Heat Conduction Equation Research Articles

A Thermo‐Mechanical Localizing Gradient Damage Model With Modified Mazars Strain

ABSTRACTThis work establishes a thermo‐mechanical coupling model for thermally induced cracks in quasi‐brittle materials within the framework of localizing gradient damage theory. The constitutive relation considering the thermal expansion effect, and the heat conduction equation integrating the volumetric strain and damage evolution, are formulated based on the Clausius–Duhem inequality to ensure thermodynamic consistency. In particular, to accommodate the tension‐compression strength asymmetry across different quasi‐brittle materials, a modified Mazars strain is proposed as the driving force for the micro equivalent strain, where the first invariant of strain tensor and an adjustable ratio of compressive strength to tensile strength () are introduced. The numerical implementation employs the generalized‐α method for time domain discretization and the staggered scheme for decoupling. Numerical simulations demonstrate a nice feasibility of the modified Mazars strain in capturing both tensile and mixed‐mode failures, and the proposed model proves capable of characterizing complex crack behaviors under thermal loading in both quasi‐static and dynamic scenarios. The quenching test simulations of ceramic plates highlight the bidirectional coupling effects on crack patterns. Unidirectional coupling models that ignore the influence of volumetric strain and damage evolution on the temperature field would underestimate the temperature gradient, resulting in larger crack spacing and fewer cracks. In the thermal shock tests of cylindrical specimens, the transition in crack patterns from spiral shear bands to an annular damage zone with increasing compression‐tension strength ratio () is precisely captured, demonstrating the generality of the proposed model for various quasi‐brittle materials with tension‐compression strength asymmetry.

A Literature Review on Numerical Simulation of Thermal Anti-Icing

This paper reviews the numerical simulation method for thermal anti-icing. Typically, the numerical study of an anti-icing system involves a coupled simulation of various physical processes: airflow, droplet flow, thin water film flow on the wall, and heat conduction within the solid wall. Airflow is commonly simulated using the Reynolds-Averaged Navier–Stokes method, while droplet flow can be modeled using either the Eulerian or Lagrangian approach. For simulating water film flow, there are three primary models: the Messinger model, the SWIM model, and the Myers model. The heat transfer process within the solid wall can be coupled with the external air/droplet and film flow using either a tight-coupling or a loose-coupling method. When simulating an electrothermal anti-icing system, methods such as the equivalent heat conductivity scheme or shell conduction method are employed to handle heat conduction in multi-layer thin walls. To improve the accuracy of thermal anti-icing simulations, additional research is still necessary, focusing on studies on rivulet flow, bead flow, and the heat convection coefficient on the system’s wall.

Machining Path Optimization of Inductively Coupled Plasma Based on Surface Heat Transfer Model.

Inductively coupled plasma (ICP), a non-contact optical processing method, has been widely used in the preparation of fused quartz. However, the thermal effect during processing inevitably affects the stability of the removal rate, reduces the processing accuracy, and restricts the further development of plasma processing. This paper analyzes the critical temperature that affects the changes in plasma removal depth, establishes a heat transfer model for plasma jet processing through simulations, derives the heat conduction equation during processing, and obtains the critical radius corresponding to the critical temperature related to the processing speed. Additionally, this work analyzes the path temperature of the grating track used in processing and obtains the path temperature variation curve. Based on the critical radius, a staggered grating track was proposed, which verified that this track can effectively control the path temperature, thereby suppressing the error caused by the thermal effect of processing. This study not only helps to gain a deeper understanding of the heat transfer process in plasma machining, but also provides a basis for achieving high-precision plasma machining path optimization schemes.

Study of Thermal Effects in Fused-Tapered Pure Passive Fibers and Signal Combiners.

This paper investigates the thermal effects in fused-tapered passive optical fibers under near-infrared absorption. The thermal effect is primarily caused by impurities, such as OH-, which absorb incident light and generate heat. Using the finite element method, the volume changes during fiber tapering were simulated, influencing power density and thermal distribution. The heat conduction equation and ray optics were employed to analyze the thermal distribution in tapered fibers and signal combiners. Results show that at 5 kW power, the temperature peak for a single fiber reaches 316.73 °C, while for bundled fibers, the temperature rises significantly as the bundle configuration increases from 7 × 1 to 61 × 1, peaking at 453.09 °C-an increase of 171.6%. Variations in tapering ratio and length also notably affect the thermal behavior. Increasing the tapering ratio from 5 to 8 results in a 52.5% temperature rise, while doubling the taper length from 25 mm to 50 mm reduces the temperature peak by 59.1%. These findings offer important insights for the design and optimization of high-power optical fiber combiners and their heat dissipation structures.

Simultaneously identifying piecewise smooth conductivity and initial value for a heat conduction equation

Simultaneously identifying piecewise smooth conductivity and initial value for a heat conduction equation

Impacts of Time‐to‐Freeze on Quality Attributes of Low‐Moisture Part‐Skim Mozzarella Cheese

ABSTRACTThis research investigated the effect of time‐to‐freeze (TTF) on key quality attributes of low‐moisture part‐skim (LMPS) Mozzarella cheese, including browning, free oil formation and meltability. Thin slabs (4‐mm) of LMPS Mozzarella were frozen at four different rates: in a still air freezer (−22°C), in a non‐isothermal blast freezer (−30°C), in a temperature‐controlled bath (−30°C), and in an ethanol‐dry ice bath (−70°C), with TTF ranging from 0.4 to 95 min. Numerical simulations of the transient heat conduction equation provided temperature profiles that were compared with the experimental data for all freezing methods. Frozen samples were stored at −22°C for 15–18 h, then thawed and analyzed for browning, free oil formation, and meltability using spectrophotometry and modified Schreiber tests. TTF variations did not significantly affect the quality attributes, but samples with a TTF of 95 min had higher levels of free surface water compared to unfrozen cheese. The formation of free surface oil increased significantly after TTF durations of 1.2, 19, and 95 min compared to unfrozen samples. Surface browning decreased after 95 min, likely due to the higher free water content. Rapid freezing methods, such as the ethanol‐dry ice bath (−70°C), are effective for quick freezing, but the still air freezer yielded better quality cheese. Cheese frozen in a still air freezer exhibited less redness (a*) and more lightness (L*), indicating reduced surface browning. Additionally, it retained more free surface water, enhancing the structural integrity and overall quality of LMPS Mozzarella.

Range and accuracy of in-plane anisotropic thermal conductivity measurement using the laser-based Ångstrom method.

High heat fluxes in electronic devices must be effectively dissipated to prevent local hotspots, which are critical for long-term device reliability. In particular, advanced semiconductor packaging trends toward thin form factor products increase the need for understanding and improving in-plane conduction heat spreading in anisotropic materials. The 2D laser-based Ångstrom method, an extension of traditional Ångstrom and lock-in thermography techniques, measures in-plane thermal properties of anisotropic sheet-like materials. This method uses non-contact infrared temperature mapping to measure the thermal response to periodic laser heating at the center of a suspended sample. The spatiotemporal temperature data are analyzed via an inverse fitting algorithm to extract thermal conductivities in the in-plane orthotropic directions that best adhere to the governing heat conduction equation. Using this algorithm, we present an approach to simultaneously fit data across multiple heating frequencies, which improves measurement sensitivity because the thermal penetration depth varies with frequency. The accuracy of this technique is assessed by tuning experimental parameters such as sample dimensions and heating frequency. A standardized workflow is proposed for measuring unknown materials and for processing the data, including filtering out regions influenced by laser absorption and heat sink boundary effects. Numerical simulations validate the method across a wide range of thermal conductivities (0.1-1000W m-1 K-1) and material thicknesses (0.1-10mm), with accuracy demonstrated for anisotropy ratios up to 1000:1. Experimental measurements on isotropic and anisotropic materials agree well with the benchmark values. Ultimately, standardization of this technique supports the development of engineered anisotropic heat-spreading materials for thermal management and packaging applications.

Stability Analysis of Gravity‐Modulated Thermohaline Convection in Viscoelastic Fluid Saturating Porous Medium

ABSTRACTThis paper investigates the modified stability analysis of thermohaline convection in a horizontal Oldroyd‐B fluid using the Darcy model, with applications in geophysics, environmental engineering, material science, and biological systems. The basic equations of motion and heat conduction in the present model are based on the Darcy model and the modified Boussinesq approximation. The perturbation equations are derived using the power series perturbation method to study two‐dimensional convective roll instability with finite amplitude disturbances. Linear stability analysis, as a first‐order stability problem, provides expressions for the Rayleigh numbers for stationary and oscillatory convection. The influence of various parameters, including the Darcy–Prandtl number, solutal Rayleigh number, stress and strain retardation parameters, and variations in the coefficient of specific heat, on the stability of the considered system, is studied numerically. It is observed that an approximate 67% increase in the coefficient of specific heat variations and a 23% increase in the solutal Rayleigh number result on average increases of 80% and 8%, respectively, in the value of the oscillatory Darcy–Rayleigh number. Conversely, an average 42% increase in the Darcy–Prandtl number leads to an average decrease of 8% in the oscillatory Darcy–Rayleigh number. In the weakly nonlinear oscillatory analysis, second‐ and third‐order stability problems are discussed, and the Landau equation is derived, which describes the amplitudes of the convection cells. The effects of the parameters on heat and mass transfer rates, as described by the Nusselt and Sherwood numbers, are studied numerically. Furthermore, the weakly nonlinear stability analysis evaluates the effects of the Darcy–Prandtl number, specific heat coefficient, stress relaxation time, salinity Rayleigh number, strain retardation time, diffusivity ratio, and gravity modulation characteristics on heat and mass transfer rates. The effects of amplitude and frequency of gravity modulation on heat and mass transport are analyzed and depicted graphically. The study establishes that heat and mass transport can be effectively controlled by a mechanism external to the system. The results are validated by comparison with previous work, which considered a special case without the velocity gradient term in the momentum equation.

Evaluation of Spatial Distribution of Crystallinity Induced by Local Heating Using Low-Frequency Raman Spectroscopy on Polyether Ether Ketone (PEEK)

Local heating was performed on a thermoplastic polymer film by contact with the tip of a soldering iron heated above the glass-transition temperature. The locally heated area was measured using microscopic Raman scattering spectroscopy, and the spatial distribution of the crystallinity was obtained from the low-frequency peak. The crystallinity distribution can be evaluated using the microscale spatial resolution. The temperature distribution around the locally heated area was calculated by applying the heat conduction equation, and good correspondence was obtained with the obtained crystallinity.

ON THE EXISTENCE OF OPTIMAL CONTROL FOR A SEMILINEAR EVOLUTION EQUATION WITH AN UNBOUNDED OPERATOR

The paper studies the problem of optimal control for an abstract firstorder semilinear differential equation in a Hilbert space, with an unbounded operator and a control linearly entering the righthand side. The objective functional is assumed to be additively separable with respect to the state and control, with a fairly general dependence on the state. A theorem on the existence of an optimal control is proved for this problem, and properties of the set of optimal controls are established. Due to the nonlinearity of the equation under study, the author further develops previous results on total preservation of unique global solvability and solution estimates for similar equations. This estimate proves essential for the investigation. As examples, a nonlinear heat conduction equation and a nonlinear wave equation are considered.

ON AN APPROACH TO IMPROVING THE ACCURACY AND EFFICIENCY OF THE ROMB METHOD FOR SOLVING THE UNSTEADY HEAT CONDUCTION EQUATION

A modification of the ROMB method is presented for numerically solving unsteady nonlinear heat conduction equations. The modified difference schemes are conservative and, for smooth solutions, approximate the model differential equations with constant coefficients with second-order accuracy in both time and space. First differential approximations are provided for the modified difference schemes, enabling the assessment of approximation errors in the schemes.

Calculation of closed combustion performance based on evolution of gun propellant burning surface

Abstract In order to enhance the combustion performance evaluation ability of irregularly shaped gun propellants and accelerate the development of propellant energy-release control technology, the present work provides the calculation method of closed combustion performance based on the evolution of gun propellant burning surface. The 3-D structure of the gun propellant should be built, and then a steady-state heat conduction equation with a source term is set, which can be considered as the Poisson equation with a diffusion term. The variable in the equation is the burned web thickness of the propellant grain. Afterwards, finite element meshing was performed on the 3-D structure to obtain a dynamic database of equivalent surfaces with burned web size. The pressure during the closed combustion process can be obtained from the mass conservation equation as a function of the burned web thickness. Furthermore, the combustion time field can be derived based on the functional relationship between burning-rate and pressure. Ultimately, the variation of gas pressure versus time in the closed bomb could be obtained. The results showed that the burnout time of the tubular propellant was 1.6 ms shorter than that of the inner pore offset 0.5 mm under the design conditions. This research can provide researchers with a new method for calculating the closed combustion performance of irregular gun propellants, which can be applied to the design of propellant shapes and the study of launch safety of propellant charges.

Conservative numerical algorithm for simulating thermoelectrical semiconductor device with unconditional optimal convergence analysis

We propose conservative type numerical method to simulate thermoelectrical semiconductor device problem, in which mixed finite element method used for electric potential equation, conservative characteristic finite element method for electron and hole concentration equations, and standard finite element method for heat conduction equation. By temporal-spatial error splitting argument, the optimal error estimates without certain time step restriction are derived, and low order convergence rate of electrostatic potential and electric field intensity will not affect the accuracy of the electron, hole density and temperature. Numerical tests are performed to validate the theoretical results and application performance of the given method.

Heat Transfer Efficiency While Cooling with a Water Spray, Air-Assisted Water Spray and Water Jet Under Boiling and Single-Phase Forced Convection Conditions

The main purpose of this paper was to determine and compare the boundary conditions of heat transfer on the cooled surface of a cylindrical sensor made of Inconel 600 alloy while cooling with a water jet, water spray and air-assisted water spray under high-temperature conditions. The inverse method for the heat conduction equation was used to determine the boundary conditions. Experimental tests were carried out, including temperature measurements at several points inside the cylinder while cooling with all the tested systems from a temperature of 900 °C for three values of water pressure: 0.05 MPa, 0.1 MPa and 0.2 MPa. Temperature measurements were used as the input data to identify the heat transfer boundary conditions. The temperature field of the axially symmetric sensor was determined using the finite element method. The boundary conditions were determined as average values of the heat transfer coefficient and heat flux and local values of the heat transfer coefficient. A comparison of the amount of thermal energy dissipating from the sensor surface as a result of boiling and a forced single-phase convection is also presented in the paper. The highest uniformity of cooling was obtained during air-assisted water spray-cooling.

Two-phase heat conduction problems with Sturm-type boundary conditions

In this work, the solution is justified using the method of separation of variables for initial-boundary value problems of the heat conduction equation with piecewise constant thermal conductivity coefficient under Sturm-type boundary conditions (separated boundary conditions). Various cases are considered. A system of eigenfunctions of the corresponding spectral problem is constructed and it is proved that this system of eigenfunctions forms a Riesz basis. The theorem of the existence of a unique classical solution to the problem is proved.

A Physics Informed Neural Network for Solving the Inverse Heat Transfer Problem in Gas Turbine Rotating Cavities

Abstract Recently, Physics-Informed Neural Networks have displayed great potential in delivering swift and accurate solutions for inverse problems. Here, a Physics-Informed Neural Network (PINN) was used to solve the inverse heat conduction problem in the rotating cavities of a aeroengine high-pressure compressor internal air system. The neural network was designed to receive experimentally captured radially distributed temperature profiles as inputs and predict the associated surface heat fluxes. The correctness of these predicted heat fluxes are assessed by numerically solving the direct heat conduction equation using a 2D Finite-Element model, thereby recovering the original temperature profiles. A comparative analysis is conducted between the predicted temperature profiles and the initial inputs. The physics informed neural network is trained using noise free synthetic data, created from a range of radial temperature curve fit coefficients, and subsequently tested on noisy experimental data at engine representative conditions. The predicted temperature values exhibit some good agreement with their respective actual counterparts. Furthermore, the sensitivity of model hyperparameters are explored to showcase the capability of the proposed approach. The results show that Physics- Informed Neural Networks exhibit reduced susceptibility to experimental uncertainties when addressing inverse problems, in contrast to inverse solution methods, and offer a possible new approach for analysis of experimental data if trained on a sufficiently large dataset.

Heat Transfer Characteristics of Cryogenically Frozen Kulfi (A Dairy Dessert)

Kulfi was frozen from concentrated milk as well as from reconstituted dry mix with solids content of 60%. The convective heat transfer coefficients during freezing of kulfi by deep freeze and cryogenic methods were determined using one-dimensional transient heat conduction equation. The heat transfer coefficient for cryogenic freezing of kulfi from milk and dry mix concentrates was 210.57 W m-2K-1 and 216.84 W m-2K-1, respectively when compared to 8.33 W m-2K-1 for conventional deep-frozen kulfi. Freezing was achieved in 267 min in deep freeze method, while it reduced to just 185-190 s under cryogenic conditions. The freezing time was predicted using empirical equations, and the prediction accuracies were compared. Cryogenic freezing was nearly 87.66 times faster than deep freezing, and Pham’s model best-predicted the freezing time. Organoleptic evaluation using fuzzy-logic revealed that the order of ranking of quality attributes of kulfi was body and texture (highly important) > melting characteristics (highly important) > flavour and taste (highly important) > colour and appearance (important), and cryogenically frozen kulfi was statistically (p<0.05) superior to conventionally frozen kulfi.

ON THE HEAT CONDUCTION EQUATION AND THE HEAT DISSIPATION FUNCTION IN ANISOTROPIC REACTING FLUID MIXTURES WITH MAGNETIC RELAXATION

In some previous papers, within the framework of the thermody namics of irreversible processes with internal variables, a linear theory for magnetic relaxation phenomena in anisotropic mixtures, consisting of n reacting fluid components, was developed. In particular, assuming that the macroscopic magnetization m can be split in two irreversible parts m = m(0) + m(1) a generalized Snoek equation was derived. In this paper we derive for these reacting anisotropic mixtures the heat conduction equation. We show that the heat dissipation function is due to the chemical reactions, the magnetic relaxation, the electric conduc tion, the viscous, magnetic, temperature fields and the diffusion and the concentrations of the n fluid components. Also, the Snoek and De Groot special cases are studied. The obtained results find applications in nuclear resonance, in biology, in medicine and other fields, where different species of molecules have different magnetic susceptibilities and relaxation times and contribute to the total magnetization.

Crystallization and crystal morphology of polymers: A multiphase-field study

In this paper, we introduce a coarse-grained model of polymer crystallization using a multiphase-field approach. The model combines a multiphase-field method, Nakamura’s kinetic equation, and the equation of heat conduction for studying microstructural evolution of crystallization under isothermal and non-isothermal conditions. The multiphase-field method provides flexibility in adding any number of phases with different properties making the model effective in studying blends or composite materials. We apply our model to systems of neat PA6 and study the impact of initial distribution of crystalline grains and cooling rate on the morphology of the system. The relative crystallinity (conversion) curves show qualitative agreement with experimental data. We also investigate the impact of including carbon fibers on the crystallization and grain morphology. We observe a more homogeneous crystal morphology around fibers. This is associated with the higher initial volume fraction of crystal grains and higher heat conductivity of the fiber (compared to the polymer matrix). Additionally, we observe that the crystalline grains at the fiber surface grow perpendicular to the surface. This indicates that the vertical growth observed in experiments is merely due to geometrical constraints imposed by the fiber surface and neighbouring crystalline regions.

A new method for temperature field characterization of microsystems based on transient thermal simulation

Microsystems face challenges such as high heat flux density and localized hot spots in the temperature field, significantly impacting their thermal reliability. Accurately and comprehensively characterizing the temperature field is a challenging problem in current research. We present a general high-order finite difference (GHOFD) algorithm for the high-accuracy numerical solution of the two-dimensional transient heat conduction equations (THCEs). The 10th-order GHOFD algorithm is accurate up to 10−7 °C. Secondly, we present a viable approach for characterizing microsystems' steady-state and transient heat conduction mechanisms. We introduce two new characterization parameters: the gradient modulus and the heat flux direction factor (HFDF). The gradient modulus can more clearly characterize the magnitude of the gradient vector and quantitatively analyze the spatial position of the temperature field change in the microsystem. The HFDF can dynamically display the heat conduction process in the temperature field. Finally, using temperature field simulation and microsystem characterization, we have validated the effectiveness of the proposed method and new parameters.