After hygroscopic aerosols are inhaled, the particles will grow in the respiratory tract, depending on their physical properties, their water content, the local relative humidity (RH) and the breathing pattern. The RH is determined by both the water vapor concentration and the temperature of the air. Hence the RH is a function of the transport of water vapor and heat, and the airflow conditions in the upper human airways. In order to obtain more information from a theoretical point of view a method of approximating the local RH has been developed and described previously. This method is used to determine conditions, whether supersaturation occurs in the lung or not. Their influence on particle growth is estimated. Based on model calculations, supersaturation of the air in respiratory tract occurs during the inhalation of cold air (less than 10 degree C) and high RH (nearly saturated). The calculated RH is too large compared with experimental data for the inhalation of air with an initial temperature of 23 degree C. So a considerable but unknown reduction of the saturation has to be expected for the inhalation of air with low temperatures.