Thermodynamic optimization of tree-shaped flow geometries

In this paper we optimize the performance of several classes of simple flow systems consisting of T- and Y-shaped assemblies of ducts, channels and streams. In each case, the objective is to identify the geometric configuration that maximizes performance subject to several global constraints. Maximum thermodynamic performance is achieved by minimization of the entropy generated in the assemblies. The boundary conditions are fixed heat flow per unit length and uniform and constant heat flux. The flow is assumed laminar and fully developed. Every geometrical detail of the optimized structure is deduced from the constructal law. Performance evaluation criterion is proposed for evaluation and comparison of the effectiveness of different tree-shaped design heat exchangers. This criterion takes into account and compare the entropy generated in the system with heat transfer performance achieved.     

Thermodynamic optimization of geometry in engineering flow systems

This review draws attention to an emerging body of work that relies on global thermodynamic optimization in the pursuit of flow system architecture. Exergy analysis establishes the theoretical performance limit. Thermodynamic optimization (or entropy generation minimization) brings the design as closely as permissible to the theoretical limit. The design is destined to remain imperfect because of constraints (finite sizes, times, and costs). Improvements are registered by spreading the imperfection (e.g., flow resistances) through the system. Resistances compete against each other and must be optimized together. Optimal spreading means spatial distribution, geometric form, topology, and geography. System architecture springs out of constrained global optimization. The principle is illustrated by simple examples: the optimization of dimensions, spacings, and the distribution (allocation) of heat transfer surface to the two heat exchangers of a power plant. Similar opportunities for deducing flow architecture exist in more complex systems for power and refrigeration. Examples show that the complete structure of heat exchangers for environmental control systems of aircraft can be derived based on this principle.     

Dendritic fins optimization for a coaxial two-stream heat exchanger

This paper documents the fundamental relation between the maximization of global performance and the maleable (morphing) architecture of a flow system with global constraints. The example is the coaxial two-stream heat exchanger with flow through a porous bed in the annular space. It is shown that the constraints force the design toward heat exchangers with finite axial length, where additional improvements are derived from installing high-conductivity fins across the porous bed. The maximization of global performance is achieved through the optimization of the configuration of plate fins. Configurations with radial fins are optimized analytically and numerically. Configurations with branched fins are optimized numerically. It is shown that the best configuration (radial vs. branched) depends on the size of the heat exchanger cross-section. When the size is small, the best is the radial pattern. When the size exceeds a certain threshold, the best configuration is the optimized branched tree of fins.     

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.     

Thermodynamic optimization of heat‐transfer equipment configuration in an environmental control system

In this paper, we show that many features of a heat transfer installation can be deduced from the maximization of the global performance of the greater system that employs the installation. The heat transfer installation is a series of two cross‐flow heat exchangers. The greater system is the environmental control system (ECS) of an aircraft. The global performance objective is the minimization of the total thermodynamic irreversibility of the ECS. Several architectural features are deduced from principle: the relative position of the two heat exchangers, their relative sizes, and all the geometric aspect ratios of the two heat exchanger cores. We find that the optimized architecture is insensitive (robust) to changes in some of the external parameters. Robustness is a useful feature because it simplifies the design work. Furthermore, one design that is built can be expected to function at near‐optimal levels when the external parameters change. The application of this method of topology optimization to more complex systems is discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Dendritic heat convection on a disc

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.     

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.     

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.     

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.     

Criteria for the decomposition of energy systems in local/global optimizations

The decomposition of an energy system into subsystems of reduced complexity, to be optimized separately, but in a way compatible with the optimum of the global system, has been recognized as a viable solution to the problem of the design optimization of highly integrated, complex energy systems. Iterative Local/Global Optimization (ILGO) and its dynamic extension (DILGO) permit the decomposition of the global problem into smaller subproblems to be optimized separately, guaranteeing in the process that the subproblem optima eventually converge after a small number of iterations to or near to the optimum of the original global problem. The aim of this paper is to analyze the criteria for energy system decomposition, in particular with regard to the formulation of the separate subproblems and to the imposition of the constraints that affect the coupling of two or more subsystems. Three general decomposition criteria are identified and discussed with simple examples to let the mathematical formulation be analyzed critically.     

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.     

Fundamentals of tree-shaped networks of insulated pipes for hot water and exergy

This paper outlines recent thermodynamic optimization work on the geometric layout of schemes for distributing hot water and exergy over a large system. Constrained are the amount of insulation material, the volume of all the pipes, and the amount of pipe wall material. Unknown are the distribution of insulation over all the links of the network, and the configuration of the network itself. The main focus is on how the geometric configuration may be selected in the pursuit of maximized global performance, and how closely a non-optimal configuration performs to the highest level. Maximum global performance means minimum heat loss to the ambient, minimum pressure loss, and minimum exergy destruction. Three configurations are optimized:

• (a)an area covered by a coiled steam, where all the users are aligned on the same stream,
• (b)a sequence of tree-shaped flows on square areas in which each area construct is made up of four smaller area constructs, and
• (c)a sequence of tree-shaped flows where each area construct is made up of two smaller area constructs.

It is shown that the tree-shaped designs (b), (c) outperform significantly the coiled stream design (a). The tree designs obtained by pairing (c) are better than the square tree constructs (b) and, in addition, they deliver water at the same temperature to all the users spread over the territory. The fundamental trade off between minimum heat loss and pressure drop, in the pursuit of minimum exergy destruction, pinpoints the optimal size of each duct and insulation shell.     

Wind plant system engineering through optimization of layout and yaw control

Recent research has demonstrated exciting potential for wind plant control systems to improve the cost of energy of wind plants. Wind plant controls seek to improve global wind plant performance over control systems in which each turbine optimizes only its individual performance by accounting for the way wind turbines interact through their wakes. Although these technologies can be applied to existing wind plants, it is probable that the maximum benefit would be derived by designing wind plants with these capabilities in mind. In this paper, we use system engineering approaches to perform coupled wind plant controls and position layout optimizations of a model wind plant. Using several cost metrics, we compare the results of this optimization to the original plant and to plants in which the control or layout is optimized separately or sequentially. Results demonstrate that the benefit of this coupled optimization can be substantial, but it depends on the particular constraints of the optimization. Copyright © 2015 John Wiley & Sons, Ltd.     

Fitting the flow regime in the internal structure of heat transfer systems

This paper explores the idea of selecting the flow regime in the internal structure of heat transfer systems in order to maximize their global performance under global constraints. In fluid distribution networks, we show that the pumping power requirement can be reduced by dividing a turbulent stream into several smaller laminar streams when the surfaces of the pipes are sufficiently rough. We also exemplify that for the packaging of heat-generating plates, in the range of Bejan number between 10⁹ and 10¹¹, spacing can be adjusted to obtain either the laminar or the turbulent regime for maximizing the heat transfer rate density. Scale analysis is used to evaluate the performance of the systems under study.     

Thermodynamic optimization of geometric structure in the counterflow heat exchanger for an environmental control system

This paper shows that the internal geometric configuration of a component can be deduced by optimizing the global performance of the installation that uses the component. The example chosen is the counterflow heat exchanger that serves as condenser in a vapor-compression-cycle refrigeration system for environmental control of aircraft. The optimization of global performance is achieved by minimizing the total power requirement or the total entropy generation rate. There are three degrees of freedom in the heat exchanger configuration, which is subjected to two global constraints: total volume, and total volume (or weight) of wall-material. Numerical results show how the optimal configuration responds to changes in specified external parameters such as refrigeration load, fan efficiency, and volume and weight. In accordance with constructal theory and design [1], it is shown that the optimal configuration is robust: major features such as the ratio of diameters and the flow length are relatively insensitive to changes in the external parameters.     

Tree-shaped networks with loops

Dendritic flow architectures are being contemplated for thermal designs that provide high heat transfer densities for the cooling of electronics. Optimized tree networks maximize the flow access between one point (source, sink) and an infinity of points (line, area, volume). This paper is a fundamental study of a new class of dendritic flow architectures for thermal design: trees combined with closed-loop structures, as in the venation of leaves. The loops provide robustness to the design: the network continues to serve its assigned area even if one or more ducts are damaged. The study documents the achievement of performance and robustness systematically, by starting from the simplest architectures and proceeding toward the more complex, namely, point-circle networks with one loop size and two loop sizes, and networks with loops without and with branching levels. It is shown that the use of loops increases the global flow resistance relative to the dendritic design without loops. Damage, or removal of a duct from the network, also leads to an increase in global flow resistance. These effects become less important as complexity increases, provided that the network is optimized. A damaged peripheral duct induces a smaller penalty than a damaged duct that is situated close to the center of the network. In summary, optimized complex flow structures are robust. Loops are an attractive design feature for maintaining a high level of global performance when the structure experiences local damage.     

Optimization of fractal-like branching microchannel heat sinks for single-phase flows

Fractal-like branching flow networks in disk-shaped heat sinks are numerically optimized to minimize pressure drop and flow power. Optimization was performed using a direct numerical search, gradient-based optimization, and genetic algorithm. A previously validated one-dimensional pressure drop and heat transfer model, with water as the working fluid, is employed as the objective function. Geometric constraints based on fabrication limitations are considered, and the optimization methodology is compared with results from a direct numerical search and a genetic algorithm.The geometric parameters that define an optimal flow network include the length scale ratio, width scale ratio, and terminal channel width. Along with disk radius, these parameters influence the number of branch levels and number of channels attached to the inlet plenum. The geometric characteristics of the optimized flow networks are studied as a function of disk radius, applied heat flux, and maximum allowable wall temperature. A maximum inlet plenum radius, minimum interior channel spacing, and ranges of terminal channel widths and periphery channel spacing are specified geometric constraints. In general, all geometric constraints and the heat flux have a significant influence on the design of an optimal flow network. Results from a purely geometrically derived network design are shown to perform within 15% of the direct search and gradient-based optimized configurations.     

Optimal operation of autonomous microgrid including wind turbines

This paper gives a novel hybrid optimization method to find optimal sitting and operation of an autonomous MG at the same time. The operation is optimized via finding the optimal droop gain parameters of DGs. The optimization problem is formulated as a multi-objective problem where the objectives are applied to minimize the fuel consumption of DGs and to improve the voltage profile and stability of MG subject to operational and security constraints. A hybrid algorithm, named HS-GA, is developed to solve the paper optimization problem. A new formulation of power flow is derived to run the proposed algorithm where the steady state frequency of system, reference frequency, reference voltage and droop coefficients of DGs, based on a droop controller, are considered as optimization variables. The performance of the paper approach is compared with other optimization and non-optimization methods in MG with 33and 69 buses using MATLAB. The performance of the proposed method is compared with a method that the parameters of DGs are pre-determined without conducting any optimization process. The results show, which optimized droop parameters improves the operation of the MG.     

Development of tree-shaped flows by adding new users to existing networks of hot water pipes

This paper describes the optimization of a tree-shaped system of insulated pipes for the distribution of a stream of hot water over an area. The area is covered uniformly by users who must receive the same flow rate of hot water. The network of pipes is developed in steps. Each step consists of attaching to an existing network an extension (one new user) that is placed in the position that maximizes the temperature of the water received by the new user. The network grows `one-by-one', i.e., by one new user at a time. Networks with up to 16 users are optimized in this manner, and their geometric features and thermo-fluid performance are documented. These one-by-one trees of hot water flows are compared with corresponding `constructal' trees that are obtained in steps of pairing (doubling), i.e., connecting together two identical area constructs of the same size. It is shown that although the constructal trees perform the best (uniform water delivery at the highest temperature), the one-by-one trees approach the same level of performance as they become more complex. It is also shown that the geometry of the insulated tree structure is relatively insensitive to how the insulation is distributed over all the pipes. The thermal performance of the structure is relatively insensitive to how finely the distribution of pipe sizes and insulation radii is optimized.     

The effect of size on efficiency: Power plants and vascular designs

In this paper we use thermodynamics to show why larger flow systems are more efficient than smaller flow systems. This trend is visible across the board, from power generation and refrigeration, to vascular design and animal design. The reason is that larger systems have larger flow passages and heat transfer surfaces, and do not strangle the flow of the currents that must flow. Three fundamental examples show how to predict this trend: a power plant with fluid friction and finite heat transfer area, a vascular body with building blocks optimized at every level of assembly, and a vascular body designed based on a duct-pairing algorithm. The examples show that the performance improves as the size increases, and that the architecture changes with the size. These constructal-design features constitute the basis for scaling up and scaling down the configurations of flow systems, from desktop models to life size installations.