Analysis of hyperbolic heat conduction in 1-D planar,cylindrical, and spherical geometry using the lattice Boltzmann method

Analyses of hyperbolic heat conduction in an 1-D planar, cylindrical, and spherical geometry are analyzed using the lattice Boltzamnn method (LBM). Finite time lag between the imposition of temperature gradient and manifestation of heat flow causes the governing energy equation to be hyperbolic one. Temporal temperature distributions are analyzed for thermal perturbation of a boundary by suddenly raising its temperature and also by imposing a constant heat flux to it. Wave-like temperature distributions in the medium are obtained when constant temperature boundary condition is used. However, when constant heat flux boundary condition is used, temperature distribution fluctuates before it becomes stable. To check the accuracy of the LBM results, the problems are also solved using the finite difference method (FDM). LBM and FDM results compare exceedingly well. LBM has computational advantage over the FDM.     

Solution of inverse heat conduction problem using the lattice Boltzmann method

In this paper the D2Q9 lattice Boltzmann method (LBM) was utilized for the solution of a two-dimensional inverse heat conduction (IHCP) problem. The accuracy of the LBM results was validated against those obtained from prevalent numerical methods using a common benchmark problem. The conjugate gradient method was used in order to estimate the heat flux test case. A complete error analysis was performed. As the LBM is attuned to parallel computations, its use is recommended in conjugation with IHCP solution methods.     

Direct and inverse mixed convections in an enclosure with ventilation ports

The numerical study presented in this work describes the direct and inverse mixed convection problems in a slot-ventilated enclosure subjected to an unknown heat flux on one side. Particularly, the interaction of internal natural convection with the cold ventilated flow leads to various flow fields depending on the Richardson number, Reynolds number, and the functional form of the imposed boundary heat flux. Fluid and heat transport structures across the enclosure are visualized by the streamlines and heatlines, respectively. Subsequently, an iterative conjugate gradient method is applied such that the gradient of the cost function is introduced when the appropriate sensitivity and adjoint problems are defined for a domain of arbitrary geometries. In this approach, no a priori information is needed about the unknown boundary heat fluxes to be determined. The accuracy of the heat flux profile solutions is shown to depend strongly on the values of Reynolds number and flux functional forms. Effects of measurement errors on the accuracy of estimation are also investigated. The present work is significant for the flow control simultaneously involving the natural convection and forced convection.     

Two-Dimensional Inverse Heat Transfer Analysis of Functionally Graded Materials in Estimating Time-Dependent Surface Heat Flux

Two-dimensional transient inverse heat conduction problem (IHCP) of functionally graded materials (FGMs) is studied herein. A combination of the finite element (FE) and differential quadrature (DQ) methods as a simple, accurate, and efficient numerical method for FGMs transient heat transfer analysis is employed for solving the direct problem. In order to estimate the unknown boundary heat flux in solving the inverse problem, conjugate gradient method (CGM) in conjunction with adjoint problem is used. The results obtained show good accuracy for the estimation of boundary heat fluxes. The effects of measurement errors on the inverse solutions are also discussed.     

The function estimation in predicting heat flux of pin fins with variable heat transfer coefficients

This article solves the two dimensional inverse problem of estimating the unknown heat flux at a pin fin base by the conjugate gradient method. In the estimating processes, no prior information on the functional form of the unknown quantity is required. The accuracy of the inverse analysis is examined by simulated exact and inexact measurements of temperature at interior locations of the pin fin. The numerical results show that good estimations on the heat flux can be obtained for all the test cases considered in this study. Furthermore such a technique can be applied to determine the heat flux acting on an internal surface, where a direct measurement is not feasible.     

Identifying heat conductivity and source functions for a nonlinear convective-diffusive equation by energetic boundary functional methods

Abstract

In the article, we solve the inverse problems to recover unknown space-time dependent functions of heat conductivity and heat source for a nonlinear convective-diffusive equation, without needing of initial temperature, final time temperature, and internal temperature data. After adopting a homogenization technique, a set of spatial boundary functions are derived, which satisfy the homogeneous boundary conditions. The homogeneous boundary functions and zero element constitute a linear space, and then a new energetic functional is derived in the linear space, which preserves the time-dependent energy. The linear systems and iterative algorithms to recover the unknown parameters with energetic boundary functions as the bases are developed, which are convergent fast at each time marching step. The data required for the recovery of unknown functions are parsimonious, including the boundary data of temperatures and heat fluxes and the boundary data of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the exact solutions with the identified results, which are obtained under large noisy disturbance.     

Inverse analysis applied to retrieval of parameters and reconstruction of temperature field in a transient conduction–radiation heat transfer problem involving mixed boundary conditions

This article deals with the application of the inverse method for simultaneous retrieval of parameters and reconstruction of the temperature field in a transient conduction–radiation problem with mixed boundary conditions. The conducting–radiating medium is absorbing, emitting and isotropically scattering. The boundaries are diffuse gray. One boundary of the planar medium is at a prescribed temperature, while the other boundary is at a prescribed heat flux. A method involving lattice Boltzmann method (LBM), the finite volume method (FVM) is used to obtain the temperature field in the mixed boundary problem which in the present work is termed as the direct method. Next, random perturbations are imposed on this exact temperature field and then simultaneous reconstruction of the same and estimation of properties are accomplished by minimizing the square of the error between the exact and guessed temperature fields. This error, that in the present work is termed as the objective function, is minimized using the genetic algorithm (GA). The impact of different genetic parameters on the accuracy of the estimation is also investigated. It is observed that subject to the proper selection of the genetic parameters, simultaneous reconstruction of the temperature field along with a reasonably good estimation of the unknown parameters can be achieved using the LBM–FVM–GA.     

Immersed boundary method for the simulation of flows with heat transfer

In the present paper a direct heat source scheme is proposed to let the temperature at the immersed boundary satisfy the temperature Dirichlet boundary condition. And the explicit interactive process of the direct heat source scheme called multi-direct heat source scheme is applied to ensure the satisfaction of the temperature Dirichlet boundary condition at the immersed boundary. The second-order spacial accuracy of the solver is confirmed by simulating the Taylor–Green vortices. The simulations of natural convection between concentric cylinders, and flow past a stationary circular cylinder are conducted to validate the accuracy of present method on solving heat transfer problems. And the computation of flow past a staggered tube bank with heat transfer is conducted to verify the capability of present method on solving complex geometries problems.     

Treatment of Transport at the Interface Between Multilayers via the Lattice Boltzmann Method

The lattice Boltzmann method (LBM) has reached maturity in many aspects for modeling incompressible, laminar flow and heat and mass transfer. However, many issues still need to be clarified. One of those is how to deal with Neumann boundary conditions (heat, momentum, and mass fluxes) at the interface between layers of different thermophysical properties, which is the topic of this work. In this work, we try to illustrate modeling of transient and steady-state heat transfer through multilayers, ensuring continuity of the flux, temperature, momentum, velocity, or species concentration at the interface, which is not a trivial issue in any of the numerical methods. Satisfying continuity conditions at the interface using the LBM needs special treatment, because the relaxation time depends on the thermophysical properties. In this work, methods to solve this issue are introduced for solving 1-D and 2-D, unsteady heat diffusion problems However, the methods can be equally applied for multilayer immiscible fluid flow and mass transfer problems. The predictions of the LBM are compared with those of the finite-volume method (FVM).     

A Principal Component Analysis and neural network based non-iterative method for inverse conjugate natural convection

Inverse Heat Transfer Problems (IHTP) are characterized by estimation of unknown quantities by utilizing any given information of the system. In this study, the inverse problem of estimation of boundary heat flux for a given temperature distribution on the walls of a two dimensional square cavity with a finite wall thickness is considered. A non-iterative method is applied utilizing Artificial Neural Network (ANN) and Principal Component Analysis (PCA) to estimate the parameters that define the boundary heat flux. The forward model is numerically solved with Fluent 6.3 for known values of a linearly varying boundary heat flux and the temperature distribution thus obtained is utilized to train the ANN for the inverse model. A parametric study is carried out to determine the effect of the thermal conductivity of the top and bottom walls on the flow and temperature distribution in the cavity. PCA analysis is carried out to reduce the dimensions of the input data set for the inverse model. These reduced dimensions are used to train the network and due to low dimensionality of the input, the effort required to train the network is considerably less. The trained networks are finally used to estimate boundary heat flux for any desired temperature distribution on the top and bottom walls. Additionally, covariance analysis is carried out in order to estimate the required number of temperatures during an experiment, on the top and bottom walls for the prediction of heat flux with a reasonable accuracy. The inverse model with covariance analysis is compared with the inverse model with PCA and both the methods are found to be equally potent.     

Two dimensional steady-state inverse heat conduction problems

In this work we estimate the surface temperature in two dimensional steady-state in a rectangular region by two different methods, the singular value decomposition (SVD) with boundary element method (BEM) and the least-squares approach with integral transform method (ITM). The BEM method is efficient for solving inverse heat conduction problems (IHCP) because only the boundary of the region needs to be discretized. Furthermore, both temperature and heat flux at the unknown boundary are estimated at the same time. The least-squares technique involves solving the equations constructed from the measured temperature and the exact solution. The measured data are simulated by adding random errors to the exact solution of the direct problem. The effects of random errors on the accuracy of the predictions are examined. The sensitivity coefficients are also presented to illustrate the effect of sensor location on the estimated surface conditions. Numerical experiments are given to demonstrate the accuracy of the present approaches.     

Two-dimensional inverse problem in estimating heat flux of pin fins

The two-dimensional inverse problem of estimating the unknown heat flux of a pin fin base has been solved using the conjugate gradient method. The advantage of the conjugate gradient method is that no information on the functional form of the unknown quantity is required beforehand. The accuracy of the inverse analysis is examined by using simulated exact and inexact measurements of temperature in an interior location of a pin fin. Numerical results show that good estimations on the heat flux can be obtained for all the test cases considered here. Furthermore, such a technique can be applied to determine the heat flux acting on an internal wall surface, where direct measurements are difficult to make.     

An inverse problem of finding the time-dependent thermal conductivity from boundary data

We consider the inverse problem of determining the time-dependent thermal conductivity and the transient temperature satisfying the heat equation with initial data, Dirichlet boundary conditions, and the heat flux as overdetermination condition. This formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data cause large errors in the output solution. The finite difference method is employed as a direct solver for the inverse problem. The inverse problem is recast as a nonlinear least-squares minimization subject to physical positivity bound on the unknown thermal conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. We investigate the accuracy and stability of results on a few test numerical examples.     

Probing the Rayleigh–Benard convection phase change mechanism of low-melting-point metal via lattice Boltzmann method

A double distribution function lattice Boltzmann method (LBM) with multirelaxation time is implemented to simulate the Rayleigh–Benard convection melting of a typical low-melting-point metal in a rectangular cavity. Typical cases frequently encountered in practice with constant heat flux/constant temperature boundary conditions are parametrically investigated, with corresponding dimensionless results outlined; the influence of inclination angle of the cavity is also clarified. The computational speed of the current LBM would reach about 40 times faster than that of conventional finite volume method as performed by commercial software Fluent. The obtained results would be valuable for guiding practical thermal design.     

A simple direct estimation of temperature-dependent thermal conductivity with kirchhoff transformation

A direct method is proposed to estimate the temperature-dependent thermal conductivity without internal measurements. In the proposed method, the steady-state nonlinear heat conduction equation is transformed into the Laplace equation via the Kirchhoff transformation. The thermal conductivity is modeled as a linear combination of known functions with unknown coefficients, which are directly determined from the imposed heat flux and measured temperatures at the boundary. Several inverse heat conduction problems are successfully introduced to confirm the validity of the proposed method.     

Iterative algorithms for solving inverse problems of heat conduction in multiply connected domains

The presented paper displays a method of solving the inverse problems of heat transfer in multi-connected regions, consisting in iterative solving of convergent series of the direct problems. For known temperature and flux values at the outer boundary of the region the temperature and flux values at the inner boundaries are sought (the cauchy problem for the Laplace equation). In case of such a formulation of the problem, the solution does not always exist, one of the conditions is met in the mean-square sense, providing the optimization criterion. The idea of the process consists in solving the direct problem in which the boundary condition is subject to iterative changes so as to attain minimum of the optimization criterion (the square functional). Two algorithms have been formulated. In the first of them the heat flux at the inner boundaries of the region, while in the other the temperature were subject to changes. Convergence of both the algorithms have been compared.The numerical calculation has been made for selected examples, for which an analytical solution is known. The effect of random disturbance of the boundary conditions on the solution obtained with iterative algorithms has been checked. Moreover, a function was defined, serving as convergence measure of the solution of the inverse problem solved with the algorithms proposed in the paper. The properties of the function give evidence that it tends to the value exceeding unity.     

A new boundary-type mesh-free method for solving the multi-domain heat conduction problem

ABSTRACT

In this article, a new high-precision boundary-type mesh-free method, called the virtual boundary mesh-free Galerkin method (VBMGM), is given for analyzing the heat conduction problem consisting of multiple media. A control equation of the proposed method is established by the the Galerkin method of the weighted residual methods. The virtual source function is approached through the radial basis function interpolation in the mesh-free method. Thus, the main feature of the proposed technique has the advantages of the mesh-free method, the boundary-element method, and the Galerkin method, a symmetrical coefficient matrix. The numerical meanings of the weighted values in the VBMGM equation, such as the partial derivatives the temperature, the heat flux, the temperature connection condition, and the heat flux connection condition, are clear. Numerical examples of a thick-walled cylinder consisting of two media and a two-layered rectangular plate are provided. The feasibility and the accuracy of the proposed technique are proved.     

Inverse estimation of spatially and temporally varying heating boundary conditions of a two-dimensional object

In many dynamic heat transfer situations, the temperature at the heated boundary is not directly measurable and can be obtained by solving an inverse heat conduction problem (IHCP) based on measured temperature or/and heat flux at the accessible boundary. In this study, IHCP in a two-dimensional rectangular object is solved by using the conjugate gradient method (CGM) with temperature and heat flux measured at the boundary opposite to the heated boundary. The inverse problem is formulated in such a way that the heat flux at heated boundary is chosen as the unknown function to be recovered, and the temperature at the heated boundary is computed as a byproduct of the IHCP solution. The measurement data, i.e., the temperature and heat flux at the opposite boundary, are obtained by numerically solving a direct problem where the heated boundary of the object is subjected to spatially and temporally varying heat flux. The robustness of the formulated IHCP algorithm is tested for different profiles of heat fluxes along with different random errors of the measured heat flux at the opposite boundary. The effects of the uncertainties of the thermophysical properties and back-surface temperature measurement on inverse solutions are also examined.     

Convection heat transfer with internal heat generation in porous media: Implementation of thermal lattice Boltzmann method

Evaluation of lattice parameters for convection heat transfer in porous media with internal heat generation from physical and macroscale properties was described. A hierarchical process was defined to implement thermal Lattice Boltzmann Method (LBM) to investigate convection heat transfer with internal heat generation in different geometries; from a simple geometry (flow channel) to complex ones (porous media). In this regard, seven different without any obstacle cases with different geometries were designed and the detailed information about how thermal LBM should be implemented for these cases are addressed. Going from one case to the next, the cases with more complex physics and/or geometries were examined. The results showed that LBM is an appropriate method to predict heat transfer with internal heat generation in porous media.     

Numerical Simulation of Radiative-Conductive Heat Transfer in an Enclosure with an Isotherm Obstacle

In this paper, the lattice Boltzmann method (LBM) and discrete ordinates method (DOM) were applied to investigate the heat transfer in a square radiative-conductive media with heat flux and temperature boundary conditions. Furthermore, an isothermal rectangular obstacle is located in the middle of participating media. The energy equation is solved using the LBM; while the radiative transfer equation is solved using DOM. The effects of various parameters such as the extinction coefficient, scattering albedo, and the conduction–radiation parameter in the presence of an obstacle are studied on temperature and heat flux distributions. It was shown that, decrease in scattering albedo value leads to decrease of the temperature field in participating media. In addition, with increase in scattering albedo value, conductive heat flux increases and radiative heat flux decreases. It was shown that increase in extinction coefficient and decrease in conduction–radiation parameter have some significant effects on increasing the temperature profile, especially in the region with longer distance from obstacle.