The rate of change of temperature is governed by three physical processes: convective heat transfer, latent heat transfer associated with mass transfer, and radiative heat transfer.

The convective heat transfer is given by:

(6–35)

where is the thermal conductivity of the fluid, and are the temperatures of the fluid and of the particle, and is the Nusselt number given by:

(6–36)

where is the specific heat of the fluid, and and are the fluid dynamic viscosity and thermal conductivity respectively.

For cases including multiphase reactions, the convective heat transfer has a blowing correction based on the rate of mass transfer from the particle:

(6–37)

where is given by

(6–38)

and is the total mass transfer rate of the particle. This modification can be omitted by using the expert parameter setting: pt heat transfer blowing correction = f.

The heat transfer associated with mass transfer is given by the relation:

(6–39)

where the sum is taken over all components of the particle for which heat transfer is taking place. The latent heat of vaporization is temperature-dependent, and is obtained directly from the MATERIALS information for the liquid in the particle and its vapor.

The radiative heat transfer, , for a particle with diameter , uniform temperature , and emissivity , is given by:

(6–40)

where is the Radiation Intensity on the particle surface at the location of the particle, is the Refractive Index of the fluid, and is the Stefan-Boltzmann constant. An equivalent amount of heat can be removed from the radiation field.

The rate of change of temperature for the particle is then obtained from:

(6–41)

where the sum in this equation is taken over all components of the particle including those not affected by mass transfer.

The energy source to the continuous fluid is obtained from:

(6–42)