Fig. 14 Variations with (a) gas phase kinetic factor and (b) liquid spray density of the rotational flow ratio for the gas-liquid phase in spray distribution.
Fig. 14a, b shows the variations of the gas-liquid phase rotational flow ratio with the liquid spray density and gas-phase kinetic factor, respectively. In Fig. 14a, the trend of the gas-phase rotational flow ratio with an increasing gas-phase kinetic energy factor is similar to that under the overflow distribution, which increases slightly and then decreases significantly. The ratio is also greater than 0.6, and the action mechanism is similar to that under the overflow distribution. The transition point of the FJF and JMF is F s = 1.6 (m/s*(kg/m3)0.5), indicating that the transition of the two flow patterns is only affected by the gas phase. However, the liquid rotational flow ratio shows a linear downward trend without a transition because, under the spray distribution, the liquid spray density is large. The liquid phase pushed by the airflow will be supplemented immediately; therefore, the transition of the rotational flow ratio in the two flow patterns is buffered, and the declining trend of the rotational flow ratio for the liquid phase is more stable.
As shown in Fig. 14b, the rotational flow ratio for the gas phase remains unchanged, above 0.6, with the increasing spray density for the liquid phase. The liquid rotational flow ratio shows a slight increase and remains below 0.4 under the spray distribution. Because of the large liquid phase volume, the gas-liquid phase load at the sieve holes are saturated. The liquid phase cannot produce more perforated flow with the increasing liquid spray density. Only rotational flow is shown, resulting in a gradual increase in the rotational flow ratio of the liquid phase. However, the gas-phase rotational flow ratio is dominated by the gas phase, and the liquid phase has minimal influence. The change is relatively stable.

3.4.3. Prediction model for the rotational flow ratio

A prediction model for the rotational flow ratio is required for the selection and regulation of suitable conditions in future engineering applications. Therefore, the mathematical modeling of the rotational flow ratio of the gas-liquid phase under overflow and spray distribution is carried out in this section.
3.4.3.1. Prediction model in overflow distribution
As previously discussed, the change of the rotational flow ratio is mainly affected by the operating conditions of the gas-liquid phase. To facilitate mathematical modeling, the gas and liquid Reynolds number and Weber number are applied to analyze the changes.
The flow of the liquid phase on the guide plate is falling film flow under the overflow distribution for the liquid phase32,33. In addition, the Reynolds number for the liquid phase can be expressed as:
, (5)
where the liquid phase density isρ l; the liquid film thickness is δ ; the average velocity for the liquid phase is u m, and the liquid phase dynamic viscosity is μ l.
In the vertical direction, one side of the liquid film is in contact with the guide plate, and the other side is in contact with the atmosphere. The boundary conditions can be expressed as follows.
Guide plate side: (6)
Atmosphere side: (7)
Here, τ is the shear stress in the flow direction, and y is the distance between the calculated point and guide plate.
According to the characteristics of falling film flow and Newton’s law of momentum conservation and shear stress, the velocity distribution equation of the liquid film can be obtained:
, (8)
where is the angle between the inclined plate and vertical direction in the liquid falling film flow. In this experiment, , and the average velocity of the liquid film can be obtained from the velocity distribution equation.