Model for heat transfer in a honey bee swarm

Abstract

During spring, it has been observed that several thousand bees leave their hive, and settle on some object such as a tree branch. Some of the scout bees search for a suitable place where a new hive can be set up, while the rest collect together to form a swarm. Heinrich (J. of Exp. Biology 91 (1981) 25; Science 212 (1981) 565; Scientific American 244:6 (1981) 147) has done some experiments with free and captive swarms. His observations are as follows. (1) The core (centre) temperature is around 35°C irrespective of the ambient temperature. (2) The mantle (outer surface) temperature exceeds the ambient temperature by 2- 3°C, provided the ambient temperature is greater than 20°C. Otherwise the mantle temperature is maintained around 17°C. (3) The temperature gradient vanishes just before take-off of the swarm. The present work attempts to predict temperature profiles in swarms and compare them with the data of Heinrich. A continuum model involving unsteady heat conduction and heat generation within the swarm is used. Heat loss from the outer surface of the swarm by free convection and radiation is accounted for approximately. To simplify the analysis, internal convection within the swarm is neglected. The energy balance equation is solved using the finite element method. The effective thermal conductivity (k) is determined by comparing model predictions with data for a swarm of dead bees. The estimated value of k is 0,20 W/m-K. Both spherical and a non-spherical axisymmetric shapes are considered. Considering axisymmetric swarms of live bees, temperature profiles are obtained using various heat generation functions which are available in literature. The effective thermal conductivity is assumed to be the same as that for the swarm of dead bees. Results based on a modified version of Southwick’s heat generation function (The Behavior and Physiology of Bees, pp. 28-47, 1991) are qualitatively in accord with the data. The predicted maximum temperature within the swarm and the temperature at the lower surface of the swarm at the ambient temperature of 5°C are 34°C and 17-20°C, respectively. These are comparable to the measured values of 36°C and 19°C. The predicted maximum temperature within the swarm and the temperature at the lower surface of the swarm at the ambient temperature of 9°C are 36.5°C and 17-22°C, respectively. These are comparable to the measured values of 35°C and 19°C. The predicted oxygen consumption rates are 2.55 ml/g/hr for a swarm of 5284 bees at an ambient temperature Ta = 5°C and 1.15 ml/g/hr for 16,600 bees at Ta = 9°C. These are of the same order as the measured values (2 ml/g/hr for 5284 bees at Ta - 4.4°C and 0.45-0.55 ml/g/hr for 5284 bees at Ta = 10°C). Omholt and L0 nvik (J. of Theoretical Biology, 120 (1986) 447) assumed a non-uniform steady state profile and used it to estimate the heat generation function. Using this function in the transient energy balance, it is found that their steady state profile is unstable.