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Mathematical model of the combustion process for turbojet engine based on fuel properties

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• Published: 28 March 2022
• Volume 13 , pages 1309–1316, ( 2022 )

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• Tomasz Białecki   ORCID: orcid.org/0000-0002-6678-5950 1  

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This paper presents the impact of the alternative fuels properties on the parameters characterizing the combustion process in a turbojet engine, expressed in the form of a mathematical model. Laboratory tests, bench tests and a regression analysis of the obtained results were conducted. The developed and published combustion process models were briefly described. It has been demonstrated that these models were insufficient in taking into account the impact of fuel properties on the course of the combustion process. The experimental data enabled developing a mathematical model of the combustion process using statistical methods. The developed model, unlike other currently known models, takes into account the chemical composition of the fuel to a greater extent, which is characterized by its physicochemical properties. Mathematical model enables predicting engine operating parameters and the emissions characteristics, based on analysing laboratory test results, and can be used as a tool verifying the environmental impact of new fuels, through predicting the exhaust gas emissions.

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Introduction

The primary fuel used to power gas turbine engines is fossil jet fuel. For decades, the chemical composition of these fuels was modifying together with technological progress and experience in exploitation [ 1 ].

The fuels are mainly mixtures of hydrocarbons, obtained from processing of crude oil (conventional fuels), supplemented with enriching additives that improve their operational parameters [ 2 ]. Jet fuel is composed primarily of paraffinic and cycloparaffinic hydrocarbons, aromas and olefins [ 3 ]. The composition depends on the used crude oil and its refining technology. For this reason, this type of fuel is not defined by the content of individual hydrocarbons, but through regulatory requirements.

Jet fuel chemical composition determines their operational properties. The content of individual hydrocarbons determines fuel properties, such as calorific value, chemical stability, combustion quality and many other operational properties. This relationship can be expressed as follows [ 4 ]:

where F p is the fuel properties and Ch f is the chemical composition of fuel.

Recently, the petroleum industry has been focused on developing technologies that utilize non-petroleum raw materials in the course of production [ 5 ]. This is due to economic, security and environmental factors. Currently, there are seven approved technologies of synthetic components originating from unconventional sources, which can be blended with fossil fuel up to 50%. They are listed in standard ASTM D7655 [ 6 ] and are approved for use in aircraft turbine engines.

The complex process of approving fuels containing synthetic hydrocarbons for use in aircraft turbine engines resulted in limiting the previously widespread diversity of hydrocarbon groups within their composition. Changing the hydrocarbon structure of the fuel, besides affecting its properties, also modifies the mechanism of reactions, making up the complex combustion process.

The papers [ 7 , 8 , 9 , 10 , 11 , 12 , 13 ] were used as a basis to present the relationships between the parameters characterizing the combustion process in a turbojet engine and the physicochemical properties of the fuel (Fig.  1 ). The physicochemical properties of fuel are quantitative variables, since they can adopt specific numerical values. However, a review of the source literature enabled only defining them as qualitative variables. The reason for this is that these relationships were collected based on experiments and observations, without a determined quantitative impact on the combustion process.

Dependence of parameters characterizing the combustion process on fuel properties (H c —neat heat of combustion, D—density, V—viscosity, T 90 —90% distillation temperature, AP—aromatic content, C f —fuel consumption, T 3 —combustion chamber temperature, E GT —exhaust gas temperature, CO—CO emission, CO 2 —CO 2 emission, NO x —NO x emission)

Combustion process models

In the case of jet engines, combustion process consists of three stages. The first one involves fuel atomization, which means using the injectors to supply the combustion chamber with a fuel mist, and consists of very finely dispersed fuel droplets [ 14 ]. During the second stage, the air stream mixes with the fuel mist (fine fuel droplets increase the evaporation volume of the injected liquid), creating a fuel-air mixture. In the case of older engine designs, fuel is supplied to the combustion chamber via evaporators, where it evaporates in contact with a hot tube (evaporator), and the vapours are lifted with air to the combustion zone. The last stage is the ignition of the fuel-air mixture, resulting from a flame (in the combustion zone), and combustion, which spreads quickly in all areas where air and fuel are mixed within flammability limits [ 15 ].

Combustion process chemistry and rate depend on the engine design [ 16 ], as well as the properties of used fuel [ 17 ]. Fuel must be injected, vaporized and mixed with air in the combustion chamber, before combustion occurs. The extent to which these processes affect combustion greatly depends on the physicochemical properties of fuel. This relationship can be expressed as follows:

where E p is the engine operating parameters and E g is the exhaust gas emission.

Mathematical modelling can be applied for solving problems involving phenomenon repeatability or similarity. However, due to the successive stages of the combustion process (atomization, evaporation and ignition), and the large number of elementary hydrocarbon oxidation reactions, an attempt to describe it using a model is not a simple task.

The following sections present the previously developed combustion process models, together with their limitations.

ARP 1533C model

Hydrocarbon fuel combustion model was presented in ARP 1533C [ 18 ]. It includes a methodology for calculating the emission indices of exhaust gas components, fuel-to-air ratio and the combustion efficiency based on carbon monoxide (CO), carbon dioxide (CO 2 ), hydrocarbon and nitrogen oxides (NO x ) measurements. The combustion process model within the procedure presents an equation for the combustion of a single mole of hydrocarbon fuel and atmospheric air.

This model reduces multi-component hydrocarbon fuels to the form of an averaged chemical formula, e.g. C 11,6 H 22 for Jet A fuel. However, hydrocarbons with the same number of carbon atoms exhibit various physicochemical properties. Hence, the conclusion that the averaged chemical formula adopted for various fuels (even petroleum-derived) is a huge simplification. Furthermore, the authors of [ 19 ] confirmed that one chemical compound with several isomers, thus the same chemical formula, is characterized by different physicochemical properties, assigned to individual isomers.

Computational fluid dynamics (CFD) modelling analyses chemical transformation of hundreds of compounds expressed by thousands of chemical reactions, taking into account the fact that the number of elementary reactions of hydrocarbon oxidation depends on their structure and falls within a range from several hundred to several thousand [ 20 ]. However, a detailed numerical simulation of actual fuel combustion is still beyond reach, when it is applied to any fuel that is not a pure component or a mix of more than several components [ 21 ].

One of the applied simplifications is the process being represented by a minor number of elementary reactions, making up a subsequent reaction chain. These so-called global mechanisms are stoichiometric relationships, for which approximated kinetic equations can be determined [ 22 ].

Another method is modelling the combustion process using surrogate fuels. They are a simplified equivalent of fuel, composed of one or more selected hydrocarbons that represent the main fuel ingredients. The idea behind minimizing the number of ingredients is obtaining a model fuel that exhibits physicochemical properties and combustion characteristics similar to the conventional jet fuel [ 7 ]. Although there are numerous models, such simplification of the mix composition that best mimics real-life fuel being tested remains a significant challenge.

Furthermore, it should be noted that CFD modelling often omits the properties of the tested fuel or takes their values from a library (averaged parameters). The authors of [ 23 ] studied the combustion of fuel and its blends with added components, as well as the emission characteristics, describing fuel properties with only two values, i.e., density and viscosity, whereas [ 24 ] describes conventional and synthetic fuels only with an average formula C m H n , the hydrogen-to-carbon ratio (information derived from the average formula C m H n ) and net heat of combustion. As highlighted before, fuel composition impacts its physicochemical properties, therefore, adopting average data from libraries or taking into account the minimum number of parameters is a very simplified approach and prevents studying the impact of fuel properties on the combustion process.

Statistical model

During literature review, one publication [ 25 ] was found which presents combustion process model for turbojet engine fuel, developed using statistical methods. However, its objective was to statistically analyse the ICAO Aircraft Engine Emissions Databank, in order to estimate the emissions in new turbofan engines. The development of the obtained statistical model did not take fuel properties into account and was based purely on a historical database. Furthermore, no statistical model of the combustion process based on the experimental studies was found.

The presented and characterized combustion process models focus on its various elements; however, in the majority of cases they assume that fuel is the element of the system, the change of which is insignificant or that is not changed at all. This is most probably due to the fact that so far, the modelling process was implemented by turbojet engine engineers. Their primary objective was to improve the engine operating parameters, which was not associated with changing the used fuel as the propulsion source. The process of introducing synthetic fuels to the aviation industry enforced the necessity to conduct work on modifying fuel physicochemical composition and stop perceiving it as a constant value in modelling the combustion process.

In the light of the above, it seems appropriate to analyse combustion process with greater emphasis on fuel, as a variable within this process, which has been defined as the objective of this paper.

3. Materials and methods

Tested fuels.

Conventional jet fuel and its blends with synthetic components form two different technologies approved by ASTM D7566 and biobutanol, which were used in this study and marked as follows:

Conventional jet fuel (Jet A-1)—Jet;

Blend of Jet A-1 with synthetic component from hydroprocessed esters and fatty acids (HEFA), feedstock: used cooking oil (75:25)—25UCO;

Blend of Jet A-1 with synthetic component from HEFA, feedstock: used cooking oil (50:50)—50UCO;

Blend of Jet A-1 with synthetic component from HEFA, feedstock: used cooking oil (25:75)—75UCO;

Blend of Jet A-1 with synthetic component from HEFA, feedstock: camelina (50:50)—50CAM;

Blending synthetic component from technology approved by ASTM D7566 (other than HEFA)—S;

Blend of Jet A-1 with synthetic component from S (75:25)—25S;

Blend of Jet A-1 with synthetic component from S (50:50)—50S;

Blend of Jet A-1 with synthetic component from S (25:75)—75S;

Blend of Jet A-1 with biobutanol (75:25)—25but;

Blend of Jet A-1 with biobutanol (50:50)—50but;

Blend of Jet A-1 with biobutanol (25:75)—75but.

HEFA and S components are specified in standard ASTM D7566 and, after blending with fossil fuel in a volume of up to 50%, can be supplied to aviation turbine engines, whereas biobutanol (n-butanol) is an alcohol that was produced through fermenting the C 5 and C 6 sugars. Choosing n-butanol, unlike other components, resulted from the need to increase the range of measured fuel properties, even going beyond the area set out by normative documents. Prepared fuel samples were subjected to selected laboratory tests (Table 1 ), determined by their impact on the combustion process.

Bench testing was conducted on a Miniature Jet Engine Test Rig (MiniJETRig) with miniature turbojet engine GTM 140 for jet fules combustion process research. The test rig is used in research and development work, mainly for testing alternative fuels for aviation [ 26 , 27 , 28 ]. Its detailed description can be found in [ 29 ].

Previous research with the use of the test rig enabled developing a procedure for testing [ 30 ]. Nonetheless, given the fact that turbojet engine operating parameters change at various rotational speeds, only one speed, namely 70000 rpm, was selected for the purpose of the paper (Fig.  2 ). This speed corresponds to 30% of the maximum thrust achieved by the engine and characterizes its lowest thermal load.

Profile of engine test

Measurement equipment

Each tested fuel was bench-tested at least twice in the course of the research. The parameter results from the last 30 s (stabilization) were taken in the case of each engine test as measurement data sets that were averaged and considered as the measurement result (for T 3 , E GT and C f ) or as a value to calculate EI (for CO, CO 2 and NO x ).

The engine tests involved recording parameters characterizing the combustion process in a turbojet engine:

Thermodynamic state, namely fuel consumption, mean temperature in the combustion chamber (measured by six thermocouples located circumferentially) and the exhaust gas temperature (measured downstream of the exhaust nozzle),

Exhaust gas emissions, namely carbon monoxide, carbon dioxide and nitrogen oxides.

The details of the equipment used to measure the exhaust concentration of CO, CO 2 and NO x are shown in Table 2 .

Emissions of gaseous exhaust gas components: CO, CO 2 and NO x converted and presented in the form of emission indices, according to [ 31 ]:

where EI i is the emission index of species i , X i / CO/CO 2 is the mole fractions of species i / CO / CO 2 , x is the number of moles of carbon in a mole of fuel, and MW i / MW f is the molecular weights of species i /fuel.

Selected properties of the fuel samples are presented in Table 3 , while Table 4 presents the results of the bench tests.

The impact of synthetic components on the fuel properties, engine operating parameters and exhaust characteristics has been presented in previous works [ 27 , 32 , 33 ]. The obtained results were used as a database for regression analysis.

Development of mathematical model

The main purpose of developing the model was an attempt to describe the relationships presented in Fig.  1 using quantitative features. Multiple regression was used to determine the effect of many independent variables on each dependent variable. During the analysis, the fuel properties were treated as independent variables and the parameters characterizing the combustion process as dependent variables (Fig.  3 ). Stepwise regression in R software was used to determine their relationship. It involves developing a model, which initially does not contain any explanatory variable, while every successive step expands the model with only such variables that significantly predict the dependent variable.

Data used for regression analysis

Data sets are analysed with a certain, predetermined likelihood level, specified by a confidence interval. A confidence interval level of 95% was adopted, which means a 95% probability that the confidence interval will cover an unknown value of the estimated parameter.

The development of a regression model for each explained variable (parameter characterizing the combustion process) took into account only explanatory variables (fuel properties) shown in Fig.  1 . Nonetheless, this figure only shows their qualitative relationships, and applying the regression will enable their quantitative correlations.

Each constructed model was rated and verified through checking the significance of all explanatory variables (using Snedecor’s F distribution), significance of partial regression coefficient (using Student’s t distribution), matching of the theoretical and experimental curves (using the R 2 determination coefficient), lack of collinearity between independent variables (using the variance inflation factor), and conformity of the residuum and normal distributions (using the Shapiro–Wilk test).

The developed and positively verified linear regression equations presented together show the model of combustion process in a turbojet engine:

This analysis did not allow to build a regression equation for NO x . This may be due to the fact that fuel is not part of the NO x formation process and the nitrogen oxide formation rate might indirectly depend on fuel properties. These oxides are formed primarily as a result of atmospheric nitrogen oxidation at very high temperatures, encountered in the combustion chamber.

The regression equations of the dependence of combustion process characterizing parameters on fuel properties are shown in Fig.  4 .

Dependence of parameters characterizing the combustion process on fuel properties from regression equations from regression equations

The regression Eqs. ( 5 )–( 9 ) allow for the verification of qualitative relationships identified by the literature review. In addition, significant independent variables can be quantified using regression coefficients. Collected Eqs. ( 5 )–( 9 ) were treated as a combustion process mathematical model, taking into account the fuel properties to a greater extent than the previously known models.

However, this model has limitations, which include:

Input data falling within a range of fuel property intervals or in its close vicinity;

Output data falling within a range of the intervals of obtained test bench results or in its close vicinity;

The ability to apply the developed model for predicting results obtained only for the GTM 140 engine (used to conduct bench tests);

The ability to apply the model for predicting results obtained for specific engine operating conditions (predefined rotational speed).

Conclusions

The objective of the research was to develop a mathematical model of combustion process using regression analysis. In order to implement the above laboratory and bench tests of conventional jet fuel and its blends with alternative components were carried out.

The obtained results were used as a database to conduct a regression analysis of data resulting in regression models covering the impact of fuel physicochemical properties on selected parameters characterizing the combustion process. Compiled regression equations were treated as a combustion process mathematical model, taking into account the fuel composition characterized by its properties to a greater extent than the previously known models. Furthermore, the created linear regression model is enabled to quantitatively describe qualitative relationships, identified by the source literature review.

The developed combustion process mathematical model enables predicting engine operating parameters and the emissions characteristics, based on analysing laboratory test results. This model can be used as a tool verifying the environmental impact of new fuels, through predicting the exhaust gas emissions. The developed model relates only the engine used in bench tests, while the use of the methodology implemented in this research allows to extend the model to other engines. However, statistical data analysis and the resulting model are aimed at facilitating and extending the scope of inference about the impact of fuel properties on the combustion process.

The author further plans to conduct laboratory and test bench tests of other alternative fuels, with their results supplementing input data, in order to constantly update the combustion process model for a GTM 140 engine. A perspective to continue the research is conducting test rig studies involving other engines, including full-sized ones, in order to verify the obtained relationships within the developed mathematical model and to attempt to develop a general model that enables assessing the impact of fuel chemical composition on the parameters characterizing the combustion process.

Data availability

All data, models and code generated or used during the study appear in the submitted article.

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Acknowledgements

This research was funded by the Ministry of Science and Higher Education (Poland) for the project financed within the framework of the statutory activity. The author would like to thank Prof. Andrzej Kulczycki DSc, PhD Eng. and Wojciech Dzięgielewski for their support as thesis advisers.

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Białecki, T. Mathematical model of the combustion process for turbojet engine based on fuel properties. Int J Energy Environ Eng 13 , 1309–1316 (2022). https://doi.org/10.1007/s40095-022-00489-2

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Received : 17 September 2021

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Published : 28 March 2022

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DOI : https://doi.org/10.1007/s40095-022-00489-2

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This report presents the results of the design of a CESSNA 550 Citation engine. It first presents an historical review of the plane then a brief description of the working principle of compressors and gas turbines used in aircrafts. To make the analysis possible, it was necessary to make some assumptions. It was assumed that at entry to the first stage of the compressor there was no guide vanes and thus the flow angle is zero; a peripheral speed of 350 m/s (12000RPM) was considered which reduces stress on the blade and an initial pressure ratio of 8. On the turbine side the principal of conservation of energy was used to find the temperature drop throughout the turbine knowing the work done by the compressor and the mechanical efficiency and further assumptions were made to calculate other important parameters of the turbine. From the investigation it was concluded that the design will comprise a single shaft, seven stages compressor and two stages turbine.

A turbine jet engine has four main parts, They are compressor, combustion chamber, turbine and exhaust nozzle. Turbine jet engine operates at an open cycle called a jet propulsion cycle. A small-scale turbine jet engine comprises of the same element as the gas-turbine engine but in a small scale. Turbine jet engines are constructed mainly for air transportation while the turbine jet engines are developed for a wider purpose, ranging for research activity to hobbyist enthusiastic. Hence, this paper encompasses the design, fabrication, and testing a turbine jet engine. The engine is derived from an automobile turbocharger, which provided the turbine and compressor component. A combustion chamber is design and fabricated. Engine support system comprised of ignition, lubrication and fuel delivery system are installed at the engine. Thermocouple K-type are installed at four different stations on the engine flow path to measure the temperature. Fuel regulators are utilized to measure the fuel flow rate. The design of the combustion chamber is developed to make primary and secondary air takes paths so as to allow a series of combustion processes that help to increase the speed of a jet engine.

Pragyan Sarma

In this research paper an attempt has been made to design and analyze a small turbojet engine using scrap automobile parts, turbocharger being the major component. The engine has been designed and analyzed in Sikkim Manipal Institute of Technology using used turbochargers that can be used in modern automobile. In this design the thrust generated from the turbine section of a turbo charger is used to drive the power turbine generating the shaft power and also an attempt has been made to convert the turbo jet into turbo prop jet so as to increase its efficiency

Journal of Thermal and Fluid Science

RAME Publishers , Neeraj Kumar

This work is based on preliminary design of Ramjet engine. Ramjet engine is simplest type of gas turbine engine used, which consists of non-moving parts for its operation. It is mainly used for power generation at supersonic speeds. This type of engine is mostly used in missiles, with a few applications in aircrafts. The present approach to preliminary design was based on mathematical equations considering ideal conditions. For the design, aero-thermodynamic equations were used starting with intake and followed by diffuser, combustor and nozzle. For the design purpose, initially Mach 2 and thrust of 10 kN was selected as desired condition. Subsequently, it was analyzed for the varying Mach numbers starting from Mach 1.5 to Mach 4 at desired thrust of 10 kN. Also, the design was analyzed for varying thrust from 6 kN to 22 kN at desired Mach number of 2. The results obtained have been reported and effects of variation of parameters have been represented graphically. The graphs were obtained using GNU Octave 6.4.0. The design achieved could be used for further steps of CAD model generation and subsequent analysis for the required purpose. It can serve as a base for detailed design of the engine.

IJIRT Journal

The growth in competitiveness and complexity in an engine manufacturing industry has called for computer based advanced tool to estimate the operating parameters of an engine. A physics based tool was used to estimate the variations of thermodynamic parameters over different conditions. These parameters were analyzed using an engine performance program written in FORTRAN programming language. A computer program designated to simulate the simple turbojet engine thermodynamically for performance analysis has been programmed and tested. The result from the program shows that by changing the operating environment, compression rate and component efficiencies, the pressure and temperature at various nodes changes which made the propulsive efficiency to get changed. This program also provides values of specific fuel consumption for a particular operating condition as it depends on pressure ratio, intake velocity, mass flow rate of air and fuel. This report includes a discussion on operation of the FORTRAN program at several cases, with input and output to show the performance of the turbojet engine. Index terms-Turbojet Engine, FORTRAN, Propulsive Efficiency. Specific Thrust I.INTRODUCTION The development of the performance analysis code starts with the understanding of the basic concepts of jet engine theory. Empirical relations were used to calculate various parameters such as temperature and pressure at different stages, losses, velocities at inlet and outlet, thrust and specific fuel consumption. The procedure followed is an iterative process of turbojet engine performance, the values obtained for each iteration was analyzed and the graphical representation is carried out. In the performance code, the values of isentropic ram and jet efficiency, mechanical efficiency, turbine and compressor efficiencies are kept constant for each iteration and iterated over a set of values which was decided from the historical data i.e. range of compressor pressure ratio, turbine inlet temperature, operating speed also the component values. Other input values include: the mass flow rate, stagnation and static properties of fluid at inlet and outlet, gas constant, specific heat, turbine inlet temperature are also provided. Together with the above input parameters the code is designed to account for providing temperature and pressure at different stages, thrust and specific fuel consumption are provided as output. The output also includes the propulsive efficiency of engine, nozzle area and fluid density at exit. Performance analysis studies deals with the thermodynamic changes of the working fluid (air and products of combustion in most cases) as it flows through the engine. It is divided into two types of analysis namely: parametric cycle analysis (also known as design-point or on-design) and engine performance analysis (also known as off-design). Parametric cycle analysis determines the performance of engines at different flight conditions and values of design choice (e.g., compressor pressure ratio) and design limit (e.g., combustor exit temperature) parameters. Engine performance analysis determines the performance of an engine at all flight conditions. II. METHODOLOGY The language chosen for this work is FORTRAN which has a linage of versions and here FORTRAN 95 is used. This is because it is suitable for science and engineering based computation. It has enhanced data type facilities. The Intel Visual FORTRAN software was used to develop, compile, run and debug FORTRAN 95 program which is a graphical Integrated Development Environment (IDE).

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JSmol Viewer

Experimental investigation on the impact of sand particle size on the jet pump wall surface erosion.

1. Introduction

2. materials and methods, 2.1. jet pump, 2.2. experiment materials and device, 2.3. experiment methods, 2.3.1. visualization experiment of sand movement in the jet pump, 2.3.2. wall wear experiment of the jet pump, 3. results and discussion, 3.1. sand distribution and movement characteristics, 3.2. macroscopic wear characteristics of sand on wall surface, 3.3. microscopic wear characteristics of sand on wall surface, 4. conclusions and outlook, author contributions, institutional review board statement, informed consent statement, data availability statement, acknowledgments, conflicts of interest.

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Qian, H.; Liu, J.; Xu, M.; Fan, C.; Duan, Z. Experimental Investigation on the Impact of Sand Particle Size on the Jet Pump Wall Surface Erosion. J. Mar. Sci. Eng. 2024 , 12 , 1390. https://doi.org/10.3390/jmse12081390

Qian H, Liu J, Xu M, Fan C, Duan Z. Experimental Investigation on the Impact of Sand Particle Size on the Jet Pump Wall Surface Erosion. Journal of Marine Science and Engineering . 2024; 12(8):1390. https://doi.org/10.3390/jmse12081390

Qian, Heng, Jian Liu, Maosen Xu, Chuanhao Fan, and Zhenhua Duan. 2024. "Experimental Investigation on the Impact of Sand Particle Size on the Jet Pump Wall Surface Erosion" Journal of Marine Science and Engineering 12, no. 8: 1390. https://doi.org/10.3390/jmse12081390

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