Minimum pumping power fluid tree networks without a priori flow regime assumption

In this paper we approach the optimization of fluid tree networks by relaxing the usual one-flow regime assumption. The pumping power requirement is minimized, under global volume constraint. Two types of constructal network geometries are investigated: (i) the fluid users are distributed uniformly on a surface and, (ii) the fluid users are located on the periphery of a disc-shaped area. In both cases, the flow regime in a given pipe of the network emerges as a result of pumping power minimization. It is shown that the individual users’ consumption and number of users dictate the transition from one optimal flow regime configuration to another. Under certain circumstances, laminar and turbulent flow regimes are present simultaneously in different pipes of an optimized network. The occurrence of turbulence at a certain level of pipes in the optimal hierarchical networks always leads to turbulence in the higher levels of pipes. The paper provides designers with basic tools for the conceptual design of fluid networks.     

Distribution of heat sources in vertical open channels with natural convection

In this paper we use the constructal method to determine the optimal distribution and sizes of discrete heat sources in a vertical open channel cooled by natural convection. Two classes of geometries are considered: (i) heat sources with fixed size and fixed heat flux, and (ii) single heat source with variable size and fixed total heat current. In both classes, the objective is the maximization of the global thermal conductance between the discretely heated wall and the cold fluid. This objective is equivalent to minimizing temperature of the hot spot that occurs at a point on the wall. The numerical results show that for low Rayleigh numbers (∼10²), the heat sources select as optimal location the inlet plane of the channel. For configuration (i), the optimal location changes as the Rayleigh number increases, and the last (downstream) heat source tends to migrate toward the exit plane, which results in a non-uniform distribution of heat sources on the wall. For configuration (ii) we also show that at low and moderate Rayleigh numbers (Ra_M ∼ 10² and 10³) the thermal performance is maximized when the heat source does not cover the entire wall. As the flow intensity increases, the optimal heat source size approaches the height of the wall. The importance to free the flow geometry to morph toward the configuration of minimal global resistance (maximal flow access) is also discussed.     

Constructal tree-shaped two-phase flow for cooling a surface

This paper documents the strong relation that exists between the changing architecture of a complex flow system and the maximization of global performance under constraints. The system is a surface with uniform heating per unit area, which is cooled by a network with evaporating two-phase flow. Illustrations are based on the design of the cooling network for a skating rink. The flow structure is optimized as a sequence of building blocks, which starts with the smallest (elemental volume of fixed size), and continues with assemblies of stepwise larger sizes (first construct, second construct, etc.). The optimized flow network is tree shaped. Three features of the elemental volume are optimized: the cross-sectional shape, the elemental tube diameter, and the shape of the elemental area viewed from above. The tree that emerges at larger scales is optimized for minimal amount of header material and fixed pressure drop. The optimal number of constituents in each new (larger) construct decreases as the size and complexity of the construct increase. Constructs of various levels of complexity compete: the paper shows how to select the optimal flow structure subject to fixed size (cooled surface), pressure drop and amount of header material.     

Dendritic constructal heat exchanger with small-scale crossflows and larger-scales counterflows

This paper describes the constructal route to the conceptual design of a two-stream heat exchanger with maximal heat transfer rate per unit volume. The flow structure has multiple scales. The smallest (elemental) scale consists of parallel-plates channels the length of which matches the thermal entrance length of the small stream that flows through the channel. This feature has two advantages: it eliminates the longitudinal temperature increase (flow thermal resistance) that would occur in fully developed laminar flow, and it doubles the heat transfer coefficient associated with fully developed laminar flow. The elemental channels of hot fluid are placed in crossflow with elemental channels of cold fluid. The elemental channel pairs are assembled into sequentially larger flow structures (first construct, second construct, etc.), which have the purpose of installing (spreading) the elemental heat transfer as uniformly as possible throughout the heat exchanger volume. At length scales greater than the elemental, the streams of hot and cold fluid are arranged in counterflow. Each stream bathes the heat exchanger volume as two trees joined canopy to canopy. One tree spreads the stream throughout the volume (like a river delta), while the other tree collects the same stream (like a river basin). It is shown that the spacings of the elemental and first-construct channels can be optimized such that the overall pumping power required by the construct is minimal. The paper concludes with a discussion of the advantages of the proposed tree-like (vascularized) heat exchanger structure over the use of parallel small-scale channels with fully developed laminar flow.     

Constructal design for cooling a disc-shaped area by conduction

This paper describes a hierarchical strategy to developing the optimal internal structure of a round heat-generating body cooled at its center with the help of optimally distributed inserts of high-conductivity material. The sequence begins with optimizing the geometry of the smallest heat generating entity - a sector-shaped elemental volume with the smallest dimension, and a single high-conductivity insert. Many such elements are assembled into disc-shaped constructs, or into sector-shaped constructs in which the elemental volumes are grouped into a formation shaped as a fan. When several sector-shaped constructs are assembled into a disc, they constitute a quasi-radial heat-flow structure in which each high-conductivity insert exhibits one branching. Every geometric detail of the optimized two-material conductive structures is determined based on principle - the minimization of global resistance subject to global constraints (total volume, total volume of high-conductivity material). The inserts of high-conductivity material form structures shape as trees. The global thermal resistance of each tree-shaped construct is reported. The minimization of global thermal resistance is the criterion for choosing between a design with radial inserts and one with branched inserts.     

Tree-shaped vascular wall designs for localized intense cooling

This paper is a proposal to embed tree-shaped vasculatures in a wall designed such that the wall withstands without excessive hot spots the intense heating that impinges on it. The vasculature is a quilt of square-shaped panels, each panel having a tree vasculature that connects the center with the perimeter. The coolant may flow in either direction, center–perimeter, or perimeter–center, although here only the center–perimeter flow direction is illustrated. Numerical simulations of conjugate heat and fluid flow in three directions show that it is possible to determine all the optimal geometric features of vasculatures with up to three levels of bifurcation (n = 3). The global performance is evaluated in terms of the overall thermal resistance, pressure difference, flow resistance and pumping power. The improvements in global performance diminish as the number of bifurcation levels increases. No flow architecture is universally superior. The dendritic designs are superior at the low and high ends of the pressure difference range. The radial designs are superior at intermediate pressure difference numbers.     

Thermodynamic optimization of geometry: T- and Y-shaped constructs of fluid streams

This paper presents a series of examples in which the global performance of flow systems is optimized subject to global constraints. The flow systems are assemblies of ducts, channels and streams shaped as Ts, Ys and crosses. In pure fluid flow, thermodynamic performance maximization is achieved by minimizing the overall flow resistance encountered over a finite-size territory. In the case of more complex objectives such as the distribution of a stream of hot water over a territory, performance maximization requires the minimization of flow resistance and the leakage of heat from the entire network. Taken together, these examples show that the geometric structure of the flow system springs out of the principle of global performance maximization subject to global constraints. Every geometric detail of the optimized flow structure is deduced from principle. The optimized structure (design, architecture) is robust with respect to changes in some of the parameters of the system. The paper shows how the geometric optimization method can be extended to other fields, e.g., urban hydraulics and, in the future, exergy analysis and thermoeconomics.     

Optimal distribution of discrete heat sources on a wall with natural convection

This paper is an application of the constructal method to the discovery of the optimal distribution of discrete heat sources cooled by laminar natural convection. The global objective is to maximize the global conductance between the wall and the fluid, or to minimize the hot-spot temperatures when the total heat generation rate and global system dimensions are specified. Two scenarios are investigated: (i) a large number of small heat sources mounted on a vertical wall facing a fluid reservoir, and (ii) a small number of finite-size heat sources mounted on the inside of the side wall of a two-dimensional enclosure. It is shown that the optimal distribution is not uniform (the sources are not equidistant), and that as the Rayleigh number increases the heat sources placed near the tip of a boundary layer should have zero spacings. In both (i) and (ii), the optimal configuration of the wall with discrete sources is generated by the pursuit of maximal global performance subject to global constraints.     

Constructal optimisation of conjugate triangular cooling channels with internal heat generation

This paper presents the development of the three-dimensional flow architecture of conjugate cooling channels in forced convection with internal heat generation within a solid. Two types of cross-section channel geometries were used. The first involved equilateral triangles with three equal legs in length and all three internal angles of 60°. The second was isosceles right triangles with two legs of equal length and internal angles of 90°, 45° and 45°. Both the equilateral triangle and isosceles right triangle are special case of triangle that can easily and uniformly be packed and arranged to form a larger constructs. The configurations were optimised in such a way that the peak temperature of the heat generating solid was minimised subject to the constraint of a fixed global volume of the solid material. The cooling fluid was driven through the channels by the pressure difference across the channel. The degrees of freedom of the channels were aspect ratio, hydraulic diameter and channel to channel spacing ratio. The shape of the channel was allowed to morph to determine the best configuration that gives the lowest thermal resistance. A gradient-based optimisation algorithm was applied in order to search for the best optimal geometric configurations that improve thermal performance by minimising thermal resistance for a wide range of dimensionless pressure difference. The effects of porosities, applied pressure and heat generation rate on the optimal aspect ratio and channel to channel spacing are reported. It was found that there are unique optimal design variables for a given pressure difference. The numerical results that were obtained were in agreement with the theoretical formulation using scale analysis and method of intersection of asymptotes. Results obtained show that the effects of applied dimensionless pressure drop on minimum thermal resistance were consistent with those obtained in the open literature.     

Vascularization with line-to-line trees in counterflow heat exchange

Here we report the heat and fluid flow characteristics of counterflow heat exchangers with tree-shaped line-to-line flow channels. The flow structures of the hot and cold sides are sequences of point-to-line trees that alternate with upside-down trees. The paper shows under what conditions the tree vascularization offers greater heat flow access than corresponding conventional designs with parallel single-scale channels. The analytical part is based on assuming fully developed laminar flow in every channel and negligible longitudinal conduction in the solid. The numerical part consists of simulations of three-dimensional convection coupled with conduction in the solid. It is shown that tree vascularization offers greater heat flow access (smaller global thermal resistance) than parallel channels when the number of pairing levels increases and the available pumping power or pressure drop is specified. When the solid thermal conductivity increases, the heat transfer effectiveness decreases because of the effect of longitudinal heat conduction. The nonuniformity in fluid outlet temperature becomes more pronounced when the number of pairing levels increases and the pumping power (or pressure drop number) increases. The nonuniformity in outlet fluid temperature decreases when the solid thermal conductivity increases.     

Vascular materials cooled with grids and radial channels

Vascularized materials are a new generation of smart-material concepts that offer novel volumetric functionalities such as self-cooling, self-healing, renewal, cleansing, and “designed” transport properties (permeability, effective thermal conductivity). In this paper, we evaluate the volumetric cooling performance of slabs with embedded flow architectures consisting of grids (G) and radial channels (R). Both flow directions are considered: inlet (I) and outlet (O) in the center of the slab. In total, four configurations (GI, RI, GO, RO) compete for high performance in three directions: (i) low peak of overheating, or low global thermal resistance, (ii) small volume fractions (σ) occupied by high temperatures, and (iii) small pumping power. The results show that grids have lower global flow resistance than radial designs while local junction losses are important. The designs with inlet in the center are attractive in having lower global flow resistance than those with outlet in the center. When objective (i) is considered, designs with outlet in the center are recommended, and the gains in performance are significant if junction losses are negligible and Reynolds numbers are small. For objective (ii), RO is attractive if Sv is greater than 10 or Be is smaller than 10⁹. When the three objectives (i)–(iii) are considered at the same time, the configurations with the outlet in the center (GO, RO) are superior when the flow system operates at low pumping power, and that GI and RI are attractive at high pumping power.     

Three-dimensional numerical optimization of a manifold microchannel heat sink

A three-dimensional analysis procedure for the thermal performance of a manifold microchannel heat sink has been developed and applied to optimize the heat-sink design. The system of fully elliptic equations, that govern the flow and thermal fields, are solved by a SIMPLE-type finite volume method, while the optimal geometric shape is traced by a steepest descent technique. For a given pumping power, the optimal design variables that minimize the thermal resistance are obtained iteratively. The procedure is robust and the optimal state is reached within six global iterations. Comparing with the comparable traditional microchannel heat sink, the thermal resistance is reduced by more than a half while the temperature uniformity on the heated wall is improved by tenfold. The sensitivity of the thermal performance on each design variable is also examined and presented in the paper. Among various design variables, the channel width and depth are more crucial than others to the heat-sink performance. The optimal dimensions and corresponding thermal resistance have a power-law dependence on the pumping power.     

Constructal design of T–Y assembly of fins for an optimized heat removal

Constructal design has been applied to a large variety of problems in nature and engineering to optimize the architecture of animate and inanimate flow systems. This numerical work uses this method to seek for the best geometry of a T–Y assembly of fins, i.e., an assembly where there is a cavity between the two branches of the assembly of fins. The global thermal resistance of the assembly is minimized by geometric optimization subject to the following constraints: the total volume, the volume of fin-material, and the volume of the cavity. Parametric study was performed to show the behavior of the twice minimized global thermal resistance. The results show that smaller cavity volume and larger fins volume improve the performance of the assembly of fins. The twice minimized global thermal resistance of the assembly and its corresponding optimal configurations calculated for the studied parameters were correlated by power laws.     

Constructal design of Y-shaped assembly of fins

This work relies on constructal design to perform the geometric optimization of the Y-shaped assembly of fins. It is shown numerically that the global thermal resistance of the Y-shaped assembly of fins can be minimized by geometric optimization subject to total volume and fin material constraints. A triple optimization showed the emergence of an optimal architecture that minimizes the global thermal resistance: an optimal external shape for the assembly, an internal optimal ratio of plate-fin thicknesses and an optimal angle between the tributary branches and the horizontal. Parametric study was performed to show the behavior of the minimized global thermal resistance. The results also show that the optimized Y-shaped structure performs better than the optimized T-shaped one.     

Enhancement of thermal performance in double-layered microchannel heat sink with nanofluids

A three-dimensional analysis aimed at enhancing the thermal performance of a double-layered microchannel heat sink by using a nanofluid and varying the geometric parameters has been conducted. A system of fully elliptic equations that govern the flow and thermal fields are solved using the finite volume method. The analysis indicates that the dominant factors determining the thermal resistance of the channel include the type of nanofluid; particle volume fraction; geometric parameters of the channel, such as the channel number, channel width ratio, channel aspect ratio; and pumping power. The results indicate that the greatest enhancement in channel cooling can be expected when an Al₂O₃–water nanofluid is used. The thermal resistance of the channel can be minimized by properly adjusting the particle volume fraction under various pumping powers; the minimum thermal resistance depends on the geometric parameters. The study also reveals that the relationship between the thermal resistance and channel number, channel width ratio, or channel aspect ratio exhibits a decrease followed by an increase. The thermal performance of the channel can usually be improved by decreasing the channel number or channel aspect ratio, or increasing the channel width ratio. Finally, increasing the pumping power reduces the overall thermal resistance. An Al₂O₃ (1%)–water nanofluid shows an average improvement in thermal performance of 26% over that of pure water for a given pumping power. However, the design’s effectiveness declines significantly under high pumping power. In particular, the thermal resistance obtained by employing nanofluids was not necessarily lower than that of water under all pumping powers, but it can be reduced by properly adjusting the geometric parameters under optimal conditions.     

Geometric optimization of shapes on the basis of Bejan''s Constructal theory

This paper documents the optimization of architecture in accordance with Bejan's Constructal theory. For illustration, we consider the optimization of a cavity that intrudes into a solid conducting wall, having internal heat generation and adiabatic conditions on the outer surfaces. The cavity is rectangular, with fixed volume and variable aspect ratio, and the solid is trapezoidal. The objective is to minimize the global thermal resistance between the volume of the entire system (cavity and solid) and the surroundings. The performance improves as the cavity becomes slender. The geometry is optimal when the cavity penetrates the conducting wall completely.     

Maximum heat transfer density with rotating cylinders aligned in cross-flow

In this paper, we study the thermal behavior of an assembly of rotating cylinders aligned in a cross-flow. The objective is to maximize the heat transfer rate density of the assembly, i.e. the overall heat transfer dissipated per unit of volume, under fixed pressure drop. A numerical model is used to solve the governing equations. Two configurations are studied: i) the cylinders rotate in the same direction, and ii) consecutive cylinders rotate in opposite directions. The spacing between consecutive cylinders is optimized in each case. The second configuration proved to be the more efficient. In that configuration, it is also possible to optimize further the architecture by using a smaller spacing for the flow passage in which the flow is in the direction of the cylinders rotation, and larger spacing for the flow passage in which the cylinders oppose the main stream.     

Combined `flow and strength' geometric optimization: internal structure in a vertical insulating wall with air cavities and prescribed strength

This paper addresses the fundamental problem of optimizing the internal structure of a vertical wall that must meet two requirements, thermal insulation and mechanical strength. The wall is a composite of solid material (e.g., brick) and parallel air caverns with varying thickness and number. It is shown that the internal structure of the wall (the number of air caverns) can be optimized so that the overall thermal resistance of the wall is maximal, while the mechanical stiffness of the wall is fixed. The maximized thermal resistance increases when the effect of natural convection in the air gaps is weaker, and when the specified wall stiffness decreases. The optimal number of air gaps is larger when the effect of natural convection is stronger, and when the specified wall stiffness is smaller. The optimal structure is such that the volume fraction occupied by air spaces decreases when the natural convection effect (the overall Rayleigh number) increases, and when the prescribed wall stiffness increases. The paper draws attention to a new class of thermal design problems, in which the system architecture is derived from a combination of heat transfer and mechanical strength considerations. This class represents an extension of the constructal design method, which until now has been used for maximizing thermofluid performance subject to size constraints.     

Optimal thermal design of microchannel heat sinks with different geometric configurations

In this study, three-dimensional models of microchannel heat sinks (MCHSs) with different geometric configurations (such as single-layered- (SL), double-layered- (DL) or tapered-(T)-channels) are constructed by an optimization procedure. This procedure integrates a direct problem solver with a simplified conjugate-gradient method as the optimizer. The overall thermal resistance of an MCHS is the objective function to be minimized with respect to geometric parameters, such as the number of channels, channel width ratio, channel aspect ratio and tapered ratios, as the search variables. The optimal thermal resistance is found to decrease in the following order: the initial guess parallel channel (IGP channel), SL-, DL- and T-channel designs. In addition, the T-channel design has the minimum temperature difference and the most uniform temperature distribution, followed by the DL-, SL- and IGP-channel designs. Moreover, the optimal thermal resistance reduces with the pumping power for the various channel configuration designs, and the lowest thermal resistance corresponds to the T-channel design. The larger the pumping power, the larger the decrement in thermal resistance. Therefore, the optimal T-channel is the best MCHS design when considering thermal resistance and temperature distribution uniformity.     

Layered porous media architecture for maximal cooling

In this paper, we address the fundamental problem of how to arrange fluid flow and solid material for minimal thermal resistance. A heat-generating board is cooled by a stack of porous layers through which a coolant flows. The stream is generated by a fixed pressure drop. The problem consists in determining the optimal porosity and material of each layer for minimizing the hot spot temperature (thermal resistance), under global mass and cost constraints. We combine a genetic algorithms (GA) toolbox with a finite volume program to optimize the design. The shape and structure of the system emerge from the global optimization, under global constraints. The optimal material to use in each layer is determined by the GA – not assumed – and is chosen from a database of four materials. The GA eliminates layers that do not contribute to the overall performance and therefore optimizes the size of the stacking. The results indicate that more solid material should be used closer to the hot plate (non-uniform distribution). Several nearly optimal configurations are found in the design space.