Thermodynamic Analysis and Performance Evaluation of Microjet Engines in Gas Turbine Education

Abstract

This paper presents a detailed study on the main parameters and performance evaluation of microjet engines, at take-off regime and at various engine working regimes, based on thermodynamic analysis of a particular engine data library, from different engine manufacturers such as JetCat and AMT Netherlands. The studied engines have the same spool design but different thrust classes ranging from 97 to 1569 N. The particular data library includes engine specifications from catalogs or data sheets as well as our own experimental data from the JetCat P80 microjet engine, obtained using the ET 796 Jet Turbine Module, a complete testing facility for gas turbine education purposes. Various ratios and differences between certain engine main parameters and performances are studied in order to calculate values through which the analyses can be performed. Even if the engines have different thrust classes, the study examines if there are close values of the ratios and differences of parameters, that can be defined as reference parameters through which the engine performance can be compared and evaluated.

1. Introduction

A micro gas turbine is a reduced-scale heat engine that operates with similar cycles and components to those found in large gas turbines, with a power range of 25 kW to 300 kW [1]. Compared to jet engines, the primary components of a micro gas turbine engine typically include a centrifugal or mixed-flow compressor, a combustion chamber, a single-stage axial-flow or radial-flow turbine, and a jet nozzle. Presently, micro turbojet engines are the most accessible and practical type of micro gas turbine. This is due to their simple design, low manufacturing and maintenance costs, and ease of operation [2].

Micro turbojet engines have a wide array of engineering applications, including both military and civil aviation, as well as power generation. These micro engines are particularly suited to such applications because of their high power-to-weight ratio, multi-fuel capability, and design simplicity [3]. In the military sector, a significant application of microjet engines is as propulsive power sources for Unmanned Aerial Vehicles (UAVs), owing to their higher energy density (Whr/kg) [3,4]. In the civilian sector, they are suitable for portable power units, radio-controlled (RC) aircraft models, and distributed power generation in combined heat and power applications [4].

Due to the technical advantages mentioned above, micro gas turbines are not only utilized for UAVs and power generation but also have significant applications in the field of gas turbine education. These engines can be installed in small laboratories without the need for complex or expensive infrastructure [2]. Consequently, they can easily become components of propulsion laboratories or small turbine engine test facilities for gas turbine education. Typically, gas turbine labs help students enhance their theoretical knowledge of gas turbines through practical experience. By conducting engine run tests, students can compare theoretical data with experimental data or with other calculated data derived from measurements. Through these experimental tests, a technical reference is established to understand the fundamentals of gas turbine theory and to validate the reasoning behind assumptions and approximations made in analytical performance evaluations [5,6].

Currently, aerospace institutes and research entities such as the German Aerospace Center (DLR) [7,8,9,10] are developing small propulsion laboratories or mobile test facilities as research infrastructure. These facilities are used to study new potential propulsion programs with the aim of improving gas turbine performance [3]. Additionally, aerospace universities are involved in various academic research projects in the field of gas turbines using micro propulsion laboratories [11,12]. The objective of these projects is to optimize performance parameters such as specific fuel consumption, noise and emissions, overall pressure ratio, gas temperature in front of the turbine, turbine and compressor efficiencies, engine thrust, and power [3,11].

Small microturbine engines are very useful for conducting experiments for both educational and research purposes because they do not require the extensive testing infrastructure needed for larger turbine engines. Using microturbine engines for educational purposes is a highly practical option for experiments serving subjects such as turbine processes, thermodynamics, acoustics, chemical pollution, and alternative fuels. Several studies present the use of small turbine engines that, in addition to their educational applications, have been used in the production of scientific papers on experimental and numerical simulation [13]. For example, the study [14] examines the transient processes that occur with various fuels and under different atmospheric conditions. Additionally, the study [15] presents methods for increasing thrust and reducing noise using an ejector for a micro engine, while another study [16,17,18] explores performance and gas emissions using alternative fuels. All these studies are conducted on various types of microturbine engines, and from the literature review on the study of microturbine engines, it was found that the experiments conducted do not provide a detailed analysis of the parameters and performance of these microturbine engines.

This paper presents a comparative study of the parameters and performance variations of micro jet engines, conducted through a detailed thermodynamic analysis and performance evaluation based on various parameter ratios and differences determined by specific calculations. The data utilized in this study are sourced from micro jet engines with varying thrusts and different manufacturers, including JetCat (Ballrechten-Dottingen, Germany) and AMT Netherlands (Geldrop, The Netherlands). The results are presented in both raw and percentage values, shown in tables and displayed in bar variation charts. This approach provides a comprehensive understanding of how the studied parameters vary for each engine and from one model to another within the JetCat and AMT micro jet ranges, as well as from idle to maximum regimes in the case of the JetCat P80.

Moreover, this paper examines a series of parameters not addressed in traditional gas turbine theory articles or other related works. No other papers or even books present the current specific coefficients, nor do they analyze such a substantial volume of data. These coefficients represent new ratios and differences between various parameters and engine performance. The findings will provide a significant contribution to calculations involving new developments or predictive analysis.

2. The Basis of Study

In the performance evaluation and analysis of micro jet engines, the initial step typically involves creating an engine data library. This library compiles data from various engine manufacturers, different engine models, and different thrust classes, making it accessible and useful for performance calculation models. These models can then be used to calculate various parameters or performance metrics. Given that the engine data are experimental results from different engine models tested with various laboratory test benches equipped with dedicated instrumentation lines and specific measuring points for pressure and temperature at particular engine cross-sections [10,19], the data must be standardized to a common reference.

The second step involves organizing the engine data library using common parameters, thereby establishing a general model of the engine’s main stations. Typically, the engine’s main stations are defined by the cross-sections of the inlets and outlets of the primary components, such as the compressor, combustion chamber, turbine, and jet nozzle. Figure 1 illustrates the general model of the micro jet engine stations for the engines studied.

In this context, station 1 refers to the compressor inlet, station 2 to the compressor outlet, station 3 to the turbine inlet, station 4 to the turbine outlet and also the jet nozzle inlet, and station 5 to the jet nozzle outlet.

Figure 1. Microjet engine main stations.

Figure 1. Microjet engine main stations.

The third step is to establish an analysis methodology, as the performance evaluation is based on a series of ratios and differences of certain parameters or engine component performance, determined through an analytical calculation model. This calculation model is based on the real Brayton cycle, depicted in Figure 2, which is the reference cycle describing the thermodynamic operation of gas turbine engines. The thermodynamic cycle is characterized by a total expansion process, meaning the jet nozzle outlet static pressure is equal to atmospheric pressure p5 = p0 = 1.01325 bar.

By following these steps, a systematic and standardized approach to performance evaluation can be achieved, ensuring consistency and reliability in the analysis of different micro jet engines.

Figure 2. Microjet engine real cycle diagram.

Figure 2. Microjet engine real cycle diagram.

3. The Particular Engine Data Library

The current data library is categorized into two segments: engine data at take-off regime for various engine models from JetCat and AMT Netherlands micro jet manufacturers [20,21] and engine data at various working regimes from the JetCat P80 model. It is noted that this data library contains data exclusively from JetCat and AMT manufacturers, as these engine models provide free access to a broader range of engine main parameters. Other popular micro jet manufacturers such as JetCentral (Ciudad de México, México), PBS Velka Bites (Velká Bíteš, Czech Republic), and Hybl Turbines (Prague, Czech Republic) [22,23,24] restrict the disclosure of comprehensive engine specifications, particularly key parameters such as air mass flow or overall total pressure ratio. Most producers typically provide only dimensional data, thrust, specific fuel consumption, exhaust gas temperature (EGT), engine speed, and fuel type. However, this limited information is insufficient to determine other critical engine parameters or performance metrics. There are numerous research papers on micro jet performance evaluation, but few present actual values of engine main parameters and performances across different working regimes. Most of these papers feature variation curves based on engine speed [25,26,27] or specific charts showing relationships between various engine parameters. One method of creating a data library involves extracting parameter values from these charts, provided the charts are clear and easy to read. However, in many cases, the charts are difficult to extract values from unless the values are explicitly shown on the curves [28]. Next, the data library is organized according to common parameters and performances in line with the general model of engine main stations.

Table 1. Engine data at take-off regime for JetCat models [29].

Engine Model	NE	TR	πCt	Maf	SFC
	rpm	N	-	kg/s	kg/(N∙h)
P80-SE	125,000	97	2.3	0.240	0.1500
P100-RX	154,000	100	2.9	0.230	0.1870
P130-RX	127,000	130	3.0	0.300	0.1850
P160-RXi-B	122,000	158	3.5	0.380	0.1600
P180-NX	126,000	175	3.5	0.380	0.1600
P200-RX	112,000	210	4.0	0.450	0.1500
P220-RXi	117,000	220	3.9	0.450	0.1580
P300-PRO	106,000	300	3.55	0.500	0.1570
P400-PRO	98,000	397	3.80	0.670	0.1579
P500-PRO GL	80,000	492	3.60	0.900	0.1510
P550-PRO GL	83,000	550	3.80	0.930	0.1440
P1000-PRO	61,500	1100	3.80	1.800	0.1270

presents the engine data at take-off regime for a series of micro jet engine models from the JetCat manufacturer [29].

In

Table 2. Engine data at take-off regime AMT models [30].

Engine Model	NE	TR	πCt	Maf	SFC
	rpm	N	-	kg/s	kg/(N∙h)
Pegasus HP	117,000	167	3.0	0.398	0.1796
Olympus HP	108,500	230	3.3	0.450	0.1670
Titan	96,000	392	3.8	0.660	0.1561
Nike	61,500	784	4.0	1.250	0.1454
Lynx	46,000	1569	4.0	2.500	0.1377

, the engine data at take-off regime are presented for several micro jet engine models from the AMT Netherlands manufacturer [30]. This table, along with Table 1, which includes data from JetCat, provides a comprehensive overview of the performance metrics and parameters for these engines at a critical operational regime. This structured data compilation allows for a thorough comparative analysis between different models and manufacturers, contributing to a deeper understanding of the performance characteristics of micro jet engines. The data from these tables will be essential for the subsequent thermodynamic analysis and performance evaluations discussed in this study.

The experimental data of the Jet-Cat P80 micro turbojet were obtained through a series of tests using the ET 796 Jet Turbine Module [10], a small mobile test bench. This test facility, depicted in Figure 3, is a specialized gas turbine laboratory equipped with a fuel system, starting system, measuring instruments, a digital acquisition module, and a user interface for engine control. The measuring system allows for the measurement of pressures and temperatures at key engine stations, as well as the engine thrust. Therefore, the measured parameters provide sufficient data to calculate both overall engine performance and the performance of individual engine components.

This setup ensures a comprehensive and detailed analysis of the Jet-Cat P80, facilitating a better understanding of its operational characteristics and performance metrics. The data gathered from these tests are critical for validating the theoretical models and assumptions made in the performance evaluation study.

In Figure 3 are presented the main components of the ET 796 module with 1—Jet engine, 2—Mixing pipe, 3—Turbine platform, 4—Force sensor for thrust measurement, 5—Gas turbine controller and 6—Displays and safety buttons.

Figure 3. ET 796 Jet Turbine Module.

Figure 3. ET 796 Jet Turbine Module.

In

Table 3. Experimental data of JetCat P80 micro jet model.

NE	TR	T3t	T2t	p3t
rpm	N	°C	°C	bar a
34,951	4.6	608.8	21.1	1.100
47,160	9.3	592.2	30.0	1.168
55,320	13.4	618.3	36.5	1.228
68,295	22.0	595.8	48.6	1.357
79,369	32.0	581.4	60.4	1.502
88,787	42.8	586.1	72.2	1.658
97,786	55.1	610.2	86.0	1.825
105,423	67.8	630.1	99.0	1.997
112,706	82.8	666.3	108.4	2.189
115,675	88.9	690.3	114.4	2.266

, the experimental data ranging from idle to maximum regime for the JetCat P80 micro turbojet are presented. These data were obtained from engine run tests conducted using the ET 796 Jet Turbine Module.

4. Calculation Model

The calculation model is founded on a series of mathematical relationships between the engine’s thermodynamic parameters. It computes various ratios and differences between key engine parameters and performance metrics to obtain actual values necessary for the analysis. To facilitate the calculation process, constant values of heat capacity at constant pressure for air and gases can be used. However, at this small scale, using a fixed value may affect the precision of the calculations. Therefore, the current model employs polynomials for the enthalpy and entropy of air and gases, which are functions of the working temperature [31,32]. This approach significantly reduces the calculation error, providing more accurate and realistic results.

In the case of air and gases (for ideal combustion), enthalpy and entropy are thus defined:

$$\begin{array}{cc}{h}_a\left(T_a\right),s_a{\left(T_a\right)}^{p0}=f\left(T_a\right)& h_{g.id}\left(T_g\right),s_{g.id}{\left(T_g\right)}^{p0}=f\left(T_g\right)\end{array}$$

(1)

where the polynomial function takes the form:

$$f\left(T\right)=a_{1.a,g}+a_{2.a,g}\cdot T+a_{3.a,g}\cdot T^2+a_{4.a,g}\cdot T^3+a_{5.a,g}\cdot T^4+a_{6.g}\cdot T^5$$

(2)

The air/gases enthalpy and entropy polynomial coefficients are presented in a previous research paper [31,32].

In case of gases (for real combustion), the enthalpy and entropy have the next forms:

$$h_g\left(T_g\right)=x_{id}\cdot i_{g.id}\left(T_g\right)+x_{a.ex}\cdot h_a\left(T_g\right)$$

(3)

$$s_g{\left(T_g\right)}^{p0}=x_{id}\cdot s_{g.id}{\left(T_g\right)}^{p0}+x_{a.ex}\cdot s_a{\left(T_g\right)}^{p0}$$

(4)

where the mass participation $x_{id},x_{a.ex}$ depending on air excess ${\alpha }_{ex}$ and air–fuel ratio ${min}_L$

$$\begin{array}{ccc}{x}_{st}={\displaystyle \frac{1+{min}_L}{1+{\alpha }_{ex}·{min}_L}}& ;& x_{st}={\displaystyle \frac{\left({\alpha }_{ex}-1\right)\cdot {min}_L}{1+{\alpha }_{ex}·{min}_L}}\end{array}$$

(5)

The most commonly used fuel for micro jet engines is Jet-A1 kerosene. Since all micro jet engine models lack a proper oil system and rely on fuel for lubrication, the fuel must be pre-mixed with 3–5% oil. For AMT engines, Aeroshell 500 turbine oil is used for the mixture, while JetCat engines use their own JetCat turbine oil [33].

As a result, the mixture of fuel and oil will have a different lower heating value (LHV) than the pure fuel. The LHV of Jet-A1 kerosene is 42,900 kJ/kg, but according to the ET 796 instruction manual [34], the LHV of the fuel–oil mix is 42,580 kJ/kg. Additionally, the air–fuel ratio for the mixture is 14.2, compared to the standard value of 14.57. This difference in LHV and air–fuel ratio needs to be considered in performance calculations to ensure accurate results.

Globally, the calculation model is based on a series of equations [35,36,37,38], as follows:

Intake and combustor chamber total pressure loss, compressor overall pressure ratio, and turbine total expansion degree:

$$\begin{array}{ccc}{\sigma }_{da.t}={\displaystyle \frac{p_{1.t}}{p_0}}& ;& \begin{array}{ccc}{\sigma }_{ca.t}={\displaystyle \frac{p_{3.t}}{p_{2.t}}}& ;& \begin{array}{ccc}{\pi }_{C.t}={\displaystyle \frac{p_{2.t}}{p_{1.t}}}& ;& {\delta }_{T.t}={\displaystyle \frac{p_{3.t}}{p_{4.t}}}\end{array}\end{array}\end{array}$$

(6)

Compressor inlet, outlet total specific enthalpy, and total specific actual work:

$$\begin{array}{ccc}{h}_{1.t}=i_a\left(T_{1.t}\right)& ;& \begin{array}{ccc}{h}_{2.t}=i_a\left(T_{2.t}\right)& ;& l_{C.t}=h_{2.t}-h_{1.t}\end{array}\end{array}$$

(7)

Turbine inlet, outlet total specific enthalpy, and total specific actual work:

$$\begin{array}{ccc}{h}_{3.t}=i_g\left(T_{3.t}\right)& ;& \begin{array}{ccc}{h}_{4.t}=i_g\left(T_{4.t}\right)& ;& l_{T.t}=h_{3.t}-h_{4.t}\end{array}\end{array}$$

(8)

Compressor and turbine total adiabatic efficiency:

$$\begin{array}{ccc}{\eta }_{C.t}={\displaystyle \frac{l_{C.t.id}}{l_{C.t}}}& ;& {\eta }_{T.t}={\displaystyle \frac{l_{T.t}}{l_{T.t.id}}}\end{array}$$

(9)

Compressor inlet, outlet total specific entropy:

$$\begin{array}{ccc}{s}_{1.t}=s_a{\left(T_{1.t}\right)}^{p0}-R_a\cdot \mathit{ln}\left({\displaystyle \frac{p_{1.t}}{p_0}}\right)& ;& s_{2.t}=s_a{\left(T_{2.t}\right)}^{p0}-R_a\cdot \mathit{ln}\left({\displaystyle \frac{p_{2.t}}{p_0}}\right)\end{array}$$

(10)

Turbine inlet, outlet total specific entropy:

$$\begin{array}{ccc}{s}_{3.t}=s_g{\left(T_{3.t}\right)}^{p0}-R_g\cdot \mathit{ln}\left({\displaystyle \frac{p_{3.t}}{p_0}}\right)& ;& s_{4.t}=s_g{\left(T_{4.t}\right)}^{p0}-R_g\cdot \mathit{ln}\left({\displaystyle \frac{p_{4.t}}{p_0}}\right)\end{array}$$

(11)

Compressor, turbine, and jet nozzle isentropic relations:

$$\begin{array}{ccc}{s}_{2.t.id}=s_{1.t}& ;& \begin{array}{ccc}{s}_{3.t}=s_{4.t.id}& ;& \begin{array}{ccc}{s}_{3.t}=s_{5p.id}& ;& s_{4.t}=s_{5.id}\end{array}\end{array}\end{array}$$

(12)

Fuel flow coefficient, air excess, and gases flow:

$$\begin{array}{ccc}{f}_{fc}={\displaystyle \frac{F_f}{M_{af}}}& ;& \begin{array}{ccc}{\alpha }_{ex}={\displaystyle \frac{1}{f_{fc}·{min}_L}}& ;& G_f=M_{af}+F_f\end{array}\end{array}$$

(13)

Inlet turbine total specific enthalpy and specific fuel consumption:

$$\begin{array}{ccc}{h}_{3.t}={\displaystyle \frac{h_{2.t}+f_{fc}·LH{V}_f·{\eta }_{ca}}{1+f_{fc}}}& ;& SFC=3600·{\displaystyle \frac{F_f}{TR}}\end{array}$$

(14)

Compressor and turbine power:

$$\begin{array}{ccc}{P}_C=M_{af}{·l}_{C.t}& ;& \begin{array}{ccc}{P}_T=G_f{·l}_{T.t}& ;& \begin{array}{ccc}{P}_c=P_T·{\eta }_m& ;& l_{T.t}={\displaystyle \frac{l_{C.t}}{\left(1+f_{fc}\right){·\eta }_m}}\end{array}\end{array}\end{array}$$

(15)

Jet nozzle thrust and outlet static pressure for a total expansion process:

$$\begin{array}{ccc}TR=G_f·C_5& ;& p_5=p_{5.id}=p_{5p.id}=p_0\end{array}$$

(16)

Jet nozzle power, fuel energy, and thermal efficiency:

$$\begin{array}{ccc}{P}_J=G_f·{\displaystyle \frac{{C_5}^2}{2}}& ;& \begin{array}{ccc}{P}_f=F_f\cdot LH{V}_f& ;& {\eta }_{te}={\displaystyle \frac{P_J}{P_f}}\end{array}\end{array}$$

(17)

Jet nozzle outlet specific enthalpies and velocity coefficient:

$$\begin{array}{cc}\begin{array}{cc}{h}_5=h_{4.t}-\left({\displaystyle \frac{{C_5}^2}{2}}\right)& ;\end{array}& \begin{array}{ccc}{h}_{5id}=h_{4.t}-\left({\displaystyle \frac{{C_{5id}}^2}{2}}\right)& ;& {\mathsf{\phi }}_J={\displaystyle \frac{C_5}{C_{5id}}}\end{array}\end{array}$$

(18)

Total specific ideal and actual enthalpy differences and power of expansion process:

$$\begin{array}{ccc}{\mathsf{\Delta }h}_{E.id}=h_{3.t}-h_{5p.id}& ;& \begin{array}{ccc}{\mathsf{\Delta }h}_E=h_{3.t}-h_5& ;& P_E=G_f\cdot {\mathsf{\Delta }h}_E\end{array}\end{array}$$

(19)

Jet nozzle ideal and actual total specific enthalpy expansion

$$\begin{array}{ccc}\begin{array}{ccc}{\mathsf{\Delta }h}_{J.p.id}=h_{4.t.p.id}-h_{5p.id}& ;& {\mathsf{\Delta }h}_{J.id}=h_{4.t}-h_{5.id}\end{array}& ;& {\mathsf{\Delta }h}_J=h_{4.t}-h_5\end{array}$$

(20)

Jet nozzle outlet heat capacity at constant pressure, adiabatic coefficient, and Mach number.

$$\begin{array}{ccc}{C}_p\left(T_5\right)={\displaystyle \frac{h_g\left(T_5\right)}{T_5}}& ;& \begin{array}{ccc}{k}_g\left(T_5\right)={\displaystyle \frac{C_p\left(T_5\right)}{C_p\left(T_5\right)-R_g}}& ;& \theta \left(M_5\right)={\displaystyle \frac{T_{5.\mathrm{t}}}{T_5}}=1+{\displaystyle \frac{k_g\left(T_5\right)-1}{2}}·{M_5}^2\end{array}\end{array}$$

(21)

Turbine and jet nozzle actual temperatures differences:

$$\begin{array}{ccc}\begin{array}{cc}{\mathsf{\Delta }T}_{T.t}=T_{3.t}-T_{4.t}& \begin{array}{cc};& {\mathsf{\Delta }T}_J=T_{4.t}-T_5\end{array}\end{array}& ;& {\mathsf{\Delta }T}_E=T_{3.t}-T_5\end{array}$$

(22)

Turbine and jet nozzle actual temperatures differences:

$$\begin{array}{ccc}{\mathsf{\Delta }T}_{T.t.id}=T_{3.t}-T_{4.t.id}& ;& {\mathsf{\Delta }T}_{J.p.id}=T_{4.t.id}-T_{5.p.id}\end{array}$$

(23)

$$\begin{array}{ccc}{\mathsf{\Delta }T}_{J.id}=T_{4.t}-T_{5.id}& ;& {\mathsf{\Delta }T}_{E.id}=T_{3.t}-T_{5.p.id}\end{array}$$

(24)

Combustor chamber, turbine, and jet nozzle actual temperatures ratio of expansion process:

$$\begin{array}{ccc}\begin{array}{cc}{dT}_{CH.t}={\displaystyle \frac{T_{2.t}}{T_{3.t}}}& \begin{array}{cc};& {dT}_{T.t}={\displaystyle \frac{T_{4.t}}{T_{3.t}}}\end{array}\end{array}& ;& \begin{array}{ccc}{dT}_J={\displaystyle \frac{T_5}{T_{4.t}}}& ;& {dT}_E={\displaystyle \frac{T_5}{T_{3.t}}}\end{array}\end{array}$$

(25)

Turbine and jet nozzle ideal temperatures ratio of expansion process:

$$\begin{array}{ccc}\begin{array}{cc}{dT}_{T.t}={\displaystyle \frac{T_{4.t.id}}{T_{3.t}}}& \begin{array}{cc};& {dT}_{J.p.id}={\displaystyle \frac{T_{5.p.id}}{T_{4.t.id}}}\end{array}\end{array}& ;& \begin{array}{ccc}{dT}_{J.id}={\displaystyle \frac{T_{5.id}}{T_{4.t}}}& ;& {dT}_{E.id}={\displaystyle \frac{T_{5.p.id}}{T_{3.t}}}\end{array}\end{array}$$

(26)

Total specific ideal and actual ratio enthalpies of expansion process

$$\begin{array}{ccc}{dl}_{T.t}={\displaystyle \frac{l_{T.t}}{{\mathsf{\Delta }h}_E}}& ;& \begin{array}{ccc}{dh}_J={\displaystyle \frac{{\mathsf{\Delta }h}_J}{{\mathsf{\Delta }h}_E}}& ;& \begin{array}{ccc}{dl}_{T.t.id}={\displaystyle \frac{l_{T.t.id}}{{\mathsf{\Delta }h}_{E.id}}}& ;& {dh}_{J.id}={\displaystyle \frac{{\mathsf{\Delta }h}_{J.p.id}}{{\mathsf{\Delta }h}_{E.id}}}\end{array}\end{array}\end{array}$$

(27)

Generally, depending on the known data and the required calculated data, specific calculation models are developed using the relevant equations. According to our data library, the known data for JetCat and AMT engines differ from the data for the JetCat P80 engine, resulting in different calculation models, though using the same equations in a different sequence. For JetCat and AMT micro jets, the particular calculation model is described by the calculated parameters in the following order:

$$\begin{array}{ccc}\begin{array}{ccc}TR,SFC& \Rightarrow & F_f\end{array}& ;& \begin{array}{ccc}\begin{array}{ccc}{M}_{af},F_f& \Rightarrow & f_{fc},{\alpha }_{ex}\end{array},G_f& ;& \begin{array}{ccc}TR,G_f& \Rightarrow & C_5\end{array}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{s}_{1.t},s_{2.t.id}& \Rightarrow & T_{2.t.id}\end{array},h_{2.t.id}& \Rightarrow & \begin{array}{ccc}\begin{array}{ccc}{l}_{C.t.id}& ;& {\eta }_{C.t},l_{C.t.id}\end{array}& \Rightarrow & l_{C.t},h_{2.t}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{f}_{fc},LH{V}_f,h_{2.t}& \Rightarrow & h_{3.t}\end{array}& ;& \begin{array}{ccc}\begin{array}{ccc}{f}_{fc},l_{C.t}& \Rightarrow & l_{T.t},\end{array}{h}_{4.t}& ;& \begin{array}{ccc}{\eta }_{T.t},l_{T.t}& \Rightarrow & l_{T.t.id}\end{array}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{h}_{3.t},l_{T.t.id}& \Rightarrow & {h_{4.t.id},T}_{4.t.id}\end{array}& ;& \begin{array}{ccc}{s}_{3.t},s_{4.t.id}& \Rightarrow & p_{4.t.id},p_{4.t}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{s}_{4.t},s_{5.id}& \Rightarrow & T_{5.id}\end{array}& ;& \begin{array}{ccc}{h}_{4.t},T_{5.id}& \Rightarrow & \begin{array}{ccc}{C}_{5.id},{\mathsf{\phi }}_J& ;& \begin{array}{ccc}{h}_{4.t},C_5& \Rightarrow & h_5, T_5\end{array}\end{array}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{s}_{3.t},s_{5p.id}& \Rightarrow & T_{5p.id,}\end{array}{i}_{5p.id,}& \Rightarrow & \begin{array}{ccc}{\mathsf{\Delta }h}_{E.id},{\mathsf{\Delta }h}_E& {{\mathsf{\Delta }h}_{J.id},\mathsf{\Delta }h}_{J.id}& P_J,\end{array}\end{array}{\eta }_{te}$$

Additionally, the calculation model requires other parameters or coefficients, which are defined by standard values. The additional parameters and coefficients used in the calculation model, along with their values, are shown in

Table 4. JetCat and AMT micro jets additional parameters and coefficients for calculation model.

σdat	σcat	ηca	ηTt	φJ	Ra [J/kg∙K]	Rg [J/kg∙K]
0.98	0.96	0.85/0.88/0.90/ 0.94/0.96	see Table 10	see Table 10	287.15	288.25

. The values of the ${\eta }_{T.t}$ and φJ are shown in Table 10.

In case of ${\eta }_{ca}$, different values are used. For AMT Pegasus, the used value is 0.85; for JetCat P100-RX, P130-RX, P160-RXi-B, P180-NX, P200-RX, and P220-RX, the used value is 0.88; for JetCat P200, P300-PRO, P400-PRO, P500-PRO GL, and P550 PRO, the used value is 0.90, the same as for AMT models Olympus (Tokyo, Japan), Titan (Bangalore, India), and Nike (Beaverton, OR, USA); for JetCat P80-SE and AMT Lyns, the used value is 0.94; and for JetCat P1000-PRO, the used value is 0.96. The values lower than 0.94 were taken following the study of several papers [3,39,40].

In case of the JetCat P80 micro jet, the particular calculation model is described by the calculated parameters in the indicated order:

$$\begin{array}{ccc}\begin{array}{ccc}{p}_{3.t},{\sigma }_{cat}& \Rightarrow & p_{2.t},\end{array}{\pi }_{C.t}& ;& \begin{array}{cc}\begin{array}{ccc}{s}_{1.t},s_{2.t.id}& \Rightarrow & T_{2.t.id}\end{array},h_{2.t.id}& \begin{array}{cc}\Rightarrow & l_{C.t.id}\end{array}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{\eta }_{C.t},l_{C.t.id}& \Rightarrow & l_{C.t},h_{2.t}\end{array}& ;& \begin{array}{ccc}{\alpha }_{ex}=f\left(T_{3.t},{\pi }_{C.t}\right)& \Rightarrow & {\alpha }_{ex}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{h}_{3.t},LH{V}_f,h_{2.t}& \Rightarrow & f_{fc}\end{array}& \Rightarrow & \begin{array}{ccc}\begin{array}{ccc}{l}_{T.t},h_{4.t}& ;& {\eta }_{T.t},\end{array}{l}_{T.t}& \Rightarrow & \begin{array}{ccc}{l}_{T.t.id}& \Rightarrow & {h_{4.t.id},T}_{4.t.id}\end{array}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{s}_{3.t},s_{4.t.id}& \Rightarrow & p_{4.t.id},p_{4.t}\end{array}& ;& \begin{array}{ccc}\begin{array}{ccc}{s}_{4.t},s_{5.id}& \Rightarrow & T_{5.id}\end{array},i_{5.id}& ;& \begin{array}{ccc}{i}_{4.t},i_{5.id}& \Rightarrow & C_{5.id}\end{array}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{C}_{5.id},{\mathsf{\phi }}_J& \Rightarrow & C_5\end{array}& ;& \begin{array}{ccc}TR,C_5& \Rightarrow & \begin{array}{ccc}{G}_f& ;& \begin{array}{ccc}{G}_f,f_{fc}& \Rightarrow & M_{af},F_f\end{array}\end{array}\end{array}\end{array}$$

$$\begin{array}{ccc}\begin{array}{ccc}{s}_{3.t},s_{5p.id}& \Rightarrow & T_{5p.id,}\end{array}{i}_{5p.id,}& \Rightarrow & \begin{array}{ccc}{\mathsf{\Delta }h}_{E.id},{\mathsf{\Delta }h}_E& {{\mathsf{\Delta }h}_{J.id},\mathsf{\Delta }h}_{J.id}& P_J,\end{array}\end{array}{\eta }_{te}$$

The additional parameters and coefficients used in the calculation model and their values are shown in

Table 5. Jet-Cat P80 micro jet additional parameters and coefficients for calculation model.

σdat	σcat	ηca	ηTt	φJ	Ra [J/kg∙K]	Rg [J/kg∙K]
0.995	0.96	0.95	0.84	0.94	287.15	288.26

.

The calculations were performed using Mathcad 14 version software, where the root function was utilized to calculate temperatures from the enthalpy and entropy polynomials.

5. Results

In the case of JetCat and AMT micro jets, the data results are obtained following the application of the first calculation model. These results are organized into tables, grouped into specific series, and present the actual values of certain key engine parameters, coefficients, and performance metrics for each micro jet engine. Given the large volume of data obtained from the calculations, it is impractical to present all the results. Therefore, only selected data are shown in the tables. This approach avoids duplication of results and provides relevant data from which other specific engine parameters can be easily determined. For the JetCat P80 micro jet, the data results are obtained following the application of the second calculation model. These results also present actual values for each working regime, covering the same parameters, ratios, differences, and performance metrics established for the JetCat and AMT jet models. This consistency allows for a comprehensive comparison and analysis of the performance across different engine models and operating conditions.

From a data sorting perspective, three sets of data results are obtained: one for JetCat models, one for AMT models, and one for the JetCat P80 model.

The first and second series are similar; however, the third series has some differences in the types of parameters due to the calculation models starting from different known data. This variation in starting data leads to differences in the calculated parameters and the results presented. The first series of data results is defined by the power of the turbine $\left(P_T\right)$, jet nozzle $\left(P_J\right)$, expansion process $\left(P_E\right)$, turbine total specific ideal and actual work $\left(l_{T.t}\right)$, $\left(l_{T.t.id}\right)$, total specific ideal and actual enthalpy of jet nozzle $\left({\mathsf{\Delta }h}_{J.p.id}\right)$, $\left({\mathsf{\Delta }h}_{J.id}\right)$, $\left({\mathsf{\Delta }h}_J\right)$, and expansion process $\left({\mathsf{\Delta }h}_{E.id}\right)$, $\left({\mathsf{\Delta }h}_E\right)$. In

Table 6. The first series of data results, at take-off regime, for JetCat and AMT micro jet models.

Engine Model	PT	PJ	PE	h3t	lTt	ΔhJ	ΔhE	lTt.id	ΔhJ.p.id	ΔhE.id	ΔhJ.id
	kW	kW	kW	kW	kW	kW	kW	kW	kW	kW	kW
P80-SE	22.912	19.277	42.190	1039.296	93.887	78.993	172.880	114.497	83.605	198.102	64.734
P100-RX	33.076	21.259	54.335	1248.755	140.631	90.389	231.020	190.042	105.248	295.290	60.303
P130-RX	44.742	27.553	72.295	1242.569	145.890	89.843	235.733	197.149	105.608	302.757	59.068
P160-RXi-B	64.353	32.251	96.605	1127.453	166.278	83.332	249.610	221.704	91.371	313.075	41.155
Pegasus HP	63.455	34.318	97.773	1179.208	156.164	84.458	240.622	208.219	96.288	304.507	48.994
P180-NX	64.353	39.488	103.841	1198.369	165.954	101.830	267.784	218.360	112.414	330.774	65.557
P200-RX	86.063	48.065	134.128	1199.415	187.603	104.774	292.377	246.846	115.783	362.629	63.217
P220-RXi	84.165	52.648	136.813	1250.528	183.103	114.539	297.642	240.926	129.283	370.209	78.269
Olympus HP	71.986	57.417	129.402	1323.616	156.264	124.639	280.903	202.940	142.946	345.886	101.657
P300-PRO	79.382	87.705	167.087	1411.256	154.715	170.938	325.653	193.394	192.821	386.215	159.647
Titan	118.998	113.489	232.487	1417.321	175.773	167.635	343.408	225.350	181.238	406.588	138.442
P400-PRO	117.664	114.639	232.302	1421.344	171.168	166.768	337.936	219.446	188.112	407.558	146.629
P500-PRO GL	148.484	131.466	279.950	1300.337	161.284	142.800	304.084	204.158	158.878	363.036	121.665
P550-PRO GL	157.199	158.876	316.076	1330.654	165.125	166.887	332.012	198.946	185.605	384.551	156.829
Nike	221.134	239.788	460.921	1398.775	172.536	187.091	359.627	210.410	205.232	415.642	173.269
P1000-PRO	286.36	329.019	615.379	1298.871	155.732	178.931	334.663	179.002	197.771	376.773	178.260
Lynx	436.807	480.813	917.620	1388.505	170.628	187.817	358.445	205.576	207.728	413.304	178.337

and

Table 7. The first series values of calculated data for JetCat P80 engine.

NE rpm	PT	PJ	PE	h3t	lTt	ΔhJ	ΔhE	lTt.id	ΔhJ.p.id	ΔhE.id	ΔhJ.id
	kW	kW	kW	kW	kW	kW	kW	kW	kW	kW	kW
34,951	0.571	0.231	0.802	916.009	12.476	5.049	17.525	14.853	5.699	20.552	5.714
47,160	1.136	0.703	1.839	905.688	18.475	11.432	29.907	21.994	12.890	34.884	12.938
55,320	1.776	1.236	3.012	935.891	24.456	17.026	41.482	29.114	19.177	48.291	19.268
68,295	3.493	2.465	5.958	909.517	35.580	25.108	60.688	42.357	28.209	70.566	28.416
79,369	5.781	4.111	9.892	892.662	46.420	33.012	79.432	55.262	36.994	92.256	37.361
88,787	8.764	6.089	14.853	897.855	58.259	40.475	98.734	69.356	45.236	114.592	45.806
97,786	11.913	8.801	20.714	925.544	69.068	51.025	120.093	82.224	56.912	139.136	57.747
105,423	15.793	11.714	27.507	948.480	80.493	59.705	140.198	95.824	66.447	162.271	67.570
112,706	19.872	15.78	35.652	990.703	91.476	72.638	164.114	108.900	80.710	189.610	82.207
115,675	21.472	17.694	39.166	1018.969	96.143	79.226	175.369	114.456	87.994	202.450	89.663

, the first series of data results are presented for the JetCat and AMT models at take-off regime, sorted by thrust. Additionally, data for the JetCat P80 model are presented at various working regimes.

The second series of data results are defined by the inlet and outlet temperatures of the combustor chamber $\left(T_{2.t}\right)$, $\left(T_{3.t}\right)$, fuel flow and air excess $\left(F_f\right)$, $\left({\alpha }_{ex}\right)$, total ideal/actual temperature differences on turbine $\left({\mathsf{\Delta }T}_{T.t}\right)$, $\left({\mathsf{\Delta }T}_{T.t.id}\right)$ and on jet nozzle $\left({\mathsf{\Delta }T}_{J.p.id}\right)$, $\left({\mathsf{\Delta }T}_{J.id}\right)$, $\left({\mathsf{\Delta }T}_J\right)$, and expansion process $\left({\mathsf{\Delta }T}_{E.id}\right)$, $\left({\mathsf{\Delta }T}_E\right)$.

Table 8. The second series of data results, at take-off regime, for JetCat and AMT micro jet models.

Engine Model	T2t	α.ex	Ff	T3t	ΔTT	ΔTJ	ΔTE	ΔTT.id	ΔTJ.p.id	ΔTE.id	ΔTJ.id
	°C	-	g/min	°C	°C	°C	°C	°C	°C	°C	°C
P80-SE	108.9	4.182	242.5	705.2	81.1	69.3	150.4	99.1	73.6	172.7	74.9
P100-RX	156.1	3.118	311.7	875.5	117.2	76.7	193.9	159.0	90.0	249.0	93.2
P130-RX	161.3	3.163	400.8	870.8	121.9	76.3	198.2	165.3	90.6	255.9	93.9
P160-RXi-B	180.9	3.811	421.3	778.6	142.2	72.7	214.9	190.4	80.6	271.0	84.4
Pegasus HP	171.3	3.363	500.0	819.5	131.9	72.7	204.6	176.6	83.7	260.3	87.1
P180-NX	180.9	3.44	466.7	836.0	140.0	87.7	227.7	184.9	97.8	282.7	101.7
P200-RX	202.1	3.622	525.0	838.0	158.7	90.7	249.4	209.8	101.4	311.2	106.2
P220-RXi	198.1	3.282	579.3	878.3	153.4	98.0	251.4	202.6	111.9	314.5	116.7
Olympus HP	171.8	2.971	640.0	935.8	129.0	104.9	233.9	168.1	121.2	289.3	125.0
P300-PRO	170.7	2.691	785.0	1003.9	126.0	142.0	268.0	157.8	161.3	319.1	165.0
Titan	191.5	2.734	1020.0	1009.3	143.2	139.6	282.8	184.2	152.1	336.3	156.8
P400-PRO	187.0	2.709	1045.0	1012.3	139.4	138.6	278.0	179.2	157.7	336.9	162.4
P500-PRO GL	176.7	3.072	1238.0	917.6	133.7	120.9	254.6	169.7	135.6	305.3	139.6
P550-PRO GL	180.6	2.977	1320.0	941.6	136.3	140.9	277.2	164.6	157.7	322.3	161.2
Nike	188.3	2.780	1900.0	994.9	141.0	156.4	297.4	172.3	172.8	345.1	176.8
P1000-PRO	171.0	3.267	2328.0	918.0	129.3	152.1	281.4	148.9	169.0	317.9	171.7
Lynx	186.1	2.934	3600.0	988.3	139.9	157.5	297.4	168.9	175.3	344.2	179.2

and

Table 9. The second series values of calculated data for JetCat P80 engine.

NE rpm	T2t	α.ex	Ff	T3t	ΔTtT	ΔTJ	ΔTE	ΔTtT.id	ΔTJ.p.id	ΔTE.id	ΔTJ.id
	°C	-	g/min	°C	°C	°C	°C	°C	°C	°C	°C
34,951	26.5	5.221	42.155	601.0	11.0	4.4	15.4	13.1	5.0	18.1	5.0
47,160	32.5	5.366	55.144	592.2	16.3	10.1	26.4	19.4	11.4	30.8	11.5
55,320	38.5	5.154	67.739	618.3	21.5	15.0	36.5	25.6	16.9	42.5	16.9
68,295	49.6	5.490	86.097	595.8	31.4	22.3	53.7	37.4	25.2	62.6	25.3
79,369	60.4	5.767	104.084	581.4	41.2	29.6	70.8	49.1	33.2	82.3	33.5
88,787	72.2	5.837	124.259	586.1	51.7	36.3	88.0	61.7	40.6	102.3	41.1
97,786	83.0	5.660	146.851	610.2	61.1	45.6	106.7	72.8	51.0	123.8	51.6
105,423	94.4	5.545	170.455	630.1	70.9	53.3	124.2	84.5	59.6	144.1	60.4
112,706	105.4	5.256	198.901	666.3	80.0	64.5	144.5	95.4	71.9	167.3	73.0
115,675	110.1	5.057	212.371	690.3	83.7	70.0	153.7	99.7	78.1	177.8	79.3

present the second series of data results at take-off regime, for JetCat and AMT models, sorted by thrust, and at each working regime for JetCat P80 model.

The third series of data results are defined by the compressor and turbine total adiabatic efficiency $\left({\eta }_{C.t}\right)$, $\left({\eta }_{T.t}\right)$, compressor outlet total pressure $\left(p_{2.t}\right)$ and specific actual enthalpy $\left(h_{2.t}\right)$, turbine total expansion degree $\left({\delta }_{T.t}\right)$ and outlet pressure $\left(p_{4.t}\right)$, gases mass flow $\left(G_f\right)$, jet nozzle outlet gas velocity $\left(C_5\right)$, Mach number $\left(M_5\right)$, velocity coefficient $\left({\mathsf{\phi }}_J\right)$, and thermal efficiency $\left({\eta }_{te}\right)$.

Table 10. The third series of data results, at take-off regime, for JetCat and AMT micro jet models.

Engine Model	ηCt	ηTt	i2t	p2t	δT	p4t	Gf	C5	ϕJ	M5	η.te
	-	-	kW	bar a	-	bar a	kg/s	m/s	-	-	-
P80-SE	0.82	0.82	382.763	2.2839	1.5335	1.4297	0.2440	397.5	0.962	0.663	11.20
P100-RX	0.72	0.74	430.619	2.8797	1.8541	1.4910	0.2352	425.2	0.908	0.658	9.61
P130-RX	0.72	0.74	435.896	2.979	1.9065	1.5000	0.3067	423.9	0.902	0.659	9.69
P160-RXi-B	0.74	0.75	455.906	3.4754	2.2378	1.4909	0.3870	408.2	0.929	0.677	10.79
Pegasus HP	0.72	0.75	446.089	3.1776	2.0547	1.4846	0.4063	411.0	0.914	0.661	9.67
P180-NX	0.74	0.76	455.906	3.4754	2.1090	1.5820	0.3878	451.3	0.929	0.728	11.92
P200-RX	0.74	0.76	477.587	3.9719	2.3464	1.6250	0.4588	457.8	0.925	0.747	12.90
P220-RXi	0.74	0.76	473.411	3.8726	2.2202	1.6745	0.4597	478.6	0.917	0.763	12.81
Olympus HP	0.74	0.77	446.617	3.2769	1.8720	1.6804	0.4607	499.3	0.917	0.763	12.64
P300-PRO	0.80	0.80	445.425	3.5251	1.7527	1.9308	0.5131	584.7	0.928	0.877	15.74
Titan	0.75	0.78	466.746	3.7733	1.9317	1.8752	0.6770	579.0	0.944	0.873	15.68
P400-PRO	0.77	0.78	462.11	3.7733	1.8931	1.9135	0.6874	577.5	0.925	0.867	15.46
P500-PRO GL	0.78	0.79	451.581	3.5747	1.8994	1.8068	0.9206	534.4	0.932	0.834	14.96
P550-PRO GL	0.80	0.83	455.59	3.7733	1.8415	1.9671	0.9520	577.7	0.936	0.900	16.96
Nike	0.80	0.82	463.387	3.9719	1.8571	2.0532	1.2817	611.7	0.941	0.935	17.78
P1000-PRO	0.85	0.87	445.747	3.7733	1.7452	2.0757	1.8388	598.2	0.942	0.946	19.92
Lynx	0.81	0.83	461.225	3.9719	1.8354	2.0775	2.5600	612.9	0.938	0.940	18.82

and

Table 11. The third series values of calculated data for JetCat P80 engine.

NE rpm	Maf	Gf	πCt	i2t	ηCt	δTt	p4t	C5	M5	η.te	ηCt∙ηTt
	kg/s	kg/s	-	kW	-	-	bar a	m/s	-	-
34,951	0.045	0.0458	1.1368	299.790	0.860	1.0612	1.0368	100.5	0.165	0.77	0.722
47,160	0.061	0.0615	1.2070	305.813	0.859	1.0932	1.0686	151.2	0.251	1.80	0.722
55,320	0.071	0.0726	1.2690	311.839	0.827	1.1217	1.095	184.5	0.304	2.57	0.694
68,295	0.097	0.0982	1.4023	322.992	0.818	1.1886	1.1419	224.1	0.378	4.03	0.687
79,369	0.123	0.1245	1.5521	333.851	0.827	1.2599	1.1924	257.0	0.443	5.57	0.695
88,787	0.148	0.1504	1.7133	345.727	0.819	1.3371	1.2401	284.5	0.494	6.90	0.688
97,786	0.170	0.1725	1.8859	356.607	0.824	1.4005	1.3032	319.5	0.552	8.44	0.692
105,423	0.193	0.1962	2.0636	368.104	0.818	1.4714	1.3574	345.6	0.596	9.68	0.687
112,706	0.214	0.2172	2.2620	379.210	0.821	1.5275	1.4332	381.2	0.650	11.18	0.690
115,675	0.220	0.2233	2.3415	383.960	0.818	1.5443	1.4675	398.1	0.672	11.74	0.687

present the third series of data results at take-off regime, for JetCat and AMT models, sorted by thrust, and at each working regime for JetCat P80 model.

To expand the volume of presented data without duplication and provide a clearer comparison of values, the following charts from Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 depict specific data results derived from the tables presented in Table 6, Table 7, Table 8, Table 9, Table 10 and Table 11. These charts offer a visual representation of key performance metrics and parameters, making it easier to compare and analyze the data across different engine models and operating conditions.

As shown in Figure 4, for $\left({dl}_{T.t}\right)$ and $\left({dl}_{T.t.id}\right)$, a random variation from an engine to another is observed between 46.5% to 66.6% in the case of $\left({dl}_{T.t}\right)$ and between 47.5% to 70.8% in case of $\left({dl}_{T.t.id}\right)$. For $\left({dh}_J\right)$ and $\left({dh}_{J.id}\right)$, a random variation between 33.4% to 53.4% is also observed in the case of $\left({dh}_J\right)$ and between 29.2% to 52.3% in the case of $\left({dh}_{J.id}\right)$.

Figure 4. Total specific ideal/actual ratio enthalpies for JetCat and AMT models.

Figure 4. Total specific ideal/actual ratio enthalpies for JetCat and AMT models.

As shown in Figure 5, for $\left({dl}_{T.t}\right)$ and $\left({dl}_{T.t.id}\right)$, a decreasing variation from the idle regime to the maximum is observed, with some random values between 71.2% to 54.8% in the case of $\left({dl}_{T.t}\right)$, and between 72.3% to 56.5% in the case of $\left({dl}_{T.t.id}\right)$. For $\left({dh}_J\right)$ and $\left({dh}_{J.id}\right)$, an increasing variation is observed with some random values between 28.8% to 45.2% in the case of $\left({dh}_J\right)$ and between 27.7% to 43.5% in the case of $\left({dh}_{J.id}\right)$.

Figure 5. Total specific ideal/actual ratio enthalpies for JetCat P80 micro jet model.

Figure 5. Total specific ideal/actual ratio enthalpies for JetCat P80 micro jet model.

As shown in Figure 6, for $\left({dT}_{CH.t}\right)$, a random variation from an engine to another is observed, between 36.4% and 40.9%. For $\left({dT}_{T.t}\right)$, $\left({dT}_J\right)$ and $\left({dT}_E\right)$ a random variation between 85.7% to 91.7% is also observed in the case of $\left({dT}_{T.t}\right)$, between 85.9% to 92.5% in the case of $\left({dT}_J\right)$, and between 76.4% to 84.5% in the case of $\left({dT}_E\right)$.

Figure 6. Actual temperatures ratio for JetCat and AMT micro jets models.

Figure 6. Actual temperatures ratio for JetCat and AMT micro jets models.

As shown in Figure 7, for $\left({dT}_{CH.t}\right)$, an increasing variation from the idle regime to the maximum is observed, but with some random values between 34.3% and 40.7%. For $\left({dT}_{T.t}\right)$, $\left({dT}_J\right)$, and $\left({dT}_E\right)$ a continuous decreasing variation is observed between 98.7% to 91.3% in the case of $\left({dT}_{T.t}\right)$, between 99.5% to 92.4% in the case of $\left({dT}_J\right)$, and between 98.2% to 84.2% in the case of $\left({dT}_E\right)$.

Figure 7. Actual temperatures ratio for JetCat P80 micro jet model.

Figure 7. Actual temperatures ratio for JetCat P80 micro jet model.

As shown in Figure 8, for $\left({dT}_{T.t.id}\right)$, a random variation from an engine to another is observed, between 81.1% and 89.9%. For $\left({dT}_{J.p.id}\right)$, $\left({dT}_{J.id}\right)$, and $\left({dT}_{E.id}\right)$, a random variation is also observed between 83.8% to 91.6% in the case of $\left({dT}_{J.p.id}\right)$ and $\left({dT}_{J.id}\right)$ and between 72.7% to 82.3% in the case of $\left({dT}_{E.id}\right)$.

Figure 8. Ideal temperatures ratio for JetCat and AMT micro jet models.

Figure 8. Ideal temperatures ratio for JetCat and AMT micro jet models.

As shown in Figure 9, for $\left({dT}_{T.t.id}\right)$ $\left({dT}_{J.p.id}\right)$, $\left({dT}_{J.id}\right)$, and $\left({dT}_{E.id}\right)$, a continuous decreasing variation from the idle to the maximum regime is observed between 98.5% to 89.6 in the case of $\left({dT}_{T.t.id}\right)$, between 99.4% to 90.9 in the case of $\left({dT}_{J.p.id}\right)$ and $\left({dT}_{J.id}\right)$, and between 97.9% to 81.5 in the case of $\left({dT}_{E.id}\right)$.

Figure 9. Ideal temperatures ratio for JetCat P80 micro jet model.

Figure 9. Ideal temperatures ratio for JetCat P80 micro jet model.

All the percentage differences between the actual and ideal values for the previously presented data are shown in

Table 12. Percentage differences for JetCat and AMT micro jet models.

Engine Model	ΔdlTt	ΔdhJ	ΔdTT	ΔdTE	ΔdTJ.id	ΔdTJ
	%	%	%	%	%	%
P80-SE	0.17	−0.17	1.50	2.91	0.03	1.65
P100-RX	−0.57	0.57	3.42	5.54	0.07	2.64
P130-RX	−1.14	1.14	3.77	5.79	0.07	2.58
P160-RXi-B	−2.57	2.57	4.64	6.05	0.08	2.06
Pegasus HP	−1.06	1.06	4.23	6.18	0.08	2.58
P180-NX	−2.43	2.43	4.09	5.70	0.08	2.26
P200-RX	−3.16	3.16	4.72	6.09	0.11	2.11
P220-RXi	−1.96	1.96	4.32	6.27	0.09	2.74
Olympus HP	−2.30	2.30	3.45	5.31	0.09	2.43
P300-PRO	−0.80	0.80	2.41	4.73	0.07	2.89
Titan	−3.88	3.88	3.56	5.13	0.11	2.20
P400-PRO	−0.79	0.79	2.92	5.53	0.08	3.30
P500-PRO GL	−0.07	0.07	2.68	5.18	0.08	3.14
P550-PRO GL	0.25	−0.25	2.20	4.70	0.06	3.12
Nike	−2.89	2.89	3.00	4.78	0.08	2.43
P1000-PRO	−1.68	1.68	1.93	3.27	0.07	1.78
Lynx	−2.08	2.08	2.42	4.02	0.08	2.15

and

Table 13. Percentage differences for JetCat P80 model.

NE	ΔdlTt	ΔdhJ	ΔdTT	ΔdTE	ΔdTJ.id	ΔdTJ
rpm	%	%	%	%	%	%
34,951	−1.15	1.15	0.23	0.30	0.00	0.07
47,160	−1.30	1.30	0.35	0.51	−0.01	0.18
55,320	−1.34	1.34	0.46	0.67	0.00	0.23
68,295	−1.40	1.40	0.69	1.01	0.00	0.36
79,369	−1.47	1.47	0.91	1.35	0.02	0.48
88,787	−1.52	1.52	1.14	1.65	0.01	0.59
97,786	−1.59	1.59	1.32	1.94	−0.01	0.75
105,423	−1.64	1.64	1.49	2.19	0.02	0.85
112,706	−1.69	1.69	1.62	2.43	0.02	1.01
115,675	−1.71	1.71	1.65	2.50	0.03	1.07

. Considering that the previous results are percentage data, percentage parameters are also used to make a direct comparison of the main engine parameters and performances for JetCat and AMT micro jets at take-off regime and for the JetCat P80 micro jet at various working regimes. For JetCat and AMT micro jets, the percentage data are calculated by comparing the data of each engine model to the Lynx engine (AMT Netherlands), which is the most powerful engine in the dataset. For the JetCat P80, the percentage data are calculated by comparing the data of each working regime to the maximum regime [41,42]. This approach ensures a consistent basis for comparison, highlighting the relative performance of each engine model and operating condition.

The following mathematical relations define the percentage parameters:

$$\begin{array}{cc}{dprm}_{eng}={\displaystyle \frac{prm}{{prm}_{Lynx}}}\cdot 100& \left[\%\right]\end{array}$$

(28)

$$\begin{array}{cc}{dprm}_{P80}={\displaystyle \frac{prm}{{prm}_{max}}}\cdot 100& \left[\%\right]\end{array}$$

(29)

Figure 10 and Figure 11 present the percentage variation of power of turbine $\left(P_T\right)$, jet nozzle $\left(P_J\right)$, and expansion process $\left(P_E\right)$, sorted by thrust, for JetCat and AMT models and the JetCat P80 model.

Figure 10. Percentage variation of powers for JetCat and AMT models.

Figure 10. Percentage variation of powers for JetCat and AMT models.

Figure 11 shows, for $\left(P_T\right)$ $\left(P_J\right)$ and $\left(P_E\right)$, a continuous increasing variation from the idle to maximum regime, starting from 2.7% in the case of $\left(P_T\right)$, from 1.31% in the case of $\left(P_J\right)$, and from 2.1 in case of $\left(P_E\right)$.

Figure 11. Percentage variation of powers for JetCat P80 model.

Figure 11. Percentage variation of powers for JetCat P80 model.

Figure 12 and Figure 13 present the percentage variation of air mass flow $\left(M_{af}\right)$, fuel flow $\left(F_f\right)$, and turbine inlet temperature $\left(T_{3.t}\right)$, sorted by thrust, for JetCat and AMT models and the JetCat P80 model.

Figure 12. Percentage variation of main parameters for JetCat and AMT models.

Figure 12. Percentage variation of main parameters for JetCat and AMT models.

Figure 13. Percentage variation of main parameters for JetCat P80 model.

Figure 13. Percentage variation of main parameters for JetCat P80 model.

Figure 14 and Figure 15 present the percentage variation of jet nozzle outlet gas velocity $\left(C_5\right)$, velocity coefficient $\left({\mathsf{\phi }}_J\right)$, and thermal efficiency $\left({\eta }_{te}\right)$, sorted by thrust, for JetCat and AMT models and the turbine total expansion degree $\left({\delta }_{T.t}\right)$ for the JetCat P80 model instead of the velocity coefficient.

Figure 14. Percentage variation of jet nozzle parameters and performance for JetCat and AMT models.

Figure 14. Percentage variation of jet nozzle parameters and performance for JetCat and AMT models.

Figure 15. Percentage variation of jet nozzle parameters and performance for JetCat P80 model.

Figure 15. Percentage variation of jet nozzle parameters and performance for JetCat P80 model.

Figure 10, Figure 12 and Figure 14 show in percentage values the comparison of main parameters and performances between the engines in order to obtain the actual thrust. These graphs show, on a scale of thrust from 97 to 1569 [N], by actual values, how many of the main parameters and performances are reported at maximum values of the Lyns engine.

In the case of the JetCat P80 engine, as seen in Figure 11, Figure 13 and Figure 15, it is observed that the engine’s main parameters and performances, when reported to the maximum regime, exhibit an increasing variation but start from different percentage values. This indicates that the parameters do not vary uniformly.

For lower thrust engines with lower thermal efficiency, it is observed that the percentage of power distribution between the turbine and jet nozzle is 67% to 33%. For higher thrust engines, this distribution shifts to 47.5% to 52.5%. This shift in power distribution reflects how different thrust levels and thermal efficiencies affect the balance of power between the turbine and the jet nozzle, providing insights into the performance characteristics of engines across different thrust ranges.

6. Discussion

Following the calculation, a significant amount of data is generated, sufficient for a thorough thermodynamic analysis at take-off regime for JetCat and AMT engine models, and at various working regimes for the JetCat P80. This study examines the differences between the actual and ideal percentage values of the main parameters, coefficients, and performance metrics for each micro jet engine to observe the limits of variation. Special attention is given to the energy distribution during the expansion process between the turbine and jet nozzle. This study aims to understand how this energy distribution varies from the smallest to the largest engine and from the minimum to maximum regime.

The comparison method is based on percentage differences. Therefore, a series of percentage differences of new coefficients are defined in relation to the turbine and jet nozzle total specific work, as well as enthalpy and temperature ratios. These differences are calculated to identify small variations, based on which the coefficients can be declared as constant. The following mathematical relations define these percentage differences of the new coefficients [41,42]:

$${\mathsf{\Delta }dl}_{T.t}={dl}_{T.t}-{dl}_{T.t.id}$$

(30)

$${\mathsf{\Delta }dh}_J={dh}_J-{dh}_{J.p.id}$$

(31)

$${∆dT}_{T.t}={dT}_{T.t}-{dT}_{T.t.id}$$

(32)

$${∆dT}_E={dT}_E-{dT}_{E.id}$$

(33)

$${∆dT}_{J.id}={dT}_{J.id}-{dT}_{J.p.id}$$

(34)

$${∆dT}_J={dT}_J-{dT}_{J.id}$$

(35)

From the Table 12, for a difference engine thrust of 1472, it is observed that the maximum differences for $\left({\mathsf{\Delta }dl}_{T.t}\right)$ and $\left({\mathsf{\Delta }dh}_J\right)$ is the same 3.0%, in case of $\left({∆dT}_{T.t}\right)$ and $\left({∆dT}_{T.t}\right)$ is 3.0% and 3.3% and in case of $\left({∆dT}_{J.id}\right)$ and $\left({∆dT}_J\right)$ is 0.08% and 1.46%. This means that the engines have almost similar thermodynamic working process with small variations.

From the Table 13 it is observed that the maximum differences for $\left({\mathsf{\Delta }dl}_{T.t}\right)$ and $\left({\mathsf{\Delta }dh}_J\right)$ is the same at 0.70%; in the case of $\left({∆dT}_{T.t}\right)$ and $\left({∆dT}_E\right)$, they are 1.4% and 2.2%; and in case of $\left({∆dT}_{J.id}\right)$ and $\left({∆dT}_J\right)$, they are 0.03% and 1.0%. The reported parameters have close variation from one regime to another, which means that they can be used as almost constant parameters, useful in model calculations.

From these values, it is observed that the engines exhibit almost similar thermodynamic working processes with minor variations. The current coefficients demonstrate close variation from one regime to another, indicating that they can be used as nearly constant coefficients in calculations for specific models where sufficient parameters are not known. One of the most valid coefficients, which has been demonstrated to be taken as constant, is $\left({∆dT}_{J.id}\right)$, as the percentage values are lower than 0.1% across regimes from idle to takeoff.

7. Conclusions

The objective of this paper is to provide reliable performance evaluation and analysis data for a series of micro jet models, based on a specific data library and analytical calculation models. This information is valuable for gas turbine engine education and micro jet experimental applications.

The main idea of this study is to observe the extent of percentage differences between the actual and ideal values of the engine’s main parameters, coefficients, and performances. If the percentage differences are very close, they can be considered constant. In such cases, they can be applied to actual values in practical calculation models when there are insufficient data available to make a complete engine performance calculation. The results are presented in various formats, including actual values, graphical representations, and percentage difference values. This multi-faceted presentation provides a better understanding of how the engine’s main parameters, coefficients, and performances vary from one engine to another for JetCat and AMT Netherlands models, and from idle to maximum regimes for the JetCat P80.

In conclusion, this paper equips students and other users with a more comprehensive understanding of the thermodynamic processes and working principles of micro jet engines. By defining these new constants and demonstrating their reliability across different operating regimes, users can incorporate them into various model calculations. This method allows for more practical and accurate computational approaches, enhancing predictive analysis and design optimization in micro jet engine technology. Ultimately, these findings significantly contribute to the advancement of engineering education and research, offering robust parameters for more efficient and effective problem-solving.