One-dimensional Heat Transfer Problem Research Articles

Propagation regimes of buoyancy-driven hydraulic fractures with solidification

This study investigates the propagation of a semi-infinite buoyancy-driven hydraulic fracture in situations when the fluid is able to solidify along the crack walls. Such problems occur when hot magma ascends from a chamber due to buoyancy forces and solidifies by interacting with colder rock. In the model, the solidification rate is calculated assuming a one-dimensional heat transfer problem, in which case it becomes mathematically equivalent to Carter’s leak-off model, which is commonly used to describe the fluid leak-off from a hydraulic fracture into a porous rock formation. In order to construct a mathematical model for a buoyancy-driven hydraulic fracture with solidification, the aforementioned thermal problem is combined with (i) linear plane-strain elasticity to ensure equilibrium of the rock surrounding the fracture, (ii) linear elastic fracture mechanics to determine the fracture propagation, (iii) lubrication theory to capture the viscous fluid flow inside the crack and to account for the effect of buoyancy, and (iv) volume balance of the magma. To address the problem, the governing equations are first rewritten in terms of one integral equation with a non-singular kernel, which significantly simplifies the analysis and the procedure for obtaining a numerical solution. The latter solution is shown to obey a multiscale behaviour near the fracture tip that is fully resolved by the numerical scheme. In order to understand the structure of the solution and to quantify the regimes of propagation (and the associated transitions), a thorough analysis of the problem has been performed. Finally, the developments are applied to investigate the non-steady propagation of a buoyancy-driven fracture that is fed by a constant flux.

Discussion on the application of entropy generation minimization to the optimizations of heat transfer and heat-work conversion

The entropy generation minimization is widely used to deal with optimization problems of heat transfer and heat-work conversion. However, it is found that the minimization of entropy generation does not always lead to the optimization of the design objectives in engineering. So, it is necessary to discuss the optimization direction and application preconditions of the entropy generation minimization. In this paper, we study this topic both theoretically and numerically. Our analyses show that the concept of entropy generation directly measures the exergy loss or the ability loss of doing work, so the optimization objective of the entropy generation minimization is to minimize the exergy loss and maximize the ability to do work for the optimized system. However, we have different design objectives in engineering, such as the maximum heat transfer rate, the maximum heat exchanger effectiveness, the minimum average temperature of the heated domain, the maximum output power, the maximum coefficient of performance of heat pump systems, the homogenization of temperature field, etc. Not all of these objectives are consistent with the optimization direction of the entropy generation minimization. Therefore, it is reasonable that the entropy generation minimization is not always applicable. Furthermore, when the relationship between entropy generation and design objective can be set up, the application preconditions of the entropy generation minimization are also discussed. When the preconditions are not satisfied, the entropy generation minimization does not always lead to the best system performance, either. Some examples are also presented to verify the theoretical analyses above. For heat transfer, a one-dimensional heat transfer problem and the entropy generation paradox in heat exchanger are analyzed. For the one-dimensional heat transfer problem, the entropy generation minimization leads to the minimum heat transfer rate when the temperature difference between the boundaries is fixed. Therefore, if our design objective is the maximum heat transfer rate, the entropy generation minimization is not applicable. When the heat transfer rate is fixed, smaller entropy generation rate leads to higher boundary temperature. Therefore, if our design objective is to reduce the boundary temperature, the entropy generation minimization is not applicable, either. For the entropy generation paradox, it is shown that the concept of entropy generation cannot describe the heat transfer performance of heat exchangers. Therefore, the paradox still exists and has not been removed to date. This is verified by the theoretical analyses and the numerical simulation for a parallel flow heat exchanger in which the irreversibility from the pressure drop can be ignored. For heat-work conversion, the energy flow and the exergy flow are analyzed. According to the analyses, we discuss the applicability of the entropy generation minimization to the heat-work conversion system in which the output power, the heat-work conversion efficiency and the thermo-economic performance are taken as the optimization objectives. It is also shown that the application of the entropy generation minimization is conditional. In a word, the discussion on the examples verifies the theoretical analyses.

An efficient numerical scheme for the simulation of parallel-plate active magnetic regenerators

A one-dimensional model of a parallel-plate active magnetic regenerator (AMR) is presented in this work. The model is based on an efficient numerical scheme which has been developed after analysing the heat transfer mechanisms in the regenerator bed. The new finite difference scheme optimally combines explicit and implicit techniques in order to solve the one-dimensional conjugate heat transfer problem in an accurate and fast manner while ensuring energy conservation. The present model has been thoroughly validated against passive regenerator cases with an analytical solution. Compared to the fully implicit scheme, the proposed scheme achieves more accurate results, prevents numerical errors and requires less computational effort. In AMR simulations the new scheme can reduce the computational time by 89%.

Modelling of a desiccant rotor by using the collocation method

Purpose – This article aims to develop a fast numerical method for solving the one-dimensional heat and mass transfer problem within a desiccant rotor. Design/methodology/approach – The collocation method is used for discretizing the axial dimension and reducing the number of dependent variables. The resulting system of equation is then solved through backward differentiation formulas. Findings – The numerical results obtained here focus on verifying the accuracy and the computation time of the proposed method with respect to the finite difference method. The proposed numerical solution method resulted faster than, and as much accurate as, the finite difference method, over a large range of operating conditions that are of interest in desiccant cooling applications. Research limitations/implications – For heat and mass transfer analysis, constant average transfer coefficients are used. The results are calculated for NTU between 2 and 15 and for Le number between 0.5 and 2. Practical implications – The results can be used in designing desiccant heat exchangers and desiccant cooling systems including complex rotor arrangements. Originality/value – Different from other simplified solution techniques, the proposed method relies on few parameters that retain physical meaning and applies also to complex rotor configurations.

Enhancement of Algorithm and Investigation of Heat Transfer Through Fins

Finite Element Method (FEM) and Differential Quadrature Method (DQM) are two very important numerical solution techniques to solve engineering and physical science problems. Usually elements are sub-divided uniformly in FEM (Conventional FEM, CFEM) to obtain temperature distribution behavior in a fin or plate. Hence, extra computational complexity is needed to obtain a fair solution with required accuracy. In this paper, non-uniform sub-elements are considered to enhance the CFEM algorithm to reduce the computational complexity. The proposed algorithm is known as Efficient FEM (EFEM).This EFEM is applied for the solution of one-dimensional heat transfer problem in convection-tip thin rectangular fin. Exact solutions are also obtained for comparison purpose. The obtained results are compared with CFEM and Efficient DQM (EDQM), with non-uniform mesh generation. It is found that the EFEM exhibits more accurate results than CFEM, EDQM and agrees with exact solution showing its potentiality.

Numerical Modeling of Ablation Heat Transfer

A unique numerical method has been developed for solving one-dimensional ablation heat transfer problems. This paper provides a comprehensive description of the method, along with detailed derivations of the governing equations. This methodology supports solutions for traditional ablation modeling including such effects as heat transfer, material decomposition, pyrolysis gas permeation and heat exchange, and thermochemical surface erosion. The numerical scheme uses a control-volume approach with a variable grid to account for surface movement. This method directly supports implementation of nontraditional models such as material swelling and mechanical erosion, extending capabilities for modeling complex ablation phenomena. Verifications of the numerical implementation are provided using analytical solutions, code comparisons, and the method of manufactured solutions. These verifications are used to demonstrate solution accuracy and proper error convergence rates. A simple example of a mechanical erosion (spallation) model is also provided to illustrate the unique capabilities of the method.

Efficient Finite Element and Differential Quadrature Methods for Heat Distribution in One-Dimensional Insulated-Tip Rectangular Fin

There are many numerical solution techniques acquainted by the computational mechanics community including the finite element method (FEM) and differential quadrature method (DQM). Usually elements are sub-divided uniformly in FEM (conventional FEM, CFEM) to obtain temperature distribution behavior in a fin. In CFEM extra computational complexity is needed to obtain a better solution with required accuracy. In this paper, non-uniform sub-elements techniques are considered for the FEM (efficient FEM, EFEM) solution to reduce the computational complexity. Then EFEM is applied for the solution of the one-dimensional heat transfer problem in an insulated-tip thin rectangular fin. The results are compared with CFEM and efficient DQM (EDQM, with non-uniform mesh generation). It is observed that EFEM exhibits more accurate results compared to CFEM and EDQM. The proposed techniques are showing the potentiality of the heat transfer related problem.

An application of Discontinuous Galerkin space and velocity discretisations to the solution of a model kinetic equation

An approach based on a Discontinuous Galerkin discretisation is proposed for the Bhatnagar–Gross–Krook model kinetic equation. This approach allows for a high-order polynomial approximation of molecular velocity distribution function both in spatial and velocity variables. It is applied to model one-dimensional normal shock wave and heat transfer problems. Convergence of solutions with respect to the number of spatial cells and velocity bins is studied, with the degree of polynomial approximation ranging from zero to four in the physical space variable and from zero to eight in the velocity variable. This approach is found to conserve mass, momentum and energy when high-degree polynomial approximations are used in the velocity space. For the shock wave problem, the solution is shown to exhibit accelerated convergence with respect to the velocity variable. Convergence with respect to the spatial variable is in agreement with the order of the polynomial approximation used. For the heat transfer problem, it was observed that convergence of solutions obtained by high-degree polynomial approximations is only second order with respect to the resolution in the spatial variable. This is attributed to the temperature jump at the wall in the solutions. The shock wave and heat transfer solutions are in excellent agreement with the solutions obtained by a conservative finite volume scheme.

Numerical Inverse Heat Transfer Analysis for Temperature-Sensitive-Paint Measurements in Hypersonic Tunnels

This paper describes a numerical inverse method used in temperature-sensitive-paint measurements for quantitative global heat flux diagnostics in hypersonic tunnels. An iterative algorithm is developed to solve the one-dimensional inverse heat transfer problem for unsteady heat flux into a polymer layer (temperature-sensitive paint) on a semi-infinite base, where the temperature dependencies of the thermal properties of materials are taken intoaccount.Surfaceheat fluxdistributionsonconemodelsarecalculatedbyusingthisalgorithmfromtemperaturesensitive-paint images acquired in hypersonic wind tunnel testing. Nomenclature a = thermal diffusivity c = specific heat I = luminescent intensity k = thermal conductivity L = polymer thickness qs = surface heat flux qs� 1D� = surface heat flux obtained by using the onedimensional method T = temperature Tin = initial temperature t = time x, z = coordinates on surface y = coordinate in normal direction to surface

Numerical properties of high order discrete velocity solutions to the BGK kinetic equation

A high order numerical method for the solution of model kinetic equations is proposed. The new method employs discontinuous Galerkin (DG) discretizations in the spatial and velocity variables and Runge–Kutta discretizations in the temporal variable. The method is implemented for the one-dimensional Bhatnagar–Gross–Krook equation. Convergence of the numerical solution and accuracy of the evaluation of macroparameters are studied for different orders of velocity discretization. Synthetic model problems are proposed and implemented to test accuracy of discretizations in the free molecular regime. The method is applied to the solution of the normal shock wave problem and the one-dimensional heat transfer problem.

On the Significance of Flow Regime for the Heat Transfer in Solid Oxide Fuel Cells

The steady state heat transfer that takes place in a hydrogen-fed tubular Solid Oxide Fuel Cell is considered here. The heat is produced due to the electrochemical reaction of the hydrogen that feds the cell with oxygen anions. An averaging technique is used to formulate a relatively simple one-dimensional heat transfer problem. The conduction-convection equation describing the heat transfer from the electrolyte’s surface to the moving gas that surrounds the cell’s cathode is solved analytically under the assumption of iso-thermal conditions. Three different cases are considered for the flow of the cathode gas: (a) plug flow, (b) fully developed incompressible laminar flow, and (c) compressible flow. Analytical expressions for the spatial distribution of gas and cell temperature along the cell's length are obtained. For constant mass flow rate, different flow regimes produce almost the similar spatial distributions for the gas temperature and, consequently, the consideration of the flow regime is of low importance in the design of fuel cell stacks.

Singular integral transforms for heat transfer problems in semi-infinite solids with coatings

This paper is devoted to singular integral transform method for non-stationary one-dimensional heat transfer problems in semi-infinite solids. Different geometries (semi-space, cylindrical channel and spherical cavity) are studied, and resulting temperature fields are compared. Special attention is given to multi-layer surfacing of solid as a means of heat defense. Kernel splitting method for these problems with non-stationary heat exchange conditions on free boundary is developed.

The generalized approximation method and nonlinear heat transfer equations

Generalized approximation technique for a solution of one-dimensional steady state heat transfer problem in a slab made of a material with temperature dependent thermal conductivity, is developed. The results obtained by the generalized approximation method (GAM) are compared with those studied via homotopy perturbation method (HPM). For this problem, the results ob- tained by the GAM are more accurate as compared to the HPM. Moreover, our (GAM) generate a sequence of solutions of linear problems that converges monotonically and rapidly to a solution of the original nonlinear problem. Each approximate solution is ob- tained as the solution of a linear problem. We present numerical simulations to illustrate and confirm the theoretical results.

Heat Distribution in Rectangular Fins Using Efficient Finite Element and Differential Quadrature Methods

Finite element method (FEM) and differential quadrature method (DQM) are among important numerical techniques used in engineering analyses. Usually elements are sub-divided uniformly in FEM (conventional FEM, CFEM) to obtain temperature distribution behavior in a fin or plate. Hence, extra computational complexity is needed to obtain a fair solution with required accuracy. In this paper, non-uniform sub-elements are considered for FEM (efficient FEM, EFEM) solution to reduce the computational complex-ity. Then this EFEM is applied for the solution of one-dimensional heat transfer problem in a rectangular thin fin. The obtained results are compared with CFEM and efficient DQM (EDQM), with non-uniform mesh generation). It is found that the EFEM exhibit more accurate results than CFEM and EDQM showing its potentiality.

Identification of temperature-dependent thermal properties of solid materials

This work proposes an experimental technique for the simultaneous estimation of temperature-dependent thermal diffusivity, α, and thermal conductivity, λ, of insulation materials. The thermal model used considers a transient one-dimensional heat transfer problem. The determination of these properties is done by using the principle of the Mixed technique. In this technique two objective functions are defined, one in the frequency domain and the other in the time domain. The objective function in the frequency domain is based on the square difference between experimental and calculated values of the phase angle, while the other objective function is the least square error function of experimental and calculated signals of temperature. The properties α and λ are obtained by using an experimental apparatus that basically consists of a Polyvinyl Chloride (PVC) sample exposed to different temperatures inside an oven. The temperature inside the oven is controlled by a PID temperature controller. The properties α and λ were estimated for 7 (seven) points of average temperature in a range from 20 oC to 66 oC. The properties were determined with an additional heating of approximately 4.5 K on the frontal surface. Analyses of sensitivity, sensors location and sample dimensions were also made. Keywords : thermal properties estimation, heat conduction, optimization, experimental methods

Estimation of internal heat transfer coefficients and detection of rib positions in gas turbine blades from transient surface temperature measurements

This paper describes a non-invasive, non-destructive, transient inverse measurement technique that allows one to determine internal heat transfer coefficients and rib positions of real gas turbine blades from outer surface temperature measurements after a sudden flow heating. The determination of internal heat transfer coefficients is important during the design process to adjust local heat transfer to spatial thermal load. The detection of rib positions is important during production to fulfill design and quality requirements. For the analysis the one-dimensional transient heat transfer problem inside of the turbine blade's wall was solved. This solution was combined with the Levenberg-Marquardt method to estimate the unknown boundary condition by an inverse technique. The method was tested with artificial data to determine uncertainties with positive results. Then experimental testing with a reference model was carried out. Based on the results, it is concluded that the presented inverse technique could be used to determine internal heat transfer coefficients and to detect rib positions of real turbine blades.

Estimation of total equivalent temperature difference values for multilayer walls and flat roofs by using periodic solution

A theoretical methodology was developed to find total equivalent temperature difference (TETD) values for multilayer flat roofs and walls of buildings. The TETD values based on time lag and decrement factor are estimated using measured hourly solar radiation on a horizontal surface and outside air temperatures. The time lag and decrement factor are obtained numerically using periodic solution of one-dimensional transient heat transfer problem for the building structures. The problem is initially formulated using a differential equation and the appropriate initial and boundary conditions. It is then converted into dimensionless form and solved by application of complex finite Fourier transform (CFFT). The solution is used to find hourly variation of temperature inside the structure, the heat gain to the building through the structure, the highest and lowest temperatures at the inner and outer surfaces of the roofs and walls, and the time periods involved in reaching these temperatures. Sol-air temperatures for different directions as well as time lag and decrement factor are obtained, and these are used to determine the TETD values of the structures.

A Numerical Study of Weight Functions, Scaling, and Penalty Parameters for Heat Transfer Applications

ABSTRACT This article describes a numerical study of weight functions, scaling, and penalty parameters for heat transfer problems. The numerical analysis is carried out using a meshless element-free Galerkin (EFG) method, which utilizes moving least-square (MLS) approximants to approximate the unknown function of temperature. These MLS approximants are constructed by using a weight function, a basis function, and a set of coefficients that depend on position. Lagrange multiplier and penalty methods are used to enforce the essential boundary conditions. MATLAB software is developed to obtain the EFG results. A new rational weight function is proposed. Comparisons are made among the results obtained using cubic spline, quartic spline, Gaussian, quadratic, hyperbolic, rational, exponential and, cosine weight functions in one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) heat transfer problems. The L2 error norm and rate of convergence are evaluated for different EFG weight functions and the finite-element method (FEM). The effect of scaling and penalty parameters on EFG results is discussed in detail. The results obtained by the EFG method are compared with those obtained by finite-element and analytical methods.

Real-time estimation of the temperature distribution and expansion of a ball screw system using an observer

Thermal deformations of ball screws directly affect the positioning errors of machine tools. In order to calibrate the thermal deformations of ball screws, an accurate estimation of the whole temperature field of the screws is required. However, it is extremely inefficient and almost impossible to acquire the whole temperature distribution by measuring the temperatures of every point. Therefore, a temperature estimator, which can estimate the whole temperature field from temperatures of just a few points, is required. In this paper, one-dimensional heat transfer problems are formulated under concepts of modal analysis and state-space design. Then an observer is designed to estimate the intensity of a heat source and the whole temperature field in real time. The reliability of the estimator is confirmed through comparisons between solutions from the proposed method and those of exact solutions. The proposed method is applied to the estimation of the intensity of the heat source and the temperature distribution of a ball screw system. It is confirmed that the proposed method estimates the whole temperature distribution of a ball screw system well without noise and time delay.

Inverse thermal analysis method to study solidification in cast iron

Solidification modelling of cast metals is widely used to predict final properties in cast components. Accurate models necessitate good knowledge of the solidification behaviour. The present study includes a re-examination of the Fourier thermal analysis method. This involves an inverse numerical solution of a one-dimensional heat transfer problem connected to solidification of cast alloys. In the analysis, the relation between the thermal state and the fraction solid of the metal is evaluated by a numerical method. This method contains an iteration algorithm controlled by an under relaxation term to achieve a stable convergence. The heat transfer problem is reduced to one-dimension to promote the practical application of the method. Thermo-physical properties such as the volumetric heat capacity tabulated in the calculation are introduced as a function of solidifying phases. Experimental equipment was developed in order to investigate the thermal behaviour of the solidifying metal. Three cylindrically shaped cast samples surrounded by different cooling materials were introduced in the same mould allowing a common metallurgical background for samples solidifying at different cooling rates. The proposed inverse thermal analysis was tested on both experimental and simulated data.