This paper addresses the fundamental question of how to position a heat-generating board inside a parallel-plate channel, where it is cooled by forced convection. It is shown that when the board substrate is a relatively good thermal conductor, the best board position is near one of the channel walls, and the worst position is in the middle of the channel. The best and worst positions switch places when the board substrate is a relatively poor conductor. The optimal spacing between a heat-generating surface (uniform temperature, or uniform heat flux) and the insulated wall that completes a parallel-plate channel is reported. Finally, it is shown under what conditions it is advantageous to divide a heat-generating board into two or more equidistant boards inside the same channel, when the total rate of heat generation of all the boards and the channel spacing are fixed.
This paper outlines the thermodynamic optimization of a combined power and refrigeration system subject to constraints. In the first part, the system operates in the refrigeration mode and is driven by a hot stream of single-phase fluid that is subsequently discharged into the ambient. The irreversibility is due to three heat exchangers and the discharging of the used stream. It is shown that the thermodynamic optimum is pinpointed by an optimal ratio between the mass flow rates of the hot stream and the stream that is heated by the hot stream, and by an optimal distribution of the heat exchanger inventory among the three heat exchangers of the installation. The second part of the paper considers the more general situation where the system delivers power and refrigeration, and where the irreversibility is due additionally to the internal parts of the system. It is shown that the thermodynamic optimum is reached by distributing optimally the heat exchanger inventory among the three heat exchangers, and that this optimum is sensitive to the total inventory and the degree of irreversibility of the internal parts. It is also shown that the optimum is robust with respect to changes in several physical parameters.
Here we show how the size of a body affects its maximum average speed of movement through its environment. The theoretical challenge was to predict that ‘outliers’ must exist, such as the cheetah for terrestrial animals and the jet fighter for airplanes. We show that during a travel that starts from rest and continues at cruising speed, the body size for minimum travel time, or maximum average speed, is not the biggest. The results are compared with extensive data for military aircraft for chase, attack and reconnaissance, in addition to data for commercial aircraft. The paper also explains why in earlier studies of flying (animals, airplanes) the airplane data deviated upward (toward greater speeds) relative to the theoretical trend followed by flying animals, and why the fastest animal flyers are one thousand times smaller than the fastest swimmers. Unlike the biggest animals and airplanes (elephant, whale, commercial jet), which move constantly, the fastest animals and airplanes spend most of their lives at rest. When judged for speed averaged over lifetime, the fastest ‘sprinters’ are in fact the slowest movers (as in Aesop’s fable ‘The Tortoise and the Hare’).
In this paper, we demonstrate that asymmetry in fluid distribution tree networks emerges from power requirement minimization, under global volume constraint. We have discovered several levels of asymmetry in optimal trees: different pipe lengths at the same level of branching, different mass flow rates at junctions or bifurcations, and different main branches to build the optimal dendrite. The emergence of asymmetry in optimal tree networks (man-made or natural) is a result of the optimization: it is not an assumption or a modeling feature. The constructal method that we used to discover asymmetry is predictive, and this distinguishes it from descriptive methods such as fractal geometry.
This paper shows that the main geometric features of a flow component can be deduced from the thermodynamic optimization of the global performance of the largest flow system that incorporates the component. This approach represents a departure from the usual approach, where a flow component is optimized in isolation. The example chosen is the counterflow heat exchanger of the environmental control system (ECS) used on modern aircraft. The heat exchanger is fitted with a diffuser and a nozzle for the ram air, and the ECS runs on the boot strap air cycle, employing an additional compressor and turbine. Two heat transfer surface types are considered, finned and smooth parallel plates. Numerical results are reported for the external geometric aspect ratios of the heat exchanger, and for the plate-to-plate spacing of the smooth-plates model. It is shown that the optimized geometry for the core with finned surfaces is nearly the same as the optimized geometry for the core with smooth plates. Several of the optimized geometric features are robust with respect to changes in external parameters that vary from one application to the next. The method illustrated in this paper – the thermodynamic (constructal) optimization of flow geometry – is applicable to any system that runs on the basis of a limited amount of fuel (exergy) installed onboard, e.g., automobiles, ships, portable tools.
This paper describes the fundamentals of melting when a shell of phase-change material rides on a heated horizontal cylinder. In the first part of the paper, contact melting theory is used to predict the history of the melting process and, in particular, the time when the remaining ice falls off the cylinder. It is shown that the melting process consists of two distinct regimes, first, an early regime when the cylinder is surrounded by ice and, second, a late regime when the cylinder cuts through the top of the ice shell. The second part describes laboratory measurements that validate the theory. The third part of the paper shows that in the complete cycle that starts with freezing the shell and ends with the contact-melting removal of the shell, there exists an optimal frozen shell thickness such that the cycle-averaged production of ice is maximized.
This paper reports a series of experiments concerning the buckling of a slender fluid layer in a state of longitudinal compression. The experiments consist of floating a layer of highly viscous oil on a pool of water and, manually, compressing the layer from the side. Photographs of the buckled layer show conclusively that the buckling wavelength is largely insensitive to either the rate of compression or the viscosity of the fluid layer. The observations suggest that the buckling wavelength is actually a characteristic length scale (a property) of the fluid layer, in contrast with the buckling theory of purely viscous layers (Buckmaster, Nachman, and Ting, [7]) where the buckling wavelength remains to be determined randomly by initial disturbances.
This article reports a numerical study of the geometric minimization of the resistance to Darcy flow between a finite-size volume and one point. The volume is two dimensional and contains materials with several permeabilities. The optimization starts with the smallest volume subsystem, and proceeds toward larger subsystems (assemblies) until the given volume is covered. It is shown that at every scale the geometric shape of the subsystem can be optimized. This principle allows us to construct the volume-to-point flow path by using assemblies of previously optimized building blocks, hence the "constructal" name for the associated theory of shape and structure formation in natural flow systems. The optimized flow architecture is such that the regions of relatively high permeability form a tree network that is completely deterministic. Every feature of this architecture is the result of a single optimization principle: the geometric minimization of flow resistance subject to constraints.
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.
Scaling arguments and numerical analysis are used to document the transient and steady-state regimes of natural convection in a triangular porous layer cooled from above (along the sloping wall). The numerical simulations are conducted in the high Rayleigh number domain, Ra = 100, 1000, where Ra is the Darcy-modified Rayleigh number based on height, H. The scale analysis predicts the existence of distinct thermal boundary layers if Ra1/2 (H/L) > 1, where H/L is the height/length geometric ratio of the attic-shaped porous layer. The numerical simulations confirm the scaling results, as well as the prediction that the flow consists primarily of an elongated horizontal counterflow driven by the cold wall. In addition, the numerical solutions show the presence of a Be´nard-type instability at high enough Rayleigh numbers. For example, if H/L = 0.2, the instability is present when Ra > 620; this critical Rayleigh number is found to increase as H/L increases.
A numerical study of two-dimensional natural convection in a horizontal water layer heated from below is reported. The density maximum associated with water at 3.98 °C and atmospheric pressure occurs inside the layer, as the top surface is maintained at 0 °C while the bottom surface temperature varies in the range 4 °C–10 °C. Three separate series of numerical simulations document the effect of Rayleigh number, bottom wall temperature and layer horizontal length on the flow pattern and on the net heat transfer rate vertically through the layer. These effects are documented numerically and graphically in the domain104<Ra<3×105, 1/6<H/L<1, where Ra is the Rayleigh number for a fluid with density maximum, and H/L is the (height/length) ratio of the water system selected for analysis. It is found that steady-state convection is present at Rayleigh numbers as low as 2.3×104 when TH =8 °C and H/L=0.5, and as low as 105 when TH =6.3 °C and H/L=0.5. These Rayleigh numbers agree very well with the corresponding critical values recommended by linear stability analysis.
In this article we show numerically that the entire flow geometry of a vertical diverging or converging channel with laminar natural convection can be optimized for maximal heat transfer rate density (total heat transfer rate per unit of flow system volume). The geometry is free to change in three ways: (1) the spacing between the walls, (2) the distribution of heating along the walls, and (3) the angle between the two walls. Numerical simulations cover the Rayleigh number range 105 ≤ RaH ≤ 107, where H is the channel height. Nonuniform wall heating is modeled as an isothermal patch of varying height H 0 (≤H) on each wall, which is placed either at the bottom (entrance) end of the channel, or at the top (exit) end. The results confirm that the use of upper unheated sections enhances the chimney effect and the heat transfer. The new aspect is that the heat transfer rate density decreases because the unheated sections increase the total volume. It is shown that for maximal heat transfer rate density it is better to place the H 0 sections at the channel entrance. It is also shown that the optimal angle between the two walls is approximately zero when Ra H is large, i.e., for maximal heat transfer rate density the walls should be parallel or nearly parallel. Finally, the optimized spacing (1) developed in the presence of (2) and (3) as additional degrees of freedom is of the same order of magnitude as the optimal spacing reported earlier for parallel isothermal walls, i.e., in the absence of features (2) and (3). The robustness of the optimized flow architecture is discussed. Additional degrees of freedom and global objectives that may be incorporated in this constructal approach are the curvature of the facing walls and the mechanical strength and stiffness of the confining walls. This work was supported by a doctoral fellowship from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and a research grant from the National Science Foundation.
Here, we show how the performance of a paddle that pushes a fluid can be increased by making parallel slits through it. The slit spacing is varied to see its effect on the drag force and the maximum stress in the paddle. The effect of water speed and paddle dimensions is documented. Designs with one or more slits are investigated. The drag force is maximum when the slit spacing matches the boundary layer thickness of the flow through the slit. Furthermore, the drag force is greater when the slit spacing is nonuniform: larger in the central slits than in the peripheral slits. The paddle with slits of one size performs almost as well as the best design with nonuniform spacings. The paddle design with slits achieves the same drag force and maximum stress with less material compared with a paddle without slits.
Heat transfer through a vertical skin surface covered with perpendicular hair strands of uniform density is investigated numerically. The heat transfer rate is the result of (1) direct heat transfer to the air that makes contact with the skin and (2) the heat conducted by each strand away from the skin. The hair strand and its surrounding air are not in local thermal equilibrium. Hair strands have the desirable effect of slowing the air that sweeps the vertical surface and the undesirable effect of acting as fins, thereby augmenting the overall heat transfer rate. Two distinct air flow models are considered: the Darcy model and the Forchheimer-Brinkman extended Darcy model. The overall heat transfer charts reported in this paper show that heat transfer rate can greatly exceed the estimate based on the traditional homogeneous porous medium model. By means of numerical examples, the Darcy model is shown to be adequate for modeling air flow through mammal hair.
The objective of the Journal of Mechanics is to provide an international forum to foster exchange of ideas among mechanics communities in different parts of world.
The main objective of this chapter is to review the fundamentals of three simple methods of problem solving and presentation of convection in porous media: scale analysis, heat lines, and the intersection of asymptotes method. Another objective is to draw attention...
No abstract is provided for this article.
La loi Constructale est une loi de la physique qui presente la tendance naturelle de tout systeme d’ecoulement (anime comme inanime) a evoluer vers des configurations offrant progressivement un acces plus facile aux ecoulements dans le temps. Les domaines scientifiques couverts par la loi Constructale sont tres vastes. Ils concernent aussi bien les ecoulements de fluide que les transferts de chaleur ou de masse et font que la loi Constructale trouve parfaitement sa place dans la thermodynamique.
