Browsing by Author "Rousseau, Pieter"

Now showing 1 - 12 of 12
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    Open Access
    A methodology to investigate the cause of quenching in once-through tower type power plant boilers
    (2020) De Klerk, Gary; Rousseau, Pieter; Jestin, Louis
    Due to the penetration of variable renewable energy (VRE) sources, conventional coal fired power plants need to operate with greater flexibility via two-shifting or low load operation whilst remaining reliable and conserving the lifetime of components. Thick sectioned components are prone to thermal fatigue cracking as a result of through-wall temperature gradients during start up and shutdown. These temperature gradients can be significantly amplified during quenching when components at high temperature are unintentionally exposed to colder liquid or steam. Such quench events are known to occur during two-shift operation of a large once-through coal fired tower type boiler, which is the subject of this study. The purpose of this study is to develop and demonstrate a methodology to determine the root cause of quenching in a once-through tower type boiler and provide information that can be used to predict the impact on thick-walled components by estimating the through-wall temperature gradients. The first modelling element in the methodology is a simplified transient heat transfer model for investigating condensation of steam in the superheater. The model is presented and verified by comparison with real plant data. The second element is a liquid tracking model that approximates the liquid level in the superheater as a function of time to predict the location and magnitude of through-wall temperature gradients. The complex geometry of the superheater was divided into a number of control volumes and a dynamic thermo-fluid process model was developed to solve the transient conservation of mass and energy equations for each volume using a semi-implicit time wise integration scheme. The liquid tracking model was verified by comparison with a similar model constructed in Flownex and also by comparison with plant data. Varying levels of discretisation were applied to a particular quench event and the results are presented. The third modelling element is a two-dimensional transient pipe wall conduction model that is used at selected localities to evaluate the temperature gradients within the pipe wall. The temperature gradients and internal heat flux were verified by temperature measurements from the outer surface of a main steam pipe undergoing quenching. The stresses associated with the temperature gradients were also briefly considered. The real plant quenching problem is analysed in detail and found to be caused by liquid overflow from the separators. A particular plant configuration creates a previously unidentified siphon of water from the separating and collecting vessel system into the superheater. This situation is not recognised by plant operators and thus persists for some time and causes flooding of the superheater. Analysis of the resultant through-wall temperature gradients show that quenching causes significant stresses which can be avoided. By understanding the causes and preventing the occurrence of quenching, the life of thick-walled high temperature components can be conserved.
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    Open Access
    A modelling methodology to quantify the impact of plant anomalies on ID fan capacity in coal fired power plants
    (2020) Khobo, Rendani Yaw-Boateng Sean; Rousseau, Pieter; Gosai, Priyesh
    In South Africa, nearly 80 % of electricity is generated from coal fired power plants. Due to the complexity of the interconnected systems that make up a typical power plant, analysis of the root causes of load losses is not a straightforward process. This often leads to losses incorrectly being ascribed to the Induced Draught (ID) fan, where detection occurs, while the problem actually originates elsewhere in the plant. The focus of this study was to develop and demonstrate a modelling methodology to quantify the effects of major plant anomalies on the capacity of ID fans in coal fired power plants. The ensuing model calculates the operating point of the ID fan that is a result of anomalies experienced elsewhere in the plant. This model can be applied in conjunction with performance test data as part of a root cause analysis procedure. The model has three main sections that are integrated to determine the ID fan operating point. The first section is a water/steam cycle model that was pre-configured in VirtualPlantTM. The steam plant model was verified via energy balance calculations and validated against original heat balance diagrams. The second is a draught group model developed using FlownexSETM. This onedimensional network is a simplification of the flue gas side of the five main draught group components, from the furnace inlet to the chimney exit, characterising only the aggregate heat transfer and pressure loss in the system. The designated ID fan model is based on the original fan performance curves. The third section is a Boiler Mass and Energy Balance (BMEB) specifically created for this purpose to: (1) translate the VirtualPlant results for the steam cycle into applicable boundary conditions for the Flownex draught group model; and (2) to calculate the fluid properties applicable to the draught group based on the coal characteristics and combustion process. The integrated modelling methodology was applied to a 600 MW class coal fired power plant to investigate the impact of six major anomalies that are typically encountered. These are: changes in coal quality; increased boiler flue gas exit temperatures; air ingress into the boiler; air heater inleakage to the flue gas stream; feed water heaters out-of-service; and condenser backpressure degradation. It was inter alia found that a low calorific value (CV) coal of 14 MJ/kg compared to a typical 17 MJ/kg reduced the fan's capacity by 2.1 %. Also, having both HP FWH out of service decreased the fan's capacity by 16.2 %.
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    Open Access
    A reduced order modelling methodology for external cylindrical concentrated solar power central receivers
    (2023) Heydenrych, James; Rousseau, Pieter; Du Sart Colin
    The use of supercritical carbon dioxide (sCO2) power cycles for concentrated solar power (CSP) applications is becoming increasingly attractive since these cycles may offer lower capital costs and increased thermal efficiency. However, there are currently no utility-scale sCO2-CSP tower plants in operation. Therefore, to aid in the design and analysis process, there is a need to develop sufficiently accurate and computationally inexpensive models for such plants. This dissertation presents a reduced order modelling methodology for external cylindrical concentrated solar power central receivers. The methodology is built on a one-dimensional thermofluid network to model the heat transfer through the tube walls, coupled to a fluid flow network of the solar salt flowing inside the tubes. This is combined with a neural network surrogate model to determine the radiative heat flux impinging upon the tube surfaces. The receiver geometry is discretized along the height and around the circumference and each increment is represented by an equivalent thermal resistance network that represents the heat transfer within the tube walls. The heat transfer network parameters are calibrated using a detailed computational fluid dynamics model, which enables the calculation of the maximum tube wall temperatures. The heat transfer network is connected to the fluid flow network that solves the mass, energy, and momentum balance equations to determine the mass flow rates, pressure drops and temperature distributions. The radiative heat flux profile impinging on the receiver is typically calculated for a specific location and specific time of the day using a tool such as SolarPILOT. However, this can be computationally expensive since the central tower is surrounded by thousands of individual heliostats that are all sources of radiative flux, which depends on the position relative to the sun and relative to the receiver, as well as the direct normal irradiation (DNI) at that location and time. To reduce the associated computational expense, a multilayer perceptron (MLP) surrogate model is developed that allows the prediction of the flux profile for a range of plant configurations and atmospheric conditions at a specific location. The application of the methodology is demonstrated via a case study. The methodology may be used in future studies where sCO2-CSP tower plants are investigated, especially those with an interest in the detail design and analysis of the central receiver.
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    Open Access
    Advanced analytics for process analysis of turbine plant and components
    (2019) Maharajh,Yashveer; Rousseau, Pieter; Mishra, Amit
    This research investigates the use of an alternate means of modelling the performance of a train of feed water heaters in a steam cycle power plant, using machine learning. The goal of this study was to use a simple artificial neural network (ANN) to predict the behaviour of the plant system, specifically the inlet bled steam (BS) mass flow rate and the outlet water temperature of each feedwater heater. The output of the model was validated through the use of a thermofluid engineering model built for the same plant. Another goal was to assess the ability of both the thermofluid model and ANN model to predict plant behaviour under out of normal operating circumstances. The thermofluid engineering model was built on FLOWNEX® SE using existing custom components for the various heat exchangers. The model was then tuned to current plant conditions by catering for plant degradation and maintenance effects. The artificial neural network was of a multi-layer perceptron (MLP) type, using the rectified linear unit (ReLU) activation function, mean squared error (MSE) loss function and adaptive moments (Adam) optimiser. It was constructed using Python programming language. The ANN model was trained using the same data as the FLOWNEX® SE model. Multiple architectures were tested resulting in the optimum model having two layers, 200 nodes or neurons in each layer with a batch size of 500, running over 100 epochs. This configuration attained a training accuracy of 0.9975 and validation accuracy of 0.9975. When used on a test set and to predict plant performance, it achieved a MSE of 0.23 and 0.45 respectively. Under normal operating conditions (six cases tested) the ANN model performed better than the FLOWNEX® SE model when compared to actual plant behaviour. Under out of normal conditions (four cases tested), the FLOWNEX SE® model performed better than the ANN. It is evident that the ANN model was unable to capture the “physics” of a heat exchanger or the feed heating process as a result of its poor performance in the out of normal scenarios. Further tuning by way of alternate activation functions and regularisation techniques had little effect on the ANN model performance. The ANN model was able to accurately predict an out of normal case only when it was trained to do so. This was achieved by augmenting the original training data with the inputs and results from the FLOWNEX SE® model for the same case. The conclusion drawn from this study is that this type of simple ANN model is able to predict plant performance so long as it is trained for it. The validity of the prediction is highly dependent on the integrity of the training data. Operating outside the range which the model was trained for will result in inaccurate predictions. It is recommended that out of normal scenarios commonly experienced by the plant be synthesised by engineering modelling tools like FLOWNEX® SE to augment the historic plant data. This provides a wider spectrum of training data enabling more generalised and accurate predictions from the ANN model.
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    Open Access
    An integrated furnace co-simulation methodology based on a reduced order CFD approach
    (2023) Rawlins, Brad; Rousseau, Pieter; R. Laubscher
    An integrated thermofluid modelling methodology for pulverised fuel fired utility-scale boilers that is computationally inexpensive, fast, and sufficiently accurate would be valuable in an industrial setting. Such a model would enable boiler operators to investigate a range of off-design operating conditions, which includes flexible operation. The aims of this study was: to develop a reduced order computational fluid dynamics (CFD) model of the furnace and radiative heat exchangers that captures all the important particulate effects while using a Eulerian-Eulerian (EE) approach; using the reduced order CFD model to generate a database of results that covers a wide range of operating conditions; to develop a data-driven surrogate model using machine learning techniques; to integrate the surrogate model with a 1-D process model of the complete boiler; and finally to demonstrate the use of the integrated model to investigate flexible operation and off-design operating conditions. The validity of the CFD modelling approach was demonstrated via application to a 2.165 [MWth] lab-scale swirl pulverised fuel burner, as well as to a 620 [MWe] utility-scale subcritical two-pass boiler, both operating at various loads. The results were compared to measured data and a detailed CFD model using the conventional Eulerian-Lagrangian (EL) approach. A computational speed enhancement of 30% was achieved. The data-driven surrogate model uses a mixture density network (MDN) to predict the heat transfer in the furnace and radiative heat exchangers, together with the uncertainty in the predicted values. The integrated model was validated against applicable measured data and then applied to a utility-scale case study boiler to investigate the optimal burner firing positions for low-load operation, as well as to investigate the effects of fuel quality on the overall boiler performance. It was shown that the integrated data-driven surrogate model and 1-D process model can predict the overall thermofluid response of the boiler and the uncertainties associated with it with good accuracy, whilst maintaining a low computational effort when compared to a conventional CFD model coupled to 1-D process model.
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    Open Access
    Dynamic process modelling of the HPS2 solar thermal molten salt parabolic trough test facility
    (2018) Temlett, Robert; Rousseau, Pieter
    In recent years power generation from renewable energy has grown substantially both in South Africa and around the world. This growth is set to continue as there is more pressure to reduce the burning of fossil fuels. However, renewable energy power generation suffers from unpredictability, which causes problems when it comes to managing power grids. Concentrated Solar Power (CSP) plants offer a practical solution to store power in the form of thermal energy storage (TES). Thus, the plant can run when there is no solar energy available, leading to a more stable power supply. Unfortunately, CSP plants cost more than other renewables such as photovoltaic and wind power. Thus, there is a need for research into how to bring down the cost of CSP plants. One of the most proven types of CSP is the parabolic trough plant. The most recent innovation is to try and use molten salt as the heat transfer fluid which would reduce the cost of the plant. However, this new technology has not been implemented on a full scale CSP plant and little testing has been done to prove the technology. The HPS2 is a test facility aimed at testing the use of molten salt as a heat transfer fluid (HTF). This test facility, located in Evora Portugal, is being developed by an international consortium led by the German DLR institute of Solar Research. It is one of the first test facilities of its kind where experiments will be conducted to demonstrate the validity of using molten salt as a HTF and a storage medium in a parabolic trough CSP plant. The HPS2 test facility is not yet operational and there is a need for a dynamic thermofluid process model to better understand and predict both its steady state and transient operational behaviour. This dissertation reports on the development of such a dynamic thermofluid process model and the results obtained from it. The process model developed primarily focuses on the steam cycle with the TES incorporated into the model. The physical geometry of each of the components are employed to construct discretized elements for which the conservation of mass, energy, and momentum are applied in a one-dimensional network approach. The economizer and evaporator combined has a helical coil geometry and uses molten salt as a heat transfer fluid, which is unique. Thus, correlations had to be adjusted for the flow characteristics found in the economizer/evaporator. Results from the steady state simulations of the steam cycle show that the molten salt mass flowrate through the steam generation system will have to be reduced from the initially expected value to meet operational requirements. Results of the dynamic simulations show that the test facility will be able to produce a constant power supply despite transient solar conditions and highlights key dynamic responses for operators to be aware of.
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    Open Access
    Heat Transfer Analysis Using Thermofluid Network Models for Industrial Biomass and Utility Scale Coal-Fired Boilers
    (2023-02-09) Rousseau, Pieter; Laubscher, Ryno; Rawlins, Brad Travis
    Integrated whole-boiler process models are useful in the design of biomass and coal-fired boilers, and they can also be used to analyse different scenarios such as low load operation and alternate fuel firing. Whereas CFD models are typically applied to analyse the detail heat transfer phenomena in furnaces, analysis of the integrated whole-boiler performance requires one-dimensional thermofluid network models. These incorporate zero-dimensional furnace models combined with the solution of the fundamental mass, energy, and momentum balance equations for the different heat exchangers and fluid streams. This approach is not new, and there is a large amount of information available in textbooks and technical papers. However, the information is fragmented and incomplete and therefore difficult to follow and apply. The aim of this review paper is therefore to: (i) provide a review of recent literature to show how the different approaches to boiler modelling have been applied; (ii) to provide a review and clear description of the thermofluid network modelling methodology, including the simplifying assumptions and its implications; and (iii) to demonstrate the methodology by applying it to two case study boilers with different geometries, firing systems and fuels at various loads, and comparing the results to site measurements, which highlight important aspects of the methodology. The model results compare well with values obtained from site measurements and detail CFD models for full load and part load operation. The results show the importance of utilising the high particle load model for the effective emissivity and absorptivity of the flue gas and particle suspension rather than the standard model, especially in the case of a high ash fuel. It also shows that the projected method provides better results than the direct method for the furnace water wall heat transfer.
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    Open Access
    Integrated network-based thermofluid model of a once-through boiler at full- and part-load
    (2022) Feng, Kai-Yu; Rousseau, Pieter; Laubscher, Ryno
    The increased penetration of renewable energy sources in South Africa requires greater operational flexibility of existing coal-fired power plants (CFPPs). Operational flexibility implies that power plants need to operate intermittently or at low load for extended periods. Existing CFPPs are designed to operate at a steady baseload. Operating at these off-design conditions increase the risk of damaging the boiler's thick-walled components, leading to reduced life expectancy and/or failure. Given that extensive experimental investigations on operating plants are impractical due to the risks, costs and complexity involved, there is a need for an integrated boiler model that has the necessary detail to study off-design and low load operations of coal-fired power plants. For that reason, a 1D quasi-steady-state thermofluid network model of a tower type once-through boiler was developed using the Flownex simulation environment. The furnace model assumes complete, infinitely fast combustion with a specified value of unburned carbon and excess air. The radiation heat transfer in the furnace is modelled using the projected area approach (Gurvich/Blokh model) together with a high ash loading model. The gas-to-steam tube bank heat exchangers are discretised pass-by-pass, and the complex heat transfer phenomena in the heat exchangers and membrane water walls are represented by equivalent thermal networks. The model results for the as-designed cases show that the ash deposition resistances suggested in the literature are not applicable for the case study boiler. For that reason, the proposed model calibration methodology was therefore applied at full-load operation (100%), and the results show good accuracy compared to real-plant data. The average error of the predicted heat exchanger heat loads is 2.0% and the maximum error is 5.2%. The calibrated model was then validated by applying it to two part-load operational states in dry mode operation, as well as a wet mode (low-load) operational state. For the 81% load case, the average error in the heat exchanger duty is 2.0% and the maximum is 4.9%, while for the 63% load case, the average error is 3.6% and the maximum is 9.7%. For the low-load wet mode case at 35% load, the average error is 10.4% and the maximum is 18.1%. The cumulative heat transfer results for all the load cases correspond closely to the measured data, with the maximum error being 0.83% for the low load case. These results suggest that the calibrated model can capture the heat distribution in the boiler with sufficient accuracy to allow suitable ash deposition resistances to be obtained from the calibration process. Furthermore, the metal temperatures predicted by the model are also shown to be sufficiently accurate, which means that it can be used to identify the heat exchanger tube passes or membrane water walls that may be at risk during operation.
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    Open Access
    Integrated process and control modelling of water recirculation in once-through boilers during low load and transient operation
    (2020) Rosslee, Pieter; Rousseau, Pieter
    Power plant stability at lower loads is becoming ever more important, highlighting the increasing requirement for the development of advanced models and tools to analyse and design systems. Such tools enable a better understanding of the thermo-fluid processes and their dynamics, which improves the ability to specify and design better control algorithms and systems. During low load operation and transients, such as start-up and shutdown, the required water flow rate through the evaporator tubes of once-though boilers must be significantly higher than the evaporation rate to protect against overheating of the tubes until once-through operation is reached. Controlling the minimum required water flow rate through the evaporator and economiser is notoriously difficult. Within industry, strong emphasis is placed on maintaining the minimum required flow through the economiser and evaporator without adequate consideration of the potential thermal fatigue damage on the economiser, evaporator and superheater components and the risk of turbine quenching incidents. The purpose of this study was to develop an integrated process and control model that can be used to study transient events. The model developed in Flownex can simulate the complex thermo-fluid processes and associated controls of the feedwater start-up system. This includes the waterrecirculation loop, and allows for detailed transient analysis of the complete integrated system. The model was validated using data from an actual power plant in steady state as well as a transient cold start-up, up to once-through operation. Transient results from the model are also compared to the power plant unit during start-up for the addition or loss of mills using the existing control strategy. The model results compare well with the actual process behaviour. A new control strategy was then proposed and tested using the model. The results indicated significant improvement in control performance and overall controllability of the start-up system, and the large temperature fluctuations currently experienced at the economiser inlet during transients were significantly reduced. The new control strategy was also implemented on a real power plant unit undergoing commissioning. During all modes of start-ups (cold, warm and hot), as well as transients, the performance of the control system showed significant improvement, with a notable decline in instabilities of the feedwater flow. As predicted in the model, the large temperature fluctuations are significantly reduced. The new model therefore enabled the development of an improved control strategy that reduces damaging thermal fatigue. The general controllability of transients is also significantly improved, thereby minimizing risks of water carry-over, quenching and unit trips during start-up.
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    Open Access
    Online boiler convective heat exchanger monitoring: a comparison of soft sensing and data-driven approaches
    (2018) Prinsloo, Gerto; Rousseau, Pieter; Gosai, Priyesh
    Online monitoring supports plant reliability and performance management by providing real time information about the condition of equipment. However, the intricate geometries and harsh operating environment of coal fired power plant boilers inhibit the ability to do online measurements of all process related variables. A low-cost alternative lies in the possibility of using knowledge about boiler operation to extract information about its condition from standard online process measurements. This approach is evaluated with the aim of enhancing online condition monitoring of a boiler’s convective pass heat exchanger network by respectively using a soft sensor and a data-driven method. The soft sensor approach is based on a one-dimensional thermofluid process model which takes measurements as inputs and calculates unmeasured variables as outputs. The model is calibrated based on design information. The data-driven method is one developed specifically in this study to identify unique fault signatures in measurement data to detect and quantify changes in unmeasured variables. The fault signatures are initially constructed using the calibrated one-dimensional thermofluid process model. The benefits and limitations of these methods are compared at the hand of a case study boiler. The case study boiler has five convective heat exchanger stages, each composed of four separate legs. The data-driven method estimates the average conduction thermal resistance of individual heat exchanger legs and the flue gas temperature at the inlet to the convective pass. In addition to this, the soft sensor estimates the average fluid variables for individual legs throughout the convective pass and therefore provides information better suited for condition prognosis. The methods are tested using real plant measurements recorded during a period which contained load changes and on-load heat exchanger cleaning events. The cleaning event provides some basis for validating the results because the qualitative changes of some unmeasured monitored variables expected during this event are known. The relative changes detected by both methods are closely correlated. The data-driven method is computationally less expensive and easily implementable across different software platforms once the fault signatures have been obtained. Fault signatures are easily trainable once the model has been developed. The soft sensors require the continuous use of the modelling software and will therefore be subject to licencing constraints. Both methods offer the possibility to enhance the monitoring resolution of modern boilers without the need to install any additional measurements. Implementation of these monitoring frameworks can provide a simple and low-cost contribution to optimized boiler performance and reliability management.
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    Steam temperature and flow maldistribution in superheater headers
    (2020) du Preez, Jean-Pierre; Rousseau, Pieter
    Heat exchangers and steam headers are at the heart of any boiler and are susceptible to a range of failures including tube leaks, ligament cracking, creep and fatigue. These common forms of header failure mechanisms can be exacerbated by local thermal stresses due to temperature and flow maldistribution at full and partial boiler load operations. The purpose of this project is to develop process models of the outlet stubbox header of a final superheater (FSH) heat exchanger in a 620MW coal-fired drum type boiler. The process models were used to assess the impact of steam flow and temperature distribution on the thermal stresses in the header material. The process models were developed using Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA). Thermocouples were installed at key locations on the stubbox headers to monitor metal temperatures and the measured metal temperatures served as boundary values and for validation of the CFD results. The thermocouple data was analysed for three different steady state boiler loads, namely full load, 80% load and 60% load. It showed that the temperature distribution across these headers was not uniform, with a maximum temperature difference across the outlet stubbox of 40℃ at full load and 43℃ at partial loads. Other relevant power plant data, such as steam pressure, was provided from the power plant's Distributed Control System (DCS) and was used as boundary conditions for the CFD models. The exact mass flow distribution across the inlet stubs of the outlet stubbox header was unknown and was estimated using a CFD model of the inlet stubbox header and steam mass flow values from power plant's DCS system. A CFD model was created for each of the three boiler loads at steady state conditions. The CFD results provided the metal temperature profile, internal steam temperature distribution and pressure distribution across the header. The CFD solid temperatures were validated using the thermocouple readings and found to be in agreement. The CFD results were exported to the FEA models, where specific displacement constraints for thermal expansion were utilised. The FEA models were used to assess the extent of thermal stresses due to thermal expansion only, as well as stresses due to thermal expansion combined with internal pressure. High local stresses were found at the borehole crotch corners of the rear outlet branch and inlet stubs. However, these are below 0.2% proof strength at elevated temperatures. The high local stresses thus did not result in local plastic deformation but contribute to exacerbate steady state failure mechanisms such as creep.
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    Open Access
    A zonal model for radiation heat transfer in coal-fired boiler furnaces
    (2015) Monnaemang, Whitney Ogalaletseng; Rousseau, Pieter
    Problems associated with boilers are a major contributor to load losses in coal-fired power plants. The boiler furnace exit temperature is a key indicator of the combustion and heat transfer processes taking place and has a profound impact on the operation of the heat exchangers downstream of the furnace. Having a model that can predict the furnace exit temperature and heat flux distributions may enable furnace performance to be predicted without having to conduct extensive experimentation. Also, comparing the results with measurements taken on the plant may enable the identification of operating problems and potential sources of losses. Thermal radiation is the dominant mode of heat transfer in the boiler furnace. The primary objective of this study is to develop and implement a radiation heat transfer network solution methodology based on the zonal method that may be applied to boiler furnace modelling. The zonal method allows for the prediction of heat flux and temperature distributions on the walls, inside, and at the exit of the furnace. Direct exchange areas are the basis of the zonal method and are a function of the furnace geometry and radiative properties of the walls and the participating medium that fills the furnace volume. The evaluation of direct exchange areas is done by discrete numerical integration, after which it needs to be smoothed to satisfy energy conservation. After evaluating two different smoothing techniques, the least squares technique using Lagrange multipliers was selected for this study. Following this, the solution of the radiation heat transfer network was implemented for an emitting-absorbing-scattering participating medium for two different scenarios, namely (i) solving surface and volume heat fluxes for known surface and medium temperatures, and (ii) solving surface heat fluxes and medium temperature distributions for known surface temperatures and volume heat source terms. Intermediate verification and validation steps throughout the development process show good agreement with other numerical techniques and correlations available in literature. In order to illustrate the applicability of the final model, a number of case studies are conducted. These include an illustration of the effect of slagging on the furnace walls, of a faulty burner and of changes in the radiative properties of the participating medium. The results of the case studies show that the trends of the heat flux and temperature distributions obtained with the new model are in agreement with those found in literature.