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This paper describes the conceptual design and performance of balanced two-stream counterflow heat exchangers, in which each stream flows as a tree network through its allotted space. The two trees in counterflow are like two palms pressed against each other. The paper develops the relationships between effectiveness and number of heat transfer units for several tree-counterflow configurations: (i) constructal dichotomous trees covering uniformly a rectangular area, (ii) trees on a disk-shaped area, and (iii) trees on a square-shaped area. In configurations (ii) and (iii) each stream flows between the center and the periphery of the area. Configurations (i) and (ii) are trees with minimal resistance to fluid flow. Configuration (iii) is designed by minimizing the length of each duct in the network. The paper reports the formula for the number of heat transfer units in each configuration. Unlike in counterflows formed by two parallel streams, in which the longitudinal temperature gradient is constant, in the counterflow formed by two trees the longitudinal temperature gradient is steeper as one approaches the periphery of the tree canopy. The application of dendritic heat exchangers to devices with maximal transport density is discussed, e.g., electronics cooling, fuel cell architectures, etc.

WOS:000342837000069; International audience; This is a treatment of the phenomenon of energy storage by melting in an enclosure where the heating is provided by an invading line of higher temperature. We show that the melting process consists of two phases: "invasion" thermal diffusion along the invading line, which is followed by "consolidation" as heat diffuses perpendicularly to the invading line. We report the duration of both phases and the evolution of the melt layer around the invading line during the two-dimensional and three-dimensional invasion. The amount of melted material increases in time according to a curve shaped as an S. Diffusion sweeps the territory faster as the complexity of the tree design increases meaning the S-curve of the history of the process becomes steeper. (C) 2014 AIP Publishing LLC.

WOS:000335643700657; International audience; This paper is a proposal to embed tree-shaped vasculatures in a wall designed such that the wall withstands without excessive hot spots and peak stresses the intense heating and pressure that impinge 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 vascular designs for volumetric cooling can be complemented by the shaping and distributing of channels for maximum strength and thermal performance at the same time. Numerical simulations of heat flow and thermal stresses in three directions show that it is possible to determine the optimal geometric features of configurations with radial channels and trees with radial and one level of bifurcations. The global performance is evaluated in terms of the overall thermal resistance and peak von Mises stresses. The dendritic design is superior under the studied thermal condition.

WOS:000313670600077; International audience; Pedestrian movement occurs in evolutionary patterns the effect of which is to facilitate the access of people into and out of living spaces (areas, volumes). In this paper, we rely on the philosophy of Constructal design as applied physics, in order to uncover two fundamental features of evolutionary design for pedestrian evacuation from rectangular areas (e. g., lecture halls with seated occupants). First, the paper shows analytically that the aspect ratio of the floor area can be selected such that the total evacuation time is minimal. Second, the shape of the floor area of each aisle can be tapered such that the total evacuation time is decreased further. These two architectural features are confirmed by means of extensive and systematic numerical simulations of pedestrian evacuation, by using two numerical packages (Simulex and FDS + Evac). (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4780612]

Here, we show that a plate heated by a moving beam can be cooled effectively by fluid that flows through a vasculature of channels embedded in the plate. The vascular designs studied are radial, grid and hybrid (radial + grid). The peak temperature of the plate changes with the path and direction of the moving beam. The strength, size and speed of the beam vary. The peak temperature increases as the beam strength and size increase and as the speed of the beam decreases. The grid and hybrid designs are robust because of loops present in the flow structure. The pressure difference that drives the fluid flow varied. The channel diameter ratios that provide greatest flow access are reported. The cooling performance of the multiscale grid structures is less sensitive to the changes in beam path than the cooling performance of the other structures studied. The effect of adding a vascular structure to the design is dramatic.

Spreading and collecting flows are united by the flow design known as the S-curve: when plotted versus time, the size of the domain that is filled or emptied has a history that is shaped as an S. Here, we show that the fastest spreading or collecting (i.e., the steepest S curve) is discovered by allowing the tree architecture to morph freely, toward greater access over time, in accord with the constructal law of design in nature. The angles between the lines of the invading flow architecture can be selected such that the overall flow proceeds the fastest, covering the greatest territory at any moment. The design is a sequence of two distinct phenomena: “invasion” by channels and branches that grow fast, and “consolidation” by slow diffusion perpendicular to the channels. Invasion and consolidation collaborate hand-in-glove to facilitate the spreading or collecting over the available finite area or volume.

Here we show that the configuration of an inhabited area controls the time required by all the pedestrians to vacate the space. From the minimization of the global evacuation time emerges the optimal configuration of the area. This is a fundamental principle for designing living spaces with efficient evacuation quality, and it is demonstrated here with several simple building blocks that can be used as components of more complex living structures: single walkway, corner, and T-shaped walkway. We show analytically and numerically that the ratio of the widths of the stem and branches of the T-shaped walkway has an optimal value that facilitates the evacuation of all the inhabitants. This result is fundamental, and is the crowd-dynamics equivalent of the Hess-Murray rule for the ratio of diameters in bifurcated ducts with fluid flow.

Here we document the effect of flow configuration on the heat transfer performance of a serpentine shaped stream embedded in a conducting solid. Several configurations with fixed volume of fluid are considered: U-shaped with varying spacing between the parallel portions of the U, serpentine shapes with three elbows, and conducting soil with several parallelepipedal shapes. We show that the spacing must be greater than a critical value in order for the heat transfer density of the stream-solid configuration to be the highest that it can be. Spacings larger than this critical value do not yield improvements in heat transfer density. We also show that even though the heat transfer is time dependent, the stream-solid configuration has an effective number of heat transfer units Ntu that is nearly constant in time. The larger Ntu values correspond to the configurations with greater heat transfer density.

This paper documents the time-dependent thermal interaction between a fluid stream configured as a plane tree of varying complexity embedded in a conducting solid with finite volume and insulated boundaries. The time scales of the convection-conduction phenomenon are identified. Two-dimensional and three-dimensional configurations are simulated numerically. The number of length scales of the tree architecture varies from one to four. The results show that the heat transfer density increases, and the time of approach to equilibrium decreases as the complexity of the tree designs increases. These results are then formulated in the classical notation of energy storage by sensible heating, which shows that the effective number of heat transfer units increases as the complexity of the tree design increases. The complexity of heat transfer designs in many applications is constrained by first cost and operating cost considerations. This work provides a fundamental basis for objective evaluation of cost and performance tradeoffs in thermal design of energy systems with complexity as an unconstrained parameter that can be actively varied over a broad range to determine the optimum system design.

In this paper, we show how vascular design controls the cooling and mechanical performance of a solid square slab heated uniformly and loaded with uniform pressure. The vascular structure is a combination of one grid and several peripheral trees. The designs of trees with grid canopies provide greater robustness than purely dendritic designs. The coolant fluid enters the slab from the center or from the rim, cools the slab to an allowable temperature level, and then exits. Numerical simulations show the flow of the fluid, heat, and stresses. The results indicate that the lowest peak temperature and lowest flow resistance can be achieved with radial channels, and the lowest peak stress can be achieved with trees with canopies. The peak stresses are the lowest when the configuration is a hybrid grid and tree design. There is an optimal ratio of the grid length divided by the slab length for each specified fraction of the fluid volume occupied by the radial channels. When the heating is concentrated in a small area, the peak temperature is smaller when the heated spot is closer to the center of the slab.

The S curve is one of the most common phenomena in nature: the spreading of populations, tumors, contaminants, innovations, economic activity. Here we show that this phenomenon can be predicted entirely by recognizing in it a flow. The flow is not by diffusion alone, rather it is a combination of tree-shaped “invasion” by convection, followed by “consolidation” by diffusion perpendicular to the invasive lines. The S curve is not unique: its scales depend on the relative magnitude of the speed of the invading lines and the diffusivity perpendicular to the lines. Tree-shaped invasion covers the territory with diffusion much faster than line-shaped invasion. The predicted S-curve flow architecture unites the designs of spreading flows and collecting flows (e.g., mining, fossil fuel extraction, Hubbert peak) in all the realms of nature: animate, inanimate, and human-made.

This work documents the design method of allocating channels that serve as heat sources or heat sinks in a conducting body. First, we develop a numerical model to find the optimum channel spacing for specified properties of the surrounding medium, the time scale of the heat transfer process, and the dimensions of the configuration. Second, we show with scale analysis that the optimal spacing (S/D) must equal τ1/2 in an order of magnitude sense, where τ is the dimensionless time scale of the process. This conclusion holds for the two heat transfer histories that were considered, exponential and top hat. We extend the method to a packing of channels of two sizes (diameters and heat source strengths). The optimal spacings and packing density obey the scaling rule determined for packing sources of a single size.

Here we show that the production and use of heating on an area must be distributed in clusters organized such that the losses associated with centers of production are balanced by the losses associated with distribution lines. The energy needs increase in time because the population density and the individual need increase. We consider only the increase in the individual need in time. We illustrate the “distributed energy systems” concept with the production and distribution of hot water on an area. Four classes of designs are analyzed and compared: (0) individual, i.e., one water heater for one user, (r) radial, i.e., N users supplied via radial pipes from a central heater, (2) dendritic network constructed by pairing N users around a central heating, and (4) dendritic network constructed by quadrupling the elemental areas occupied by the users. We show that there is an optimal cluster size (N) as a tradeoff between central losses and distributed losses. We also discover that several distinct (abrupt) de...

Here we show that the configuration of a slender enclosure can be optimized such that the radiation heating of a stream of solid is performed with minimal fuel consumption at the global level. The solid moves longitudinally at constant rate through the enclosure. The enclosure is heated by gas burners distributed arbitrarily, in a manner that is to be determined. The total contact area for heat transfer between the hot enclosure and the cold solid is fixed. We find that minimal global fuel consumption is achieved when the longitudinal distribution of heaters is nonuniform, with more heaters near the exit than the entrance. The reduction in fuel consumption relative to when the heaters are distributed uniformly is of order 10%. Tapering the plan view (the floor) of the heating area yields an additional reduction in overall fuel consumption. The best shape is when the floor area is a slender triangle on which the cold solid enters by crossing the base. These architectural features recommend the proposal to organize the flow of the solid as a dendritic design, which enters as several branches, and exits as a single hot stream of prescribed temperature. The thermodynamics of heating is presented in modern terms in the Sec. VII (exergy destruction, entropy generation). The contribution is that to optimize “thermodynamically” is the same as reducing the consumption of fuel.

When solid material is removed in order to create flow channels in a load carrying structure, the strength of the structure decreases. On the other hand, a structure with channels is lighter and easier to transport as part of a vehicle. Here, we show that this trade off can be used for benefit, to design a vascular mechanical structure. When the total amount of solid is fixed and the sizes, shapes, and positions of the channels can vary, it is possible to morph the flow architecture such that it endows the mechanical structure with maximum strength. The result is a multifunctional structure that offers not only mechanical strength but also new capabilities necessary for volumetric functionalities such as self-healing and self-cooling. We illustrate the generation of such designs for strength and fluid flow for several classes of vasculatures: parallel channels, trees with one, two, and three bifurcation levels. The flow regime in every channel is laminar and fully developed. In each case, we found that it...

The generation of heat in high-density electronics is concentrated in spots, not uniform. Here we account for this fundamental aspect as a phenomenon of heat conduction in a flat body, from a concentrated heat source to surface dissipation of heat to an external flow. To facilitate conduction through the body, the heat source is connected to the rest of the body (the blade) with a vein of considerably higher thermal conductivity. The architecture resembles a botanical leaf, in which the heat source deposits the heat current at the root of the vein, and the upper and lower surfaces of the leaf dissipate the heat current to the surrounding fluid flow. The optimal dimensions of this leaflike architecture are determined analytically and numerically: the blade length, width, and thickness, the vein cross-sectional area, and the global thermal conductance of the entire multiscale body. The optimized features are nondimensionalized and reported as power laws that show explicitly the design effect of external convection and the thermal conductivities of the vein and the blade.

Here we report the transient cooling performance of a body vascularized with tree-shaped channels supplied with coolant, which flows from one side of the body to the other. The vasculature consists of trees that alternate with upside down trees. Heat is generated volumetrically through the body at t=0. A time delay td separates the start of the flow of coolant from the start of heating. Three-dimensional simulations of conduction and convection in the solid-fluid composite show the formation and evolution of hot spots in the material. If the delay is not short enough, the maximum temperature of the body overshoots the maximum allowable level. The paper shows how to design the cooling delay time such that the hot-spot temperature does not exceed the safe level represented by the maximum temperature in the limit of steady state operation. The critical delay time is determined as a function of the applied pressure difference and the complexity of the dendritic flow architecture.

This paper describes a proposal to position electrodes optimally in a porous material with the objective of distributing ionic species over a targeted territory. The ionic species are supplied by an external reservoir and are transported through the porous network to a targeted zone by means of an electrical potential difference. The constructal theory of generation of flow access configuration is used to discover the optimal configuration of electrodes. An optimal number of electrode layers is proposed for the example studied. The new concept of ionic lines is proposed as a means of visualizing, explaining, and designing ionic transport.

Nature shows that fluids bathe the animal body as trees matched canopy to canopy. The entering streams invade the body as river deltas and the reconstituted streams sweep and exit the body as river basins. Why should this be so? Why is animal vascularization not based on arrays of parallel channels, as in modern heat exchangers? In this paper, we rely on constructal theory to show that the flow architecture that provides greatest access from point to volume and from volume to point is the three-dimensional compounding of trees matched canopy to canopy. This three-dimensional tree architecture is deduced, not assumed. Its flow performance is evaluated at every step relative to the performance of equivalent architectures with parallel channels. This paper also shows that the dendritic design must become more complex (with more levels of branching) as the volume inhabited by the flow design increases. The transition from designs with p branching levels to p+1 levels occurs abruptly as the available flow volu...

This article describes the generation of dendritic flow architectures that provide maximum side-to-side flow access, across a vascularized body. The flow architecture consists of trees that alternate with upside-down trees (a,b,a,b,…). If in tree “a” the flow is from root to canopy, in tree “b” the flow is from canopy to root. This means that the flow proceeds in the same direction through all the trees, i.e., the flow is oriented side-to-side, or line-to-line. The channel cross sections are rectangular, and they all have the same depth. The article shows under what conditions the tree vascularization offers greater flow access than parallel single-scale channels oriented perpendicularly to the two parallel lines. The analytical part of the work is based on the assumption of fully developed laminar flow in every channel. The numerical part consists of simulations of three-dimensional laminar flow through the entire tree architecture and its many bifurcations. It is shown that tree vascularization is more ...

The current literature on ionic diffusion through porous media teaches that the diffusion coefficient is a complicated function depending on concentration, concentration gradient, and electrical potential gradient. This paper documents how natural diffusion tests and migration tests (electrically enhanced transport) lead to the measurement of a unique diffusion coefficient for a given ionic species and a given material. Natural diffusion tests for chloride and a ceramic of TiO2 were implemented at two different concentration levels. The experiments were designed to emphasize the impact of the membrane potential in the pore solution on the chloride flux. By accounting for the membrane potential it is shown that the chloride diffusion coefficient is unique for a given material. An iterative method based on a numerical model solving the continuity equations and the current law is proposed to determine the diffusion coefficient. The approach is applied with success to published results on a cement-based mater...

This is a fundamental study of how to vascularize a self-healing composite material so that healing fluid reaches all the crack sites that may occur randomly through the material. The network of channels is built into the material and is filled with pressurized healing fluid. When a crack forms, the pressure drops at the crack site and fluid flows from the network into the crack. The objective is to discover the network configuration that is capable of delivering fluid to all the cracks the fastest. The crack site dimension and the total volume of the channels are fixed. It is argued that the network must be configured as a grid and not as a tree. Two classes of grids are considered and optimized: (i) grids with one channel diameter and regular polygonal loops (square, triangle, hexagon) and (ii) grids with two channel sizes. The best architecture of type (i) is the grid with triangular loops. The best architecture of type (ii) has a particular (optimal) ratio of diameters that departs from 1 as the crack length scale becomes smaller than the global scale of the vascularized structure from which the crack draws its healing fluid. The optimization of the ratio of channel diameters cuts in half the time of fluid delivery to the crack.