Constructal blade shape in nanofluids
 Chao Bai^{1} and
 Liqiu Wang^{1}Email author
DOI: 10.1186/1556276X6240
© Bai and Wang; licensee Springer. 2011
Received: 6 December 2010
Accepted: 21 March 2011
Published: 21 March 2011
Abstract
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 cylindershaped and regularrectangularprismshaped building blocks of the bladeconfigured heat conduction systems (using nanofluids as the heat conduction media) to find the optimal crosssectional shape for the nanoparticle blade under the same composing materials, composition ratio, volumetric heat generation rate, and total building block volume. The regulartriangularprismshaped blade has been proven to perform better than all the other three kinds of blades, namely, the regularrectangularprismshaped blade, the regularhexagonalprismshaped blade, and the cylindershaped blade. Thus, the regulartriangularprismshaped 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 cylinderregulartriangularprism building block performs better than the constructal regularrectangularprismregulartriangularprism building block.
Introduction
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 heattransfer 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 [18]. This is what one sees naturally in river basins, lung structure, and the percolation threshold effect [19]. Our previous studies have proved that the blade configuration of nanofluids is much better than the dispersed configuration for the two kinds of diskshaped 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 highconductivity blades in two kinds of building blocks of the total bladeconfigured heat conduction systems. This study is inspired by the need for the optimization of the cross section of duct for minimum flow resistance [11]. 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 bladeconfigured 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 cylindercylinder 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 cylindercylinder 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 onedimensional routesradial conduction inside crosssectional planes of the base fluid region and axial conduction along the blade.
At , the bestperforming cylindercylinder 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 bestperforming slenderness .
Numerical calculation for all the eight kinds of building blocks
The actual heat conduction in the cylindercylinder building block is of course not a simple combination of two onedimensional 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 [22] 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.
Gridindependence check (cylindertriangularprism building block, R_{0}/L_{0} = 0.25)
Number of grids 
 Changing of 

2500  0.271978  0.010181 × 10^{0} 
20120  0.269209  8.803569 × 10^{4} 
167360  0.268972  3.346073 × 10^{5} 
1248160  0.268963 
Perimeters of four blade cross sections having the same area of π
Cross section shape  Perimeter 

Triangle  = 8.080642 
Rectangle  = 7.089816 
Hexagon  = 6.597817 
Circle  2π = 6.283186 
Thus, the constructal cylindershaped heat conduction building block performs better than the rectangularprismshaped building block.
Conclusions
Inspired by the duct cross section optimization for minimum flow resistance, the shape of the nanoparticle blade is optimized for the cylindershaped and rectangularprismshaped building blocks of the bladeconfigured 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 cylindercylinder 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 buildingblock slenderness. The constructal slendernesses leading to minimum system overall temperature differences (system thermal resistances) are 0.25 and 0.3, respectively, for the cylinderseries and rectangularprismseries building blocks. For both the cylinderseries and rectangularprismseries building blocks, the triangularprismshaped 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 cylindertriangularprism building block is proved to perform better than the constructal rectangularprismtriangularprism building block at the same composing materials, composition ratio, volumetric heat generation rate and total buildingblock volume.
Abbreviations
 CFD:

computational fluid dynamics.
Declarations
Acknowledgements
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.
Authors’ Affiliations
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