The tree of convective heat streams: its thermal insulation function and the predicted 3/4-power relation between body heat loss and body size

Two parallel fluid streams in counterflow and in close thermal contact convect an energy current longitudinally, in the direction of the warmer stream. This paper describes the flow of convective heat currents through two identical tree networks superimposed, and oriented in counterflow. The convective currents form a single tree. The fluid flow is laminar through straight tubes. The ratios of successive tube diameters and tube lengths are deduced from the minimization of flow resistance subject to two constraints, total volume and total tube volume. It is shown that the total heat current convected by the double tree is proportional to the total volume raised to power 3/4. The conclusion is that the resistance to the loss of body heat governs the relation between metabolic rate and body size in all animals.