves westward because evaporation is enhanced to the west of increased convection. Turning on nonlinear advection leads to disturbances of wavenumber 10 that translate eastward via advection of dry air by Rossby gyres. Combining nonlinear WISHE and nonlinear advection leads to gross moist instability and prohibits long-term numerical integration, which suggests that QTCM must be refined to fully describe the nonlinear dynamics of the moisture mode.

Stability of direct current transfer to thermionic cathodes: II. Numerical simulation

Spectra of perturbations of steady-state current transfer to cathodes of high-pressure argon arcs are computed in the framework of the model of nonlinear surface heating. The following pattern of stability has been established for a current-controlled arc on a cylindrical cathode on the basis of the numerical results and trends derived previously by means of an approximate analytical treatment: the diffuse mode is stable beyond the first bifurcation point and unstable at lower currents; steady-state modes with more than one spot are unstable; the axially symmetric mode with a spot at the centre of the front surface of the cathode is unstable; the 3D steady-state mode with a spot at the edge is unstable between the bifurcation point and the turning point and stable beyond the turning point; the transition between the latter mode and the diffuse mode is non-stationary and accompanied by hysteresis. Theoretical results are in agreement with trends observed in the experiment.

Effect of a near-cathode sheath on heat transfer in high-pressure arc plasmas

Numerical simulation of a high-pressure arc discharge has been performed with a self-consistent modelling of most of the components, including the electrodes, and the interactions between them. In particular, the arc column and the cathodic part of the discharge are simulated by means of a two-temperature hydrodynamic model and of a model of nonlinear surface heating, respectively. Simulation results are given for a free-burning arc in atmospheric-pressure argon in the range of arc currents from 10 to 200 A. It is found that the electric power deposited into the near-cathode layer is transported not only to the cathode but also to the arc column, an effect that cannot be described by the local thermodynamic equilibrium (LTE) model. The electron enthalpy transport substantially exceeds the net contribution of thermal conduction by the electrons and heavy particles and is thus the dominating mechanism of energy transfer from the near-cathode layer to the arc column. The predicted gas temperatures along the arc axis in the arc column using the LTE model are much higher than the calculated electron and heavy-particle temperatures (∼1000–2000 K or higher) under the same operation conditions using the non-equilibrium model with the consideration of the near-cathode sheath for the cases studied in the present paper. Studies on the influences of the cathode shapes and metal vapour contaminations on the arc characteristics will be conducted in future work.

Stability of direct current transfer to thermionic cathodes: I. Analytical theory

Stability of different modes of steady-state current transfer to cathodes of high-pressure arc discharges is investigated in the framework of the model of nonlinear surface heating. An eigenvalue problem governing perturbations has been formulated and its properties analysed. It is shown that the spectrum of this problem is real, therefore the change of stability can occur only at bifurcation points, in which different modes of steady-state current transfer join each other. Analytical expressions describing behaviour of the increment in the vicinity of bifurcation points have been derived. A general pattern of stability of the diffuse mode of current transfer to axially symmetric cathodes and of 3D spot modes branching off from the diffuse mode is established. Theoretical predictions are found to be in agreement with trends observed in the experiment and it is shown that the recently published comparison of the theory and the experiment on the spot mode needs to be revisited.

Dynamic mode changes of cathodic arc attachment in vertical mercury discharges

Cathodic arc attachment after a steady state anodic phase is investigated in a vertically operated 0.7 MPa mercury high intensity discharge lamp with pure tungsten electrodes. Experimental evidence of a dynamic cathodic mode change of arc attachment influenced by convective effects is documented. Finite element simulation of the temperature field in the cathode body in two and three dimensions is performed. Nonlinear surface heating using Neumann's and Benilov's models is employed. The presheath voltage drop as a function of electron temperature is proposed as a compact input parameter. For Benilov's model two alternatives are compared. All models have the ability to show transient mode changes. The models show no sharp distinction. There are strong indications that the transient spot possesses a nonzero minimum life time when it abruptly emerges at a distinct threshold. The model fairly predicts the thresholds observed in experiments for the upper and lower electrodes depending on the convection heating. The evidence is more in favour of a lower presheath voltage drop, i.e. lower electron temperatures during the spot mode. The results are applicable for the design of dimming modes of HID lamps.

An analysis of effective thermal properties of thermally thick materials

The standard methods used to derive effective thermal properties of thermally thick materials based on bench-scale radiant exposure tests are reviewed and analyzed. These methods are compared with numerical calculations for the same boundary conditions. These comparisons show that the standard analytical methods for predicting surface temperature histories are not accurate because they either ignore heat losses from the surface or do not adequately treat the highly nonlinear reradiative surface heat loss term. A method is presented for determining more accurate values for the thermal inertia based on effective values for this term published widely in the literature. It is found that actual thermal inertias tend to be lower by about a factor of 1.3–2.7 when compared with reported effective values for a wide range of conditions. This can have a significant effect on flame spread predictions for models that rely on accurate values for the thermal inertia.

Cathodic arc attachment in a HID model lamp during a current step

Changes in the cathodic arc attachment mode during a current step are studied both experimentally under well-defined and reproducible conditions, and by finite element calculations. A transition from diffuse to spot and back to diffuse attachment is observed as the response to the step. Spot formation is identified experimentally by a sharp peak in the light intensity in front of the cathode and a steep decrease in the lamp voltage. Numerical three-dimensional solutions of the thermal-conduction equation in the cathode body exposed to nonlinear surface heating give good quantitative agreement of the time course of the observed changes in attachment modes. The model that describes the heating by the near cathode layer uses a simpler equation for the ion current density than other models in the recent literature. The model describes how an excess energy flux is created in the layer in front of a spot, which cannot be transported back to the cathode by the ion current and must, therefore, be supplied to the arc. The calculated integral excess power shows a time course similar to the observed light intensity.