This paper describes an analytical study of laminar natural convection on both sides of a vertical conducting wall of finite height separating two semi-infinite fluid reservoirs of different temperatures. The countercurrent boundary layer flow formed on the two sides is illustrated via representative streamlines, temperature and heat flux distributions. The net heat transfer between reservoirs is reported for the general case in which the wall thermal resistance is not negligible relative to the overall reservoir-to-reservoir thermal resistance.
Why is size so important? Why are “economies of scale” a universal feature of all flow systems, animate, inanimate, and human made? The empirical evidence is clear: the bigger are more efficient carriers (per unit) than the smaller. This natural tendency is observed across the board, from animal design to technology, logistics, and economics. In this paper, we rely on physics (thermodynamics) to determine the relation between the efficiency and size. Here, the objective is to predict a natural phenomenon, which is universal. It is not to model a particular type of device. The objective is to demonstrate based on physics that the efficiencies of diverse power plants should increase with size. The analysis is performed in two ways. First is the tradeoff between the “external” irreversibilities due to the temperature differences that exist above and below the temperature range occupied by the circuit executed by the working fluid. Second is the allocation of the fluid flow irreversibility between the hot and cold portions of the fluid flow circuit. The implications of this report in economics and design science (scaling up, scaling down) and the necessity of multi-scale design with hierarchy are discussed.
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 article is a principle-based review of a growing body of fundamental research that documents the opportunity for optimizing geometrically the cooling of spaces (e.g., electronics packages) that generate heat volumetrically. The chief result of geometric optimization is the identification of an optimal internal structure—optimal spacings between components (e.g., plates and fins), optimal sizes and aspect ratios for cooling channels, and optimal frequencies for pulsating flows. The origin of these optimal geometric features—the construction of the system—lies in the global effort to use every infinitesimal volume to the maximum, i.e., to pack the volume not only with the most heat generating components, but also with the ‘most’ coolant, in such a way that every fluid packet is engaged effectively in cooling. The optimal aspect ratio for ducts with forced and natural convection corresponds to the special geometry and flow conditions where boundary layers meet just as the coolant exits the channel. This “constructal” design principle is illustrated by several classes of examples: laminar forced and natural convection, and various internal arrangements (parallel plates, staggered plates, cylinders in cross flow, square pins with impinging flow). General trends (scaling laws) of optimal geometric form are revealed by the optimal-structure results, this, in spite of the diversity of the optimized configurations.
The geometric form of the tree network is deduced from a single mechanism. The discovery that the shape of a heat-generating volume can be optimized to minimize the thermal resistance between the volume and a point heat sink, is used to solve the kinematics problem of minimizing the time of travel between a volume (or area) and one point. The optimal path is constructed by covering the volume with a sequence of volume sizes (building blocks), which starts with the smallest size and continues with stepwise larger sizes (assemblies). Optimized in each building block is the overall shape and the angle between constituents. The speed of travel may vary from one assembly size to the next, however, the lowest speed is used to reach the infinity of points located in the smallest volume elements. The volume-to-point path that results is a tree network. A single design principle – the geometric optimization of volume-to-point access – determines all the features of the tree network.
In this paper we develop the optimal tree-shaped flow paths for cooling a disc-shaped body by convection. Heat is generated uniformly over the disc area. The coolant enters through the center of the disc, and exits through ports positioned equidistantly along the perimeter. The unknown is the flow architecture. The constraints are the disc size and the total volume occupied by the ducts. It is assumed that the ducts are narrow enough so that the flow is hydrodynamically and thermally fully developed. The ultimate goal is to determine flow architectures that reach simultaneously two objectives: (i) minimal global fluid flow resistance (or pumping power), and (ii) minimal global thermal resistance. When the architecture is optimized for (i), the result is a dendritic structure in which every geometric feature is uniquely determined. The corresponding thermal resistance decreases as the total mass flow rate and the pumping power increase. When the objective is (ii), the optimal architecture has radial ducts, not dendrites. The corresponding fluid-flow resistance increases as the flow rate increases and the global thermal resistance decreases. Put together, these geometric results show that methods (i) and (ii) lead to nearly the same combined performance (thermal and fluid). Examined more closely, the dendrites produced by method (i) perform progressively better as the length scales become smaller. Optimized increasing complexity is the route to high thermal and fluid-flow performance in the limit of decreasing scales.
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.
This paper introduces a model and a structured procedure to optimize the internal structure (relative sizes, spacings) and external shape (aspect ratios) of a unit PEM fuel cell so that net power is maximized. The optimization of flow geometry is conducted for the smallest (elemental) level of a fuel cell stack, i.e., the unit PEM fuel cell, which is modeled as a unidirectional flow system. The polarization curve, total and net power, and efficiency are obtained as functions of temperature, pressure, geometry and operating parameters. The optimization is subjected to fixed total volume. There are two levels of optimization: (i) the internal structure, which basically accounts for the relative thicknesses of two reaction and diffusion layers and the membrane space, and (ii) the external shape, which accounts for the external aspect ratios of a square section plate that contains all unit PEM fuel cell components. The available volume is distributed optimally through the system so that the net power is maximized. Temperature and pressure gradients play important role, especially as the fuel and oxidant flow paths increase. Numerical results show that the optimized internal structure is “robust” with respect to changes in external shape. The optimized internal structure and external shape are a result of an optimal balance between electrical power output and pumping power required to supply fuel and oxidant to the fuel cell through the gas channels. Directions for future improvements at the PEM fuel cell stack level in flow architecture (constructal design) are discussed.
In recent years there has been interest in research related to hyperthermia combined with radiation and cytotoxic drugs to enhance the killing of tumors. When heating the tumor tissue, the crucial problem is keeping the temperature of the normal tissue surrounding the tumor below a certain threshold so as not to cause damage to the tissue. Hence, in order to control the process, it is important to obtain a temperature field of the entire treatment region. Recently, we have developed a numerical method for obtaining an optimal temperature distribution in a triple-layered skin structure. In this article, we extend our study to a triple-layered skin structure embedded with multilevel blood vessels, and develop a numerical method for obtaining an optimal temperature distribution.
No abstract is provided for this article.
This study explores various uses of engineering as a tool for solving problems and improving the quality of life. The experiments in the article show that competitive athletes swim with their fingers spread slightly, because this configuration generates greater speed, the research being based on a principle known as constructional law. The constructional law has been applied to predict all the key features of the design of animal locomotion, which includes human running and swimming. In engineering, the discovery expands a domain of constructal-design results that has been growing fast. Bodies that generate heat volumetrically are endowed with maximum heat transfer density when the spacing between the solid surfaces internal to the volume have certain sizes that are smaller in forced convection than in natural convection. The volumetric cooling of future electronics, avionics, and self-cooling materials rests on this class of constructal designs. The swimming with spread fingers is the corresponding design of a body for maximum momentum transfer density.
By a porous medium, we mean a material consisting of a solid matrix with an interconnected void. We suppose that the solid matrix is either rigid (the usual situation) or it undergoes small deformation. The interconnectedness of the void (the pores) allows the flow...
Here, we show that the most basic features of tree and forest architecture can be put on a unifying theoretical basis, which is provided by the constructal law. Key is the integrative approach to understanding the emergence of “designedness” in nature. Trees and forests are viewed as integral components (along with dendritic river basins, aerodynamic raindrops, and atmospheric and oceanic circulation) of the much greater global architecture that facilitates the cyclical flow of water in nature (Fig. 1) and the flow of stresses between wind and ground. Theoretical features derived in this paper are: the tapered shape of the root and longitudinally uniform diameter and density of internal flow tubes, the near-conical shape of tree trunks and branches, the proportionality between tree length and wood mass raised to 1/3, the proportionality between total water mass flow rate and tree length, the proportionality between the tree flow conductance and the tree length scale raised to a power between 1 and 2, the existence of forest floor plans that maximize ground-air flow access, the proportionality between the length scale of the tree and its rank raised to a power between −1 and −1/2, and the inverse proportionality between the tree size and number of trees of the same size. This paper further shows that there exists an optimal ratio of leaf volume divided by total tree volume, trees of the same size must have a larger wood volume fraction in windy climates, and larger trees must pack more wood per unit of tree volume than smaller trees. Comparisons with empirical correlations and formulas based on ad hoc models are provided. This theory predicts classical notions such as Leonardo's rule, Huber's rule, Zipf's distribution, and the Fibonacci sequence. The difference between modeling (description) and theory (prediction) is brought into evidence.
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.
