Constant Temperature Boundary Condition Research Articles

Numerical Investigation of Flow Inside a Channel with Elastic Vortex Generator and Elastic Wall for Heat Transfer Enhancement

In the present investigation, a detailed numerical investigation of the flow and heat transfer characteristics of a channel with an elastic fin (vortex generator) and an elastic wall has been carried out using finite element method. The Fluid-Structure Interaction (FSI) model is used to capture the interaction between the fluid and the solid structure. A sinusoidal time dependent velocity profile has been imposed at the inlet of the channel and the right half of the upper wall of the channel is heated and exposed to constant temperature boundary condition. Due to the sinusoidal velocity profile at the inlet, the elastic fin oscillates periodically and act as a vortex generator, which causes more turbulence in the flow. The obtained results showed that the Nusselt number over the heated wall is affected by the position of the flexible fin, height of flexible fin and elasticity modulus of elastic fin. Moreover, due to the elasticity of the elastic wall and sinusoidal behavior of the inlet velocity, the elastic wall oscillates periodically upward and downward. The Nusselt number values over the heated wall are increased with decrease of the elastic modulus value of the elastic wall. However, the decrease in elastic modulus value of the elastic wall contributes to an increase in the pressure drop inside the channel. It should be added that the interplay between the fluid motion and the deformable structures leads to enhanced turbulence, as the flexible fin and elastic wall introduce additional disturbances and fluctuations into the flow regime. Consequently, this heightened turbulence level has profound implications for heat transfer processes within the system.

Numerical study of uranium hexafluoride desublimation process based on Eulerian two-phase model

This paper presents a numerical study on the condensation process of uranium hexafluoride (UF6) in the supply and extraction system of a uranium enrichment plant. The numerical study was conducted using a standard UF6 receiving container with a nominal diameter of 127 mm and a volume of 8L under constant temperature boundary conditions. A phase change heat and mass transfer model for the desublimation process of uranium hexafluoride gas was established, and this model was combined with the Euler two fluid model to predict the distribution of key parameters such as temperature, pressure, and volume fraction of each phase during the UF6 cooling process, as well as the gas flow characteristics inside the container. In addition, this study also investigated the influence of various factors on the cooling rate. The research results indicated that UF6 gas initially condensed on the container wall and gradually diffused inward, with the highest solid density observed on the wall. Furthermore, increasing the inlet gas temperature, inlet mass flow rate, and reducing the cooling temperature, initial pressure inside the container could all enhance the desublimation rate to various degrees. The optimization plan proposed based on the above influencing factors could increase the yield of UF6 solid by 64.20% in 2 h, compared to the basic operating conditions.

Numerical analysis of magnetohydrodynamic free stream and radiative heat transfer flow of nanofluid in a permeable medium over a linearly stretched cylinder with Arrhenius activation energy

This research article explores the transport characteristics of magnetohydrodynamic Newtonian nanofluid flowing past a stretching cylinder in a permeable medium, considering the activation energy effects using Buongiornos mathematical model. Unlike conventional studies employing constant temperature, concentration and density boundary conditions, convective boundary conditions are utilized in this investigation. The analysis incorporates various significant parameters such as magnetic field, joule heating, viscous dissipation, thermophoresis, Brownian motion, Arrhenius activation energy, and chemical reaction. The governing equations in the form of ODEs are obtained by employing similarity variables.The MATLAB built-in solver bvp4c is employed for obtaining the solutions. The impacts of these parameters on the velocity profile, temperature profile, concentration profile, density profile, skin friction coefficient, Sherwood number, and Nusselt number are elucidated through graphical and tabular representations. The results indicate that an increase in the chemical reaction parameter leads to higher nanoparticle concentration distributions, while the activation energy parameter exhibits the opposite trend. This study holds relevance for various practical applications in applied sciences, including nuclear reactor cooling, geothermal reservoirs, chemical engineering, and thermal oil recovery.

Investigation of the Effect of Duct Geometry on Drying Air Flow in Conventional Grain Dryers by Porous Media Approach

It is a traditional practice to store many agricultural products after drying, ensuring that they are used all year round. Mixed counter flow air drying is one of the most common and traditional methods in the bulk drying process. In this application, the air flow produced by the air channels placed in the dryer bed is forced to flow through the grains in the opposite direction to the grain flow. The moisture contained in the grains is thrown out of the dryer through forced convection. However, it is expected that the air ducts installed in the dryers should not obstruct the flow of grain and provide the best possible drying performance. In this study, computational fluid dynamics (CFD) modeling for a counter-flow grain dryer was performed and the effect of the geometry of the drying channels on the process was investigated. Fluent 2020 R2 commercial software was used for 2-D flow modeling through the dryer. The airflow in the grain zone, modeled as porous media, was included in the calculation for three different geometries of the dryer air ducts (circular, angular, and straight). A constant temperature boundary condition (37°C) was applied for the air ducts in which the drying air circulated without mixing with the grain. As an output of the analysis, the dryer outlet temperature and differential pressure variation along the flow were calculated for 5 different inlet velocities (between 0.005-0.25) to determine the behavior of different air flow rates in the drying process. The increase in the inlet velocity increased the pressure difference and consequently the stability of the flow for all models. The outlet temperature decreased by about 2.5 °C with a 5-fold increase in velocity. The results showed that the sufficient outlet air for moisture removal depends on the structure of the porous medium and the flow geometry. For this analysis, the best flow was found to be for the circular cross-section model and the outlet temperature could be at acceptable levels.

Investigation of the role of charge injection and Coulomb force during the melting of phase-change materials under constant temperature boundary conditions

This paper presents an investigation of the melting of dielectric material in a rectangular cavity under the effect of electrohydrodynamics (EHD). First, phase-change modeling is implemented to simulate the melting performance of paraffin wax without EHD under constant temperature boundary conditions until a steady-state condition is achieved. Next, the whole set of coupled EHD equations is introduced to the model, with the Coulomb force using a Heaviside function for charge injection being the only electrical body force considered. Finally, the numerical model is implemented using the finite element method to solve for the electric field, flow field, temperature field, and charge transport. The numerical results show that, under the effect of EHD, melting continues due to the generation of electroconvection cells in the liquid phase-change material and the flow field manifests as two symmetric rotational cells generated between every two successive electrodes. The flow field causes the redistribution of the temperature field in the liquid bulk, which enhances the heat transfer. Melting continues until a steady-state condition is almost reestablished after about one hour. The enhancement factor, defined as the ratio of the EHD melt thickness to the steady-state melt thickness without EHD, is 2.33 at 6 kV applied voltage.

Numerical study on flow and heat transfer characteristics of rectangular mini-channel of interpolated double S turbulators.

In this study, we propose a new type of small-channel plug-in, the double S turbulators, for passive heat transfer enhancement to improve the flow and heat transfer performance of the fluid in the channel. In the range of Reynolds number 254.51~2545.09, under constant wall temperature heating conditions, the effects of interpolated double S turbulators with different long axial radii (1mm, 1.5mm, 2mm) on the average Nusselt number, pressure drop, total thermal resistance and field synergy number in the rectangular mini-channel were studied. The simulation results show that compared with the smooth rectangular mini-channel, after interpolating double S turbulators with different long axial radii (1mm, 1.5mm, 2mm), the average Nusselt number increased by 81.74%~101.74%, 71.29%~94.06%, 67.16%~88.48%, the total thermal resistance decreased by 45.1%~50.72%, 41.72%~48.74%, 40.28%~47.2%, and the number of field synergies increased by 85.58%~111.65%, 74.1%~102.6%, 69.64%~96.12%. At present, there are few studies on the boundary condition of constant wall temperature, and this paper supplements the research on this aspect. At the same time, the heat transfer performance of the rectangular mini-channel of the interpolated double S turbulators is stronger than that of the ordinary smooth rectangular mini-channel, which not only provides a new idea for the manufacture of micro heat dissipation equipment, but also improves the heat transfer performance of micro heat dissipation equipment and improves its work efficiency. According to the simulation data, the prediction formula of average Nusselt number and pressure drop was established by nonlinear regression method, which can be used to predict the flow and heat transfer characteristics of the rectangular mini-channel of the interpolated double S turbulators.

Numerical thermal performance assessment of phase change process in a PCM/foam-fins enclosure under various thermal conditions

ABSTRACT The low ability of phase change material (PCM) to transfer thermal energy has created a serious challenge for the development of latent heat storage (LHTES) units. Embedding foam with high thermal conductivity material like copper has attracted significant attention. Besides, high porosity, lightweight, and easy usage are other important benefits of copper. However, its higher cost and decreasing effective volume of PCM led to the creation of amendments in the foam design. Employment of the foam in the form of fin or foam fin has emerged as an alternative method to benefit from the advantages of porous material and overcome its defects. In this study, various LHTES units with different numbers of foam fins have been simulated numerically and compared with pure PCM and fully foam PCM enclosures. Two practical boundary conditions including constant temperature and constant heat flux were examined for the designed systems. The obtained results demonstrated that foam fins have a remarkable effect on the thermal performance enhancement of LHTES under constant temperature boundary condition. For LHTES with six foam fins, 42% and 30% reductions in melting time were achieved for both types of boundary conditions including constant temperature and heat flux, respectively. At the beginning of charging process, a significant enhancement of input heat rate was proved for LHTES with 6 foam fins and fully foam up to 112% and 155%, respectively. In addition, the foam fins method provided some free volume of PCM to permit the falling out of natural convection of molten PCM.

Investigating heat transmission in a wellbore for Low-Temperature, Open-Loop geothermal systems

Heat transfer processes in geothermal wellbores are an essential component to overall system performance, but can be overlooked in subsurface modelling studies that tend to focus on reservoir response. This study addresses heat transfer in the wellbore through a comprehensive modelling study of 10 different parameters on OpenGeoSys software designed to evaluate heat losses during open-loop production conditions for typical low-temperature (<100 °C) single phase geothermal systems. Models were set-up to focus on wellbore effects only, using a single production well with constant fluid inlet temperature boundary condition. Results indicate that under base case conditions for a 2-km deep well surrounded by rock formations with a thermal conductivity of 2.5 W/(m.K) and bottom-hole temperature of 60 °C, the difference in inlet (reservoir) and production (wellhead) temperature at the end of a 40-year production period is 2.06 °C. This corresponds to minimum heat losses into the surrounding formations of −63.3 W/m, which is ∼ 7 % of the thermal power recorded at the wellhead (assuming a rejection temperature of 30 °C) and a 3.4 % difference to bottom-hole temperature. These losses in heat can be significant, particularly when in combination with surface losses through the heat exchanger or in the reservoir through thermal breakthrough. Wellbore insulation can reduce losses, but it would appear this only impacts the short-term. Wellbore performance can also be improved, with heat losses minimized, when developing wells above the reservoir interval in low thermal conductivity rock with high geothermal gradients.

Numerical simulation of fluid flow and heat transfer in steady state laminar natural convection in horizontal circular cylindrical enclosures

Heat transfer by natural convection often occurs in a closed enclosure in various applications. Different geometries of the cross section of a two-dimensional enclosure, such as triangle, rectangle, and trapezoid have already been investigated. In this research, a regular polygon with twelve sides is considered. The study focuses on the rate of heat transfer through the enclosure, using one side of the polygon as the hot surface and another side as the cold surface. The dodecagon can serve as a model for circular cylindrical enclosures under constant temperature boundary conditions over a specified length. The Nusselt number of the enclosure is obtained for each side as the hot surface. Both the lowest and highest Nusselt numbers are evaluated based on the position of the cold side. The findings reveal that the highest Nusselt number does not necessarily correspond to configurations where the hot and cold sides are directly opposite each other. In situations where convection heat transfer through the enclosure is the predominant mode of heat transfer, the Nusselt number is a power function of the Rayleigh number. Conversely, a logarithmic relationship is observed in cases of conduction heat transfer within the enclosure.

Thermal-hydraulic and load following performance analysis of a heat pipe cooled reactor

Heat pipe cooled reactors have gained attention as a potential solution for nuclear power generation in space and deep sea applications because of their simple design, scalability, safety and reliability. However, under complex operating conditions, a control strategy for variable load operation is necessary. This paper presents a two-dimensional transient characteristics analysis program for a heat pipe cooled reactor and proposes a variable load control strategy using the recuperator bypass (CSURB). The program was verified against previous studies, and steady-state and step-load operating conditions were calculated. For normal operating condition, the predicted temperature distribution with constant heat pipe temperature boundary conditions agrees well with the literature, with a maximum temperature difference of 0.4 K. With the implementation of the control strategy using the recuperator bypass (CSURB) proposed in this paper, it becomes feasible to achieve variable load operation and return the system to a steady state solely through the self-regulation of the reactor, without the need to operate the control drum. The average temperature difference of the fuel does not exceed 1 % at the four power levels of 70 %,80 %, 90 % and 100 % Full power. The output power of the turbine can match the load change process, and the temperature difference between the inlet and outlet of the turbine increases as the power decreases.

Modeling study on the geometric optimization of thermoelectric modules

Abstract The performance of thermoelectric power generators (TEG) primarily depends on the properties of the thermoelectric materials employed. For conventional thermoelectric modules (TEM) utilizing the same material, the geometric parameters also play a significant role in determining TEM performance. As such, optimizing the geometry of TEM can lead to improved performance. In this study, TEM were modeled, designed, fabricated, and tested to investigate the effects of different geometric parameters on their performance. Numerical simulations were conducted under both constant temperature and constant flow boundary conditions, and the results were validated through experimental testing. The simulation results under constant flow boundary conditions exhibited good agreement with the experimental results. The effects of thickness, cross-sectional area, and filling ratio of thermoelectric legs on TEM performance were investigated through numerical simulations and compared with findings from previous studies. It was observed that increasing the cross-sectional area of the thermoelectric legs led to a decrease in the power output of TEM. Conversely, increasing the filling ratio effectively enhanced the TEM's performance. Furthermore, an optimal thermoelectric leg thickness was identified through the numerical simulations that could yield the maximum power output of TEM. The underlying mechanism behind this observation was explained, shedding light on why different reports have identified different optimal thicknesses. Optimizing the thermoelectric leg thickness can help maintain a high effective temperature difference and low internal resistance, which can vary based on the specific type of TEM and the thickness and thermal conductivity of the insulating substrates and copper sheets.

A Non-Isothermal Pore Network Model of Primary Freeze Drying.

In this work, a non-isothermal pore network (PN) model with quasi-steady vapor transport and transient heat transfer is presented for the first time for the application of primary freeze drying. The pore-scale resolved model is physically based and allows for the investigation of correlations between spatially distributed structure and transport conditions. The studied examples were regular PN lattices with a significantly different structure, namely a spatially homogeneous PN, also denoted as monomodal PN, and a PN with significant structure variation, referred to as bimodal PN because of its bimodal pore size distribution. The material properties selected for the solid skeleton in this study are equivalent to those of maltodextrin. The temperature ranges applied here were -28 °C to -18 °C in the PN and -42 °C in the surrounding environment. The environmental vapor pressure was 10 Pa. The PNs were dried with constant temperature boundary conditions, and heat was transferred at the top side by the vapor leaving the PN. It is shown how the structural peculiarities affect the local heat and mass transfer conditions and result in a significant widening of the sublimation front in the case of the bimodal PN. The possibility of spatially and temporally resolved front structures is a unique feature of the PN model and allows the study of situations that are not yet described by classical continuum approaches, namely heterogeneous frozen porous materials. As demonstrated by the thin layers studied here, the pore-scale simulations are of particular interest for such situations, such as in lyomicroscopes or collagen scaffolds, where a length-scale separation between dry and ice-saturated regions is not possible.

Magnetohydrodynamic natural convection of In-Ga-Sn alloy in a horizontal concentric annulus with tree-shaped internal fins

The use of low melting point liquid metals for convective cooling has seen a recent increase in interest in the literature. Few studies have explored the use of such metals in concentric annuli with fins to enhance heat transfer. In this study, numerical simulations are conducted using COMSOL Multiphysics® to investigate the magnetohydrodynamic (MHD) natural convection heat transfer inside a concentric annulus with tree-shaped fins. The fluid in the enclosure is eutectic In20.5Ga67Sn12.5. The tree-shaped fins feature three geometric parameters: (i) length of branches a, (ii) fin spacing angle θ, and (iii) branch angle α. Constant temperature boundary conditions are used. The average Nu at the outer walls is used to gauge the heat transfer. Transient simulations are performed to observe stability of the flow profile over time due to the low Pr of the In-Ga-Sn. Steady-state simulations are performed to sweep the fin parameters. A uniform magnetic field is added to investigate the effects on the heat transfer. This is the first study to use the tree-shaped fins in the annulus with liquid metal and in the presence of an external magnetic field. The fins are generated using Constructal theory, in that each pair of branches (or construction) is sequentially built from the previous pair of branches. The optimum case with the tree-shaped fins and no magnetic field is found to increase Nu by 789.7%. With the magnetic field, the “intermediate” tree-shaped fins produced the highest Nu (8.66) of the fin geometries explored at Ha = 20.

A wearable flexible battery using human body temperature differences

Ionic battery materials are widely used in wearable devices, especially ionic gel materials. In the face of a variety of ionic thermoelectric materials, it is a key issue to select the appropriate thermoelectric ones for different artificial heart power supply needs. In this study, it is aimed to establish a database of ionic thermoelectric materials for existing wearable devices, compare the battery performance under different boundary conditions and different materials based on the physical model of the battery, and select the optimal battery material suitable for an artificial heart. 72 kinds of ionic thermoelectric materials were identified through a literature review. The properties of ionic thermoelectric materials were comprehensively considered from the aspects of the Seebeck coefficient, electric conductivity, and power factor. The materials were determined by a thermoelectric figure of merit (ZT). The physical model of the battery was established by Solidworks. The three-dimensional physical field of thermoelectric materials was analyzed by COMSOL, so as to clarify the temperature and potential change characteristics of thermoelectric materials under different boundary conditions and different material types. Finally, the thermoelectric material with the largest electric potential energy was selected. Under a constant temperature boundary condition of 33 ℃ and 5 ℃, the maximum potential produced was 23.21 mV. By changing the materials and keeping other conditions unchanged, it was found that the potential generated by the six ion gel materials is mostly mV.

A fully coupled thermo-poro-elastic model predicting the stability of wellbore in deep-sea drilling. Part A: Analytic solutions

Wellbore stability is important to deep-sea drilling (DSD) for exploiting offshore oil and gas resources. Accurate prediction of the mechanical behaviors of rock formation around the wellbore can effectively prevent wellbore failure via adjusting mud weight. In this paper, a fully coupled thermo-poro-elastic (THM) model is developed by combining fluid circulation in the wellbore and rock deformation in the reservoir. Transient heat transfer in the wellbore/reservoir and heat exchange between the tubing, annulus, riser, casing, cement, seawater and rock formation are considered. By decomposing the problem into axisymmetric and antisymmetric sub-problems, the analytical solutions for transient temperature, pressure and stress in the rock formation and fluid temperature changes in the drill pipe, annulus and seawater near the riser are obtained and validated against with the numerical solutions, existing fluid circulation model, field measurement data and the existing model without considering fluid circulation. After that, a typical case study is carried out to show the mechanical behavior around the wellbore. It is found that the proposed fully coupled THM model has a high accuracy with a relative error of around 1.2%. There is a large difference in the predicted bottomhole temperature (BHT) and thus the temperature in the reservoir between the models with and without considering fluid circulation in the wellbore. An assumption of constant temperature boundary condition may lead to overestimated or underestimated results depending on the selection of the assigned wellbore temperature. In the DSD case studied, the rock formation at the upper part of the reservoir (Z < 0.40) is continuously heated and the rock formation at the lower part (Z > 0.6) of the reservoir is continuously cooled during fluid circulation, which thus results in a complex heat transfer and thus mechanical behavior in the wellbore/reservoir system, especially near the wellbore. The bottomhole temperature (BHT) decreases by 38.9 K due to cold fluid circulation from 3.68 min to 25.68 days, which can lead to 14.59 MPa tensile stress that cannot be ignored for wellbore stability analysis. The proposed analytical model provides a useful tool to predict the mechanical behaviors such as the transient temperature, pressure, displacement and stresses in the wellbore/reservoir system during fluid circulation in DSD. It runs efficiently in terms of seconds per case study in a portable laptop computer and can also work a benchmark for other numerical methods. However, the fully coupled THM model and its analytical solution are based on several assumptions, such as the heat generated by the drill bit rotation is ignored and only the liquid phase is considered, which requires further investigation.

Numerical studies of natural convection phenomena for a vertical cylinder with multiple lateral baffles in triangular and hexagonal enclosures

Numerical case studies for natural convection were conducted for triangular and hexagonal enclosures as a follow-up to a previous study of natural convection for a baffled cylinder in a square enclosure. For a Prandtl number of 0.71, a range of the length ratio corresponding to the spatial ratio was varied from 0.4 to 0.9, and Rayleigh numbers from 2.5 × 103 to 5.0 × 107 was considered. To model the buoyancy force, the Boussinesq approximation was adopted. The boundary condition of constant wall temperature was applied to the cylinder and enclosure surfaces. Overall trends of the heat transfer performance were separated into conduction- and convection-dominated regimes, similar to the previous study. Because of the wider space of the triangular enclosure, however, an advantage of the convection effect was observed in the triangular enclosure. Due to the short distance between the cylinder and enclosure, the hexagonal enclosure had a more conduction-dominated characteristic than that of the triangular enclosure. A strong correlation was identified between the location of the high-energy vortex and the Nusselt number distribution. Using the calculation results, an improved average Nusselt number correlation is proposed to include triangular, square and hexagonal enclosures.

Experimental Evaluation of Thermoelectric Generator Performance under Different Heat Conduction Boundary Conditions

Heat conduction boundary conditions play a crucial role in the performance of thermoelectric generators (TEG). The TEG output voltage and power were measured under constant temperature boundary and heat flux conditions to evaluate the TEG performance under different heat conduction boundary conditions. External loading pressure and thermal interface material (TIM) were applied to reduce the interfacial thermal contact resistance. In our measurement setup, a fast-response electronic load was used for the rapid current-voltage scan, which can eliminate the thermal drift caused by the Peltier effect. A guard heater arrangement is used to minimize heat loss. In constant temperature boundary conditions, reducing the thermal contact resistance can increase the effective temperature drop across the TEG module and significantly improve the output voltage and power. But in the constant heat flux conditions, since the heat flux flow through the TEG is unchanged, the temperature drop across the TEG was unaffected by the thermal contact resistance. As a result, the TEG performance was lightly influenced by the thermal contact resistance.

Numerical study of the effect of heat-flow coupling on interface instability based on front tracking

The interfacial instability of two-phase immiscible fluids in inclined tubes at constant temperature boundary conditions is numerically investigated by the front-tracking method. By analyzing the effects of inclination angle φ, Marangoni number (Ma), and Rayleigh number (Ra) on the interfacial instability, the interaction law between unstable interface fluctuation and heat transfer is studied. The results show that the larger the inclination angle, the easier the interface is destabilized, and the heat transfer at the interface will also decrease. In the comparison of inclination angles of 0°, 30°, 45°, and 60°, the heat transfer is more stable at 45°. The heat flux between fluids decreases with the increase of Ma number, and Ma number has little effect on the interfacial fluctuation. It mainly affects the interfacial morphology by changing the surface tension gradient at the interface, which is mainly reflected in the end of the convolution interface. The larger the Ma number, the more inward the interface develops. Ra number has an obvious inhibitory effect on the interfacial instability. The effect of Ra number on heat-flow transfer at the interface shows alternating changes. In the initial stage, the heat transfer between fluids in the inclined tube is greater than that in the horizontal tube, but in the later stage, the heat transfer is less than that in the horizontal tube.

Controlling the freezing process of nano-enhanced phase change materials with internal fins in the latent thermal energy storage system

In this article, the freezing process of phase change material (PCM) in a latent energy thermal storage system with new internal fins was numerically discussed. Two PCMs: RT35 and RT50 are used. The finite volume method and the enthalpy-porosity model are employed. The effects of seven new internal fins, the addition of aluminum oxide nanoparticles in two volume fractions of 2.5 % and 5 %, and changes in the temperature boundary condition on the freezing process of the PCM are explored and the results were compared with the case without fins. The outcomes determine that with the addition of nanoparticles, the freezing time decreases for all states with fins compared to the state without fins, and the percentage of these changes is around 29–62 %. Also, the outcomes show that the reduction in freezing time compared to the case without fins for case C3 is about 69.43 % and 71.82 % for RT35 and RT50, respectively, and this mode performed best among the considered cases. The spatial changes in the temperature of the active walls of the chamber along the y, and z-directions affect the freezing rate and increase the freezing rate compared to the constant temperature boundary condition.

Thermo-bioconvection performance of nanofluid containing oxytactic microorganisms inside a square porous cavity under constant and periodic temperature boundary conditions

Thermo-bioconvection performance of nanofluid containing oxytactic microorganisms inside a square porous cavity under constant and periodic temperature boundary conditions