Isothermal Phase Change Research Articles

A novel liquid fraction iteration methodology for addressing oscillatory issues in the total enthalpy method

The total enthalpy method (TEM) has been proposed and employed for several decades to address both isothermal and gradual phase change problems. However, recent investigations into isothermal phase changes have revealed that the phase interface predicted by the TEM oscillates with variations in spatial and temporal discretizations, due to the inconsistency between liquid fraction and enthalpy. To address this issue, and drawing inspiration from the liquid fraction iteration methodology used in the widely adopted heat source method (HSM) to ensure consistency between liquid fraction and temperature, this paper introduces a novel liquid fraction iteration methodology specifically tailored for the TEM, referred to as the iterative TEM (ITEM). This approach is validated against an experimental benchmark involving the melting of a pure substance. Grid and time-step dependence studies confirm that the ITEM effectively eliminates oscillations and exhibits convergence with respect to both grid size and time step. Moreover, the ITEM achieves accuracy comparable to that of the HSM. Finally, the computational costs associated with the ITEM are examined, revealing that costs increase rapidly once the grid Fourier number (Fo) exceeds one. Maintaining the grid Fo number below approximately 0.5 and ensuring the ratio of the relaxation factor to the grid Fo number to approach one significantly improve computational efficiency. The relaxation factor is a crucial parameter within the liquid fraction iteration scheme.

Bifunctional energy materials based on cellulose ionic complexes toward low-grade heat and photon energy storage

Discovering energy materials for low-grade heat and photon energy storage would advance the energy utilization from natural resources. Here, the ionic complexes based on cellulose and azobenzene-containing surfactant are presented as a new class of phase change materials for achieving this objective. Such materials could accomplish the energy charging in two different ways. Under UV light, the ionic complexes store photon energy and low-grade heat via the photoisomerization of azobenzene and the induced isothermal phase change at room temperature. While, without light condition, they could be employed for storing low-grade heat through a gradual phase change below 100 °C, by taking advantage of the intermolecular interaction changes. In the two different energy charging ways, the gravimetric energy densities for low-grade heat storage upon heating and photoinduced energy storage are up to 202 J/g and 152 J/g, respectively. The stored energy could be fast released as heat under Vis light irradiation. In addition, the fabrication of gold nanoparticles is proved to be a highly effective method for the fast energy charging under UV light. This work provides a new physicochemical principle for the development of energy storage materials based on biomolecules, which would also advance the economic value of cellulose as biomass resource.

Numerical investigation on flow and heat transfer characteristics in honeycomb-like microchannel heat sink encapsulated with phase change material

The present study proposes a novel model to numerically investigate the flow and heat transfer characteristics of a honeycomb-like microchannel heat sink (MCHS) encapsulated with phase change material (PCM). Compared with a single MCHS, the hybrid MCHS combines the advantages of microchannels in terms of micro efficiency with the isothermal phase change effect of PCM. The finite volume method is used to discretize the three-dimensional flow and heat transfer processes within the hybrid MCHS. Special attentions are paid to the effects of PCM, geometric parameters and local hot spots on the temperature uniformity and flow heat transfer performance. The results show that the honeycomb inflow structure design and the thermal management characteristics of PCM phase change cooling exhibit better temperature uniformity and cooling performance. The overall performance of the heat sink is best at microchannel height of 0.09 mm and inlet number of 5 under the range of parameters studied. Adjusting the channel height from 0.06 mm to 0.09 mm at a volume flow rate of 10 mL∙min−1 can reduce the thermal resistance by 51 % and the pumping power by nearly 20 %. Increasing the number of microchannel inlets results in better heat transfer performance, but this improvement comes at the expense of larger pressure drop. When a local hotspot occurs, the hybrid MCHS can reduce the temperature peak by 2–5 K, achieving a smoother temperature distribution.

Phase Change Thermal Storage Materials for Interdisciplinary Applications

Functional phase change materials (PCMs) capable of reversibly storing and releasing tremendous thermal energy during the isothermal phase change process have recently received tremendous attention in interdisciplinary applications. The smart integration of PCMs with functional supporting materials enables multiple cutting-edge interdisciplinary applications, including optical, electrical, magnetic, acoustic, medical, mechanical, and catalytic disciplines etc. Herein, we systematically discuss thermal storage mechanism, thermal transfer mechanism, and energy conversion mechanism, and summarize the state-of-the-art advances in interdisciplinary applications of PCMs. In particular, the applications of PCMs in acoustic, mechanical, and catalytic disciplines are still in their infancy. Simultaneously, in-depth insights into the correlations between microscopic structures and thermophysical properties of composite PCMs are revealed. Finally, current challenges and future prospects are also highlighted according to the up-to-date interdisciplinary applications of PCMs. This review aims to arouse broad research interest in the interdisciplinary community and provide constructive references for exploring next generation advanced multifunctional PCMs for interdisciplinary applications, thereby facilitating their major breakthroughs in both fundamental researches and commercial applications.

Adoption of CO2-based binary mixture to operate transcritical Rankine cycle in warm regions

This paper investigates the potential of CO2-based binary mixtures to operate a Rankine cycle with low-grade heat sources in warm regions. Utilizing low-grade heat sources in warm regions is a challenging task due to lower turbine inlet temperatures and higher pump inlet temperatures. A transcritical Rankine cycle with heat recuperation is implemented to minimize the heat losses linked with the isothermal phase change process of the working fluid. A proper selection of the working fluid suitable for a given scenario is one of the crucial tasks to maximize the thermal efficiency of the cycle. Energy and exergy analyses are presented for the cycle running with pure CO2 and CO2-based binary mixtures as a working fluid. Five different organic fluids (n-pentane, cyclopentane, cyclohexane, R1336mzz, and R600) are selected as additives to make a binary mixture. The effect of additive concentration in the mixture on the thermodynamic performance of the cycle is investigated. The cycle is found to perform significantly better if operated with a binary mixture than pure CO2. Binary mixtures of CO2 with n-pentane, cyclopentane, and cyclohexane perform substantially better than the rest with an increase in the thermal efficiency by more than 30% in comparison to the thermal efficiency of the cycle operating with pure CO2. The results indicate that the adoption of the CO2-based binary mixture for a transcritical Rankine cycle not only improves the overall thermodynamic performance but also reduces the operating pressures of the cycle which may lead to less expensive materials needed for the various components of the plant. Exergy analysis reveals that the irreversibility losses in the system during the heat transfer process are much larger than in the turbomachinery.

Pore-Scale Study on Convective Drying of Porous Media.

In this work, a numerical model for isothermal liquid-vapor phase change (evaporation) of the two-component air-water system is proposed based on the pseudopotential lattice Boltzmann method. Through the Chapman-Enskog multiscale analysis, we show that the model can correctly recover the macroscopic governing equations of the multicomponent multiphase system with a built-in binary diffusion mechanism. The model is verified based on the two-component Stefan problem where the measured binary diffusivity is consistent with theoretical analysis. The model is then applied to convective drying of a dual-porosity porous medium at the pore scale. The simulation captures a classical transition in the drying process of porous media, from the constant rate period (CRP, first phase) showing significant capillary pumping from large to small pores, to the falling rate period (FRP, second phase) with the liquid front receding in small pores. It is found that, in the CRP, the evaporation rate increases with the inflow Reynolds number (Re), while in the FRP, the evaporation curves almost collapse at different Res. The underlying mechanism is elucidated by introducing an effective Péclet number (Pe). It is shown that convection is dominant in the CRP and diffusion in the FRP, as evidenced by Pe > 1 and Pe < 1, respectively. We also find a log-law dependence of the average evaporation rate on the inflow Re in the CRP regime. The present work provides new insights into the drying physics of porous media and its direct modeling at the pore scale.

Designing and thermodynamic optimization of a novel combined absorption cooling and power cycle based on a water-ammonia mixture

The efficiency of generating power and cooling can be enhanced by using cogeneration systems. To simultaneously generate cooling and power, a novel ammonia-water cogeneration system driven by waste heat recovery is presented. In the proposed system, the boiler and condenser work at the non-isothermal and isothermal phase changes, respectively. To evaluate the novel cycle, firstly, for heat source temperature of 473K, the effect of maximum pressure of cycle and ammonia concentration is investigated. Considering the results, increasing ammonia concentration decreases the ratio of net power to cooling capacity while increasing maximum pressure increases the net power to cooling capacity ratio. At various ammonia concentrations and the constant pressure of 25 bar, the maximum exergy efficiency occurs at a concentration of 89%. Also, at a constant concentration of 85% and various maximum pressures, the optimum exergy efficiency occurs at 34 bar. To optimize the proposed system, the selected objective function is the total exergy efficiency. So, by a genetic algorithm, at a heat source temperature of 473K, the concentration and pressure are optimized at the values of 88.28% and 31.1 bar, respectively. Eventually, the optimum values of pressure and concentration for various heat source temperatures are presented in a table.

Thermal storage performance of eutectic sugar alcohols applied to buildings and enhancement of crystallization

Sugar alcohols are suitable for compact and high-density building thermal energy storage systems because of the high enthalpy, low price, and non-toxicity. But most sugar alcohol materials have severe supercooling characteristic, which limits their application as phase change materials (PCMs) in thermal energy storage systems. In this study, xylitol, sorbitol, erythritol, mannitol and dulcitol were selected as PCMs. Based on the five single sugar alcohols, ten multicomponent eutectic sugar alcohols were obtained which melting temperatures is between 60 °C and 90 °C applied to buildings. The thermal parameters of sugar alcohols were tested by differential scanning calorimetry (DSC), the supercooled eutectic sugar alcohols were stimulated by means of mechanical agitation to accelerate crystallization. The crystallization after excitation was monitored by microscope, and the supercooled crystallization phenomenon of eutectic sugar alcohols was investigated by isothermal phase change tests. The results showed that only three eutectic sugar alcohols could crystallize by stirring, and the melting temperatures of three eutectic sugar alcohols are about 80 °C. After multiple repetitive experiments, three eutectic sugar alcohols still have good crystallization characteristics. The results of the research provide a reference value in practical aspects for the design and selection of phase change materials in buildings.

Increasing the rating performance of paraffin up to 5000 cycles for active latent heat storage by adding high-density polyethylene to form shape-stabilized phase change material

The unstable phase transition of pure paraffin limits its application, particularly for active latent heat storage (ALHS). In passive latent heat storage, the addition of high-density polyethylene (HDPE) can decrease paraffin's leakage rate, which is associated with unstable phase transition and increase its durability after a large thermal number. This study aims to discover the possibility to use HDPE as a shape-stabilizer to increase the paraffin's performance for the ALHS system. Pure paraffin and composite paraffin/HDPE (80:20 wt%) are prepared as the reference material. The evaluation is focused on thermal properties, stability and performance during charging and discharging under different thermal cycles. Thermal cycle treatment uses temperature as a working boundary where the sample is heated from 30°C to 150°C, then cooled back to 30°C and repeated until 5000 cycles. Three different cycle references are chosen (1, 1000 and 5000 cycles) for each evaluation. The melting temperature of paraffin is decreased from 61.6°C to 59.8°C after 5000 cycles where Paraffin/HDPE slightly decreased from 60.4°C to 60.1°C. The heat fusion is decreased by 8.7% and 2.9% for paraffin and Paraffin/HDPE, respectively. The isothermal phase change is observed for paraffin/HDPE even after 5000 cycles with a higher rating performance by 30% faster to complete one cycle than pure paraffin. The other thermal performance is discussed in detail within the article, including the phase transition model during charging and discharging for the sample.

VarRhoTurbVOF 2: Modified OpenFOAM volume of fluid solvers with advanced turbulence modeling capability

varRhoTurbVOF contains a set of OpenFOAM volume of fluid (VOF) solvers for turbulent isothermal multiphase flows, which are variable-density incompressible. Unlike their official counterparts, where Favre-averaged and Reynolds-averaged velocities coexist in different equations, new solvers use Favre-averaged velocities consistently in all equations. This major update introduces three main improvements to the previous version of varRhoTurbVOF. First, the implementation is extended to VOF solvers for isothermal and non-isothermal phase change two-phase flows, where the flow is no longer incompressible. Second, in order to introduce backward compatibility and to avoid code duplication, the turbulence model construction procedure is redesigned such that solvers can determine whether the variable-density effect is considered or not in the turbulence modeling part based on the input file at run time. Third, the Egorov turbulence damping model for ω-based turbulence models is implemented with its most recent developments. Plus, an extension to ϵ-based turbulence models is developed and implemented. Program summaryProgram Title: varRhoTurbVOFCPC Library link to program files:http://dx.doi.org/10.17632/4t8z8vzyvs.2Developer’s repository link:https://github.com/wenyuan-fan/varRhoTurbVOF_2Licensing provisions: GPLv3Programming language: C++Journal reference of previous version: Wenyuan Fan, Henryk Anglart, 2020. varRhoTurbVOF: A new set of volume of fluid solvers for turbulent isothermal multiphase flows in OpenFOAM, Computer Physics Communications, 247, 106876.Does the new version supersede the previous version?: YesReasons for the new version: Implementation of new solvers and models; code redesign to maintain backward compatibility.Summary of revisions: The updated version mainly has three new features in comparison with the previous version: •The issue addressed in [1] exists in all OpenFOAM VOF solvers where the variable-density effect is neglected in the turbulence modeling. In order to solve this issue for phase change VOF solvers, two additional solvers, namely, varRhoInterPhaseChangeFoam and varRhoInterCondensatingEvaporatingFoam, are implemented based on their official counterparts.•The procedure of constructing the turbulence model inside individual solvers is redesigned. Using the new design, solvers will construct turbulence models with or without the variable-density effect being considered based on the user input. When the latter mode is activated, solvers will behave just like their official counterparts. In such a way, the backward compatibility of the code is achieved without introducing code duplications.•The Egorov turbulence damping model [2], which is a popular approach to phenomenologically modify turbulence behaviors near a two-phase interface, is implemented. In addition to the original model, recent modifications, e.g., a new length scale calculation and asymmetric damping treatments [3], are implemented as well. The Egorov model only works with ω-based turbulence models. In the new release, the model is extended to ϵ-based turbulence models following the idea proposed in [4] such that the turbulence damping model is applicable to commonly used turbulence models.Nature of problem: Within the VOF framework, the mixture has a density changing with phases. However, in quite a few OpenFOAM VOF solvers, the variable-density effect is considered in the momentum equation but neglected in the turbulence modeling. As a result, Favre-averaged velocities and Reynolds-averaged velocities are used in the momentum equation and turbulence quantity equations, respectively and simultaneously, introducing a severe self-inconsistency. Solutions to this problem have been proposed in [1] with no backward compatibility support. Also, no phase change solvers are supported in the previous version. Another issue with turbulent VOF simulations is that, even with the correct implementation, certain modifications are still needed to correct the turbulence behavior around the interface.Solution method: The extension to phase change VOF solvers is straightforward according to [1]. Regarding the construction of turbulence models, two additional fields, namely, rhoTurb and rhoPhiTurb, are created and used in turbulence models. If the user chooses to use the variable-density turbulence models, real two-phase density rho and mass flux rhoPhi will be assigned to rhoTurb and rhoPhiTurb, respectively. Otherwise, rhoTurb and rhoPhiTurb will use unity and volume flux phi, respectively. This run time selection design enables solvers to operate in two modes, which guarantees the backward compatibility and avoids code duplications. The implementation of the turbulence damping model also takes the backward compatibility into account. At run time, the code will automatically determine whether the mixture density should be included in the turbulence damping model by checking the dimension of the corresponding equation.Additional comments: A manual is provided to introduce how the turbulence damping modeled is formulated and how to use the model. It should be emphasized that the Egorov damping model is a phenomenological model. Model parameters should be carefully selected for any given mesh and flow condition.

Numerical investigation drag coefficient of micro-encapsulated roughened PCM particles in laminar flow

Phase Change Materials (PCM) have received considerable attention in recent years in many thermal energy storage applications, due to the isothermal phase change process and large storage capacities. This study deals with the hypothetical structure of micro encapsulated phase change materials of matrix type and the particle’s surface roughness influence on its laminar drag coefficient. The investigation has been performed by numerical simulation using commercially available software. Results show that the drag coefficient decreases as particles’ surface roughness increases, even at low Reynolds numbers.

Driving a Liquid Crystal Phase Transition Using a Photochromic Hydrazone.

The dynamic manipulation of the properties of soft matter can lead to adaptive functional materials that can be used in advanced applications. Here we report on a new chiral dopant, built on an isosorbide scaffold attached to two bistable hydrazone-based light switches that can be used to control the self-assembly, and hence photophysical properties, of nematic liquid crystals (LCs). The bistability of the switch allows kinetic trapping of various helical assemblies as a function of the photostationary states, resulting in the reflection of different wavelengths of light. Surprisingly, doping 5CB with the chiral switch, followed by irradiation with blue light, triggers an isothermal phase change from the helical cholesteric phase to the untwisted lamellar smectic A* phase. This transition was used to modulate the transparency of a LC film, resulting in a light-gated optical window.

Microencapsulated n-alkane eutectics in polystyrene for solar thermal applications

Some n-alkanes (n-tetracosane: C24, n-eicosane: C20, n-nonadecane: C19, n-octadecane: C18, and n-heptadecane: C17) and their eutectic mixture have been encapsulated in polystyrene (PS) as phase change materials (PCMs) in order to benefit from solar energy. The eutectic melting points and combination ratios of C17-C18, C20-C17, C20-C19, and C20-C24 eutectic mixtures (EMs) were found out theoretically and proven experimentally before the encapsulation processes. Microencapsulated eutectic mixtures (MEEMs) were structurally proven using a Fourier Transform Infrared (FTIR) Spectroscopy Analysis and morphologically investigated using a Scanning Electron Microscopy (SEM), a Polarized Optical Microscopy (POM), and a Particle Size Distribution (PSD) analyzer. Differential Scanning Calorimetry (DSC) analysis was primarily considered for characterization. MEEMs had reversible and mostly isothermal phase change temperatures between 21.0 and 35.9 °C and enthalpy between 61.2 and 146.1 J/g. These values were intentially suitable for solar thermal applications. Thermogravimetry Analysis (TGA) was performed to reveal thermal stability limits of the MEEMs. Accelarated thermal cycling experiments were performed to investigate stability of MEEMs.

A comprehensive benchmark of fixed-grid methods for the modeling of melting

Numerical simulation of melting processes is known to be tricky yet due to new application fields they raise again the attention of both, researchers and engineers. Several approaches were developed to solve this moving boundary problem, fixed grid methods being the most widespread ones. However, up to now no holistic and quantitative comparison, even of the most common enthalpy methods (some of these being included in famous commercial software), exists. Therefore, within this work, an exhaustive study is performed to evaluate the corresponding accuracy of the five most used macroscopic energy formulations with a strong coupling between temperature and enthalpy. In addition to pure conductive cases with analytical solution, an experimental test including natural convection is considered. Thus, the influence of the time step, of the grid and of the tolerance within the energy equation are investigated. In the same way several thermodynamical modelings are considered: either isothermal or non-isothermal phase change, several temperature ranges being used in this later case. From the more than 2500 simulations obtained, painstaking quantitative error analysis are conducted and quality thresholds are defined. Generally, approaches formulated in terms of enthalpy appear to be more robust than the ones using temperature formulations instead. The popular effective heat capacity method (with iterative correction) leads to the largest errors when considering the complete enthalpy in the convection term.

Multi-Objective Optimization of Organic Rankine Cycle Power Plants Using Pure and Mixed Working Fluids

For zeotropic mixtures, the temperature varies during phase change, which is opposed to the isothermal phase change of pure fluids. The use of such mixtures as working fluids in organic Rankine cycle power plants enables a minimization of the mean temperature difference of the heat exchangers, which is beneficial for cycle performance. On the other hand, larger heat transfer surface areas are typically required for evaporation and condensation when zeotropic mixtures are used as working fluids. In order to assess the feasibility of using zeotropic mixtures, it is, therefore, important to consider the additional costs of the heat exchangers. In this study, we aim at evaluating the economic feasibility of zeotropic mixtures compared to pure fluids. We carry out a multi-objective optimization of the net power output and the component costs for organic Rankine cycle power plants using low-temperature heat at 90 °C to produce electrical power at around 500 kW. The primary outcomes of the study are Pareto fronts, illustrating the power/cost relations for R32, R134a and R32/R134a (0.65/0.35mole). The results indicate that R32/R134a is the best of these fluids, with 3.4 % higher net power than R32 at the same total cost of 1200 k$.

Thermally conductive, dielectric PCM-boron nitride nanosheet composites for efficient electronic system thermal management.

Phase change materials (PCMs) possessing ideal properties, such as superior mass specific heat of fusion, low cost, light weight, excellent thermal stability as well as isothermal phase change behavior, have drawn considerable attention for thermal management systems. Currently, the low thermal conductivity of PCMs (usually less than 1 W mK-1) greatly limits their heat dissipation performance in thermal management applications. Hexagonal boron nitride (h-BN) is a two-dimensional material known for its excellent thermally conductive and electrically insulating properties, which make it a promising candidate to be used in electronic systems for thermal management. In this work, a composite, consisting of h-BN nanosheets (BNNSs) and commercialized paraffin wax was developed, which inherits high thermally conductive and electrically insulating properties from BNNSs and substantial heat of fusion from paraffin wax. With the help of BNNSs, the thermal conductivity of wax-BNNS composites reaches 3.47 W mK-1, which exhibits a 12-time enhancement compared to that of pristine wax (0.29 W mK-1). Moreover, an 11.3-13.3 MV m-1 breakdown voltage of wax-BNNS composites was achieved, which shows further improved electrical insulating properties. Simultaneously enhanced thermally conductive and electrically insulating properties of wax-BNNS composites demonstrate their promising application for thermal management in electronic systems.

A Lagrangian–Eulerian finite element algorithm for advection–diffusion–reaction problems with phase change

This paper presents a particle-based Lagrangian–Eulerian algorithm for the solution of the unsteady advection–diffusion–reaction heat transfer equation with phase change. The algorithm combines a Lagrangian formulation for the advection + reaction problem with the Eulerian-based heat source method for the diffusion + phase change problem. The coupling between the Lagrangian and Eulerian subproblems is achieved with a phase change detector scheme based on a local latent heat balance and a consistent/conservative interpolation technique between Lagrangian particles and the Eulerian grid. This technique makes use of an auxiliary (finer) Eulerian grid that provides a simple and efficient method of tracking internal heterogeneities (e.g. phase boundaries), allows the use of higher order integration quadratures, and facilitates the implementation of multiscale techniques. The performance of the proposed algorithm is compared against one- and two-dimensional benchmark problems, i.e. pure rigid-body advection, isothermal and non-isothermal phase change, two-phase advective heat transfer and chemical reactions coupled with diffusion and advection. The numerical results confirm that the proposed solution method is accurate, oscillation-free and useful for and applicable to a wide range of fully coupled problems in science and engineering.

Phase Change Materials Applications to Optimize Cooling Performance of Buildings in the Mediterranean Area: A Parametric Analysis

Building integrated thermal energy storage systems cover a wide range of techniques and solutions depending on technology applications and aims. They however all have in common the concept behind: being able to store energy for later use in order to reduce the time mismatch between energy availability and demand. In this context, Phase Change Materials (PCMs) fit the above description, since they would allow for mostly isothermal phase change within normal thermal comfort range. In order to face the typical challenges of the Mediterranean climate, the following concept was elaborated: the idea is to use the phase change mechanics as a substitute to the thermal inertia of massive walls to obtain a similar effect in lightweight structures. A simulated test room was arranged and through parametric analysis the potential of PCM for energy efficiency in connection to natural ventilation strategies was assessed. Simulation results vary according to the configuration analysed but in all cases, cooling consumption reduction reached at least 40% with the use of PCMs. Air Temperature during peak hours in summer can be reduced by more than 7-8°C.

An enrichment scheme for solidification problems

A new enriched finite element formulation for solving isothermal phase change problems is presented. We propose a fixed mesh method, where the discontinuity in the temperature gradient is represented by enriching the finite element space through a function whose definition includes a gradient discontinuity. Generally, in these types of formulations, the enrichment location (the location of the solidification front) is determined through a level set auxiliary scheme. In this work, this position is determined implicitly by constraining the temperature at the phase change boundary to be equal to the melting temperature. Several numerical examples are presented to show the application of the method.

What is the most suitable fixed grid solidification method for handling time-varying inverse Stefan problems in high temperature industrial furnaces?

This paper presents a simple inverse heat transfer method for predicting the time-varying bank thickness of the phase change protective layer inside a high temperature furnace. The direct problem is handled with the single phase method which is neglecting the latent heat. The inverse method rests on the Adjoint Problem and the Conjugate Gradient Method. It is shown that the iterative inverse procedure based on the single phase method predicts a virtual incident heat flux in the liquid zone that yields accurate time-varying bank predictions. Results also indicate that the benefits of the virtual iterative approach are twofold: the CPU time required for solving the inverse problem is reduced and the lagging effect inherent to the inversion is diminished for non isothermal phase change processes. For typical melting furnace operating conditions, it is shown that the virtual approach doubles the allowable diagnostic frequency for predicting the time-varying bank thickness.