Transient thermal model for the hot chamber injection system in the pressure die casting process

This paper describes a three-dimensional numerical model that is used to predict the transient thermal behaviour of the metal injection system of a hot chamber pressure die casting machine. The behaviour of the injection system is considered in conjunction with that of the die. The Boundary Element Method (BEM) is used to model the transient thermal behaviour of the injection system elements and the die blocks. A perturbation approach is adopted. By adopting this approach, only those surfaces over which a significant transient variation in temperature occurs need be considered. The model assumes that a corresponding steady-state analysis has first been performed so that time-averaged thermal information is available. A finite element based technique is used to model the phase change of the liquid metal in the die cavity and in the injection system. At injection the nozzle and die are assumed to be instantly filled with liquid metal, however, a procedure is presented that attempts to model the heat transfer associated with the flow through the nozzle, gate, and runner regions during injection. Model predictions are compared against thermocouple readings and thermal images obtained from experimental tests. Good agreement is obtained between predicted and measured temperatures. The transient thermal behaviour of an existing hot chamber injection system is investigated in detail and recommendations for improved performance are made. In an attempt to improve the solidification pattern of the casting and the thermal behaviour of the injection system, a redesign of the experimental die is considered. The numerical predictions indicate that the redesign will have a beneficial effect on the solidification pattern of the casting, and on the performance of the injection system.     

Objective Functions for the Shape Optimization of Temperature Control Channels in High-Pressure Die Casting

In high-pressure die casting, the quality of the produced cast parts highly depends on the die quality. One die component are temperature control channels. The state-of-the art approach for the temperature control channels are boreholes drilled from the sides into the die that are afterwards formed into a closed flow channel by inserting plugs. Since additive manufacturing methods are available to manufacture the dies, an obvious step is to construct more advanced temperature control channels. The shape of these free-form channels can be obtained by numerical optimization. To find the optimal shape of the channels, appropriate objective functions are required. This contribution is focused on such objective functions. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A solution methodology for contacting domains in pressure die casting

Multi-domain elastostatic problems can often be efficiently solved using coupled single domain iterative techniques. A particular difficulty however is the decrease in convergence rate associated with the increase in number of solution domains. Further convergence difficulties are encountered for multi-domain problems where contacting domains suffer variable contact conditions. Problems of this type are evident in pressure die casting with die blocks coupled together prior and subsequent to thermal loading. Particular interest in this paper is the development of an efficient solution methodology for prediction of gaps at block interfaces in pressure die casting.     

Spatial-resolution,lumped-capacitance thermal model for cylindrical Li-ion batteries under high Biot number conditions

The time-efficient yet accurate thermal modeling of the battery cells for electric and hybrid electric vehicles is essential improving the performance, safety, and lifetime of the battery system. This paper presents a spatial-resolution, lumped-capacitance (LC) thermal model for cylindrical battery cells under high Biot number (Bi ? 1) conditions where the classical LC thermal model is generally inapplicable because of a significant temperature variation in the cell volume. The spatial-resolution LC model was formulated using zero- and first-order Hermite integral approximations. For model validation, a one-dimensional, transient analytical (exact) solution using Green functions was obtained for a cylindrical Li-ion battery cell with uniform volumetric battery heat generation of Joule and entropic heating under convective cooling boundary conditions. It was found from the comparison of the results that the spatial-resolution LC thermal model can accurately and quickly predicts the cell temperatures (core, skin and area-averaged) under various dynamic battery duty cycles even for high Biot numbers due to highly convective conditions such as liquid cooling.     

A matrix perturbation method for computing the steady-state probability distributions of probabilistic Boolean networks with gene perturbations

Modeling genetic regulatory interactions is an important issue in systems biology. Probabilistic Boolean networks (PBNs) have been proved to be a useful tool for the task. The steady-state probability distribution of a PBN gives important information about the captured genetic network. The computation of the steady-state probability distribution involves the construction of the transition probability matrix of the PBN. The size of the transition probability matrix is 2ⁿ×2ⁿ where n is the number of genes. Although given the number of genes and the perturbation probability in a perturbed PBN, the perturbation matrix is the same for different PBNs, the storage requirement for this matrix is huge if the number of genes is large. Thus an important issue is developing computational methods from the perturbation point of view. In this paper, we analyze and estimate the steady-state probability distribution of a PBN with gene perturbations. We first analyze the perturbation matrix. We then give a perturbation matrix analysis for the captured PBN problem and propose a method for computing the steady-state probability distribution. An approximation method with error analysis is then given for further reducing the computational complexity. Numerical experiments are given to demonstrate the efficiency of the proposed methods.     

Thermal explosion for a slab with spatially varying surface temperature

Summary The thermal stability of reacting slabs whose surface temperature is a function of position is investigated. It is assumed that the variation is not large so that a perturbation approach may be adopted. Conditions for criticality are sought both for periodic distributions and others. A related problem where the boundary is perturbed but maintained at constant temperature is also considered.

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Zusammenfassung Die thermische Stabilität von chemisch reagierenden Platten mit örtlich variierender Oberflächentemperatur wird untersucht. Es wird angenommen, daß die Temperaturunterschiede gering sind. Dann kann eine Perturbationsmethode angewendet werden. Die kritischen Bedingungen für periodische und aperiodische Verteilungen werden angegeben. Ebenso wird ein verwandtes Problem behandelt, bei dem der Verlauf des Plattenrandes leicht gestört, aber auf einer konstanten Temperatur gehalten wird.

     

Numerical exploration of a system of reaction–diffusion equations with internal and transient layers

A reaction pathway for a classical two-species reaction is considered with one reaction that is several orders of magnitudes faster than the other. To sustain the fast reaction, the transport and reaction effects must balance in such a way as to give an internal layer in space. For the steady-state problem, existing singular perturbation analysis rigorously proves the correct scaling of the internal layer. This work reports the results of exploratory numerical simulations that are designed to provide guidance for the analysis to be performed for the transient problem. The full model is comprised of a system of time-dependent reaction–diffusion equations coupled through the non-linear reaction terms with mixed Dirichlet and Neumann boundary conditions. In addition to internal layers in space, the time-dependent problem possesses an initial transient layer in time. To resolve both types of layers as accurately as possible, we design a finite element method with analytic evaluation of all integrals. This avoids all errors associated with the evaluation of the non-linearities and allows us to provide an analytic Jacobian matrix to the implicit time stepping method. The numerical results show that the method resolves the localized sharp gradients accurately and can predict the scaling of the internal layers for the time-dependent problem.     

Numerical investigation of the deformation characteristics and heat generation in pneumatic aircraft tires Part II. Thermal modeling

A numerical procedure to determine the temperature rise in aircraft tires under free rolling conditions is presented in this article. Energy dissipation from cyclic inelastic deformation is considered the main heat generation source. This modeling considers the deformation process of the tire to be a steady-state problem, where all concurrent cycles are assumed to be the same as the first. The inelastic energy is determined by imposing a phase lag between the strain and the stress fields. The phase lag is assumed to be frequency independent in the range of interest, in keeping with the experimental observations in aircraft tire materials. It is further assumed that the inelastic energy is completely converted into volumetric heat input for a transient thermal conduction analysis. A conduction model is described and results are compared against thermocouple data recorded by Clark and Dodge [1].     

Short shots and industrial case studies: Understanding fluid flow and solidification in high pressure die casting

The geometric complexity and high fluid speeds involved in high pressure die casting (HPDC) combine to give strongly three dimensional fluid flow with significant free surface fragmentation and splashing. A simulation method that has proved particularly suited to modelling HPDC is Smoothed Particle Hydrodynamics (SPH). Materials are approximated by particles that are free to move around rather than by fixed grids, enabling more accurate prediction of fluid flows involving complex free surface motion. Three practical industrial case studies of SPH simulated HPDC flows are presented; aluminium casting of a differential cover (automotive), an electronic housing and zinc casting of a door lock plate. These show significant detail in the fragmented fluid free surfaces and allow us to understand the predisposition to create defects such as porosity in the castings. The validation of flow predictions coupled with heat transfer and solidification is an important area for such modelling. One powerful approach is to use short shots, where insufficient metal is used in the casting or the casting shot is halted part way through, to leave the die cavity only partially filled. The frozen partial castings capture significant detail about the order of fill and the flow structures occurring during different stages of filling. Validation can occur by matching experimental and simulated short shots. Here we explore the effect of die temperature, metal super-heat and volume fill on the short shots for the casting of a simple coaster. The bulk features of the final solid castings are found to be in good agreement with the predictions, but the fine details appear to depend on surface behaviour of the solidifying metals. This potentially has significant implications for modelling HPDC.     

轴对称金属模具电磁热裂纹止裂中热应力场的分析

为求解金属模具脉冲放电止裂瞬间裂纹尖端附近的热应力场,选择具有半埋藏环形裂纹的金属凹模为研究对象,采用复变函数方法求解了凹模内外环面均匀通入强脉冲电流放电止裂时的热应力场．理论分析结果证实:由于放电瞬间脉冲电流的绕流集中效应,使金属凹模内部环形裂纹尖端附近金属迅速升温,金属熔化形成堆焊,并由于瞬间温升产生热压应力场．研究结果表明:应用电磁热效应止裂技术可以减小裂纹尖端的应力集中,形成的热压应力场有效地阻止金属模具中干线裂纹源的开裂趋势,达到了裂纹止裂目的．     

Measuring heat transfer during spray cooling using controlled induction-heating experiments and computational models

This paper presents a methodology combining experimental measurements with computational modeling to find the heat flux extracted during spray cooling of a metal surface. Controlled experiments are performed to impinge air-mist spray onto a metal probe surface while applying induction heating to follow a desired temperature history. A transient axisymmetric computational model of induction heating which couples electromagnetics and heat conduction has been developed and validated with a test problem. The model is calibrated to match transient dry measurements and then used to simulate a steady-state air-mist spray cooling experiment in order to quantify the heat extracted from the probe surface by the boiling water droplets. A detailed example is presented to illustrate this approach.     

Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction

An analytical study for the problem of unsteady mixed convection with thermal radiation and first-order chemical reaction on magnetohydrodynamics boundary layer flow of viscous, electrically conducting fluid past a vertical permeable plate has been presented. Slip boundary condition is applied at the porous interface. The classical model is used for studying the effect of radiation for optically thin media. The non-linear coupled partial differential equations are solved by perturbation technique. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of chemical reaction, thermal stratification and magnetic field. It is observed that the effect of thermal radiation and magnetic field decreases the velocity, temperature and concentration profiles in the boundary layer. Also, the effects of the various parameters on the skin-friction coefficient and the rate of heat transfer at the surface are discussed.     

Simulation of thermal disturbances with finite wave speeds using a high order method

The classical heat diffusion theory based on the Fourier’s model breaks down when considering transient heat flow, for short times, extreme thermal gradients or at low temperatures. The hyperbolic heat conduction equation based on the Cattaneo model for the heat flux incorporates a relaxation mechanism in order to gradually adjust to a change in the temperature gradient. A spectral element method is applied for solving the hyperbolic system treating the heat flux as an independent variable in addition to temperature. The numerical solution is based on the time-space least squares spectral method. Numerical examples are included for discussing the effects of the thermal waves.     

Modelling thermal front dynamics in microwave heating

The formation and propagation of thermal fronts in a cylindricalmedium that is undergoing microwave heating is studied in detail.The model consists of Maxwell's wave equation coupled to a temperaturediffusion equation containing a bistable nonlinear term. When the thermal diffusivity is sufficiently small the leading-ordertemperature solution of a singular perturbation analysis isused to reduce the system to a free boundary problem. This approximationis then used to derive predictions for the steady-state penetrationand profiles of the temperature and electric fields. These solutionsare valid for arbitrary values of the electric conductivity,and thus extend the previous (small conductivity) results foundin the literature. A quasi-static approximation for the electric field is thenused to obtain an ordinary differential equation for the relaxationdynamics to the steady state. This equation appears to accuratelydescribe the time scale of the electric field's evolution bothwith and without the presence of a strongly coupled temperaturefront, and may be of wider interest than the model for microwaveheating studied here.     

Optimal Die Stock Control at Firth Vickers Foundry

This case study emphasises the growing importance of management sciences, operational research, and computing in the decision-making processes of production management in a dynamic high-technology company. The study is concerned with a computer system for die stock control and selection developed in conjunction with the management of Firth Vickers Foundry, Sheffield, following an initial diagnostic survey to define the problem and outline a solution to it. This article describes the effect on production costs (melting, spinning, machining and die costs) of the system which is designed to enable management to exercise optimal control of die stocks in the centri-spinning section of the foundry. The approach to the problem and development of logic are, however, applicable to all fluid-to-solid process industries where differently sized regular moulds or dies are stocked to meet order patterns of variable dimensions.     

Some problems of mathematical modeling in thermomechanics of bodies of various transparencies subjected to thermal irradiation

Features of the statement of problems of thermomechanics for solids of various transparencies for thermal radiation are presented. Approximate approaches to the calculation of temperature in semitransparent solids are analyzed. We investigate the influence of the effects of radiation and transfer of thermal energy on temperatures and stresses in semitransparent and opaque solids on a model problem for an irradiated layer.     

Thermal runaway in microwave heating: a mathematical analysis

A study is made of the solution of a differential equation modelling the heating of a layer of material specimen by microwave radiation. Depending on the microwave power bistable steady-state temperatures may be expected. When changing the power, a switch from one stable branch to another one may arise. The sudden increase of temperature, known as thermal runaway, is studied from the differential equation using asymptotic methods. Such analysis reveals distinct stages in the process of thermal runaway. At the moment the solution leaves a branch, and becomes unstable a particular type of behaviour is observed (onset of runaway). The most specific element at this stage is a time shift delaying the rapid change in temperature. For this shift a simple expression in terms of the parameters of the system is given. Next it is shown that the rapid transition from one branch to the other can be put in a mathematical formula that smoothly matches the two steady state solutions.     

多孔介质平板通道发展传热中非局部热平衡时的温度分布特征

研究了多孔介质平板通道中,Darcy流体发展传热强迫对流非局部热平衡下,固相骨架和孔隙流体的温度分布特征．考虑流体流动方向的热传导以及固相和流相相互作用的粘性耗散,根据非局部热平衡的两能量方程模型,得到了常壁温度时多孔介质固相骨架温度和孔隙流体温度的解析解．证明了当两相间的热交换系数趋于无穷大时,两能量方程的温度解趋于局部热平衡时一能量方程的温度解．针对不同的无量纲参数,给出了固相和流相的温度分布状态,通过参数研究,揭示了非局部热平衡强迫对流时温度对无量纲参数的依赖关系．     

An analytical solution for the unidirectional solidification problem

The extraction of heat from a molten casting is resisted by an imperfect thermal contact at the mold-casting interface. The nature of the contact varies throughout the casting process and has the effect of increasing the thermal resistance at the interface. This can be modelled by incorporating a gaseous gap at the mold-casting interface that grows with increasing time.

This paper is concerned with an analytical solution of the unidirectional solidification problem, which incorporates movement of the casting at the interface. The derivation of the analytical solution requires the simultaneous solution of the transient heat equations, for the mold, gaseous gap, and solid and liquid parts of the melt. The analytical solution is extended so that contamination layers on the mold and casting can be incorporated as well as an initial gap. This is achieved by introducing virtual layers of mold, gas, and casting. Using the extended solution, the effects of interfacial resistance, air conductivity, and gap variation on solidification rates are examined.     

Dynamic modelling of die melt temperature profile in polymer extrusion: Effects of process settings,screw geometry and material

Extrusion is one of the major methods for processing polymeric materials and the thermal homogeneity of the process output is a major concern for manufacture of high quality extruded products. Therefore, accurate process thermal monitoring and control are important for product quality control. However, most industrial extruders use single point thermocouples for the temperature monitoring/control although their measurements are highly affected by the barrel metal wall temperature. Currently, no industrially established thermal profile measurement technique is available. Furthermore, it has been shown that the melt temperature changes considerably with the die radial position and hence point/bulk measurements are not sufficient for monitoring and control of the temperature across the melt flow. The majority of process thermal control methods are based on linear models which are not capable of dealing with process nonlinearities. In this work, the die melt temperature profile of a single screw extruder was monitored by a thermocouple mesh technique. The data obtained was used to develop a novel approach of modelling the extruder die melt temperature profile under dynamic conditions (i.e. for predicting the die melt temperature profile in real-time). These newly proposed models were in good agreement with the measured unseen data. They were then used to explore the effects of process settings, material and screw geometry on the die melt temperature profile. The results showed that the process thermal homogeneity was affected in a complex manner by changing the process settings, screw geometry and material.