Constructal dendritic configuration for the radiation heating of a solid stream

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. (exergy destruction, entropy generation). The contribution is that to optimize "thermodynamically" is the same as reducing the consumption of fuel. © 2010 American Institute of Physics.

Plantar Loading During Cutting While Wearing a Rigid Carbon Fiber Insert.

Context : Stress fractures are one of the most common injuries in sports, accounting for approximately 10% of all overuse injuries. Treatment of fifth metatarsal stress fractures involves both surgical and nonsurgical interventions. Fifth metatarsal stress fractures are difficult to treat because of the risks of delayed union, nonunion, and recurrent injuries. Most of these injuries occur during agility tasks, such as those performed in soccer, basketball, and lacrosse. Objective : To examine the effect of a rigid carbon graphite footplate on plantar loading during 2 agility tasks. Design :  Crossover study. Setting : Laboratory. Patients or Other Participants : A total of 19 recreational male athletes with no history of lower extremity injury in the past 6 months and no previous metatarsal stress fractures were tested. Main Outcome Measure(s) :  Seven 45° side-cut and crossover-cut tasks were completed in a shoe with or without a full-length rigid carbon plate. Testing order between the shoe conditions and the 2 cutting tasks was randomized. Plantar-loading data were recorded using instrumented insoles. Peak pressure, maximum force, force-time integral, and contact area beneath the total foot, the medial and lateral midfoot, and the medial, middle, and lateral forefoot were analyzed. A series of paired t tests was used to examine differences between the footwear conditions (carbon graphite footplate, shod) for both cutting tasks independently (α = .05). Results : During the side-cut task, the footplate increased total foot and lateral midfoot peak pressures while decreasing contact area and lateral midfoot force-time integral. During the crossover-cut task, the footplate increased total foot and lateral midfoot peak pressure and lateral forefoot force-time integral while decreasing total and lateral forefoot contact area. Conclusions : Although a rigid carbon graphite footplate altered some aspects of the plantar- pressure profile during cutting in uninjured participants, it was ineffective in reducing plantar loading beneath the fifth metatarsal.