Showing papers in "Numerical Heat Transfer Part B-fundamentals in 1998"
TL;DR: In this paper, the finite volume method is extended to compute radiation in axisymmetric geometries using unstructured polyhedral meshes and conservative discretization of the angular redistribution term is proposed.
Abstract: The finite-volume method is extended to compute radiation in axisymmetric geometries using unstructured polyhedral meshes. Control angle overhangs, resulting from the nonalignment of the angular discretization with cell faces, are accounted for in the differencing. Conservative discretization of the angular redistribution term is proposed, and allows the solution of axisymmetric problems on the same two-dimensional mesh as the fluid flow. The method is shown to be equivalent to the mapping procedure of Chui and Railhby. The scheme is applied to a number of benchmark problems and is shown to perform satisfactorily.
84 citations
TL;DR: In this article, a finite-volume solution procedure for radiative heat transfer in a three-dimensional nonorthogonal enclosure containing participating medium is proposed with geometric relations commonly adopted in computational fluid dynamics.
Abstract: A finite-volume solution procedure for radiative heat transfer in a three-dimensional nonorthogonal enclosure containing participating medium is proposed with geometric relations commonly adopted in computational fluid dynamics. A general discretization equation is formulated by using the directional weight and the step scheme for spatial differencing. The present approach is validated through comparison with the problems of hexahedral enclosure, annular sector, and three-dimensional combustion chamber, in which the solution accuracy as well as computation efficiency required have been examined for various cases. Alt of the results presented here support its accuracy as well as moderate efficiency in computation time in the nonorthogonai three-dimensional radiation calculation. Finally, the present method is applied to a kidney-shaped combustion chamber as a three-dimensional test case.
82 citations
TL;DR: In this paper, the dual reciprocity method (DRM) for the modified Helmholtz operator and the Laplace transform has been developed to solve diffusion-type and diffusion-reaction problems.
Abstract: A ‘mesh-fre’ and ‘time-free’ numerical method, based on the method of fundamental solutions, the dual reciprocity method ( DRM) for the modified Helmholtz operator, and the Laplace transform, has been developed lo solve diffusion-type and diffusion-reaction problems. Our proposed method has achieved both excellent accuracy and efficiency, as illustrated by two numerical examples, in two dimensions and three dimensions. Another attractive feature of our proposed method is the easy implementation for the 3D problems.
68 citations
TL;DR: In this article, a curvilinear level set formulation was developed for highly deformable free surface problems. And the level set function is solved based on a finite-difference method using the second-order essentially nonoscillatory (ENO) scheme.
Abstract: A curvilinear level set formulation has been developed for highly deformable free surface problems. In this new scheme, the grid lines follow the irregular domain generated by the multizone adaptive grid-generation (MAGG) scheme [1] and free surfaces are captured by level set functions among the curvilinear grids. Navier-Stokes equations are discretized and solved based on a multiphase curvilinear finite-volume method [2], and the level set function is solved based on a finite-difference method using the second-order essentially nonoscillatory (ENO) scheme [3]. The scheme is capable of accurately and efficiently representing the deformation, oscillation, merger, and separation of free surfaces. The effectiveness and robustness of the algorithm are demonstrated by using it for problems involving merger of bubbles, mold filling, and the spreading and solidification of molten droplets on a cold substrate where both free surface and solidification interfaces move and the mass of the liquid phase is continuous...
48 citations
TL;DR: In this paper, a finite-difference algorithm is presented for the solution of non-Fourier one-dimensional heat conduction, where the accuracy of the solution is limited only by the selection of the mesh width in the space and time directions.
Abstract: A simple and concise finite-difference algorithm is presented for the solution of non-Fourier one-dimensional heat conduction. The numerical algorithm is developed by applying the Godunov scheme on the characteristic equations that govern thermal waves within the medium. A vigorous investigation of the order of accuracy and stability of the procedure is presented. Several heat conduction problems are tested with the scheme, and the predicted temperature field is verified by comparing with results available in the literature. The accuracy of the solution is limited only by the selection of the mesh width in the space and time directions. The solution is free of oscillation, while the dissipation error associated with the numerical scheme in the wave front is controlled by specifying the proper Courant-Fredrichs-Lewy (CFL) condition. The algorithm provides a convenient, accurate, and efficient approximation to the hyperbolic heat conduction equation. It is evident that the procedure is beneficial for the st...
47 citations
TL;DR: In this article, an adaptive weighting input estimation algorithm that efficiently and robustly on-line estimates time-varied thermal unknowns is presented, which is based on a recursive least-squares estimator weighting by an adaptive forgetting factor.
Abstract: This work presents an adaptive weighting input estimation algorithm that efficiently and robustly on-line estimates time-varied thermal unknowns. While providing for the adaptivity, the Kalman filter allows us to derive a regression equation between the bias innovation and the thermal unknown. Based on this regression model, a recursive least-squares estimator weighting by an adaptive forgetting factor is proposed to extract the unknowns that are defined as the inputs. The maximum-likelihood-type estimator( M estimator) combining the Huber psi function is used to construct the adaptive weighting forgetting factor as a function of biased innovation at each time step, thereby allowing us to estimate the unknown in a system involving measurement noise, modeling error, and unpredictable time-varying changes of the unknowns. In addition, Ike superior capabilities of the proposed algorithm are demonstrated in several time-varying estimate cases and two benchmark performance tests in one-dimensional inverse heat...
45 citations
TL;DR: In this article, a boundary-element method was proposed to identify the strength of line heat sources when their position is known, and a regularization procedure over some future time steps was used to solve the ill-posed problem correctly.
Abstract: This article deals with an inverse problem relative to point heat source identification in homogeneous solids. Using the boundary-element method, it is proposed to identify the strength of line heat sources when their position is known. To solve the inverse problem, some internal temperatures at prescribed locations are given. A regularization procedure over some future time steps is used to solve the ill-posed problem correctly. To test the method, some disrupt data provided by numerical simulation are used. Different solicitations are shown, and an example involving two line sources is presented.
43 citations
TL;DR: In this article, the radial velocity at a center is given by the average of adjacent circumferential directional velocity components located at ± 0.5π rod and those velocity profiles through a cylinder center are smoothly represented.
Abstract: A technique was invented to circumvent computational singularity at a radial center of cylindrical and spherical coordinate systems. The idea is as follows. The radial velocity at a center is given by the average of adjacent circumferential directional velocity components located at ± 0.5π rod. Sample computations are presented for natural convection in a vertical cylinder with two different thermal boundary conditions. Those velocity profiles through a cylinder center are smoothly represented.
35 citations
TL;DR: In this paper, a numerical procedure for solving two-dimensional boundary inverse heat conduction problems is presented, where a boundary condition (surface temperatures and surface heat fluxes) of a body is estimated.
Abstract: A numerical procedure for solving two-dimensional boundary inverse heat conduction problems is presented in this article. A boundary condition (surface temperatures and surface heat fluxes)of a body is estimated. The major advantage of this work is that the solution can be obtained using limited information about the problem. Another advantage is that the formulation of the problem is completely general. The proposed procedure does not need any stabilization method when exact data are used for solving the problem. A digital filter method is used to stabilize the inverse algorithm by smoothing the real (noisy) data. Then, we apply the control-volume formulation for the filtered data. Accuracy and stability of the solution method is verified by utilizing a solution of a direct problem. The influence on the numerical solution of the time step, digital filter coefficients, and measurement errors is also considered.
35 citations
TL;DR: In this paper, the finite element method is used to solve fully developed convection problems in spatially periodic domains, where symmetric and antisymmetric periodicity in temperature is imposed in an original way that allows for different thermal boundary conditions at the walls.
Abstract: The finite-element method is used to solve fully developed convection problems in spatially periodic domains. Symmetric and antisymmetric periodicity in temperature is imposed in an original way that allows for different thermal boundary conditions at the walls. The formulation is first validated by comparing the numerical results with the analytical solutions for fully developed velocity and temperature distributions in a parallel-plate channel. Afterward, the accuracy and the capabilities of the procedure are demonstrated by two examples involving laminar flow and heat transfer in a periodic corrugated channel and in a parallel-plate channel with staggered fins.
34 citations
TL;DR: The integrated space-time (IST) finite- volume method extends the finite-volume principle to both space and time, thereby satisfying the geometric conservation law (GCL) implicitly and lends itself well to moving-boundary problems.
Abstract: A new finite-volume method for moving-mesh problems, called the integrated space-time (IST) finite-volume method, is presented. This concept extends the finite-volume principle to both space and time, thereby satisfying the geometric conservation law (GCL) implicitly. Consequently, the method lends itself well to moving-boundary problems. An additional feature of the IST is the unification of the transient and advection terms, so that the same second-order discretization can be used for both. The IST framework is implemented using a cell-centered method, which includes a new linearly-exact discretization for diffusion terms that is equally applicable to isotropic and anisotropic continua and collapses to classical computational molecules on orthogonal meshes
TL;DR: In this article, the effect of temperature correction on the velocity correction is considered during the derivation of the pressure-linked equation, and a modification to the SIMPLE algorithm, SIMPLET, is proposed and two cases are tested.
Abstract: The essence of the SIMPLE method lies in its coupling between the momentum and continuity equations. Almost all the algorithms of the SIMPLE family are based on one precondition, that is, the corrected velocity is obtained from the corrected pressure only. However, in buoyancy-driven flows, there are two major factors driving the fluid movement: the temperature gradients and the pressure (including kinetic pressure) gradients. In this article, the effect of the temperature correction on the velocity correction is considered during the derivation of the pressure-linked equation. A modification to the SIMPLE algorithm, SIMPLET, is proposed and two cases are tested. It is shown that this modification provides a better convergence rate and more robust results.
TL;DR: The pressure correction algorithm introduced for solution of incompressible Navier-Stokes equations on a nonstaggered grid is extended to prediction of compressible flows with and without shocks, showing results that compare extremely favorably with previous ones obtained using a staggered grid.
Abstract: The present author recently devised a pressure correction algorithm for solution of incompressible Navier-Stokes equations on a nonstaggered grid [6]. This algorithm introduced the notion of smoothing pressure correction to overcome the problem of checkerboard prediction of pressure. In this article, the algorithm is extended to prediction of compressible flows with and without shocks. The predictions show that the algorithm yields results that compare extremely favorably with previous ones [6] obtained using a staggered grid. Accurate shock capturing on coarse grids, however, requires use of total variation diminishing ( TVD) discretization of the covective terms coupled with measures for stabilisation of the iteration process.
TL;DR: In this article, a boundary element method (BEM)-based inverse algorithm utilizing the iterative regularization method, i.e., the conjugate gradient method (CGM), is used to solve the inverse heat conduction problem (IHCP) of estimating the unknown boundary temperature in a multidimensional steady state problem with arbitrary geometry.
Abstract: A boundary-element-method (BEM)-based inverse algorithm utilizing the iterative regularization method, i.e., the conjugate gradient method (CGM), is used to solve the inverse heat conduction problem (IHCP) of estimating the unknown boundary temperature in a multidimensional steady-state problem with arbitrary geometry. The results obtained by the CGM are compared with that obtained by the standard regularization method (RM). The error estimation based on the statistical analysis is derived from the formulation of the RM. A 99% confidence bound is thus obtained. Finally, the effects of the measurement errors on the inverse solutions are discussed. The present technique can be easily extended to the transient heat conduction problem. Results show that the advantages of applying the CGM in the inverse calculations lie in that (I) the major difficulties in choosing a suitable form of quadratic norm, determining a proper regularization order, and determining the optimal smoothing (or regularization) coefficien...
TL;DR: In this article, a comparison between the diffuse approximation method (DAM) and the control-volume finite element method (CVFEM) is made for several test cases between the two methods.
Abstract: This article presents a comparison for several test cases between the diffuse approximation method (DAM) and the control-volume finite element method (CVFEM) The diffuse approximation method is a relatively new numerical approach that does not need a finite-element mesh to solve fluid flow and heat transfer problems in complex geometries. It appears that the two methods provide similar accurate results. However, when some manipulations of the grids are needed during the calculation process, such as a local refinement, the use of a simple set of nodes appears to be less time consuming, as shown in the last example.
TL;DR: A novel enthalpy formulation is applied to the Stefan problem in various types of domains, including cylindrical and spherical geometries, annuli, and two-dimensional square domains as mentioned in this paper.
Abstract: A novel enthalpy formulation is applied to the Stefan problem in various types of domains, including cylindrical and spherical geometries, annuli, and two-dimensional square domains. The results are compared with exact solutions; when exact solutions are unavailable, comparison is made between the enthalpy method and the heat balance integral method (HBIM), a front-tracking method that has been applied successfully in simple domains. The enthalpy formulation provides a simple way of tracking the phase front and is shown to be accurate even with relatively coarse grids.
TL;DR: In this article, a one-dimensional investigation is performed to compare the performance of velocity-based and momentum-based procedures, which are formulated based on a control-volume approach with pressure as a dependent variable.
Abstract: In this work, a one-dimensional investigation is performed to compare the performance of velocity-based and momentum-based procedures. Both procedures are formulated based on a control-volume approach with pressure as a dependent variable. The related integration point operators are derived based on incorporating the correct physical influence of the flow and other relevant couplings. The formulations are free from employing any explicit artificial viscosity and damping mechanism. They are applied to highly compressible flow by testing the shock tube problem in order to investigate two aspects of linearization and constant mass flux advantage within the developed procedures. The results show that the proposed momentum-based procedure (MBP) is more stable and accurate than the velocity-based procedure (VBP) without damping.
TL;DR: In this article, a harmonic sine approximation method is used to obtain approximate solutions to two-dimensional steady-state heat conduction problems with singularities and semi-infinite domains and Dirichlet boundary conditions.
Abstract: In this article, a recently derived harmonic sine approximation method is used to obtain approximate solutions to two-dimensional steady-state heat conduction problems with singularities and semi-infinite domains and Dirichlet boundary conditions. The first problem is conduction in a square geometry, and the second one involves a semi-infinite medium with a rectangular cavity. In the case of square geometry, results show that the harmonic sine approximation method performs better than the finite-difference and multigrid methods everywhere within the computational domain, especially at points close to the singularity at the upper left and right corners of the square. The results from the harmonic sine approximation method for the semi-infinite domain problem with a very shallow rectangular cavity agree well with the analytical solution for a semi-infinite domain without the cavity. The results obtained from the harmonic sine approximation also agree well with the results from the finite-element package ANS...
TL;DR: In this article, a two-equation heat transfer model was constructed with the aid of the most up-to-date direct numerical simulation (DNS) data for wall turbulence with heat transfer.
Abstract: A rigorous two-equation heat transfer model has been constructed with the aid of the most up-to-date direct numerical simulation (DNS) data for wall turbulence with heat transfer. The DNS data indicate that the near-wall profile of the dissipation rate, ϵ1, for the temperature variance, kt (= t2¯ /2), is completely different from the existing model predictions. In this study, we performed a critical assessment of existing ϵ1 and ϵ1, equations for both two-equation and second-order closure models. Based on these assessments, we propose a new dissipation rate equation for temperature variance, taking into account all the budget terms in the exact ϵ1 equation. Also, the proposed ϵ1 equation is linked with a similarly refined kt equation to constitute a new two-equation heat transfer model. Comparisons of the present model predictions with the DNS data for a channel flow with heat transfer have shown excellent agreement for the profiles of kt and ϵ1 themselves and of the budget in the kt and ϵ1 equations. The...
TL;DR: In this paper, a very high-resolution (AVHR) scheme based on the normalized variable and space formulation methodology is presented, which is composed of the high resolution SMART scheme and a wide stencil scheme.
Abstract: A new, efficient, adaptive very-high-resolution (AVHR) scheme based on the normalized variable and space formulation methodology is presented in this article. The scheme is composed of the high-resolution SMART scheme and a wide stencil scheme based on a bounded seventh-order interpolation profile denoted here by BSEVENTH. The normalized weighting factor (NWF) acceleration technique and the deferred-correction procedure are used to implement the SMART and BSEVENTH schemes, respectively. The AVHR scheme switches to the seventh-order profile near a discontinuity or in the presence of a high gradient and to the SMART scheme in smooth regions. A new normalized switching criterion is developed, and its accuracy is illustrated for a number of scalar profiles. Numerical results for four purely convective test problems show that while it preserves the accuracy of the BSEVENTH scheme, the new AVHR scheme reduces the cost by more than 400% if compared to the traditional DC implementation of the BSEVENTH scheme and by 50% when compared to a new accelerated NWF-DC implementation of the BSEVENTH scheme
TL;DR: In this article, an economical method for the determination of the uncertainty domain of the numerical results compared with a computational fluid dynamics (CFD) code due to data uncertainties is proposed, based on the two-dimensional modeling of laminar airflow in natural convection.
Abstract: The aim of this article is to put forward an economical method for the determination of the uncertainty domain of the numerical results compared with a computational fluid dynamics (CFD) code due to data uncertainties In the first part of the study, we compare two methods of determining the domain of uncertainty: the quasi-Monte Carlo (QMC) method, and the finite-differences differential analysis (FDDA) method. This comparison is based on the two-dimensional modeling of laminar airflow in natural convection, in a small, thermally driven enclosure. The FDDA method is shown to be much more economical, in terms of computing time, than the QMC method.
TL;DR: In this article, a finite-difference-based method is presented for the solution of fluid mechanics and convective heat transfer problems related to the flow of purely viscous non-Newtonian fluids with or without a yield stress.
Abstract: A finite-difference-based method is presented for the solution of fluid mechanics and convective heat transfer problems related to the flow of purely viscous non-Newtonian fluids with or without a yield stress. The method is based on a block-implicit formulation of the discretited governing equations in which the continuity, momentum, and energy equations are all solved simultaneously along grid lines. A number of flow problems have been solved involving both attached and separated flows of power-law, Bingham, and Herschel-Bulkley fluids, demonstrating a robust algorithm with the ability to solve such problems efficiently and accurately.
TL;DR: In this article, the authors report on measures to improve the robustness and performance of a three-dimensional pressure-based multiblock algorithm and demonstrate the implementation of the treatment at block interfaces for arbitrary orientations of the blocks and investigate the effects of applying different boundary conditions on the pressure correction equation at block boundaries.
Abstract: We report on measures to improve the robustness and performance of a three-dimensional pressure-based multiblock algorithm. In particular, we demonstrate the implementation of the treatment at block interfaces for arbitrary orientations of the blocks and investigate the effects of applying different boundary conditions on the pressure-correction equation at block boundaries. It is shown that the pressure corrections in different blocks need to be strongly coupled in order to achieve better convergence. The performance of the algorithm is also shown to be significantly affected by the way in which the multiblock solution scheme is organized for the individual equations, which are solved sequentially. The method developed is then applied to two practical flow problems involving turbulent and rotational effects in complex geometries.
TL;DR: A finite-volume method for the solution of a scalar transport equation in a 2D incompressible fluid flow is presented and the evolution of the error norm slope confirms that the ULSS scheme is second-order accurate to solve 2D scalar Transport equations in hybrid structured / unstructured grids.
Abstract: A finite-volume method for the solution of a scalar transport equation in a 2D incompressible fluid flow is presented. The 2D transport equation is solved in a computational domain divided into polygon triangles and quadrilaterals allowing the general use of hybrid structured / unstructured grids. Convection discretization is performed with a new second-order-accurate upwind least-squares scheme (ULSS), and the system of algebraic equations is solved either by the biconjugate gradient stabilized method (BI-CGSTAB) or by the generalized minimal residual method (GMRES). The program language C++ was selected in order to manipulate the mesh data structure easily. Numerical solutions obtained using several different existing convection discretization schemes are compared with analytical solutions of several 2D test cases. The evolution of the error norm slope, as a Junction of the mesh parameters, confirms that the ULSS scheme is second-order accurate to solve 2D scalar transport equations in hybrid structured...
TL;DR: Numerical results for select benchmark compressible and incompressible steady-state Navier-Stokes problem definitions are presented, confirming theoretical prediction for attainment of monotone solutions devoid of excess numerical diffusion on minimal-degree-of-freedom meshes.
Abstract: A nonlinear subgrid embedded (SGM) finite-element basis is established for generating monotone solutions via a CFD weak statement algorithm. The theory confirms that only the Navier-Stokes dissipative flux vector term is appropriate for implementation of the SGM, which thereafter employs element-level static condensation for efficiency and nodal-rank homogeniety. Numerical results for select benchmark compressible and incompressible steady-state Navier-Stokes problem definitions are presented, confirming theoretical prediction for attainment of monotone solutions devoid of excess numerical diffusion on minimal-degree-of-freedom meshes.
TL;DR: In this article, a new family of adaptive very high resolution (AVHR) schemes is proposed for convection-diffusion type problems. But the adaptive schemes are accelerated by using in tandem the normalized weighting factor method to implement the HR scheme and the deferred-correction (DC) procedure to implement skew scheme.
Abstract: The family of skew very high resolution (VHR) schemes is adaptively combined with the family of high-resolution (HR) schemes to yield a new family of adaptive very high resolution (AVHR) schemes. A new simple adaptive switching criterion is devised. For convection-diffusion type problems the adaptive schemes are accelerated by using in tandem the normalized weighting factor method to implement the HR scheme and the deferred-correction (DC) procedure to implement the skew scheme. For flow problems the DC procedure is used to implement both types of schemes. Numerical results for the new family of AVHR schemes are compared in terms of accuracy and computation cost against those generated using the VHR base family of schemes by solving four problems: (I) pure convection of a step profile in an oblique velocity field, (2) driven flow in a skew cavity, (3) laminar sudden expansion of an oblique velocity field in a rectangular cavity, (4) and turbulent sudden expansion of an oblique velocity field in a rectangular cavity
TL;DR: In this paper, two commonly used quantities for deciding numerical iteration convergence of elliptic and parabolic flow fields are assessed: the maximum local value of the relative change, over two consecutive iterations, of the dependent variable is somewhat more appealing than the nondimensionalized sum of the local residual magnitudes.
Abstract: Two commonly used quantities for deciding numerical iteration convergence of elliptic and parabolic flow fields are assessed. Three 2-D test case problems, which are representative of many problems of momentum and/or heat transport, are used. The numerical model employed is the pressure-based finite-volume algorithm SIMPLE to solve the steady, incompressible-flow Navier-Stokes equations in full elliptic form. Considerable insight is gained by comparing the variation of the two quantities with the true mean relative error for the velocity components, the local mass imbalance, and the temperature. It is concluded that, except when using low underrelaxation factors, use of the maximum local value of the relative change, over two consecutive iterations, of the dependent variable is somewhat more appealing than the nondimensionalized sum of the local residual magnitudes.
TL;DR: The parallel version of the radiation model is evaluated by analyzing timing data for the computation of four test problems where the major parameters controlling numerical accuracy and computational cost are systematically varied.
Abstract: An efficient strategy for the implementation of the discrete transfer radiation model on a parallel computer architecture is presented. The parallel version of the radiation model is evaluated by analyzing timing data for the computation of four test problems where the major parameters controlling numerical accuracy and computational cost are systematically varied. A semiempirical model for the parallel run time, calibrated using the timing data, is described. The model's range of application is restricted to the IBM SP2 computer architecture; however, it could be recalibrated for any parallel computer.
TL;DR: A diagonal Cartesian method is proposed for the simulation of incompressible fluid flows over complex boundaries in Cartesian coordinates, which utilizes cell-centered nodes on a nonstaggered grid and uses boundary velocity information to avoid the specification of pressure values on boundaries.
Abstract: A diagonal Cartesian method is proposed for the simulation of incompressible fluid flows over complex boundaries in Cartesian coordinates. A structured grid is utilized for the sake of simplicity. The method approximates complex boundaries using both Cartesian grid lines and diagonal line segments. The grid is generated automatically, and the geometry approximation is shown to be more accurate than the traditional sawtooth method. Mass conservation on complex boundaries is enforced with an appropriate pressure boundary condition. The method, which utilizes cell-centered nodes on a nonstaggered grid, uses boundary velocity information to avoid the specification of pressure values on boundaries, An enlarged control-volume method is introduced for mass conservation and pressure boundary conditions on complex boundaries. The conservation of momentum on complex boundaries is enforced through the finite analytic (FA) method, using nine-point and five-point FA elements. Velocity boundary conditions at moving bou...
TL;DR: In this paper, an approximate solution for the one-dimensional Stefan problem with time-dependent boundary conditions was developed by reducing the governing partial differential equation (PDE) to two ordinary differential equations (OOE), and then simultaneously solving them along with the ODE for the moving-boundary interface condition.
Abstract: An approximate—locally analytic in the time variable and globally analytic in the space variable—solution is developed for the one-dimensional Stefan problem with time-dependent boundary conditions. Because of local analyticity in time, the solution is accurate over time intervals that are much larger than the step size permitted by most numerical schemes. The solution is developed by reducing the governing partial differential equation (PDE) to two ordinary differential equations (OOE), and then simultaneously solving them along with the ODE for the moving-boundary interface condition. The resulting scheme requires the solution of only a single transcendental (algebraic) equation at each time interval for the time-step-averaged temperature gradient at the moving boundary. The position of the moving boundary and the temperature distribution at the end of the time interval are then simply evaluated using approximate analytical expressions. Good agreement with reference solutions is obtained.