Optimization of tree‐shaped flow distribution structures over a disc‐shaped area

In this paper, we review the fundamental problem of how to design a flow path with minimum overall resistance between one point (O) and many points situated equidistantly on a circle centred at O. This is a fundamental problem in energy engineering: the distribution of fluid, energy, electric power, etc., from points to surrounding areas. This problem is also fundamental in heat transfer and electronics cooling: how to bathe and cool with a single stream of coolant a disc‐shaped area or volume that generates heat at every point. This paper outlines, first, a direct route to the construction of effective tree‐shaped flow structures. The starting point is the optimization of the shape of each elemental area, such that the length of the flow path housed by the element is minimized. Proceeding towards larger and more complex structures—from elements to first constructs, second constructs, etc.—the paper develops tree‐shaped flow structures between one point and a straight line, as an elemental problem, and a circle and its centre. We also consider the equivalent tree‐shaped networks obtained by minimizing the pressure drop at every step of the construction, in accordance with geometric constraints. The construction method is applied to a fluid flow configuration with laminar fully developed flow. It is shown that there is little difference between the two methods. The minimal‐length structures perform very close to the fully optimized designs. These results emphasize the robustness of optimized tree‐shaped flows. Copyright © 2003 John Wiley & Sons, Ltd.     

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 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.     

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.     

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.     

Optimization of tree-shaped fluid networks with size limitations

In this paper, we show how to minimize the pumping power requirement in a fluid tree-shaped network under different size constraints (volume, surface, length). The Lagrange multiplier method is applied to obtain a problem formulation in which the pipe diameters do not appear explicitly. It is found that such a formulation exists for both volume and surface constrained networks. In Y-shaped junctions, optimal angles of branching and diameter ratios are determined. A different approach aiming at minimizing the network global cost (summation of size and pumping costs) is presented. It is showed that the geometrical features of the network are the same when one minimizes the global cost rather than minimizing pumping power under constraint. An optimal allocation of cost between pumping and size limitation was found. Finally, we extend the global cost minimization approach to the design of a porous architecture. This article provides fundamental tools for the designers of fluid tree-shaped networks.     

Optimal design of tree-shaped water distribution network using constructal approach: T-shaped and Y-shaped architectures optimization and comparison

This paper presents an original approach for the optimization of drinking water distribution networks. A multi-scale design of point-to-points networks using the Lagrangian function coupled with the constructal approach is performed. Two tree-shaped networks (T- and Y-) are optimized and compared by taking diameters of pipes and bifurcation angles in the network as selected design variables. The objective is to minimize the total head losses (factor of energy consumption) subject to the overall water residence time (factor of water quality). Y-shaped structures are found better than T-shaped structures.     

Optimal tree-shaped networks for fluid flow in a disc-shaped body

In this paper we consider the fundamental problem of how to design a flow path with minimum overall resistance between one point (O) and many points situated equidistantly on a circle centered at O. The flow may proceed in either direction, from the center to the perimeter, or from the perimeter to the center. This problem is an integral component of the electronics cooling problem of how to bathe and cool with a single stream of coolant a disc-shaped area or volume that generates heat at every point. The smallest length scale of the flow structure is fixed (d), and represents the distance between two flow ports on the circular perimeter. The paper documents a large number of optimized dendritic flow structures that occupy a disc-shaped area of radius R. The flow is laminar and fully developed in every tube. The complexity of each structure is indicated by the number of ducts (n₀) that reach the central point, the number of levels of confluence or branching between the center and the perimeter, and the number of branches or tributaries (e.g., doubling vs. tripling) at each level. The results show that as R/d increases and the overall size of the structure grows, the best performance is provided by increasingly more complex structures. The transition from one level of complexity to the next, higher one is abrupt. Generally, the use of fewer channels is better, e.g., using two branches at one point is better than using three branches. As the best designs become more complex, the difference between optimized competitors becomes small. These results emphasize the robustness of optimized tree-shaped networks for fluid flow.     

Constructal tree-shaped flow structures

This paper is an introduction to a new trend in the conceptual design of energy systems: the generation of flow configuration based on the “constructal” principle that the global performance is maximized by balancing and arranging the various flow resistances (the irreversibilities) in a flow system that is free to morph. The paper focuses on distribution and collection, which are flows that connect one point (source, or sink) with an infinity of points (volume, area, curve). The flow configurations that emerge from this principle are tree-shaped, and the systems that employ them are “vascularized”. The paper traces the most recent progress made on constructal vascularization. The direction is from large-scale applications toward microscales. The large-scale tree-shaped designs of electric power distribution systems and networks for natural gas and water are now invading small-scale designs such as fuel cells, heat exchangers and cooled packages of electronics. These flow configurations have several properties in common: freedom to morph, multiple scales, hierarchy, nonuniform (optimal) distribution of scales through the available volume, compactness and finite complexity.     

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.     

Flow boiling in constructal tree-shaped minichannel network

Flow boiling in constructal tree-shaped minichannel network with an inlet diameter of 4 mm is numerically investigated using a one-dimensional model, taking into consideration the minor losses at junctions. The pumping power requirement, pressure drop, temperature uniformity and coefficient of performance of the constructal tree-shaped minichannel network are all evaluated and compared with those of the corresponding traditional serpentine channel, and the fluid stream undergoes a phase change from saturated liquid to saturated vapor. The effects of the length dimension and top view area (i.e. the path length) on saturated gas–liquid two-phase flow boiling heat transfer in tree-shaped minichannel networks are all analyzed and discussed. The results indicated that, the tree-shaped network configured with length dimension of two is able to maximum flow access; the path length plays a significant role in the determination of flow boiling in tree-shaped minichannel networks. In particular, compared to the traditional serpentine channel, flow boiling in constructal tree-shaped minichannnel network possesses less pressure drop, lower pumping power requirement, better temperature uniformity and higher coefficient of performance (COP).     

Thermal characteristics of tree-shaped microchannel nets with/without loops

Several tree-shaped microchannel networks with/without loops are numerically examined and compared for application in cooling of electronic components. The physical model of microchannel electronic cooling system is set up with tree-shaped networks. The tree-shaped microchannel nets are embedded in a disk-shaped heat sink, which is attached to a chip to remove the heat dissipated by a chip. The effects of total branching level and loops on the thermal and flow performances of heat sink system are investigated numerically. Results show that tree-shaped nets with loops provide a great advantage when the structure experiences accidental damage to one or more channel segments since the loop assures continuity of coolant flow. Under blockage of some branches, the channel networks only experience an increase of pressure drop while maintaining the capability to remove the heat generated by the chip.     

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.     

Svelteness, freedom to morph, and constructal multi-scale flow structures

This paper reviews recent progress on constructal theory and design. The emphasis is on the development of multi-scale, nonuniformly distributed flow structures that offer increased compactness (e.g., heat transfer density). Examples are counterflow heat exchangers with tree-shaped hot and cold streams, and tree architectures on a disc. Every flow system has a property called svelteness (Sv), which is the ratio between its external (global) length scale and its internal length scale (V^1/3), where V is the volume occupied by all the ducts. Emphasis is placed on the development of simple strategies for decreasing the computational cost required by the development of such structures. The generation of multi-scale flow configurations is a process that can be projected on a diagram having global performance on the abscissa and degrees of freedom on the ordinate. This process rules the development (evolution) of all flow configurations for systems with global objective, global constraints and freedom to morph.     

Energy and Mass Transfer Phenomena in Natural Draft Cooling Towers

In this paper, the development of natural draft cooling towers diagnostics is presented. Diagnostic method is based on measurements of velocity and temperature fields of the airflow in the entire surface area of cooling tower and the raised phenomenological model of heat and mass transfer in a selected reference vertical segment of cooling tower. Velocity and temperature fields of the airflow were measured with the aid of a remote control mobile robot unit that was developed to enable measurements in an arbitrary measurement point above the spray zones over the entire cooling tower area. Topological structures of the humid air velocity profiles and temperature profiles above the spray zones were obtained at constant integral parameters of a power plant. Measurement results of temperature and mass flow characteristics of the air and water flows in a selected reference vertical segment of cooling tower are presented in the form of phenomenological dependence. Phenomenological dependence links local cooling tower efficiency, geometrical characteristics of spray elements, and air and water flow rates. In the concluding part, both methods are applied together on a selected segment of cooling tower, and local and integral cooling tower efficiency can be determined.     

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.     

Performance analysis of a proton exchange membrane fuel cell using tree-shaped designs for flow distribution

The performance of a proton exchange membrane (PEM) fuel cell is directly associated to the flow channels design embedded in the bipolar plates. The flow field within a fuel cell must provide efficient mass transport with a reduced pressure drop through the flow channels in order to obtain a uniform current distribution and a high power density. In this investigation, three-dimensional fuel cell models are analyzed using computational fluid dynamics (CFD). The proposed flow fields are radially designed tree-shaped geometries that connect the center flow inlet to the perimeter of the fuel cell plate. Three flow geometries having different levels of bifurcation were investigated as flow channels for PEM fuel cells. The performance of the fuel cells is reported in polarization and power curves, and compared with that of fuel cells using conventional flow patterns such as serpentine and parallel channels. Results from the flow analysis indicate that tree-shaped flow patterns can provide a relatively low pressure drop as well as a uniform flow distribution. It was found that as the number of bifurcation levels increases, a larger active area can be utilized in order to generate higher power and current densities from the fuel cell with a negligible increase in pumping power.     

Conduction tree networks with loops for cooling a heat generating volume

Tree-shaped networks are now being considered as small-scale architectures for high-densities in electronics cooling and fuel cells design. This paper documents the optimization of tree-shaped inserts of high thermal conductivity. The new feature is the presence of loops in the tree canopy. Every feature of the tree-with-loops architecture is optimized numerically. Two classes of trees with loops are considered: loops with one size, and loops with two sizes. The performance of trees with loops is compared with that of trees without loops and designs with purely radial inserts. It is shown that dendrites and loops are features that become attractive as scales decrease and complexity increases. In the same direction, the robustness of tree-with-loops architectures increases.     

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.     

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.