Cylindrical Heat Source Research Articles

Oriented magnetic field effect on mixed convective flow of nanofluid in a grooved channel with internal rotating cylindrical heat source

In this paper, the flow and heat transfer characteristics for the influence of oriented magnetic field on mixed convection flow of water based nanofluid inside a grooved channel with a rotating heat source are numerically investigated. The channel is cooled from the grooved as well as vertical walls and heated from the bottom walls as well as rotating heat source while the remaining walls are thermally insulted. The channel is permeated by an inclined magnetic field of uniform strength, and a modified model of effective thermal conductivity is used to improve the overall thermal conductivity of nanofluids. The governing partial differential equations representing the flow model are solved with Galerkin weighted residual finite element method. A complete parametric study is carried out based on numerical results to show the variations of flow and temperature fields in terms of streamlines, isotherms, velocity and temperature profiles, average Nusselt number and average temperature for the effects of pertinent parameters including Reynolds number, Hartmann number, volume fraction of nanoparticles and inclination angle of the magnetic field. It is observed that average heat transfer rate enhances noticeably with the increase in Reynolds number and volume fraction and reduces for increasing Hartmann number. It is also found that the augmentation of heat transfer due to higher inclination angle of magnetic field becomes significant when the value of Hartmann number is sufficiently large. Moreover, the effects of governing parameters on the fluid flow and heat transfer behaviors are affected remarkably with the presence of rotating heat source and the direction of rotation as well. Comparisons of the present results with the previous published results are performed and excellent agreement is found. The outcome of this study can be applied to design engineering equipments such as high performance heat exchangers, cooling of electronic devices and circuit boards, cooling of nuclear reactors and biomedical equipments, etc.

Thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source

We develop a new numerical model based on a precise integration method to investigate the coupled thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source. To obtain the relational matrices of the extended precise integration method, we first convert the governing equations of the problem into ordinary differential matrix equations through the Laplace–Hankel transform. Then, the cylindrical heat source is divided into a series of plane heat sources, and the plane temperature load term is added to the state vector between layer elements. By combining the layer elements, we build a layered transversely isotropic numerical model containing a cylindrical heat source in the transformed domain. Finally, we solve the model in the transformed domain and obtain the solution of the problem in the real domain through the Laplace–Hankel transform inversion. The accuracy of this method is verified by comparing the solutions with the results of the analytical method and the finite element method. Then, we study the influence of the anisotropy of thermal parameters, the embedded depth, the length/radius ratio, the type of heat source and the stratification of the medium on the thermo-mechanical coupled performance.

High-performance and integrated design of thermoelectric generator based on concentric filament architecture

A concentric thermoelectric filament structure used for radioisotope thermoelectric generator (RTG) is designed to satisfy low-power and high-voltage demands for aerospace microelectronic devices. The thermoelectric filament is proven to have a better output voltage than the traditional thermoelectric structure by using COMSOL. The unbreakable thermoelectric filament and corresponding devices are prepared by employing a simple and easy brush coating process. A single thermoelectric filament obtains an open-circuit voltage (Voc) of 6.0 mV and a maximum power output (Pmax) of 2.0 μW at 398.15 K hot surface temperature. An array of thermoelectric filaments is fabricated for the radioisotope heat source of planar heat surface, which obtains Voc of 83.5 mV and Pmax of 32.1 μW at the hot surface temperature of 398.15 K, thereby verifying the practicability of the concentric filament structure in devices. Another radial thermoelectric filament device is designed for the cylindrical heat source. This device exports 84.5 mV Voc and 42.5 μW Pmax at the hot surface temperature of 398.15 K. These thermoelectric devices could gain larger electrical performance in small space through further series connection and fabrication, which will be a commendable design scheme for the radioisotope thermoelectric generator.

TEMPERATURE DISTRIBUTION IN A SEMI-INFINITE MEDIUM DUE TO A HEAT SORCE OF DIFFERENT SHAPE

Thermographic methods based on IR images are actively used in medical practice for the early diagnosis of diseases, in clinical procedures and surgical operations. One of the weak points of the above methods is that the images are noisy due to the thermal influence of a tissue layer located between a human organ under examination and a thermal imager and they therefore carry only indirectly information about the temperature regime of internal organs. In order to improve the accuracy of remote thermographic methods, as applied to the human skin, the method of temperature estimation due to the presence of heat sources of different origin in the tissue has been developed. This method is based on using the linear systems theory approach and the calculation results of a point source. Sources in the shape of a sphere, a cylinder of different orientation, straight lines, and a circle are considered. Specific features of thermal fields from heat sources are indicated. Simple computational models are obtained for a temperature from a spherical source inside and outside it in an infinite medium. Temperatures from a heated line of finite and infinite length are compared. Using the example of a cylindrical heat source, the temperature inside and outside the source is analyzed in detail as a function of thermal physical parameters, occurrence depth and sizes. Particular attention is paid to comparing the results of the temperature distribution on the surface of the skin and its sizes. The given results can be used for temperature correction in thermographic studies.

Effect of geometrical parameters on natural convection in a porous undulant-wall enclosure saturated by a nanofluid using Buongiorno's model

The paper refers to investigate the effect of geometrical parameters on natural convection of nanofluid over a cylindrical heater located inside of a porous undulant-wall cavity using local thermal non-equilibrium (LTNE) condition. Two phase-Buongiorno's model was applied to consider the thermal and flow indexes under geometrical variations. The validity of the numerical procedure was examined by contrasting the present and the previous works. The effect of governing parameters of the cavity, including horizontal and vertical displacement of cylindrical heat source (x and y respectively), the amplitude of the lateral wave walls (λ) on streamlines, isotherms of two porous medium phases and also concentration of nanoparticles has been inquired. Overall, the results show that temperature distribution is different for the two phases in the porous medium, making it inevitable to utilize the non-equilibrium model for porous media. It is obvious that the flow is weakened as the heat source is elevated, while by the horizontal deviation of the flow toward a certain direction, it becomes stronger in the region opposite to the deviation and weakened at the deviated side. When the cylindrical heat source is located at the bottom of the cavity, the nanofluid becomes more homogeneous as a result of the agitation by strong vortices. As the corrugation amplitude of the lateral walls is increased, the nanofluid flow becomes stronger, leading to a more uniform distribution of nanoparticles around the cavity. At lower corrugation amplitudes, the average Nusselt number of the fluid is reduced as the heater is elevated. On the other hand, at higher amplitudes, the variations of the average Nusselt number of the fluid shows an increasing-decreasing trend as the cylinder is elevated.

Study on the influence of the identification model on the accuracy of the thermal response test

A 3-D numerical heat transfer model of a ground-source heat pump thermal response test is established in this paper and verified by testing. The numerical model is used to simulate the thermal response test. The soil thermo-physical parameters are estimated based on the cylindrical heat source and the linear heat source theory and contrasted with the values setting in the numerical heat transfer model. The effect of the identification model of the in-situ thermal response on the accuracy of the identification results is studied under different soil thermal properties. The results show that there is an optimal time to identify the soil thermo-physical parameters based on the two models. Identifying the physical parameters at the non-optimal time points will produce a large error. Under the same conditions, the optimal identification time of the cylindrical heat source model is earlier than that of the linear heat source model and is also related to the soil physical properties. Therefore, the test time should be selected to ensure the accuracy of the results. Furthermore, there is no evident influence of the initial soil temperature on the identification accuracy of the two models. This paper offers a reference for improving the thermal response test of ground source heat pumps.

Flow characteristics in occupied zone – An experimental study with symmetrically located thermal plumes and low-momentum diffuse ceiling air distribution

Airflow interaction between thermal plumes and vertical air distribution may cause significant effects on airflow characteristics such as velocity and temperature fields, turbulence intensity and fluctuation frequency. The flow interaction creates a random flow motion, vortical structures and turbulent mixing that can further yield a draught discomfort in an occupied zone. The main objective was to investigate large-scale airflow patterns and fluctuations as a result of interaction of buoyancy flows and diffuse ceiling flow. Experiments were performed in a test room of 5.5 m (length) x 3.8 m (width) x 3.2 m (height) with symmetrical set-up of cylindrical heat sources that gave a thermal load of 40–80 W/floor-m2. The ventilation air was supplied through a diffuse ceiling with 0.5% degree of perforation. The observations indicate that the mean air speed and the airflow fluctuation increase with thermal load. Furthermore, the results show that a range of length scales increases with thermal load and with mean air speed. The results indicate that it can be difficult to fulfill the standard air velocity criteria for highly occupied spaces, where the maximum allowable mean air velocity is relatively low, i.e. 0.15–0.20 m/s. This is because the buoyancy flows from heat sources accelerate locally the flow field.

Transient forced convection from an infinite cylindrical heat source in a saturated Darcian porous medium

This paper deals with the problem of an infinite cylindrical heat source embedded into a saturated porous medium and subject to a cross-axial Darcian flow. Only forced convection is considered. We derived the transient dimensionless solution through a combined analytical – numerical method consisting of four steps: (a) a preliminary dimensional analysis of the constitutive equations of the problem in order to find the dimensionless groups governing the solution; (b) the identification of the validity range of the model as a function of the just-mentioned dimensionless groups; (c) the numerical resolution of the problem; (d) the synthesis of the numerical results in a general dimensionless form. Specifically, we provide several dimensionless maps of the 2D thermal field evolution for six different orders of magnitude of the Péclet number (10-3–102). The evolution of the temperature of the heat source is fully illustrated and discussed through plain dimensionless criteria. Then, we discuss the time, space and fluid velocity scales in which the solution is practically equivalent to the ones given by a linear heat source and a purely conductive model. We conclude that the present model has to be employed to evaluate the temperature in proximity of the heat source when the reference Péclet number is greater than 0.5. On the contrary, the linear model can be successfully used for radial distances 5–10 times greater that the heat source radius, depending on the reference Péclet number.

Features of a blue-sky transition in an autonomous convective flow

The nonlinear dynamics of a buoyant air flow in the gap formed by a horizontal cylindrical heat source and a concentric isothermally cooled square enclosure have been recently studied, in the limits of a two-dimensional approximation, for different values of the system aspect ratio, defined as the ratio between the cavity side length and the minimum gap width. Accurate analyses have revealed that, for low values of the aspect ratio, the dominant pattern is the swaying motion of a buoyant plume, which undergoes a period-doubling cascade for increasing values of the Rayleigh number. On the other hand, a substantially different behaviour has been preliminarily observed for higher values of the aspect ratio, since, in a thinner gap, the flow exhibits Rayleigh-Bénard-like cellular patterns above the cylinder.The aim of this communication is to describe the peculiar route to chaos which has been detected for this second type of geometry. The scenario consists in the progressive winding of a periodic limit cycle on an attractor whose topology resembles that of a French horn, which is a typical signature of the codimension-2 Shilnikov bifurcation from periodic dynamics to chaos. A similar feature, known as the blue-sky catastrophe, has never been reported so far for confined convection of pure fluids.

Short-term fluid temperature variations in either a coaxial or U-tube borehole heat exchanger

This paper uses composite cylindrical heat-source (CCS) models and typical thermal response test procedures to investigate two full-scale borehole heat exchangers (BHE); where one is a U-tube BHE, and the other, a pipe-in-pipe (coaxial) BHE. A previously developed CCS model is compared to a simplified infinite line-source (ILS) model. A time-varying heat-flux term is verified for the U-tube case, noting the error found when using the CCS model. A model is developed using a similar approach accounting for a coaxial configuration showing a root mean square error (RMSE) of less than 0.1°C over the duration of the test.

Thermo-mechanical coupling response of a layered isotropic medium around a cylindrical heat source

Based on the governing equations of the thermo-elastic problem, the analytical layer-elements of a finite layer and an underlying half-space are obtained using the Laplace-Hankel transform and the characteristic value method. The cylindrical heat source is divided into several micro cylindrical heat sources, which can be approximately simulated by plane heat sources. Then, the global stiffness matrix for the problem is assembled and solved in the transformed domain, and a Laplace-Hankel transform inversion is taken to obtain the real solution. Finally, the influence of heat source types, division numbers, embedded depths and layered properties on the thermo-mechanical coupling response is investigated.

Dimensionless Maps for the Validity of Analytical Ground Heat Transfer Models for GSHP Applications

This article provides plain and handy expressions to decide the most suitable analytical model for the thermal analysis of the ground source in vertical ground-coupled heat pump applications. We perform a comprehensive dimensionless analysis of the reciprocal deviation among the classical infinite, finite, linear and cylindrical heat source models in purely conductive media. Besides, we complete the framework of possible boreholes model with the “hollow” finite cylindrical heat source solution, still lacking in the literature. Analytical expressions are effective tools for both design and performance assessment: they are able to provide practical and general indications on the thermal behavior of the ground with an advantageous tradeoff between calculation efforts and solution accuracy. This notwithstanding, their applicability to any specific case is always subjected to the coherence of the model assumptions, also in terms of length and time scales, with the specific case of interest. We propose several dimensionless criteria to evaluate when one model is practically equivalent to another one and handy maps that can be used for both design and performance analysis. Finally, we found that the finite line source represents the most suitable model for borehole heat exchangers (BHEs), as it is applicable to a wide range of space and time scales, practically providing the same results of more complex models.

Analytical solutions for two pile foundation heat exchanger models in a double-layered ground

In the energy pile technology, pipes of the heat exchanger are installed in the concrete pile. When the pipes are configured in the form of spiral coils, they can usually be described by the solid cylindrical heat source model or the ring-coil heat source model. This paper investigates the temperature responses of these two heat source models in a double-layered ground, and the corresponding analytical solutions are established using the Green's function method. The Green's function, which delineates the temperature response induced by an instantaneous ring source in a double-layered ground, is obtained by the separation of variables technique, and in order to solve the nonlinear equations of the eigenvalues, a constrained Newton method is also developed. The analytical solutions are verified by two means. After that, computational examples are presented. The difference between the temperature response in a layered ground and that in a homogenous ground is discussed, and the computational error of using the homogeneous model to describe the temperature response of a layered ground is also studied.

Flat surface versus present-day topography for cylindrical and spherical sources in temperature field simulation models: The Cerritos Colorados geothermal field, Jalisco, Mexico

Most thermal simulation studies assume a flat upper surface, which is not strictly valid for geothermal areas and volcanic centers. Further, even those thermal simulations that have taken into account the actual topography did not objectively evaluate the simulated temperature differences between the flat surface and topography models. We provide the first 3-D simulations of the Cerritos Colorados geothermal field of Mexico for the flat surface and actual topography models as well as for cylindrical and spherical primary heat sources whose upper surface was assumed to be located at the same subsurface depths. The simulated subsurface temperatures were reported at 9 different subsurface depths from +1275m a.s.l. to −1725m a.s.l. The flat surface models always overestimated the subsurface temperatures as compared to the respective topography models by about 27% near the surface (at +1275m a.s.l.) to only about 1% near the heat source (at −1725m a.s.l.) in the central part of the modelled region, and about 18–70% near the surface to 2–16% near the heat source at eight other selected locations within the area. The percent temperature differences between the flat surface and topography models showed a statistically significant linear dependence for both cylindrical and spherical heat sources on the topographic contrast.

On the bimodal nature of a confined buoyant plume. Part II: Flow structure echoes in state space

This work represents the second part of a twofold study aiming at a detailed characterisation of the low order dynamics of a buoyant plume arising from an enclosed horizontal cylindrical heat source, for which chaos unfolds through a period doubling cascade preluded by a window of quasiperiodicity. In part I, this behaviour has been found to be related to the inherent bimodal nature of the flow, and an analytical characterisation of the asymptotic flows in the vicinity of the bifurcation points has been carried out.In this second part, the thermo-fluid dynamic origin of this bimodal flow is investigated. A novel approach is proposed based on the observation of the existing correspondences between the recurrent formation of cellular flow structures, and the global structure of the system dynamics, as described by the attractor in an appropriate state space. In particular, the two modes are shown to correspond to the existence of two alternate and distinct sequences of flow patterns, and the locking between them corresponds to the synchronisation of analogous events in the evolution of the flow structures. Moreover, reported results show that, within the same framework, the period doubling cascade generated by the extinction of the quasiperiodic flow can be seen as a consequence of the modulation of the cellular flow structures.In light of the remarkable similarities with experimental evidences of other buoyancy-driven flows, the proposed approach may represent a useful paradigm for the analysis of transitional, confined convection problems.

Transient heat conduction in an infinite medium subjected to multiple cylindrical heat sources: An application to shallow geothermal systems

In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are formulated for two types of boundary conditions: a time-dependent Neumann boundary condition, and a time-dependent Dirichlet boundary condition. We solve the initial and boundary value problem for a single heat source using the modified Bessel function, for the spatial domain, and the fast Fourier transform, for the temporal domain. For multiple heat sources, we apply directly the superposition principle for the Neumann boundary condition, but for the Dirichlet boundary condition, we conduct an analytical coupling, which allows for the exact thermal interaction between all involved heat sources. The heat sources can exhibit different time-dependent signals, and can have any distribution in space. The solutions are verified against the analytical solution given by Carslaw and Jaeger for a constant Neumann boundary condition, and the finite element solution for both types of boundary conditions. Compared to these two solutions, the proposed solutions are exact at all radial distances, highly elegant, robust and easy to implement.

Effect of Borehole Material on Analytical Solutions of the Heat Transfer Model of Ground Heat Exchangers Considering Groundwater Flow

Groundwater flow is one of the most important factors for the design of a ground heat exchanger (GHEX) since the thermal environment of the ground around the buried GHEX is significantly affected by heat convection due to the groundwater flow. Several preceding studies have been conducted to develop analytical solutions to the heat transfer model of GHEX with consideration of groundwater flow. One of these solutions is the combined heat transfer model of conduction and convection. However, the developed combined analytical models are inapplicable to all of the configurations of ordinary GHEXs because these solutions assume that the inner part of the borehole is thermally inert or consists of the same material as that of the surrounding ground. In this paper, the applicability of the combined solid cylindrical heat source model, which is the most suitable to energy piles until now, was evaluated by performing a series of numerical analyses. In the numerical analysis, the inner part of the borehole was modeled as two different materials (i.e., permeable ground formation and impermeable fill such as concrete) to evaluate applicability of the analytical solution along with different diameter-length (D/L) ratios of borehole. In a small value of the D/L ratio, the analytical solution to the combined heat transfer model is in good agreement with the result of numerical analysis. On the other hand, when increasing the D/L ratio, the analytical solution significantly overestimates the effect of groundwater flow on the heat transfer of GHEXs because the analytical solution disregards the existence of the impermeable region in the borehole. Consequently, such tendency is more critical in the GHEX with a large D/L ratio such as large-diameter energy piles.

Visualization of isotherms around a heated horizontal cylinder embedded in a porous medium

Experiments reported here confirm experimentally the existence of the characteristic planar buoyant plumes that emerge as a consequence of the presence of a cylindrical heat source embedded in an homogeneous air-saturated and non-consolidated porous medium. The resultant buoyant plumes were visualized with IR thermography considering two important cases. First, under the action of natural convection and second, under the action of forced convection due to the passage of a constant air stream directed perpendicularly towards the horizontal cylindrical heat source buried in the porous medium. Such streams were induced at different tilt angles. For each case, the temperature distribution is shown and correspondence is found with the isotherms distribution predicted by the theoretical models reported in the literature. The use of infrared techniques appears as a helpful tool that allows us to witness all the process of rise and growth of the buoyant plumes until the phenomenon reaches steady state.

Temperature response factors at different boundary conditions for modelling the single borehole heat exchanger

Design and simulation of borehole heat exchangers rely on the solution of the transient conduction equation. The typical approach for predicting the ground temperature variations in the short and long term is to recursively apply basic thermal response factors available as analytical functions or as pre-estimated tabulated values. In this paper a review of the existing response factor models for borehole heat exchangers (BHE) analysis is presented and a numerical model, built in Comsol environment is employed for calculating the temperature distribution in time and space around a single, finite length, vertical cylindrical heat source also taking into account its position with reference to the ground surface (effects of the adiabatic length or “buried depth” D). The temperature values are recast as dimensionless response factors in order to compare them with analytical solutions where available. Furthermore new temperature response factors suitable for describing the single Finite Cylindrical Source (FCS) under different operating modes (i.e. boundary conditions) are generated. Boundary conditions include imposed heat transfer rate, imposed temperature and a combination of both conditions, where spatially uniform temperature at the BHE interface is attained while also keeping constant the applied heat transfer rate.

Constructal optimization of cylindrical heat sources with forced convection based on entransy dissipation rate minimization

Based on constructal theory and entransy theory, the optimal designs of constant- and variable-cross-sectional cylindrical heat sources are carried out by taking dimensionless equivalent resistance minimization as optimization objective. The effects of the cylindrical height, the cylindrical shape and the ratio of thermal conductivity of the fin to that of the heat source are analyzed. The results show that when the volume of the heat source is fixed, there exists an optimal ratio of the center-to-centre distance of the fin and the heat source to the cylinder radius which leads to the minimum dimensionless equivalent thermal resistance. With the increase in the height of the cylindrical heat source and the ratio of thermal conductivity, the minimum dimensionless equivalent thermal resistance decreases gradually. For the heat source model with inverted variable-cross-sectional cylinder, there exist an optimal ratio of the center-to-centre distance of the fin and the heat source to the cylinder radius and an optimal radius ratio of the smaller and bigger circles of the cylindrical fin which lead to a double minimum dimensionless equivalent thermal resistance. Therefore, the heat transfer performance of the cylindrical heat source is improved by adopting the cylindrical model with variable-cross-section. The optimal constructs of the cylindrical heat source based on the minimizations of dimensionless maximum thermal resistance and dimensionless equivalent thermal resistance are different. When the thermal security is ensured, the optimal construct of the cylindrical heat source based on minimum equivalent thermal resistance can provide a new alternative scheme for the practical design of heat source. The results obtained herein enrich the work of constructal theory and entransy theory in the optimal design field of the heat sources, and they can provide some guidelines for the designs of practical heat source systems.