Development of Euler-Lagrange Code Based on Particle Tracking and Single Particle Evaporation Method

Abstract

In this research, the goal is to evaporate a drop of injected fuel in a hot static fluid. All the forces on the particle have been applied correctly and we have also obtained very good results in the validation. Things like the dimensional analysis of forces, the development of the Euler-Lagrangian code, the curve of the forces applied to the droplet in terms of time and also in terms of location, the curve of changes in the Reynolds number of the drops have been fully investigated.

1. Introduction

Evaporation of liquid drops in the form of spray is used in technology such as fuel spray in combustion chambers, evaporative coolers, separation of solid particles from gas, in furnaces, medical engineering, firefighting systems, in cooling nuclear reactors, etc. To model evaporation, there are two droplet models: limited conduction and unlimited conduction. Considering that there is no obvious difference between these two models for small drops, the unlimited conduction model is used.

The discussion of modeling droplet evaporation has been ongoing for decades. Generally, two models of limited conductivity (the temperature inside the droplet is not the same throughout the droplet) and unlimited conductivity (the temperature inside the droplet is assumed to be the same) are used, but in most industrial applications, due to the low importance of the thermal resistance inside the droplet and due to the small size of the droplet, the unlimited conduction model is used.

2. Methodology & Solution

The assumptions used in this model are as follows:

• The droplet is assumed to be a symmetrical sphere.
• The thermodynamic characteristics of the droplet and surrounding hot gas are uniform.
• Effects of gravity, thermal gradient, viscosity losses, etc., are ignored.
• The droplet surface temperature changes with time.

Reynolds number is calculated using equation (1):

(1) \( Re = \frac{2 \rho_{\infty} |U - U_{\infty}| R_s}{\mu_g} \)

The modified Nusselt is calculated from equation (2):

(2) \( Nu^* = 2 + \frac{Nu_0 - 2}{F_T} \)

The exchanged heat is calculated using equation (3):

(3) \( Q_L = \dot{m} \left( \frac{\bar{C}_p F (T_{\infty} - T_s)}{B_T} - L(T_s) \right) \)

3. Flow Simulation and Validation Results

The equations governing the particle are solved using the 2nd order Rang-Kutta method. All the forces acting on the particle are calculated and finally applied in the form of C++ code and verified. Errors were very low.

In order to investigate the effect of the initial velocity of the drop on its movement, two velocities higher than the reference state and two velocities lower than the reference (15 m/s) with a step of 2.5 m/s have been used. As the speed increases, the heat transfer coefficient increases, and it is expected that with the increase in the heat transfer rate, the rate of evaporated mass from the droplet surface will also increase, and the droplet will need less time to evaporate.

In summary, the results are as follows:

• With the increase in the diameter of the droplets, the maximum level of evaporation is increased. For this reason, in applications related to combustion and especially cooling, air liquid should be injected in the form of sprays and drops so that it has the maximum level for evaporation. With the increase in the droplet diameter, although the evaporation increases, the shelf life of the droplet is prolonged. Therefore, in the spray, one should seek to create many droplets with small size so that the maximum level of evaporation increases and the evaporation time decreases.
• With the increase of time and moment, the drag force decreases because with the increase of time, the diameter of the drop and also its speed decrease, which should also decrease with the decrease of these two factors, and so on.