Influence of the main geometrical parameters on the design and performance of mixed inflow turbines

Citation: Chelabi, M.A.; Dobrotvorskiy, S.; Basova, Y.; Aleksenko, B.A.; Edl, M.; Zdebor, J.; Machado, J. Influence of the Main Geometrical Parameters on the Design and Performance of Mixed Inflow Turbines. Appl. Sci. 2022,12, 12165. https://doi.org/10.3390/ app122312165 Academic Editor: JoséAntónio Correia Received: 29 October 2022 Accepted: 24 November 2022 Published: 28 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Influence of the Main Geometrical Parameters on the Design and Performance of Mixed Inflow Turbines Mohammed Amine Chelabi 1, Sergey Dobrotvorskiy 2, Yevheniia Basova 2, Borys A. Aleksenko 2, Milan Edl 3, Jan Zdebor 4and JoséMachado 5,* 1FERTIAL SPA Company, Industrial Zone SPA BP 40, Arzew 31200, Algeria 2Department of Mechanical Engineering Technology and Metal-Cutting Machines, National Technical University “Kharkiv Polytechnic Institute”, 2, Kyrpychova St., 61002 Kharkiv, Ukraine 3Department of Industrial Engineering and Management, Faculty of Mechanical Engineering, University of West Bohemia, 301 00 Plzen, Czech Republic 4Department of Power System Engineering, Faculty of Mechanical Engineering, University of West Bohemia, 301 00 Plzen, Czech Republic 5MEtRICs Research Center, Campus of Azurém, University of Minho, 4800-058 Guimarães, Portugal *Correspondence: [email protected] Featured Application: This study consists of a contribution to the design of mixed inflow turbines with the view to improve their performance when integrated in internal combustion engines. Abstract: The blade shape is of great interest to hybrid turbine designers, due to its significant and direct impact on turbine performance. The inlet and outlet diameters of the vane affect the size of the rotor, which is limited because of the small space available in internal combustion engines. The effect of the ratio of the average inlet diameter and the average exducer inlet diameter on the performance of a mixed inlet turbine will be the focus of this study, which consists of two cases included herein for the purpose of illustrating the means of improving rotor performances and controlling the flow mass rate. In the first case, we achieved this by changing the average diameter of the exducer inlet, while, in the second one, we achieved this by changing the average inlet diameter. Additionally, the angles of the inlet and outlet blades were recalculated to preserve the same blade profile and to eliminate the effect of curvilinearity. It was noted that the shape of the blade was very sensitive to changes in the ratio of the investigated diameters, and—in both cases—interesting results were obtained. First, an increase in output work and in total static isentropic efficiency by 2.16% and 2.15%, respectively, was generated, with a saving of 3.52% of the used mass flow and a lighter rotor compared to one that used to take up the same space by using fixed average inlet diameter blades. In the second case, there was an increase in the output work by 3.31%, and in the total static isentropic efficiency by 3.34%, but the rotor became heavier and required an increase in the mass flow used. Since inter-blade flows are very complex, three-dimensional and viscous—featuring various types of secondary and eddy flows—the CFX.15-CFD code was used in all models to solve the averaged Navier–Stokes equations. Keywords: mixed flow turbine; blade; exducer average root diameter; camberline 1. Introduction The issues of intensifying technological processes and increasing the efficiency of mixed-flow turbines, which are important components of internal combustion engines, are a priority in modern mechanical engineering. The peculiarity of such turbines lies in the simultaneous presence of an axial and a radial flow, with none of these flows being negligible. To increase the power of an engine, the quantity of air and fuel admitted into the combustion chamber must be increased. For this purpose, turbocharging is used—a technique that uses a turbocharger for the purpose of expanding the exhaust gases along Appl. Sci. 2022,12, 12165. https://doi.org/10.3390/app122312165 https://www.mdpi.com/journal/applsci Appl. Sci. 2022,12, 12165 2 of 21 a mixed or radial turbine to drive them to a centrifugal compressor interposed between the air inlet and the intake manifolds. In this process, a heat exchanger (an intercooler) is generally added to lower the temperature. In this light, due to the great importance of using mixed and radial inflows turbines in the internal combustion engine field to raise their energy, they have attracted the attention of many researchers who are trying to find ways to improve their performance. To this end, Sawada et al. [ 1 ] presented a method of performance estimation for radial inflow turbines by basing the analysis on the so-called one-dimensional flow. Other, such as Milton et al. [ 2 ] studied the effect of reducing rotor blade inlet diameter on turbine performance. The results of this investigation indicated that maximum efficiencies were obtained with a Clearance ratio of 0.075 for the turbine, in which case the Clearance ratio was defined as the difference between the diameter at the stator blade trailing edge and the rotor blade inlet diameter divided by the diameter at the stator blade trailing edge. Other authors also had important contributions. Baines [ 3 ] presented the development of the primary and secondary flows in the rotors of radial-inflow turbines. Richard [ 4 ] studied radial turbine cooling to produce high specific work output at safe rotor stress levels. Anthony [ 5 ] designed and tested a small, high-pressure ratio radial turbine and described the aerodynamic rotor design. Takamura et al. [ 6 ] analyzed the influence of blade aerodynamic loading on the efficiency of the radial-inflow turbine. Chen et al. [ 7 ] studied the effect of blade loading in radial and mixed-flow turbines. Rodgers [ 8 ] analyzed the aerothermodynamic, structural, and economic factors which influence the performances of small gas turbines in the 50–100 kw class. Wallace et al. [ 9 ] developed a unified approach to the one-dimensional analysis and design of radial and mixed-flow turbines. Carlos et al. [ 10 ] realized a preliminary design and performance estimation of a radial inflow turbine (an automated approach). Zeng et al. [ 11 ] studied the effects of the squealer geometry of the turbine blade tip on the tip leakage flow and loss. Pesiridis et al. [ 12 ] analyzed an experimental evaluation of active flow control mixed-flow turbines for automotive turbocharger applications. Copeland et al. [ 13 ] combined experimental and computational results to study the various timescales associated with pulsed turbine operation. Hagen et al. [ 14 ] presented equation-oriented methods for the design optimization and performance analysis of radial inflow turbines. Dadone et al. [ 15 ] clarified a method for evaluating the off-design performance of a radial inflow turbine with comparison experiments. Wasserbauer et al. [16] programmed the Fortran program for predicting the off-design performance of radial-inflow turbines. Rodgers [ 17 ] established an advanced radial inflow turbine rotor program design with dynamic testing and studied direct conduction cooling at turbine inlet temperatures from 1478 K (2660 ◦ R) to 1700 K (3060 ◦ R). Lauriau et al. [ 18 ] implemented preliminary design considerations for variable geometry radial turbines with multipoint specifications. Alawadhi et al. [ 19 ] studied the design and optimization of a radial turbine to be used in a Rankine cycle operating with an OTEC System. Zahed et al. [ 20 ] presented the radial turbine design process. Rodgers [ 21 ] studied radial turbine-blade number and reaction effects. Doran et al. [ 22 ] presented an experimental performance evaluation of a 99.0 mm radial inflow nozzle turbine with varying shroud profiles; the results have helped to confirm some of the trends noted in the earlier tests. Gao et al. [ 23 ] validated a mean line performance prediction method for radial and mixed-flow turbines. Palfreyman et al. [ 24 ] carried out a numerical study of the internal flow field characteristics in mixed flow turbines. Chou et al. [ 25 ] treated the design and testing of a mixed-flow turbine for turbochargers. Minegishi et al. [ 26 ] developed a small mixed-flow turbine for automotive turbochargers. Wallace et al. [ 27 ] presented a design construction and testing of a mixed-flow gas turbine. Rajoo et al. [ 28 ] examined an experimental study on the performance of a variable geometry mixed flow turbine for an automotive turbocharger. Pesiridis et al. [ 29 – 31 ] realized an experimental evaluation of an active flow control mixed-flow turbine for automotive turbocharger applications. Lee et al. [32] analyzed a tilted turbine housing volute design under pulsating inlet condi- Appl. Sci. 2022,12, 12165 3 of 21 tions, and Ketata et al. [ 33 ] examined a numerical study of a vanned mixed flow turbine operating in various steady flow conditions. Ali et al. [ 34 ] studied the number of blade effects on the performance of a mixed turbine rotor, and Meghnine et al. [ 35 ] presented the influence of the volute cross-sectional shape on mixed inflow turbine performances. Hamel et al. [ 36 ] investigated a twin entry mixed flow turbine volute and its benefits for the eco-system. Morrison et al. [ 37 ] demonstrated the effects of flow conditions at the rotor inlet on mixed-flow turbine performance for automotive applications; Padzillah et al. [ 38 ] analyzed an experimental and numerical investigation of flow angle characteristics of an automotive mixed-flow turbocharger turbine. Udayakumar et al. [ 39 ] presented the impact of mixed-flow turbines on the efficiency of automotive turbocharger applications. Karamanis [ 40 ] examined the steady and unsteady performance of mixed-flow turbines for automotive turbochargers. Zhang et al. [ 41 ] created a 3D inverse design method for the optimization of radial and mixed-inflow turbines. Leonard et al. [ 42 ] analyzed the design and performance of mixed-flow turbine rotors with extended blade chords. Chelabi et al. [ 43 – 45 ] studied the effects of the cone, inlet, and deviation blade angles on mixed inflow turbine performances and analyzed the three-dimensional accelerating flow in a mixed turbine rotor. A smaller number of blades was suggested for a cone angle of 20 ◦ in the case of parallel surfaces [ 43 ] because of parameters, such as a large surface, weight, inertia, and a wide range of operation, for maximum efficiency. Also, the machine experienced [ 44 ] a 3.71% and 3.67% increase in work output and efficiency, respectively. It has been established [ 45 ] that for larger absolute exit kinetic energies, for values of deviation blade angle between − 10 ◦ and − 20 ◦ , an exhaust diffuser is recommended to recover a part of it into a greater expansion ratio. Kononenko et al. [ 46 ] examined the deflections and frequency in the milling of thin-walled parts with variable low stiffness; Dobrotvorskiy et al. [ 47 ] developed an optimum thin-walled parts milling parameters calculation technique. This study will present the effect of inlet average diameter to exducer mean root diameter ratio on mixed inflow turbine performances in two cases. The first is by fixing the average inlet diameter and the second is by fixing the average exducer root diameter while preserving: •The same ratio values for the two cases discussed. • The same dawn profile needed to eliminate the effect of the camberline shape on the turbine performance, resulting in the need to calculate the new outlet blade angles in the first case and the inlet blade angle in the second case. • The inlet and outlet blade height needed to eliminate the flow convergence effect at 2d, resulting in the need to calculate the new hub and shroud radius at the outlet in the first case, and the new hub and shroud radius at the inlet in the second case. 2. Materials and Methods Due to the complexity and importance of the blade shape and its impact on the turbine’s efficiency, it is necessary to use an easy mathematical model that enables control of all angles and dimensions of the blade. To this end, we used the Bezier polynomial model of four degrees to determine the meridian plane and the camberline profile. 2.1. Initial Rotor Design 2.1.1. Meridian Plane The blade meridian plane (Figure 1) is characterized by its shroud and hub profiles; the relations below of the Bezier polynomial are respected (Equations (1) and (2)) to determine de meridional plan: r=(1−u)4r0+4u(1−u)3r1+6u2(1−u)2rc+4u3(1−u)r2+u4r3, (1) x=(1−u)4x0+4u(1−u)3x1+6u2(1−u)2xc+4u3(1−u)x2+u4x3(2) where u is acoefficient between 0 and 1. Appl. Sci. 2022,12, 12165 4 of 21 Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 24 r=(1−u)4r0+4u(1−u)3r1+6u2(1−u)2rc+4u3(1−𝑢)r2+u4r3, (1) x=(1−u)4x0+4u(1−u)3x1+6u2(1−u)2xc+ 4u3(1−𝑢)x2+u4x3 (2) where u is acoefficient between 0 and 1. (a) (b) (с) (d) Figure 1. The mixed turbine rotor dimension’s view: (a) Intel view; (b) Outlet view; (c) Rotor length view; (d) Rotor in 3D view. For the hub, the items of 0 and 3 are defined as follows (Equations (3) and (4)): 𝒙𝟎=𝟎; 𝒓𝟎=𝟏𝟐(𝑫𝟐−𝒃𝟐𝐬𝐢𝐧(𝛅𝟐)), (3) 𝒙𝟑=𝑿𝟏; 𝒓𝟑=𝟏𝟐𝑫𝟑𝒉. (4) and, for the shroud, the items of 0 and 3 are defined as follows (Equations (5) and (6)): 𝒙𝟎=𝒃𝟐𝐜𝐨𝐬(𝜹𝟐); 𝒓𝟎=𝟏𝟐(𝑫𝟐+𝒃𝟐𝐬𝐢𝐧(𝜹𝟐)) (5) 𝒙𝟑=𝑿𝟏; 𝒓𝟑=𝟏𝟐𝑫𝟑𝒔. (6) 2.1.2. Camberline Profile The camberline consists of a leading edge and a trailing edge; the first one is obtained based on the following relations (Equations (7) and (8)): 𝜃=𝜃𝑟𝑒𝑓+1 sin (𝛿2)∫tan (𝛽2𝑏)𝑑𝑥 𝑟 𝑥 𝑥𝑟𝑒𝑓 , (7) 𝑟=𝑟0ℎ+(𝑥−𝑥0ℎ)tan(𝛿2). (8) The rotor type A has a constant blade angle, and by merging Equations (7) and (8), the equation of the leading edge camberline becomes (Equation (9)): Figure 1. The mixed turbine rotor dimension’s view: ( a ) Intel view; ( b ) Outlet view; ( c ) Rotor length view; (d) Rotor in 3D view. For the hub, the items of 0 and 3 are defined as follows (Equations (3) and (4)): x0=0;r0=1 2(D2−b2sin(δ2)), (3) x3=X1;r3=1 2D3h. (4) and, for the shroud, the items of 0 and 3 are defined as follows (Equations (5) and (6)): x0=b2cos(δ2);r0=1 2(D2+b2sin(δ2)) (5) x3=X1;r3=1 2D3s. (6) 2.1.2. Camberline Profile The camberline consists of a leading edge and a trailing edge; the first one is obtained based on the following relations (Equations (7) and (8)): θ=θre f +1 sin(δ2) x Z xre f tan(β2b)dx r, (7) r=r0h+(x−x0h)tan(δ2). (8) Appl. Sci. 2022,12, 12165 5 of 21 The rotor type A has a constant blade angle, and by merging Equations (7) and (8), the equation of the leading edge camberline becomes (Equation (9)): θ=θre f +tan(β2b) sin(δ2)tan(δ2)Ln"(r0h−x0h tan(δ2)) + xtan(δ2) (r0h−x0h tan(δ2)) + xre f tan(δ2)#. (9) To complete the remainder of the camberline, it is necessary to calculate the trailing edge profile (Figure 2); this is where the Bezier polynomial is used. The following formulas are applied (Equations (10) and (11)): x=(1−u)4x0+4u(1−u)3x1+6u2(1−u)2xb+4u3(1−u)x2+u4x3, (10) θ=(1−u)4θ0+4u(1−u)3θ1+6u2(1−u)2θb+4u3(1−u)θ2+u4θ3. (11) Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 24 𝜃=𝜃𝑟𝑒𝑓+tan(𝛽2𝑏) 𝑠𝑖𝑛(𝛿2) tan (𝛿2) 𝐿𝑛[(𝑟0ℎ− x0h tan (𝛿2)) + 𝑥 tan (𝛿2) (𝑟0ℎ− x0h tan (𝛿2))+𝑥𝑟𝑒𝑓 tan (𝛿2) ]. (9) To complete the remainder of the camberline, it is necessary to calculate the trailing edge profile (Figure 2); this is where the Bezier polynomial is used. The following formulas are applied (Equations (10) and (11)): x=(1−u)4x0+4u(1−u)3x1+6u2(1−u)2xb+4u3(1−u)x2+u4x3, (10) 𝜃=(1−u)4𝜃0+4u(1−u)3𝜃1+6u2(1−u)2𝜃b+4u3(1−u)𝜃2+u4𝜃3. (11) Figure 2. The camberline blade view. 2.2. Numerical Simulation 2.2.1. Numerical Method Applied The task of obtaining solutions to the governing equations of turbomachinery flow represents one of the most challenging problems in science and research. In most instances, the mathematical formulation equations of the momentum, continuity, and energy are expressed as partial differential equations (PDE). Second-order partial differential equations—which must be solved within an irregular domain subject to various initial and boundary conditions—arise frequently. In general, these equations do not admit analytical solutions, except in very simplified cases. Therefore, resorting to digital resolution methods called CFD is necessary. The ANSYS-CFD is used in this project—where it is based on the finite volume methods in its calculation—because it is well suited to conservative problems. The first stage is to identify discrete locations (that generate a grid), in which the variables are to be calculated and defined by the numerical grid—which is essentially a discrete representation of the geometric domain on which the problem is to be solved. It divides the solution domain into a finite number of subdomains. This study used the unstructured hexahedral grid because it adapts well to turbomachine simulations. In the second and the final stages, the transport equations are solved after simplifying them in line with the nature of the flow. To reach more precise results, the grid corresponding to the surface located in the vicinity of the walls of the object were refined to obtain more exact thermodynamic results. Figure 2. The camberline blade view. 2.2. Numerical Simulation 2.2.1. Numerical Method Applied The task of obtaining solutions to the governing equations of turbomachinery flow represents one of the most challenging problems in science and research. In most instances, the mathematical formulation equations of the momentum, continuity, and energy are expressed as partial differential equations (PDE). Second-order partial differential equations—which must be solved within an irregular domain subject to various initial and boundary conditions—arise frequently. In general, these equations do not admit analytical solutions, except in very simplified cases. Therefore, resorting to digital resolution methods called CFD is necessary. The ANSYS-CFD is used in this project—where it is based on the finite volume methods in its calculation—because it is well suited to conservative problems. The first stage is to identify discrete locations (that generate a grid), in which the variables are to be calculated and defined by the numerical grid—which is essentially a discrete representation of the geometric domain on which the problem is to be solved. It divides the solution domain into a finite number of subdomains. This study used the unstructured hexahedral grid because it adapts well to turbomachine simulations. In the second and the final stages, the transport equations are solved after simplifying them in line with the nature of the flow. To reach more precise results, the grid corresponding to the surface located in the vicinity of the walls of the object were refined to obtain more exact thermodynamic results. Appl. Sci. 2022,12, 12165 6 of 21 2.2.2. Boundary Conditions The rotor type A of a mixed inflow turbine (with a constant inlet blade angle) with dimensions recorded in Table 1is under examination in this section. The general calculation conditions applicable are summarized as follows: the simulation domain was an interblading channel (as per the periodicity condition), the mesh used was an unstructured one in the shape of a hexahedron, the fluid used was an ideal air gas (ideal gas), the conditions at the channel inlet were the total pressure of 2.91 bars, the total temperature was 920 ◦ K, and the absolute flow angle was − 13 ◦ . The conditions at the channel outlet included the static pressure of 1 bar, the rotation speed of 98,000 rpm and the use of kε was a model of turbulence. In Figure 3, the different geometrical zones are regrouped and depicted. Table 1. The geometrical parameter values (in mm). b2D2R0h R0s X1 b3D3h D3-rms D3s δ2(◦)θ3(◦)β2b(◦)β3(◦) 17.99 83.58 36 47.57 40 25.79 27.07 55.7 78.65 40 −25 20 −52 Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 24 2.2.2. Boundary Conditions The rotor type A of a mixed inflow turbine (with a constant inlet blade angle) with dimensions recorded in Table 1 is under examination in this section. The general calculation conditions applicable are summarized as follows: the simulation domain was an inter-blading channel (as per the periodicity condition), the mesh used was an unstructured one in the shape of a hexahedron, the fluid used was an ideal air gas (ideal gas), the conditions at the channel inlet were the total pressure of 2.91 bars, the total temperature was 920 °K, and the absolute flow angle was −13°. The conditions at the channel outlet included the static pressure of 1 bar, the rotation speed of 98,000 rpm and the use of k-ε was a model of turbulence. In Figure 3, the different geometrical zones are regrouped and depicted. Figure 3. The different geometrical zones. Table 1. The geometrical parameter values (in mm). b2 D2 R0h R0s X1 b3 D3h D3-rms D3s 𝜹𝟐 (°) 𝜽𝟑 (°) 𝜷𝟐𝒃 (°) 𝜷𝟑 (°) 17.99 83.58 36 47.57 40 25.79 27.07 55.7 78.65 40 −25 20 −52 2.2.3. Grid Solution Dependency To find the optimal grid condition that indicated the smallest number of elements without generating a difference in the numerical results based on the evaluation of various grid conditions, grid independence was mandatory. Four grids of 107,244, 233,844, 333,372, and 415,030 elements were examined for the rotor type A. The analysis of the graphs revealed that the number of elements did not have any effect on the parameters of torque and the mass flow rates (Figure 4). The report also noted a negligible impact on the static pressure distribution around the rotor blade (Figure 5a,b). The efficiency graph reached stability at 333,372 elements. According to this indicator, the grid parameter corresponded to 333,372 elements—which were then applied in the numerical simulations. Figure 3. The different geometrical zones. 2.2.3. Grid Solution Dependency To find the optimal grid condition that indicated the smallest number of elements without generating a difference in the numerical results based on the evaluation of various grid conditions, grid independence was mandatory. Four grids of 107,244, 233,844, 333,372, and 415,030 elements were examined for the rotor type A. The analysis of the graphs revealed that the number of elements did not have any effect on the parameters of torque and the mass flow rates (Figure 4). The report also noted a negligible impact on the static pressure distribution around the rotor blade (Figure 5a,b). The efficiency graph reached stability at 333,372 elements. According to this indicator, the grid parameter corresponded to 333,372 elements—which were then applied in the numerical simulations. Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 24 Streamwise at radius mean 1.0 0.8 0.60.4 0.2 0.0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Static pressure (bar) Number of elements 107.244 233.844 333.372 415.030 Figure 4. Effect of element number on static pressure around the blade for an average radius. 400.000 300.000 200.000 100.000 number of elements 0.0 0.5 1.5 1.0 Torque(N.M) (a) 400.000 300.000 200.000 100.000 number of elements 0.000 0.015 0.030 0.045 0.060 mass flow (Kg/s) 400.000 300.000 200.000 100.000 number of elements 0.8350 0.8355 0.8360 0.8365 0.8370 0.8375 0.8380 total to static isen efficiency (b) (c) Figure 5. Effect of the number of elements on the result of the study: (a) torque, (b) mass flow rate, (c) efficiency. Figure 4. Effect of element number on static pressure around the blade for an average radius. Appl. Sci. 2022,12, 12165 7 of 21 Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 24 Streamwise at radius mean 1.0 0.8 0.60.4 0.2 0.0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Static pressure (bar) Number of elements 107.244 233.844 333.372 415.030 Figure 4. Effect of element number on static pressure around the blade for an average radius. 400.000 300.000 200.000 100.000 number of elements 0.0 0.5 1.5 1.0 Torque(N.M) (a) 400.000 300.000 200.000 100.000 number of elements 0.000 0.015 0.030 0.045 0.060 mass flow (Kg/s) 400.000 300.000 200.000 100.000 number of elements 0.8350 0.8355 0.8360 0.8365 0.8370 0.8375 0.8380 total to static isen efficiency (b) (c) Figure 5. Effect of the number of elements on the result of the study: (a) torque, (b) mass flow rate, (c) efficiency. Figure 5. Effect of the number of elements on the result of the study: ( a ) torque, ( b ) mass flow rate, (c) efficiency. 2.2.4. Numerical Model Validations The used numerical model was validated by the experimental work of Chen and Abidet [ 7 ], who found an agreement between the experimental and numerical results (Figure 6). Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 24 2.2.4. Numerical Model Validations The used numerical model was validated by the experimental work of Chen and Abidet [7], who found an agreement between the experimental and numerical results (Figure 6). Figure 6. Change in pressure distribution along the turbine canal length. Below, the expansion pressure ratio defined as the inlet stagnation pressure over the exit static pressure was represented along the blade axial direction. 2.3. New Design Theory It is known that the shape of the rotor is important in the design process of a mixedflow turbine because it affects the efficiency of the turbocharger. However, the space occupied by the structure is also important because of the difficulty in accommodating a large structure in the internal combustion engine structure. The influence of the ratio between the average diameter of the inlet and the average diameter of the outlet pipe on the characteristics of the mixed turbine is investigated in the two cases illustrated below. 2.3.1. A Fixed Value of the Inlet Means Diameter Case In the first stage of research, the average inlet diameter was chosen as constant, and the average diameter of the exducer root varied. In this stage, the average inlet diameter was preserved and—at the same time—the exducer average root diameter was varied. An iterative calculation described by Figure 7 was carried out to determine the value of the diameter ratio and to keep the same degree of the meridian plane. The ratio between the average inlet diameter and the average exducer root diameter is presented in Figure 8, and it was obtained from the following relation (Equations (12) and (13)): 𝑅= 𝐷2 𝐷3−𝑟𝑚𝑠, (12) 𝐷3−𝑟𝑚𝑠=√1𝑛(∑𝐷3,𝑖 2) 𝑛 𝑖=1 . (13) Figure 6. Change in pressure distribution along the turbine canal length. Appl. Sci. 2022,12, 12165 8 of 21 Below, the expansion pressure ratio defined as the inlet stagnation pressure over the exit static pressure was represented along the blade axial direction. 2.3. New Design Theory It is known that the shape of the rotor is important in the design process of a mixedflow turbine because it affects the efficiency of the turbocharger. However, the space occupied by the structure is also important because of the difficulty in accommodating a large structure in the internal combustion engine structure. The influence of the ratio between the average diameter of the inlet and the average diameter of the outlet pipe on the characteristics of the mixed turbine is investigated in the two cases illustrated below. 2.3.1. A Fixed Value of the Inlet Means Diameter Case In the first stage of research, the average inlet diameter was chosen as constant, and the average diameter of the exducer root varied. In this stage, the average inlet diameter was preserved and—at the same time—the exducer average root diameter was varied. An iterative calculation described by Figure 7was carried out to determine the value of the diameter ratio and to keep the same degree of the meridian plane. The ratio between the average inlet diameter and the average exducer root diameter is presented in Figure 8, and it was obtained from the following relation (Equations (12) and (13)): R=D2 D3−rms , (12) D3−rms =s1 n( n ∑ i=1 D2 3,i). (13) Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 24 Figure 7. Iterative calculation of the meridian plane based on the 𝑫𝟑−𝒓𝒎𝒔 verification. Figure 8. The inlet mean diameter and exducer mean root diameter presentation. Figure 7. Iterative calculation of the meridian plane based on the D3−rms verification. Appl. Sci. 2022,12, 12165 9 of 21 Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 24 Figure 7. Iterative calculation of the meridian plane based on the 𝑫𝟑−𝒓𝒎𝒔 verification. Figure 8. The inlet mean diameter and exducer mean root diameter presentation. Figure 8. The inlet mean diameter and exducer mean root diameter presentation. In order to study the same values of diameter ratio (1.4–1.45–1.5–1.55–1.6) and keep the same blade height at the outlet blade to eliminate the flow convergence effect at 2D, it is obligatory to calculate the new hub and shroud radius at the outlet. The outlet relative angle (Figure 9) was recalculated according to Equation (14), which maintained the same camberline slope at the rotor exit in order to preserve the same camberline shape and to eliminate its effect. a3=dθ dx x3 =2·tan(β3) D3 (14) Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 24 In order to study the same values of diameter ratio (1.4–1.45–1.5–1.55–1.6) and keep the same blade height at the outlet blade to eliminate the flow convergence effect at 2D, it is obligatory to calculate the new hub and shroud radius at the outlet. The outlet relative angle (Figure 9) was recalculated according to Equation (14), which maintained the same camberline slope at the rotor exit in order to preserve the same camberline shape and to eliminate its effect. 𝑎3=[𝑑𝜃 𝑑𝑥]𝑥3=2·tan (𝛽3) 𝐷3 (14) Figure 9. The outlet blade angle presentation. The new meridian plan geometries—with its camberline angle obtained and the different blade shapes—are presented respectively in Figures 10 and 11; all its dimension values are regrouped in Table 2. 010 20 30 40 5 10 15 20 25 30 35 40 45 50 D2 fixed case Radial lenght (mm) Axial length (mm) R=D2/D3-rms R = 1.4 R = 1.45 R = 1.5 R = 1.55 R = 1.6 010 20 30 40 5 0 5 10 15 20 25 _ _ _ _ _ D2 fixed case Cambrure angle (? Axial length (mm) R=D2/D3-rms R = 1.4 R = 1.45 R = 1.5 R = 1.55 R = 1.6 (a) (b) Figure 10. The new meridian plane geometries: (a) with the same camberline shape; (b) for a fixed value of average inlet diameter. Figure 9. The outlet blade angle presentation. The new meridian plan geometries—with its camberline angle obtained and the different blade shapes—are presented respectively in Figures 10 and 11; all its dimension values are regrouped in Table 2. Table 2. The meridian plan geometrical parameter values for a fixed value of average inlet diameter (in mm). b2D2R0h R0s β2b(◦) b3D3h D3-rms D3s β3(◦)A (mm2)R 17.99 83.58 36.008 47.57 20.00 25.79 38.7970 59.7 90.3770 −53.90 1046 1.4 17.99 83.58 36.008 47.57 20.00 25.79 32.9157 57.6 84.4957 −52.94 1029 1.45 17.99 83.58 36.008 47.57 20.00 25.79 27.0700 55.7 78.6500 −52.00 1013 1.5 17.99 83.58 36.008 47.57 20.00 25.79 21.1200 53.9 72.7000 −51.08 998 1.55 17.99 83.58 36.008 47.57 20.00 25.79 14.9810 52.2 66.5610 −50.18 986 1.6 Appl. Sci. 2022,12, 12165 16 of 21 Appl. Sci. 2022, 12, x FOR PEER REVIEW 18 of 24 (a) (b) Figure 22. Static pressure contours on extrados: (a) With the relative Mach number contours near the extrados; (b) For a fixed value of the average inlet diameter. Figure 22. Static pressure contours on extrados: ( a ) With the relative Mach number contours near the extrados; (b) For a fixed value of the average inlet diameter. Appl. Sci. 2022,12, 12165 17 of 21 Appl. Sci. 2022, 12, x FOR PEER REVIEW 19 of 24 (a) (b) Figure 23. Static pressure contours on extrados: (a) With the relative Mach number contours near the extrados; (b) For a fixed value of the average exducer root diameter. Figure 23. Static pressure contours on extrados: ( a ) With the relative Mach number contours near the extrados; (b) For a fixed value of the average exducer root diameter. Appl. Sci. 2022,12, 12165 18 of 21 The choice of the rotor material depends on the highest static temperature over the blades. A good distribution of the static temperature over the blade could result in a well-balanced thermal stress in the rotor. It may also indicate a lower temperature loss occurring near the blade. Figure 24 shows the static temperature contours on the exducers in the two cases. From the contours, one can see a drop in temperature from the inlet to the outlet, due to the acceleration of the flow through a converging passage. By comparing the contours of the two cases, it is obvious to notice that all blades have compatible thermal stress. Appl. Sci. 2022, 12, x FOR PEER REVIEW 20 of 24 The choice of the rotor material depends on the highest static temperature over the blades. A good distribution of the static temperature over the blade could result in a wellbalanced thermal stress in the rotor. It may also indicate a lower temperature loss occurring near the blade. Figure 24 shows the static temperature contours on the exducers in the two cases. From the contours, one can see a drop in temperature from the inlet to the outlet, due to the acceleration of the flow through a converging passage. By comparing the contours of the two cases, it is obvious to notice that all blades have compatible thermal stress. (a) (b) Figure 24. Static temperature contours on extrados: (a) For a fixed value of the average inlet diameter; (b) For a fixed value of the average exducer root diameter. Figure 24. Static temperature contours on extrados: ( a ) For a fixed value of the average inlet diameter; (b) For a fixed value of the average exducer root diameter. Appl. Sci. 2022,12, 12165 19 of 21 4. Conclusions The change in the complex of physical, gas-dynamic parameters and the efficiency of a mixed turbine depends on the ratio between the average diameter of the rotor at the inlet and the average diameter of the inlet of the exducer. A mathematical model for controlling the three-dimensional geometric dimensions of the blade was used. It was described by a Bezier polynomial of the fourth degree, with the definition of the meridian plane and the Cumberline profile. An iterative calculation was used to compare the characteristics of a rotor with a given ratio of diameters. The inlet and outlet blade angles were recalculated to keep the blade profile unchanged. It was found that—with a change in the exducer average root diameter and a fixed value of the average diameter of the rotor at the inlet—the output work and the total static isentropic efficiency were directly proportional to the ratio between the blade diameters. At the same time, the output work increased by 2.16%, along with a mass flow saving of 3.52%, which ultimately led to an increase in the isentropic efficiency by 2.15%. It should also be noted that the weight of the rotor was reduced by 31.98% while maintaining the size required to fit in the casing. In the second scenario, with a change in the average diameter at the inlet and a fixed value of the exducer average root diameter, the work at the exit and the total static isentropic efficiency was also directly proportional to the diameter ratio, but higher than in the first case, since they reached values of 3.31% and 2.34%, respectively. However, this was accompanied by an increase in the mass flow rate by 4.07% and an increase in the dimensions of the rotor by 6.63%. An analysis of the static temperature distribution, static pressure, and relative Mach number revealed dangerous low-pressure areas on the blade surface, accompanied by a high relative Mach number. The most rational choice would be to use the diameter ratio (R) in diapason from 1.5 to 1.55. The higher probability of blade surface erosion with a diameter ratio of 1.6 in both cases should be expected. In general, a more accurate determination of the economic efficiency of a turbine in the stages of design and while running is a complex multi-parameter problem that requires an optimization solution, which the authors plan to approach in future studies. Author Contributions: Conceptualization, M.A.C. and Y.B.; methodology, M.A.C.; software, M.A.C., S.D., M.E. and Y.B.; validation, S.D., J.M. and J.Z.; formal analysis, S.D. and B.A.A.; investigation, M.A.C. and Y.B.; resources, Y.B. and M.A.C.; data curation, J.M.; writing—original draft preparation, M.A.C. and Y.B.; writing—review and editing, S.D. and J.M.; visualization, M.A.C., Y.B. and B.A.A.; supervision, M.E. and J.Z; project administration, M.A.C., Y.B. and J.M.; funding acquisition, J.M. All authors have read and agreed to the published version of the manuscript. Funding: The authors are grateful to FCT—Fundação para a Ciência e Tecnologia (Portugal)—who partially financially supported this work through the RD Units Project Scope: UIDP/04077/2020 and UIDB/04077/2020. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The general approach has been partially developed within the research project “Development of a methodology for optimal design and manufacture highly efficient, highly reliable turbomachines, taking into account various operating modes” (State reg. No. 0121U107511). Conflicts of Interest: The authors declare no conflict of interest. Abbreviations b2Height at the inlet of the rotor D2Means diameter at rotor inlet D3rms Exducer mean root diameter Appl. Sci. 2022,12, 12165 20 of 21 D3H Exducer hub diameter D3S Exducer shroud diameter iIteration rRadial polar variable RRatio r0hThe radius at hub inlet rotor RcThe calcul ratio X1Length of the rotor x0hThe axial distance for the initial hub xre f Reference axial distance of the blade β2bInlet balde angle θCamber angle θre f Reference camber angle δ2Cone angle at the inlet blade References 1. Sawada, T.; Nishi, A. Investigation of Radial Inflow Turbine: 2nd Report A Method of Performance Estimation for Variable Geometry Radial Inflow Turbine. Trans. Jpn. Soc. Mech. Eng. 1969,35, 2401–2411. [CrossRef] 2. Kofskey, M.G.; Haas, J.E. Effect of Reducing Rotor Blade Inlet Diameter on the Performance of a 11.66-Centimeter Radial-Inflow Turbine; Work of the US Gov. Public Use Permitted; NASA: Washington, DC, USA, 1973. 3. Baines, N.C. Flow development in radial turbine rotors. In Proceedings of the 1996 International Gas Turbine and Aeroengine Congress & Exhibition, Birmingham, UK, 10–13 June 1996. [CrossRef] 4. Roelke, R.J. Radial Turbine Cooling; Work of the US Gov. Public Use Permitted; NASA: Washington, DC, USA, 1992. 5. Jones, A.C. Design and test of a small, high pressure ratio radial turbine. J. Turbomach. 1996,118, 362–370. [CrossRef] 6. Takamura, T.; Nishiguchi, F. Influence of blade aerodynamic loading on efficiency of radial-inflow turbines. In Proceedings of the ASME 1992 International Gas Turbine and Aeroengine Congress and Exposition, GT 1992, Cologne, Germany, 1–4 June 1992. [CrossRef] 7. Chen, H.; Abidat, M.; Baines, N.C.; Firth, M.R. The Effects of Blade Loading in Radial and Mixed Flow Turbines. In Proceedings of the ASME 1992 International Gas Turbine and Aeroengine Congress and Exposition, GT 1992, Cologne, Germany, 1–4 June 1992. [CrossRef] 8. Rodgers, C. Turbochargers to Small Gas Turbines? In Proceedings of the ASME 1997 International Gas Turbine and Aeroengine Congress and Exhibition, Orlando, FL, USA, 2–5 June 1997. [CrossRef] 9. Wallace, F.J.; Baines, N.C.; Whitfield, A. A Unified Approach to the One-Dimensional Analysis and Design of Radial and Mixed Flow Turbines. In Proceedings of the ASME 1976 International Gas Turbine and Fluids Engineering Conference, New Orleans, LA, USA, 21–25 May 1976. [CrossRef] 10. Ventura, C.A.M.; Jacobs, P.A.; Rowlands, A.S.; Petrie-Repar, P.; Sauret, E. Preliminary design and performance estimation of radial inflow turbines: An automated approach. J. Fluids Eng. Trans. ASME 2012,134, 031102. [CrossRef] 11. Zeng, F.; Zhang, W.; Wang, Y.; Cao, X.; Zou, Z. Effects of Squealer Geometry of Turbine Blade Tip on the Tip-Leakage Flow and Loss. J. Therm. Sci. 2021,30, 1376–1387. [CrossRef] 12. Pesiridis, A.; Martinez-Botas, R.F. Experimental evaluation of active flow control mixed-flow turbine for automotive turbocharger application. J. Turbomach. 2007,129, 44–52. [CrossRef] 13. Copeland, C.D.; Newton, P.; Martinez-Botas, R.F.; Seiler, M. A comparison of timescales within a pulsed flow turbocharger turbine. In Proceedings of the Institution of Mechanical Engineers—10th International Conference on Turbochargers and Turbocharging 2012, Savoy Place, London, UK, 12–16 May 2012; pp. 389–404. [CrossRef] 14. Hagen, B.A.L.; Agromayor, R.; Nekså, P. Equation-oriented methods for design optimization and performance analysis of radial inflow turbines. Energy 2021,237, 121596. [CrossRef] 15. Dadone, A.; Pandolfi, M. A method for evaluating the off-design performance of a radial inflow turbine and comparison with experiments. Int. J. Mech. Sci. 1969,11, 241–252. [CrossRef] 16. Wasserbauer, C.A.; Glassman, A.J. FORTRAN Program for Predicting Off-Design Performance of Radial-Inflow Turbines; Work of the US Gov. Public Use Permitted; NASA: Washington, DC, USA, 1975. 17. Rodgers, C. Advanced Radial Inflow Turbine Rotor Program:Design and Dynamic Testing; NASA-CR-135080, SOLAR-ER-2519; NASA: Washington, DC, USA, 1976. 18. Lauriau, P.-T.; Binder, N.; Cros, S.; Roumeas, M.; Carbonneau, X. Preliminary design considerations for variable geometry radial turbines with multi-points specifications. Int. J. Turbomach. Propuls. Power 2018,3, ijtpp3040022. [CrossRef] 19. Alawadhi, K.; Alhouli, Y.; Ashour, A.; Alfalah, A. Design and optimization of a radial turbine to be used in a rankine cycle operating with an OTEC system. J. Mar. Sci. Eng. 2020,8, 855. [CrossRef] 20. Zahed, A.H.; Bayomi, N.N. Radial turbine design process. Isesco J. Sci. Technol. 2015,11, 9–22. 21. Rodgers, C. Radial turbines—Blade number and reaction effects. In Proceedings of the ASME Turbo Expo 2000: Power for Land, Sea, and Air, Munich, Germany, 8–11 May 2000. [CrossRef] Appl. Sci. 2022,12, 12165 21 of 21 22. Doran, W.J.; Spence, S.W.T.; Artt, D.W. Experimental performance evaluation of a 99.0 mm radial inflow nozzled turbine at larger stator-rotor throat area ratios. Proc. Inst. Mech. Eng. Part A J. Power Energy 1999,213, 205–218. [CrossRef] 23. Gao, Y.; Petrie-Repar, P. Validation of meanline performance prediction method for radial and mixed flow turbine. In Proceedings of the Institution of Mechanical Engineers—13th International Conference on Turbochargers and Turbocharging 2018, London, UK, 16–17 May 2018; pp. 357–372. 24. Palfreyman, D.; Martinez-Botas, R.F. Numerical Study of the Internal Flow Field Characteristics in Mixed Flow Turbines. In Proceedings of the Turbo Expo 2002: Power for Land, Sea, and Air, Amsterdam, The Netherlands, June 3–6 2002; Volume 5, pp. 455–472. [CrossRef] 25. Chou, C.; Gibbs, C.A. The Design and Testing of a Mixed-flow Turbine for Turbochargers. SAE Tech. Pap. Ser. 1989 , 890644. [CrossRef] 26. Minegishi, H.; Matsushita, H.; Sakakida, M.; Koike, T. Development of a Small Mixed-Flow Turbine for Automotive Turbochargers. Volume 2: Aircraft Engine; Marine; Microturbines and Small Turbomachinery. In Proceedings of the International Gas Turbine and Aeroengine Congress and Exposition, Houston, TX, USA, 5–8 June 1995. [CrossRef] 27. Wallace, F.J.; Pasha, S.G.A. Design construction and testing of a mixed flow gas turbine. In Proceedings of the Second International JSME Symposium on Fluid Machinery and Fluids, Tokyo, Japan, 4–9 September 1972; pp. 213–214. 28. Rajoo, S.; Martinez-Botas, R. Experimental study on the performance of a variable geometry mixed flow turbine for automotive turbocharger. In Proceedings of the 8th International Conference on Turbochargers and Turbocharging, London, UK, 17–18 May 2006; pp. 183–192. [CrossRef] 29. Pesiridis, A.; Martinez-Botas, R.F. Experimental evaluation of active flow control mixed-flow turbine for automotive turbocharger application. In Proceedings of the ASME Turbo Expo 2005—Gas Turbie Technology: Focus for the Future, Reno, NV, USA, 6–9 June 2005; Volume 6, pp. 881–895. [CrossRef] 30. Pesiridis, A.; Martinez-Botas, R. Active control turbocharger for auto-motive application an experimental evaluation. In Proceedings of the IMechE International Conference on Turbochargers and Turbocharging, London, UK, 17–18 May 2006; pp. 223–232. 31. Pesiridis, A. Turbocharger Turbine Unsteady Aerodynamics with Active Control. Ph.D. Thesis, Imperial College London, London, UK, 2007. 32. Lee, S.P.; Jupp, M.L.; Nickson, A.K.; Allport, J.M. Analysis of a tilted turbine housing volute design under pulsating inlet conditions. In Proceedings of the ASME Turbo Expo 2017, Charlotte, NC, USA, 26–30 June 2017. [CrossRef] 33. Ketata, A.; Driss, Z. Numerical Study of a Vanned Mixed Flow Turbine Operating in Various Steady Flow Conditions. Int. J. Mech. Appl. 2017,7, 24–30. [CrossRef] 34. Ali, L.S.; Mohammed, H.; Kamel, H.M. The number of blade effects on the performance of a mixed turbine rotor. Eng. Rev. 2017 , 37, 349–360. 35. Meghnine, M.A.; Hamidou, M.K.; Hamel, M. Influence of the volute cross-sectional shape on mixed inflow turbine performances. Adv. Mech. Eng. 2017,9, 1–15. [CrossRef] 36. Hamel, M.; Bencherif, M.M.; Hamidou, M.K. Investigation of a twin entry mixed flow turbine volute, benefits with regard to the eco-system. Mater. Phys. Mech. 2017,32, 31–42. 37. Morrison, R.; Spence, S.; Kim, S.; Filsinger, D.; Leonard, T. Investigation of the effects of flow conditions at rotor inlet on mixed flow turbine performance for automotive applications. In Proceedings of the International Turbocharging Seminar 2016, Tianjin, China, 22–23 September 2016. 38. Padzillah, M.H.; Rajoo, S.; Martinez-Botas, R.F. Experimental and numerical investigation on flow angle characteristics of an automotive mixed flow turbocharger turbine. J. Teknol. 2015,77, 7–12. [CrossRef] 39. Udayakumar, R.; Pipada, Y.V. Impact of mixed flow turbines on the efficiency of automotive turbocharger applications. Int. J. Mech. Eng. Robot. Res. 2020,9, 791–796. [CrossRef] 40. Karamanis, N.; Martinez-Botas, R.F. Mixed-flow turbines for automotive turbochargers: Steady and unsteady performance. Int. J. Engine Res. 2002,3, 127–138. [CrossRef] 41. Zhang, J.; Zangeneh, M. Multidisciplinary and multi-point optimisation of radial and mixed-inflow turbines for turbochargers using 3D inverse design method. In Proceedings of the 14th International Conference on Turbochargers and Turbocharging— Proceedings of the International Conference on Turbochargers and Turbocharging 2021, London, UK, 11-12 May 2020; pp. 263–277. 42. Leonard, T.; Spence, S.; Filsinger, D.; Starke, A. Design and performance analysis of mixed flow turbine rotors with extended blade chord. J. Turbomach. 2020,142, 121003. [CrossRef] 43. Chelabi, M.A.; Hamidou, M.K.; Hamel, M. Effects of cone angle and inlet blade angle on mixed inflow turbine performances. Period. Polytech. Mech. Eng. 2017,61, 225–233. [CrossRef] 44. Chelabi, M.A.; Saga, M.; Kuric, I.; Basova, Y.; Dobrotvorskiy, S.; Ivanov, V.; Pavlenko, I. Effects of deviation blade angle on mixed inflow turbine performances. Appl. Sci. 2022,12, 3781. [CrossRef] 45. Chelabi, M.A.; Basova, Y.; Hamidou, M.K.; Dobrotvorskiy, S. Analysis of the three-dimensional accelerating flow in a mixed turbine rotor. J. Eng. Sci. 2021,8, D1–D7. [CrossRef] 46. Kononenko, S.; Dobrotvorskiy, S.; Basova, Y.; Gasanov, M.; Dobrovolska, L. Deflections and frequency analysis in the milling of thin-walled parts with variable low stiffness. Acta Polytech. 2019,59, 283–291. [CrossRef] 47. Dobrotvorskiy, S.; Kononenko, S.; Basova, Y.; Dobrovolska, L.; Edl, M. Development of Optimum Thin-Walled Parts Milling Parameters Calculation Technique. In Proceedings of the 4th International Conference on Design, Simulation, Manufacturing: The Innovation Exchange, DSMIE 2021, Lviv, Ukraine, 8–11 June 2021; Volume 2021, pp. 343–352. [CrossRef]