Explanation:

In a mass transfer process, the temperature and concentration profiles represent the distribution of temperature and concentration across the system. These profiles are influenced by various factors, including the properties of the fluids involved and the transport mechanisms.

In the given scenario of diffusion of hot smoke in cold air, the temperature and concentration profiles will become identical when the Lewis number is equal to 1.

Lewis Number:
The Lewis number represents the ratio of thermal diffusivity to mass diffusivity. It is defined as:

Lewis number (Le) = thermal diffusivity / mass diffusivity

The thermal diffusivity (α) is a measure of how quickly heat can be conducted through a material, while the mass diffusivity (D) is a measure of how quickly mass can be transported through a fluid.

Explanation of Options:
a) Prandtl number - 1: The Prandtl number (Pr) represents the ratio of momentum diffusivity to thermal diffusivity. It is not directly related to the temperature and concentration profiles in this scenario.

b) Nusselt number = 1: The Nusselt number (Nu) represents the ratio of convective to conductive heat transfer. It is not directly related to the temperature and concentration profiles in this scenario.

c) Lewis number = 1: As mentioned earlier, the Lewis number represents the ratio of thermal diffusivity to mass diffusivity. When the Lewis number is equal to 1, it means that the rate of heat transfer is equal to the rate of mass transfer. In other words, the thermal and concentration profiles will become identical.

d) Schmidt number = 1: The Schmidt number (Sc) represents the ratio of momentum diffusivity to mass diffusivity. It is not directly related to the temperature and concentration profiles in this scenario.

Conclusion:
The temperature and concentration profiles in the diffusion of hot smoke in cold air will become identical when the Lewis number is equal to 1. This indicates that the rate of heat transfer is equal to the rate of mass transfer.