Analytical Heat Transfer Model Research Articles

Digital twinning of temperature fields for modular multilayer multiphase pipeline structures

The temperature field of oil and gas wells in the field of petroleum engineering presents a core problem and challenge in the digital twin framework due to its ultra-long-distance and highly variable structural characteristics. The varying wellbore cross-sectional structures with depth make it difficult to establish an effective and generalized analytical model for heat transfer. In this study, we propose, for the first time, a method to automate the construction of multi-layered and multi-component heat transfer models by using a general computational model based on non-steady-state single-phase structural modules. This method enables the automated generation of complex multi-layered and multi-component heat transfer models, thereby achieving the construction of a generalized model for temperature field characterization with varying wellbore cross-sectional structures over ultra-long distances. Utilizing this modeling approach, we validate the proposed method through case studies using actual wellbore temperature field data. The results demonstrate the lightweight and efficient computational analysis of temperature field information under non-steady-state conditions.

Analytical solution model of heat transfer for energy soldier piles during excavation to backfilling

This paper presents a two-dimensional heat transfer model for energy soldier piles during excavation and after backfilling. Based on Green's function and separation of variables method, this model considers the heat transfer process from various angles during excavation and after backfilling, deriving a series of analytical solutions for the heat transfer processes. It is more suitable for solving the heat transfer problem of energy soldier piles. The accuracy of the analytical solution was verified by numerical simulation and classical analytical solutions. Then, the effects of air convection coefficient and backfill materials' thermal diffusivity on heat transfer are analyzed. The results show that the temperature rise of both the soil and the pile in the non-excavated case (ordinary energy pile) is lower than that of the various convective coefficient cases, and the convective heat transfer contributes to the heat transfer capacity of energy soldier piles. After backfilling, materials with higher thermal diffusivity have smaller temperature rise. Conversely, materials with lower thermal diffusivity have higher temperature rise. Under the long-term operation strategy, the heat flux within the soil and backfill is higher and favorable for heat transfer by selecting the backfill with lower thermal diffusivity.

Enhancing ground source heat pump system design optimization: A stochastic model incorporating transient geological factors and decision variables

In this study, a cost minimization model of a ground source heat pump (GSHP) was performed to obtain its best design configurations based on system reliability. To implement a high-fidelity optimization, the uncertainties of decision parameters and geological parameters' distributions were considered. The GSHP's lifespan costs were functionalized based on an analytical borehole heat transfer model. Multiple constraints were established to serve as reliability goals to ensure a turbulent ground pipe flow without extreme temperature variations. The ground thermal conductivity, groundwater velocity, and unit price of electricity were considered as three random variables. Their dynamic characteristics were incorporated by assigning different increasing rates to the average magnitudes of them. Additionally, their transient fluctuations were accommodated by integrating probabilistic uncertainties into the optimization process, all of which followed normal distributions. The results demonstrated that the time-dependent characteristics of these uncertain variables could significantly affect the system investment of GSHP by reducing the total costs by 20.7% and 21.1% when the increasing rates of groundwater velocity and ground thermal conductivity reach 5% and 4% respectively. The study also provided optimized design variables under an 85% confidence level in various predicted scenarios, considering a wide range of dynamic geological parameters and energy costs.

Ground Heat Flux Reconstruction Using Bayesian Uncertainty Quantification Machinery and Surrogate Modeling

AbstractGround heat flux (G0) is a key component of the land‐surface energy balance of high‐latitude regions. Despite its crucial role in controlling permafrost degradation due to global warming, G0 is sparsely measured and not well represented in the outputs of global scale model simulation. In this study, an analytical heat transfer model is tested to reconstruct G0 across seasons using soil temperature series from field measurements, Global Climate Model, and climate reanalysis outputs. The probability density functions of ground heat flux and of model parameters are inferred using available G0 data (measured or modeled) for snow‐free period as a reference. When observed G0 is not available, a numerical model is applied using estimates of surface heat flux (dependent on parameters) as the top boundary condition. These estimates (and thus the corresponding parameters) are verified by comparing the distributions of simulated and measured soil temperature at several depths. Aided by state‐of‐the‐art uncertainty quantification methods, the developed G0 reconstruction approach provides novel means for assessing the probabilistic structure of the ground heat flux for regional permafrost change studies.

Thermal and structural analysis of TCABR vacuum vessel during baking process

TCABR is a small-sized tokamak located at the University of São Paulo, in Brazil. It is planned to undergo an extensive upgrade, which includes a baking system. However, this tokamak wasn't designed considering baking as a possible wall conditioning technique. This paper analyzes whether a baking system with ohmic heating could in principle be installed on TCABR, considering the thermal stresses that it might produce on the vacuum vessel. An analytical steady-state heat transfer model was used to estimate the heat fluxes required to reach a target temperature of 200 °C. Thermal and structural transient finite element method models were developed using ANSYS Workbench®. The temperature distribution was used to calculate the thermal stresses, also considering the effects of gravity and vacuum pressure. The stresses produced in the vacuum vessel by baking were below the limit defined by the ASME stress code, but the graphite tiles’ support rails reached stress values above their material's yield strength. The resulting temperature distribution was not uniform, with some regions far from the 200 °C target. A redesign of the baking system will be required to reach a more uniform temperature field and avoid stresses above yield strength for some components.

An overview of the Transient Hot Wire and preliminary studies for its implementation

The opportunity to design a new portable system for measuring thermal conductivity in fluids arose from the need to measure the phase state (liquid and/or vapour). As is well known, thermal conductivity varies in relation to the phase state of the fluid. The Transient Hot Wire Method was used to measure thermal conductivity. First, the work includes innovations in the analytical model of heat transfer, progressing over time to contemporary numerical models used in standards. The applications in the bibliography span a wide range, including gases, liquids, and the latest advances in the field of nanofluids. Secondly, the study delves into the transient model of heat exchange, focusing on conduction. The authors used standardized correction equations for the prototype, ensuring accuracy and relevance. Finally, the work includes the design and construction of the transducer. This process begins with a Monte Carlo simulation, predicting the expected results. This simulation helps to define measurement system components based on their performance characteristics and in-depth analysis.

A mechanistic explanation of shrinkage porosity in laser powder bed fusion additive manufacturing

This work focuses on the occurrence of shrinkage porosity and its manifestation in Alloy 718 during laser-powder bed fusion (PBF-LB) processing. Shrinkage porosity is a common defect in metal castings that degrades part performance. In traditional metal castings, the Niyama criterion is a reliable heuristic for predicting the formation of shrinkage porosity. However, the Niyama criterion’s applicability in the PBF-LB process remains unexplored, and there is no known and evaluated heuristic to predict shrinkage porosity in the PBF-LB manufactured parts. This work employs microstructure characterization and an analytical heat transfer model to develop a mechanistic explanation for the formation of shrinkage porosity in the PBF-LB process. The results show that the Niyama criterion could not effectively predict the occurrence of shrinkage porosity. Further, the formation of shrinkage porosity is primarily driven by secondary dendrite arm growth in the solidifying microstructure, where the transition to cellular growth at high cooling rates during solidification mitigates porosity by removing locations for pore formation. This means a heuristic based on solidification cooling rate could reliably predict the occurrence of shrinkage porosity. For practical use, shrinkage porosity process maps are presented for use in process design and control to directly aid shrinkage porosity mitigation in process planning. The process-shrinkage porosity relationship results also indicate that trends in PBF-LB manufacturing toward higher deposition temperatures and higher throughput are likely to aggravate the conditions for shrinkage porosity formation and further elevate the importance of the mitigation strategies presented in this work.

Preliminary analysis of maximum hypothetical accident for a solid core space nuclear reactor power system

In order to study the safety characteristics of the solid core space nuclear reactor power (SNRP) system under the maximum hypothetical accident, a two-dimensional entire core transient heat transfer analysis model was established, and the key parameters response characteristics of the ultra-small lithium-cooled SNRP under two maximum hypothetical accidents, namely, loss of heat sink (LOHS) and loss of coolant accident (LOCA), were calculated and analyzed. In the maximum hypothetical accidents, the coolant cooling capacity is lost, thus the core decay power is discharged into space only through the radiation of the residual heat removal system and the side surface of the reactor vessel. During the accidents, the heat transfer of the core deteriorated, and the fuel temperature may rise to the melting point, resulting in radioactive leakage. The results show that: (1) In the LOHS accident, the maximum fuel temperature reaches 2150 K at 550 s, and the pressure of the primary system volume accumulator continues to increase to the set system pressure safety limit of 2 MPa, resulting the primary loop overpressure failure. And the fuel matrix temperature is close to the set cladding limit temperature of 2200 K; (2) In the LOCA, the deterioration of heat transfer in the core makes the temperature increase rapidly, reaching a maximum of 3016 K at about 630 s, which is very close to the melting temperature of UN fuel 3123 K. As the decay power decreases, the maximum core temperature decreases to less than 1600 K after 24 h of the accident. The auxiliary cooling system of the solid core SNRP system under the maximum hypothetical accident is optimized, and the design parameters of the auxiliary cooling system meeting the safety requirements are obtained.

Post-fire performance of circular steel tube confined steel-reinforced concrete slender columns: Testing, simulation and design

Steel tube confined steel-reinforced concrete (STCSRC) stub columns offer good residual bearing capacity, great ductility, and large stiffness after fire exposure because inner steel profiles are unaffected by exterior temperature. Nonetheless, instability issues exist in STCSRC slender columns, and study into the mechanical characteristics of post-fire slender specimens is critical. At present, no relevant study on the axial performance of STCSRC slender columns after fire exposure has been reported. In this paper, nine circular STCSRC slender columns that had been exposed to fire were tested under axial load. The failure modes, axial load–displacement relationship, and lateral deflections of STCSRC columns were discussed. Furthermore, heat transfer analysis models and mechanical analysis models were established, and the effects of crucial variables on the buckling responses of STCSRC slender columns were explored. Among the parameters studied, sectional diameters and steel profile to concrete area ratio have a significant impact on buckling capacity. To calculate the buckling capacity of the post-fire STCSRC slender column, a simplified calculation method according to JGJ/T 471–2019 was proposed.

Modeling of heat conduction processes in porous absorber of open type of solar tower stations

An analysis of existing methods for calculating heat and mass transfer processes in porous absorbers of receivers of tower solar power plants is carried out. It is shown that the resulting thermophysical properties of the material are influenced not only by the porosity but also by the location of the pores in the material volume. The criterion of the dislocation vector is proposed as a mathematical indicator of various porous structures. The shortcomings of the existing dependences of the effective thermal conductivity of a material on the type of porosity are shown. The most reliable dependences for determining the thermophysical parameters of a porous medium are also determined and independent factors are proposed on which the mathematical model of heat and mass transfer in open-type solar receivers should be based.The current state of research on the effective thermal conductivity of the porous structure of solar receivers is described in detail. A new formula for calculating the effective thermal conductivity of a porous structure with regard to the dislocation vector and a method for calculating the processes of heat transfer in open solar receivers based on the proposed formula are proposed. The proposed equation has been tested. It is determined that for simple channel structures it is sufficient to use the existing equations to calculate the thermal conductivity coefficient, while for more complex porous structures, such as the StepRec absorber, it is better to use the proposed equation.Among the strengths of this study is a new calculation formula that allows us to build an analytical model of heat transfer in a porous medium. The use of the analytical model can significantly reduce the complexity of modern calculations of heat transfer processes in a porous absorber and will help improve the quality of optimization models of solar receivers.

Analytical model of heat transfer of U-shaped borehole heat exchanger under convolution theory and its optimal heat extraction analysis

Middle deep borehole heat exchanger (DBHE) has become widely known as one of the main ways of geothermal energy development and utilization, and the use of middle DBHE system for heating buildings has received much attention. More accurate analytical models for a coaxial borehole heat exchanger (CBHE) have been built using the convolution theorem, however, a U-shaped deep borehole heat exchanger (U-DBHE) has not been reported. Therefore, in this paper, to obtain more accurate fluid temperature distribution, an analytical model of middle DBHE considering geological stratification phenomenon is developed by using convolution theorem. The relative error of the analytical model is verified within 5% by numerical simulation. At the same time, the interaction between the three factors was analyzed and studied by response surface method: mass flow, inlet temperature and geothermal gradient on heat extraction and heat loss rate. The results reveal that for thermal extraction, the significance levels of the three factors are from large to small: geothermal gradient > inlet temperature > mass flow, for heat loss rate, the two factors of flow rate and inlet temperature are significant, and the impact of geothermal gradient is not significant. In this engineering project, when the inlet water temperature is 10 °C, the flow rate is 26.65 kg/s and the geothermal gradient is 4.5 °C/hm, there is a maximum heat extraction of 2424.16 kW.

Thermal Assessment of Repository Footprint Requirement for Advanced Fuel Cycles

Decay heat is an important constraint for repository size and design because it can drive processes that affect performance and compromise critical materials. This paper investigates how compliance with repository thermal limits is affected by three decay heat management strategies: waste package loading, waste package spacing, and surface storage time. In particular, this paper focuses on how repository area, a result of package spacing, is impacted by waste loading and surface storage time. A two-part analytical heat transfer model is presented and executed iteratively to determine the minimum allowable repository area. The analysis considers three generic close-contact repository designs along with the wastes generated from the 40 fuel cycle analysis examples used to generate metric data for the Fuel Cycle Evaluation and Screening study. Detailed results are presented for two fuel cycles: the once-through use of low-enriched uranium fuel in light water reactors and the continuous recycling of U and Pu in sodium fast reactors. Two limits for surface storage time are identified: the time required for disposal to be possible at all and the time at which further surface storage time yields no gains. The impact of waste loading is also diminished with increasing surface storage time. In general, the generic salt repository is most flexible to accept high-heat-generating wastes with less surface storage time than other repository environments. Limited-recycle fuel cycles are shown to pose a disposal challenge because of elevated, sustained decay heat generation in the waste.

Partial crystallization in a Zr-based bulk metallic glass in selective laser melting

Metals and alloys in amorphous state are promising for the use as structural and functional materials due to their superior properties related to the absence of such defects as grain boundaries and dislocations. Obtaining the amorphous state requires quenching from liquid state with a high cooling rate. In conventional technologies, this is the reason for a considerable size limitation hindering application of amorphous alloys. Additive manufacturing (AM) is free of the size limitation. The so-called bulk metallic glass (BMG) alloys have extremely low critical cooling rates and can be used in AM. Recent studies on AM from Zr-based BMGs by selective laser melting (SLM) has proved the possibility of attaining the amorphous state and revealed partial crystallization to be the principal drawback of this process. The present work aims to analyze the conditions for partial crystallization and attempts to control it by optimizing the process parameters. A comprehensive parametric analysis is accomplished in single-track SLM experiments with Zr-based BMG alloy Vit 106. The observed microstructures are related to the temperature fields and thermal cycles estimated by an analytic heat-transfer model in the laser-impact zone. In the cross section of a laser track, a central bright domain is identified as the remelted zone. An annular darker crystallization zone encircles the remelted zone. The model fits the experimentally obtained dependencies of the remelted zone size versus laser power and scanning speed and indicates that 43 ± 2% of the incident laser energy is transferred into the substrate thermal energy. The principal energy losses are reflection of the incident laser radiation and material evaporation. Primary crystalline inclusions existing in the substrate before laser processing dissolve at the laser melting. The mean cooling rate in the remelted zone is up to 4 orders of magnitude greater than the critical cooling rate. Therefore, homogeneous nucleation is not expected. Nevertheless, the theoretically estimated crystallization times are sufficient for a considerable crystal growth in the heat-affected zone where primary crystalline inclusions and nuclei are not completely dissolved.

Analytical interpretation and numerical analysis of multiple energy pile thermal response tests

Thermal response tests (TRT) are an essential in situ test for the sizing of ground source heat pump systems. Its application on energy piles is becoming an established practice, involving key differences in relation to testing boreholes. Recently, group TRTs emerged as an alternative to testing single short energy pile elements. This work assesses the execution and interpretation of those tests through different analytical heat transfer models implemented in a unit-response calculation methodology. To assess the suitability of the method, a comprehensive parametric analysis using a validated numerical model is undertaken, considering different energy pile sizes, thermal properties for both concrete and soil and group pile arrangements. The analytical methodology is validated against experimental and numerical results, and then tested to interpret the numerically simulated TRTs using a curve fitting algorithm. The broad parametric investigation and the evaluation of the capacity of different analytical methods on modelling each scenario provide directions to execute and interpret group TRTs. The soil effective thermal conductivity and energy pile thermal resistance are obtained with less than 5% precision error for most scenarios and less than 10% considering all scenarios. These results can be further improved with specific individual G-functions for the pile elements, that can be implemented on the proposed methodology.

Analiza toplotnih efekata u postupcima frikcionog zavarivanja mešanjem

This paper describes the formation of an analytical model of heat generation and transfer during friction stir welding. By solving differential equations of heat transfer, adapted for friction stir welding, the foundations for the formation of an analytical model were laid. In order to vary the technological parameters of welding that affect the generated heat, certain approximations were performed and the stress state of the material was observed at elevated temperatures. The aim was to simulate heat transfer for thin and thick plates, by obtaining a planar and spatial temperature image for certain welding parameters. The formation of the analytical model was performed in the Python programming language.

Analytical heat transfer model for coaxial heat exchangers based on varied heat flux with borehole depth

In the classical analytical coaxial borehole heat exchanger (CBHE) models, the variations of the specific heat flux exchanged vertically along the depth of CBHE are ignored, which induces considerable prediction errors of the ground temperature. Based on the infinite line source model (ILSM), an improved varied heat flux model for the CBHE is developed using the superposition method, taking the vertical inhomogeneity of the specific heat flux into account. The mathematical expressions of the specific heat flux and the temperatures of the fluids and ground at different working conditions are obtained. The model is validated analytically and numerically. Finally, the influence of the mass flow rate, the outer-to-inner pipe radius ratio, and the thermal influence radius are investigated for optimal design. The results show that the specific heat flux is significantly related to the borehole depth, operation time, and fluid circulation direction. Compared with the finite line source method (FLSM), the relative errors of the ground temperature predicted by the proposed model can be reduced maximumly by 8.9% and 10.3% for heat extraction and injection, respectively. In addition, using higher mass flow rates and larger outer-to-inner pipe radius ratios can eliminate thermal interference to increase thermal efficiency. In general, the improved analytical model can provide an accurate and efficient method for this inhomogeneous problem and a valuable tool for designing and optimizing CBHEs for actual engineering applications.

Analytical modeling of mass transfer dynamics, velocity, heat transfer and enthalpy in a gas-liquid combustible mixture

The article studied the distribution of the droplet and temperature over the volume of the combustion chamber when liquid fuel was injected into it. The study showed that the portable mass in the combustion chambers and temperatures are directly proportional. Compiled systems of dynamic equations describing mass transfer, motion, heat transfer and enthalpy at the beginning inside the chamber phase and during combustion inside the chamber. Mathematically modeled a gasoline-hydrogen-air mixture with the mass transfer method. With this simulation, pressures are determined at the beginning inside the chamber phase and during combustion inside the chamber, taking into account the newly introduced enthalpy coefficient of gas kHg and steam kHp phase media.

An analytical heat transfer model for the mid-deep U-shaped borehole heat exchanger considering groundwater seepage

Ground source heat pump systems consisting of deep boreholes and heat pumps can be used for heating buildings. It is known as an effective way to utilize geothermal resources, however, there is a problem with high development costs. Therefore, it is important to establish a precise heat transfer model to develop a high-efficiency geothermal heat exchanger to reduce the development cost of mid-deep geothermal energy. Most of the existing heat transfer models for deeply buried pipes only consider thermal conductivity and treat the geotechnical soil as a homogeneous medium, or do not consider the stratification and groundwater seepage phenomena, which make the models inaccurate. This study established an analytical heat transfer model considering geotechnical stratification and seepage developed for the U-shaped deep borehole heat exchanger (U-DBHE), and the analytical solution for temperature response is obtained. Using the developed analytical model, we plotted the trends that reveal the relationship between the thermal performance of U-DBHE and various parameters and explored the influence of seepage phenomena on heat extraction performance. The model can also serve as a reference for the development of heat transfer models for buried pipe heat exchangers with groundwater seepage. Given that the proposed analytical model is time-consuming and stable, and can be an efficient tool for U-DBHE design and optimization.

Simulation of thermal behavior evolution in lithium-ion pouch cells using a coupled reaction-thermal-expansion model

Conventional lithium-ion battery thermal simulation methods establish a coupled heat generation/transfer model to calculate the temperature evolution for hard-pack cells. However, the thermal behavior evolution of a pouch cell includes multi-processes such as heat generation, gas production, heat transfer, and pouch expansion. There is currently a lack of computational models for predicting the multi-state parameter evolution in temperature and pressure. In this paper, an efficient and accurate thermal behavior simulation algorithm for lithium-ion pouch cells is developed. A heated/exothermic two-stage hypothesis for the thermal behavior evolution in pouch cells was established under thermal abuse conditions. Analytical models of exothermic reaction, heat transfer and pouch expansion were formulated and integrated into a unified thermal behavior analysis framework. The proposed algorithm simulated the thermal behavior evolution in four commercial lithium-ion pouch cells. The computational and experimental results showed the calculation errors of the cell temperature and gas volume for the four samples were less than 3% and 5%, respectively. In terms of efficiency, the calculation time of the algorithm is within milliseconds. The accuracy and efficiency of the proposed algorithm are verified.

Analytical and numerical modeling and simulation of heat transfer through raffia vinifera bamboo

The permanent search for materials that can improve climatic conditions and thermal comfort in buildings remains a major concern for engineers. The choice of material is motivated by its behavior to the thermal stresses. To predict the distribution of temperature within the material requires mathematical models. In this paper, the heat transfer through raffia vinifera bamboo is modeled using the equation of energy conservation. Raffia vinifera bamboo consists of cork and shell and is treated as two layers material. The thermophysical properties of each layer are constant. The equation is solved analytically by the separation of variables method and numerically by the finite difference method. A numerical simulation with CASTEM software is also performed. A constant temperature is imposed at the external surface of the material while the continuity of the temperature and flux are applied at the cork/shell interface. To test the computational results, the thermophysical properties of the two layers were set equal. The analytical solution in this case of single layer was used as the reference result and the computational methods developed in this work were compared. Results demonstrated that the absolute value of the relative error at the center of material decreases abruptly with time and a maximum value less than 8% occurred for all the developed methods in the paper. Results also indicated that at a given time, the relative error increases slowly to a maximum value less than 8% close the surface. These results showed that the methods presented were correct. The case of two layers with different thermophysical properties was second examined and the raffia vinifera bamboo was viewed as heterogeneous material. The temperature profile at a given time decreases from the surface to the center of the material with a peak occurring at the interface cork/shell. At time t=9750.336s, the temperature at the center increases only by 4.75 °C. Thus, the raffia vinifera bamboo can be used as thermal insulation in buildings.