Efficiency Optimization of an Annular-Nozzle Air Ejector under the Influence of Structural and Operating Parameters

Abstract

The efficiency of annular-nozzle ejectors serving as components of complex technical systems interacting with high-temperature media in engines and in the field of energy technologies is not linearly related to the gas-dynamic characteristics of the flows formed in the device. In this paper, we have analyzed the results of numerical and experimental studies of gas jets in an annular-nozzle air ejector. The regression equations built according to the circumscribed central composite design described the relationship between a pressure drop and the structural parameters of the nozzle with the speed and mass flow rates of the airflows, including error rates of no more than 15 percent. A two-factor optimization based on Harington’s generalized desirability function was performed to obtain a relatively accurate estimate of the ejector efficiency under the influence of the structural and operating parameters. An optimization method based on the combination of response surface methodology and the desirability function approach, allowing simultaneous consideration of all responses, made it possible to simultaneously optimize multiple conflicting objectives.

1. Introduction

An ejector is a type of vacuum pump that produces a vacuum by means of the Venturi effect when a working fluid flows through a jet nozzle. The high velocity of the jet flow results in a low pressure in it due to Bernoulli’s principle, thus generating a vacuum and entraining the fluid in a flooded space. Annular-nozzle ejectors have a ring-shaped working nozzle. There are many areas where this ejector is used. One of the most important applications of an annular-nozzle ejector is to spray liquids. This scheme provides better conditions for phase interaction. Currently, ejector devices are key components in many complex systems that are used to intensify energy and mass exchange in the production of energy obtained from renewable sources, in air conditioning and refrigeration equipment, in heat recovery and energy storage systems intended for gas turbine engines, in integrated energy systems, and in other fields [1,2,3,4,5]. Ejectors have a simple structure and high efficiency. The absence of moving parts increases the reliability and durability of devices in aggressive and high-temperature environments. The ejector’s performance is significantly affected by the operating and structural conditions. For certain conditions, there is only one nozzle configuration [6,7]. Hence, in energy conversion techniques, the most promising ones are ejectors equipped with adjustable nozzles, which can be adjusted to a design point to obtain high performance under variable operating conditions [8,9]. The adjustable cross-sectional area of the nozzle allows controlling the gas-dynamic characteristics of the flows.

Many theoretical and experimental studies on the characteristics of the primary and secondary flows occurring in the ejector and the interaction of the flows with the external environment have been undertaken [10,11,12,13,14]. The work [10] presented an investigation of the performance characteristics of a variable area ratio ejector based on the analysis of the experimental data. It proposed a dependence prediction of the critical area ratio in terms of primary and secondary pressure. In [11], the author presented the results of an experimental investigation of the entrainment and thrust augmentation characteristics of ejectors that incorporate annular nozzles. Another study [12] summarized the main findings and trends found in the area of heat-driven ejectors and ejector-based machines used in refrigeration systems. It highlighted the predominance of modeling over experimental work due to the high cost of experimental data. In [13], a semi-empirical model for an effective liquid-gas ejector design was developed based on its hydraulic analysis and experimental data.

Currently, a unified theory that generalizes the results of studying the processes in flows formed by ejectors is still being developed. Therefore, attempts are being made to study these processes using modern numerical analysis tools, mainly to optimize the structural parameters of ejectors and achieve the highest entrainment ratio [14,15,16,17,18,19,20,21,22,23,24,25]. In [14,15], the authors proposed Huang’s model and Chen’s model. In both cases, there is some hypothetical effective throat where the speed of the secondary flow reaches its maximum value. Just beyond this section of the throat, the mixing process of the primary and secondary streams begins under uniform pressure. Ghadai et al. [16] performed an extensive numerical study on the performance of the cutting fluid flow through gradual and sudden converging nozzles in order to obtain their optimum performance and the best effective length. In [17], using computational fluid dynamics (CFD), a jet ejector was optimized by creating a model and analyzing the multi-phase flow model of steam and water, considering the consequences of a possible borderline environment. In [18], the authors proposed a 1D mathematical model that uses an iterative process to obtain the entrainment ratio. Further, a regression model was suggested that combines the operating and geometric parameters involving the entrainment ratio. In [19], a 1D unsteady model was proposed for supersonic single-phase ejectors based on a pipeline analogy having two inlets and one outlet. A characteristic-based junction model combined 1D gas-dynamic equations across computational domains. Hence, the model admits a solution in the critical and sub-critical regimes, including backflows. The authors in [20] presented the numerical results for modeling gas-dynamic processes in an impulse ejector to determine optimal geometric parameters for the given flow rate characteristics. Gas-dynamic equations were solved during the axisymmetric formulation. Since the adjustable nozzle in the varied geometry ejector (VGE) significantly affects the performance of the ejector, theoretical models are being developed to evaluate the VGE’s performance. The authors of [21] proposed a two-dimensional theoretical model based on an adjustable-nozzle theory. The method of characteristics was employed to accurately predict the driving-flow development in the mixing section. In [22,23,24] and in many other studies, thermodynamic and analytical models solve the integral form of the conservation equations to provide operation and structural variables. In [25], the authors established a computational fluid dynamic model of a hydrogen recirculation ejector in order to consider the influence of the ejector’s structural parameters on the boundary layer separation effect and the performance of the ejector. The simulation of the separation phenomenon of the boundary layer inside the ejector provided a more relevant optimization of the ejector geometry. Therefore, analytical and numerical models constitute the two main approaches for the theoretical investigation of the ejectors. The analytical approach is useful for providing suggestions on the preliminary design. The numerical approach is necessary for the local flow structure assessment to refine structural parameters and operating conditions. The focus of most of these studies was the central ejector, in which the primary gas flow is supplied along the axis of the secondary flow. These studies put much emphasis on the effect of various geometrical aspects on the ejector’s performance. However, in some complex technical systems that use high-temperature fluids, such as ramjet engines, gas turbines, and some other applications, the use of annular-nozzle and multi-nozzle ejectors is promising for reducing the length of the mixing chamber and enhancing the entrainment performance [26,27,28,29,30].

The complex process of turbulent mixing of high-speed flows in ejectors during their interaction complicates experimental and theoretical studies aiming to optimize the gas-dynamic characteristics of ejectors. Due to the complexity of the involved flow physics, there is no unified theory of flows occurring in ejectors. Dependencies associating their design with operating conditions remain mostly empirical or semi-empirical. Energy and mass transfer processes occurring in pneumohydraulic systems are characterized by the flow rate and pressure drop of the primary gas flow, which are directly related to the efficiency of such systems. When studying these dependencies, the linear response model is considered inadequate, which necessitates using mathematical models of a higher order. Simultaneously, it is promising to use elements of the orthogonal compositional design theory, which provides for the construction of quadratic models of the hydro-gas-dynamic apparatus characteristics and their optimization.

Nowadays, researchers often use meta-heuristic algorithms based on stochastic search approaches to solve different optimization problems. For instance, in [31], the authors propose a variant of the well-known swarm-based algorithm, the Particle Swarm Optimization, to solve constrained problems using a hybrid meta-heuristic approach to the classical penalty function technique. Moreover, the Pareto algorithm is often used to solve multi-objective optimization problems [32]. Computing the set of all Pareto-optimal solutions is a common task in multi-objective optimization to exclude unreasonable advantages and disadvantages.

However, the mathematical approach to multi-objective optimization based on the Harrington desirability function (HDF) provides a systematic and quantitative method for evaluating and comparing solutions. This function assigns weightages to different objectives based on their importance and calculates a desirability score for each potential solution. The optimal solution is then determined by maximizing the desirability score. This holistic approach makes it advantageous over traditional single-objective optimization techniques, including meta-heuristic algorithms. HDF allows for more flexibility in incorporating subjective judgment and preferences into the optimization process by assigning weightages to different objectives; HDF contributes to explicitly balancing the conflicting objectives, ensuring a more practical and realistic solution. Compared to Pareto optimization, HDF is often considered simpler and easier to implement. It avoids complex Pareto front calculations and allows for focusing on the qualitative interpretation of desirability scores.

The Harrington desirability function offers a practical and scientifically rigorous approach to multi-objective optimization. While Pareto optimization and meta-heuristic algorithms have their own benefits and applications in various optimization problems, the ability of the desirability function to consider multiple response variables holistically makes it a preferred choice.

The present study aims to evaluate the effect of the operational and structural parameters of the annular-nozzle air ejector on its efficiency using quadratic models of the gas-dynamic flow characteristics to perform their multi-objective optimization.

2. Ejector Description and Operating Principle

The model used in this study consists of the lower body and upper body having an entry nozzle, whose inner surfaces form a settling chamber connected to an annular convergent nozzle (Figure 1). The inclination angle of the generatrixes of the conical surfaces is α = 10°.

The possibility of mutual displacement of the upper and lower parts of the body of the annular-nozzle air ejector (ANAE) provides control of geometric parameters, i.e., the gap width of the annular working nozzle, which determine the volumetric flow rate of the energy carrier. Simultaneously, the gap width of the working nozzle is related to the magnitude of the linear displacement of the body parts along their common axis by a linear relationship whose coefficients depend on the taper angle of the working nozzle. The gap width determines the critical section area of the annular nozzle and, accordingly, the flow rate of the energy carrier. We have empirically established that the largest displacement of the body parts can be limited to 0.4 of the smallest diameter of the entry nozzle D.

We equipped the ejector with a tool for adjusting the critical section area of the nozzle, which ensures the constancy and repeatability of the structural configuration of the annular working nozzle. We controlled the nozzle area by moving the upper body of the ejector relative to the lower body along their common axis. Before starting the work, we set the relative displacement of parts (1) and (2) connected using thread (3), providing the necessary dimensions of the annular working nozzle (4). The set position is fixed by aligning the holes (8) and (9) on the upper and lower bodies of the housing with the locking screw (10). The location of the holes allows fixing the linear displacement of the upper and lower parts of the body along the axis every 1/48 of the thread (3) pitch. After adjusting the nozzle geometry, compressed air is supplied through port (6) and distributed in prechamber (5). This air is discharged through the annular slot (4) formed by the conical port of the body (2) and the lip of the entry nozzle (1), through which atmospheric air is drawn by the entrainment (7). The mixture of primary and secondary streams is discharged against atmospheric pressure at the end of the conical port of body (2). In the lower part of body (2), there is a thread used for connecting a mixing chamber. The port for supplying the energy carrier is made radially in the lower body, and the presence of a settling chamber ensures a uniform outflow of the compressed air from the working annular nozzle. Hence, the device design does not comprise complex units that are labor-intensive to manufacture.

3. Collecting Data

According to the mass conservation law, the total gas flow rate at the exit of the annular-nozzle air ejector (G_Σ) comprises the flow rates of the active gas (G₁), the ejected secondary gas (G₂), and the gas entrained by a free stream from the flooded space (G₃).

$$G_{\Sigma }=G_1+G_2+G_3,$$

(1)

In the ejector devices containing mixing chambers, the gas flow (G₃) entrained from outer space equals zero in the balance of mass (1). However, since the gas exhausts freely into the flooded space, the gas jet entrains the surrounding air and thus reduces the volume of the ejected secondary gas (Figure 2). Importantly, the value of the secondary flow is a key characteristic of the ejector operation, and thus it should remain positive.

The gas volume of the secondary and total flow rates (G₂ and G_Σ) was calculated based on the respective flow velocity profiles according to the equation:

$$G_i={\displaystyle \int u dF}.$$

(2)

The G_Σ flow rate in the cross-sections of the mixed flow was calculated according to the experimentally determined flux velocity profiles.

The conical walls of adjacent parts (1) and (2) of the ejector body (Figure 3) form the ejector nozzle, which is a conical convergent channel. The area of the critical section of this nozzle depends on the value of their mutual displacement (h), the angle of inclination of the generating cones (α), and the inner diameter (d₁), corresponding to the position of the critical section A–A.

Preliminary experiments show that the range of 1–4 mm should limit mutual displacement of the ejector body parts along the longitudinal axis.

3.1. Processing the Primary Experimental Data

We used the contact (probe) method to measure the velocity of the air flowing freely from the ejector. The flow velocity was calculated based on the measured total and static pressures [33]. According to the experimental results, the gas jets are completely mixed, and the flow loses its annular shape in the section L = 2.8D.

To investigate the velocity of the airflow generated by the ejector, we used a special experimental setup [34]. The setup consists of ANAE and systems of supply, measurement, and regulation of the air flow. We controlled the flow rate of the active gas by adjusting the area of the critical section of the annular convergent nozzle by changing the value of the mutual axial displacement of parts 1 and 2 of the ejector (Figure 3). The air was supplied through a pipeline having such characteristics as P_p = 15 bar and T₀ = 293 K. To monitor the pressure, we used an exemplary pressure gauge. We controlled the gas supply with a shut-off valve and a reducer. To investigate the airflow velocity, we used a probe method with a pressure sensor (Type RPD-I) and a Pitot tube of a circular cross-section having a diameter of 1 mm. The measurement error was 3–5%. The positioning of the sensor was provided by a coordinate device with an error of no more than 0.1 mm. The information obtained from the sensor was fed to an analog-to-digital converter (ADC). The ADC channels were scanned at a frequency of 100 Hz. The sensor was displaced in the cross-section plane of the jet at a constant speed of 5 mm/s and was controlled by a stepper motor connected to a screw drive shaft. We measured the overpressure in the jet cross-section at a distance of L = 2.8D from the nozzle exit under a pressure drop in the prechamber equal to Npr = 2, 3, and 4 and the area of the critical section of the nozzle corresponding to the axial displacement of the parts of h = 1.0, 2.5, and 4.0 mm. The values of the levels and intervals of variation of the listed factors were determined based on the results of the analysis of preliminary one-factor experiments; they corresponded to the points of the factor space of the circumscribed central composite design. The airflow velocity was determined according to the energy conservation equation (Bernoulli) formulated for a compressible adiabatic flow with an error of no more than 10%.

In order to automate the processing of large arrays of primary experimental data, we have developed a special software program called “Flow Velocity Profile.v.1.6” (Tomsk, Russia). The image of the main window of the program is shown in Figure 4. The software performs the mathematical and statistical processing of the experimental data based on gas-dynamic dependencies, taking into account the compressibility of the gas flow [35], as well as calculation profiles of the total pressure and velocity of the gas flow and volumetric flow rates of ANAE gas flows through the primary and secondary nozzles, including the entrainment ratio.

The initial data for operating the program are measurement data registered by a PC and presented as a text file in the CSV format, which, after loading, are displayed in a tabular and graphical form in the working window of the program. In the “Measurement parameters” dialog box, the coefficients of the polynomial calibration dependence connecting the measured values with the pressure value are entered, including the sampling frequency of the analog-to-digital converter channels (n) and the velocity of the sensor movement in the cross-section of the gas flow jet (v). The sensors were calibrated by a statistical method using a calibration pump both during the experiment and after it. The calibration coefficients in different series differed by no more than 1%. Let us use the first-order polynomial regression as a calibration dependence in the study, although special software allows using up to a fourth-order polynomial regression.

In the “Properties of the working gas” dialog box, we can enter the parameters of the gas used for the calculations, such as a specific gas constant (R), gas density (ρ_g), temperature (T), ambient pressure (p₀), and adiabatic exponent (χ). Furthermore, in the main window, we should enter the effective cross-sectional area of the nozzle and the pressure in the prechamber of the ANAE when the measurements were made.

According to the collected data, the program automatically determines the position of the axis of the gas jet as well as its radius during the forward and reverse passage of the sensor through the flow. In this case, this sensor is assumed to move at a constant speed along a straight line in the cross section of the jet. It is possible to manually filter the collected values.

When significant ranges are selected, the measurement data obtained during the forward and backward passes are combined and statistically processed, which also includes the option of non-linear smoothing of the data array by seven points. A procedure for averaging the experimental data using an interpolation polynomial of the third degree is implemented, which aids in obtaining an updated value y(i) for a given value y(i) and a number of nearby values (..., y(i − 1), y(i), y(i + 1), ...) known to have a random error. Based on the calculation results, the pressure profile in the considered jet section is constructed (Figure 5).

Next, the program calculates the velocity profile, gas flow rates, and an entrainment ratio. The gas flow velocity over the surface of the jet section (2) is integrated numerically by the trapezoidal method.

It is possible to output the primary data and calculation results in numerical and graphical form, as well as export the data in CSV format for further processing in specialized software packages. The use of the developed software can significantly reduce the computer time cost and the complexity of performing routine computational operations, which is especially important in the presence of a significant number of measurements.

3.2. Gas-Dynamics Model of the Ejector

The experimental data allowed for validation of the results of the numerical modeling. The flow rate of the secondary flow (G₂) was determined using the gas velocity fields obtained through numerical modeling. Furthermore, this value was used as a criterion to optimize ANAE modes [36].

The dynamics of gas flows were considered in the approximation of a continuous medium and described by the system of Navier-Stokes equations in the case of a compressible flow:

The flow equation:

$$\mathsf{\rho }\left(\frac{\partial \upsilon }{\partial t}+\upsilon .\nabla \upsilon \right)=-\nabla p+\nabla \nu .$$

(3)

The continuity equation:

$$\frac{\partial \mathsf{\rho }}{\partial t}+\nabla \left(\mathsf{\rho }\upsilon \right)=0.$$

(4)

The energy equation:

$$\frac{\partial E}{\partial t}+\nabla \left[\left(E+p\right)u\right]=\nabla \left(\nu .\upsilon \right)+\nabla \left(\nabla T.\lambda \right).$$

(5)

Here p is the pressure, Pa; ρ is the density, kg/m³; υ is the velocity, m/s; t is the time, s; λ is the thermal conductivity coefficient, W/m·K; T is the flow temperature, K; E is energy, J; ν is the viscous stress tensor.

The elements of the ejector are bounded by rotation surfaces; thus, the design itself and the generated gas flows have an axial symmetry. Air is used as a working gas, whose speed can reach high values. Hence, we can reasonably make the following assumptions:

-

Gas flows are considered to be steady and can be described by equations in the axisymmetric formulation.

-

The working gas is air, as described by the ideal gas model, taking into account its compressibility.

-

For a more convenient presentation, the model may be shown as if rotated by 90°.

In a two-dimensional axisymmetric formulation, the flow equation can be represented as:

$$\begin{array}{l}\frac{\partial }{\partial t}\left(\mathsf{\rho }{\upsilon }_x\right)+\frac{1}{r}\frac{\partial }{\partial x}\left(\mathsf{\rho }{\upsilon }_x^2\right)+\frac{1}{r}\frac{\partial }{\partial r}\left(r\mathsf{\rho }{\upsilon }_r{\upsilon }_x\right)=\\ -\frac{\partial p}{\partial x}+\frac{1}{r}\frac{\partial }{\partial x}\left[r\mathsf{\mu }\left(2\frac{\partial {\upsilon }_x}{\partial x}-\frac{2}{3}\left(\nabla \cdot \overline{\upsilon }\right)\right)\right]+\frac{1}{r}\frac{\partial }{\partial r}\left[r\mathsf{\mu }\left(2\frac{\partial {\upsilon }_x}{\partial r}+\frac{\partial {\upsilon }_r}{\partial x}\right)\right]+F_x,\end{array}$$

(6)

$$\begin{array}{l}\frac{\partial }{\partial t}\left(\mathsf{\rho }{\upsilon }_r\right)+\frac{1}{r}\frac{\partial }{\partial x}\left(r\mathsf{\rho }{\upsilon }_x{\upsilon }_r\right)+\frac{1}{r}\frac{\partial }{\partial r}\left(r\mathsf{\rho }{\upsilon }_r^2\right)=-\frac{\partial p}{\partial r}+\frac{1}{r}\frac{\partial }{\partial x}\left[r\mathsf{\mu }\left(2\frac{\partial {\upsilon }_r}{\partial x}+\frac{\partial {\upsilon }_x}{\partial r}\right)\right]+\\ +\frac{1}{r}\frac{\partial }{\partial r}\left[r\mathsf{\mu }\left(2\frac{\partial {\upsilon }_r}{\partial r}-\frac{2}{3}\left(\nabla \cdot \overline{\upsilon }\right)\right)\right]-2\mathsf{\mu }\frac{{\upsilon }_r}{r^2}+\frac{2}{3}\frac{\mathsf{\mu }}{r}\left(\nabla \cdot \overline{\upsilon }\right)+\mathsf{\rho }\frac{{\upsilon }_z^2}{r}+F_r,\end{array}$$

(7)

where

$$\nabla \cdot \overline{\upsilon }=\frac{\partial {\upsilon }_x^{}}{\partial x}+\frac{\partial {\upsilon }_r}{\partial r}+\frac{{\upsilon }_r}{r},$$

(8)

and the continuity equation:

$$\frac{\partial \mathsf{\rho }}{\partial t}+\frac{\partial }{\partial x}\left(\mathsf{\rho }{\upsilon }_x\right)+\frac{\partial }{\partial r}\left(\mathsf{\rho }{\upsilon }_r\right)+\frac{\mathsf{\rho }{\upsilon }_r}{r}=0$$

(9)

where x and r are axial and radial coordinates; υ_x and υ_r are the axial and radial velocity components.

A perfect gas equation for a compressible flow:

$$p={\mathsf{\rho }}_g\mathrm{R}T,$$

(10)

where p is the pressure, Pa; ρ_g is the density of the gas, kg/m³; R is the specific gas constant, J/kg K; T is the temperature, K.

The jet streams were assumed to be steady and were described by the Reynolds-averaged Navier-Stokes equations. The turbulence model k-ε is adopted to describe the turbulent characteristics of flows.

The equations were discretized by the finite volume method; second-order discretization was used. Let us apply a pressure-based solution of the equations using a coupled algorithm. The commercial software programs SolidWorks.v.2020 (Waltham, MA, USA) and ANSYS Workbench.v.2020 (Canonsburg, PA, USA) were used for calculations. When performing test calculations, the dimensions and shape of the computational domain were optimized to exclude their influence on the resulting fields of gas flow velocities. The influence of the grid dimension on the calculation results was studied by comparing the velocity profiles in the cross-section of the gas jet obtained using grids having different dimensions. The simulation results were validated by comparing the experimental and calculated velocity profiles. The calculation results were found to demonstrate a good agreement with the experimental data; the discrepancy was 9–21%.

4. Optimization of Ejector Operating Parameters

When considering the gas dynamics, the value of the aerodynamic momentum determined by the velocity and gas flow rate has the predominant effect on the operating characteristics of the ejector. Therefore, the objective of optimizing the operating modes of the considered ejector is to establish the range of values of the cross-sectional area of the annular nozzle, defined by its mutual displacement (h) and the pressure drop (Npr). They have the highest values of the gas-dynamic pressure on the jet axis (u) and the secondary gas flow rate (G₂) in combination with a high total flow rate (G_Σ) and a minimum pressure drop (Npr). This combination of characteristics of the annular-jet device is best suited for its use in most applications.

Complex gas-dynamic processes occurring in the ejector require optimizing several experimental characteristics simultaneously. This becomes especially relevant for ejector systems with adjustable nozzle geometry. In this case, a rational combination of nozzle parameters and compressed gas pressure is necessary to ensure the best efficiency of the system as a whole. The ejector operation can be optimized based on the results of numerous experiments. Through trial and error, intuition, and accumulated experience, based on the obtained data, we can reach a certain optimality level. Linear models and one-parameter optimization methods do not provide adequate results when describing complex systems. Therefore, considering the possibility of using alternative analytical procedures that can control several parameters is important [37,38]. The methods of mathematical planning for an experiment are well known and are used in engineering practice. However, the potentialities of these methods have not yet been fully discovered, especially in the case of optimizing complex technical systems. We used the Response Surface Methodology (RSM) to improve the efficiency of multivariate experimental studies of ejector flows. The objective of RSM is to identify the effect of the nozzle geometry and pressure drop on the gas velocity, secondary flows, and mixed flows of the mass flow rate. Most optimization approaches rely on mathematical differentiation to find the optimal solution, which is not applicable to discrete functions since they do not have continuous derivative properties. We have used RSM to approximate discrete dependencies. The mathematical model RSM is a polynomial equation of the second degree; its advantage is its ease of use for approximating the experimental response. We reduced the number of experiments using the composition design developed by Box and Wilson [39]. The second-order response function model is a surface. The determination of the curvature of such a surface requires varying the factors at least at three levels. We used the regression model as a full quadratic polynomial with regression coefficients b to describe the dependence of responses on the factors affecting the characteristics of jet streams. To obtain information about the degree of each factor’s influence on the responses, we used a circumscribed central composite design (CCCD), which allows calculating the regression coefficients independently of each other.

In the case of the factor space, we compiled a second-order central design (

Table 1. Points of the factor space of CCCD type 2². Coding the factors.

Factor Name	Factor	Coded Factor Notation	Level of Variation	Level
				−1	0	1
Mutual axial displacement of ANAE parts, mm	h	x1	1.5	1.0	2.5	4.0
Pressure drop	Npr	x2	1	2	3	4

). We tested three different nozzle areas, which were obtained by a mutual displacement of the ejector body parts along the longitudinal axis in the range of 1–4 mm when the variation level was 1.5 mm, and three different pressure drops in the range of 2–4 when the variation level was 1.

We have compiled the design so that the algebraic sum of the elements of any column vector of the design matrix, as well as the sum of the products of the elements of any two-column vectors of the matrix, is equal to zero, thus providing the conditions for the symmetry and orthogonality of the design (11) and (12). The conditions for symmetry and orthogonality of the design matrix are:

$${\displaystyle \sum _{l=1}^N{x}_{sl}=0,}$$

(11)

$${\displaystyle \sum _{l=1}^N{x}_{sl}{x}_{ql}=0},$$

(12)

Here, the numbers of the design matrix columns are s ≠ q, q = 1 − (N − 1).

The above conditions were fulfilled by choosing the shoulder value of star points d and a linear transformation of the quadratic variable by shifting it by the value φ (i.e., instead of the variable x_i², the variable x_i² − φ is considered). The boundaries of the study area were the lower and upper levels of the relevant factors. The CCCD matrix for two factors consists of the design core, one central point, and four star points, which are located on the axis of each factor symmetrically with respect to the origin (

Table 2. Matrix of the circumscribed central composite design.

l	x0	x1	x2	x12 − φ	x22 − φ	x1x2	$\overline{\mathit{y}}$
a	b	c	d	e	f	g	h
1	+1	−1	−1	1 − φ	1 − φ	+1	$\overline{y}$1
2	+1	+1	−1	1 − φ	1 − φ	−1	$\overline{y}$2
3	+1	−1	+1	1 − φ	1 − φ	−1	$\overline{y}$3
4	+1	+1	+1	1 − φ	1 − φ	+1	$\overline{y}$4
5	+1	−d	0	d2 − φ	−φ	0	$\overline{y}$5
6	+1	+d	0	d2 − φ	−φ	0	$\overline{y}$6
7	+1	0	−d	−φ	d2 − φ	0	$\overline{y}$7
8	+1	0	+d	−φ	d2 − φ	0	$\overline{y}$8
9	+1	0	0	−φ	−φ	0	$\overline{y}$9

).

Here l is the number of points in the factor space, d is the shoulder of star points, and φ is the shift parameter.

Each row vector ranging from 1 to 9 corresponds to one point in the factor space. Row vectors ranging from 1 to 4 contain the data of the experiments in which all the limiting combinations of factor levels are realized. Row vectors 5 through 8 contain the data for star points, and row vector 9 represents the experiment carried out at the center point. The column vector (a) contains a number of points in the factor space. Column vectors (b–g) contain the values of the input factors of their squares and products. The column vector (h) contains the average values of the response function calculated based on the results of parallel experiments conducted for each point in the factor space.

5. Results and Discussions

Let us determine the response values for all the levels of CCCD factors: a flow velocity on the jet axis u, a total gas flow rate G_Σ, and its components G₁, G₂, and G₃ in section L (

Table 3. CCCD response values for the factor space points.

l	x1	x2	u, m/s	G1, 103·m3/s	G2, 103·m3/s	G3, 103·m3/s	GΣ, 103·m3/s
1	−1	−1	85.55	3.30	4.67	24.43	32.4
2	+1	−1	199.31	8.87	6.45	53.49	68.8
3	−1	+1	129.0	6.60	6.43	33.14	46.17
4	+1	+1	275.16	17.73	4.76	79.66	102.15
5	−1	0	113.62	4.95	5.73	28.38	39.06
6	+1	0	299.77	13.30	7.27	57.50	78.07
7	0	−1	155.37	6.64	6.79	40.69	54.13
8	0	+1	233.68	13.29	6.61	58.00	77.9
9	0	0	214.25	9.96	7.90	44.86	62.73

).

The analysis of the given data showed that an increase in the primary flow led to an increase in the total gas flow rate, and the amount of air entrained from the flooded space significantly exceeded the mass flow rates of the primary and secondary flows. At the lower level of factors, the minimum flow rate of the primary flow corresponds to the lowest flow rate of all the flows and the low speed of the mixed flow. However, when the levels of factors increase, the flow rate velocity of the mixed flow grows non-linearly, while the contribution of the primary, secondary, and entrained flows to the mixed flow balance changes in a complex way. We chose the total airflow rate, the flow velocity on the jet axis, and the airflow rate of the secondary flow as a set of criteria for improving the ejector efficiency according to the analysis of gas-dynamic characteristics. To find the optimal values of the factors at which the responses reach maximum values, we built a regression equation in the form of a complete quadratic polynomial:

$$y=b_0+{\displaystyle \sum _{i=1}^n{b}_i{x}_i}+{\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=i+1}^n{b}_{ij}{x}_i{x}_j}+{\displaystyle \sum _{i=1}^n{b}_{ii}\left(x_i^2-\mathsf{\phi }\right),}}$$

(13)

where b is the coefficients of the equation, determined independently of each other by the general formula.

$$b_m=\frac{{\displaystyle \sum _{l=1}^N{x}_{ml}{\overline{y}}_l}}{{\displaystyle \sum _{l=1}^N{x}_{ml}^2}},$$

(14)

where m is the conditional number of the variable, and

$$b_0=b_0^{\ast }{x}_0-\mathsf{\phi }{\displaystyle \sum _{i=1}^k{b}_{ii}}.$$

(15)

The values of the coefficients of the regression equation, determined according to (14), are given in

Table 4. Coefficients of the regression equation.

	b0 *	b0	b1	b2	b11	b22	b12
GΣ	62.38	62.49	21.90	11.82	−3.81	3.64	4.90
u	189.52	220.79	74.35	32.94	−17.37	−29.54	8.10
G2	6.290	7.777	0.275	−0.018	−1.215	−1.015	−0.863

*—intermediate value of the parameter at m = 0.

.

We calculated the errors of the regression equations for each point of the factor space (

Table 5. Calculation errors.

l	GΣ		G2		u
	δ, 103·m3/s	Δ, %	δ, 103·m3/s	Δ, %	δ, m/s	Δ, %
1	1.10	3.29	0.24	5.48	10.84	14.51
2	1.29	1.91	0.25	3.77	7.89	3.81
3	1.17	2.48	0.31	5.14	4.62	3.71
4	1.22	1.21	0.18	3.66	14.11	4.88
5	2.28	6.19	0.56	8.85	15.46	11.98
6	2.51	3.12	0.43	6.34	22.00	7.92
7	0.19	0.35	0.01	0.15	2.95	1.86
8	0.05	0.06	0.13	1.98	9.49	4.23
9	0.24	0.38	0.12	1.59	6.54	2.96
max	1.10	3.29	0.56	8.85	10.84	14.51

):

$$\mathrm{Absolute} \mathrm{error}: \mathsf{\delta }=\left|y-y^{\prime }\right|;$$

$$\mathrm{Relative} \mathrm{error}: \mathrm{\Delta }=\frac{\delta }{y^{\prime }}\cdot 100\%.$$

Testing the significance of the regression equations using Fisher’s F-criterion showed the adequacy of the mathematical models and the significance of all their coefficients. In view of this, the airflow velocity and total gas flow rate were determined when a confidence interval was ±5%; the secondary gas flow rate was calculated when the error was no more than 15%. The pressure drop was measured when the error was no more than 1% and the approximation error was no more than 15%. Therefore, the compiled regression equations can be used to optimize the gas-dynamic characteristics of the ejector. Figure 5 shows a graphic interpretation of the response surfaces of the gas-dynamic characteristics selected for analyzing the factor space coordinates.

The maximum value of the mixed flow velocity (Figure 6a) is physically limited by the sound velocity in the flooded space since the primary gas flow discharges freely and expands uncontrolled. The presented data permit concluding that shock waves reduce the velocity of the primary flow to subsonic values up to cross-section L. The unimodal shape of the velocity profile of the gas jet also indicates the formation of a mixed flow in this section.

The intensity of the vortex formed behind the nozzle exit increases along with an increase in the active flow velocity; thus, it limits the flow in the central channel. The maximum of the G₂ regression function is reached when x₁ = 0.14 and x₂ = −0.07, which is close to the central point of CCCD (Figure 6b).

The mixed gas flow rate grows significantly with an increase in the levels of factors x₁ and x₂ (Figure 6c). However, a further increase in these parameters and going beyond the boundaries of the studied ranges would limit the total flow value to the capacity of the compressed air supply system.

Moving on to physical variables, the regression equations can be represented as follows:

$$G_{\Sigma }{}^{\prime }=37.18+13.28h-18.17Npr+3.26h\cdot Npr-1.69h^2+3.64Np{r}^2;$$

(16)

$$u^{\prime }=-275.47+71.95h+196.65Npr+5.40h\cdot Npr-7.72h^2-29.54Np{r}^2;$$

(17)

$$G_2{}^{\prime }=-9.45+4.61h+7.51Npr-0.58h\cdot Npr-0.54h^2-1.02Np{r}^2.$$

(18)

A simultaneous consideration of all the responses is necessary to establish optimal combinations of factor levels. To achieve a satisfactory compromise, we used the desirability approach as a tool for the multi-objective optimization of the investigating system. This approach is based on converting all the responses received on different scales into a scale-free value. The values of the desirability functions are between 0 and 1. The value 0 is assigned when the factors provide an undesirable response, and the value 1 corresponds to the optimal combination of the studied factors.

To evaluate the efficiency of the ejector in the presence of different factors, we used the exponential desirability function [40]. The equation associating the value of the partial response (d_i) on a psychophysical scale from 0 to 1 with its dimensionless representation (y_i′) can be presented as:

$$d_i=e^{-e^{-y_i\prime }},$$

(19)

where

$$y_i\prime =\frac{y_i-y_{i0}}{\mathrm{\Delta }{y}_i},$$

(20)

The value of the generalized response was calculated as the geometric mean of partial responses according to:

$$Y=\sqrt[n]{{\displaystyle \prod _{i=1}^n{d}_i}}.$$

(21)

Table 6. Values of optimization criteria and responses d_i and Y.

GΣ, 103·m3/s	u, m/s	G2, 103·m3/s	Npr	d1	d2	d3	d4	Y
32.4	85.6	4.7	2	0.07	0.07	0.07	0.95	0.129
68.8	199.3	6.5	2	0.71	0.72	0.74	0.95	0.777
46.2	129.0	6.4	4	0.29	0.30	0.74	0.07	0.255
102.2	275.2	4.8	4	0.95	0.92	0.09	0.07	0.267
39.1	113.6	5.7	3	0.16	0.20	0.48	0.69	0.319
78.1	299.8	7.3	3	0.82	0.95	0.90	0.69	0.834
54.1	155.4	6.8	2	0.46	0.48	0.82	0.95	0.643
77.9	233.7	6.6	4	0.82	0.84	0.78	0.07	0.434
62.7	214.3	7.9	3	0.62	0.78	0.95	0.69	0.752

shows the values of partial responses on the psychophysical scale, the assessment of their desirability, and the magnitude of the generalized response. The optimization criteria comprised the total flow rate (G_Σ), the flow velocity on the jet axis (u), the secondary gas flow rate (G₂), and the pressure drop (Npr). The optimization task was to achieve the highest values of the first three criteria in the presence of the lowest Npr.

The generalized response was thus calculated using the regression equation:

$$Y^{\prime }=-2.53+0.85h+1.47Npr-0.11h\cdot Npr-0.08h^2-0.22Np{r}^2.$$

(22)

Figure 7 shows that the generalized response in the factor space reaches its maximum when Npr = 2.5 and h = 3.65 mm. The level of 0.73 corresponds to the preferred rating on the desirability scale, while the level of 0.37 is the lower limit of the satisfactory rating.

The atomization of liquids containing a gas is widely used in modern technology for burning liquid fuels, cooling hot gases, obtaining a vapor-gas mixture, creating coatings, and other applications. During spraying, gas and liquid interact in a complex way. To date, there is no unified theory of gas atomization of liquids, especially in the case of rheologically complex systems. Therefore, the study of the atomization processes of such liquids containing gases is one of the key factors in developing the annular nozzle-ejector intended for various purposes [41,42,43,44]. One of the research fields is the optimization of the ANAE efficiency under the influence of structural and operational parameters. The most optimal conditions for spraying the liquids can be obtained in areas of preferable or satisfactory assessment of the generalized response.

6. Conclusions

We performed a two-factor optimization based on Harington’s generalized desirability function to obtain a relatively accurate estimate of the ejector efficiency. The efficiency of the ejector was evaluated by a set of criteria, including the total airflow rate, the flow velocity on the jet axis, the airflow rate of the secondary flow carried from the environment depending on the pressure drop, and the throat area of the annular nozzle. According to a set of criteria that provide the highest value of the dynamic gas pressure, we have established the area of preferred values of the factor space coordinates. The most favorable conditions for spraying liquids having different viscosities can be obtained in areas of preferable or satisfactory assessment of the generalized response.

The developed model is designed to be a good physical representation of the behavior of the atomizing gas jet under practical operating conditions to provide the best possible atomization ranges. This technique can be used to optimize ejectors with different nozzle geometries for other applications.

Hence, the developed algorithm of multi-objective optimization based on the combination of the response surface methodology and the desirability function approach, allowing a simultaneous consideration of all the responses, makes it possible to optimize the annular-nozzle air ejector efficiency under the influence of structural and operating parameters.

The proposed algorithm for the primary experimental data processing allows calculating the flow velocity based on the measured total and static pressures. The developed software performs the mathematical and statistical processing of the experimental data, proceeding from gas-dynamic dependencies obtained for the compressible gas flow, calculation profiles of the total pressure and gas velocity, flow rates, and the entrainment ratio.

The use of the circumscribed central composite design made it possible to reduce the number of experiments as well as establish approximating surfaces in factor space coordinates for all the optimized characteristics. The regression equations built according to the circumscribed central composite design describe the relationship of the pressure drop and structural parameters of the nozzle with the speed and mass flow rates of the airflows with an accuracy of 15 percent.

In view of the above-mentioned, the combination of experimental design techniques, surface response methods, and Harington’s desirability function is promising for optimizing complex systems.