Analytical Heat Conduction Model Research Articles

Thermal Management of Serpentine Flexible Heater Based on the Orthotropic Heat Conduction Model.

Flexible heaters can perfectly fit with undevelopable surfaces for heating in many practical applications such as thermotherapy, defogging/deicing systems and warming garments. Considering the requirement for stretchability in a flexible heater, certain spacing needs to be retained between serpentine heat sources for deformation which will inevitably bring critical challenges to the thermal uniformity. In order to reconcile these two conflicting aspects, a novel method is proposed by embedding the serpentine heat source in orthotropic layers to achieve comprehensive performance in stretchability and uniform heating. Such a scheme takes advantage of the ability of orthotropic material to control the heat flow distribution via orthotropic thermal conductivity. In this paper, an analytical heat conduction model with orthotropic substrate and encapsulation is calculated using Fourier cosine transform, which is validated by finite element analysis (FEA). Meanwhile, the effects of the orthotropic substrate or encapsulation with different ratios of thermal conductivity and the geometric spacing on the thermal properties are investigated, which can help guide the design and fabrication of flexible heaters to achieve the goal of uniform heating.

Blackbox optimization for approximating high-fidelity heat transfer calculations in metal additive manufacturing

Heat transfer simulations play an important role in predicting solidification conditions that determine microstructure properties of additive manufactured products. High-fidelity heat transfer models, e.g., Truchas [1], provide a numerical framework capable of resolving many desired physics (e.g., thermal radiation, elemental vaporization, gas-solid-liquid interactions, etc.), but they are often too computationally expensive to simulate the solidification conditions at the AM process-scale. Recently, Stump et al. [2] proposed a low-fidelity analytical heat conduction model capable of simulating solidification conditions on the process-scale. This model makes the problem analytically tractable by neglecting many important physics that may have an impact on the accuracy of the predicted solidification conditions. An effective way of improving the accuracy of the analytical with little computational effort is to determine an optimal set of parameters that minimize the discrepancy between the low-fidelity analytical model and a high-fidelity numerical model. In this work, a model approximation problem is formulated which can be solved by leveraging blackbox optimization methods. Multiple heat sources are used in the analytical model to improve the calibration performance in a flexible framework. Two blackbox optimization methods, i.e., Bayesian optimization and directional Gaussian smoothing methods, are employed to address the challenge that the gradient of the loss function is inaccessible during optimization. The results show that the single-beam analytical model has limitations in fitting the high-fidelity model, but the proposed multi-beam analytical model provides satisfactory approximations to the high-fidelity temperature and melt pool fields.

Particle–Substrate Transient Thermal Evolution During Cold Spray Deposition Process: A Hybrid Heat Conduction Analysis

A hybrid analytical heat conduction model was developed to predict the transient thermal evolution at the particle–substrate interface during the deposition of cold spray process. First, three-dimensional heat conduction model based on classical diffusion approach was developed to predict the transient surface temperature of the substrate. To that end, the traveling wave solution technique was utilized in order to take into account the effect of the movement of the cold spray nozzle. The results of the analytical model were validated with the experimentally measured surface temperature which was obtained by employing a low-pressure cold spray unit to generate the impingement of a compressed air jet on a flat substrate. The analytical model was further utilized to investigate the effect of the non-dimensional characteristic velocity of the traveling heat source on the surface temperature profile of the substrate. It was found that both the maximum surface temperature and the spatial variation of surface temperature profile of the substrate decreased as the non-dimensional characteristic velocity increased. The mathematical model was further extended by developing a one-dimensional hyperbolic (non-Fourier) heat conduction model to predict the temperature rise at the particle–substrate interface during the cold spray deposition process. In order to validate the results of the hybrid model, a three-dimensional finite element model was developed in ABAQUS to simulate the thermal and dynamic behavior of a single particle upon impact. The results of the hybrid analytical model for the temperature at the particle–substrate interface were compared to the results of the numerical model, and good agreement was found. It was concluded that by coupling the classical diffusion theory and hyperbolic heat conduction approach, the proposed hybrid analytical model can be utilized to predict the transient temperature of the particle–substrate interface during the cold spray deposition process a priori before experimentation.

Spatiotemporal parallelization of an analytical heat conduction model for additive manufacturing via a hybrid OpenMP + MPI approach

The ability to do thermal simulations for entire additive manufacturing builds is a key computational problem facing the additive manufacturing community; however, complex numerical models considering multiple physical phenomena currently do not have the capacity for simulations at this scale. To this end, conduction only analytic models offer a viable approach due to the massive drop in computational expense. Here, we extend an existing implementation which uses a governing equation which can be evaluated at any point in space and time. This implementation already utilizes OpenMP with a spatial decompositions scheme stemming from a melt pool tracking algorithm. We then combine this with a parallel in time (PinT) approach to make the problem highly parallelizable. The new scheme, which uses MPI for internode communication and OpenMP for intranode communication, is shown to scale very well across multiple computational nodes. This approach results in the ability to simulate the 3D solidification conditions for entire layers of additively manufactured parts in minutes making part scale thermal simulations more practical.

Analytical heat conduction model of particle reinforced tertiary composite materials based on complete spatial randomness of fillers in base matrix and its application in the development of cryosorption pump

In this article, we propose an analytical heat conduction model within a stochastic frame work which estimates the thermal conductivity (TC) value of particle reinforced composite materials comprising of three parent elements i.e. a base matrix along with two different filler element particles randomly distributed in it. The spatial distribution of the filler particles in a sample of specific dimension has been estimated by applying bivariate Poisson distribution. This distribution is then used to arrive at the TC value of the composite. This concept has been applied to predict the TC of the tertiary composite comprised of epoxy as the base matrix, aluminium and zinc particles as filler elements. The TC values obtained from this model for different volume fractions of fillers were extensively compared with experimental results. The model is found to predict the results fairly well with less aberrations up to the total filler volume fraction of ∼20%. The developed model for TC prediction has been used in the design of high efficiency cryosorption pump where the adhesive material used is Epoxy-Aluminium -Zinc composite.

The Radial Temperature Gradient in the Gleeble® Hot-Torsion Test and Its Effect on the Interpretation of Plastic-Flow Behavior

The radial temperature gradient developed via direct-resistance heating of round-bar hot-torsion specimens in a Gleeble® machine and its effect on the interpretation of plastic-flow behavior were established using a suite of experimental, analytical, and numerical-simulation tools. Observations of the microstructure variation developed within a γ′-strengthened nickel-base superalloy were used to infer the temperature gradient as well as differences between the temperature at the outer diameter and that indicated by thermocouples welded to the surface. At temperatures of the order of 1375 K (1102 °C), the radial variation of temperature was typically ~20 K (~20 °C). Such variations were in agreement with an analytical heat-conduction model based on the balance of input thermal energy and radiation heat loss at the free surface. Using a constitutive model for LSHR, the effect of the radial temperature gradient on plastic flow during hot torsion was assessed via numerical integration of the torque as a function of radial position for such cases as well as that corresponding to a uniformly-heated sample. These calculations revealed that the torque generated in the non-uniform case is almost identical to that developed in a sample uniformly preheated to a temperature corresponding to that experienced at a fractional radial location of 0.8 in the former case.

Analytical heat conduction model of a composite material based on complete spatial randomness of filler in base matrix

In this work, we propose a novel analytical model to obtain the thermal conductivity of a composite material comprising of filler particles randomly distributed in a base matrix. The model assumes complete spatial randomness of the filler particles in the matrix and this can be described by Poisson's distribution. This concept has been applied to arrive at the thermal conductivity of a composite material with Aluminium powder as the filler in the Epoxy matrix. The results obtained by the proposed model have been compared with the experimental results as well as with the other established models such as Maxwell model, Bruggeman model, Lewis - Nielsen model, Russell model and Rule of Mixtures. This comparison reveals that the proposed model predicts the experimental results more accurately than the others. Also the proposed model is able to predict the experimental results of thermal conductivity of Aluminium-Epoxy composite over the entire temperature range from 4.2 K to 300 K. When applied to other published experimental data of similar composites, the model is found to predict the results fairly well with minimal aberrations. Due to the non-empirical nature of the proposed model, perhaps it may be useful for the prediction of other properties of composite materials involving the filler in a base matrix.

Analytical heat conduction modelling for shaped laser beams

Conduction mode laser spot welding and laser spot hardening usually employ Gaussian or top-hat-like beam modes. One main requirement of these techniques is the avoidance of overheating in the centre of the laser-material interaction zone. Process flexibility can be improved by spatially and/or temporally shaping the beam, which can enable higher process quality, robustness or speed. A desired spatial beam shape can be achieved by a suitably designed diffractive optical element. However, the prediction of a suitable beam shape for a particular process can be complex. A simplified analytical heat conduction model has been developed that can rapidly calculate the temperature field and cooling behaviour for almost any spatial and temporal beam shape. The potential and limits of the model are demonstrated and discussed by calculating and analysing temperature profiles for several cases of multi-spot welding.

Validation of a heat conduction model for finite domain, non-uniformly heated, laminate bodies

Infrared thermographic validation is shown for a closed-form analytical heat conduction model for non-uniformly heated, laminate bodies with an insulated domain boundary. Experiments were conducted by applying power to rectangular electric heaters and cooled by natural convection in air, but also apply to constant-temperature heat sources and forced convection. The model accurately represents two-dimensional laminate heat conduction behaviour giving rise to heat spreading using one-dimensional equations for the temperature distributions and heat transfer rates under steady-state and pseudo-steady-state conditions. Validation of the model with an insulated boundary (complementing previous studies with an infinite boundary) provides useful predictions of heat spreading performance and simplified temperature uniformity calculations (useful in log-mean temperature difference style heat exchanger calculations) for real laminate systems such as found in electronics heat sinks, multi-ply stovetop cookware and interface materials for supercooled salt hydrates. Computational determinations of implicit insulated boundary condition locations in measured data, required to assess model equation validation, were also demonstrated. Excellent goodness of fit was observed (both root-mean-square error and R2 values), in all cases except when the uncertainty of low temperatures measured via infrared thermography hindered the statistical significance of the model fit. The experimental validation in all other cases supports use of the model equations in design calculations and heat exchange simulations.

Thermographic validation of a novel, laminate body, analytical heat conduction model

The two-region fin model captures the heat spreading behaviour in multilayered composite bodies (i.e., laminates), heated only over a small part of their domains (finite heat source), where there is an inner layer that has a substantial capacity for heat conduction parallel to the heat exchange surface (convection cooling). This resulting heat conduction behaviour improves the overall heat transfer process when compared to heat conduction in homogeneous bodies. Long-term heat storage using supercooling salt hydrate phase change materials, stovetop cookware, and electronics cooling applications could all benefit from this kind of heat-spreading in laminates. Experiments using laminate films reclaimed from post-consumer Tetra Brik cartons were conducted with thin rectangular and circular heaters to confirm the laminate body, steady-state, heat conduction behaviour predicted by the two-region fin model. Medium to high accuracy experimental validation of the two-region fin model was achieved in Cartesian and cylindrical coordinates for forced external convection and natural convection, the latter for Cartesian only. These were conducted using constant heat flux finite heat source temperature profiles that were measured by infrared thermography. This validation is also deemed valid for constant temperature heat sources.

Real-time infrared thermography detection of magnetic nanoparticle hyperthermia in a murine model under a non-uniform field configuration

Purpose: Magnetic nanoparticle hyperthermia consists of an increase of the temperature of magnetic nanoparticles (heat centres) due to the interaction of their magnetic moments with an alternating magnetic field. In vivo experiments using this method usually use a few fibre-optic thermometers inserted in the animal body to monitor the heat deposition. As a consequence, only a few points of the 3D temperature distribution can be monitored by this invasive procedure. It is the purpose of this work to show that non-invasive infrared thermography is able to detect, in real time, magnetic nanoparticle hyperthermia as well as monitor the harmful field-induced eddy currents in a murine model with a subcutaneous tumour. This surface temperature measurement method has the potential to give information about the intratumoral temperature. Materials and methods: The non-invasive magnetic hyperthermia experiments were performed at 300 kHz in non-uniform field configuration conditions in healthy mice and murine tumour induced by sarcoma S180. A soft ferrite-based biocompatible magnetic colloid consisting of manganese–ferrite nanoparticles surface-coated with citric acid were used in the experiments, which were extensively characterised by several techniques (transmission electron microscopy (TEM), X-ray diffraction (XRD), vibrating sample magnetometer (VSM)). The amplitude of the alternating magnetic fields was obtained from measurements using an AC field probe at similar experimental conditions. The temperature measurements were obtained from an infrared thermal camera and a fibre-optic thermometer. Results: Three-minute magnetic hyperthermia experiments revealed surface temperature increase as high as 11 °K in healthy and (5 °K in S180 tumour) animals when injecting subcutaneously 2 mg of magnetic nanoparticles (86 μL of magnetic fluid), in contrast to around 1.5 °K (for healthy) and 2.5 °K (for cancerous) animals in experiments without the colloid due to field-induced eddy currents at the animal surface. The thermographic temperature measurements were found to agree with the fibre-optic measurements within a 5% error, and were associated with the skin emissivity angle of dependence in the experimental set-up. On the other hand, a 30-min magnetic nanoparticle hyperthermia revealed surface temperature increases as high as 12 °K close to the injection site, while above 2–3 cm no significant temperature increase was observed. Curiously, the intratumoral temperature, monitored by a fibre-optic sensor, was found to be almost the same as the thermal camera surface temperature after achieving an equilibrium temperature regime. From the observed isotherms at the animal surface, using an analytical heat conduction model, taking into account surface conductance, we estimate a magnetic heating power of 0.45 W/cm3 and a specific loss power (SLP) of 85 W/g for a field of the order of only 10 kA/m at the injection site region. Conclusions: The results indicate that infrared thermography may be a promising tool for both early cancer detection and for hyperthermia treatment (at least for subcutaneous tumours), since the method permits access to information about the intratumoral temperature during a real-time magnetic hyperthermia as well as to estimate the in vivo nanoparticles SLP.

Contribution to dynamic characteristics of the cutting temperature in the drilling process considering one dimension heat flow

Abstract The machining of hardened steels has always been a great challenge in metal cutting, particularly for drilling operations. Generally, drilling is the machining process that is most difficult to cool due to the tool’s geometry. The aim of this work is to determine the heat flux and the coefficient of convection in drilling using the inverse heat conduction method. Temperature was assessed during the drilling of hardened AISI H13 steel using the embedded thermocouple technique. Dry machining and two cooling/lubrication systems were used, and thermocouples were fixed at distances very close to the hole’s wall. Tests were replicated for each condition, and were carried out with new and worn drills. An analytical heat conduction model was used to calculate the temperature at tool–workpiece interface and to define the heat flux and the coefficient of convection. In all tests using new and worn out drills, the lowest temperatures and decrease of heat flux were observed using the flooded system, followed by the MQL, considering the dry condition as reference. The decrease of temperature was directly proportional to the amount of lubricant applied and was significant in the MQL system when compared to dry cutting.

Thermal contact resistance between plasma-sprayed particles and flat surfaces

Plasma-sprayed molybdenum and yttria-stabilized zirconia particles (38–63 μm diameters) were sprayed onto glass and Inconel 625 held at either room temperature or 400 °C. Samples of Inconel 625 were also preheated for 3 h, and then air-cooled to room temperature before spraying. Photographs of the splats were captured by using a fast charge-coupled device (CCD) camera. A rapid two-color pyrometer was used to collect thermal radiation from the particles during flight and spreading to follow the evolution of their temperature. The temperature evolution was used to determine the cooling rate of spreading particles. An analytical heat conduction model was developed to calculate the thermal contact resistance at the interface of the plasma-sprayed particles and the surfaces from splat cooling rates. The analysis showed that thermal contact resistance between the heated or preheated surfaces and the splats was more than an order of magnitude smaller than that on non-heated surfaces held at room temperature. Particles impacting on the heated or preheated surfaces had cooling rates that were significantly larger than those on surfaces held at room temperature, which was attributed to smaller thermal contact resistance.