The air freshener shown in the figure below is used to deliver a pleasant vanilla fragrance in a room. The device consists of a rod 1 cm in diameter and 5 cm in length. The rod is composed of a solid mixture containing a volatile aromatic substance that smells like vanilla.
A heating element in the center of the rod, 0.5 cm in diameter, maintains a constant surface temperature of 40°C to evaporate the aromatic substance. Air at 25°C flows perpendicular to the rod at a velocity of 0.2 m/s, and the aroma vapors are immediately diluted to a very low concentration.
The density of the gel is 1:1 g/cm3, and the initial loading of the aromatic compound in the solid is 40 mol%. The diffusion coefficient of the volatile substance generating the aroma, which approximates the properties of benzaldehyde, is 0:08 cm2/s in air at 40°C. The vapor pressure of this substance is
428 Pa at 408°C, and its molecular weight is 106 g/gmol. Assume that the release of the aromatic compound is controlled by the convective transport across the gas film surrounding the rod, and that the diameter of the rod remains constant. Neglect internal diffusion of the aroma substance within the rod itself.
a. What is the initial rate of aroma delivery to the surrounding in grams/hour?
b. How long can the air freshener provide fragrance?