Enthalpy Formulation Research Articles

Influence of natural convection during melting and solidification of paraffin in a longitudinally finned shell-and-tube latent thermal energy storage on the applicability of developed numerical models

In this paper, melting and solidification processes in the shell-and-tube type latent thermal energy storage (LTES) with longitudinal fins have been numerically studied in order to determine an accurate and time-effective model for describing charging and discharging processes. Two mathematical models, one including natural convection in the liquid phase of the phase change material (PCM) and one neglecting it, have been developed. They both describe the transient three-dimensional problem of fluid flow and phase change heat transfer in the LTES. The computations, carried out using ANSYS Fluent 18.2 software, are based on an iterative, finite volume numerical procedure that incorporates enthalpy formulation for simulation of the phase change phenomenon. Results from both numerical models have been compared with data obtained by experimental investigations during melting and solidification, performed on the test LTES tank with paraffin RT 25 as the PCM and water as the heat transfer fluid (HTF). Numerical results of the model with natural convection have shown good agreement with experimental measurements, which indicates that the models could be used to accurately simulate phase change problems, while results obtained by model without natural convection have shown good agreement for solidification processes. Since numerical results obtained by both models are similar for solidification and significantly different for melting, it can be concluded that the model neglecting natural convection, with notably shorter calculation time, could be used for solidification modeling, while the model including natural convection in the liquid phase of the PCM should be used for melting modeling.

Impact of surface roughness parameterizations on tropical cyclone simulations over the Bay of Bengal using WRF-OML model

The impact of different surface roughness schemes in air-sea coupling on simulating tropical cyclones (TCs) over the Bay of Bengal is analyzed using Weather Research and Forecasting-Ocean Mixed layer (WRF-OML) modeling system. The sensitivity of three surface roughness schemes is tested by conducting three experiments for seven TCs namely Phailin (2013), Lehar (2013), Hudhud (2014), Vardah (2016), Gaja (2018), Fani (2019) and Amphan (2020). The first experiment (Opt0) is configured with surface drag and moist enthalpy from Garratt formulations, the second (Opt1) uses surface drag from Donelan and constant moist enthalpy formulations and the third experiment (Opt2) employs the modified moist enthalpy from Garratt formulations along with Donelan drag. Results of predicted track, intensities, precipitation and structure of the cyclones are highly sensitive to the surface exchange coefficients formulated through the roughness parameterization. The Opt2 followed by Opt1 experiments captured the deepening and mature phases of the storm close to the observed estimates for both pre- and post-monsoon TCs, by improving the ocean-atmosphere feedback of surface energy fluxes. Comparison of simulated moisture convergence, transport and precipitation with observations for all cyclones suggest that the Opt2 provides an improved representation of surface enthalpy fluxes playing a crucial role in the moisture transport and cloud microphysical processes. The improved simulation of TCs with Opt1 and Opt2 over Opt0 is attributed to the realistic simulation of mixed layer deepening and sea surface temperature cooling effects in the model. An interesting result found from the WRF-OML simulations with three roughness schemes is the maximum moisture transport is mainly concentrated on the right sectors while the moisture convergence and precipitation are distributed on the left sectors of the cyclones.

Combined close-contact and convective melting in a vertical cylindrical enclosure

The use of phase change materials (PCM) for latent heat thermal energy storage (LHTES) is receiving considerable amount of attention in recent years. Experimental findings have demonstrated that the so-called ‘close contact melting’ (CCM) enhances heat transfer during the melting process remarkably. Yet, the commercially-available numerical schemes are not suitable for the modeling of CCM. Therefore, in this study a new numerical model for combined convective and close-contact melting in an axisymmetric cylindrical geometry is devised. A full set of the governing conservation equations is solved using finite differencing framework, integrated with an advanced immersed boundary method for the fluid-solid interaction and enthalpy formulation for the phase change process into an original in-house code. First, the model is validated carefully for each physical phenomenon with known benchmarks. Then, a numerical study is conducted to elucidate the melting process in a vertical cylindrical enclosure heated isothermally from the bottom and side wall. Twelve study cases are considered in order to reveal the transient phase-change sequence dependence on the aspect ratio and the excess temperature. Detailed data on the flow and temperature fields are obtained. The overall results are generalized using a dimensional analysis, which includes the Fourier, Stefan and Archimedes numbers and the enclosure aspect ratio. A correlation for the melt fraction, suitable for all cases studied, is suggested.

A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation

Abstract. The accurate simulation of heat transfer with phase change is a central problem in cryosphere studies. This is because the non-linear behaviour of enthalpy as function of temperature can prevent thermal models of snow, ice, and frozen soil from converging to the correct solution. Existing numerical techniques rely on increased temporal resolution in trying to keep corresponding errors within acceptable bounds. Here, we propose an algorithm, originally applied to solve water flow in soils, as a method to solve these integration issues with guaranteed convergence and conservation of energy for any time step size. We review common modelling approaches, focusing on the fixed-grid method and on frozen soil. Based on this, we develop a conservative formulation of the governing equation and outline problems of alternative formulations in discretized form. Then, we apply the nested Newton–Casulli–Zanolli (NCZ) algorithm to a one-dimensional finite-volume discretization of the energy–enthalpy formulation. Model performance is demonstrated against the Neumann and Lunardini analytical solutions and by comparing results from numerical experiments with integration time steps of 1 h, 1 d, and 10 d. Using our formulation and the NCZ algorithm, the convergence of the solver is guaranteed for any time step size. With this approach, the integration time step can be chosen to match the timescale of the processes investigated.

Transient behaviour of heat transfer with complete evaporation process in Porous Channel with localised heating using non-Darcian flow and LTNE model

This study numerically presents the transient behaviour of complete evaporation process inside porous channel. Based on the modified enthalpy formulation under non-Darcy flow and Local Thermal Non-Equilibrium (LTNE) conditions, solutions have been obtained utilizing the finite volume method for various relevant parameters. The results have been validated with the experiments and they show good agreement between them. The results demonstrate that LTNE model should be employed while modelling the transient complete evaporation process in porous media. The non-Darcy effects have considerable influences on the locations of the initiation and termination of the phase change processes as compared to Darcy flow model. Therefore, the behaviours of the two-phase and the superheated vapour zones, when the steady-state solutions is reached, are substantially changed for non-Darcy flow as compared to Darcy model due this assumption provides an additional mechanism for heat transfer that represented by the heat exchange between the solid and fluid phases. It has been observed that the two-phase zone is considerably extended in the axial direction and this zone has not been occupied a large size towards transverse direction due to the axial velocity is noticeably higher than the transverse velocity, caused by reducing the diffusive energy through the two-phase region. The results indicate that non-Darcian effects leads to considerably extend the size of superheated region towards the outlet of the channel, which is not the case for Darcy model. The non-Darican effects become more pronounced for high Reynolds number and heat flux. The influences of varying in Reynolds number, heat flux, porosity, and solid thermal conductivity are substantial near the heated surface, whereas Darcy number has a minor impact on the solution of complete evaporation process. It can be emphasised from the present predictions that the non-Darcy flow along with LTNE condition is indispensable model for the complete evaporation process, especially under high mass flowrate as well heat flux.

3D numerical simulation and experimental investigation of pure tin solidification under natural and forced convection

The numerical simulation of the horizontal solidification of pure tin under natural convection and under forced convection induced by Electromagnetic Stirring (EMS) is presented and compared with experimental results obtained by the ‘AFRODITE’ benchmark setup, described in several previous publications [6–10]. The experiment consists in solidifying a rectangular ingot (100 × 10 × 60 mm) using two lateral heat exchangers which allow the application of a controlled horizontal temperature difference. The experimental temperature difference between the two lateral sides of the sample DT = 40 K and the cooling rate CR = −0.03K/s. Under these conditions the solidification front is planar throughout the experiment. Enthalpy formulation based on fixed-grid techniques is used for the numerical simulations of the phase-change problems, accounting for buoyancy convection and forced convection created by Lorentz forces generated by an external Traveling Magnetic Field (TMF). The temperature distribution obtained by numerical simulation is demonstrated to effectively reproduce the temperature maps obtained from the experimental measurements. The proposed 3D numerical model has demonstrated its effectiveness in predicting the effect of EM stirring on the solidification process in terms of thermal field, dynamic field and the shape and localization of the solidification front.

Computational analysis of copper@paraffin composite in a cylindrical cavity for enhanced thermal energy storage system

Abstract The present study shows a comparative behavior and thermal performance of the paraffin and copper-doped paraffin composite in a cylindrical cavity system with a heat source. The primary objective is to improve the heat storage capacity using metal (copper) particles due to their high thermal conductivity. A computational fluid dynamic model with physical enthalpy formulation has been used to predict paraffin-copper composite behavior as phase change material to absorb and store heat energy at given operating conditions. The progression of temperature and liquid liquefaction fraction for the cylindrical cavity filled with different PCM’s at an applied temperature (Th = 350 K) has been carried out during the study. Temperature and liquid fraction measurement were studied, and it was observed that configuration with copper-doped paraffin composite showed enhanced heat transfer as compared to pure paraffin and reduced its liquefaction time from 5000 s to 2400 s. Reduction in liquefaction time signifies improved heat transfer between the base fluid and PCM and may serve as next generation PCM for energy storage.

Extended enthalpy formulations in the Ice-sheet and Sea-level System Model (ISSM) version 4.17: discontinuous conductivity and anisotropic streamline upwind Petrov–Galerkin (SUPG) method

Abstract. The thermal state of an ice sheet is an important control on its past and future evolution. Some parts of the ice sheet may be polythermal, leading to discontinuous properties at the cold–temperate transition surface (CTS). These discontinuities require a careful treatment in ice sheet models (ISMs). Additionally, the highly anisotropic geometry of the 3D elements in ice sheet modelling poses a problem for stabilization approaches in advection-dominated problems. Here, we present extended enthalpy formulations within the finite-element Ice-Sheet and Sea-Level System model (ISSM) that show a better performance than earlier implementations. In a first polythermal-slab experiment, we found that the treatment of the discontinuous conductivities at the CTS with a geometric mean produces more accurate results compared to the arithmetic or harmonic mean. This improvement is particularly efficient when applied to coarse vertical resolutions. In a second ice dome experiment, we find that the numerical solution is sensitive to the choice of stabilization parameters in the well-established streamline upwind Petrov–Galerkin (SUPG) method. As standard literature values for the SUPG stabilization parameter do not account for the highly anisotropic geometry of the 3D elements in ice sheet modelling, we propose a novel anisotropic SUPG (ASUPG) formulation. This formulation circumvents the problem of high aspect ratio by treating the horizontal and vertical directions separately in the stabilization coefficients. The ASUPG method provides accurate results for the thermodynamic equation on geometries with very small aspect ratios like ice sheets.

Numerical simulation of a polymer melting process using solar energy

In this work, the melting process of a polymeric material is numerically investigated. In general, the heat required for melting plastics is obtained throughout electrical resistances or by burning fossil fuels. The heat transfer mechanisms of these common practices correspond to conduction and convection, respectively. This work explores the feasibility of using radiation as the primary mechanism of energy supply, which has not been widely studied. The energy to achieve the phase change from solid to liquid can be obtained by concentrated solar energy radiation. The total energy required is calculated solving the energy equation using the enthalpy formulation. An explicit formulation with an enthalpy linearization was implemented in the Mathematica programming language and compared with the solution in the commercial softwares Ansys Fluent and COMSOL Multiphysics showing a good agreement. Based upon numerical predictions, it is examined the effects of the relevant parameters, such as incident radiation and convective heat transfer coefficient, on the melting process. It is observed that under weather conditions commonly attained in different cities worldwide, with a radiation of 1000 W/m$^2$ and low convective losses with $h=8$ W/m$^2\cdot$K, the melting process of a cylindrical rod of 3/4 inches diameter can be carried out in around 2 hours.

A two-dimensional, higher-order, enthalpy-based thermomechanical ice flow model for mountain glaciers and its benchmark experiments

Understanding the dynamics of glaciers is essential for the knowledge of global sea-level rise, local freshwater resources in high mountain and arid regions, and the potential glacial hazards. In this paper, we present a two-dimensional thermomechanically coupled ice flow model named PoLIM (Polythermal Land Ice Model). The velocity solver of PoLIM is developed based on the Blatter–Pattyn approximation of Stokes flow. It uses an enthalpy formulation of the energy balance, an approach that is suitable for modeling the polythermal glaciers. PoLIM also includes a scheme for gravity-driven drainage of water in temperate ice, a subglacial hydrology model coupled with ice dynamics, and multiple basal sliding laws. The model has been verified by standard benchmark problems, including the ISMIP-HOM experiments, the enthalpy benchmark experiments, and the SHMIP experiments. PoLIM shows good performances and agrees well with these benchmark results, indicating its robust capability of simulating the thermomechanical behaviors of glaciers.

Microwave assisted freezing part 1: Experimental investigation and numerical modeling

The aim of this study was to develop an innovative process dedicated to enhancing the quality of frozen food using microwaves. A microwave assisted freezing device was designed at the laboratory scale to perform experiments in controlled conditions. Small samples of methylcellulose gels were frozen using nitrogen gas in a TE10 waveguide, where microwaves at 2.45 GHz were emitted intermittently or continuously. A numerical model was also developed to obtain results difficult to measure such as the local electric field and the corresponding energy density. The phase change part of the model was based on an enthalpy formulation and on the growth of spherical ice crystals. An original “hybrid 2D-3D” solving methodology was used to reduce the duration of the simulations. Favourable comparisons between the predicted temperatures and the experimental data highlighted the relevance of the models used for the thermophysical and dielectric properties. The analysis of the interactions between microwaves and matter, performed with numerical simulations, revealed the role of the freezing front as a boundary. The strong influence of sample size and of dielectric properties on the power density distribution were also illustrated when comparing our results with those published previously. The scientific knowledge obtained through this study and the original structure of the numerical model will be used to optimize microwave assisted freezing and link the process parameters to the reduction of ice crystal size highlighted in the companion paper. Industrial relevanceMicrowave assistance during freezing can improve frozen product quality by reducing ice crystal size. This innovative and promising process has not yet been given much attention. Developing an accurate model which describes microwave – matter interactions during phase change permits numerical simulations that can facilitate the design of industrial equipment, and determine optimal product dimensions.

Influences of Latent Heat on Temperature Field, Weld Bead Dimensions and Melting Efficiency During Welding Simulation

The present study aims to evaluate the influence of latent heat effect during phase change in a three-dimensional finite element numerical simulation of arc welding. A mathematical model is developed to incorporate the effect of latent heat of phase transformation using enthalpy formulation. The accuracy of the developed model was initially validated using available results in the literature and found consistent prediction. The validated numerical model is used to predict the temperature distribution, weld width, depth of penetration, melting efficiency with and without incorporating the effect of latent heat of fusion, solidification vaporization. The results obtained from a developed model incorporating the latent heat effect are in good agreement with the experimentally measured values, whereas a considerable deviation is observed between the predicted values and experimental values without considering the effect of latent heat. It is found that without considering the latent heat effect in the model, the mean error is 14%, 24.6%, 22% and 29.6% for peak temperatures, weld width, depth of penetration and melting efficiency respectively. Therefore, it can be concluded that the effect of latent heat of fusion, solidification, and vaporization is significant for analysis involving phase change and must be accounted for during numerical simulations.

Numerical investigation of two-phase flow in a horizontal porous evaporator with localised heating using non-Darcian flow and two equations model

In this article, a new mathematical model, based on the modified enthalpy formulation of Two-Phase Mixture Model (TPMM), is developed by considering non-Darcy flow and Local Thermal Non-Equilibrium (LTNE) conditions to describe numerically the two-dimensional problems of the complete evaporation process inside an asymmetrically heated porous evaporator. The governing equations have been discretised using Finite Volume Method (FVM) on a fixed non-staggered grid layout. The in-house code has been validated against experimental data and the results show good agreement between them. Three different models have been employed for the partitioning of the wall heat flux. Only the effect of the heat flux has been analysed, while all other parameters and properties have been kept fixed during the present investigation. The numerical simulations indicated that the locally high heat flux significantly influences the working fluid flow performances, and lead to extend the superheated vapour region towards the exit of the evaporator. The results also show that the two-phase zone significantly expanded in the axial direction and does not occupy a large portion inside the channel due to the energy transport through the two-phase region is significantly higher in the flow direction as compared to the transverse direction, caused by the decrease in the effective diffusion coefficient in this region and higher axial velocity. Numerical analysis reveal that the different models for the partitioning of the wall heat flux have almost no influence on the overall predictions, except in the vicinity of the wall. The results clearly demonstrate that the numerical simulations with the modified h-formulation by considering non-Darcy law and LTNE model are indispensable for the problem of complete evaporation process, particularly under the high both heat flux and inlet velocity conditions.

Transient thermal analysis of multilayer pipeline with phase change material

Based on thermal energy storage concept, multilayer composite pipeline with phase change material (PCM) has been proposed recently as an effective method to meet flow assurance challenges in deep water environment by releasing latent heat and extending cool-down time under shut-in condition. This work presents a mathematical model for the start-up and cool-down processes in a multilayer pipeline with an inner steel pipe, an intermediate PCM layer, and an external thermal insulation layer. With enthalpy formulation adopted for the PCM layer, the governing equations for heat conduction in the composite pipe walls and for energy transport in the produced fluid are solved numerically by explicit finite difference methods. The effects of the PCM thickness ratio, the phase change temperature, and the thermal conductivity ratio between the thermal insulation layer and phase change material are investigated. The effect of PCM hysteresis on the transient heat transfer process is also evaluated. It is found that there is an optimal PCM thickness ratio for a given combination of PCM and thermal insulation materials that maximizes the overall cool-down time. The cool-down time and the optimum PCM thickness ratio increase when the phase change temperature decreases. For a given PCM, the optimum PCM thickness ratio decreases for increasing thermal conductivity ratio. There is an interval of thermal conductivity ratio between 0.2 and 0.8 within which the PCM layer with optimum thickness ratio provides a significant increase in the cool-down time.

Processing of pre-cooked frozen Brussels sprouts: Heat transfer modelling as related to enzyme inactivation and quality stability

The finite element numerical method was applied to simulate heat transfer processes (heating, cooling and freezing) in irregular domains during the production of precooked frozen Brussels sprouts; the effect of temperature on thermo-physical properties in the partial differential equation, and experimental heat transfer coefficients were considered. During precooking, heat transfer was coupled to the inactivation kinetics of thermo-labile and thermo-resistant fractions of peroxidase. The temperature distribution, and enzyme activity in the vegetable domain were modeled; these simulations were validated, obtaining a satisfactory agreement between the predicted values and: (i) the experimental time-temperature curves, (ii) the residual enzyme activity of peroxidase measured in the vegetable.The simulation of the time-temperature curve during the freezing process is a complex nonlinear problem that requires knowledge of the thermo physical properties changing with temperature. A change of variables based on Enthalpy and Kirchhoff formulations, was adopted for the numerical solution. The effects of precooking times (3,6,10min) on quality parameters of the frozen vegetable (Browing Index, textural properties, residual peroxidase activity, ascorbic acid content) were measured in the samples after freezing and storage during 4 months at −20°C. The pre-cooking time was optimized to get acceptable quality parameters avoiding peroxidase reactivation during frozen storage.

Modeling of macrosegregation benchmarks using a stabilized finite element algorithm based on a semi-implicit pressure correction scheme

Purpose The purpose of this paper is to simulate two macrosegregation benchmarks with a newly developed stabilized finite element algorithm based on a semi-implicit pressure correction scheme. Design/methodology/approach A streamline-upwind/Petrov–Galerkin (SUPG) stabilized finite element algorithm is developed for the coupled conservation equations of mass, momentum, energy and species. A semi-implicit pressure correction method combined with SUPG stabilization technique is proposed to solve the convection flow during solidification. An analytically derived enthalpy method is adopted to solve the energy conservation equation. The nonlinearities of the energy and species equations are tackled by Newton–Raphson method. Two macrosegregation benchmarks considering the solidification of an Al-4.5 per cent Cu alloy and a Sn-10 per cent Pb alloy are simulated. Findings A very good agreement is achieved by comparison with the classical finite volume method and a novel meshless method for the Al-4.5 per cent Cu alloy solidification benchmark. Moreover, a unique reference numerical solution has been obtained. Besides, it is demonstrated that the stabilized finite element algorithm can capture the flow instability and channel segregation during solidification of the Sn-10 per cent Pb alloy. Originality/value A semi-implicit pressure correction method combined with SUPG stabilization technique is adopted to develop robust stabilized finite element algorithm for the macrosegregation model. A new enthalpy formulation for heat transfer problems with phase change is derived analytically.

Numerical analysis of phase change material (PCM) for concentrated solar power (CSP) application

This paper demonstrates the movement of the phase change front using the explicit numerical method applied to the one-dimensional enthalpy formulation of the Stefan equations. The numerical solution is then applied to several types of phase change materials found in the literature to know their behaviours. The obtained outcomes are interesting; however, each type of material is presenting its own advantages and disadvantages in terms of thermophysical properties. Implementation of eutectic compounds could, however, overcome some difficulties and lead to optimised results.

An enthalpy-based finite element method for solving two-phase Stefan problem

Stefan problem is a problem involving phase transition from solid to liquid or vice versa where boundary between solid and liquid regions moves as function of time. This paper presents numerical solution of one-dimensional two-phase Stefan problem by using finite element method. The governing equations involved in Stefan problem consist of heat conduction equation for solid and liquid regions, and also transition equation in interface position (moving boundary). The equations are difficult to solve by ordinary numerical method because of the presence of moving boundary. As consequence, the equations is reformulated into the form of internal energy (enthalpy). By the enthalpy formulation, solution of the heat conduction equations is no longer concerning the phase state of material. The advantage of the enthalpy formulation is that, finite element method can be easily implemented to solve Stefan problem. Numerical simulation of interface position, temperature profile, and temperature history has good agreement with the exact solution. The approximation of interface position using finite element method was found that it is more accurate than the approximation by using Godunov method. The simulation results also reveal that the finite element method for solving Stefan problem have smaller mean absolute error than the Godunov method.

Analysis of variable porosity, thermal dispersion, and local thermal non-equilibrium on two-phase flow inside porous media

This work involves the numerical simulations of two-phase flow inside porous media and the associated phase change processes in order to investigate the effects of thermal dispersion and variable porosity near the wall boundary. The modified enthalpy formulation has been employed. The mathematical model for the conservation of energy is based on the assumption of Local Thermal Non-Equilibrium (LTNE) condition. The relevant parameters such as free stream porosity, pore diameter, thermal conductivity of the solid phase, heat flux and Reynolds number have been used to analyse the significance of the above mentioned effects. The governing equations have been discretised using the Finite Volume Method (FVM). The developed two-dimensional code has been validated against experimental data, which definitely display that there is a good agreement between them. The numerical results indicated that considering the effects of thermal dispersion and variable porosity is significant, particularly near the heated wall. It is also observed that those two effects become more pronounced for the large values of pore diameter, free stream porosity and Reynolds number. The effect of variations in the free stream porosity, Reynolds number and the pore diameter becomes substantial near the wall when thermal dispersion and variable porosity effects are included, whereas the effect of variations in the heat flux and thermal conductivity of the solid phase have minor influence on the temperature distribution near the heated wall. It is evident that adding the effect of variable porosity and thermal dispersion have significant impact on the initiation and termination of phase change process, especially in the superheated vapour region as compared to the results that obtained by excluding those two effects. In general, it has been found that the present model provides a lower maximum fluid temperature near the wall as compared to that obtained by excluding those effects. Therefore, excluding these effects may lead to inaccurate predictions of two-phase flow with phase change process inside porous media. Most importantly, the thermal dispersion in the transverse direction plays an important role in the dispersion phenomenon inside porous media owing to the thermal boundary layer growth is more dependent on the transverse diffusion as compared to the axial diffusion.

Experimental and numerical characterization of an impure phase change material using a thermal lattice Boltzmann method

Of the phase change materials (PCMs) that regulate ambient temperature while reducing energy consumption, Octadecane is a good candidate because of its transparency properties and its adequate melting temperature. This study aims to characterize, through an approach combining numerical simulation and experiment, the behavior and thermo-physical properties of n-Octadecane. The approach takes into consideration the natural convection and the use of PCM's experimentally-obtained enthalpy-temperature curve that includes the supercooling and soluble impurities effects. The model uses the thermal lattice Boltzmann method based both on a partial bounce-back and an enthalpy formulation while including the experimental relationships. The numerical and experimental results exhibit good agreement. The approach adopted allows to highlight the behavior of the PCM to better characterize its thermo-physical properties.