Radiative Heat Exchange Research Articles

Quantitative evaluation of radiative heat transfer from reactor surface to multiphase reaction medium in a supercritical water gasification reactor for coal

Supercritical water gasification (SCWG) of coal is a clean coal utilization technology, which converts organic matter in coal into hydrogen-rich gases through a series of thermo-chemical reactions. In this process, thermal radiation is an important heat transfer mechanism and significantly affects thermal efficiency and gasification performance. This paper aims to develop a comprehensive numerical model integrating with solving radiative transfer to precisely simulate the process of SCWG of coal, and quantitatively evaluate non-grey radiative heat exchange between reactor surface and multiphase reaction medium. The simulation results are validated against experimental data. It is shown that under practical operating conditions, thermal radiation accounts for 30 % – 42 % of total heat exchange in the reactor. More than 94 % of radiant energy lies in the spectral range of 1.99 – 50.0 μm, which just covers the three most important radiation bands of supercritical water. Because both coal particles and supercritical water participate in radiative heat transfer, enhancing radiative heat transfer can effectively improve reaction medium temperature and gasification rate. On average, for every 20 % increase in the proportion of radiative heat transfer in the reactor, fluid medium temperature increases by around 7.1 K, and the gasification reaction rate rises by 9.5 %. These findings would provide valuable guidance for reactor design and optimization in practice.

Niepewność pomiarów średniej temperatury promieniowania za pomocą termometrów kulistych = Uncertainty of mean radiant-temperature measurements using globe thermometers

In the analysis of the human heat balance, the biggest problem is to precisely determine the amount of energy absorbed by a human organism under dynamically changing solar conditions and while being surrounded by many different surfaces that can absorb, emit or reflect solar radiation in different ways. Therefore, in thermal comfort studies some simplifications of the radiation field around human body are necessary and thus the concept of mean radiant temperature (Mrt) is most often used. Mrt values can be derived either from direct instrumental measurements or calculated by one of the microclimate models. The most accurate way to determine the Tmrt outdoors is by integral radiation measurements and the calculation of angular factors (i.e. the proportion of radiation received by the human body from different directions). This method involves simultaneous measurements of long- and short-wave radiation flux densities in six directions, which requires three sets of net radiometers. Although this method of Mrt determination describes the radiation environment accurately and independently of the variable wind field, it requires the use of expensive and complex measurement devices. Cheap and easy to operate alternative offers globe thermometer. In steady state conditions, readings from the air temperature sensor inside the globe thermometer (tg) will reflect the convective and radiative heat exchange around the globe thermometer. Knowing tg, as well as globe emissivity and diameter, the Mrt may be calculated. In this research three different globe thermometers were used to determine Mrt: 15-cm standard globe thermometer (SGT) and two 4-cm custom made acrylic globe thermometers (AGT) painted in black and grey. The measurements were conducted in the summer in Warsaw, on two selected green areas, under sunny and warm weather conditions. In each location Mrt measurements were taken in two spots – in the shade of trees and in the open grassy spaces, with high SVF. Our results demonstrate, that globe thermometers, although being relatively cheap and less complex instruments for determining Mrt, can be used to a limited extend, due to high instability of their indications under significant radiative loads.

Effect of surface radiation on double diffusion convection in a low Mach number compressible gaseous mixture (H2–air)

This study aims to numerically investigate the effect of surface radiation on double diffusion convection in a low Mach number compressible gaseous mixture, specifically focusing on the H2–air system in a square cavity. The cavity is subjected to low horizontal temperature and concentration gradients and isolated from its flat walls. The mathematical model's equations were discretized using the finite volume method based on the semi-implicit method for pressure-linked equations revised algorithm for the pressure–velocity coupling. The radiosity method is employed to calculate the radiative heat exchange between the internal walls of the cavity. This work addresses a significant research gap by exploring the interplay between surface radiation, compressibility effects, and double diffusion convection. It makes a novel contribution to the field and has implications for combustion, astrophysics, and industrial heat exchange processes. Comparing the results with those obtained for a light gaseous mixture, the findings demonstrate that the presence of radiation considerably modifies the thermal, dynamic, and mass fields, as well as the thermophysical properties of the gaseous mixture in the case of heavy gaseous mixtures. These modifications can reach up to 20%.The outcomes of this study provide a foundation for further research and experimentation, with applications ranging from engineering to astrophysics.

Revealing patterns in the behavior of a reinforced concrete slab in fire based on determining its stressed and deformed state

The object of research is the fire resistance of reinforced concrete ribbed slabs. The subject of research is the effect of the level of mechanical load on the fire resistance of the studied reinforced concrete ribbed slab under the influence of fire. Currently, the assessment of the fire resistance of such structures using the tabular method is significantly limited by certain geometric parameters of these plates. It is also not considered possible to apply the zone method, since reinforced concrete ribbed slabs consist of components that receive thermal effects according to various scenarios and geometric parameters, which is not taken into account in Eurocode 2. Experimental calculations carried out in the current work using a refined procedure involving the finite element method allow solving the actual scientific and technical problem related to the determination of the dependence of the fire resistance of these structures on the level of load applied. The calculation of the temperature spread over a reinforced concrete ribbed slab under the influence of the standard temperature regime of fire was carried out using convection and radiant heat exchange, which is recommended by Eurocode 2. The iterative implicit Newton-Raphson method was used to solve the mechanical problem. Evaluation of fire resistance of reinforced concrete ribbed slabs was carried out according to the onset of signs of the limit state of fire resistance due to the loss of load-bearing capacity. According to the results, it was established that at 100 % load level of the structure under investigation, the critical deflection of more than 268 mm and the rate of growth of deformation exceeding 18 mm/min were recorded simultaneously on minute 43.9. According to the results of the research, the regularity of the limit of fire resistance of reinforced concrete ribbed slabs from the level of the applied mechanical load was established. This will make it possible to design and build buildings and structures using the specified building structures with guaranteed fire resistance classes, which improves the level of safety for people at the facilities

Investigation of the heat transfer through down clothing with quilting structure

Down clothing is commonly adopted as cold protective clothing to maintain human thermal comfort. However, its surface temperature distribution, heat transmission and thermal insulation are still not fully understood due to the complex heat and mass transfer caused by the quilting structure and varying mass density. In this work, we comprehensively explored the thermal behavior of down clothing with quilting structures. Objective tests were carried out by the thermal manikin and infrared thermal imager in a climate room at a fixed temperature of 10 ± 1 °C and relative humidity of 40 ± 5%. Four different down clothing were examined to investigate the effect of down lattice structure and mass density on heat flux, surface temperature distribution and thermal resistance. The results revealed a distinct valley in the surface temperature along the down lattice due to the quilting structure. A space-averaged method to estimate the thermal insulation of down clothing would neglect variation among different local positions. Besides, the thermal insulation performance significantly improved with the higher down mass density. Because the average clothing thickness as well as the thickness near quilting position both increased, resulting in an additional reduction of heat dissipation. Furthermore, the surface convective and radiative heat exchange decreased with the higher down mass density as a result of lower surface temperature. The lowest comfort temperature for clothing with various down mass density under activity level 2 met and 4 met was also estimated. The results can help to improve the design of down clothing's thermal performance and enhance human comfort in cold environments.

Micropolar Hydromagnetic Fluid Over a Vertical Surface in Darcian Regime: An Analytical Approach

In the present paper, the researcher investigates the mutual impact of radiative heat and mass exchange on hydromagnetic micropolar fluid moving along an infinite vertical surface in a porous regime. The goal of the research is to investigate the impact of convective temperature and mass flow on hydromagnetic motion of micropolar fluid across a vertical plate ingrained in a porous regime. The conservation equations with appropriate boundary conditions are resolved analytically by assuming a convergent series solution and thus obtained the analytical solutions for velocity, angular velocity (microrotation), temperature and molar-concentration. The novelty of the current work is that it takes heat transfer into account while considering for the impacts of chemical reaction in a micropolar fluid flow of reactive diffusing species. The influence of different physical variables on temperature, molar-concentration, velocity and angular velocity of the fluid molecules have been presented graphically for dual solutions. It is seen that the micropolar parameter and porosity of the medium play a significant behaviour over the momentum and thermal boundary layers. This investigation may involve with various disciplines of chemical engineering, bio-mechanics and medical sciences. The outcomes of the present study have significant applications in MHD generators and geothermal resource extraction.

Conditions for use of long-wave infrared camera to measure the temperature of the sky

During thermovision measurements, it is necessary to know the radiative temperature of surroundings of the tested object. For objects exposed to an open atmospheric space, such as buildings, the surroundings consists of the ground and the sky. The article concerns the measurement of sky temperature using an infrared camera with a long-wave spectral range of operation. This temperature is needed for the thermovision measurement of the real temperature of walls or roofs of buildings. It is also useful for calculating the emission of the sky over the entire spectral range, what is useful during the radiation heat exchange calculations. The use of an infrared camera for this purpose is a very convenient method, as the thermographer has this device at his disposal when performing object diagnostics. However, this measurement result is exposed to a systematic inaccuracy resulting from the radiation distribution of atmosphere inside the atmospheric window and the shape of infrared camera detector and lens technical characteristics. In addition, there is often a problem with measuring the relatively low temperature of the sky. The work contains an analysis and a proposal of the aforementioned problems solution.

Conduction–Radiation Coupling between Two Distant Solids Interacting in a Near-Field Regime

In the classical approach to dealing with near-field radiative heat exchange between two closely spaced bodies, no coupling between the different heat carriers inside the materials and thermal photons is usually considered. Here, we provide an overview of the current state of research on this coupling between solids of different sizes while paying specific attention to the impact of the conduction regime inside the solids on the conduction–radiation coupling. We describe how the shape of the solids affects this coupling, and show that it can be located at the origin of a drastic change in the temperature profiles inside each body and the heat flux exchanged between them. These results could have important implications in the fields of nanoscale thermal management, near-field solid-state cooling, and nanoscale energy conversion.

Actual sky visibility under shading by trees with different crown densities

Introduction. The development of green infrastructure is an important task in urban planning strategies to reduce urban heat island effect and improve urban ecosystem. The aim of this study is to determine the actual visibility of the skyline when shaded by trees of different crown densities. The scientific novelty is that for the first time a study of the actual sky visibility with trees shading as complex heterogeneous systems has been conducted to assess the effect of mitigating of urban heat island. The practical significance lies in the establishment of the actual visibility of the firmament under the shading of trees with different crown density, which enables the correction of simulation models to analyze the parameters of the microclimate of the urban environment under given climatic conditions.
 
 Materials and methods. Various approaches to determining the sky visibility are considered. Direct measurements were made by photographing with a special fisheye lens and a smartphone. The research was carried out in a temperate continental climate (according to the international climate classification Köppen — type Dfa). Several tree species most commonly found in landscaping under the climate conditions in question and having different crown densities were selected: elm, oak, maple and poplar. For each, several samples were selected as required for data averaging. A total of 27 samples were studied.
 
 Results. A diagram of measured averaged values of the sky visibility factor was plotted, taking into account shading by trees with different crown densities in different periods of the year. To protect trees with dense spreading crowns from the direct solar radiation in the summer daytime, it is better to use trees with dense spreading crowns to intensify the radiant heat exchange of the urban surface with the sky at night — trees with less dense crowns. In case of the need to soften the urban heat island during the day, hybrid solutions are the most effective.
 
 Conclusions. The actual assessment of the sky visibility, made by the authors taking into account shading by trees, allows us to make corrections in simulation models to analyze the parameters of the microclimate of the urban environment under the given climatic conditions. The results of the study make it possible to select the type of tree providing the required shading and urban heat island’s mitigation effect.

A Review of Radiative Heat Transfer in Fixed-Bed Particle Solar Receivers

A highly efficient receiver is required because re-radiation loss increases dramatically with increased working temperature. Among a large number of receivers, the fixed-bed Particle Solar Receiver (PSR) represents a new pathway to high temperature with maximum overall thermal efficiency. The incoming solar radiation can penetrate deeper into the fixed-bed PSR filled with semi-transparent quartz and ceramic particles (spheres or Raschig rings), resulting in an increased volumetric effect. Reports show that an optimized PSR can realize overall receiver efficiency of around 92% at outlet temperatures above 1000 K, and achieve the annual temperature above 1000 K over 65% annual operating hours integrated with a concentrated solar power (CSP) system. To fully understand radiative heat transfer characteristics and provide deep insight into thermal efficiency, radiation energy is classified as incident solar radiation and radiative heat exchange in two parts. The transfer mechanism, the solution method and the progress of the investigation for each section are summarized and discussed in detail. Then, challenges and future directions, including an innovative design method, an improved experimental approach and an effective simulation method are proposed to put forward this receiver to be a preferred substitute in advanced, high-temperature power cycles.

Combined numerical approach for the evaluation of the energy efficiency and economic investment of building external insulation technologies

The objective of this paper is to examine the energy efficiency and economic feasibility of different coating technologies used to improve the thermal insulation of external walls in buildings. The comparison is made between traditional coat insulation and ventilated façade, evaluating their impact on energy consumption to maintain a constant indoor temperature throughout the year by means of a combined numerical approach. . Furthermore, the paper investigates the effect of opening and closing the air gap in ventilated façades on thermal insulation during the winter and summer seasons. Energy efficiency calculations are employed to estimate the economic investment required for implementing the different insulation solutions.The paper proposes an innovative combined approach to determine the performance of the building insulation technologies. Firstly, a computational fluid dynamics (CFD) simulation is carried out on the full three-dimensional geometry of the building during two reference days representing extreme temperature and sun radiation conditions during the summer and winter. This modeling includes the effects of solar radiative heat transfer during the day: for a chosen date, time, and geographical location, the model computes sun altitude and azimuthal angles, along with the corresponding direct and diffuse solar fluxes. In addition, the model uses a multiband thermal radiation approach to capture the different nature of radiative heat exchange according to the light wavelengths. The total heat transfer coefficient of the building walls in each scenario is calculated through the computational fluid dynamic analysis and implemented in an in-house developed library based on the open source Open-Modelica platform to simulate the energy requirement of the building throughout the year. This combined numerical approach provides a comprehensive performance analysis of the studied technologies in terms of electric energy and fuel consumption required for maintaining a constant indoor temperature and internal ambient comfort.The results of the simulations demonstrate that by adopting the two proposed solutions, there was a potential to save approximately 37% of the fuel required for the heating system and more than 51% of electric energy required for the air conditioning systems. Finally, the payback period for each scenario is calculated, and it was found that coat insulation offers the best balance between thermal performance improvement and economic effort, with a payback time close to 20 years.

Heat transfer and thermal analysis in a semi-spherical fin with temperature-variant thermal properties: an application of Probabilists’ Hermite collocation method

Finned surfaces are utilized in several industrial and technological applications including radiators in cars, air conditioning systems, and computer CPU heat sink to dissipate heat through convection or radiation. In this regard, the heat transfer enhancement utilizing the fins has received more attention in the scientific field. Motivated by this, the present research explicates temperature variation through a convective-radiative semi-spherical fin (SSF) with internal heating by taking into consideration of linear and nonlinear temperature-dependent heat transfer coefficient and thermal conductivities. Furthermore, the efficiency and heat transfer aspects of the fin are elaborated. The Rosseland approximation is used in this study to analyze radiation heat exchange. The developed energy equation is converted to a nonlinear ordinary differential equation (ODE) using dimensionless terms, and the probabilists’ Hermite collocation method (PHCM) is used to solve this problem. The governing parameters have been varied in their appropriate ranges to illustrate their impact on the thermal profile, and the results have been portrayed graphically. Some notable key findings of the current analysis are that as the scale of the convection–conduction and the radiation number upsurges, the thermal distribution through SSF drops. On the other hand, thermal distribution escalates for heat generation and thermal conductivity parameters.

INVESTIGATION OF INFLUENCE OF FACTORS ON FIRE RESISTANCE OF PARTITIONS MADE OF SANDWICH PANELS

The authors describe the results of studies of thermal processes in partitions made of sandwich panels under one-sided influence of fire. Analysis of the thermal state of multilayer partitions with mineral wool plates and development of recommendations for the design of buildings from such structures is possible using the simulation of thermal processes that occur during fire exposure to the studied samples. There is a certain number of models of the thermal state of multilayer partitions with mineral wool plates of different levels of complexity and purpose. However, in these models there are parameters that are individual for each partition and are known with insufficient accuracy. The authors provided an improved mathematical model of heat conduction in partitions, the difference of which is the application of heat transfer coefficients from the unheated surface of the partition in boundary conditions. A comparison of the fire resistance limits for a partition made of a sandwich panel indicates a strong influence of the accuracy of setting the heat transfer coefficient on the dependence of the thickness of the partition on the fire resistance limit. The accuracy of the task of the heat transfer coefficient on the dependence of the thickness of the partition on the fire resistance limit was scientifically substantiated, which showed a strong influence of the boundary condition describing the processes of heat transfer on the surface of the partition, which is not heated, on the accuracy of determining the limits of fire resistance of the structure under study. The influence of the accuracy of the assignment of the coefficient of heat transfer from the object to the environment on the fire resistance of the investigated building structures, which during the test are subject to both natural (mobility) of air and forced blowing, as, for example, during tests of air ducts with fire protection, must be taken into account convectively radiation heat exchange, which will allow to obtain the most accurate result regarding the determination of their fire resistance characteristics. Keywords: fire-resistance, methodological support, parameters, modeling.

Directly irradiated liquid metal film in an ultra-high temperature solar cavity receiver. Part 1: Concepts and a quasi-steady-state analysis

Solar thermal energy has the theoretical potential to deliver heat at ultra-high temperatures (>1300 K), which can enable integration with state-of-the-art thermal energy storage systems and unlock new applications, including advanced power cycles and thermal processes. Liquid metals are prospective heat transfer fluids for such systems, given their favourable thermo-physical properties, while their aggressive corrosiveness is shown to be mitigated using compatible refractory containment materials. The conventional approach of collecting concentrated solar energy typically involves intermediate solid absorbers, in form of tubes or porous structures, which are prone to thermomechanical and chemical failure under high solar radiation. This paper investigates the use of directly irradiated liquid metal (tin) film, operating between 800 and 1673 K, in two possible cavity configurations: A ‘reflective cavity’ and an ‘absorptive cavity’. The former employs cavity walls as internal reflectors to entrap radiation by secondary reflections until directly absorbed by the liquid metal. In the latter, the directly irradiated film is used to moderate the initial shot of concentrated solar radiation before diffusively reflecting them to the absorptive cavity walls, which perform as a radiative heat exchanger used to preheat the liquid metal. The concept performance is evaluated using an approximate quasi-steady-state energy model of the receiver. The reflective cavity performance is found strongly dependent on the optical properties of its internal surfaces, which resulted in poor efficiencies (<40 %) without special treatments. The absorptive cavity demonstrated higher efficiencies (>70 %) with greater insensitivity to the optical properties, hence, promoting its consideration in future developments of this concept.

DESIGN AND IMPLEMENTATION OF A HOT WATER HEATING SYSTEM

The heating system in winter is very necessary, especially in cold areas, and there are several ways, including the use of fuel, and because of the increase in the use of fossil fuel energy and the high cost, and the risks of toxic gases resulting from fuel combustion, and this negatively affects human health. Where a heating system was designed based on the domestic heater instead of a boiler and the use of a radiator heat exchanger, a fan that enhances the air temperature distribution system in the room, and a water pump to push water into the system with an electronic control, a system equipped with digital sensors to measure the temperature. After operating the system, the temperature of the hot water ranged from 80°C to 90°C, and the atmosphere inside the test room ranged from 17°C, the finding are good from where of energy consumption and heating in a record time at an average of 25.8°C.

A novel nonlinear radiative heat exchanger for molten-salt applications

One of the main difficulties associated with using molten salts as heat transfer fluids (HTFs) in practical applications such as in the solar and nuclear energy industries is the possibility of solidifying molten salts. Since the salts proposed for Generation-IV nuclear reactors (MSRs) have higher melting points, this is especially true for those reactors. The consequences of accidental freezing within the direct reactor auxiliary cooling system (DRACS) of MSRs could be catastrophic. DRACS is a passive safety system designed to remove decay heat from the reactor core during an emergency. The present study examines, as a case study, the freezing problem in the most vulnerable component of DRACS, namely the natural draft heat exchanger (NDHX). For this application, a unique and innovative heat exchanger design termed radiative heat exchanger (RHX) is proposed, with nonlinear heat transfer characteristics. Specifically, by suppressing convection and exploiting thermal radiation as the primary heat transfer mechanism, the RHX concept reduces the heat rejection/removal rate passively to allow safe operation when the molten salt coolant temperature drops and is near its melting point, while maintaining a high capacity for decay heat extraction in case the coolant temperature rises during an accident and the reactor becomes overheated; this moderation of the heat removal rate is accomplished without the requirement for external (manual or automatic) control. A thermohydraulic model is developed and applied to evaluate the proposed RHX with the existing conventional NDHX. Compared to the conventional design, the RHX has the potential to reduce heat removal by approximately 30 % at low temperatures and near-solidification conditions. In the case of the prototypical 20 MW reactor with a decay rate of 0.2 MW, RHX extends the freezing time of molten salt by three times, to approximately 18 h. During high-temperature events, such as the initial stages of a reactor scram, the RHX can improve decay heat extraction by 17 %. Other applications of molten salts with similar challenges can also be addressed using the proposed design approach.

Radiative heat exchange driven by acoustic modes between two solids at the atomic scale

In the near field (for separation distances smaller than the thermal wavelength, of the order of some microns at ambient temperature), the radiative heat flow between two solids at different temperatures can exceed the blackbody limit by several orders of magnitude. Furthermore, at the atomic scale, close to contact, the vibrational modes of the crystal lattice are expected to play an important role in the heat exchange. While the contribution of acoustic phonons tunnel-ing due to Van der Waals forces and electrostatic interactions near the surface has been investigated [1–5], the radiative contribution of the acoustic modes has been neglected so far. Under the local assumption (wavevector k ≈ 0), optical modes are independent from the acoustic modes, and are the sole responsible for the thermal radiation for distances larger than a few nanometers. However at subnanometer distances, the electromagnetic response of solids is nonlocal (k ≠ 0) and both acoustic and optical modes are coupled, contributing together to the radiative heat exchange. In order to demonstrate the role of this contribu-tion, we calculate [6] the nonlocal dielectric permittivity of magnesium oxide (MgO) using molecular dynamics. In conjunction with fluctuational electrody-namics calculations, we are able to highlight the role of radiation between two polar crystals produced by acoustic modes compared to the expected results from the local theory. We show that this additional contribution can become the dominant channel for radiative heat exchanges at atomic scale in the cryo-genic regime (below 100 K). Since the acoustic vibration modes can be excited with the help of piezoelectric transducers, our work opens the possibility to the control of radiative heat exchanges at atomic scale using external mechanical actuation.

Radiant capillary heat exchangers – power calculation for optimal heating and cooling

The issue of switching to renewable energy sources becomes very actual and it is important not only to change the energy source, but also the reduce the final energy needs by improving the energy efficiency of buildings and usage of efficient heating systems. Heat pumps as the most popular renewable energy source are widely used, but their energy efficiency is depending on temperature of the supplied energy carrier. The most efficient are radiant capillary heat exchangers with a large surface area and a low temperature, which typically does not exceed 30°C. Another advantage of radiant capillary heat exchangers is the possibility to operate them in both - heating and cooling modes. Unlike the underfloor heating solution, where the role of thermal convection is very important, the built-in radiant capillary heat exchanger systems provide the energy mainly due to thermal radiation. This study explores two modelling approaches for determination of required power and corresponding area of radiant capillary heat exchangers to be installed in a room to provide heating and cooling: simplified approach, which allows to create the heat balance with a minimum amount of input data and a precise standard-based approach. Calculations were made for three different rooms with variable glazing area and spatial orientation using both approaches. Analysis of the calculation results shows the limits of the simplified method, which overestimates heating need and underestimates cooling need, and the main reason for such differences is simplification of room orientation and subsequent solar heat gains. As the calculated cooling power is less than heating power, therefore the heating estimation is sufficient to estimate the amount of radiant capillary heat exchangers in small/medium rooms for providing both heating and cooling in the climatic conditions of Riga. The use of complex, comprehensive modelling approaches is necessary for rooms with large glazed areas, where the simplified method gives incorrect estimations.

Study on radiant heat exchange between human body and radiant surfaces under asymmetric radiant cooling environments

Study on radiant heat exchange between human body and radiant surfaces under asymmetric radiant cooling environments

Численное моделирование процесса выращивания монокристаллов методом Чохральского в квазистационарном приближении

We consider the steady-state growth of a single crystal by the liquid encapsulated Czochralski method. The mathematical model accounts for heat transfer in the crystal, melt and encapsulant, radiative heat exchange between the crucible and the surrounding furnace, formation of the growth interface and the melt/encapsulant interface meniscus, release of the latent heat during the phase transition. The time history of the growth is reconstructed from a sequence of steady state calculations with decreasing melt depth. The numerical method is based on explicit interface tracking. The employed computational procedure ensures global conservation of thermal energy. The response of prototype growth system for changes in pull rate or heater temperature is examined numerically.