One-dimensional Steady-state Heat Conduction Research Articles

Geothermal resource evaluation of the Middle Permian Qixia-Maokou Formation in the southern Sichuan Basin, China

The southern Sichuan Basin is an abnormal area of high present geothermal fields in the Sichuan Basin. It is a favorable area for the development of geothermal resources in the Sichuan Basin. But its evaluation of geothermal resources is lagging, which has severely affected the development of geothermal resources in the southern Sichuan Basin. In the paper, firstly, based on the temperature data of wells, the geothermal gradients and heat flows of typical wells in the study area were calculated, and the temperatures at the top and bottom of the reservoir in the Qixia-Maokou Formation were calculated using a one-dimensional steady-state heat conduction equation. Then, geothermal resource intensities and reserves of the Middle Permian Qixia-Maokou Formation in the southern Sichuan Basin were evaluated by using a method for rapidly calculating geothermal resources in sedimentary basins by multi-data fusion. Finally, suggestions for geothermal development were proposed. The results show that the geothermal gradient and heat flow values in the southern Sichuan Basin range from 15.6 to 33.70 °C/km and 49 to 85 mW/m2, respectively, and the temperature of the reservoir in the Qixia-Maokou Formation ranges from 50 to 120 °C. The geothermal resources are 3.09 × 1021 J, equivalent to 105 billion tons of standard coal, and the recoverable resources are 6.18 × 1020 J, equivalent to 21 billion tons of standard coal. The development of demonstration projects will focus on the mid- to low-temperature geothermal power generation, geothermal grain drying and geothermal agriculture in the southwestern region of the southern Sichuan Basin. The research results can provide a resource basis for the development of geothermal resources in the southern Sichuan Basin.

A high geothermal setting in the Linyi geothermal field: Evidence from the lithospheric thermal structure

The lithospheric thermal structure has a profound indicative significance for the potential evaluation and exploration of geothermal resources. The Linyi geothermal field, located in the southern section of the Yishu fault in Shandong Province, boasts abundant geothermal resources. However, its origin is still unclear. This study analyzed the characteristics of terrestrial heat flow using the temperature logging of geothermal wells and measurements of thermal conductivity and heat production. Combined with the geophysical information of the Xiangshui (Jiangsu Province)-Mandula (Nei Mongol) geoscience transect and the lithology revealed by the geothermal wells, this study built a conceptual model of the lithospheric thermal structure using the one-dimensional steady-state heat conduction equation. Based on this model, the heat flow is 64.9 mW/m2, indicating a high geothermal setting in the study area. Furthermore, the ratio of crustal to mantle heat flow is < 1 (0.67), implying that surface heat flow originates predominantly from the mantle. The Yishu fault probably acts as a pathway for heat transfer from the mantle. The temperature of the Moho boundary in the study area was estimated to be 614.8 °C. The Curie depth was calculated to be 29.5 km (585 °C), which is consistent with the depth estimated using the aeromagnetic data. In sum, the Linyi geothermal field has a high geothermal setting, which contributes to the formation of geothermal resources.

Characteristics of geothermal field and evaluation of geothermal resource potential in the Yingjiang Basin

Geothermal resource, a green and sustainable energy resource, plays an important role in achieving ‘emission peak’ and ‘carbon neutrality’ targets. The Yingjiang Basin is located in the eastern branch of the Mediterranean-Himalayan high-temperature geothermal belt and exhibits considerable potential for geothermal resources. However, current investigations into the distribution of deep geothermal resources in this region are somewhat limited. In this paper, the transient plane source (TPS) method is used to measure the thermal conductivity parameters of 31 rock samples within the study area. Additionally, the one-dimensional steady-state heat conduction equation is employed to calculate the deep geothermal field, considering the constraints of rock thermal properties and terrestrial heat flow in the study area. Furthermore, the “stripping method” is used to determine the contribution rate of sedimentary layer to terrestrial heat flow, while the volume method is applied to estimate the geothermal resources at burial depths of 3000–5000 m. The results show that (1) The heat generation rate of granite is the highest with an average value of 4.52 μW/m3, followed by gneiss with an average value in the range of 2.0–3.5 W/(m·K), mudstone and sandstone being the lowest with an average value between 1.0 and 2.0 W/(m·K). (2) The main contributor of terrestrial heat flow in the study area is mantle heat flow, and the contribution of sedimentary layers to terrestrial heat flow only accounts for about 2%. (3) The geothermal resources in Yingjiang Basin within the depth range of 3000–5000 m is 93.6 × 1015 kJ, or 3.2 × 109 tonnes standard coal equivalent (SCE).

Temperature Modeling of AZ31B Alloy Plate during open-Roller Conveying Process Considering Air-Cooling Characteristics

In the process of open-roller conveying, magnesium alloy plates easily produce temperature drops and uneven temperatures. Ignoring the heat dissipation caused by the contact between the plate and the roller table, the process can be described as a process of air cooling. The present study aimed to investigate the temperature distribution and evolution of a magnesium alloy plate during the process. The air-cooling process of the AZ31B magnesium alloy plate was investigated and analyzed in detail under different initial temperatures and plate thicknesses, with a specific focus on the temperature distribution along both the width and thickness directions. The results show that the temperature-difference curves between the end face and the center under different air-cooling conditions appeared to have four stages: rapid increase, slow increase, basic stable and slow decline. To facilitate the establishment of the temperature model, the whole air-cooling process was approximately divided into two independent one-dimensional heat-conduction processes in the thickness direction and the width direction. Subsequently, one-dimensional steady-state heat-conduction models were developed in various directions, based on the fundamental principles of heat transfer and assuming that the adjacent temperature-drop layer satisfied the quadratic function distribution. Through the superposition of cooling with time in two directions, the temperature evolution at different positions in the process of air cooling can be solved accurately.

Evolution Characteristics through Thermo-Rheological Lithosphere of the Liaonan Metamorphic Core Complex, Eastern North China Craton

Metamorphic core complexes are developed in crustal activity belts at the continental margins or within continents, and their main tectonic feature is that the ductile middle crust is exhumed at the surface. The deformation properties are closely related to the geodynamic process affecting the continental crust. However, the evolution of the metamorphic core complexes after their formation is still unclear. The Cretaceous Liaonan metamorphic core complex developed in the eastern North China craton provides an ideal environment to study its evolution. In this study, we estimate the paleo-temperature and paleo-stress at the time of formation of the metamorphic core complex dynamical recrystallization of quartz and calculate the thermo-rheological structure of the present Liaonan metamorphic core complex by one-dimensional steady-state heat conduction equation and power-creep law. The results show that compared with the Cretaceous period, the geothermal heat flow value of the present Liaonan metamorphic core complex decreases from 70–80 mW/m2 to 49.4 mW/m2, the thermal lithosphere thickness increases from 59–75 km to 173 km, and the brittle transition depth increases from 10–13 km to about 70 km, showing coupling of the crust–mantle rheological structure. We speculate that the evolution of the thermo-rheological structure of the Liaonan metamorphic core complex is possibly caused by rapid heat loss or lithospheric mantle flow in the Bohai Bay Basin.

Accurate determination of anisotropic thermal conductivity for ultrathin composite film

Highly anisotropic thermal conductive materials are of significance in thermal management applications. However, accurate determination of ultrathin composite thermal properties is a daunting task due to the tiny thermal conductance, severely hindering the further exploration of novel efficient thermal management materials, especially for size-confined environments. In this work, by utilizing a hybrid measuring method, we demonstrate an accurate determination of thermal properties for montmorillonite/reduced graphene oxide (MMT/rGO) composite film with a thickness range from 0.2 μm to 2 μm. The in-plane thermal conductivity measurement is realized by one-dimensional (1D) steady-state heat conduction approach while the cross-plane one is achieved via a modified 3ω method. As-measured thermal conductivity results are cross-checked with different methods and known materials, revealing the high measurement accuracy. A high anisotropic ratio of 60.5, independent of composite thickness, is observed in our measurements, further ensuring the negligible measurement error. Notably, our work develops an effective approach to the determination of ultrathin composite thermal conductivity, which may promote the development of ultrathin composites for potential thermal-related applications.

Geothermal potential and genetic mechanisms of the geothermal system in the mountainous area on the northern margin of north china

Geothermal resources as clean and renewable energy can be utilized for agriculture, tourism, and industry. The assessment of geothermal potential and the study of genetic mechanism of the geothermal system is an essential part of geothermal resource development. In this study, 16 steady-state temperature logs are obtained in the mountainous area on the northern margin of North China. Thermal conductivity and heat production rates are tested or collected from more than 200 rock samples of these wells and outcrops around the study area. Based on these data, for the first time, the detailed delineations of temperature distributions, genetic mechanisms of geothermal systems, and resource potential of Hot Dry Rock in the study area are achieved. The heat flow map indicates a low heat flow state with an average value of 53.1 mW/m2 in the study area, which is lower than the average value of 62.5 mW/m2 in mainland China. The distribution of hot springs in the area is mainly controlled by fault systems. Heat flow only exhibits a minor effect on the temperature of hot springs and geothermal wells. On this basis, the deep temperature distribution within 3–10 km depths of the study area is calculated using the one-dimensional steady-state heat conduction equation. With it, the reservoir depths of hot springs are estimated to be 3–5 km with temperatures ranging from 70°C to 110°C. Furthermore, a conceptual model for the geothermal system in the study area is derived. According to the results, Northeastern Chengde and northern Beijing exhibit the highest temperatures at all depths. Similar patterns are observed in the temperature distribution maps and the heat flow map, which suggest that the deep temperature distribution is mainly controlled by regional heat flow. With the depth increases, the temperature shows larger variation at each depth level, which is possibly caused by the heterogeneity of crustal composition. According to our resource assessment by volumetric method, the exploitable potential of Hot Dry Rock within the depth of 7–10 km of the study area is equivalent to about 3.1 × 1011 tons of standard coal, but the barrier is still existing for development under the current technical and economic conditions.

Influence of insulation paper on the hot spot temperature of oil-immersed transformer winding

The oil-immersed transformer is a crucial piece of equipment in the power system. Operating at the specified temperature is necessary to ensure the normal operation of the transformer. The insulation paper on the winding surface has a significant impact on the actual temperature of the transformers, which is often overlooked by researchers. The one-dimensional steady-state heat conduction model of the transformer is established by analyzing the heat diffusion process of winding to transformer oil. Atomic force microscope was used to observe the microsurface structure of insulation paper and copper. According to the experiment, the heat transfer resistance in the series process of heat transfer at [Formula: see text]C is 0.0138 m2 K/W. Space thermal circuit model of transformer is established by thermoelectricity analogy method, and the simulation circuit is optimized according to the boundary conditions set up in the actual environment. The results show that the error of the hot spot temperature is closer to the measured temperature and decreases by 2.5% when considering the thermal resistance of insulation paper.

Inversion of Geothermal Heat Flux under the Ice Sheet of Princess Elizabeth Land, East Antarctica

Antarctic geothermal heat flux is a basic input variable for ice sheet dynamics simulation. It greatly affects the temperature and mechanical properties at the bottom of the ice sheet, influencing sliding, melting, and internal deformation. Due to the fact that the Antarctica is covered by a thick ice sheet, direct measurements of heat flux are very limited. This study was carried out to estimate the regional heat flux in the Antarctic continent through geophysical inversion. Princess Elizabeth Land, East Antarctica is one of the areas in which we have a weak understanding of geothermal heat flux. Through the latest airborne geomagnetic data, we inverted the Curie depth, obtaining the heat flux of bedrock based on the one-dimensional steady-state heat conduction equation. The results indicated that the Curie depth of the Princess Elizabeth Land is shallower than previously estimated, and the heat flux is consequently higher. Thus, the contribution of subglacial heat flux to the melting at the bottom of the ice sheet is likely greater than previously expected in this region. It further provides research clues for the formation of the developed subglacial water system in Princess Elizabeth Land.

Study of the castable selection for blast furnace blowpipe

The contradictory properties required of castable refractories makes selecting castable refractories for industrial applications challenging. This paper seeks to describe the material selection for a blast furnace blowpipe application that is subjected to sudden temperature changes and must prevent heat loss. Three commercial high alumina castables containing andalusite or mullite from different manufacturers were characterized. Thermal shock damage resistance was evaluated using thermal shock damage resistance theory and experiments. The castables’ coefficient of thermal expansion was estimated using quantitative X-ray diffraction. Crack propagation resistance was measured using the work-of-fracture technique. Thermal shock damage was experimentally evaluated by measuring the modulus of elasticity and rupture prior to and after thermal cycles. Ultimately, the microstructure of the castables was related to the thermal shock damage behavior by estimating the aggregate size and the fracture toughening mechanisms using light optical and scanning electron microscopes. Heat loss was evaluated by calculating the blowpipe shell temperature using a one-dimensional steady-state heat conduction model. The best commercial castable refractory for blowpipe showed high thermal shock damage resistance and low thermal conductivity. The results in this study agreed with thermal shock damage resistance parameters and showed a correlation between coarse microstructure with large aggregate and higher thermal shock damage resistance.

A steady-state heat conduction problem of a nonhomogeneous conical body

Numerous studies and textbooks deal with the steady-state thermal conduction of radially nonhomogeneous circular cylinder. In contrast, there are relatively few studies on the thermal conduction problems of conical solid bodies. This study is intended as a modest contribution to the solution of thermal conductance problems of nonhomogeneous conical bodies. A one-dimensional steady-state heat conduction in nonhomogeneous conical body is considered. The thermal conductivity of the hollow conical body in a suitable chosen spherical coordinate system depends on the polar angle but is independent of the radial coordinate and azimuthal angle coordinate. A functionally graded type of material inhomogeneity is considered. All results of the paper are based on Fourier’s theory of heat conduction in solid bodies.

Construction of a Code Verification Matrix for Heat Conduction With Finite Element Code Applications

Abstract When establishing the pedigree of a simulation tool, code verification is used to ensure that the implemented numerical algorithm is a faithful representation of its underlying mathematical model. During this process, numerical results on various meshes are systematically compared to a reference analytic solution. The selection of analytic solutions can be a laborious process, as it is difficult to establish adequate code confidence without performing redundant work. Here, we address this issue by applying a physics-based process that establishes a set of reference problems. In this process, code simulation options are categorized and systematically tested, which ensures that gaps in testing are easily identified and addressed. The resulting problems are primarily intended for code verification analysis but may also be useful for comparison to other simulation codes, troubleshooting activities, or training exercises. The process is used to select fifteen code verification problems relevant for the one-dimensional steady-state heat conduction equation. These problems are applicable to a wide variety of simulation tools, but, in this work, a demonstration is performed using the finite element-based nuclear fuel performance code BISON. Convergence to the analytic solution at the theoretical rate is quantified for a selection of the problems, which establishes a baseline pedigree for the code. Not only can this standard set of conduction solutions be used for verification of other codes, but also the physics-based process for selecting problems can be utilized to quantify and expand testing for any simulation tool.

Beyond phonon hydrodynamics: Nonlocal phonon heat transport from spatial fractional-order Boltzmann transport equation

Spatially convoluting formulations have been used to describe nonlocal thermal transport, yet there is no related investigation at the microscopic level such as the Boltzmann transport theory. The spatial fractional-order Boltzmann transport equations (BTEs) are first applied to the description of nonlocal phonon heat transport. Constitutive and continuity equations are derived, and two anomalous behaviors are thereafter observed in one-dimensional steady-state heat conduction: one is the power-law length-dependence of the effective thermal conductivity, κeff∝Lβ with L as the system length, and the other is the nonlinear temperature profile, Tx−Tx=0∼x1+η. A connection between the length-dependence and nonlinearity exponents is established, namely, β=−η. Furthermore, we show that the order of these BTEs should be restricted by the ballistic limit. In minimizing problems, the nonlocal models in this work give rise to different results from the case of Fourier heat conduction, namely that the optimized temperature gradient is not uniform.

Nonlinear finite element analysis of temperature-dependent functionally graded porous micro-plates under thermal and mechanical loads

In the present study, a displacement based nonlinear finite element model for functionally graded porous micro-plates is developed based on the general third-order shear deformation plate theory and the modified couple stress theory. The developed finite element model accounts for von Kármán nonlinear strains, a power-law variation of two material constituents through the plate thickness, and different distributions of porosity with a constant volume of voids on static bending of micro-plates are analyzed. The length scale dependency is captured using a single parameter of the modified couple stress theory. A power-law distribution is assumed to model the variation of the two material constituents, while the porosity distributions vary according to cosine functions. The temperature-dependent properties are obtained using a cubic-spline interpolation from experimental data. A one-dimensional steady-state heat conduction problem is solved using the effective thermal conductivity of the porous material based on the Maxwell–Eucken model to obtain temperature distribution through the plate thickness. The Newton–Raphson iteration scheme is used to solve the nonlinear system of equations. A parametric study is conducted to demonstrate the effects of material and porosity parameters, temperature and length scale dependencies, and boundary conditions on the deflections and stress distributions.

Thermo-mechanical analysis of porous sandwich S-FGM plate for different boundary conditions using Galerkin Vlasov's method: A semi-analytical approach

abstract In the present paper, for the first time, an attempt has been made to obtain the solution for bending and stress under the thermal environment using Galerkin Vlasov's method. A non-polynomial based higher-order shear deformation theory with inverse tangent hyperbolic shape function is used to define the displacement field. The formulation is performed for the author's recently developed sandwich plate using a new modified sigmoid function-based functionally graded material (S-FGM) plate of different symmetric and non-symmetric configurations. Using a one-dimensional steady-state heat conduction equation, a new temperature distribution through the thickness based on modified sigmoid law is proposed. Three different types of porosity are considered viz. even, uneven but symmetric and uneven but non-symmetric. A new uneven non-symmetric porosity model is used in which micro-voids are varied in accordance with material property variation in the thickness direction in order to capture the accurate distribution of voids on the plate. The principle of virtual work is employed to derive equilibrium equations. An exact solution is obtained using the assumed solution with shape functions satisfying the edge boundary conditions. From the present study on static analysis, it is deduced that more refined and accurate results for plate in thermal environment, it is necessary to include the thermal effect on the stiffness of the plate in addition to the initial deflection of the plate. The effect of boundary conditions on stress and deflection distribution along the surface of the porous plate is studied, and it is observed that the distribution is prominently affected by the type of symmetric or asymmetric boundary conditions. The considerable increase in deflection and stress can be seen for even porosity distribution (P-1) in comparison to uneven symmetric (P-2) or uneven non-symmetric (P-3) porosity distribution. In addition, the maximum transverse shear stresses are offset more from the center of the sandwich plate with an increase in temperature differences with a maximum offset at ΔT = 300 K and minimum offset at ΔT = 0 K. Different examples are considered to check the accuracy and validation of the present formulation. The calculated outcomes and interpretations can be useful as a validation study for the imminent investigation of sandwich S-FGM plates having porosities in the thermal environment.

Steady-state thermomechanical analysis of composite laminated plate with damage based on extended layerwise method

A steady-state thermomechanical analysis model is established for the composite laminated plate with transverse cracks and delaminations based on the extended layerwise method. The one-dimensional weak and strong discontinuous functions are employed to simulate the interlaminar interfaces and the delaminations, respectively, while the transverse crack is simulated in the in-plane displacement fields by using the standard extended finite element method. The temperature field in the thickness direction is obtained by using a one-dimensional steady-state heat conduction equation and constructed with the linear Lagrange interpolation functions in the thickness direction. In several numerical examples, the proposed model is verified for the laminated plates with/without transverse crack and/or delaminations against the classic laminated plate theory, first-order shear deformation theory, Reddy’s layerwise theory or three-dimensional elastic model.

Practical Jointed Approach to Thermal Performance of Functionally Graded Material Annular Fin

Using the complementary functions method, a general solution for the one-dimensional steady-state heat conduction equation for the fin together with the boundary conditions made of functionally graded material is presented. Two cases for the variation of the functionally graded material wall thermal conductivity were considered. These are 1) power and 2) exponential types of variation of the thermal conductivity varying with the radial coordinate. In the present paper, a semi-analytical iterative technique, one of the most efficient unified methods, is employed to solve the heat conduction equation. For different values of the inhomogeneity constant, distributions of temperature, heat transfer rate, fin efficiency, and fin effectiveness, as a function of radial direction, are obtained. Various material models from the literature are used and corresponding temperature distributions are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases, and virtually exact results are obtained.

Thermal regime of the lithosphere and geothermal potential in Xiong'an New Area

Investigation into the thermal regime of lithosphere is an essential part of geothermal research. The thermal state of lithosphere can be expressed as the vertical distribution of temperature. It has been found that the thermal regime of lithosphere can control the heat flow distribution and the geothermal mechanism of local geothermal fields. In this work, equilibrium temperature logs were obtained from 27 wells, and thermal conductivity and heat generation data were collected from 148 rock samples from different wells. Besides, 55 high-quality terrestrial heat flow values were extracted. Based on these data, the distribution of heat flow across Xiong'an New Area was mapped. Later, the thickness of the thermal lithosphere in Xiong'an and the deep crustal temperature were analyzed using the one-dimensional steady-state heat conduction equation. The crustal structure beneath this area was derived from the seismic wave velocity profile, and it was then used to illustrate the connection between the geothermal fields’ heat sources and regional tectonic setting. The results indicate that high heat flow is mainly distributed around the basement uplifts, such as Niutuozhen uplift and Rongcheng uplift. The average heat flux in the study area is 70.5 mW/m2, higher than the 61.5 mW/m2 in mainland China. The temperature-depth profiles show great temperature variation across the new area. At the depth of 40 km, the crustal temperature ranges from about 750°C to 1100°C. Despite the large temperature variation, this area shows high thermal state. The temperature variation with depth may be attributed to the heat flow variability and the high thermal state may be caused by the large residual heat flow from the lithosphere boundary. The average reduced heat flow in the new area is high at around 44.35mW/m2, which is associated with the high thermal state of this area.

Comparative Assessment of the Finite Difference, Finite Element, and Finite Volume Methods for a Benchmark One-Dimensional Steady-State Heat Conduction Problem

The finite difference (FD), finite element (FE), and finite volume (FV) methods are critically assessed by comparing the solutions produced by the three methods for a simple one-dimensional steady-state heat conduction problem with heat generation. Three issues are assessed: (1) accuracy of temperature, (2) accuracy of heat flux, and (3) satisfaction of global energy conservation. It is found that if the order of accuracy of the numerical discretization schemes is the same (central difference for FD and FV, linear basis functions for FE), the accuracy of the temperature produced by the three methods is similar, except close to the boundaries where the FV method outshines the other two methods. Consequently, the FV method is found to predict more accurate heat fluxes at the boundaries compared to the other two methods and is found to be the only method that guarantees both local and global conservation of energy irrespective of mesh size. The FD and FE methods both violate energy conservation, and the degree to which energy conservation is violated is found to be mesh size dependent. Furthermore, it is shown that in the case of prescribed heat flux (Neumann) and Newton cooling (Robin) boundary conditions, the accuracy of the FD method depends in large part on how the boundary condition is implemented. If the boundary condition and the governing equation are both satisfied at the boundary, the predicted temperatures are more accurate than in the case where only the boundary condition is satisfied.

Heat transfer analysis through high-strength borated concrete for use as nuclear reactor biological shield

Heat transfer describes the exchange of thermal energy, between physical systems depending on the temperature and pressure, by dissipation of heat. Addition of borax (Na2B4O7) in the range of 0.5-1% by wt. to concrete improves the gamma radiation shielding capabilities of concrete by around 80% and may also influence the various heat transfer parameters. This work deals with the analysis of the heat transfer through a test specimen comprising a rectangular block of high strength borated concrete with Con Plast 430 as a suitable super-plasticizer, and thereby compute the average value of co-efficient of thermal conductivity (K) for it. Also one-dimensional steady state heat conduction analysis is carried out using empirical formulae and the corresponding results are verified using FEA methods applying ANSYS software.