## 1 Introduction
Applying the aforementioned conditions forced growth in the content of the liquid fraction present as droplets in the core of the flow, which is a desirable effect. A uniform coverage of the surface of the throat cross-section with droplets implies an increase in cleansing efficiency [1]. The multiphase flow in the Venturi nozzle has been studied empirically and analytically for several decades, as per the available literature [5; 6; 7; 8; 10; 11; 12; 13; 14]. The main objective of the theoretical and empirical studies being conducted by researchers is to examine the efficiency of gas cleaning when employing Venturi scrubbers. In the construction of Venturi scrubbers, there are three different methods of liquid injection: as a liquid film, as a jet, and as a spray. An ejector Venturi gas scrubber employs a high-speed scrubbing liquid spray to effectively remove dust, chemicals, and other particulates from the gas [15; 16; 17]. Adsorption liquid injection is usually used as a jet in the Pease-Anthony Venturi scrubber [6; 18; 19; 20; 21; 22]. The investigation on the Venturi scrubbers with liquid injection at the confusor inlet, either by liquid film or jet, has also been carried out in the following works [5; 21; 23; 24; 25; 26]. A mathematical description of the trajectory of the injected liquid was presented by [PERSON] et al. [6]. The model takes into account how the liquid stream penetrates the gas stream and then separates. By injecting the liquid through a hole that was perpendicular to the throat, it was possible to study how the penetration of the liquid stream affects droplet dispersion. By ensuring the liquid is properly distributed as small droplets, it is possible to achieve greater gas dedusting efficiency with minimal water consumption. In addition, the experiments were conducted on throat lengths that varied. Despite a wide application of the mathematical model in different types of scrubbers, it remains not applicable to scrubbers applied for gas dedusting. The focus of the article [18] was to analyze the droplet density distribution in the horizontal Venturi scrubber with a rectangular cross-section through both theoretical and experimental techniques. In addition, the Venturi Pease-Anthony scrubber \((VPAS)\), which is used to remove micron-sized particles, was considered. To address the problem, the mathematical representation of the motion of the liquid jet was adopted from [6]. The Venturi scrubbers, where the liquid stream is sprayed perpendicular to the throat with a subsonic gas flow \((VPAS)\), are applicable in various industrial processes such as combustion, evaporative cooling, or in medical and agricultural devices. The topic of liquid atomization, droplet size distribution, and droplet existence in the core of the throat flow is important in the purification process and is still being researched. The scrubbers with fluctuating droplet concentrations are less efficient than in the opposite scenario. The spatial distribution of the droplets and their sizes were noted to have a significant impact on the efficiency of pollutant removal as per the article [13]. The pressure drop is a measure of the dynamics of mechanisms during the gas dedusting process in the Venturi scrubber, regardless of the method of the liquid injection. One of the simplest models for predicting the pressure drop was proposed by [PERSON] [27]. A one-dimensional, incompressible, and adiabatic flow is the model's assumption, and the atomized liquid is characterized by a uniform droplet size distribution. In addition, the [PERSON] model assumes that the droplet velocity is equal to the gas velocity at the end of the throat while neglecting the gas acceleration and frictional pressure drop. [PERSON] [28] made improvements to the [PERSON] model by assuming that the droplet velocity at the end of the throat would not be equal to the gas velocity. [PERSON] [29] added droplet deceleration in a converging channel to enhance the model addressed.