Entransy Dissipation Rate Minimization Research Articles

Improvement of Constructal Optimization for “Volume-Point” Heat Conduction Based on Uniformity Principle of Temperature Difference Fields

The uniformity principle of temperature difference fields (TDFs) is applied in this study to improve the constructal optimization for “volume-point” heat conduction based on entransy dissipation rate (EDR) minimization without the premise of an optimal last-order construct, and the constructal optimization algorithm based on EDR minimization is simplified in this paper. The results further prove that the uniformity principle of TDF is consistent with entransy theory. The constructal optimization of “volume-point” heat conduction based on EDR minimization is conducted not only to lower the average temperature but also to obtain a more uniform TDF distribution. Through comparing the optimal results based on EDR minimization without the premise of an optimal last-order construct with those based on maximum temperature difference (MTD) minimization, some criteria and formulas for designing conductivity paths based on EDR minimization and MTD minimization are proposed, and the idea and method of improving constructal optimization via the variational principle are proposed.

Constructal optimization of heat conduction with minimum entransy dissipation rate for leaf-shaped quadrilateral heat-generating body

<p indent=0mm>In this work, a constructal study of a leaf-shaped quadrilateral heat-generating body based on the bionic quadrilateral of the common leaf vein shape was performed by combining the constructal and entransy theories, thus allowing the optimal construct of the leaf quadrilateral heat-generating body to be derived. The effect of the number of intervals, i.e., the number of branches containing high thermal conductivity materials, <italic>n</italic>, and angle <italic>θ</italic> between the high thermal conductivity material in the branch and the center, on the optimal construction of the heat-generating body was also studied. The constructal differences of the heat-generating body were compared under the minimum entransy dissipation rate and minimum temperature difference. The results show that the mean heat transfer temperature difference of the heat-generating body is the smallest in the optimal constructal when the entransy dissipation rate is the lowest. With the increase in the number of intervals (<italic>n</italic>≥20), the optimal aspect ratio and minimum mean temperature difference of the heat-generating body decreased slightly. Therefore, increasing the complexity of the high heat conduction passage in the heat-generating body can decrease the mean heat transfer temperature difference. Different optimal constructs of the heat-generating body were obtained using the entransy dissipation rate and maximum temperature difference as the optimization objective; both provide more optimal solutions when the high thermal conductivity material of the branch and center are perpendicular. Furthermore, minimizing the entransy dissipation rate allowed for a smaller mean temperature difference to be obtained. Therefore, using the minimum entransy dissipation rate as the optimization objective can reduce the mean heat transfer temperature difference more effectively in a leaf-shaped quadrilateral heat-generating body.

Optimization design of grooved evaporator wick structures in vapor chamber heat spreaders

This article proposes a novel optimization method for designing the layout of grooved evaporator wick structures in vapor chamber heat spreaders. Firstly, the design is transformed into a general ‘Area-to-Line’ (AL) constructal problem with conductive heat transfer in a disc configuration. Then, constructal optimization of the first order high conductivity channels (HCCs) with strict length scale is carried out based on entransy dissipation rate (EDR) minimization. According to the optimal constructs, the uniformly heated disc is divided into a series of identical sectors whose perimeter is set as the heat sink and other boundaries are all adiabatic. Based on this, a new growth simulator is developed from the inspiration of natural branching phenomena, and applied to obtain the full constructal layouts of the HCCs. In such simulation, the HCCs are described by parameterized level-set surfaces, which start at the heat sink line and grow toward the disc center according to the design sensitivities. The evaporator wick structure derived by growth simulation shows a unique topology exhibiting dichotomous, hierarchical and branching characteristics, which make the capillary pressure improve, and the temperature gradient more homogeneous. The proposed method contributes to the vapor chamber heat spreaders with better thermal performance.

Constructal entransy dissipation rate minimization for X-shaped vascular networks

Combining with entransy theory, constructal designs of the X-shaped vascular networks (XSVNs) are implemented with fixed total tube volumes of the XSVNs. The entransy dissipation rates (EDRs) of the XSVNs are minimized, and the optimal constructs of the XSVNs are derived. Comparison of the optimal constructs of the XSVNs with two optimization objectives (EDR minimization and entropy generation rate (EGR) minimization) is conducted. It is found that when the dimensionless mass flow rate (DMFR) is small, the optimal diameter ratio of the elemental XSVN derived by EDR minimization is different from that derived by EGR minimization. For the multilevel XSVN, when the DMFR is 100, compared the XSVN with the corresponding H-shaped vascular network (HSVN), the dimensionless EDRs of the elemental, second and fourth order XSVNs are reduced by 26.39%, 15.34% and 9.81%, respectively. Compared with the entransy dissipation number (EDN) of the second order XSVN before angle optimization, the EDN after optimization is reduced by 26.15%, which illustrates that it is significant to conduct angle optimization of the XSVN. Entransy theory is applied into the constructal design of the vasculature with heat transfer and fluid flow in this paper, which provides new directions for the vasculature designs.

Constructal optimizations for “+” shaped high conductivity channels based on entransy dissipation rate minimization

Constructal optimizations of the elemental and first order “+” shaped high conductivity channels (HCCs) in the square bodies are carried out based on entransy dissipation rate (EDR) minimization. The optimal shapes of the HCCs and minimum EDRs of the bodies are derived. The results show that there exist optimal geometrical parameters of the first order “+” shaped HCC which lead to triple minimum dimensionless EDR of the body. The optimal shape of the first order “+” shaped HCC derived by EDR minimization makes the global heat conduction performance (GHCP) of the square body improve, and the temperature gradient more homogeneous. The optimal design scheme of the “+” shaped HCC with minimum EDR can be adopted in the heat dissipation problem of the electronic device to improve its GHCP.

New thermodynamic analysis and optimization of performance of an irreversible diesel cycle

In this study, an irreversible Diesel cycle is investigated based on entransy method and optimization conducted by considering various scenarios. Entransy expressed as a heat transfer potential of a subject and it has begun to investigate as a new thermodynamic assessment parameter. This paper presents thermodynamical study of an irreversible Diesel cycle in order to optimize its performance. Moreover, four different strategies in the process of multiobjective optimization are proposed, and the outcomes are assessing separately. The first strategy is proposed to minimize the entransy dissipation rate and maximize the ecological coefficient of performance along with the maximum available work. Furthermore, the second one is suggested to minimize the entransy dissipation rate and maximize the ecological function along with the exergetic performance criteria. The third strategy is proposed to minimize ethe entransy dissipation rate and maximize the ecological function along with the thermal efficiency. Moreover, the fourth strategy is suggested to achieve the minimum entransy dissipation rate and maximum ecological function along with the entransy efficiency. Finally, the values of objective functions were obtained at the optimum conditions of each scenario; afterwards, compared with other scenarios to select the most appropriate one. Based on obtained results, maximizing ecological coefficient of performance is the most appropriate approach for optimization, since it leads to the maximum values of ecological function, exergetic performance criteria and thermal efficiency. In addition, in the fourth scenario the lowest entransy dissipation rate was achieved. © 2017 American Institute of Chemical Engineers Environ Prog, 37: 1475–1490, 2018

Constructal entransy dissipation rate minimization of a rectangular body with nonuniform heat generation

Based on construtal theory, a nonuniform heat generation problem in a rectangular body is investigated in this paper. Entransy dissipation rate (EDR) is taken as the optimization objective. The optimal body shapes with constant and variable widths of the high conductivity channel (HCC) are derived. For the rectangular first order assembly (RFOA) with constant cross-section HCC, the shape of the RFOA and width ratio of the HCCs are optimized, and the double minimum EDR is obtained. The heat transfer performance of the RFOA becomes worse when the nonuniform coefficient increases. For the RFOA with variable cross-section HCC, the EDR of the RFOA can be minimized for four times. Compared the optimal construct based on minimum EDR of the RFOA with that based on minimum maximum temperature difference, the shape of the former optimal construct is tubbier, and the average temperature difference is lower. In the practical design of electronic devices, when the thermal safety is ensured, the constructal design scheme of the former optimal construct can be adopted to improve the global heat transfer performance of an electronic device.

Constructal design for blast furnace wall based on the entransy theory

Based on the entransy dissipation extremum principle for thermal insulation process, constructal design of blast furnace wall is performed in this paper. Optimal construct of the cooling stave in the furnace wall is obtained by using finite element method. The results show that there exists twice optimal axial diameter ratio and optimal cooling water velocity which makes the entransy dissipation rate of the blast furnace wall reach its critical minimum. Moreover, the optimal cross-section shape of the cooling pipe is approximately round. Within the discussed variation range of the axial diameter ratios, the minimum entransy dissipation rate is 2.73% less than the maximum entransy dissipation rate when the cooling water velocity is preset. Within the discussed variation range of the cooling water velocities, the minimum entransy dissipation rate is 2.09% less than the maximum entransy dissipation rate. Besides, increasing inlet temperature of cooling water appropriately is propitious to improve the global thermal insulation performance of the blast furnace wall while it cannot reduce the cooling water. The global thermal insulation performance of the furnace wall with approximately round cooling pipe is higher than that with composite-oblate cooling pipe.

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.

Experimental study on + shaped high conductivity constructal channels based on entransy theory

Based on constructal theory and entransy theory, an experimental study on + shaped high conductivity channels in a square body is carried out. Heat conduction performance comparisons of the bodies based on different optimization objectives and different layouts of the high conductivity channels are performed. In the experiment, the materials of the square body and high conductivity channel are epoxy resin and brass, respectively; the brass channel is embedded in the square body. Two square heating boards, closed at the upper and lower sides of the square body, are used to uniformly heat itself. The internal heat generation of the square body is approximately simulated by this method. The square body is placed in a thermal insulation box to reduce the heat dissipation caused by heat convection. The heat generated by the heating boards is absorbed by the outside refrigerator device. A measurement window is set at the front side of the thermal insulation box. The temperature field of the square body is measured by the infrared thermal imager. The peak temperature, average temperature difference, and entransy dissipation rate of the body can be calculated by the measured results, respectively. Experimental results are compared to those obtained by numerical calculations; the results show that for the + shaped high conductivity channels in a square body, the maximum temperatures are located between the two branches of the + shaped high conductivity channels for both experimental result and numerical calculation. The errors in the average temperature and entransy dissipation rate of the body based on the experimental result and numerical calculations are within the acceptable range. The two results verify their validity of the heat conduction constructal optimization. Compared the square body with H shaped high conductivity channel, the entransy dissipation rate of the body caused by heat conduction is reduced by adopting the first order + shaped high conductivity channel. Compared with the optimal constructs of the first order + shaped high conductivity channels based on the minimizations of entransy dissipation rate and maximum temperature difference, the entransy dissipation rate caused by heat conduction of the former construct is reduced by 5.98%, but the maximum temperature difference is increased by 3.57%. The aim of maximum temperature difference minimization helps to improve the thermal safety of a body, while that of the entransy dissipation rate helps to improve the global heat conduction performance of a body. When the thermal safety is permitted, the optimal construct based on entransy dissipation rate minimization can be adopted in the design of practical electronic device to improve its global heat conduction performance.

Improvement of constructal tree-like network for “volume-point” heat conduction with variable cross-section conducting path and without the premise of optimal last-order construct

The constructal “tree-like” network has been proved an effective way for “volume-point” heat conduction. This paper aims to further improve the constructal “tree-like” network with the optimization objectives of minimizations of maximum thermal resistance and entransy dissipation rate respectively. By removing the constraint that the last-order construct must be optimal, and using variable cross-section conducting path, the maximum thermal resistance and entransy dissipation rate can be greatly reduced. It is also found that the optimal construct for minimum peak temperature is similar to the optimal one for minimum entransy dissipation rate. They hold the same shape and configuration, and the main difference of them is the shape of high-conductivity path.

Constructal entransy dissipation rate minimization for solid–gas reactors with heat and mass transfer in a disc-shaped body

A heat and mass transfer process for solid–gas reactors in a disc-shaped body is investigated in this paper. Radial- and branched-pattern integrated collectors are embedded in the body to enhance heat and mass transfers. The distributions of the integrated collectors in the two pattern discs are optimized using constructal theory and entransy theory. The results indicate that the optimal elemental volume fractions obtained by the minimizations of entransy dissipation rate and entropy generation rate are evidently different. There exists a critical radius, which decides whether the radial- or branched-pattern design is used. The decrease in the thermal conductivity ratio and increase in the first order volume fraction of the integrated collector lead to a better performance of the solid–gas reactor. The model with pure heat conduction or mass transfer is special case of the model in this paper. The results obtained in this paper can provide some design guidelines for the solid–gas reactors.

Constructal entransy dissipation rate minimization for helm-shaped fin with inner heat sources

A model of three-dimensional helm-shaped body composed of a helm-shaped fin and inner heat sources is built in this paper. For the specified volumes of the body, fin and heat source, the constructal optimizations of the body with single and multiple inner heat sources are implemented. The entransy-dissipation-rate-based equivalent thermal resistance (ETR) is minimized in the optimizations. It shows that for the helm-shaped body with multiple inner heat sources, there exist an optimal ratio of the heat source distance to the radius of the extended fin and a twice optimal radius ratio of the centre fin to the extended fin which lead to the double minimum dimensionless ETR. Comparing the optimal result of the body with helm-shaped fin with that with annular fin, the radius of the centre fin and the distance between the heat source and the center of the body are decreased, and the ETR is decreased by 9.57%. Essentially, the temperature gradient field of the helm-shaped body is more homogenous, and its global heat transfer performance is improved.

Constructal entransy dissipation rate minimization for triangular heat trees at micro and nanoscales

The “volume-point” heat conduction model with triangular heat trees is considered in this paper. Several high conductivity channels are distributed in the heat generating body, and the size effects are present in these channels. The minimization of the entransy dissipation rate (EDR) is chosen as the optimization objective. The optimal constructs of the triangular heat trees at different scales are obtained by using the local optimization and global optimization constructal design methods, respectively. The results show that the optimal construct at micro and nanoscales is evidently different from that at convectional scale. The heat conduction performance of the triangular heat trees based on local optimization constructal design method can be further improved compared with that based on the global one. The optimal constructs of the triangular second order assembly (TSOA) with minimum EDR and minimum maximum temperature difference (MTD) are evidently different, and the same conclusion can be obtained for the comparison between the optimization objectives of EDR and entropy generation rate (EGR).

Constructal optimization of variable cross-section insulation layer of steel rolling reheating furnace wall based on entransy theory

Based on the entransy dissipation extremum principle for thermal insulation process, the constructal optimizations of a variable cross-sectional insulation layer of the steel rolling reheating furnace wall with convective and compound heat transfer (mixed convective and radiative heat transfer) boundary conditions are carried out. An optimal construct of the insulation layer with minimum entransy dissipation rate can be obtained. Results show that the global thermal insulation performance of the variable cross-sectional insulation layer at minimum entransy dissipation rate is better than that of the constant cross-sectional one. The optimal constructs of the insulation layer obtained based respectively on the minimizations of the entransy dissipation rate and heat loss rate are different. The optimal construct of the insulation layer at minimum heat loss rate leads to a reduction of the energy loss, and that at minimum entransy dissipation rate leads to an improvement of the global thermal insulation performance. The difference between the optimal constructs of the variable cross-sectional insulation layer based on the minimizations of the entransy dissipation rate and the maximum temperature gradient is small. This makes the global thermal insulation performance and thermal safety of the insulation layer improved simultaneously. The constructal optimization of the insulation layer based on entransy theory can provide some new guidelines for the optimal designs of the insulation systems.

Constructal optimization of complex fin with convective heat transfer based on entransy dissipation rate minimization

Based on the constructal theory, the constructal optimization of a complex fin is carried out by taking the minimum equivalent thermal resistance, which is defined according to entransy dissipation rate, as the optimization objective. Optimal constructal of the complex fin is obtained by tsking into consideration the entransy dissipation performance caused by heat conduction and heat convection. Comparisons between the optimal constructal with different shapes and optimization objectives of the fins are performed. Results show that there exist the optimal ratios of the height to the length of the elemental fin, central cavity and fin tip which lead to the triple minimum equivalent thermal resistance of the complex fin. By comparing the optimal constructal of the complex fin with that of the T-Y shaped fin, the structure of the complex fin will greatly improve its global heat transfer performance. When the heat transfer of the fin is two-dimensional and the root of the fin is broader, the more non-uniform the temperature at the fin root, the bigger difference of the optimal constructs the complex fin obtains, based on the minimizations of the equivalent thermal resistance and maximum thermal resistance. For the optimal design of the fin in pracuice, when the thermal safety of the fin is ensured, the constructal design scheme of the fin with minimum equivalent thermal resistance can be adopted to reduce temperature difference in the average heat transfer and improves the global heat transfer performance. This paper provides some guidelines for the optimal design of the complex fin from the point of view of heat transfer optimization.

Comparative study on constructal optimizations of rectangular fins heat sink based on entransy dissipation rate minimization and maximum thermal resistance minimization

Constructal optimization of a rectangular fin heat sink with two-dimensional heat transfer model is carried out through using numerical simulation by finite element method, in which the minimized maximum thermal resistance and the minimized equivalent thermal resistance based on entransy dissipation are taken as the optimization objectives, respectively. The optimal constructs based on the two objectives are compared. The influences of a global parameter (a) which integrates convective heat transfer coefficient, overall area occupied by fin and its thermal conductivity, and the volume fraction (φ), on the minimized maximum thermal resistance, the minimized equivalent thermal resistances and their corresponding optimal constructs are analyzed. The results show that there does not exist optimal thickness of fins for the two objectives when the shape of the heat sink is fixed, and the optimal constructs based on the two objectives are different when the shape of the heat sinks can be changed freely. Besides, the global parameter has no influence on the optimal constructs based on the two objectives, but the volume fraction does. The increases of the global parameter and the volume fraction reduce the minimum values of the maximum thermal resistance and the equivalent thermal resistance, but the degrees are different. The reduce degree of the global parameter to the minimized equivalent thermal resistance is larger than that to the minimized maximum thermal resistance. The minimized equivalent thermal resistance and the minimized maximum thermal resistance are reduced by 40.03% and 41.42% for a= 0.5, respectively, compared with those for a = 0.3. However, the reduce degree of the volume fraction to the minimized maximum thermal resistance is larger than that to the minimized equivalent thermal resistance. The minimized equivalent thermal resistance and the minimized maximum thermal resistance are reduced by 59.69% and 32.80% for φ= 0.4, respectively, compared with those for φ= 0.3. As a whole, adjusting the parameters of the heat sink to make the equivalent thermal resistance minimum can make the local limit performance good enough at the same time; however, the overall average heat dissipation performance of the heat sink becomes worse when the parameters of the heat sink are adjusted to make the maximum thermal resistance minimum. Thus, it is more reasonable to take the equivalent thermal resistance minimization as the optimization objective when the heat sink is optimized.

Performance optimization of two-stage latent heat storage unit based on entransy theory

In order to enhance the performance of the latent heat storage (LHS) process and provide the criterion for the selection and match of the multistage PCMs, the effects of PCMs melting temperatures on the heat storage rate, entransy dissipation rate and heat storage quality were numerically analyzed based on the entransy theory. For the single stage LHS unit, although decreasing the PCM melting temperature can augment the heat storage rate, the lower melting temperature causes larger entransy dissipation and reduces the heat storage quality. The larger heat storage rate results in the larger entransy dissipation rate, which is accordant with the entransy dissipation extremum theory. For the two-stage LHS unit with reasonably matching the PCMs melting temperature, the heat storage rate can be augmented and the entransy dissipation rate can be reduced. Then the optimization for the match of the two-stage PCMs melting temperatures was performed based on the entransy theory. The results show that there is an optimal match of the two-stage PCMs melting temperatures to achieve the maximum heat transfer rate or the minimum entransy dissipation rate. And the formulas for the optimum two-stage PCMs temperatures were presented, which can provide the criterions for the selection and match of the PCMs.

Constructal entransy dissipation rate minimization for “volume-point” heat conduction at micro and nanoscales

A “volume-point” heat conduction model with rectangular element at micro and nanoscales is re-optimized by taking the minimization of ‘entransy’ dissipation rate as optimization objective. The optimal constructs of the model with local optimization and global optimization constructal design methods under the effect of size effect are obtained. The results show that the size effect has remarkable effects on the optimal construct of the model. For the local optimization constructal design method, the optimal constructs of the rectangular second order assembly based on the minimizations of the entransy dissipation rate and hot spot temperature are different. The optimal construct of the rectangular second order assembly based on the minimization of entransy dissipation rate can reduce the average temperature difference better than that based on the minimization of the hot spot temperature, and improves the heat transfer ability simultaneously. The global heat transfer performances of the rectangular assemblies are evidently improved by adopting the global optimization constructal design method. The rectangular heat conduction model at micro and nanoscales is more generalized, and the results obtained by the constructal optimization of this model include those not only at micro and nanoscales but also at conventional scale.

Constructal entransy optimizations for insulation layer of steel rolling reheating furnace wall with convective and radiative boundary conditions

Based on the entransy dissipation extremum principle for thermal insulation process, the constructal optimizations for a plane insulation layer of the steel rolling reheating furnace wall with convective and radiative boundary conditions are carried out by taking the minimization of entransy dissipation rate as optimization objective. The optimal construct of the plane insulation layer is obtained. The results show that for the convective heat transfer boundary condition, the optimal constructs of the insulation layer obtained based on the minimizations of the entransy dissipation rate and heat loss rate are obviously different. Comparing the optimal construct obtained based on the minimization of the entransy dissipation rate with that based on the minimization of the heat loss rate, the entransy dissipation rate is reduced by 5.98 %, which makes the global thermal insulation performance of the insulation layer improve. For the combined convective and radiative heat transfer boundary condition, compared the insulation layer having an increasing thickness with that having constant thickness and a decreasing thickness, the entransy dissipation rates are reduced by 16.59 % and 39.72 %, respectively, and the global thermal insulation performance of the insulation layer is greatly improved. There exits an optimal constant coefficient \( a_{{2,{\text{opt}}}} \) which leads to the minimum dimensionless entransy dissipation rate of the insulation layer. The difference between the optimal constant coefficients \( a_{{2,{\text{opt}}}} \) obtained based on the minimizations of the entransy dissipation rate and the maximum temperature gradient of the insulation layer is small. This makes the corresponding thermal stress obtained based on the minimum dimensionless entransy dissipation rate also be small, and the global thermal insulation performance and thermal safety of the insulation layer are improved simultaneously. The results obtained can provide some guidelines for the optimal designs of the insulation layers.