15.4.1 Inert Heating or Cooling (Law 1/Law 6)

The inert heating or cooling laws (Laws 1 and 6) are applied when the particle temperature is less than the vaporization temperature that you define, $T_{\rm vap}$, and after the volatile fraction, $f_{v,0}$, of a particle has been consumed. These conditions may be written as

Law 1:

Law 6:

where $T_p$ is the particle temperature, $m_{p,0}$ is the initial mass of the particle, and $m_p$ is its current mass.

Law 1 is applied until the temperature of the particle/droplet reaches the vaporization temperature. At this point a noninert particle/droplet may proceed to obey one of the mass-transfer laws (2, 3, 4, and/or 5), returning to Law 6 when the volatile portion of the particle/droplet has been consumed. (Note that the vaporization temperature, $T_{\rm vap}$, is an arbitrary modeling constant used to define the onset of the vaporization/boiling/volatilization laws.)

When using Law 1 or Law 6, ANSYS FLUENT uses a simple heat balance to relate the particle temperature, $T_p (t)$, to the convective heat transfer and the absorption/emission of radiation at the particle surface:

where
	$m_p$	=	mass of the particle (kg)
	$c_p$	=	heat capacity of the particle (J/kg-K)
	$A_p$	=	surface area of the particle (m $^2$)
	$T_{\infty}$	=	local temperature of the continuous phase (K)
	$h$	=	convective heat transfer coefficient (W/m $^2$-K)
	$\epsilon_p$	=	particle emissivity (dimensionless)
	$\sigma$	=	Stefan-Boltzmann constant (5.67 x 10 $^{-8}$ W/m $^2$-K $^4$)
	$\theta_R$	=	radiation temperature, $(\frac{G}{4\sigma})^{1/4}$

Equation  15.4-3 assumes that there is negligible internal resistance to heat transfer, i.e., the particle is at uniform temperature throughout.

$G$ is the incident radiation in W/m $^2$:

where $I$ is the radiation intensity and $\Omega$ is the solid angle.

Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model dialog box.

Equation  15.4-3 is integrated in time using an approximate, linearized form that assumes that the particle temperature changes slowly from one time value to the next:

As the particle trajectory is computed, ANSYS FLUENT integrates Equation  15.4-5 to obtain the particle temperature at the next time value, yielding

where $\Delta t$ is the integration time step and

and

ANSYS FLUENT can also solve Equation  15.4-5 in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See this section in the separate User's Guide for details.

The heat transfer coefficient, $h$, is evaluated using the correlation of Ranz and Marshall [ 284, 285]:

where

	$d_p$	=	particle diameter (m)
	$k_{\infty}$	=	thermal conductivity of the continuous phase (W/m-K)
	Re $_d$	=	Reynolds number based on the particle diameter and
			the relative velocity (Equation  15.2-3)
	Pr	=	Prandtl number of the continuous phase ( $c_p \mu / k_{\infty}$)

Finally, the heat lost or gained by the particle as it traverses each computational cell appears as a source or sink of heat in subsequent calculations of the continuous phase energy equation. During Laws 1 and 6, particles/droplets do not exchange mass with the continuous phase and do not participate in any chemical reaction.