Mathematical models of evaporation drops in the process of movement in the air volume on the premises

Abstract

Humidification of air by spray water is widely used in agroindustrial production for air conditioning in industrial premises, cooling of air and plants in the hothouse and animals and poultry in premises during periods of stable and high temperature of atmospheric air. When choosing a device for wetting by sawing, it is important to determine which model (type of apparatus) best implements specific agro-and zootechnical requirements. From the production experience, it is known that the best results can be achieved when choosing not just an economic model, but optimal in terms of providing a comfortable mode, the possibility of automation and the accuracy of humidity control. Therefore, the tasks of determining the dynamic characteristics of the process and water separation devices remain relevant.

Today, the problem of sputtering the fluid, the movement and evaporation of dispersed droplets is sufficiently highlighted in modern scientific publications and in individual publications]. The processes of mass exchange when "disassembled" dispersed droplets in a stationary air environment are given little attention. But, the lack of transporting air (concomitant) flow intensifies the process of evaporation of droplets and humidifying the air, which ensures the full evaporation of droplets.

For the formulation of a mathematical model, the motion of a single drop in the form of a material point of mass decreases in the process of displacement (flight) with intensity which depends on the speed of motion and the parameters of the air environment in which the process takes place. To simplify the description, such assumptions are adopted, idealizing the process, but not distorting the general physical picture:

- the drop has the form of a ball and in the process of evaporation it does not deform (the form does not change); drops do not interact with each other;

- the change in mass and, accordingly, the size of the drop occurs by evaporation into the environment, whose parameters do not change with the evaporation of the droplet; the motive force of evaporation accepted the difference in mass concentration of moisture over the surface of the drop (concentration of saturated vapor at a drop temperature);

- the resistance of the movement of the droplet in the air medium depends on the velocity of the drop has a quadratic dependence;

- the temperature of the drop in the process of evaporation does not change and is equal to the temperature of the wet thermometer.

Let's assume that the initial velocity of a drop is equal to the circumferential velocity of the spray disk. The origin of the coordinates on the edge of the disk at the point of departure of the drop from the disk, the angle of inclination of the initial velocity is arbitrary. The axis 0x is directed horizontally, and the axis 0y is vertically down.

The motion of a drop taking into account the mass change by evaporation in an accepted ordinate system in projections on the corresponding axis is described by the differential equations. Let's inspect.

The analysis of the change in the value of the Reynolds number in the flight of spherical particles with a diameter of 10 ... 250 μm and an initial velocity of up to 60 m / s, has shown that the most reliable results in the transition region are the approximation of the coefficient of aerodynamic resistance by the formula L. Klyachka.

The value of the vertical component of the velocity of the droplet, much smaller than the horizontal component and with the Reynolds numbers: expression can be replaced by a simplified dependence.

The results of numerical analysis of the obtained mathematical model carried out in the computer environment MathCAD are represented by the dependences of the change in the size of drops in time (kinetics of evaporation), and the trajectories of motion (displacement of droplets in two-dimensional space) when sprayed with a device with a vertical disk nozzle  are shown in Figure

For an approximate determination of the parameters of the motion of the front of the jet of the spray liquid (velocity, distance, time), the method of motion of an isolated drop, with systems of equations  under simplicity assumed: analytical methods can be obtained . Expressions for projections of speed and displacement by coordinates.

Thus, the resulting dependencies allow you to select the parameters of the spray: the dispersion, the departure angle and the initial velocity that provide uniform humidity of air or the required surface (plane).

Conclusions

1. On the basis of the theoretical analysis of the mechanic-technological bases, a mathematical model of drop ballistics evaporating during flight in the indoor air environment has been formulated. The model takes into account the change in the drag coefficient in the process of moving the drop, which significantly improves the accuracy of calculations.

2. To analyze the calculations of the trajectories of motion, you can use simplified dependencies.