3.1. Problem statement
In this paper a direct numerical simulation is undertaken to obtain a correlation for fluid flow over a spherical particle. More precisely, we will develop an equation for Nusselt number (Nu) in terms of Reynolds number (Re) and Prandtl number (Pr).
Introducing corrective factors for the Nusselt number, to take account of all these effects, the Nusselt number is often written in the following form:
With:
In which, µ, and are, respectively, the plasma density, dynamic viscosity, and heat capacity of the plasma, and is the velocity of the plasma approaching the particle.
The coefficient a, c, m, n and w are numerically calculated in respect to the fluid and the flow geometry. For low Reynolds numbers and for spherical particles their values remain preserved [18].
A model with the appropriate settings has been proposed for correlation. The particle diameter is 50 µm, the plasma mid-temperature is in the range of 1100-9100 K, and the particle surface temperature remains constant (300 K). The inlet velocity and inlet temperature are variable parameters in the model and calculate the heat flow rate; from the particle to the ambient fluid.
The values of Nu, Re, and Pr are then calculated for each case, and the parameters C, m, n and w are determined by a fit.