Mathematical Modelling of the Entrainment Ratio of High Performance Supersonic Industrial Ejectors

: For many years now, manufacturers have been producing supersonic ejectors with a high entrainment ratio for the chemical, oil, and food industries. In the present work, mathematical modelling of the entrainment ratio of such industrial ejectors is carried out, in which a variation of the diffuser efﬁciency is also assumed to be a function of the Mach number of the motive gas. To determine this unknown relationship, the mathematical modelling was overturned by inserting the entrainment ratios of ten different high-performance industrial ejectors, as identiﬁed through an experimental investigation. The mathematical modelling, completed through the use of the relationship between the diffuser efﬁciency and the Mach number of the motive gas, was applied to sixty-eight ejectors, built and tested experimentally over the last twenty years as part of research aimed at the development of thermal ejector refrigeration systems (ERSs), to obtain the entrainment ratios proposed by the manufacturers (industrial entrainment ratios). A comparison of the experimental entrainment ratios with respect to the industrial ones demonstrated that the former were always lower, ranging from a minimum of − 17% to a maximum of − 82%. These results indicate that the lab-built ejectors for ERS prototypes can be improved. Therefore, in the future, researchers should apply numerical analysis iteratively, starting from a given geometry of the ejector, and modifying it until the numerical analysis provides the industrial value of the entrainment ratio.


Introduction
An ejector, or jet pump, is an apparatus (Figure 1) of which the task is the compression of a fluid-called the induced fluid-by molecular entrainment and diffusion, through a jet produced by the expansion of another fluid-called the motive fluid [1,2]. Fluids can be liquid or gas and, in this case, are in supersonic motion. In this work, supersonic ejectors are discussed, as they are widespread in the chemical, oil, and food industries, where they are used in the following: -Vacuum pumps to eliminate incondensable gases (sucked fluid) in various treatment plants, where the motive fluid is steam from a boiler; or -Steam compressors in concentration and distillation plants. The vapor resulting from boiling liquid is thermodynamically requalified and then reused to heat the boiling liquid itself. This ejector-heat exchanger system is, therefore, a heat pump, and the ejector is also called a thermo-compressor. Furthermore, in this case, the motive fluid is steam from a boiler.
Various studies focused on the use of supersonic ejectors as thermo-compressors in heat-driven ejector refrigeration systems (ERSs), where both fluids are refrigerating gases, have been underway for many years. This type of system, which has been known for more than a century [3], is now being re-proposed for environmental reasons, as they can operate on solar energy [4]. Various studies focused on the use of supersonic ejectors as thermo-compressors in heat-driven ejector refrigeration systems (ERSs), where both fluids are refrigerating gases, have been underway for many years. This type of system, which has been known for more than a century [3], is now being re-proposed for environmental reasons, as they can operate on solar energy [4].
The peculiarities of these ejectors are as follows: The absence of moving parts and, therefore, reliability and durability; simplicity of construction and, therefore, relatively low investment cost; operational rigidity, as they operate correctly only in design conditions; and the complexity of the phenomena of mixing and diffusion of the two fluids operating in transonic and supersonic conditions, therefore leading to difficulty in the mathematical modelling and design of high-performance ejectors. In confirmation of this last point, it should be noted that there are no more than a dozen manufacturers in the world capable of producing high-performance supersonic ejectors.
The performance of an ejector can be identified by the entrainment ratio ω, defined as the ratio between the mass flow rate of the induced gas Gi to that of the motive gas Gm. Its calculation-and, therefore, prediction of the performance of an ejector-can be carried out with the aid of a one-dimensional gas-dynamics theory. Keenan and Neumann [5] used such a one-dimensional approach with the hypothesis of ideal gas behavior considering the equations of conservation of mass, momentum, and energy; however, they excluded frictional losses and diffusion modelling. In a subsequent work, Keenan, Neumann, and Lustwerk [6] introduced these two phenomena; however, the mathematical description of the dissipative phenomenon of shock waves occurring in the transition from supersonic to subsonic motion of the gas mixture was lacking.
A later work by Lukasiewicz [7] on diffuser ducts highlighted the presence of several curved or oblique shock waves. Furthermore, Shapiro [8], in his 1953 textbook, referred to a series of bifurcated normal shocks in diffuser ducts. Matsuo [2] has called this phenomenon a shock train, but other expressions are also in use, besides those mentioned by Shapiro and Lukasiewicz: a series of shocks [9], Lambda foot shock system [10], Bifurcated normal shock waves [11], a series of oblique shocks [12], multiple normal shocks [13], and multiple shocks [14].
Munday and Bagster [15] introduced a mathematical formulation of the phenomenon of diffusion in the ejector from supersonic to subsonic stream using the Prandtl-Meyer equation, which predicts the presence of a frontal shock wave. Furthermore, they proposed that the induced gas, in the first part of the mixing chamber (Figure 1), remained The peculiarities of these ejectors are as follows: The absence of moving parts and, therefore, reliability and durability; simplicity of construction and, therefore, relatively low investment cost; operational rigidity, as they operate correctly only in design conditions; and the complexity of the phenomena of mixing and diffusion of the two fluids operating in transonic and supersonic conditions, therefore leading to difficulty in the mathematical modelling and design of high-performance ejectors. In confirmation of this last point, it should be noted that there are no more than a dozen manufacturers in the world capable of producing high-performance supersonic ejectors.
The performance of an ejector can be identified by the entrainment ratio ω, defined as the ratio between the mass flow rate of the induced gas G i to that of the motive gas G m . Its calculation-and, therefore, prediction of the performance of an ejector-can be carried out with the aid of a one-dimensional gas-dynamics theory. Keenan and Neumann [5] used such a one-dimensional approach with the hypothesis of ideal gas behavior considering the equations of conservation of mass, momentum, and energy; however, they excluded frictional losses and diffusion modelling. In a subsequent work, Keenan, Neumann, and Lustwerk [6] introduced these two phenomena; however, the mathematical description of the dissipative phenomenon of shock waves occurring in the transition from supersonic to subsonic motion of the gas mixture was lacking.
A later work by Lukasiewicz [7] on diffuser ducts highlighted the presence of several curved or oblique shock waves. Furthermore, Shapiro [8], in his 1953 textbook, referred to a series of bifurcated normal shocks in diffuser ducts. Matsuo [2] has called this phenomenon a shock train, but other expressions are also in use, besides those mentioned by Shapiro and Lukasiewicz: a series of shocks [9], Lambda foot shock system [10], Bifurcated normal shock waves [11], a series of oblique shocks [12], multiple normal shocks [13], and multiple shocks [14].
Munday and Bagster [15] introduced a mathematical formulation of the phenomenon of diffusion in the ejector from supersonic to subsonic stream using the Prandtl-Meyer equation, which predicts the presence of a frontal shock wave. Furthermore, they proposed that the induced gas, in the first part of the mixing chamber (Figure 1), remained separate from the motive one and was forced to expand up to the sonic conditions. This was due to widening of the motive gas jet, leading to a reduction in the section available for the induced gas. From the sonic condition of the induced gas, the real mixing hypothesized at constant pressure then started.
With the renewed interest in ejectors used as thermo-compressors in refrigeration cycles, Eames et al. [16] verified this one-dimensional analytical approach for various ejectors inserted in a heat-driven ejector refrigeration system (ERS) operated with steam, noting that the experimental entrainment ratio was always less than that predicted theoretically. Similar differences have also been identified by Huang et al. [17], considering different ejectors inserted in an ERS with refrigerant fluid R141b. Rogdakis and Alexis [18] also experimentally verified the theory with an ejector operating in an ERS with refrigerant fluid R717, always identifying a more optimistic theoretical entrainment ratio.
Therefore, in order of a better comprehension of the phenomena of mixing and diffusion, the computer fluid dynamics CFD modelling is useful. Consequently, for about 20 years, computational studies on the operation and design of ejectors have continued to be produced, focused on both on-design and off-design conditions . Most of the results of the numerical analyses presented in these works have been experimentally validated by the authors: in many cases, the entrainment ratio in the on-design condition predicted by the numerical analysis has presented a negligible difference, compared with the experimental one.
Parallel to this incessant scientific commitment-which began over 80 years ago and has accelerated in the last twenty years due to the availability of numerical analysis software-manufacturers of supersonic ejectors for various industrial sectors (i.e., chemical, oil, and food) have been present on the market for over a century.
For example, in Germany, Koerting Gebrueder began manufacturing supersonic ejectors in 1920 and, in 1964, began using computers for the fluid dynamic design of supersonic jet pumps (as ejectors are also called). Therefore, it is very likely that the design guidelines adopted by Koerting AG were not and (even more so today) are not purely empirical. Moreover, in Germany, another company was founded in the 1940s, Wiegand (which has recently been absorbed by GEA AG), by the researcher Joachim Wiegand, who in 1940, published a 24-page booklet full of theory and experimental data on the design of ejectors [47]. In Europe, other relevant manufacturers, such as the French SCAM, Kinetic-therm, and LVI and the Spanish Equirepsa, subsequently established themselves, while in the United States [1], Croll-Reynolds, Fox-Valve, Graham Manufacturing, Jetvac technologies, and Nash-kinema have long been known. It is likely that some names have been unintentionally forgotten, but it should be noted that there are only a dozen manufacturers internationally, and not all of them produce high-performance ejectors (i.e., with high entrainment ratio = industrial entrainment ratio).
In the first part of this work, we detail an experimental investigation carried out to determine the entrainment ratio values obtainable from high-performance supersonic ejectors designed under different boundary conditions (i.e., pressure and temperature of the motive gas, the induced gas, and the mixed gases at the outlet). These ejectors offer high-performance (optimal entrainment ratio) because they are built by a big European manufacturer that has been present on the international market for 80 years. In the second part, mathematical modelling is carried out, which, starting from the Wiegand equations [47], correlates the entrainment ratio of the aforementioned ejectors of the European manufacturer to the gas properties, the boundary conditions (pressures and temperatures), and the isentropic efficiency of the supersonic nozzle (motive gas), suction chamber (induced gas), and diffuser (mixed gases). In the third part of the work, the mathematical model is used to obtain the values of the entrainment ratio as a function of the gas properties and the boundary conditions (pressures and temperatures) of ejectors built and tested in the laboratory. These ejectors are considered benchmarks, as they are used to test the results of the numerical analyses proposed in many papers.
A comparison between the industrial entrainment ratios predicted from the mathematical model elaborated in the second part and the entrainment ratios obtained experimentally from the benchmark ejectors highlights important differences; namely, the experimental values are always lower than the industrial ones. This is due to the fact that all of the numerical analyses conducted on the ejectors are characterized by computational thermo-fluid dynamics investigations, which provide entrainment ratio values for given geometries of the elements of the ejector, and all the validations made in the laboratory were used to experimentally check the entrainment ratio predicted by the numerical analysis, but not to improve the geometry in order to maximize the entrainment ratio. This is because numerical analysis does not allow for derivation of the geometry by imposing the value of the entrainment ratio; that is, it does not allow for a direct design [48]. In other words, numerical analyses can estimate, with great precision, the entrainment ratio for a given ejector component geometry without having to build it; however, if one wishes to obtain a high-performance ejector, it is necessary to analyze various geometries, until one having a suitably high entrainment ratio is obtained. On the other hand, maximizing the entrainment ratio involves maximizing the coefficient of performance COP [19] of the ejector refrigeration system ERS. Therefore, it is necessary to know the maximum values of the entrainment ratio that international manufacturersincluding that of the ejectors investigated in this work-can obtain and which is the result of thousands of industrial tests and millions of ejectors that have been built and sold.

High Performance of Industrial Ejectors
At an Italian company for the construction of food evaporators equipped with ejectors/ thermo-compressors purchased from a European company that has manufactured ejectors for 80 years, a survey was conducted to determine the performance, represented by the entrainment ratio, of various industrial ejectors installed in the evaporators. Eight ejectors were identified with a wide variability of boundary conditions-that is, in terms of the pressure of the motive gas p m , pressure of the induced gas p i , and pressure of the mixed gases p 4 at the outlet ( Figure 1)-to constitute a heterogeneous population.
In Table 1, the operating data of these eight industrial ejectors (ejectors No. 1-8) are presented, including the pressures and stagnation temperatures of the motive gas (p m and T m , respectively), those of the induced gas (p i and T i , respectively), the outlet pressure (p 4 ), the mass flow rates of the motive gas (G m ) and the induced gas (G i ), and the diameters of the intake pipes (D i ) and outlet (D 4 ). All of the ejectors had motive gas consisting of dry saturated steam and induced gas also consisting of dry saturated steam. Finally, Table 1 shows the industrial entrainment ratios ω ind declared by the European manufacturer. They are to be considered optimal, as these ejectors have been improved to have the maximum value of the entrainment ratio ω. The European manufacturer has also declared that their ejectors are sized for a critical pressure p c that is 5% higher than the outlet pressure p 4 ( Figure 2). This is a safety margin to avoid the risk of operating in off-design mode, in which the entrainment ratio ω would drop drastically, stopping the operation of the evaporator in which the ejector is installed.  To operate correctly-that is, keeping the entrainment ratio constant and equal to the design value (on-design)the pressure p4 must be equal to or lower than the critical pressure pc.
The survey was conducted with an inspection of the evaporators installed in Italy. This survey allowed for verification of the regular operation of the evaporators and, therefore, of the installed ejectors. It could not be otherwise, as this European manufacturer is a company that is known internationally and has operated for eighty years in the construction of ejectors.
In addition, ejectors No. 9 and 10 are two ejectors from the same European company, subjected to tests at the Food Processes Engineering Laboratory at the TESAF Dept. of To operate correctlythat is, keeping the entrainment ratio constant and equal to the design value (on-design)-the pressure p 4 must be equal to or lower than the critical pressure p c .
The survey was conducted with an inspection of the evaporators installed in Italy. This survey allowed for verification of the regular operation of the evaporators and, therefore, of the installed ejectors. It could not be otherwise, as this European manufacturer is a company that is known internationally and has operated for eighty years in the construction of ejectors. In addition, ejectors No. 9 and 10 are two ejectors from the same European company, subjected to tests at the Food Processes Engineering Laboratory at the TESAF Dept. of University of Padova. The two ejectors had motive gas consisting of dry saturated steam produced by a boiler operating at p m = 5 bar and G m = 128 kg h −1 (Figure 3a), and the induced gas consisting of air at pressure of 0.3 and 0.5 bar, respectively, due to the use of a multiple-orifice device (flute) built according to the guidelines of Power [1]. The tests (Figure 3b) confirmed the performances-represented by the entrainment ratio ω inddeclared by the manufacturer, as shown in Table 1. Below, a mathematical model is developed to determine the entrainment ratio, with

Calculation of the Pressure at the Outlet of the Suction Chamber and the Diffuser Efficiency of High-Performance Industrial Ejectors
Below, a mathematical model is developed to determine the entrainment ratio, with respect to the pressure and temperature of the motive gas (p m and T m , respectively), the pressure and temperature of the induced gas (p i and T i , respectively), the pressure of the mixed gases at the outlet (p 4 ), the physical properties of the gases themselves, and the isentropic efficiencies, which take into account the frictional effects during expansion and diffusion. Figure 1 presents the schematic of an ejector, in which an appropriate acceleration of the fluid induced in the suction chamber is provided to reduce the speed difference between the two gases in the mixing chamber [49], therefore reducing irreversibility. In this way, an increase in the entrainment ratio is achieved.
The momentum theorem applied to the mixing chamber [47,50,51], neglecting pressure variations and friction with the mixing chamber wall, gives the following: where G m is the mass flow rate of the motive gas (kg s −1 ), w 1 is the supersonic velocity of the motive stream at the inlet of the mixing chamber (m s −1 ), G i is the mass flow rate of the induced gas (kg s −1 ), and w 2 is the subsonic velocity of the induced stream at the inlet of the mixing chamber (m s −1 ). It should be noted that, unlike Eames [3] and other subsequent researchers, the mixing efficiency η m is not included in Equation (1); that is, following the old German school, η m is considered unitary. By defining the entrainment ratio as ω = G i /G m , from Equation (1), we obtain the following: The three speeds inside the ejector (w 1 , w 2 , and w 3 ) can be obtained from the energy equation (first law of thermodynamics) [8,50] applied to the irreversible adiabatic steady flow, as follows:

1.
Motive stream in the nozzle:
Mixed stream in the mixing chamber: where ∆h 1 , ∆h 2 , and ∆h 3 , are the isentropic difference in enthalpies of the motive stream in the nozzle, induced stream in the suction chamber, and mixed stream in the diffuser, respectively; η E1 is the isentropic efficiency in the nozzle; η E2 is the isentropic efficiency in the suction chamber; and η D is the isentropic efficiency of the diffuser, which accounts for the whole loss during the pressure gain process due to the shock train and subsonic diffuser section.
In Equations (3)-(5), the velocities of the gases in the m, i, and 4 sections of the ejector (i.e., w m , w i , and w 4 ; see Figure 1) have been omitted, as the corresponding kinetic energies are smaller than 2% of those corresponding to the velocities inside the ejector (i.e., w 1 , w 2 , and w 3 ).  (2), we obtain the following:

By introducing Equations (3)-(5) into Equation
The isentropic difference in enthalpies for the ideal gas-a valid model for the gases operating in the ejector-are as follows:
Induced stream in the suction chamber:

6.
Mixed stream in the diffuser: where k m is the ratio of specific heats c p /c v of the motive gas; k i is the ratio of specific heats c p /c v of the induced gas; k 4 is the ratio of specific heats c p /c v of the mixed gases; R m is the specific constant of the motive gas; R i is the specific constant of the induced gas; R 4 is the specific constant of the mixed gases; p 1 is the pressure of the motive gas at the exit of the nozzle, which is equal to p 2 ; p 2 is the pressure of the induced gas at the exit of the suction chamber, which is equal to the pressure at the inlet of mixing chamber; p m is the pressure of the motive gas at the inlet of the nozzle; p i is the pressure of the induced gas at the inlet of the suction chamber; p 3 is the pressure of the mixed gas at the end of mixing chamber, which is supposed to be equal to the pressure at the inlet of mixing chamber p 2 (isobaric process of mixing); p 4 is the pressure of the mixed gases at the exit of the diffuser; and T m is the absolute stagnation temperature (K) of the motive gas, which is supposed to be equal to the absolute temperature at the inlet of nozzle (t m is the same temperature in • C); T i is the absolute stagnation temperature of the induced gas, which is supposed to be equal to the absolute temperature at the inlet of the suction chamber; and T 4 is the absolute stagnation temperature of the mixed gases, which is supposed to be equal to the absolute temperature at the exit of the diffuser.
The equations for the calculation of the specific heat at constant pressure c p4 and that at constant volume c v4 , the ratio of specific heats k 4 , the absolute stagnation temperature T 4 , and the specific constant of the mixed gases R 4 are provided in Appendix A.
By combining Equations (6)-(9), we obtain the following: In this equation, considering the appropriate design of the nozzle, the high Reynolds number, and the accelerating flow with the consequence of a very thin boundary layer compared to the nozzle section, the isentropic efficiency of the nozzle η E1 has limited variation, ranging from 0.95 to 0.99 [8,19]. Furthermore, the isentropic efficiency of the suction chamber η E2 , before the mixing chamber, also has limited variation (from 0.92 to 0.99) [8,19]. For caution, here, η E1 is assumed to have a constant value of 0.95 and η E2 has a constant value of 0.92, as the wall of the suction chamber is considered to be rough.
In the diffusion process from the last part of the mixing chamber to the ejector outlet, the isentropic efficiency η D is not constant, due the presence of a shock train. In a similar case considering a cylindrical diffuser with initial supersonic motion, Shapiro [8] indicated that η D is dependent of the inlet Mach number. In this work, a dependence of η D on the Mach number of the motive stream at the nozzle exit, M m−2 , was found; this Mach number is as follows [52]: Equation (10) shows that the entrainment ratio, ω, depends on two important variables: the efficiency of the diffuser η D and the pressure of induced gas at the exit of suction chamber p 2 , which is equal to the pressure at the nozzle exit p 1 (Figure 1).
The diagram shown in Figure 4 represents the function of the entrainment ratio ω with respect to the pressure p 2 and the isentropic efficiency of the diffuser η D ; that is, The diagram shows that, for each diffusion efficiency value η D , the function ω = f (p 2 ) has a maximum ( Figure 5). Therefore, the entrainment ratio ω has a maximum value corresponding to a precise value of the pressure p 2 , which accelerates the induced gas such that the lower speed difference between the motive gas and the induced gas reduces the frictional effects in the mixing chamber, while avoiding an excessive compression ratio in the diffuser which, instead, negatively affects the entrainment ratio.   (10)) of the entrainment ratio of an ejector (N. 1, Table 1) vs. pressure at the exit of the suction chamber p2 and the diffuser efficiency ηD: = ( , ). The red curve represents the function: = ( ). The yellow curve represents the function: = ( ). The p2 pressure value that maximizes the entrainment ratio ω is also the one that minimizes the diffuser efficiency ηD.
The diagram shows that, for each diffusion efficiency value ηD, the function = ( ) has a maximum ( Figure 5). Therefore, the entrainment ratio ω has a maximum value corresponding to a precise value of the pressure p2, which accelerates the induced gas such that the lower speed difference between the motive gas and the induced gas reduces the frictional effects in the mixing chamber, while avoiding an excessive compression ratio in the diffuser which, instead, negatively affects the entrainment ratio.  (10)) of the entrainment ratio of an ejector (N. 1, Table 1) vs. pressure at the exit of the suction chamber p 2 and the diffuser efficiency η D : ω = f (p 2 , η D ). The red curve represents the function: ω = f (p 2 ). The yellow curve represents the function: η D = f (p 2 ). The p 2 pressure value that maximizes the entrainment ratio ω is also the one that minimizes the diffuser efficiency η D . The diagram shows that, for each diffusion efficiency value ηD, the function = ( ) has a maximum ( Figure 5). Therefore, the entrainment ratio ω has a maximum value corresponding to a precise value of the pressure p2, which accelerates the induced gas such that the lower speed difference between the motive gas and the induced gas reduces the frictional effects in the mixing chamber, while avoiding an excessive compression ratio in the diffuser which, instead, negatively affects the entrainment ratio. The values of the entrainment ratio ωind presented in Table 1 concern high-performance industrial ejectors, in which the manufacturer has also foreseen the correct acceleration of the induced stream in the suction chamber to obtain the high entrainment ratio

Diffuser Efficiency η
The values of the entrainment ratio ω ind presented in Table 1 concern high-performance industrial ejectors, in which the manufacturer has also foreseen the correct acceleration of the induced stream in the suction chamber to obtain the high entrainment ratio (i.e., ω ind ). If these ω ind values are introduced into (10), the equation is left with two unknowns: The pressure p 2 and the diffuser efficiency η D . It seems to be a problem without solution, but studying the function ω = f (p 2 ), which is parametric in η D (shown in Figure 5), as well as the function η D = f (p 2 ), which is parametric in ω (shown in Figure 6) we find that when the former, ω = f (p 2 ), presents its maximum, the latter, η D = f (p 2 ), presents its minimum.
Processes 2022, 10, 88 10 of 28 (i.e., ωind). If these ωind values are introduced into (10), the equation is left with two unknowns: The pressure p2 and the diffuser efficiency ηD. It seems to be a problem without solution, but studying the function = ( ), which is parametric in ηD (shown in Figure  5), as well as the function = ( ), which is parametric in ω (shown in Figure 6) we find that when the former, = ( ), presents its maximum, the latter, = ( ), presents its minimum. Therefore, it is possible to determine the value of p2 that minimizes the isentropic efficiency of the diffuser, by zeroing the derivative of the function = ( , ), providing the parameter ω with the values ωind in Table 1 for the high-performance industrial ejectors. The solution provides p2 values that minimize the diffuser efficiency while also maximizing the entrainment ratio. The function = ( , ) is obtained from (10): Therefore, it is possible to determine the value of p 2 that minimizes the isentropic efficiency of the diffuser, by zeroing the derivative of the function η D = f (p 2 , ω), providing the parameter ω with the values ω ind in Table 1 for the high-performance industrial ejectors. The solution provides p 2 values that minimize the diffuser efficiency while also maximizing the entrainment ratio. The function η D = f (p 2 , ω ind ) is obtained from (10): Taking the derivative of (12) and equating it to zero, we have the following: where F m , F i , and F 4 are defined as follows: Equation (13), which can be solved an iterative method, provides the value of p 2 that improves the ejector, as it is also the one that maximizes the entrainment ratio ω.
The value of p 2 , inserted into Equation (12), allows us to obtain the efficiency of the diffuser η D in high-performance industrial ejectors.
As mentioned above, it is to be expected that η D depends on the Mach number of the motive gas in the second section, M m−2 , at the inlet of the mixing chamber. As already said, the calculation method of p 2 described by Equation (13) and that of η D through Equation (12) can be applied to the high-performance industrial ejectors detailed in Table 1. The obtained p 2 values also allow for obtaining the Mach number, M m−2 , through Equation (11). Therefore, the relation η D = f (M m−2 ) can be easily found, as discussed in the results section.

Calculation of the Entrainment Ratio for Ejectors to Be Improved
The calculation of the Mach number M m−2 of the motive stream passing from the pressure p m to the best pressure p 2 requires determination of p 2 , which is still unknown.
This means that an iterative calculation procedure must be implemented; however, this can be avoided if the Mach number M m−2 is replaced by the Mach number M m−i of the motive stream at the final pressure equal to the suction pressure p i (Figure 1). In fact, p i is known as it is a boundary condition of the ejector. As detailed in the results, the equation η D = f (M m−2 ) obtained by regression (Figure 7) by the procedure of Section 2.2.1, has a good coefficient of determination R 2 , but the function η D = f (M m−i ) (Figure 8) has a higher R 2 . This is a good reason to always use the relationship η D = f (M m−i ). Therefore, the procedure for calculating the entrainment ratio ω, as outlined below, begins with the calculation of M m−i : Processes 2022, 10, 88 12 of 28   Setting the derivative of Equation (10) equal to zero allows us to obtain the value of the best pressure p 2 which maximizes the entrainment ratio ω: where F m , F i , and F 4 are given by Equations (14)- (16), respectively. Equation (18), similar to Equation (13), also requires solution through an iterative method, for example, by a spreadsheet.
Finally, the pressure value p 2 obtained from Equation (18) allows for obtaining the entrainment ratio ω ind maximized through Equation (10).
The solution of Equations (18) and (10) requires knowledge of the values of k 4 , R 4 , and T 4 which, according to Equations (A1)-(A5), presented in Appendix A, depend on the unknown value of the entrainment ratio ω.
If the induced gas is equal to the motive gas, then k m = k i = k 4 and R m = R i = R 4 , but the need remains to determine the stagnation temperature T 4 through Equation (A5), starting from the stagnation temperatures T m and T i and from the entrainment ratio ω. The absolute stagnation temperatures T 4 , T m , and T i practically coincide with the absolute temperatures as (see Figure 1), in Section 4, m, and i, the gases generally have velocities lower than 50 m s −1 , often around 30 m s −1 . The problem arises of needing to know ω a priori to calculate T 4 which, in turn, serves to calculate the same ω by Equation (18). The problem can be overcome by inserting into the spreadsheet, before Equations (18) and (10), a cell containing an initial value of ω (experience suggests ω = 1) and cells containing Equations (A1)-(A5), which provide the values of c p4 , c v4 , R 4 , k 4 , and T 4 resulting from this initial ω. Additionally, it is necessary to enter an initial value for p 2 (experience suggests p 2 = 0.99·p i ).
Therefore, the procedure includes the following: (1) Execution of the iterative method (e.g., by spreadsheet) to search for the ω value associated to p 2 = 0.99·p i , using Equation (10); (2) Execution of the iterative method (e.g., by spreadsheet) to search for the best value of p 2 , using Equation (18); (3) Final execution of the iterative method (e.g., by spreadsheet) to recalculate the final industrial ω value, now for the best p 2 obtained in step 2, by Equation (10); and (4) Final execution of the iterative method (e.g., by spreadsheet) to recalculate the final best p 2 value, using Equation (18).
Only two iterations of steps (1) and (3) are sufficient to maximize ω and only two for steps (2) and (4) are required to improve p 2 , as the solution method is strongly convergent.

Experimental Entrainment Ratio of Benchmark Ejectors Built and Tested to Use in ERSs
Benchmark ejectors built and tested by various researchers in their laboratories over the past twenty years, for use in ERS prototypes, have been considered. Details on an initial series of such benchmark ejectors, in terms of their geometric characteristics, boundary conditions, and performance (represented by the entrainment ratio measured in the laboratory) have been presented in the work of Besagni [19] exactly in its Tables 6, 7 and 11.
To check whether these ejectors were improved (i.e., if they were high-performance ejectors), the mathematical model for calculating the industrial high entrainment ratio ω ind proposed in this work was applied.
For some of these ejectors, only the experimental values of the boundary conditions and the entrainment ratio corresponding to the critical point were available while, for others, these values were available both in on-design and off-design contexts (Figure 2). For the latter, using the results reported by [19], the value of the entrainment ratio ω and the corresponding outlet pressure p 4 of the critical point were identified. The critical point was not always uniquely identifiable. Therefore, in these cases, two pairs of ω-p 4 values were taken into consideration, that is, those that could be closest (one to the right and one to the left) to the ω-p 4 pair of the critical point.
Therefore, Table 2 summarizes the values of the boundary conditions and the experimental entrainment ratio ω exp corresponding to the critical point for each ejector, except in some cases, in which there were two critical pseudo-points, marked with the symbol *. The alphanumeric codes reported, which distinguish each ejector (e.g., A1f), are the same as those used in [19]. A second series of benchmark ejectors, with their geometric characteristics, boundary conditions, and performance represented by the entrainment ratio measured in the laboratory, was found in the works [62][63][64][65][66], as summarized in Table 3. Table 3. Operating data of the second series of benchmark ejectors built and tested experimentally by the authors indicated in the first column.

High-Performance Industrial Ejectors
For the ten high-performance industrial ejectors produced by the European manufacturer, Equation (13) was applied, together with Equations (A1)-(A5), in order to determine the best values of the pressure p 2 (Table 4) of the induced gas at the exit of the suction chamber. It should be noted that the value of the specific heat ratio of motive steam, k m , used in the calculations, was not equal to 1.3 (typical of superheated steam), but was equal to an average empirical value of 1.14 (which corresponds to wet steam) [67]. In fact, the motive steam, during the expansion in the nozzle starting from the dry saturated condition, assumes the wet condition. The k i value was also equal to 1.14 for the first eight ejectors, which intake dry saturated steam which, when expanded in the suction chamber, became wet steam. Meanwhile, for ejectors 9 and 10, which intake air, k i was set to 1.4.  Equation (12) was applied to the same ten high-performance industrial ejectors, in order to find the corresponding values of the diffuser efficiency η D . Table 4 also reports the values of the Mach number M m−2 calculated using Equation (11), as well as the values of the Mach number M m−i calculated using Equation (17). The values of η D vs. M m−2 are plotted in Figure 7, which also shows the regression line (R 2 = 0.9725).
As mentioned in Section 2.2.2, determining the Mach number M m−2 requires prior knowledge of the pressure p 2 . To overcome this problem, Equation (17) was used to calculate the Mach number M m−i reached by the motive gas at the outlet of the nozzle with the pressure p i of the induced gas, as if this were not accelerated in the suction chamber by w i to w 2 (Figure 1). The resulting diagram (Figure 8) shows that the correlation between the efficiency of the diffuser η D and the Mach number M m−i was better than the previous one, with a coefficient of determination R 2 equal to 0.9741. Therefore, the relationship found η D = f (M m−i ), as a result of linear regression, as follows:

Validation of the Mathematical Modelling of the Entrainment Ratio vs. the Values of Industrial Ejectors
To validate the results of the mathematical modelling of the industrial entrainment ratio, represented by Equations (18) and (10), it is necessary to compare the calculated entrainment ratios from Equation (10) with those declared by the manufacturer of the ten high performance ejectors (Table 1). This comparison produced an average error only of 1.7%, due to the linear relationship presented in Equation (19), characterized by R 2 = 0.9741 < 1.

Calculation of the Industrial Entrainment Ratio on Benchmark Ejectors Built and Tested to Use in ERSs
Following the method indicated in Section 2.2.2, Equation (17) allowed for calculation of the Mach Number M m−i . Then, Equation (19) made it possible to calculate the efficiency of the diffuser η D . Consequently, Equation (18) allowed us to calculate the best pressure p 2 , and finally, Equation (10) allowed us to calculate the industrial entrainment ratio ω ind .
This method was applied to each of the benchmark ejectors presented in Tables 2 and 3.  The calculation results are shown in Tables 5 and 6, respectively, together with the experimental entrainment ratio value ω exp and the relative percentage difference: δ = ω exp −ω ind ω ind ·100. Equations (10), (17) and (18), as well as the auxiliary Equations (14)- (16), contain the quantities k m , k i , k 4 , R m , R i , and R 4 . In all ejectors presented in Tables 2 and 3, the induced gas was the same as the motive gas. Therefore, k m = k i = k 4 = k and R m = R i = R 4 = R. Tables 5 and 6 shows the values of the gas constant R calculated for each gas by the following formula: where m m is the molecular mass found in [19] and R is the universal gas constant equal to 8314 (J kmole −1 K −1 ).  Tables 5 and 6 also show the values of the ratio of specific heat k. As the bibliography provided disparate values, they were calculated as follows: For each group of ejectors characterized by the same gas, the average of the pressures p m and temperatures T m (Figure 1 and Tables 2 and 3) and the average of the pressures p i were calculated, noting that the S.D. was limited. The average value of the pressure p m and temperature T m of each gas made it possible to trace the point of the boundary condition m (Figure 1) on the p-h diagram of the gas. Considering the isentropic expansion of the motive gas in the nozzle, the mean value of p i made it possible to trace, on the p-h diagram, the point of isentropic conditions at the outlet of the nozzle. In correspondence to the two points (m and i) the p-h diagram provided the enthalpy values h m and h i-is and, therefore, the isentropic difference in enthalpies ∆h m−is = (h m − h i−is ). As the temperature T m-is can be read from the p-h diagram, and as ∆h m−is = c p ·(T m − T i−is ), the specific heat at constant pressure c p can therefore be obtained as follows: Meyer's relationship provides the specific heat at constant volume c v : Finally, k, the ratio of the two specific heats, is obtained as follows: Processes 2022, 10, 88

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After having obtained the best pressure p 2 with the method in Section 2.2.2, Equation (8), with k i = k and R i = R, provides the value of the isentropic difference in the enthalpies ∆h 2 . Plotting of the points i and 2is on the p-h diagram made it possible to measure the exact value of ∆h 2 , which was found to be superimposable to the value provided by Equation (8), with an error of less than 2%.
Therefore, the ratio of specific heats k obtained from the expansion of the motive gas was also usable for the expansion of the induced gas and, according to Equations (A1)-(A3), also for the diffusion of the mixed gases.
On the other hand, this method for calculating the specific heat ratio k is the same as that adopted previously [67] to quantify the k of the wet steam result equal to 1.14. It is very different from the value of 1.3 for dry steam, as the specific heat c p of the wet steam calculated by Equation (21) considers the latent heat released during the partial condensation and which is present in the quantification of Differences between the experimental and industrial entrainment ratios presented in Tables 5 and 6 are also depicted in the histograms of Figures 9 and 10, respectively.
Processes 2022 9, x FOR PEER REVIEW 22 of 29 Differences between the experimental and industrial entrainment ratios presented in Tables 5 and 6 are also depicted in the histograms of Figures 9 and 10, respectively.   (Table 6).

Discussion and Conclusions
The long experience, extending for over a century, of the manufacturers of ejectors for the chemical, oil, and food industries, has led some of them to improve their products, where improving an ejector means maximizing the entrainment ratio ω. For this reason, these high entrainment ratios are called here: industrial entrainment ratios.  (Table 5).
Processes 2022 9, x FOR PEER REVIEW 22 of 29 Differences between the experimental and industrial entrainment ratios presented in Tables 5 and 6 are also depicted in the histograms of Figures 9 and 10, respectively.  (Table 5).  (Table 6).

Discussion and Conclusions
The long experience, extending for over a century, of the manufacturers of ejectors for the chemical, oil, and food industries, has led some of them to improve their products, where improving an ejector means maximizing the entrainment ratio ω. For this reason, these high entrainment ratios are called here: industrial entrainment ratios.  (Table 6).

Discussion and Conclusions
The long experience, extending for over a century, of the manufacturers of ejectors for the chemical, oil, and food industries, has led some of them to improve their products, where improving an ejector means maximizing the entrainment ratio ω. For this reason, these high entrainment ratios are called here: industrial entrainment ratios.
However, Chen, in his 1997 work [63], wrote: "Although ejector manufacturers possess much valuable experimental data, they are understandably reluctant to publish this data". Almost a quarter of a century has passed since then, and Chen's statement is still valid. It is even impossible to know the entrainment ratio values with respect to the boundary conditions unless the ejector is purchased.
The first studies conducted in Germany in the 1930s led to a mathematical model for calculating the entrainment ratio ω based on the momentum theorem and on the identification of isentropic efficiencies of expansion and diffusion. The same German studies showed that it was necessary to accelerate the induced gas before it entered the mixing chamber (Figure 1). The acceleration in the suction chamber-and, therefore, the reduction in the pressure p 2 at the inlet of the mixing chamber-cannot be indiscriminate but should be designed to reduce the speed difference between the motive gas and the induced gas in the mixing chamber, while avoiding an excessive compression ratio in the diffuser which, instead, negatively affects the entrainment ratio ω.
The best value of this acceleration-and, therefore, the best value of the discharge pressure from the suction chamber of the induced gas p 2 -is usually determined [68] by searching for the maximum of the function represented by Equation (10), assuming the value of the diffuser efficiency η D to be constant.
In this work, the diffuser efficiency η D was instead considered to be variable, depending on the Mach number of the motive gas at the inlet in the mixing chamber.
Therefore, Equation (10) was studied analytically as a function of two variables: p 2 and η D (Figures 4-6). Two equations were produced as a result: Equation (13), of which the solution provides the best values of p 2 with respect to the industrial entrainment ratio ω ind , and Equation (12), which provides the efficiency of the diffuser η D always with respect to the industrial entrainment ratio ω ind .
If the entrainment ratio ω ind inserted into Equations (13) and (12) is that of a highperformance ejector, such as those produced by the most competitive manufacturers mentioned at the beginning of this discussion, then Equation (13) gives the relative value of p 2 and Equation (12) provides the efficiency of the diffuser η D of this ejector.
This calculation method was applied to ten high-performance industrial ejectors from a major European company, which has operated in the global ejector market for eighty years. This choice was made in order to have a heterogeneous population of boundary conditions (i.e., p m , p i , and p 4 ). To circumvent the problem of scarce data availability for industrial ejectors, as stated by Chen [63], a manufacturer of evaporators for the food industry was asked to make available their data archive of the ejectors purchased over the course of twenty years from a big European company.
For each ejector, in addition to p 2 and η D , the Mach number of the motive gas M m−i was also calculated, using Equation (17). Comparison between the efficiency of the diffuser η D and the Mach number M m−i produced Equation (19), which is a linear regression function characterized by a high coefficient of determination R 2 , confirming that the hypothesis In the second part of the present work, comprising mathematical modelling, from Equation (10), the function (18) was obtained, of which the solution gives the value of the pressure p 2 which maximizes the entrainment ratio. Equation (18) contains the quantity η D which depends, through Equation (19), on the Mach Number M m−i which, in turn, according to Equation (17), is only dependent on the known boundary conditions p m and p i . Finally, Equation (10) can provide the maximized value of entrainment ratio, that is industrial entrainment ratio ω ind . This procedure was applied to a series of 68 ejectors built and tested experimentally over the last twenty years or so (Tables 2 and 3), as a part of research activities aimed at developing heat-driven ejector refrigeration systems (ERSs); for this reason, they were called benchmark ejectors [19]. The renewed interest in this type of refrigeration system, mainly due to environmental concerns (as they can operate on solar energy), has activated the interest of many researchers towards the study of the functioning of supersonic ejectors through computational methods. Such numerical investigations have produced more and more refined results, are now able to simulate the functioning of the ejectors and to predict, with relative precision, the experimental entrainment ratios ω exp [19] of the benchmark ejectors.
Again, with reference to the second part of this work, the results of the procedure for calculating the industrial entrainment ratio ω ind , applied to the series of 68 benchmark ejectors, are presented in Tables 5 and 6 (as well as in Figures 9 and 10), where they were compared with the experimental entrainment ratio values ω exp .
The experimental entrainment ratios ω exp were always lower than the industrial ones ω ind , with a minimum difference of −17% for the ejector of [55] and a maximum difference of about −80% for the ejector of [60]. Limited differences of about −30% were also found for the ejectors of [56,58,59] and only −25% for the ejector of [64]. For many other ejectors, the differences were consistent. This means that, while numerical analysis in this field has shown promise, it must be extended to the search for the best geometry of the ejector elements, first ensuring that the suction chamber has an outlet section which provides the best pressure p 2 , to be obtained a priori, for example, using the analytical method proposed herein. Second, it is necessary to pay close attention to the shape and size of the mixing chamber, wherein the shock train phenomena are very complex. Therefore, researchers who have the objective of designing a high-performance ejector should repeat the numerical analysis, which has only a checking character, with simulated variations in the shape and size of the mixing chamber, in an attempt to reach the high value of the entrainment ratio of industrial ejector. The industrial value can be calculated, for example, using the method proposed in this work, and corresponds to those of high-performance ejectors already on the market.
Finally, Equation (19) proposed in this work, which indicates the influence of the Mach number of motive gas at the inlet of mixing chamber M m−i on the diffuser efficiency η D with good precision, allows for calculation of the latter in the extreme situations encountered during the survey on 68 laboratory ejectors (Tables 5 and 6). These are the ejectors [61] with a minimum Mach number M m−i of about 1, for which Equation (19) proposes a diffuser efficiency η D of 0.870 and, excluding the ejector of [54], the ejector of [62] with the relative maximum M m−i of almost four, for which the diffuser efficiency η D is 0.710. It must be remembered that, in this work, the diffuser efficiency η D takes into account the whole loss during the pressure gain process due to shock train and subsonic diffuser section and, as a unitary mixing efficiency was assumed in Equation (1), any energy losses during mixing are also represented by η D , due to the adopted mode of calculation of η D .
In conclusion, it is useful to say that in this work, the objective was not for the design method of ejectors to have the optimal entrainment ratio, but rather for an algorithm to determine the maximum value of the entrainment ratio, built through the application of Calculus [69][70][71]. This algorithm can be used to determine the maximum (therefore optimal) value of the entrainment ratio ω of the ejector operating at the expected boundary conditions represented by the pressures and temperatures of the motive gas (p m , T m ), of the induced gas (p i , T i ), and of the pressure of the mixed gases (p 4 ). The maximum values, therefore optimal, of entrainment ratios obtained with the algorithm are those (ω ind ) corresponding to the industrial ejectors of the big European manufacturer chosen because it is the one with the longest experimental research history and with the greatest diffusion in the world market of ejectors.
Starting from the best value of entrainment ratio ω = ω ind , obtainable with the algorithm of this work, the researchers with experience in CFD applied to ejectors, will have to use the computational methods repeatedly, varying the dimensions especially of the mixing chamber, until the CFD provides the value of the entrainment ratio equal to ω ind . This is a similar method adopted in machine design with the FEA [72].