Minimization Of Maximum Temperature Difference Research Articles

Constructal design of a rectangular porous fin considering minimization of maximum temperature difference and pumping power consumption

Constructal design of a rectangular porous fin considering minimization of maximum temperature difference and pumping power consumption

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.

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.

Constructal Entransy Dissipation Rate Minimization for Gas-turbine Blade Cooling

Based on constructal theory, the construct of an elemental gas-turbine blade is optimized by taking minimum entransy dissipation rate as optimization objective. The optimal construct of the elemental gas-turbine blade is obtained. The results show that for the fixed total cross-sectional area and cavity faction of the elemental gas-turbine blade, there exists an optimal ratio of the cavity diameter to the thickness of the elemental blade which leads to the minimum dimensionless equivalent thermal resistance of the elemental blade; but there do not exist the optimal ratios of the cavity edge distance to the thickness, as well as the length to the thickness of the elemental blade which lead to the minimum dimensionless equivalent thermal resistance of the elemental blade. On the premise that the structural strength of the blade is allowed, the increase of the cavity fraction, the decrease of the cavity edge distance, and the slenderness of the elemental blade will help to improve the global heat transfer performance of the elemental blade. The optimal constructs of the elemental blade based on the minimizations of entransy dissipation rate and maximum temperature difference are different. The average heat transfer temperature difference of the elemental blade based on the minimization of entransy dissipation rate can be more effectively reduced compared with that based on the minimization of maximum temperature difference, and its global heat transfer performance is obviously improved. Moreover, the global heat transfer performance of the elemental blade can also be obviously improved by using the multi-scale constructal optimization of the cooling cavity of the elemental blade.

Constructal optimization for “disc-point” heat conduction at micro and nanoscales

Based on constructal theory, the construct of a “disc-point” heat conduction model at micro and nanoscales is optimized by taking maximum temperature difference minimization as optimization objective, and the optimal constructs of the radial-pattern and first order branched-pattern discs under the effect of size effect are obtained. The results show that the size effect has an obvious influence on the optimal construct of the disc. The minimum dimensionless maximum temperature difference of the first order disc for structure form nn is relative to the number of elemental tributaries, but those for structure forms nb and bb are independent of the number of elemental tributaries. There exist the critical dimensionless radiuses, which determine whether the radial-pattern design or branched-pattern design for the high conductivity channels is adopted. The critical dimensionless radiuses of the first order branched-pattern discs for the structure forms nn, nb and bb are 1.25, 1.72 and 2.18, respectively. With the increase in the product of thermal conductivity ratio and the square of elemental high conductivity material fraction, the minimum dimensionless thermal resistance of the first order disc for structure form nn decreases, but that for structure form nb is undetermined. The optimal “disc-point” construct at micro and nanoscales improves the heat transfer performance of the disc.

Constructal entransy dissipation rate minimization the problem of constracting “disc-point” cooling channels

Based on configucation theory, the construction of a “disc-point” heat transfer with cooling channels can be optimized by taking minimum entransy dissipation rate. Thus an optimal construction of the disc-shaped assembly with cooling channels is obtained. The results show that there exists an optimal aspect ratio of the elemental sector which leads to the minimum dimensionless equivalent thermal resistance of the elemental sector at the fixed pumping power; there also exists an optimal width ratio of the elemental and first-order cooling channel to the optimal dimensionless radius of the elemental sector, which leads to the minimum dimensionless equivalent thermal resistance of the first-order branched-pattern disc at the fixed total pumping power. Moreover, the optimal width ratio of the elemental and first-order cooling channels is only relative to the number of elemental tributaries. When the radius of the central disc tends to zero, the branch-pattern disc is simplified into a radial-pattern disc, and the radius of the first-order branch-pattern disc becomes the critical radius at this point. When the radius of the branch-pattern disc is higher than the critical radius, the branch-pattern design should be adopted, otherwise the radial-pattern design should be adopted. There exists an optimal number of elemental tributaries which lead to the minimum dimensionless equivalent thermal resistance of the first-order branch-pattern disc, which is obviously different from the results of the “disc-point” heat conduction constructional optimization with high-conductivity channels. The optimal constructions of the first-order branch-pattern disc based on the minimizations of entransy dissipation rate and maximum temperature difference are different. The dimensionless equivalent thermal resistance of the disc with cooling channels based on the minimization of entransy dissipation rate is greatly reduced as compared with that based on the minimization of maximum temperature difference, and its global heat transfer performance is obviously improved simultaneously. Therefore, the combination of the entransy dissipation extremum principle and the heat convection constructional optimization further illustrates the advantages of minimization of entransy dissipation rate for heat transfer optimizations.

T-shaped assembly of fins with constructal entransy dissipation rate minimization

T-shaped assembly of fins is optimized by adopting an analytical method and taking the minimum mean thermal resistance in terms of entransy as optimization objective. The optimal construct of T-shaped assembly is obtained and compared with the optimal construct with maximum temperature difference minimization. The results show that for the two optimization objectives the thicknesses of the optimized fins are almost equal, but the lengths are different obviously: the former length of the elemental fin is almost half the value of the latter length, and the former length of the first-order fin is almost twice as large as the latter length. Compared with the latter optimal construct, the dimensionless mean thermal resistance corresponding to the former optimal construct decreases by 16.4%, and the corresponding dimensionless maximum thermal resistance increases by 9.74%. Both the minimizations of the EDR and the maximum temperature difference (MTD) should be combined to consider the efficiency and the temperature limitation simultaneously for designing the fins.

“Volume-point” heat conduction constructal optimization based on entransy dissipation rate minimization with three-dimensional cylindrical element and rectangular and triangular elements on microscale and nanoscale

Based on constructal theory, the constructs of three “volume-point” heat conduction models with three-dimensional cylindrical element and rectangular and triangular elements on microscale and nanoscale are optimized by taking minimum entransy dissipation rate as optimization objective. The optimal constructs of the three “volume-point” heat conduction models with minimum dimensionless equivalent thermal resistance are obtained. The results show that the optimal constructs of the three-dimensional cylindrical assembly based on the minimizations of dimensionless equivalent thermal resistance and dimensionless maximum thermal resistance are different, which is obviously different from the comparison between those of the corresponding two-dimensional rectangular assembly based on the minimizations of these two objectives. The optimal constructs based on rectangular and triangular elements on microscale and nanoscale when the size effect takes effect are obviously different from those when the size effect does not take effect. Because the thermal current density in the high conductivity channel of the rectangular and triangular second order assemblies are not linear with the length, the optimal constructs of these assemblies based on the minimization of entransy dissipation rate are different from those based on the minimization of maximum temperature difference. The dimensionless equivalent thermal resistance defined based on entransy dissipation rate reflects the average heat transfer performance of the construct. The studies on “volume-point” heat conduction constructal problems at three-dimensional conditions and microscale and nanoscale by taking minimum entransy dissipation rate as optimization objective extend the application range of the entransy dissipation extremum principle.

Constructal entransy dissipation rate minimization for leaf-like fins

Based on constructal theory, the constructs of the leaf-like fins are optimized by taking minimum entransy dissipation rate (for the fixed total thermal current, i.e., the equivalent thermal resistance) as optimization objective. The optimal constructs of the leaf-like fins with minimum dimensionless equivalent thermal resistance are obtained. The results show that there exists an optimal elemental leaf-like fin number, which leads to an optimal global heat conduction performance of the first order leaf-like fin. The Biot number has little effects on the optimal elemental fin number, optimal ratios of length and width of the elemental and first order leaf-like fins; with the increase of the thermal conductivity ratio of the vein and blade, the optimal elemental fin number and optimal ratio of the length and width of the elemental leaf-like fin increase, and the optimal shape of the first order leaf-like fin becomes tubbier. The optimal construct based on entransy dissipation rate minimization is obviously different from that based on maximum temperature difference minimization. The dimensionless equivalent thermal resistance based on entransy dissipation rate minimization is reduced by 11.54% compared to that based on maximum temperature difference minimization, and the global heat conduction performance of the leaf-like fin is effectively improved. For the same volumes of the elemental and first order leaf-like fins, the minimum dimensionless equivalent thermal resistance of the first order of the leaf-like fin is reduced by 30.10% compared to that of the elemental leaf-like fin, and the global heat conduction performance of the first order leaf-like fin is obviously better than that of the elemental leaf-like fin. Essentially, this is because the temperature gradient field of the first order leaf-like fin based on entransy dissipation rate minimization is more homogenous than that of the elemental leaf-like fin. The dimensionless equivalent thermal resistance defined based on entransy dissipation rate reflects the average heat transfer performance of the leaf-like fin, and can provide some guidelines for the thermal design of the fins from the viewpoint of heat transfer optimization.

Constructal entransy dissipation rate minimization for a heat generating volume cooled by forced convection

The geometry of a heat generating volume cooled by forced convection is optimized by applying the entransy dissipation extremum principle and constructal theory, while the optimal spacing between the adjacent tubes and the optimal diameter of each tube are obtained based on entransy dissipation rate minimization. The results of this work show that the optimal constructs based on entransy dissipation rate minimization and maximum temperature difference minimization, respectively, are clearly different. For the former, the porosity of the volume of channels allocated to the heat generating volume is 1/2; while for the latter, the larger the porosity is, the better the performance will be. The optimal construct of the former greatly decreases the mean thermal resistance and improves the global heat transfer performance of the system compared with the optimal construct of the latter. This is identical to the essential requirement of the entransy dissipation extremum principle that the required heat transfer temperature difference is minimal with the same heat transfer rate (the given amount of heat generated in the heat generating volume) based on the entransy dissipation extremum principle.

Constructal optimization for “disc-to-point” heat conduction without the premise of optimized last-order construct

Based on constructal theory, branch-patterned disc with first- (or higher-) order assembly is optimized based on constructal minimization of maximum temperature difference without the premise that elemental sectors that assemble the perimeter of the branch-patterned disc are optimal. Results show that the optimal constructs without the premise and that with the premise are different obviously with the same conditions, maximum temperature difference of heat transfer without the premise decreases by 49.3%, and every optimal branch-patterned disc with higher-order assembly reduces to a branch-patterned disc with first-order assembly. The optimal construct obtained in this paper decreases the maximum thermal resistance greatly.

Constructal entransy dissipation rate minimization for “disc-to-point” heat conduction

Based on constructal theory, “disc-to-point” heat conduction is optimized by minimizing the entransy dissipation rate whereby a critical point is determined that distributes the high-conductivity material according to optimized radial or branch patterns. The results show that the critical point is determined by the product of the thermal conductivity ratio of the two materials and the volume fraction of the high-conductivity material allocated to the entire volume. The notion of optimal heat transfer performance can be attributed to the disc based on the entransy dissipation extremum principle. Comparing the results based on EDR minimization (entransy dissipation rate minimization) with those based on MTD minimization (maximum temperature difference minimization), one finds that the performance derived from the two optimization procedures are different. When the product of the thermal conductivity ratio and volume fraction is 30, the critical point of the former procedure is that for which the nondimensional radius of the disc equals 1.75, while that of the latter procedure is that for which this radius of the disc equals 2.18. Comparing heat transfer performances from the two procedures, the mean heat transfer temperature difference is decreased more for the former procedure thereby receiving an improved performance quota.

Constructal entransy dissipation rate minimization of a disc

Disc cooling problem is optimized by taking entransy dissipation rate minimization as optimization objective. The non-dimensional mean temperature difference of the disc cooling model with radial high conducting fins inserted is deduced. The effects of the fin geometry, the fin aspect ratio, the ratio between the high conductivity and low conductivity, the relative amount of high conductivity material and the number of high conducting fins on the entransy dissipation rate of disc cooling are analyzed. The optimization results show that the high conducting fin should be extended to the centre of circle as the heat transfer effect of the high conducting fins is improved, and there exists an optimal fin aspect ratio corresponding to minimum entransy dissipation rate for different high conducting effects of the fin, and the number of high conducting fins has a slight effect on the entransy dissipation rate. Comparison with those for maximum temperature difference minimization shows that the constructs based on entransy dissipation rate minimization are different from those based on maximum temperature difference minimization, but the optimal constructal shape changing potentials of the number of fins and the relative amount of high conductivity material are similar.

Constructal entransy dissipation minimization for “volume-point” heat conduction based on triangular element

By taking equivalent thermal resistance, which reflects the average heat conduction effect and is defined based on entransy dissipation, as optimization objective, the “volume to point” constructal problem based on triangular element of how to discharge the heat generated in a fixed volume to a heat sink on the border through relatively high conductive link is re-analyzed and re-optimized in this paper. The constructal shape of the control volume with the best average heat conduction effect is deduced. For the same parameters, the constructs based on minimization of entransy dissipation and the constructs based on minimization of maximum temperature difference are compared, and the results show that the constructs based on entransy dissipation can decrease the mean temperature difference better than the constructs based on minimization of maximum temperature difference. But with the increase of the number of order, the mean temperature difference does not always decrease, and there exists some fluctuations. Because the idea of entransy describes heat transfer ability more suitably, the optimization results of this paper can be put to engineering application of electronic cooling.

Constructal entransy dissipation minimization of an electromagnet

How to determine the optimal distribution of high conductivity material through the given volume such that the heat generated at every point was transferred most effectively to its boundary has become the focus of attention in the current constructal theory literature. In general, the minimization of maximum temperature difference in the volume is taken as the optimization objective. A physical quantity, entransy, has been identified as a basis for optimizing heat transfer processes in terms of the analogy between heat and electrical conduction recently. Heat transfer analyses show that the entransy of an object describes its heat transfer ability, as the electrical energy in a capacitor describes its charge transfer ability. Entransy dissipation occurs during heat transfer processes, as a measure of the heat transfer irreversibility with the dissipation related thermal resistance. Based on entransy dissipation, the mean temperature difference of solenoid (electromagnet) with high thermal conductivity material inserted is deduced, which can be taken as the fundamental for heat transfer optimization using the extremum principle of entransy dissipation. Then, the electromagnet working at steady state (constant magnetic field, constant heat generating rate per unit volume) is optimized for entransy dissipation minimization (i.e., mean temperature difference minimization) with and without volume constraint. The variation of minimum mean temperature difference with volume and magnetic field and the corresponding optimal constructs are obtained. The effect of high thermal conductivity material on the magnetic field is analyzed. The optimization results show that for fixed G parameter and ϕ, the minimum mean temperature difference decreases as the number of cooling disks n increases but the decreasing magnitude of minimum mean temperature difference is relatively decreased. As n increases, the radius of the solenoid decreases, the length of the solenoid increases, and the volume decreases. The solenoid optimized based on minimization of entransy dissipation with fixed magnetic induction is considerably larger than the winding optimized solely from the electromagnetic point of view. The mean temperature difference decrease as the volume increases.

“Volume-Point” heat conduction constructal optimization with entransy dissipation minimization objective based on rectangular element

By taking equivalent thermal resistance, which reflects the average heat conduction effect and is defined based on entransy dissipation, as optimization objective, the “volume to point” constructal problem of how to discharge the heat generated in a fixed volume to a heat sink on the border through relatively high conductive link is re-analyzed and re-optimized in this paper. The constructal shape of the control volume with the best average heat conduction effect is deduced. For the elemental area and the first order construct assembly, when the thermal current density in the high conductive link is linear with the length, the optimized shapes of assemble based on the minimization of entransy dissipation are the same as those based on minimization of maximum temperature difference, and the mean temperature difference is 2/3 of the maximum temperature difference. For the second and higher order construct assemblies, the thermal current densities in the high conductive link are not linear with the length, and the optimized shapes of assemble based on the minimization of entransy dissipation are different from those based on minimization of maximum temperature difference. For the same parameters, the constructs based on minimization of entransy dissipation and the constructs based on minimization of maximum temperature difference are compared, and the results show that the constructs based on entransy dissipation can decrease the mean temperature difference better than the constructs based on minimization of maximum temperature difference. But with the increase of the number of the order, the mean temperature difference does not always decrease, and there exist some fluctuations. Because the idea of entransy describes the heat transfer ability more suitably, all of the heat conduction constructal problems may be re-optimized based on it.