What is the \(d^{2}\)-law for droplet evaporation? What are the assumptions that have to be satisfied for this law to be valid?

The \(d^2\)-law states that the evaporation rate of a droplet is proportional to the square of its diameter, i.e., \( \frac{d^2}{dt} = -k \), where \(d\) is the droplet diameter, \(t\) is the time, and \(k\) is the evaporation rate constant.

The \(d^2\)-law provides a simple way to describe droplet evaporation dynamics which is important in various industrial processes, such as spray drying, combustion, and cloud formation. It offers a useful quantitative measure of how the droplet size changes over time due to evaporation.

There are several assumptions that must be satisfied for the \(d^2\)-law to be valid. These include: 1. The evaporation process is diffusion-limited, meaning that the vapor concentration at the droplet surface is in equilibrium with the liquid, and the main resistance to mass transfer is in the surrounding gas phase. 2. The droplet's temperature remains constant throughout the evaporation process (i.e. it is isothermal). 3. The droplet shape remains spherical, and its density and surface tension do not change significantly during evaporation. 4. The surrounding gas is stagnant, and there are no external forces (e.g., gravity, electric fields) acting on the droplet. 5. The mass transfer of vapor away from the droplet's surface is not influenced by any chemical reactions, condensation, or coalescence with other droplets. In conclusion, the \(d^2\)-law for droplet evaporation states that the evaporation rate is proportional to the square of the droplet's diameter. It is a simplified model based on several assumptions, which if satisfied, provide a useful description of the evaporation process in various applications.