Nonlinear Heat Transfer Equations Research Articles

Heat and mass transfer analysis of s-PTT nanofluid in microchannels under combined electroosmotic and pressure-driven flows with wall slip using the homotopy perturbation method

The heat and mass transfer of the electroosmotic flow in microchannel transporting viscoelastic nanofluid is investigated considering Brownian motion of nanoparticles and slip boundary conditions. The simplified Phan-Thien-Tanner model is employed to describe the rheological behavior of fluid and the nonlinear Navier model with non-zero slip critical shear stress is considered at walls. The governing nonlinear momentum, mass, and heat transfer equations are solved using the Homotopy Perturbation Method. The study reveals that increasing the fluid elasticity, nanoparticle concentration, and size significantly enhances the flow rate, heat and mass transfer. Additionally, elasticity and Reynolds number decrease the friction factor. Reducing the double-layer thickness and increasing the Reynolds number lead to higher flow rates and fluid velocities. Notably, the findings emphasize the critical role of the slip conditions on the Sherwood and Nusselt numbers.

Towards a theory of steady-state solidification process with a quasi-equilibrium two-phase region

AbstractThe process of directional crystallization in the presence of a quasi-equilibrium two-phase region located between the solid material and the liquid phase is studied theoretically. The mathematical model of the process is based on heat and mass transfer equations in the solid, liquid and two-phase regions, as well as boundary conditions at the phase interfaces “solid phase” – “two-phase region” and “two-phase region” – “liquid phase”, which are moving with a constant velocity. The process of directional crystallization is given by fixed temperature gradients in the solid and liquid phases, which determine a constant velocity of melt solidification. An exact analytical solution of the nonlinear problem with two moving boundaries of phase transformation is obtained, which is based on the transition to a new independent variable, the solid phase fraction, when integrating the nonlinear heat and mass transfer equations in the two-phase region. As a result of solving the problem, the distributions of temperature and concentration of dissolved impurity, the solid phase fraction in a two-phase region, the laws and velocities of motion of its interphase boundaries are determined. It is analytically shown that the impurity concentration and temperature in the two-phase region are only the functions of solid phase fraction, which, in turn, depends on the spatial coordinate. Analysis of the obtained solutions shows that the solid phase fraction in a two-phase region can be both a decreasing and increasing function of the spatial coordinate, which is directed from the solid material to the melt. This determines the internal structure of two-phase region, its permeability, average interdendritic spacing, distribution of dissolved impurity, crystallization velocity and laws of two-phase region boundaries.

Thermosonication process of sour cherries: Microbial inactivation kinetics, experimental and computational fluid dynamics simulation

In this study, the inactivation of Listeria monocytogenes PTCC 1302 and Shigella sonnei PTCC 1974 of sour cherries by thermosonication (37 kHz, 300 W; 50and 60°C; 0–12 min) was evaluated. Several statistical models including polynomial, Weibull, and biphasic linear models were implemented to obtain the variation of microorganism population through time. The coupled non-linear Westervelt and heat transfer equations were solved to determine the acoustic pressure and thermal fields using computational fluid dynamics (CFD). According to the obtained results, average inactivation improvements of 15.47 and 17.13% were obtained when the temperature level was increased for both S. sonnei and L. monocytogenes, respectively. S. sonnei was more affected by the sonication procedure at the initial stages, while similar behavior was also found for L. monocytogenes but at the final stages. The maximum and minimum pressure level was obtained at ∼380 and −230 kPa, respectively. In the beginning, the hot spot was formed near the bottom of the simulated sour cherry, where the acoustic intensity was higher. Although, with the elapse of time, the temperature field became more uniform, and the hot spot moved toward the center of the sour cherry.

GRID-STEP OPTIMIZATION OF THE PARAMETERS OF THERMAL IMPACT ON THE TECHNICAL SYSTEM

The article solves applied problems of optimizing the parameters of technical systems under the influence of concentrated, discrete sources of thermal load. To improve the accuracy and speed of solving applied optimization problems, the authors propose a generalized procedure for finding the optimal thermal load parameters. This procedure consists of well-known computational methods for solving boundary value problems and ensuring the optimization of the objective function and its parameters. To calculate the values of the temperature field, it is necessary to solve a nonlocal boundary value problem of a system of nonlinear differential heat transfer equations. In order to find the conditions for the correctness of boundary value problems, the authors used the theory of pseudo-differential operators in the space of generalized slow power functions. This made it possible to guarantee the correctness of computational and applied optimization mathematical models. The procedure for grid-step optimization of the parameters of thermal action on a multilayer material proposed by the authors is based on a grid approach with discretization of the main parameters of thermal load and grinding of a large uniform grid in the vicinity of the selected node of the large grid. The directed search for local extremes of the mesh model is carried out by the nodes of the crushed mesh. The search for local extrema is performed by selecting the next node of a large grid. This approach will increase the accuracy of solving applied optimization problems by allowing further mesh refinement and refinement of the values of local extremes, and, consequently, refinement of the values of the global extremum of the objective function. The optimization of the thermal impact parameters was carried out according to the criterion of reducing damage to the test material. This makes it possible to increase the accuracy of the technological process of thermal action on a multilayer material.

A new solution to a weakly non-linear heat conduction equation in a spherical droplet: Basic idea and applications

A new analytical solution to a non-linear heat transfer equation in a spherically-symmetric droplet is suggested. All thermophysical properties inside the droplet are considered to be close to their average values. This allows us to consider the non-linearity of this equation as weak. The solution is presented as T=T0+T1, where T0 is the solution to a linear heat conduction equation, and T1≪T0. The equation for T1 is presented as a linear heat conduction equation with a source term depending on the distribution of T0 and its spatial derivatives inside the droplet. The latter equation is solved analytically alongside the linear equation for T0, and the final solution is presented as T=T0+T1. The predictions of the numerical code in which this solution was implemented are verified based on a comparison of those predictions with the predictions of COMSOL Multiphysics code using input parameter values that are typical for nanofluid (water and SiO2 nanoparticles) droplet evaporation in atmospheric conditions. It is demonstrated that for these experiments T1≪T0 which justifies the applicability of the linear heat conduction equation used for the analysis of this process. Small differences in the temperatures predicted by both non-linear and linear models lead to a much more noticeable difference in integral characteristics such as time before the start of the formation of the cenosphere when the mass fraction of nanoparticles at the droplet surface reaches about 40%.

Flexural buckling and post-buckling analysis of tapered columns in transient fire

This paper presents time-varying flexural buckling and post-buckling responses of tapered columns under a transient fire of which the temperature is non-uniform along the length. First, the transient temperature field within the column is determined by solving a dimensionally reduced transient heat transfer problem considering the convection-radiation boundary conditions and temperature-dependent (TD) material properties. Subsequently, the von Kármán geometric nonlinear theory is applied to obtain the nonlinear equilibrium equation. The nonlinear heat transfer and equilibrium equations are solved by the differential quadrature (DQ) method. According to a yield criterion, the ultimate load at which the material failure occurs is predicted. The influences of a transient localised fire on the temperatures, buckling and post-buckling responses of tapered steel tubular columns are discussed.

Iterative Solutions for the Nonlinear Heat Transfer Equation of a Convective-Radiative Annular Fin with Power Law Temperature-Dependent Thermal Properties

The temperature distribution in a conductive-radiative rectangular profiled annular fin with internal heat generation is scrutinized in the present investigation. The nonlinear variation of thermal conductivity and heat transfer coefficient governed by the power law is considered. The analytical approximation for the non-dimensional temperature profile is obtained using the differential transform method (DTM)-Pade approximant. The nondimensionalization of the governing energy equation using dimensionless terms yields a nonlinear ordinary differential equation (ODE) with corresponding boundary conditions. The resulting ODE is analytically solved with the assistance of the DTM-Pade approximant procedure. Furthermore, the impact of thermal parameters on the temperature field and thermal stress is elaborated with graphs. The important results of the report divulge that temperature distribution greatly enhances with an augmentation of the heat generation parameter, but it gradually reduces with an increment in the magnitude of the thermogeometric and radiative-conductive parameter.

Fast heat transfer simulation for laser powder bed fusion

Accurate and fast modeling of the temperature distribution and phase transitions in laser powder bed fusion is a major milestone in achieving its quality assurance. Commonly referred to as digital twin technology, the goal is to find agile, fast-to-compute but also sufficiently accurate simulators that can replicate the 3D printing process while enhancing the quality of its outcomes. In this work, we propose a surrogate model for the nonlinear heat transfer equation coupled with subspace projection and randomized sketching, that exploits the accuracy and explainability of finite element time-domain simulation with the computational efficiency of Monte Carlo sampling, applied to the modality of laser powder bed fusion. Focusing on tackling the high-dimensionality imparted from the finite element approximation and the nonlinearity in the governing equations, our surrogate relies on low-dimensional projection with subspace selection and subsequently sub-samples the Picard iterations utilized to solve the projected non-linear system of equations. The projection bases are generated in the process of simulation by combining previous temperature profiles and locally deployed anisotropic Gaussian functions, while the sketching process utilizes efficient sampling without replacement based on approximate optimal sampling distributions. Both the projection and the sketching are designed to implement alongside the printing process, which makes the proposed surrogate capable of handling different process parameters without requiring prior computations offline. A series of numerical experiments are presented to validate the surrogate’s accuracy and reduction in compute time compared to high-fidelity finite element simulations. Although the achieved speed-up can be as a high ten, computational times are still orders of magnitude away from what would be required for real-time computations. The presented methodology allows to handle different printing attributes (laser power and scan speed) and arbitrary thermal conductivity anisotropy.

Thermo-mechanical buckling analysis of non-uniformly heated rectangular plates with temperature-dependent material properties

This paper examines the thermo-mechanical buckling behavior of a rectangular plate with temperature-dependent (TD) material properties exposed to non-uniform heating. The buckling analysis was performed by using the differential quadrature (DQ) method, which mainly includes three steps: (i) a two-dimensional (2D) non-linear heat transfer equation was proposed to obtain the temperature distribution in the plate considering the convective–radiative conditions on the plate surface; (ii) the in-plane stress distribution of the plate was obtained by solving the thermoelastic equilibrium equation; (iii) the critical load was determined by solving the stability equation based on the stress distribution in the previous step. The comparison study shows that the present results were in good agreement with the finite element (FE) solutions and the results reported in the literature. The effects of the surrounding temperature fields and in-plane load cases on the thermo-mechanical behavior of the plate are analyzed in detail.

Расчет температур стенок жаровой трубы камеры сгорания прямоточного воздушно-реактивного двигателя при завесном охлаждении

Along with the turbojet engines, cruise missiles are using the ramjet engines, Features of the ramjet rockets use in most cases are characterized with the combustion chamber operating for significant time at the maximum heating, which makes it a rather heat-loaded unit that requires solving complex issues of the heat resistance. Therefore, calculations are relevant to determine thermal state of the combustion chamber elements, which makes the basis for making rational design decisions. This paper determines temperatures of the burner liner wall of the ramjet engine main combustion chamber affected by combustion products and cooling air. Certain similarity in the design of the burner liners of turbojet and ramjet engine on liquid fuel made it possible to adapt calculation methods worked out for a turbojet engine to the ramjet engine combustion chambers. Main sections of the combustion chamber and their characteristic features were identified. A system of non-linear heat transfer equations was solved using empirical coefficients for both cooled and uncooled sections. Graphs of temperature distribution in sections of the burner liner along the entire length of the combustion chamber were plotted. Qualitative analysis of the obtained results showed that the applied method could be used to determine the thermal state of the burner liner walls as the first approximation, without taking into account local design features, but using only the aerial vehicle flight conditions and the liquid fuel parameters.

Convective-radiative moving porous fin with temperature-dependent thermal conductivity, heat transfer coefficient and wavelength-dependent surface emissivity

PurposeIn this paper, temperature distribution and fin efficiency in a moving porous fin have been discussed. The heat transfer equation is formulated by using Darcy's model. Heat transfer coefficient and thermal conductivity vary with temperature. The surface emissivity of the fin varies with temperature as well as with wavelength. Thermal conductivity is taken as a linear and quadratic form of temperature. The entire analysis of the paper is presented in non-dimensional form.Design/methodology/approachIn this study, a new mathematical model is investigated. The novelty of this model is surface emissivity which is considered temperature and wavelength dependent. Another interesting point is the addition of porous material. The Legendre wavelet collocation method has been used to solve the nonlinear heat transfer equation. Numerical simulations are carried out in MATLAB software.FindingsAn attempt has been made to discuss temperature distribution in the presence of porosity and wavelength-temperature-dependent surface emissivity. The effect of various parameters on temperature has been discussed, including thermal conductivity, emissivity, convection-radiation, Peclet number, sink temperature, exponent “n” and porosity. Fin efficiency is also calculated for some parameters. According to the study, heat transfer rate increases with higher radiation-convection, emissivity, wavelength and porosity parameters.Originality/valueThe numerical results are carried out by using the Legendre wavelet collocation method, which has been compared with exact results in a particular case and found to be in good agreement. The percent error is calculated to find the error between the current method and the exact result. A comparison of the obtained results with the previous data is presented to validate the numerical results.

Buckling and post-buckling analysis of restrained non-uniform columns in fire

This paper investigates the buckling and post-buckling behavior of restrained non-uniform columns in fire where the temperature surrounding the column varies along the longitudinal direction. The material properties of the column are considered to be temperature-dependent (TD). First, according to the convective–radiative condition on the surfaces of the column, a dimensionally reduced nonlinear heat transfer equation is proposed and is solved by the differential quadrature (DQ) method. Then, considering the geometric imperfection, the nonlinear governing equation is derived based on the von-Kármán theory when the column is subjected to an axial compression load at the column end. An improved DQ method is applied to the buckling and post-buckling analysis of the non-uniform column under the fire. Considering the yield strength of material at elevated temperature, the ultimate load of the column is obtained. A tapered H-section steel column surrounded by localized fire is taken as an example. The present results are verified by comparing with those in the literature and those obtained from the finite element (FE) method. The parameter analysis reveals the temperature distribution, the buckling and post-buckling behavior of the tapered column in the localized fire.

Significance of Convection and Internal Heat Generation on the Thermal Distribution of a Porous Dovetail Fin with Radiative Heat Transfer by Spectral Collocation Method.

A variety of methodologies have been used to explore heat transport enhancement, and the fin approach to inspect heat transfer characteristics is one such effective method. In a broad range of industrial applications, including heat exchangers and microchannel heat sinks, fins are often employed to improve heat transfer. Encouraged by this feature, the present research is concerned with the temperature distribution caused by convective and radiative mechanisms in an internal heat-generating porous longitudinal dovetail fin (DF). The Darcy formulation is considered for analyzing the velocity of the fluid passing through the fin, and the Rosseland approximation determines the radiation heat flux. The heat transfer problem of an inverted trapezoidal (dovetail) fin is governed by a second-order ordinary differential equation (ODE), and to simplify it to a dimensionless form, nondimensional terms are utilized. The generated ODE is numerically solved using the spectral collocation method (SCM) via a local linearization approach. The effect of different physical attributes on the dimensionless thermal field and heat flux is graphically illustrated. As a result, the temperature in the dovetail fin transmits in a decreasing manner for growing values of the porosity parameter. For elevated values of heat generation and the radiation-conduction parameter, the thermal profile of the fin displays increasing behavior, whereas an increment in the convection-conduction parameter downsizes the thermal dispersal. It is found that the SCM technique is very effective and more conveniently handles the nonlinear heat transfer equation. Furthermore, the temperature field results from the SCM-based solution are in very close accordance with the outcomes published in the literature.

Influence of Convective and Radiative Cooling on Heat Transfer for a Thin Wire with Temperature-Dependent Thermal Conductivity

In this study, a numerical prediction of temperature profiles in a thin wire exposed to convective, radiative and temperature-dependent thermal conductivity is carried out using a finite-difference linearization approach. The procedure involves a numerical solution of a one-dimensional nonlinear unsteady heat transfer equation with specified boundary and initial conditions. The resulting system of nonlinear equations is solved with the Newton-Raphson’s technique. However unlike the traditional approach involving an initial discretization in space then in time, a different numerical paradigm involving an Euler scheme temporal discretization is applied followed by a spatial discretization. Appropriate numerical technique involving partial derivatives are devised to handle a squared gradient nonlinear term which plays a key role in the formulation of the Jacobian matrix. Tests on the numerical results obtained herein confirm the validity of the formulation.

Distributed control of nonlinear conductivity heat transfer equation in a thick functionally graded plate

In this paper, distributed control of nonhomogeneous nonlinear conductivity heat transfer equation on a single plate is studied. The inhomogeneity of the equation is due to the use of functionally graded materials. In the proposed method, the governing equation of the system, which is a nonlinear partial differential equation, is represented as some linear partial differential equation using the Takagi-Sugeno linearization technique; so that their fuzzy combination accurately retrieve the original nonlinear equation. This process is done to prevent the phenomenon of frequency disorder or aliasing. Then, the modified wavenumber method is utilized to convert the obtained linear partial differential equations to a set of ordinary differential equations. In this way, a combination of fast Fourier transform and finite difference methods is implemented. Consequently, by combining Takagi-Sugeno and the modified wavenumber strategies, several ordinary differential equations are obtained for each grid point. In this regard, the parallel distributed compensation method is used to stabilize each grid point. The second method of Lyapunov is implemented to study the stability of the proposed control strategy. Numerical Simulations have been used to demonstrate the efficacy of the proposed control scheme and also its relative robustness with respect to measurement noises and parametric uncertainties.

Differential Transformation Method for Solving Nonlinear Heat Transfer Equations

In this paper, Differential Transformation Method (DTM) is applied for solving the nonlinear differential equations arising in the field of heat transfer. The solution is considered as an infinite series expansion where converges rapidly to the exact solution. The nonlinear convective–radioactive cooling equation and nonlinear equation of conduction heat transfer with the variable physical properties are chosen as illustrative examples and solutions have been found for each case. Results by DTM with other results calculated by Homotopy Perturbation Method are compatible.

Tangent stiffness for nonlinear heat transfer and its application to additive manufacturing

Prediction-correction method is commonly employed in the solution of nonlinear heat transfer equation with temperature dependent thermophysical parameters arising in various engineering problems such as additive manufacturing of metal components. However, the stiffness of such conventional method is secant rather than tangent and thus it only results in suboptimal rate of convergence in Newton's iteration. In this paper, tangent stiffness for nonlinear heat transfer is proposed through the linearization of the weak form and an additional term presents which disappears in the conventional prediction-correction method. Numerical results show that the solution with the proposed tangent stiffness converges much faster than that with secant stiffness and approximately only half of the iteration steps are needed. Especially, in the case that the thermal conductivity varies dramatically with the temperature, the conventional method diverges whereas the tangent stiffness still shows quite good convergence. Application of the proposed method to the modeling of heat transfer in additive manufacturing (AM) process is presented and the obtained temperature results agree quite well with the existing experimental studies. It is worth mentioning that almost half of CPU time is saved due to the application of the proposed tangent stiffness. This saving is of great significance for modeling the AM process of complicated components, which is usually very time-consuming.

Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient

A new approach to the study of multidimensional singularly perturbed problems of nonlinear heat conduction is proposed, based on the further development and use of asymptotic analysis methods. We study the question of the existence of classical Lyapunov stable stationary solutions with boundary and internal transition layers (stationary thermal structures) of the nonlinear heat transfer equation. We suggest the efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use the principle of comparison.We consider the application of the results of asymptotic analysis to solving inverse problem of reconstructing the temperature dependence of the thermal conductivity coefficient from a known position of the internal layer of thermal structure.

Modelling of Applied Magnetic Field and Thermal Radiations Due to the Stretching of Cylinder

In this study, a numerical approach was adopted in order to explore the analysis of magneto fluid in the presence of thermal radiation combined with mixed convective and slip conditions. Using the similarity transformation, the axisymmetric three-dimensional boundary layer equations were reduced to a self-similar form. The shooting technique, combined with the Range–Kutta–Fehlberg method, was used to solve the resulting coupled nonlinear momentum and heat transfer equations numerically. When physically interpreting the data, some important observations were made. The novelty of the present study lies in finding help to control the rate of heat transfer and fluid velocity in any industrial manufacturing processes (such as the cooling of metallic plates). The numerical results revealed that the Nusselt number decrease for larger Prandtl number, curvature, and convective parameters. At the same time, the skin friction coefficient was enhanced with an increase in both slip velocity and convective parameter. The effect of emerging physical parameters on velocity and temperature profiles for a nonlinear stretching cylinder has been thoroughly studied and analyzed using plotted graphs and tables.

Holistic analysis and optimization of distributed energy system considering different transport characteristics of multi-energy and component efficiency variation

Different natures and nonlinear transport characteristics of electric, thermal and gas energies make it so complex for analyzing distributed energy system (DES). Many researches usually ignored nonlinear heat transfer equations of heat exchange devices and assumed constant efficiencies of energy conversion devices to construct a simplified electric-analogy model of DES. This contribution develops a heat current model of thermal system to consider heat transfer constraints and their influences on energy conversion efficiencies, applies a linearized DistFlow model to describe electric power flow in radial distribution power grid, and then offers a holistic multi-energy transport model of whole DES. On this basis, a hierarchical iteration method is developed for system operation optimization under different working conditions. Results show that the proposed method fully reflects the influences of energy transport constraints on efficiencies of energy conversion devices and further the optimal operation strategy of DES. Specifically, under the consideration of time-of-use electric tariff, the heat generation rate of heat pump has a considerable decrease at 4:00 compared with the results without considering efficiencies variations and power transmission constraints. Besides, the total released heat of heat storage device from 10:00 to 18:00 has a 36.53% reduction if ignoring solar collector efficiency variations.