Constructal blade shape in nanofluids
© Bai and Wang; licensee Springer. 2011
Received: 6 December 2010
Accepted: 21 March 2011
Published: 21 March 2011
Blade configuration of nanofluids has been proven to perform much better than dispersed configuration for some heat conduction systems. The analytical analysis and numerical calculation are made for the cylinder--shaped and regular-rectangular-prism--shaped building blocks of the blade-configured heat conduction systems (using nanofluids as the heat conduction media) to find the optimal cross-sectional shape for the nanoparticle blade under the same composing materials, composition ratio, volumetric heat generation rate, and total building block volume. The regular-triangular-prism--shaped blade has been proven to perform better than all the other three kinds of blades, namely, the regular-rectangular-prism--shaped blade, the regular-hexagonal-prism--shaped blade, and the cylinder--shaped blade. Thus, the regular-triangular-prism--shaped blade is selected as the optimally shaped blade for the two kinds of building blocks that are considered in this study. It is also proven that the constructal cylinder--regular-triangular-prism building block performs better than the constructal regular-rectangular-prism--regular-triangular-prism building block.
Nanofluids are mixtures of nanoparticles and base fluids, which have different thermal conductivities [1–5]. They were identified and proposed as a result of people's persistent pursuit for more and more efficient heat-transfer media. It should be noted that conventional heat transfer fluids have normally very low thermal conductivity, thus destroying much exergy during heat transport. At present, a great amount of attention is paid to studies on nanofluids, with the aim of addressing many unsolved issues [6–10].
Constructal theory is a novel thought for nature and society [11–17]. It tries to explain phenomena based on optimization, or natural selection in biological terms. One of its viewpoints is that two flow mechanisms are better than one . This is what one sees naturally in river basins, lung structure, and the percolation threshold effect . Our previous studies have proved that the blade configuration of nanofluids is much better than the dispersed configuration for the two kinds of disk-shaped heat conduction systems with different boundary conditions [20, 21]. It is believed that the continuous nanoparticle blades with higher thermal conductivity serve as the second conduction mechanism, and its optimized cooperation with the base fluid of low thermal conductivity leads to the much better performance. In this study, the blade configuration of nanofluids is considered in detail by studying the influence of the shapes of high-conductivity blades in two kinds of building blocks of the total blade-configured heat conduction systems. This study is inspired by the need for the optimization of the cross section of duct for minimum flow resistance . By treating heat as a flow medium flowing in blades, one can also find the optimal shape for the blades, which offers minimum thermal resistance.
Optimal blade shape for two kinds of building blocks of blade-configured heat conduction systems
It is assumed that ϕ ≪ 1, and the thermal conductivity ratio of nanoparticle material and base fluid material is fixed and large. The thermal contact resistance is not considered.
In order to study the influence of the blade shape, the materials of the base fluid and nanoparticle, volumetric heat generation rate, and volumes of the eight kinds of building blocks are also fixed, besides the volume fraction and thermal conductivity ratio; however, the slenderness is free to vary to achieve the constructal system (building block) overall temperature difference. Here, the slenderness refers to the ratio of the radius to length for the cylinder building block, and the ratio of the circumscribing cylinder radius to length for the rectangular prism building block. For the simplest cylinder--cylinder building block, analytical analysis can be made; the system overall temperature difference, the constructal system overall temperature difference, and the constructal slenderness can be obtained analytically. Based on a slenderness range predicted by the analytic result, the numerical calculation is then conducted for all the eight kinds of systems to obtain, as accurately as possible, the results for comparison among different blade shapes and different building block shapes.
Analytical analysis for cylinder--cylinder building block
Owing to the much higher thermal conductivity of nanoparticle material, heat conduction inside this kind of building block can be considered to consist of two one-dimensional routes-radial conduction inside cross-sectional planes of the base fluid region and axial conduction along the blade.
At , the best-performing cylinder--cylinder building block can be obtained. If one specifies ϕ = 0.05 and = 641.6667, then the optimal slenderness will be 0.240618, and the nondimensional constructal system overall temperature difference will be 0.296778, which is the lowest point in Figure 2. Similarly, for the other kinds of building blocks considered here, there also exists such a best-performing slenderness .
Numerical calculation for all the eight kinds of building blocks
The actual heat conduction in the cylinder--cylinder building block is of course not a simple combination of two one-dimensional conductions. For the other kinds of building blocks, the flow of heat is even more complex. In order to have as accurately as possible results for the comparison, a finite volume computational fluid dynamics (CFD) code  is used for obtaining numerical results for all the eight kinds of building blocks.
Note that is exactly the nondimensional system overall temperature difference, , as shown in Equation (9), where is the maximal nondimensional temperature in the heat conduction building blocks.
Grid-independence check (cylinder--triangular-prism building block, R0/L0 = 0.25)
Number of grids
-0.010181 × 100
-8.803569 × 10-4
-3.346073 × 10-5
Perimeters of four blade cross sections having the same area of π
Cross section shape
2π = 6.283186
Thus, the constructal cylinder--shaped heat conduction building block performs better than the rectangular-prism--shaped building block.
Inspired by the duct cross section optimization for minimum flow resistance, the shape of the nanoparticle blade is optimized for the cylinder--shaped and rectangular-prism--shaped building blocks of the blade-configured heat conduction systems (blade configuration of nanofluids) based on the same composing materials, composition ratio, volumetric heat generation rate, and total building block volume. The four kinds of blade shapes are triangular prism, rectangular prism, hexagonal prism, and cylinder. For the cylinder--cylinder building block, analytical analysis can be conducted. Explicit expressions for the system overall temperature difference, constructal system overall temperature difference, and constructal slenderness can be obtained. Then, based on the slenderness range predicted by the analytical result, numerical calculations are performed for the eight kinds of building blocks to obtain as accurately as possible results for comparison. One specifies that ϕ = 0.05 and = 641.6667 for the numerical calculation.
The performances of the eight kinds of building blocks depend strongly on the building-block slenderness. The constructal slendernesses leading to minimum system overall temperature differences (system thermal resistances) are 0.25 and 0.3, respectively, for the cylinder--series and rectangular-prism--series building blocks. For both the cylinder--series and rectangular-prism--series building blocks, the triangular-prism--shaped blade performs the best among all the four kinds of blades considered. This is explained by the size of interfacial area sustained by the four kinds of blades with a fixed volume. Also, the constructal cylinder--triangular-prism building block is proved to perform better than the constructal rectangular-prism--triangular-prism building block at the same composing materials, composition ratio, volumetric heat generation rate and total building-block volume.
computational fluid dynamics.
The financial support from the Research Grants Council of Hong Kong (GRF718009 and GRF717508) and the CRCG of the University of Hong Kong (Project 10400920) is gratefully acknowledged.
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