- Email: [email protected]
Design of double curvature radial turbine blades for a micro gas turbine
Accepted Manuscript
Design of Double Curvature Radial Turbine Blades for a Micro Gas Turbine Sagar Pakle , Kyle Jiang PII: DOI: Reference:
S0307-904X(18)30512-2 https://doi.org/10.1016/j.apm.2018.10.020 APM 12508
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
4 May 2018 22 August 2018 22 October 2018
Please cite this article as: Sagar Pakle , Kyle Jiang , Design of Double Curvature Radial Turbine Blades for a Micro Gas Turbine, Applied Mathematical Modelling (2018), doi: https://doi.org/10.1016/j.apm.2018.10.020
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Highlights Design of a radial turbine for the 20kW micro gas turbine. Radial turbine through-flow analysis and comparison of 1D and CFD results. Optimization of the turbine geometry using an inverse design method. Performance improvement by introducing a unique double curvature turbine blade. Turbine mechanical qualification and aerodynamic performance estimation.
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Design of Double Curvature Radial Turbine Blades for a Micro Gas Turbine Sagar Pakle1* and Kyle Jiang2 1
2 School of Engineering, The University of Birmingham Edgbaston, Birmingham, UNITED KINGDOM, B15 2TT [email protected]
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School of Engineering The University of Birmingham, Edgbaston, Birmingham, UNITED KINGDOM, B15 2TT [email protected]
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Abstract The paper reports a study on the design of a radial turbine for a portable micro gas turbine engine. A unique design of blades with double curvatures is proposed which demonstrates the considerable advantage in terms of efficiency and power output. The design process largely consists of design, optimization, and analysis of radial turbine closely based on low mass flow rate and high-pressure ratio. The intention of the development is to achieve reductions both in fuel consumption and emissions. The micro gas turbine engine is designed to generate 20 kW net power output. As a 110mm centrifugal compressor takes 56 kW power in the Brayton cycle, a 113mm inlet diameter radial turbine is designed to produce 76kW power. Optimization of the initial turbine geometry is carried out using an inverse design technique to improve the overall turbine stage performance. The original turbine blade meridional frame was used as an input to the optimization. The optimized 3-D blade geometry have a unique double curvature blade sections which is unlikely to be achieved using the conventional design approach. The performance of the optimized geometry has been compared with that of the initial geometry. Computational fluid dynamic and finite element analysis techniques were used for examining the aerodynamic performance and structural integrity respectively. The rotational speed of the turbine was set up to 101400 RPM. The result of computational dynamic analysis indicates that the optimized geometry provides an approximately 6% increase in total-to-static efficiency on average and 6 kW increase in power output at high expansion ratio as compared to the original design. Stress analysis shows that the maximum stress is lower than the yield strength of the material, which verifies that the turbine design is mechanically viable. Keywords
Micro gas turbine, radial turbine, 1-D analysis, inverse design optimization, computational fluid dynamics, stress analysis, modal analysis.
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b C0 , C s
Blade width/height Spouting velocity
C 𝐶𝑚 f h H i KB m Mrel N, N R Q r 𝑟𝑉̅𝜃 s o U V W Z Z R θ ω 𝛺 𝛿𝑝 ρ
Tangential velocity Meridional velocity Blade wrap angle Static enthalpy Total enthalpy Incident angle Blockage factor
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Nomenclature
Meridional distance, mass flow Relative Mach number Number of blades
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Volume flow radius Pitchwise-average swirl velocity Blade pitch Blade passage throat width Blade tip speed Velocity Relative velocity Axial coordinate Rotor axial length
σ β θ 2 c
Slip factor Blade angle Polar angle Diffuser divergence angle
s
Velocity ratio
ns
Specific speed(total-to-static)
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s
Tangential co-ordinate Rotational speed Vorticity Periodic delta function density Static efficiency
Subscripts bl m 0 Superscripts ̅ + -
at the blade Meridional direction Total condition Pitchwise mean value Relative to the upper blade surface Relative to the lower blade surface
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Introduction In present days, micro gas turbines are considered to be one of the most sophisticated portable devices to generate electricity power owing to its attribute of lower emission and compactness. Such devices are expected to reduce fuel consumption and alleviate environmental pollution, as the pressure on CO2 reduction is mounting and low fuel consumption is ever more demanding [1, 2]. Although micro gas turbines have issues to be addressed in terms of reliability, efficiency, and initial set–up cost, the intrinsic advantage of large power to weight ratio and lower emissions make them attractive to be used as portable power units [3-8]. The application of micro gas turbines covers many domains such as unmanned aerial vehicles, auxiliary power units, distributed power generator, mini combined heat and power units, and range extenders for electric vehicles. The broad range of the applications have encouraged industry and research organizations to study this area in order to develop efficient and reliable micro gas turbines. Table 1 is a list of academic organizations which have carried out micro gas turbine development and the corresponding engine specifications reported so far [9-11]. Table 1 Activity of micro gas turbine development and specification of the engines. Rotational speed/rpm 2,400,000 1,200,000 800,000 1,170,000 870,000 930,000 400,000 250,000 35,000 140,000 500,000 176,500
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Diameter/mm 4 6 12 8 10 10 8.4 10 10 12 16 30
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Type radial radial axial-radial axial-radial axial-radial radial radial radial axial-radial radial axial-radial radial disk
Output/W 60 52 300 475 485 500 44 10 10 100 800
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Organization MIT(US) MIT(US) Stanford University (US) University of Tokyo (Japan) Tohoku University (Japan). XJTU (China) SIMTech (Singapore) KIMM (Korea) KUL (Europe) AIT (Europe) ICL (Europe) ETH Zurich (Europe) WUT (Europe)
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A micro gas turbine follows the same thermodynamic cycle as the large sized ones. The average efficiency for a simple Brayton cycle system is reported to be 17%. However, it can be improved up to 30% by incorporating the recuperator [12]. In a recuperative cycle, hot exhaust gas is directed to the recuperator to heat compressed air before it enters the combustion chamber. By reusing the heat energy from the exhaust, the engine with a recuperator can improve the overall thermal efficiency[10]. The schematic of the micro gas turbine with recuperator is shown in Figure 1
Figure 1: The configuration and internal flows of a gas turbine with a recuperator[13] .
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The performance of the compressor and the turbine has significant influence on the overall performance of the gas turbine engine. Scaling a large diameter rotating component to a smaller one to meet design requirements is not appropriate. This is due to the fact that scaling results in a large change in the Reynolds number and heat transfer properties of the component [9, 14]. Therefore turbomachinery components of smaller dimensions need to be redesigned to meet the design specification. This paper introduces the design of a 113mm radial turbine for a 20 kW micro gas turbine using conventional design approach and optimization of the turbine geometry through an inverse design method. Subsequently, the resultant design is validated for performance and structural integrity using CFD and FEA. Interestingly, an inverse design method is confirmed to be very effective to optimize the existing design in a short period of time. The optimization through this method has resulted in double curvature turbine blade which is noteworthy. In practice, modifying conventionally designed turbomachinery device to improve the overall aerodynamic performance is a tedious job and sometimes attempts in such direction could further deteriorate the aerodynamic performance. However, a three-dimensional inverse design method ensures optimized aerodynamic performance which makes it a rational approach for optimization of radial turbo components.
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Design of a Radial Turbine In the design, it is important to ensure that the radial turbine will generate enough power to meet the needs of both 56 kW power that 110 mm centrifugal compressor consumes and 20kW output power for the generator. It was found that a 113mm radial turbine would meet the requirement. The geometry design process is based on the design approach proposed by Aungier [19]. A preliminary or mean-line design is essential in order to get an overview of the dimensions and aerodynamic performance of the turbine rotor. The success of any turbine design is mainly based on the accuracy of the mean-line design [1, 20]. A preliminary design of the radial turbine stage for this specific requirement is started with the basic design specifications which can be seen in Table 2. Table 2 Preliminary design input specification. Specification
Fluid
Air
Dimensionless specific speed
0.447
Static Adiabatic efficiency
0.8581
Velocity ratio
0.6274
Inlet total pressure (kPa)
420
Inlet total temperature (k)
1173
Mass flow rate (kg/s)
0.183583
Total-to-Static Pressure ratio
5.25
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Parameter
Ideally, the specification of the velocity ratio s and total-to-static efficiency is a function of the specific speed ns . The dimensionless specific speed can be given as [19], √
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(1)
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The chosen value of efficiency and velocity ratio for the design condition is shown in Figure 2. Although a preferred value for specific speed lies in the range of 0.45 to 0.75 [19, 21], the geometry dimension and rotational speed impose the constraint to choose a value, which marginally differs from this range.
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Figure 2 Generalized stage performance correlation.
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The outline of the stage considered for this analysis is shown in Figure 3. In general, 65 to 80 degree of vane outlet flow angle is recommended [18, 19]. Therefore, in line with the best practice, the inlet absolute angle about 72 degrees is considered for the design.
Figure 3 Turbine stage layout.
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In fact, nozzle vanes are considered to be significant components in the turbine stage, as they are intended to collect flow from a radial direction and direct it towards the rotor inlet [24]. Furthermore, these vanes are the main components that control the swallowing capacity of the turbine stage. Therefore, vanes blade angles and flow angles are considered to be a more significant parameter and need proper attention [25]. In the present case, a nozzle vane is designed with a parabolic-arc camber line proposed by Aungier [19]. The last part in the stage is a diffuser section. The diffuser section with better static pressure recovery coefficient has a positive effect on the total–to-static pressure ratio and efficiency [26]. Here, diffuser with 11-degree divergence angle is implemented as the divergence angle governs the diffuser performance [19]. It is imperative to account for the rotor passage losses in a preliminary design stage to 6
ACCEPTED MANUSCRIPT arrive at the turbine design with accurate performance estimation [15, 20, 21, 27]. In the present design endeavor, various losses are considered during the preliminary design and analysis. These losses include blade clearance loss, incidence loss, disk friction loss, blade loading and profile loss [15, 19, 21]. These losses are systematically employed in a preliminary analysis and details of which can be found in reference [19].
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After geometric modeling, the through-flow analysis was carried out to understand the flow behavior in the turbine and ANSYS Vista-TF commercial software was used in the process. Based on Mach number distribution, the end wall contours were altered to achieve a better flow distribution. The initial and modified meridional sketch is shown in Figure 4.
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Figure 4 Rotor meridional sketch.
Hub
Hub
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Furthermore, Mach number distribution with initial and modified contour has been analyzed as shown in Figure 5, which indicates the flow behavior is obviously improved particularly in the turbine exducer region.
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Mid-span
Mid-span
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Figure 5 Mach number distribution ( Left: Initial; Right: Modified).
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Finally, the turbine stage is modeled as 19 modified rotor blades, 30 nozzle vanes and diffuser passage downstream of the rotor blades. This stage geometry will be treated as the original geometry for further analysis. An overview of the rotor and vane geometry can be seen in Figure 6.
Figure 6 Three dimensional Rotor and stator geometry.
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It is indeed valuable to compare the one-dimensional and CFD result to get a confidence in the design approach. Here, the comparison for aerodynamic performance is made between one-dimensional analysis result of initial stage geometry and CFD result of modified stage geometry as shown in Figure 7, Figure 8, andFigure 9. Although the performance trend appears to be encouraging, the quantitative deterioration in performance can be attributed to the higher frictional loss. This loss is due to a narrow flow passage in the rotor inlet and nozzle vane section of the turbine stage. Therefore, in order to improve the performance, the turbine blades need to be optimized. The process of optimization is described in the next section.
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Figure 7 Comparison of 1-D analysis and CFD result for expansion ratio.
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Figure 8 Comparison of 1-D analysis and CFD result for efficiency.
Figure 9 Comparison of 1-D analysis and CFD result for power output.
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ACCEPTED MANUSCRIPT Creation of Double Curvature Turbine Blades As demonstrated in the previous section, an original turbine design appears to have lower aerodynamics performance than expected; therefore, the scope for improvement in design is obvious. This improvement in design and its performance can be achieved by optimization of turbine blades. The inverse design method is one of the sophisticated methods which can provide a platform to efficiently optimize the turbine blade in order to improve the overall performance of the turbine stage.
N rV 2
2 / N
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A procedure of inverse design was proposed and described by Zangeneh [17]. The fundamental idea of this method is to represent the blade by sheets of vorticity having its strength directly related to the specified bound circulation 𝑟𝑉̅𝜃 ; here 𝑟𝑉̅𝜃 can be given as (2)
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where N is a number of blades. In this method, a camber line of the blade is represented by a single sheet of vorticity and the blade blockage effect is included in continuity equation of the mean flow by means of stream surface thickness parameter. Hence, the bound vorticity on the blade can be presented by, V (rV ) p ( )
(3)
where α is the blade surface and can be represented as
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2 N
(4)
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V (rV )
(5)
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In this method, an initial blade shape can be obtained by using one-dimensional velocity derived from the specified mass flow rate and meridional geometry prescribed with mean tangential velocity. A precaution need to be taken to align the flow with blade surface by means of applying the inviscid slip condition which can be represented by a first order hyperbolic partial differential equation as shown below
Wbl 0
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f f V Vr bl z r r
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here is a vector normal to the blade surface and is the relative velocity at the blade surface , where and are the velocities on the upper and lower surface of blades. The equation (30) needs to be integrated along the meridional projection of streamlines on the blade surface to find the blade shape. The requirement of initial condition for this integration can be satisfied by specifying the wrap angle ‘f’ which is called as ‘stacking condition of the blade’. Finite difference discretization scheme with Crank-Nicholson numerical technique can be used to solve this equation. After obtaining the blade shape the 10
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h h
rV 2 Wmbl N m
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Secondary flow in a radial turbomachinery is an important phenomenon which needs to be controlled to achieve good performance. This can be readily achieved through threedimensional inverse design method by controlling the blade pressure loading distribution on the hub and shroud independently. When the turbomachinery is compressible, the pressure/enthalpy loading can be represented in terms of meridional derivatives of 𝑟𝑉̅𝜃 or blade loading as, (7)
where superscript + and – represents either side of the blade, 𝑚 is meridional velocity at the blade and m is the direction of streamlines in the meridional plane [18].
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Ideally, a loading distribution or 𝑟𝑉̅𝜃 distribution has a significant effect on the blade surface pressure (Mach number distribution on the blade) [18]. This loading distribution can be represented by three segment curve which starts with a parabolic curve, then straight line section and end with a parabolic curve. The distribution of this curve can be manipulated depending on the work requirement of the blade. Once an optimum distribution of 𝑟𝑉̅𝜃 at hub and shroud section is arranged, it can be integrated to find the 𝑟𝑉̅𝜃 value at hub and shroud. Further, it can be linearly interpolated from hub to shroud and overall distribution of 𝑟𝑉̅𝜃 on a meridional plane can be achieved [17, 18].
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Another factor that affects the pressure (Mach number) distribution is the stacking condition [28]. This is analogous to the lean feature in the conventional turbomachinery design. The main effect of a lean or stacking is to distribute the blade forces in the spanwise direction. In this case, the blade can be lean linearly against or in the direction of rotation which has the corresponding effects on the aerodynamic performance and mechanical stress of the blade. If the blades are leaned against the direction of the rotation making the blade hub to lead to the blade shroud, then the consequent blade force would augment the pressure at the shroud end wall and subsequently lowers the pressure at the hub end wall [28]. Ultimately, care should be taken to apply the stacking in conjunction with 𝑟𝑉̅𝜃 in order to achieve the design which results in the suppression of secondary flow while complying with structural limitations [18, 28].
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Nevertheless, based on the CFD analysis of the modified geometry, it is evident that the turbine geometry needs to be optimized to achieve the improved performance. Optimizing the turbine geometry with the conventional design approach is tedious and time-consuming with limited or no certainty of improvement in performance. In such a scenario, an optimized turbine blade can be achieved by adopting the inverse design approach. The inputs for this method can be listed as:
Meridional sketch of geometry
Initial flow condition
Mass flow rate
Blade thickness
Blade numbers 11
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Rotational speed
Stacking condition
Specific work
Blade loading distribution (meridional derivatives of 𝑟𝑉̅𝜃 )
Input
Objective function
Objectives
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Optimization
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Figure 10 indicates the workflow of the inverse design method. In general, constraints are based on the design conditions. The objective function which in this case is a blade loading distribution can be optimized to accomplish the objectives. The objectives referred here are resultant geometry and performance attributes. A classification of constraints, objectives and objective function are also shown in Figure 10.
Constraints
Objective function
Objectives
Higher expansion ratio. Higher efficiency. Higher power output. lower secondary flow.
Y=f(Xi,…,Xn) Blade loading distribution = f (NCH,NDH,NCS,NDS,SLOPE_H,SLOPE_S)
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Meridional geometry. Initial flow condition. Mass flow rate. Blade thickness. Blade numbers. Rotational speed. Stacking condition. Specific work.
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Output
Figure 10 Inverse design optimization workflow.
Most importantly, the objective function depends on the parameters such as NCH, NDH, NCS, NDS, SLOPE_H, and SLOPE_S which governs the distribution of blade loading. The impact of these parameters on blade loading is depicted in Figure 11. It must be noted that the optimization of the turbine blade through the conventional design approach needs to be done by tuning several parameters and the combination of this alteration could worsen the design performance. However, through inverse design method, the optimized geometry can be easily achieved by optimizing a few parameters depicted in Figure 11. 12
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Figure 11 Dependent variables of blade loading distribution.
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Therefore, the vital part, in this case, is to get optimum blade loading distribution by optimizing dependent variables as indicated in Figure 11. These variables remarkably influence the shape of blade geometry and performance of the radial turbine. Ideally, optimizing the magnitude and location of dependent variables is an iterative process which is based on the designer’s experience and blade loading requirements.
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In the present case, optimization of streamwise blade loading is resulted in front loading of hub and shroud as shown in Figure 12. In particular, peak loading on hub and shroud can be observed at 15% and 40% of the meridional length respectively. The blade loading at hub thereafter decreases consistently till the trailing edge of the blade. Likewise, the shroud loading decreases gradually to 90 % of the meridional length and thereafter steeply reduces towards the trailing edge of the blade. One of the reasons behind imposing the zero loading at trailing edge hub and shroud is to reduce the losses generated by variation of spanwise blade loading. In addition, zero value of blade loading at the leading edge (m=0) and trailing edge (m=1) is used to satisfy zero incident at the inlet and Kutta condition at blade outlet [17].
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Another important input to be specified is 𝑟𝑉̅𝜃 distribution or specific work in a spanwise direction. The 𝑟𝑉̅𝜃 value of 0.99 normalized by blade tip speed and inlet tip radius was specified at the inlet and zero at the turbine trailing edge based on turbine work requirement.
Figure 12 Blade loading distribution
Furthermore, the wrap angle or stacking condition was set at the blade inlet quasiorthogonal. Primarily for stress consideration, the wrap angle is applied in a manner that shroud leads to hub in the direction of rotation. The wrap angle at hub is set to zero whereas at shroud a positive 2 degrees wrap angle is applied. Interestingly, it has been observed that 13
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the inverse design can generate more complex shapes which are unlikely to be generated using the conventional design approach.
Figure 13 Optimized blade angle distribution.
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As shown in Figure 13, the distribution of blade angle profile is appeared to be highly curved with double curvatures. This distribution of blade angle is a result of the optimized blade loading shown in Figure 12. This fact indicates that the blade angle distribution is govern by the distribution of blade loading which is a crucial part of this optimization technique. In present study, the optimization is resulted in double curvature blade which is shown in Figure 14. Therefore, it is evident from this design optimization that a double curvature blade shape can be achieved by means of optimizing the blade loading distribution to enhance the performance. This type of turbine blade does not exist in the industry. Thus, an insight of such design and its design approach can pave a way for the development of new types of turbines.
Figure 14 (a) Conventional design (b) Optimized design.
In this study, no modifications were done on vane geometry and only turbine blade was optimized through the inverse design method for aerodynamic performance improvement. As mentioned previously, a same meridional geometry with normal thickness was used without any blockage factor. An ideal gas equation with specific heat ratio of 1.333 was used in flow 14
ACCEPTED MANUSCRIPT properties and inlet temperature of 1173K along with velocity profile was imposed as inlet condition. The rotational speed of 86000 RPM with 19 rotor blades was considered during the design iterations. Computational Fluid Dynamics
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A three-dimensional viscous flow analysis was carried out for original and optimized radial turbine stage while the nozzle vane geometry remains unchanged in both cases. For this purpose, commercial CFD software CFX16.0 was used. A mesh with around 340 thousand nodes and a first cell height of 5e-06m was used on both the turbine blade channels. Special care has been taken to keep the mesh size approximately similar in both cases for meaningful comparison. A single passage computational domain consists of a nozzle vane, radial turbine blade followed by the conical diffuser. The outline of this configuration is shown in Figure 15.
Figure 15 CFD simulation domain.
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In the CFD analysis, a mixing plane interface was used between vane and rotor domains whereas the rotor and diffuser have frozen rotor interface. The same tip clearance is used in both configurations. Moreover, for this flow analysis, compressible RANS equations were used with the SST turbulence model. This turbulence model has the capability to predict the flow in the vicinity of the solid wall with better accuracy [29]. The formulation of the SST turbulence is presented in equations (32) and (33).
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( k ) ( u j k ) k P * k [( k t ) ] t x j x j x j
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k ( ) ( u j ) P 2 [( t ) ] 2(1 F1 ) 2 t x j t x j x j x j x j
(8)
(9)
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here k is a turbulent kinetic energy, is dissipation rate, P a production term, t turbulent viscosity, and t turbulent kinematic viscosity. The primary objective of this viscous flow analysis is to estimate the overall aerodynamic performance in terms of mass flow rate, expansion ratio, efficiency and power output. To accomplish this objective, flow in the turbine stage was simulated at the speed of 86000, 93000 and 100000 RPM. Inlet temperature of 1173K, outlet static pressure of 84 kPa was set in the simulations and the total pressure was varied at the inlet. The results indicate that the optimized design turbine stage has a better total-to-static efficiency at high expansion ratio (lower U/Cs ratio) which can be seen in total-to-static efficiency against U/Cs ratio (Ratio of blade tip seed to spouting velocity) plot in Figure 16. Furthermore, Figure 17 shows that the improvement in expansion ratio is marginal and cannot be accounted as a significant 15
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improvement. However, based on efficiency plots, it can be concluded that optimized turbine geometry provides around 6% on average increase in total-to-static efficiency which can be considered as a significant gain in efficiency.
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Figure 16 Total-to-Static efficiency vs U/C: (a) 86000 (b) 93000 (c) 100000.
Figure 17 Expansion ratio Vs Mass flow rate.
Furthermore, as shown in Figure 18, over three rotational speeds, on average 6 kW improvement in turbine output power at high mass flow was observed. At a lower mass flow, there is hardly any obvious improvement in output power. However, at higher mass flow, the improvement in output power is considered to be significant.
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Figure 18 Power output (kW) : (a) 86000 RPM (b) 93000 RPM (c) 100000 RPM.
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In addition, contours of relative Mach number (Mrel) can be seen on the meridional plot in Figure 19. The distribution of relative Mach number is observed to be improved significantly in the optimized design. Likewise, the same pattern of relative Mach number can be seen on the blade-to-blade contour plots for shroud section shown in Figure 20 and for hub section shown in Figure 21. Indeed, the distribution of flow is largely improved in the optimized design where the flow separation, particularly at tip section, has been reduced substantially. The location of the maximum relative Mach has been shifted to trailing edge region in the optimized blade, which contributes to the reduction in rotor passage loss and enhances the aerodynamic performance of the turbine stage.
Figure 19 Contours of Mrel on meridional view (Left: Original; Right: Optimized).
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Figure 20 Blade to Blade view of Mrel contours at shroud (Left: Original; Right: Optimized).
Figure 21 Blade to Blade view of Mrel contours at the hub (Left: Original; Right: Optimized).
Structural Integrity
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A radial turbine in a micro gas turbine engine operates at high speeds. Therefore, an evaluation of strength, particularly at highest rotational speed, is of paramount importance [30, 31]. In the current design, a radial turbine is designed for maximum tip speed of 600m/s i.e. a rotational speed of 101400 RPM. Therefore, strength evaluation at this rotational speed was conducted.
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Stress analysis of optimized designed turbine was performed using the ANSYS structural software. An unstructured mesh on single blade sector was generated as shown in Figure 22.
Figure 22: FEA unstructured mesh.
A cyclic boundary condition is applied on the periodic surfaces of the blade sector with a fixed support at hub region. The material used for the turbine is Inconel 718 which is hightemperature resistance and high strength material with yield strength up to 1100 MPa. As stress analysis indicates, the maximum von-Mises stress on the blade appears at the blade root particularly at the trailing edge region which is shown in Figure 23.
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Figure 23 Equivalent (von-Mises) stress.
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The magnitude of stress is much less than the yield strength of the material which makes design safe for operation with 25% of safety margin. Furthermore, blade fillet is a crucial feature in the turbine blade modeling along with hub disk and has a significant impact on blade stress level. Therefore, in order to alleviate the centrifugal stress on the blade, a fillet of 0.7 mm radius is applied at the blade root section. As the turbine blades undergo a centrifugal load, a deformation of the structure is inevitable. As shown in Figure 24, the turbine blade deforms by 0.239 mm at blade exducer tip under centrifugal load which seems to be reasonable.
Figure 24 Total deformation under centrifugal load.
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Frequency mode evaluation is another important aspect of the rotor design which needs to be done to ensure that the resonance would not happen in the operating regime [21]. In order to verify it, a modal analysis was carried out using the single blade sector domain with cyclic boundary condition imposed at blade sector periodic surfaces.
Figure 25 1st frequency mode shape.
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A radial turbine design is developed for a 20 kW micro gas turbine engine. A comparison between the one-dimensional analysis result and CFD result was made to get the confidence in the design approach. Further, an optimization was carried out through an inverse design method to improve the efficiency and output power of a radial turbine. Optimization has resulted in a double curvature blade shape. Such types of blades currently do not exist in the industry and cannot be achieved using conventional design approach. The aerodynamic performance of the optimized geometry was subsequently evaluated using CFD simulations. It was found that an optimization results in a 6% increase in total-to-static efficiency on average and 6 kW additional total power output, particularly at higher expansion ratio and higher mass flow rate. Stress and frequency of the optimized three-dimensional rotor blade were analyzed. The results indicate that the maximum computed stress is much lower than the yield strength of the material and offer a 25% safety margin. A modal analysis shows the magnitude of first mode frequency is more than 4 times higher than the blade vibration frequency, which shows that the designed blade will work outside the resonance condition.
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Overall, this research demonstrates that the optimization using three-dimensional inverse design method is an effective way for achieving non-conventional turbine geometry. This optimization results in significant improvement in turbine aerodynamic performance while complying with the structural integrity of the turbine blade.
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Acknowledgments
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The research was jointly supported by European Horizon 2020 project 644971, 2017 T-TRIG project of the UK Department for Transport, and Innovate UK project 104021. The Authors wish to thank Advanced Design Technology Limited for their help and guidance.
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References 1. 2. 3. 4. 5. 6.
Rahbar, K., S. Mahmoud, and R.K. Al-Dadah, Mean-line modeling and CFD analysis of a miniature radial turbine for distributed power generation systems. International Journal of Low-Carbon Technologies, 2016. 11(2): p. 157-168. Li, Y., X.S. Li, and X.D. Ren, Aerodynamic optimization of a high-expansion ratio organic radialinflow turbine. Journal of Mechanical Science and Technology, 2016. 30(12): p. 5485-5490. Lim, H.S., et al., Secondary flow stabilization of 100 kW-class micro gas turbine. Journal of Mechanical Science and Technology, 2017. 31(4): p. 1753-1761. Liguori, V., Numerical investigation: Performances of a standard biogas in a 100 kWe MGT. Energy Reports, 2016. 2: p. 99-106. Kim, M.J., J.H. Kim, and T.S. Kim, Program development and simulation of dynamic operation of micro gas turbines. Applied Thermal Engineering, 2016. 108(Supplement C): p. 122-130. Fikri, M., M. Ridzuan, and H. Salleh, Preliminary study of Low-Cost Micro Gas Turbine. International Engineering Research and Innovation Symposium (Iris), 2016. 160.
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Mizuki, S., Development of compressor for ultra micro gas turbine. Journal of Thermal Science, 2007. 16(1): p. 19-27. Isomura, K., et al., Development of micromachine gas turbine for portable power generation. Jsme International Journal Series B-Fluids and Thermal Engineering, 2004. 47(3): p. 459-464. Peirs, J., et al., Micropower generation with microgasturbines: A challenge. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2007. 221(4): p. 489-500. Marco Antônio Rosa do Nascimento, L.d.O., et al., Micro Gas Turbine Engine: A Review, Progress in Gas Turbine Performance. InTech, 2013(http://dx.doi.org/10.5772/54444). Fu, L., Z.P. Feng, and G.J. Li, Experimental investigation on overall performance of a millimeter-scale radial turbine for micro gas turbine. Energy, 2017. 134: p. 1-9. Xiao, G., et al., Recuperators for micro gas turbines: A review. Applied Energy, 2017. 197: p. 83-99. HROTE. Croatian Energy Market Operator (HROTE). 2017; Available from: http://www.hrote.hr/default.aspx?id=226. Van den Braembussche, R., Micro Gas Turbines—A Short Survey of Design Problems. Micro Gas Turbines, 2005: p. 1-1. Nithesh, K.G., et al., Design and performance analysis of radial-inflow turboexpander for OTEC application. Renewable Energy, 2016. 85: p. 834-843. Fiaschi, D., G. Manfrida, and F. Maraschiello, Thermo-fluid dynamics preliminary design of turboexpanders for ORC cycles. Applied Energy, 2012. 97(Supplement C): p. 601-608. Zangeneh, M., A compressible three-dimensional design method for radial and mixed flow turbomachinery blades. International Journal for Numerical Methods in Fluids, 1991. 13(5): p. 599624. Zangeneh, M., A. Goto, and H. Harada, On the role of three-dimensional inverse design methods in turbomachinery shape optimization. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 1999. 213(1): p. 27-42. Aungier, R.H., Turbine aerodynamics : axial-flow and radial-inflow turbine design and analysis. 2006, New York, N.Y. (ASME, Three Park Avenue. New York, NY 10016): American Society of Mechanical Engineers. Al Jubori, A.M., R. Al-Dadah, and S. Mahmoud, Performance enhancement of a small-scale organic Rankine cycle radial-inflow turbine through multi-objective optimization algorithm. Energy, 2017. 131: p. 297-311. Jung, H.C. and S. Krumdieck, Meanline design of a 250 kW radial inflow turbine stage using R245fa working fluid and waste heat from a refinery process. Proceedings of the Institution of Mechanical Engineers Part a-Journal of Power and Energy, 2016. 230(4): p. 402-414. Rohlik, H.E., Analytical Determination of Radial Inflow Turbine Design Geometry for Maximum Efficiency. NASA Technical Note 1968(NASA TN D-4384). Dixon, S.L. and C.A. Hall, Chapter 8 - Radial-Flow Gas Turbines, in Fluid Mechanics and Thermodynamics of Turbomachinery (Seventh Edition). 2014, Butterworth-Heinemann: Boston. p. 319360. Moustapha, H. and M.F. Zelesky, Axial and Radial Turbines. 2003: Concepts NREC. Whitfield, A. and N.C. Baines, Design of radial turbomachines. 1990: Longman Scientific & Technical. Keep, J.A., A.J. Head, and I.H. Jahn, Design of an efficient space constrained diffuser for supercritical CO2 turbines. 1st International Seminar on Non-Ideal Compressible-Fluid Dynamics for Propulsion & Power, 2017. 821. Fiaschi, D., G. Manfrida, and F. Maraschiello, Design and performance prediction of radial ORC turboexpanders. Applied Energy, 2015. 138(Supplement C): p. 517-532. Zangeneh, M., A. Goto, and H. Harada, On the Design Criteria for Suppression of Secondary Flows in Centrifugal and Mixed Flow Impellers. Journal of Turbomachinery, 1998. 120(4): p. 723-735. Menter, F.R., Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 1994. 32(8): p. 1598-1605. Baines, N.C., Fundamentals of Turbocharging. 2005: Concepts NREC. Fu, L., et al., Experimental validation of an integrated optimization design of a radial turbine for micro gas turbines. Journal of Zhejiang University-Science A, 2015. 16(3): p. 241-249.
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Design of Double Curvature Radial Turbine Blades for a Micro Gas Turbine Sagar Pakle , Kyle Jiang PII: DOI: Reference:
S0307-904X(18)30512-2 https://doi.org/10.1016/j.apm.2018.10.020 APM 12508
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
4 May 2018 22 August 2018 22 October 2018
Please cite this article as: Sagar Pakle , Kyle Jiang , Design of Double Curvature Radial Turbine Blades for a Micro Gas Turbine, Applied Mathematical Modelling (2018), doi: https://doi.org/10.1016/j.apm.2018.10.020
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Highlights Design of a radial turbine for the 20kW micro gas turbine. Radial turbine through-flow analysis and comparison of 1D and CFD results. Optimization of the turbine geometry using an inverse design method. Performance improvement by introducing a unique double curvature turbine blade. Turbine mechanical qualification and aerodynamic performance estimation.
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Design of Double Curvature Radial Turbine Blades for a Micro Gas Turbine Sagar Pakle1* and Kyle Jiang2 1
2 School of Engineering, The University of Birmingham Edgbaston, Birmingham, UNITED KINGDOM, B15 2TT [email protected]
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School of Engineering The University of Birmingham, Edgbaston, Birmingham, UNITED KINGDOM, B15 2TT [email protected]
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Abstract The paper reports a study on the design of a radial turbine for a portable micro gas turbine engine. A unique design of blades with double curvatures is proposed which demonstrates the considerable advantage in terms of efficiency and power output. The design process largely consists of design, optimization, and analysis of radial turbine closely based on low mass flow rate and high-pressure ratio. The intention of the development is to achieve reductions both in fuel consumption and emissions. The micro gas turbine engine is designed to generate 20 kW net power output. As a 110mm centrifugal compressor takes 56 kW power in the Brayton cycle, a 113mm inlet diameter radial turbine is designed to produce 76kW power. Optimization of the initial turbine geometry is carried out using an inverse design technique to improve the overall turbine stage performance. The original turbine blade meridional frame was used as an input to the optimization. The optimized 3-D blade geometry have a unique double curvature blade sections which is unlikely to be achieved using the conventional design approach. The performance of the optimized geometry has been compared with that of the initial geometry. Computational fluid dynamic and finite element analysis techniques were used for examining the aerodynamic performance and structural integrity respectively. The rotational speed of the turbine was set up to 101400 RPM. The result of computational dynamic analysis indicates that the optimized geometry provides an approximately 6% increase in total-to-static efficiency on average and 6 kW increase in power output at high expansion ratio as compared to the original design. Stress analysis shows that the maximum stress is lower than the yield strength of the material, which verifies that the turbine design is mechanically viable. Keywords
Micro gas turbine, radial turbine, 1-D analysis, inverse design optimization, computational fluid dynamics, stress analysis, modal analysis.
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b C0 , C s
Blade width/height Spouting velocity
C 𝐶𝑚 f h H i KB m Mrel N, N R Q r 𝑟𝑉̅𝜃 s o U V W Z Z R θ ω 𝛺 𝛿𝑝 ρ
Tangential velocity Meridional velocity Blade wrap angle Static enthalpy Total enthalpy Incident angle Blockage factor
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Nomenclature
Meridional distance, mass flow Relative Mach number Number of blades
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Volume flow radius Pitchwise-average swirl velocity Blade pitch Blade passage throat width Blade tip speed Velocity Relative velocity Axial coordinate Rotor axial length
σ β θ 2 c
Slip factor Blade angle Polar angle Diffuser divergence angle
s
Velocity ratio
ns
Specific speed(total-to-static)
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Tangential co-ordinate Rotational speed Vorticity Periodic delta function density Static efficiency
Subscripts bl m 0 Superscripts ̅ + -
at the blade Meridional direction Total condition Pitchwise mean value Relative to the upper blade surface Relative to the lower blade surface
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Introduction In present days, micro gas turbines are considered to be one of the most sophisticated portable devices to generate electricity power owing to its attribute of lower emission and compactness. Such devices are expected to reduce fuel consumption and alleviate environmental pollution, as the pressure on CO2 reduction is mounting and low fuel consumption is ever more demanding [1, 2]. Although micro gas turbines have issues to be addressed in terms of reliability, efficiency, and initial set–up cost, the intrinsic advantage of large power to weight ratio and lower emissions make them attractive to be used as portable power units [3-8]. The application of micro gas turbines covers many domains such as unmanned aerial vehicles, auxiliary power units, distributed power generator, mini combined heat and power units, and range extenders for electric vehicles. The broad range of the applications have encouraged industry and research organizations to study this area in order to develop efficient and reliable micro gas turbines. Table 1 is a list of academic organizations which have carried out micro gas turbine development and the corresponding engine specifications reported so far [9-11]. Table 1 Activity of micro gas turbine development and specification of the engines. Rotational speed/rpm 2,400,000 1,200,000 800,000 1,170,000 870,000 930,000 400,000 250,000 35,000 140,000 500,000 176,500
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Diameter/mm 4 6 12 8 10 10 8.4 10 10 12 16 30
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Type radial radial axial-radial axial-radial axial-radial radial radial radial axial-radial radial axial-radial radial disk
Output/W 60 52 300 475 485 500 44 10 10 100 800
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Organization MIT(US) MIT(US) Stanford University (US) University of Tokyo (Japan) Tohoku University (Japan). XJTU (China) SIMTech (Singapore) KIMM (Korea) KUL (Europe) AIT (Europe) ICL (Europe) ETH Zurich (Europe) WUT (Europe)
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A micro gas turbine follows the same thermodynamic cycle as the large sized ones. The average efficiency for a simple Brayton cycle system is reported to be 17%. However, it can be improved up to 30% by incorporating the recuperator [12]. In a recuperative cycle, hot exhaust gas is directed to the recuperator to heat compressed air before it enters the combustion chamber. By reusing the heat energy from the exhaust, the engine with a recuperator can improve the overall thermal efficiency[10]. The schematic of the micro gas turbine with recuperator is shown in Figure 1
Figure 1: The configuration and internal flows of a gas turbine with a recuperator[13] .
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The performance of the compressor and the turbine has significant influence on the overall performance of the gas turbine engine. Scaling a large diameter rotating component to a smaller one to meet design requirements is not appropriate. This is due to the fact that scaling results in a large change in the Reynolds number and heat transfer properties of the component [9, 14]. Therefore turbomachinery components of smaller dimensions need to be redesigned to meet the design specification. This paper introduces the design of a 113mm radial turbine for a 20 kW micro gas turbine using conventional design approach and optimization of the turbine geometry through an inverse design method. Subsequently, the resultant design is validated for performance and structural integrity using CFD and FEA. Interestingly, an inverse design method is confirmed to be very effective to optimize the existing design in a short period of time. The optimization through this method has resulted in double curvature turbine blade which is noteworthy. In practice, modifying conventionally designed turbomachinery device to improve the overall aerodynamic performance is a tedious job and sometimes attempts in such direction could further deteriorate the aerodynamic performance. However, a three-dimensional inverse design method ensures optimized aerodynamic performance which makes it a rational approach for optimization of radial turbo components.
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Design of a Radial Turbine In the design, it is important to ensure that the radial turbine will generate enough power to meet the needs of both 56 kW power that 110 mm centrifugal compressor consumes and 20kW output power for the generator. It was found that a 113mm radial turbine would meet the requirement. The geometry design process is based on the design approach proposed by Aungier [19]. A preliminary or mean-line design is essential in order to get an overview of the dimensions and aerodynamic performance of the turbine rotor. The success of any turbine design is mainly based on the accuracy of the mean-line design [1, 20]. A preliminary design of the radial turbine stage for this specific requirement is started with the basic design specifications which can be seen in Table 2. Table 2 Preliminary design input specification. Specification
Fluid
Air
Dimensionless specific speed
0.447
Static Adiabatic efficiency
0.8581
Velocity ratio
0.6274
Inlet total pressure (kPa)
420
Inlet total temperature (k)
1173
Mass flow rate (kg/s)
0.183583
Total-to-Static Pressure ratio
5.25
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Parameter
Ideally, the specification of the velocity ratio s and total-to-static efficiency is a function of the specific speed ns . The dimensionless specific speed can be given as [19], √
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(1)
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The chosen value of efficiency and velocity ratio for the design condition is shown in Figure 2. Although a preferred value for specific speed lies in the range of 0.45 to 0.75 [19, 21], the geometry dimension and rotational speed impose the constraint to choose a value, which marginally differs from this range.
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Figure 2 Generalized stage performance correlation.
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The outline of the stage considered for this analysis is shown in Figure 3. In general, 65 to 80 degree of vane outlet flow angle is recommended [18, 19]. Therefore, in line with the best practice, the inlet absolute angle about 72 degrees is considered for the design.
Figure 3 Turbine stage layout.
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In fact, nozzle vanes are considered to be significant components in the turbine stage, as they are intended to collect flow from a radial direction and direct it towards the rotor inlet [24]. Furthermore, these vanes are the main components that control the swallowing capacity of the turbine stage. Therefore, vanes blade angles and flow angles are considered to be a more significant parameter and need proper attention [25]. In the present case, a nozzle vane is designed with a parabolic-arc camber line proposed by Aungier [19]. The last part in the stage is a diffuser section. The diffuser section with better static pressure recovery coefficient has a positive effect on the total–to-static pressure ratio and efficiency [26]. Here, diffuser with 11-degree divergence angle is implemented as the divergence angle governs the diffuser performance [19]. It is imperative to account for the rotor passage losses in a preliminary design stage to 6
ACCEPTED MANUSCRIPT arrive at the turbine design with accurate performance estimation [15, 20, 21, 27]. In the present design endeavor, various losses are considered during the preliminary design and analysis. These losses include blade clearance loss, incidence loss, disk friction loss, blade loading and profile loss [15, 19, 21]. These losses are systematically employed in a preliminary analysis and details of which can be found in reference [19].
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After geometric modeling, the through-flow analysis was carried out to understand the flow behavior in the turbine and ANSYS Vista-TF commercial software was used in the process. Based on Mach number distribution, the end wall contours were altered to achieve a better flow distribution. The initial and modified meridional sketch is shown in Figure 4.
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Figure 4 Rotor meridional sketch.
Hub
Hub
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Furthermore, Mach number distribution with initial and modified contour has been analyzed as shown in Figure 5, which indicates the flow behavior is obviously improved particularly in the turbine exducer region.
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Mid-span
Mid-span
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Figure 5 Mach number distribution ( Left: Initial; Right: Modified).
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Finally, the turbine stage is modeled as 19 modified rotor blades, 30 nozzle vanes and diffuser passage downstream of the rotor blades. This stage geometry will be treated as the original geometry for further analysis. An overview of the rotor and vane geometry can be seen in Figure 6.
Figure 6 Three dimensional Rotor and stator geometry.
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It is indeed valuable to compare the one-dimensional and CFD result to get a confidence in the design approach. Here, the comparison for aerodynamic performance is made between one-dimensional analysis result of initial stage geometry and CFD result of modified stage geometry as shown in Figure 7, Figure 8, andFigure 9. Although the performance trend appears to be encouraging, the quantitative deterioration in performance can be attributed to the higher frictional loss. This loss is due to a narrow flow passage in the rotor inlet and nozzle vane section of the turbine stage. Therefore, in order to improve the performance, the turbine blades need to be optimized. The process of optimization is described in the next section.
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Figure 7 Comparison of 1-D analysis and CFD result for expansion ratio.
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Figure 8 Comparison of 1-D analysis and CFD result for efficiency.
Figure 9 Comparison of 1-D analysis and CFD result for power output.
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N rV 2
2 / N
rV d 0
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A procedure of inverse design was proposed and described by Zangeneh [17]. The fundamental idea of this method is to represent the blade by sheets of vorticity having its strength directly related to the specified bound circulation 𝑟𝑉̅𝜃 ; here 𝑟𝑉̅𝜃 can be given as (2)
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where N is a number of blades. In this method, a camber line of the blade is represented by a single sheet of vorticity and the blade blockage effect is included in continuity equation of the mean flow by means of stream surface thickness parameter. Hence, the bound vorticity on the blade can be presented by, V (rV ) p ( )
(3)
where α is the blade surface and can be represented as
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V (rV )
(5)
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In this method, an initial blade shape can be obtained by using one-dimensional velocity derived from the specified mass flow rate and meridional geometry prescribed with mean tangential velocity. A precaution need to be taken to align the flow with blade surface by means of applying the inviscid slip condition which can be represented by a first order hyperbolic partial differential equation as shown below
Wbl 0
or
VZ
f f V Vr bl z r r
(6)
here is a vector normal to the blade surface and is the relative velocity at the blade surface , where and are the velocities on the upper and lower surface of blades. The equation (30) needs to be integrated along the meridional projection of streamlines on the blade surface to find the blade shape. The requirement of initial condition for this integration can be satisfied by specifying the wrap angle ‘f’ which is called as ‘stacking condition of the blade’. Finite difference discretization scheme with Crank-Nicholson numerical technique can be used to solve this equation. After obtaining the blade shape the 10
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h h
rV 2 Wmbl N m
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Secondary flow in a radial turbomachinery is an important phenomenon which needs to be controlled to achieve good performance. This can be readily achieved through threedimensional inverse design method by controlling the blade pressure loading distribution on the hub and shroud independently. When the turbomachinery is compressible, the pressure/enthalpy loading can be represented in terms of meridional derivatives of 𝑟𝑉̅𝜃 or blade loading as, (7)
where superscript + and – represents either side of the blade, 𝑚 is meridional velocity at the blade and m is the direction of streamlines in the meridional plane [18].
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Ideally, a loading distribution or 𝑟𝑉̅𝜃 distribution has a significant effect on the blade surface pressure (Mach number distribution on the blade) [18]. This loading distribution can be represented by three segment curve which starts with a parabolic curve, then straight line section and end with a parabolic curve. The distribution of this curve can be manipulated depending on the work requirement of the blade. Once an optimum distribution of 𝑟𝑉̅𝜃 at hub and shroud section is arranged, it can be integrated to find the 𝑟𝑉̅𝜃 value at hub and shroud. Further, it can be linearly interpolated from hub to shroud and overall distribution of 𝑟𝑉̅𝜃 on a meridional plane can be achieved [17, 18].
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Another factor that affects the pressure (Mach number) distribution is the stacking condition [28]. This is analogous to the lean feature in the conventional turbomachinery design. The main effect of a lean or stacking is to distribute the blade forces in the spanwise direction. In this case, the blade can be lean linearly against or in the direction of rotation which has the corresponding effects on the aerodynamic performance and mechanical stress of the blade. If the blades are leaned against the direction of the rotation making the blade hub to lead to the blade shroud, then the consequent blade force would augment the pressure at the shroud end wall and subsequently lowers the pressure at the hub end wall [28]. Ultimately, care should be taken to apply the stacking in conjunction with 𝑟𝑉̅𝜃 in order to achieve the design which results in the suppression of secondary flow while complying with structural limitations [18, 28].
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Nevertheless, based on the CFD analysis of the modified geometry, it is evident that the turbine geometry needs to be optimized to achieve the improved performance. Optimizing the turbine geometry with the conventional design approach is tedious and time-consuming with limited or no certainty of improvement in performance. In such a scenario, an optimized turbine blade can be achieved by adopting the inverse design approach. The inputs for this method can be listed as:
Meridional sketch of geometry
Initial flow condition
Mass flow rate
Blade thickness
Blade numbers 11
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Rotational speed
Stacking condition
Specific work
Blade loading distribution (meridional derivatives of 𝑟𝑉̅𝜃 )
Input
Objective function
Objectives
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Figure 10 indicates the workflow of the inverse design method. In general, constraints are based on the design conditions. The objective function which in this case is a blade loading distribution can be optimized to accomplish the objectives. The objectives referred here are resultant geometry and performance attributes. A classification of constraints, objectives and objective function are also shown in Figure 10.
Constraints
Objective function
Objectives
Higher expansion ratio. Higher efficiency. Higher power output. lower secondary flow.
Y=f(Xi,…,Xn) Blade loading distribution = f (NCH,NDH,NCS,NDS,SLOPE_H,SLOPE_S)
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Meridional geometry. Initial flow condition. Mass flow rate. Blade thickness. Blade numbers. Rotational speed. Stacking condition. Specific work.
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Figure 10 Inverse design optimization workflow.
Most importantly, the objective function depends on the parameters such as NCH, NDH, NCS, NDS, SLOPE_H, and SLOPE_S which governs the distribution of blade loading. The impact of these parameters on blade loading is depicted in Figure 11. It must be noted that the optimization of the turbine blade through the conventional design approach needs to be done by tuning several parameters and the combination of this alteration could worsen the design performance. However, through inverse design method, the optimized geometry can be easily achieved by optimizing a few parameters depicted in Figure 11. 12
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Figure 11 Dependent variables of blade loading distribution.
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Therefore, the vital part, in this case, is to get optimum blade loading distribution by optimizing dependent variables as indicated in Figure 11. These variables remarkably influence the shape of blade geometry and performance of the radial turbine. Ideally, optimizing the magnitude and location of dependent variables is an iterative process which is based on the designer’s experience and blade loading requirements.
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In the present case, optimization of streamwise blade loading is resulted in front loading of hub and shroud as shown in Figure 12. In particular, peak loading on hub and shroud can be observed at 15% and 40% of the meridional length respectively. The blade loading at hub thereafter decreases consistently till the trailing edge of the blade. Likewise, the shroud loading decreases gradually to 90 % of the meridional length and thereafter steeply reduces towards the trailing edge of the blade. One of the reasons behind imposing the zero loading at trailing edge hub and shroud is to reduce the losses generated by variation of spanwise blade loading. In addition, zero value of blade loading at the leading edge (m=0) and trailing edge (m=1) is used to satisfy zero incident at the inlet and Kutta condition at blade outlet [17].
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Another important input to be specified is 𝑟𝑉̅𝜃 distribution or specific work in a spanwise direction. The 𝑟𝑉̅𝜃 value of 0.99 normalized by blade tip speed and inlet tip radius was specified at the inlet and zero at the turbine trailing edge based on turbine work requirement.
Figure 12 Blade loading distribution
Furthermore, the wrap angle or stacking condition was set at the blade inlet quasiorthogonal. Primarily for stress consideration, the wrap angle is applied in a manner that shroud leads to hub in the direction of rotation. The wrap angle at hub is set to zero whereas at shroud a positive 2 degrees wrap angle is applied. Interestingly, it has been observed that 13
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the inverse design can generate more complex shapes which are unlikely to be generated using the conventional design approach.
Figure 13 Optimized blade angle distribution.
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As shown in Figure 13, the distribution of blade angle profile is appeared to be highly curved with double curvatures. This distribution of blade angle is a result of the optimized blade loading shown in Figure 12. This fact indicates that the blade angle distribution is govern by the distribution of blade loading which is a crucial part of this optimization technique. In present study, the optimization is resulted in double curvature blade which is shown in Figure 14. Therefore, it is evident from this design optimization that a double curvature blade shape can be achieved by means of optimizing the blade loading distribution to enhance the performance. This type of turbine blade does not exist in the industry. Thus, an insight of such design and its design approach can pave a way for the development of new types of turbines.
Figure 14 (a) Conventional design (b) Optimized design.
In this study, no modifications were done on vane geometry and only turbine blade was optimized through the inverse design method for aerodynamic performance improvement. As mentioned previously, a same meridional geometry with normal thickness was used without any blockage factor. An ideal gas equation with specific heat ratio of 1.333 was used in flow 14
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A three-dimensional viscous flow analysis was carried out for original and optimized radial turbine stage while the nozzle vane geometry remains unchanged in both cases. For this purpose, commercial CFD software CFX16.0 was used. A mesh with around 340 thousand nodes and a first cell height of 5e-06m was used on both the turbine blade channels. Special care has been taken to keep the mesh size approximately similar in both cases for meaningful comparison. A single passage computational domain consists of a nozzle vane, radial turbine blade followed by the conical diffuser. The outline of this configuration is shown in Figure 15.
Figure 15 CFD simulation domain.
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In the CFD analysis, a mixing plane interface was used between vane and rotor domains whereas the rotor and diffuser have frozen rotor interface. The same tip clearance is used in both configurations. Moreover, for this flow analysis, compressible RANS equations were used with the SST turbulence model. This turbulence model has the capability to predict the flow in the vicinity of the solid wall with better accuracy [29]. The formulation of the SST turbulence is presented in equations (32) and (33).
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( k ) ( u j k ) k P * k [( k t ) ] t x j x j x j
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(8)
(9)
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here k is a turbulent kinetic energy, is dissipation rate, P a production term, t turbulent viscosity, and t turbulent kinematic viscosity. The primary objective of this viscous flow analysis is to estimate the overall aerodynamic performance in terms of mass flow rate, expansion ratio, efficiency and power output. To accomplish this objective, flow in the turbine stage was simulated at the speed of 86000, 93000 and 100000 RPM. Inlet temperature of 1173K, outlet static pressure of 84 kPa was set in the simulations and the total pressure was varied at the inlet. The results indicate that the optimized design turbine stage has a better total-to-static efficiency at high expansion ratio (lower U/Cs ratio) which can be seen in total-to-static efficiency against U/Cs ratio (Ratio of blade tip seed to spouting velocity) plot in Figure 16. Furthermore, Figure 17 shows that the improvement in expansion ratio is marginal and cannot be accounted as a significant 15
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improvement. However, based on efficiency plots, it can be concluded that optimized turbine geometry provides around 6% on average increase in total-to-static efficiency which can be considered as a significant gain in efficiency.
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Figure 16 Total-to-Static efficiency vs U/C: (a) 86000 (b) 93000 (c) 100000.
Figure 17 Expansion ratio Vs Mass flow rate.
Furthermore, as shown in Figure 18, over three rotational speeds, on average 6 kW improvement in turbine output power at high mass flow was observed. At a lower mass flow, there is hardly any obvious improvement in output power. However, at higher mass flow, the improvement in output power is considered to be significant.
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Figure 18 Power output (kW) : (a) 86000 RPM (b) 93000 RPM (c) 100000 RPM.
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In addition, contours of relative Mach number (Mrel) can be seen on the meridional plot in Figure 19. The distribution of relative Mach number is observed to be improved significantly in the optimized design. Likewise, the same pattern of relative Mach number can be seen on the blade-to-blade contour plots for shroud section shown in Figure 20 and for hub section shown in Figure 21. Indeed, the distribution of flow is largely improved in the optimized design where the flow separation, particularly at tip section, has been reduced substantially. The location of the maximum relative Mach has been shifted to trailing edge region in the optimized blade, which contributes to the reduction in rotor passage loss and enhances the aerodynamic performance of the turbine stage.
Figure 19 Contours of Mrel on meridional view (Left: Original; Right: Optimized).
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Figure 20 Blade to Blade view of Mrel contours at shroud (Left: Original; Right: Optimized).
Figure 21 Blade to Blade view of Mrel contours at the hub (Left: Original; Right: Optimized).
Structural Integrity
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A radial turbine in a micro gas turbine engine operates at high speeds. Therefore, an evaluation of strength, particularly at highest rotational speed, is of paramount importance [30, 31]. In the current design, a radial turbine is designed for maximum tip speed of 600m/s i.e. a rotational speed of 101400 RPM. Therefore, strength evaluation at this rotational speed was conducted.
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Stress analysis of optimized designed turbine was performed using the ANSYS structural software. An unstructured mesh on single blade sector was generated as shown in Figure 22.
Figure 22: FEA unstructured mesh.
A cyclic boundary condition is applied on the periodic surfaces of the blade sector with a fixed support at hub region. The material used for the turbine is Inconel 718 which is hightemperature resistance and high strength material with yield strength up to 1100 MPa. As stress analysis indicates, the maximum von-Mises stress on the blade appears at the blade root particularly at the trailing edge region which is shown in Figure 23.
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Figure 23 Equivalent (von-Mises) stress.
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The magnitude of stress is much less than the yield strength of the material which makes design safe for operation with 25% of safety margin. Furthermore, blade fillet is a crucial feature in the turbine blade modeling along with hub disk and has a significant impact on blade stress level. Therefore, in order to alleviate the centrifugal stress on the blade, a fillet of 0.7 mm radius is applied at the blade root section. As the turbine blades undergo a centrifugal load, a deformation of the structure is inevitable. As shown in Figure 24, the turbine blade deforms by 0.239 mm at blade exducer tip under centrifugal load which seems to be reasonable.
Figure 24 Total deformation under centrifugal load.
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Frequency mode evaluation is another important aspect of the rotor design which needs to be done to ensure that the resonance would not happen in the operating regime [21]. In order to verify it, a modal analysis was carried out using the single blade sector domain with cyclic boundary condition imposed at blade sector periodic surfaces.
Figure 25 1st frequency mode shape.
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ACCEPTED MANUSCRIPT The modal analysis indicates that the first mode shape appears at blade exducer tip as shown in Figure 25, with a frequency magnitude more than 4 times of blade vibration frequency which is sufficient to avoid resonance [21]. Hence, based on overall results, it is clear that the optimized design complies with the structural integrity and is suitable for engine operations. Conclusions
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A radial turbine design is developed for a 20 kW micro gas turbine engine. A comparison between the one-dimensional analysis result and CFD result was made to get the confidence in the design approach. Further, an optimization was carried out through an inverse design method to improve the efficiency and output power of a radial turbine. Optimization has resulted in a double curvature blade shape. Such types of blades currently do not exist in the industry and cannot be achieved using conventional design approach. The aerodynamic performance of the optimized geometry was subsequently evaluated using CFD simulations. It was found that an optimization results in a 6% increase in total-to-static efficiency on average and 6 kW additional total power output, particularly at higher expansion ratio and higher mass flow rate. Stress and frequency of the optimized three-dimensional rotor blade were analyzed. The results indicate that the maximum computed stress is much lower than the yield strength of the material and offer a 25% safety margin. A modal analysis shows the magnitude of first mode frequency is more than 4 times higher than the blade vibration frequency, which shows that the designed blade will work outside the resonance condition.
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Overall, this research demonstrates that the optimization using three-dimensional inverse design method is an effective way for achieving non-conventional turbine geometry. This optimization results in significant improvement in turbine aerodynamic performance while complying with the structural integrity of the turbine blade.
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Acknowledgments
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The research was jointly supported by European Horizon 2020 project 644971, 2017 T-TRIG project of the UK Department for Transport, and Innovate UK project 104021. The Authors wish to thank Advanced Design Technology Limited for their help and guidance.
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