Filippo Valentini

(1)

Numerical Modelling of a Radial Inflow Turbine with and without

Nozzle Ring at Design and Off-Design Conditions

Filippo Valentini

Master of Science Thesis EGI_2016-094 MSC EKV1169

KTH School of Industrial Engineering and Management Machine Design

(2)

Master of Science Thesis EGI_2016-094 MSC EKV1169

Numerical Modelling of a Radial Inflow Turbine with and without Nozzle Ring at Design and

Off-Design Conditions Filippo Valentini Approved Examiner Paul Petrie-Repar Supervisor Jens Fridh

Commissioner Contact person

Abstract

The design of a radial turbine working at peak efficiency over a wide range of operating conditions is nowadays an active topic of research, as this constitutes a target feature for applications on turbochargers. To this purpose many solutions have been suggested, including the use of devices for better flow guidance, namely the nozzle ring, which are reported to boost the performance of a radial turbine at both design and off-design points. However the majority of performance evaluations available in literature are based on one-dimensional meanline analysis, hence loss terms related to the three-dimensional nature of real flows inside a radial turbine are either approximated through empirical relations or simply neglected.

In this thesis a three-dimensional approach to the design of a radial turbine is implemented, and two configurations, with and without fixed nozzle ring, are generated. The turbine is designed for a turbocharging system of a typical six-cylinder diesel truck engine, of which exhaust gas thermodynamic properties are known. The models are studied by means of a CFD commercial software, and their performance at steady design and off-design conditions are compared.

(3)

ACKNOWLEDGEMENTS

I would like to express my gratitude to my supervisor, Jens Fridh, and my examiner, Paul Petrie-Repar, for their availability and support, and without whom this work would have never been realised.

Sincere gratitude to my Italian examiner Alessandro Talamelli, who followed the project of Dual Degree between UniBo and KTH since its inception: it is also thanks to him that I ended up at KTH, a circumstance which I will never regret.

I would like to thank my parents, who believed in me and allowed me to complete my academic iter, providing full moral and material support.

A special mention goes to my Italian friends: Pietro, who was always ready to support me with a Skype call and welcome me back during my short visits in Italy, and Simone, who, no matter where he is, always finds a way to be present in the most crucial moments.

(4)

iii

LIST OF FIGURES

NUM. TITLE PAG.

Figure 1 Thermodynamics of a radial turbine (Nguyen-Schäfer, [16], adapted) 1 Figure 2 Scheme of a 90° IFR turbine. Left: frontal view. Right: side view (Ventura,

[13])

2 Figure 3 Flow velocity triangles within a radial turbine (Dixon, [6]) 3 Figure 4 Thermodynamic diagram of the process through a 90° IFR turbine (Dixon,

[6])

4 Figure 5 Nominal design configuration (Saravanamuttoo, [19], adapted) 5 Figure 6 Behaviour of loss terms as function of incidence angle (Yahya, [33]) 6

Figure 7 Secondary flow in a blade passage (Yahya, [33]) 7

Figure 8 Full process iter in turbine design (Khader, ,[9]) 8 Figure 9 Schematic diagram of a vaneless volute casing (Whitfield, [31]) 16 Figure 10 Left – velocity profile across the centre-line of the volute section. Right –

variation of centroid radius at two subsequent azimuth positions (Whitfield, [31], adapted)

18

Figure 11 Theoretical distribution of centroid radius, cross section area and flow angle with azimuth location

20 Figure 12 Distribution of centroid radius, cross section area and flow angle with

azimuth location. Comparison between theoretical and implemented solution, vaneless case

21

Figure 13 Distribution of centroid radius, cross section area and flow angle with azimuth location. Comparison between theoretical and implemented solution, vaned case

22

Figure 14 3D geometrical model of the volute casing. Left: vaneless. Right: vaned (no nozzle)

23 Figure 15 Nozzle vane geometry definition (Rajoo & Martinez-Botas, [18]) 24 Figure 16 Nozzle ring. Left: sketch in the frontal plane. Right: shape of the blade 24

Figure 17 3D geometrical model of the nozzle ring 25

Figure 18 Sketch of the velocity triangles at rotor inlet and outlet (Saravanamuttoo, [19], adapted)

26 Figure 19 Rotor views. Left: r-z (or meridional) plane. Right: θ-z (or blade to blade)

plane

27 Figure 20 Bezier curve of degree 3. Left: basis of vector space. Right: control points

(Floater, [7], adapted)

28 Figure 21 Rotor: θ-distribution (top), β-distribution (middle), thickness distribution

(down)

29 Figure 22 Rotor: distribution of wrap angle (top left), flow angle (top right) and

thickness (bottom) in the meridional plane

30 Figure 23 Rotor: variation from inlet to outlet of channel cross-section area (left) and

lean angle (right)

31

Figure 24 3D geometrical model of the rotor 31

Figure 25 Geometrical definition of the problem 32

Figure 26 Pressure distribution in a crosswise section. Effect of streamwise pressure gradient (left), effect of blade-to-blade pressure gradient (middle),

ensemble (right) (Van den Braembussche, [28])

33

Figure 27 Conical diffuser. Left: 2D sketch. Right: lines of appreciable stall for given geometrical configuration (Blevins, [4], adapted)

34

(7)

vi

Figure 29 Elements of a 3D mesh - tetrahedron, hexahedron, prism, pyramid 35 Figure 30 Velocity profile in a turbulent boundary layer (Bakker, [3]) 36 Figure 31 Non-dimensional velocity as function of 𝑦₊ in the inner region (Kundu,

[11])

37 Figure 32 Stretching of a quadrilateral element. Nominal shape (left), deformed

shape (right)

38

Figure 33 Orthogonal quality on a 2D quadrilateral cell 39

Figure 34 Mapping of an hexahedral element (Bucki, [5]) 39

Figure 35 Example of setup of 𝐴𝑁𝑆𝑌𝑆®_{Meshing (volute)} ₄₀

Figure 36 Mesh of the volute (section) 42

Figure 37 Mesh of the diffuser (ensemble) 42

Figure 38 Mesh of the nozzle ring (one blade) 43

Figure 39 Topology for the rotor blade. The blade (blue) is surrounded by meshing blocks

43

Figure 40 Mesh of the rotor (portion) 44

Figure 41 Mesh statistics for the rotor 45

Figure 42 Illustration of interfaces and boundary conditions for vaned configuration 48 Figure 43 Choice of representative off-design points (Mora, [1], adapted) 49 Figure 44 Spanwise distribution of 𝛽₂, comparison at design point 51 Figure 45 Spanwise distribution of 𝛼₃, comparison at design point 51 Figure 46 Rotor blade loading comparison at 10% span (top, left), 50% span (top,

right), 90% span (bottom), design point

52 Figure 47 Static entropy around the blade at 90% span. Left: vaneless. Right: vaned 52 Figure 48 Mach distribution around the blade at 10% span. Top: vaneless.

Bottom:vaned

53 Figure 49 Mach distribution around the blade at 50% span. Left: vaneless. Right:

vaned

53 Figure 50 Mach distribution around the blade at 90% span. Left: vaneless. Right:

vaned

53 Figure 51 Velocity contour around nozzle ring blade, 50% span, design point 54 Figure 52 Spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 1 55

Figure 53 Velocity distribution around the blade at 50% span. Left: vaneless. Right: vaned

55 Figure 54 Velocity distribution around the blade at 90% span. Left: vaneless. Right:

vaned

55 Figure 55 Spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 4 56

Figure 56 Static entropy in meridional lane, circumferential average, off-design point 4. Left: vaneless. Right: vaned

56 Figure 57 Velocity distribution around the blade (left) and blade loading (right) at

90% span, off-design point 4

56 Figure 58 Spanwise distribution of 𝛽₂ (left) and 𝛼₃ (right), off-design point 5 57 Figure 59 Static entropy in meridional plane, circumferential average, off-design

point 5. Left: vaneless. Right: vaned

57 Figure 60 Velocity contour around nozzle ring blade, 50% span, off-design point 5 57 Figure 61 Spanwise distribution of 𝛽₂ (left) and 𝛼₃ (right), off-design point 7 58 Figure 62 Rotor blade loading, comparison at 10% span (top, left), 50% span (top,

right), 90% span (bottom), off-design point 7

58 Figure 63 spanwise distribution of 𝛼₂. From top to bottom: design point, off-design 1,

off-design 4, off-design 5, off-design 7

(8)

vii

LIST OF TABLES

NUMBER TITLE PAGE

Table 1 Meanline design parameters for nozzle ring 19

Table 2 Meanline design parameters for vaneless and vaned volutes 20 Table 3 Relative angle at rotor outlet under nominal design condition 26

Table 4 Meanline design parameters for the rotor 28

Table 5 Meanline design parameters for the diffuser 33

Table 6 Estimation of first layer thickness for turbine components 41

Table 7 Mesh statistics for turbine components 41

Table 8 Thermodynamic properties of the studied off-design points 49

Table 9 Performance comparison, design point 50

Table 10 Comparison between mean velocity triangles at rotor inlet (top) and at rotor outlet (bottom), design point

50 Table 11 Comparison of performance (top) and mean flow angles (bottom) at

off-design points

(9)

viii

NOMENCLATURE

ABBREVIATIONS

ATM Automated Topology and Meshing CAD Computer Aided Design

CFD Computational Fluid Dynamics DNS Direct Numerical Simlation FEA Finite Element Analysis FEM Finite Element Method IFR Inflow Radial

JR Jacobian Ratio LE Leading Edge

PDE Partial Differential Equation RANS Reynolds Averaged Navier-Stokes RPM Revolutions per Minute

OQ Orthogonal Quality SBP Single Blade Passage SST Shear Stress Transport TE Trailing Edge

T-T Total-to-Total

1-2-3D One-Two-Three Dimensional

GREEK SYMBOLS

α Absolute flow angle Constant (k- ω model) β Relative flow angle Blade angle (rotor) Constant (k- ω model) γ Specific heat ratio

δ Boundary layer thickness

ε Dissipation of turbulent kinetic energy η Efficiency

θ Wrap angle (rotor)

Non-dimensional mass flow rate (volute) Deformation angle (mesh statistics) λ Angle (general notation)

μ Dynamic viscosity μ_𝑡 Turbulent viscosity ν Kinematic viscosity ξ General coordinate ρ Density ζ Constant (k- ω model) η Shear stress

Φ, θ Azimuth angle (volute) ψ Diffusion angle (diffuser) Ω, ω Angular velocity

Dissipation rate of turbulent kinetic energy (RANS model)

(10)

ix a Speed of sound

A Area

b Passage width (volute)

Vane axial chord (nozzle ring) c Absolute flow velocity

Vane true chord (nozzle ring) 𝐶𝑓 Skin friction coefficient

g Gravitational acceleration h Enthalpy

Blade height (volute) i,j,k Indexes (general notation) k Constant (volute)

Turbulent kinetic energy (RANS model) 𝑘𝑎 Axial length (rotor)

l Axial length (diffuser) L Work exchange

m Meridional coordinate Vortex exponent (volute) Mass (general notation) 𝑚 Mass flow

M Mach number

Moment of external forces (general notation) n Rotational speed

Coordinate normal to streamline (rotor) o Nozzle throat P, p Pressure Constant (volute) q Heat exchange R, r Radius 𝑅𝑒 Reynolds number s Entropy

Nozzle pitch (nozzle ring)

S Angular momentum ratio (volute) T Temperature

u Velocity (general notation) U Blade speed

Velocity (general notation) 𝑢∗ Friction velocity

w Relative flow velocity W Work exchange X, x General coordinate y General coordinate

𝑦+ Non-dimensional coordinate normal to a wall

𝑍_𝑣 Number of blades (nozzle ring) 𝑍_𝑏 Number of blades (rotor)

SUBSCRIPTS (*)

0 Stagnation property 1 At volute inlet

(11)

x 3 At rotor outlet, At diffuser inlet

4 At diffuser outlet

a Relative to axial coordinate b Relative to the blade

e Relative to ideal case (mesh statistics) h Relative to blade hub (rotor)

i,j,k Relative to indexes i,j,k

m Relative to meridional coordinate n Relative to the coordinate n nr At nozzle ring

r Relative to radial coordinate

R Recirculating at volute tongue (volute) ref Relative to a reference quantity s Along isentropic transformation, Static t Stagnation property

Relative to blade tip (rotor)

Relative to the tangential coordinate (rotor) ts Total-to-static

tt Total-to-total

x Relative to the coordinate x y Relative to the coordinate y

θ Relative to circumferential coordinate

(12)

1

1 – INTRODUCTION

In an internal combustion engine a turbocharger is a system composed by a radial turbine and a centrifugal compressor mounted on a common shaft. The turbine converts part of the enthalpy of the exhaust gases into kinetic energy delivered to the shaft. The shaft drives the compressor which increases the air pressure sent to the combustion chamber at each piston cycle, allowing an increase in power output: for this reason turbocharging is often considered in applications where power demand is a priority, such as in race car, heavy duty vehicle and marine engines. Turbocharging is also beneficial from the point of view of overall engine efficiency. As noted by Mora [14], more power available per unit cycle leads to smaller engines, with consequent decrease of mechanical losses, and the higher density of air at the combustion chamber improves volumetric efficiency. Moreover the performance of a supercharged engine is only minimally affected by variations in ambient pressure, since pressure is a design parameter of the turbocharger itself at steady conditions: consequently the engine operates at more constant regime.

The absolute performance of a turbocharger is limited by the energy content of the working fluid, defined by its total enthalpy. As shown in Fig.1, the work that the turbine can extract is the difference between the total enthalpy at rotor inlet and the total enthalpy at rotor outlet, i.e. without taking into account the contribution of flow speed. Whatever velocity component the flow still has at outlet, infact, is “wasted” from a thermodynamic point of view; in the ideal case the turbine should be able to expand the flow so to reach outlet pressure at zero velocity, and without introducing losses (i.e. isentropically).

Figure 1: thermodynamics of a radial turbine (Nguyen-Schäfer, [16], adapted)

The ratio between the actual work and the total available energy defines the efficiency of the turbine, which is a key driver in the design of the turbomachinery.

(13)

2

physical aspects of power generation. Then the main dissipative phenomena arising in a real-case turbomachinery are presented, since their knowledge is crucial for design purpose. The subsequent section illustrates the steps of the design process, which also the present work is based on. The introduction is concluded by a literature review highlighting the results obtained so far in the design of radial turbines.

1.1 The radial turbine

For turbocharger applications a 90° IFR turbine is generally employed, as it guarantees compactness, good efficiency within a wide range of operating conditions and high structural strength. A sketch of such a turbine is shown in Fig.2.

Inside the volute the flow is accelerated and at the outlet it is delivered uniformly with a desired outflow angle. At this stage the flow can enter directly into the rotor or pass through a nozzle ring (stator): the former configuration is called vaneless, the latter vaned. The nozzle ring, when present, aims at guiding the flow into the rotor inlet, and since the passage between blades is convergent it further increases its speed. Inside the rotor (also called impeller) the flow exerts an aerodynamic force on the blades, thus transferring part of its energy to the rotating device: the goal of the impeller is to extract the highest possible amount of work from the working fluid. The flow exits the impeller with a mainly axial velocity component and passes through a diffuser, where it is slowed down and part of its static pressure is recovered before being discharged.

Figure 2: scheme of a 90° IFR turbine. Left: frontal view. Right: side view (Ventura, [13]) The process can be described from a thermodynamic point of view, which is useful to highlight the physical quantities affecting the performance of the turbine and to deduce preliminary considerations about its design.

(14)

3 𝑑𝑕 + 𝑐𝑑𝑐 + 𝑔𝑑𝑧 = 𝑑𝑞 − 𝑑𝐿 𝑑𝑞 =0; 𝑑𝑧=0; −𝑑𝐿=𝑑𝑊 𝑑𝑕 + 𝑐𝑑𝑐 𝑑 𝑕+𝑐2₂ = 𝑑𝑊 𝑕0≝𝑕+ 𝑐2 2 𝑑𝑕₀ = 𝑑𝑊 (1.1)

The work per unit mass exerted by the flow on the turbine shaft is linked to the rate of change of angular momentum that the flow itself undergoes inside the rotor, and may be expressed by the so called “Euler turbine equation”, here in differential form (for the exact derivation see Mora [14])

𝑑𝑊 = 𝑑 𝑈𝑐_𝜃 (1.2) The combination of eqn.(1.1) and eqn.(1.2) yields to eqn.(1.3):

𝑑(𝑕 ₀− 𝑈𝑐_𝜃)

𝑟𝑜𝑡 𝑕𝑎𝑙𝑝𝑦

= 0 (1.3)

which states that in the thermodynamic process through the impeller the rotational total enthalpy 𝐼 ≝ 𝑕0− 𝑈𝑐𝜃 (also called rothalpy) is constant. The same expression is re-arranged according

to the velocity triangle in Fig.3, which relates the velocity vectors in the fixed and the rotating frame of reference:

𝐼 = 𝑕 +1₂𝑐2_{− 𝑈𝑐}

𝜃 = 𝑕 +1₂ 𝑤2 + 𝑈2+ 2𝑈𝑤𝜃 − 𝑈 𝑤𝜃 + 𝑈 = 𝑕 +1₂𝑤2−1₂𝑈2 (1.4)

Figure 3: flow velocity triangles within a radial turbine (Dixon, [6])

Through eqn.(1.4) it is possible to express the variation of static enthalpy between inlet and outlet of the rotor (denoted by indexes 2 and 3, Fig.4), thus from the combination of eqn.(1.1) with eqn.(1.4) the total specific work on the turbine is:

∆𝑊2−3 = 𝑕 +𝑐 2 2 ₂₋₃ = 1 2[ 𝑈2 2_{− 𝑈} 32 − 𝑤22− 𝑤32 + 𝑐22− 𝑐32 ] (1.5)

(15)

4

a) In the case of an IFR turbine the term 1₂ 𝑈22− 𝑈32 is always positive because 𝑈2> 𝑈3; the

same does not hold for an axial turbine, where inlet and outlet radiuses are equal. As direct consequence, the work per unit mass that can be extracted with a single turbine stage is higher in the former case, and so the efficiency, which justifies the exclusive employment of radial turbines in turbochargers.

b) A positive contribution to the specific work is obtained when 𝑤3 > 𝑤2. This is achieved

if the channels between blades are convergent.

Eqn.(1.4) shows that the higher is 𝑤₃ the higher is the static enthalpy jump 𝑕2 − 𝑕3

across the rotor (and consequently the specific work) but the lower is the static pressure 𝑝3 at rotor outlet (see Fig.4). However 𝑝3 cannot be lower than the atmospheric pressure,

otherwise the discharge would not be possible. The diffuser is used to allow 𝑝3 < 𝑝𝑎𝑡𝑚

(with a benefit for efficiency) and to recover part of the static pressure afterwards so that at turbine outlet the condition 𝑝4 ≥ 𝑝𝑎𝑡𝑚 is fulfilled.

Moreover “accelerating the relative velocity through the rotor is a most useful aim of the designer as this is conducive to achieving a low loss flow” (Dixon, [6]).

c) Since the specific work is proportional to the quantity 1₂(𝑐22− 𝑐32) the absolute velocity

should be large at impeller inlet, which is achieved by means of the volute.

The volute (as well as the diffuser) is a static component (𝑑𝑊 = 0) and from eqn. (1.1) the total enthalpy is conserved (see also Fig.4) which implies that the higher is the drop in static enthalpy𝑕₁− 𝑕₂ the higher is the absolute velocity seen by the rotor at its inlet.

Figure 4: thermodynamic diagram of the process through a 90° IFR turbine (Dixon, [6]) With reference to point c) above, a deeper analysis is needed. Eqn.(1.2) can be evaluated between points 2 and 3, leading to eqn.(1.6)

∆𝑊₂₋₃ = 𝑈₂𝑐_𝜃2− 𝑈₃𝑐_𝜃3 (1.6)

(16)

5

 a high value of 𝑐2is not the only requirement because what matters is the projection of 𝑐2

along the circumferential direction, 𝑐𝜃2. The ideal condition would be 𝑐2 ≡ 𝑐𝜃2 but this is

not physical since there would be no inflow at rotor inlet: the absolute flow angle is chosen such that the relative velocity has only radial component.

A good volute design must therefore take into account the orientation of the absolute velocity at rotor inlet, which may be accomplished by using vaned stators.

 the specific work is increased if the absolute velocity of the flow at rotor outlet is axial, i.e. 𝑐𝜃3 = 0.

The two conditions mentioned above constitute the so called nominal design (Fig.5).

Figure 5: nominal design configuration (Saravanamuttoo, [19], adapted)

1.2 Sources of losses

Nominal design is a theoretical condition which sets an upper bound to the maximum specific work that a radial turbine can extract.

In real cases, however, this limit is never reached because dissipative phenomena arise from the interaction between flow and solid surfaces of the turbomachinery: these phenomena (take Fig.1 as reference) increase the level of entropy and hence lower the jump in total enthalpy across the turbine, which is linked to the amount of work by eqn.(1.1).

Several loss models have been developed and are now available in literature (Ventura, [13]) since careful evaluation of losses is crucial for performance estimation: however most of them are based on empirical relations. In this section a qualitative description of the main sources of losses in radial turbines is presented, and observations are made on how to limit them.

 Channel losses

(17)

6  Incidence losses

Here are considered, without further distinctions, all losses which arise from the non-ideal orientation of the velocity vector with respect to the rotor blades (and blades of the nozzle ring, when present). For a 90° IFR turbine incidence losses originate when the flow does not enter the rotor radially or leave it axially, as stated in the nominal design. Incidence losses (also called shock losses, even if they are not related to compressibility effects) grow almost parabolically with incidence angle (Dixon, [6]): the specific behaviour depends on the loss model adopted but more in general Fig.6 shows that this term contributes heavily to overall losses at off-design incidence angles.

Figure 6: behaviour of loss terms as function of incidence angle (Yahya, [33])

A well known solution to reduce incidence losses is to achieve better flow guidance at the exit of the volute by means of a nozzle ring.

 Secondary losses

Secondary flow generally denotes a portion of fluid which does not follow the streamlines of the main flow.

As an example let us consider a channel between rotor blades (Fig.7). Suction and pressure sides of adjacent blades create a pressure gradient orthogonal to the streamlines: this induces a crossflow which develops a boundary layer in the local cross section planes of the channel. A direct consequence is the onset of shear stresses because of the velocity gradient (recall 𝜏 = 𝜇 𝜕𝑈_𝜕𝑦

𝑤𝑎𝑙𝑙); moreover the vorticity injected in the flow creates

(18)

7

Figure 7: secondary flow in a blade passage (Yahya, [33])  Tip clearance losses

Tip clearance is a gap between blade shroud and casing which is designed to allow the expansion of the rotor under thermal effect and centrifugal force. Across this gap leakage occurs between the pressure and the suction side of the blade, driven by the pressure gradient: the behaviour is the same of a wing of finite span.

This secondary flow through the clearance does not contribute to the work done on the rotor, causing a reduction of the work output compared to the condition of designed mass flow. Moreover tip clearance flow is responsible for the formation of vortices which may also interact with other secondary flows (Siggeirsson et al., [22]).

Tip clearance losses depend on the relative size of the gap compared to the blade height.

1.3 The design process

In section 1.1 an ideal design condition was derived from studying the thermodynamics of the expansion process inside the radial turbine. As this approach does not require any knowledge about the internal flow pattern in terms of velocity and pressure fields, but only the average physical quantities at specific sections of the turbomachinery, it is a common tool at early stages of the design process: such approach is called meanline design.

Meanline analysis consists of treating the working fluid as a one dimensional flow at the mean radius of the turbine while the flow parameters are assumed as reasonable average values across the full span (Wei, [29]).

(19)

8

Finally an accurate performance estimation is done on the 3D geometrical configuration by simulating the internal flow with CFD techniques. A feasibility check is needed as well in order to assess structural robustness under operating loads: FEA is commonly employed in this case.

Figure 8: full process iter in turbine design (Khader, ,[9])

“Turbine design is an iterative process” (Khader, [9]). Fig.8 shows the correlation between different steps: in particular the initial 3D geometry is modified according to the outcomes of the analysis step. The latter are continuously compared with the results from the meanline design (which is an idealized design, thus represents the target) and the final configuration is reached when sufficient convergence exists between the two steps.

1.4 Design and analysis: state of the art

“There are two problems of interest to designers of turbomachinery” (Yang, [34]). In the so called “inverse” (or design) problem the overall geometry is defined, based on assumptions about the flow pattern and other existing specifications, to yield desired performance characteristics, while in the “direct” (or analysis) problem the performance of a given geometrical configuration is assessed.

The direct problem is commonly tackled by means of CFD tools: despite the approximations linked to numerically solving NS equations this method allows the simulation of three-dimensional flows on complex geometries, which explains the presence of a vaste literature on the analysis of radial turbines including comparative studies among different geometrical configurations. On the other hand “research activities pertaining to the inverse design problem has not been extensive” (Tjokroaminata, [26]).

Since both three dimensional design and performance analysis (second and third step of the design process, respectively) are topics of the present work, a review of the main results of public domain in these fields is now presented.

1.4.1 Inverse problem: the volute

(20)

9

thesis. Firstly they developed a practical method, suitable for implementation in a program code, based on simple hypotheses of incompressible flow and free vortex law. Since the outflow angle is specified as input parameter the aim of Whitfield’s method is to design distributions of centroid radius and cross section area which ensure the flow to be delivered uniformly and at the appropriate angle to the impeller. A subsequent experimental investigation on such obtained geometry showed that the free vortex pattern is only a fair approximation over the first 180° of azimuth angle, while as the tongue position is approached the variation of the tangential component of velocity with radius reduces considerably (Whitfield et al., [32]). The free vortex law was therefore modified accordingly by means of a so called vortex exponent.

The method gives satisfactory results, at least at first attempt, but relies on several purely empirical considerations such as the shape of the vortex exponent or the variation of Mach number with azimuth angle. Moreover no indications are given about the optimal shape of the volute cross sections. The latter aspect is studied by Shah et al. [20] who, after the comparison between trapezoidal, Bezier-trapezoidal and circular cross sections, suggested that the circular cross section will give a better efficiency.

Abidat et al. [1] suggest to design the distribution of centroid radius by means of a Bezier polynomial and the distribution of cross section area by assuming a linear reduction of mass flow with azimuth angle. The employment of parametric curves makes this method flexible for the designer and allows her to easily take into account other possible constraints (for example on the radius at volute outlet) but has somehow an empirical basis. Moreover there is no careful design of the critical region around the tongue.

1.4.2 Inverse problem: the rotor

Yang [34] pointed out that the complex nature of the flow through a real turbomachine would make a three-dimensional design procedure difficult if not impossible: for this reason lot of effort was put on the development of approximate two-dimensional methods in the hub-to-shroud plane.

An example of such method is presented in the work by Smith & Hamrick [24]. They prescribed as input parameters blade shape, hub shape and velocity distribution along the hub, then they introduced an estimated streamline from inlet to outlet of the rotor and checked for continuity of flow through the annular streamtube within the hub and the streamline. If continuity is not established, the streamline spacing is adjusted accordingly and another annular streamtube is constructed over the first, following the same criterion: in this way the final streamline determines the shape of the impeller shroud.

This method relies on the assumptions of isentropic, steady and non-viscous flow, but its real limitation is due to the arbitrary choice of the input parameters, which has no theoretical basis although strongly affects the final solution.

A turbomachinery blading design method in three-dimensional invscid flow was suggested by Yang [34]. Here the blade is represented by a sheet of bound vorticity, i.e. bound to the solid surface of the blade. Under the assumptions of steady inviscid and irrotational flow the only vorticity in the flow field is that generated on the blade surfaces, which is related to the circulation on a closed path around the axis of rotation: by carefully specifying the mean swirl distribution (which, if integrated, gives the total amount of circulation) the distribution of bound vorticity is specified as well, and the blade surface geometry is obtained as that location of the bound vortex sheet in which the normal velocity vanishes. The condition of non-penetrating flow must infact hold on the solid surface of the blade.

(21)

10

 “for a wide variety of swirl distributions there always exists a region of inviscid reversed flow on the pressure surface of the blade [...] which may result in flow separation” (Tjokroaminata, [26])

 Blades obtained with such design tend to be highly twisted, which may lead to structural problems especially in applications where the rotational speed of the impeller is high

1.4.3 Inverse problem: the remaining components

Diffuser and fixed stator are relatively simple components and their design is not investigated extensively.

Siggeirsson & Gunnarsson [22] report that “when the diffusion angle [(defined as the slope of the wall of the divergent pipe)] is large the diffusion rate is rapid and can cause boundary layer separation resulting in flow mixing and stagnation pressure losses. On the other hand if the diffusion rate is too low the required length of the diffuser will be very large and the fluid friction losses increase”. This opinion is shared by other authors such as Dixon [6], who sets to 7° - 8° the value of the diffusion angle which gives an optimal rate of diffusion.

Khader [9] relied on a direct approach for the design of a fixed-geometry nozzle ring. Straight symmetric blades with rounded LE and TE are chosen, and an initial 3D geometry of the component is guessed. A following analysis with CFD is performed, and modifications on the incidence angle of the blades are made until no separation of the flow is detected. The method followed by Khader, i.e. the iteration of the analysis step in order to reach an optimized design, may be rather time consuming, but allows to end up with a configuration in which incidence losses are minimized.

1.4.4 Direct problem: performance analysis

Simpson [23] performed a CFD analysis of existing test turbine geometries in both vaned and vaneless configurations. A total of six geometries were analysed and the results compared with measured turbine performance data. “Steady state predictions showed good agreement with the experimental trends confirming the vaneless stators to yield higher efficiencies across the full operating range” [23]

According to the author, vaned stators lead to a higher level of losses because of the wake detaching from vane trailing edges, boundary layer growth and secondary flows.

Spence et al. [25] tested three pairs of vaneless and corresponding vaned stators within a range of pressure ratios and flow rates. For each pair of stators the rotor was the same and the operating conditions were identical. “The vaneless volutes delivered consistent and significant efficiency advantages over the vaned stators over the complete range of pressure ratios tested. At the design operating conditions, the efficiency advantage was between 2% and 3.5%” (Spence et al, [25]).

(22)

11

2 – MOTIVATION AND OBJECTIVE

2.1 Motivation

It is estimated (ATRI, [27]) that fuel-related cost accounts for around 35% of overall operational costs of trucking, which justifies the need to improve engine efficiency and nowadays’ extensive research on turbochargers for applications on heavy-duty vehicles.

Despite attempts to find optimal solutions have been done (and have been summarized in the literature review), the problem of designing the turbine of a turbocharger so to guarantee the best achievable performance is still open. The reason is that the complexity of the flow within a turbomachinery requires the introduction of simplifying assumptions about its mean motion, which may lead the designer to neglect some dynamics in the flow pattern (typically secondary flows) and misevaluate losses.

Moreover, even assuming an ideal geometrical configuration, this would be valid for just one design point. During a combustion cycle, however, pressure and temperature of the exhaust gases vary over a considerable range, thus the performance of that configuration should be optimal also at off-design points. Variable angle stators try to achieve the latter condition by continuously adapting the inflow angle at rotor inlet, hence reducing incidence losses, but the implementation of such devices is by now considered unfeasible due to high mechanical stresses and vibrations.

A comparative study performed by Mora [14] pointed out that a turbine with fixed nozzle ring has both higher efficiency and power output with respect to a vaneless turbine throughout all the combustion cycle. This statement is based on a meanline analysis, but in order to be tested a three-dimensional design of the two turbine configurations should be developed and an investigation should be performed by means of CFD techniques so to describe the internal flow in details and identify sources of losses otherwise undetectable.

2.2 Objective

In light of the previous research carried out by Mora, and considering the existence of different opinions reported in literature, the purpose of this paper is to present a comparative

performance assessment between two turbine configurations, vaneless and with static nozzle

(23)

12

3 – METHODOLOGY AND TOOLS

3.1 Methodology

The strategy used in the present work is meant to be an implementation of the design process which was described in the introductory section. In order to achieve the objective several steps were followed

 Literature review

A literature review was carried out firstly. The purpose was to collect background information about existing configurations of radial turbines and studies of their performance, as well as techniques for the design of specific components, when available.

 Analysis of the meanline design

The reference meanline design was analysed in details, with specific focus on identifying the average flow parameters at inlet and outlet of all components of the turbine and potential geometrical constraints. This phase defines the inputs which enter the subsequent step.

 3D design of components

A theory-based procedure was developed for the 3D design of the volute and the impeller. Diffuser and static nozzle ring were not designed with an inverse approach but as first attempt geometries due to their relatively simple shapes. All the components were drawn by means of a CAD software (for the rotor a dedicated software, allowing a high flexibility in the design, was employed).

 CFD simulation setup

All components were meshed separately. Structured hexahedral mesh was used for the rotor blade passage while coarser tetrahedral mesh for the remaining components (with a refinement close to the walls, so to better describe the boundary layer). After meshing, vaneless and vaned configurations were assembled: the connection of different meshes was obtained by specifying suitable interfaces on the contact surfaces. Fluid thermodynamic properties and boundary conditions were specified as well, and an appropriate turbulence model was chosen for the solution of RANS. At the end of this step the two turbine configurations were ready for a CFD steady simulation.

 Final design

The results of CFD simulations on both vaneless and vaned cases were compared and the original 3D design was modified until the average velocity vector of the flow at rotor inlet was the same for both configurations (since rotor and diffuser are also the same, this guarantees a “fair” comparison between the two turbines, which only differ by the presence/absence of a static nozzle ring: every difference pointed out in a subsequent performance analysis may thus be attributed to that component). At the end of this iterative phase the final geometrical configurations were obtained and the design process is concluded.

 Analysis of results

(24)

13

3.2 Tools – Software

The methodology contains several sub-objectives to be reached before the final comparative analysis can be done: drawing the 3D model of a component, meshing a blade passage... For each intermediate task a set of computer programs is employed.

 𝑴𝑨𝑻𝑳𝑨𝑩®

MATLAB®_{is a numerical environment suitable for matrix computation and}

implementation of algorithms. Here it is used to implement the 3D design procedure of the volute and to generate plots from vectors of data.

 𝑺𝑶𝑳𝑰𝑫𝑾𝑶𝑹𝑲𝑺®

SOLIDWORKS®_{is the CAD software which was used to build the complete geometrical}

models of volute, nozzle ring and diffuser. It is flexible because it allows to specify relative constraints between parts of a sketch and adapt it when a modification is done, so that it is most suitable for the iterative phases of the design process.

Moreover files generated in SOLIDWORKS® can be easily transferred to ANSYS® package for a subsequent analysis (CFD, FEM...).

 𝑫𝑬𝑺𝑰𝑮𝑵𝑴𝑶𝑫𝑬𝑳𝑬𝑹®

DESIGNMODELER®_{is a CAD software within}_ANSYS®_{package. It mainly handles}

external geometry models, usually created for manufacturing purpose, and allows modifications (for example the suppression of non meaningful details) so that the model is ready for meshing and simulation. DESIGNMODELER® may also be used to draw geometries from scratch.

In this thesis the program imports geometry files from SOLIDWORKS® and edits them when needed (by rotating or translating components with respect to the frame of reference) so that the turbine model is assembled correctly after meshing.

 𝑩𝑳𝑨𝑫𝑬𝑮𝑬𝑵®

BLADEGEN®_{is an} _ANSYS®_{software specific for the design of rotative machinery}

blades. The designer is allowed to specify the evolution of some representative sections of the blade in a cylindrical frame of reference and the thickness distribution: the program automatically generates the CAD model of the machine and monitors several key parameters such as the cross-section area of the flow channel, the flow angle distribution...

The design of the impeller of the radial turbine was done with BLADEGEN®_.

 𝑻𝑼𝑹𝑩𝑶𝑮𝑹𝑰𝑫®

TURBOGRID®_{is an}_ANSYS®_{software, specific for rotating machinery, which creates}

high quality hexahedral meshes. The program imports the model of the impeller from BLADEGEN®_{and meshes the blade passage: when ATM default feature is enabled, the}

program chooses the optimal topology for a given blade geometry and allows to create a good quality mesh in a highly automated way and with minimal effort, with no need for control point adjustment.

 𝑴𝑬𝑺𝑯𝑰𝑵𝑮®

The tool ANSYS MESHING® imports the geometry of a component from DESIGNMODELER®_{and allows the creation of a mesh in a guided and automated way.}

(25)

14

mesh close to a wall to capture the boundary layer. Once fixed the setup parameters, mesh generation is program-controlled, thus the process is fast and robust, and subsequent modifications are directly implementable.

Volute, diffuser and nozzle ring are meshed with this software.  𝑪𝑭𝑿®

CFX®_{is a general-purpose commercial CFD software which is commonly used in}

problems involving rotating machinery (turbines, pumps, fans...). CFX® has a preprocessor in which the setup for the simulation is defined: this includes the definition of fluid properties, boundary conditions, turbulence model, target of accuracy... All components of the turbine, once meshed, are imported in CFX® preprocessor and assembled there to reach the final configuration (either vaneless or vaned).

The final solution is considered achieved when some monitoring parameters, typically the residuals of mass and momentum equations, have converged below a minimal threshold value. CFX®_{also includes a postprocessor which allows to analyse the solution}

(26)

15

4 – LIMITATIONS

 Uncertainties of CFD computation

The present work consists of a numerical performance evaluation of a radial turbine, as the internal flow is modelled with a CFD commercial software. However, “due to modelling, discretization, and computation errors, the results obtained from CFD simulations have a certain level of uncertainty. It is important to understand the sources of CFD simulation errors and their magnitudes to be able to assess the magnitude of the uncertainty in the results” [35].

Such uncertainty is mainly linked to the mathematical model describing the flow (accuracy of the turbulence model, model for Reynolds stress when solving RANS, wall functions...), to the method for the discretization of PDE’s, to the quality of the computational grid and computer round-off errors. Moreover the validity of the numerical solution is jeopardized wherever large regions of separated flow exist in the domain, which may be the case for certain turbine configurations, especially at off-design points. These sources of uncertainty constitute a limitation for the study because a certain set of data (for example a performance difference between vaneless and vaned turbine configurations at given working condition) may fall inside the uncertainty range, and since experimental results are not available and cannot be used to validate the numerical model, those data would lead to misleading interpretations.

 Simplifications in the computational domain

In the following chapters it will be seen that the mesh of a full radial turbine may be rather “heavy”, especially if low values of 𝑦+ are set as requirement. A simplifying

approach present in literature [22] and also adopted in this work is to mesh a single rotor blade passage (SBP) and assume symmetry of the flow around the rotational axis: in this way, however, the non-uniformities of the flow within different rotor blade-to-blade channels are neglected.

 Simplifications in the 3D geometrical model

The design of simplified geometrical models may lead to neglecting or misevaluating certain features of the flow pattern, therefore it constitutes a limitation to the validity of the results obtained in this study. This is particularly true for the rotor, where tip clearance, scalloping and other geometrical details associated with the onset of secondary flows are not accurately modelled at a first-attempt design.

Moreover the design of all components of the turbine is based on modelling assumptions which introduce further simplifications (details in the following chapter)

(27)

16

5 – DESIGN OF COMPONENTS

5.1 Design of the volute

In a radial inflow turbine the volute has the purpose to decrease the static pressure of the working fluid and to increase its speed (conversion to kinetic energy) in order to reach desired values of velocity and flow angle at rotor inlet. Moreover the volute must distribute the flow uniformly along the azimuth direction and perform the energy conversion as efficiently as possible, that is with a minimum loss in stagnation pressure.

With reference to Fig.9, the constraints on the volute design are the following:

 Radius at outlet (𝑅₂) and passage width (𝑏₂). These are set by the rotor geometry and define the volute discharge area (𝐴₂).

 Mach number (𝑀2) and absolute flow angle (𝛼2) at outlet. These requirements are

imposed by the designed performance of the rotor.

 Thermodynamic flow conditions at volute inlet (𝑃₀₁, 𝑇₀₁, γ). These parameters are set by the working point of the engine.

 Mach number (𝑀₁) at inlet, linked to the velocity of the exhaust gases.

The goal of the preliminary 3D design is to size the flow passage in terms of the variation, with azimuth angle (φ), of cross-section area and centroid radius. The solving strategy illustrated in the following section is a modified version of the procedure whose original development is due to Whitfield [31], and which is based on the assumptions of adiabatic incompressible flow and conservation of angular momentum. Notice that incompressibility is a rough assumption, as the flow undergoes a variation of Mach number inside the volute: however, according to the design parameters (Tab.2), the flow regime is low subsonic, moreover the resultant design turns out to be acceptable at first attempt, thus the assumption is ultimately justified by experience.

(28)

17

5.1.1 Theoretical procedure

Newton’s second law of motion applied to moment forces reads: “For a system of mass m, the vector sum of the moments of all external forces acting on the system about some arbitrary axis A–A fixed in space is equal to the time rate of change of angular momentum of the system about that axis” (Dixon, [6])

𝑀 = _𝑑𝑡𝑑 (𝑅𝐶𝜃) (4.1)

In a purely ideal case no external forces, so moments, are applied on a fluid particle moving from the volute inlet to the volute outlet. In practice viscous shear forces, even if weak (due to high temperature of the exhaust gas its viscosity is low), are not absent, and the angular momentum is not fully conserved. Eqn.(4.1), evaluated from inlet to outlet, then becomes

𝑅2𝐶𝜃2 = 𝑆𝑅1𝐶𝜃1 (4.2)

where S is the angular momentum ratio across the volute.

For an adiabatic flow eqn.(4.2) can be developed in terms of absolute Mach numbers and flow angles to give the volute radius ratio as

𝑅1 𝑅2 = 𝑀2 sin⁡𝛼2 𝑆𝑀1 sin⁡𝛼1 1 + 𝛾 − 1 /2 𝑀₁2 1 + 𝛾 − 1 /2 𝑀₂2 1 2 4.3

The area ratio is derived from the conservation of mass between sections 1 and 2 (Fig.9) 𝐴₁ 𝐴2 = 𝑃₀₂ 𝑃01 𝑀₂cos 𝛼₂ 𝑀1sin 𝛼1 1 + 𝛾 − 1 /2 𝑀₂2 1 + 𝛾 − 1 /2 𝑀₁2 −_{2 𝛾 −1} 𝛾+1 (4.4)

where the stagnation pressure at outlet 𝑃02 is a function of the target efficiency of the volute (in

ideal case 𝑃02 = 𝑃01).

It should be noticed that eqns.(4.3) and (4.4) depend on 𝛼₁, which, in the original procedure (Whitfield, [31]), is given as input parameter. However the inflow angle at inlet is unknown at this stage, because it is linked to the orientation of section 1 (take Fig.9 as reference) which in turn depends on the tangent to the centroid locus in section 1, not yet determined. Thus 𝛼₁ must be either entred as guess parameter or estimated.

Here 𝛼₁ is derived by equating the expressions of non-dimensionalised mass flow rate 𝜃 =_𝜌 𝑚

0𝑎0𝐴

between sections 1 and 2, expressed in terms of Mach numbers

(29)

18

Once the overall volute geometry is defined in terms of inlet-to-outlet radius and area ratio the next step is to extend the analysis to the derivation of geometric parameters as function of the azimuth angle. To this purpose the following hypotheses are introduced:

 Mach number at passage centroid increases linearly with θ → 𝑀_𝑦 = 𝑀₁ + _2𝜋Φ 𝑀₂− 𝑀₁

 Mass flow rate decreases linearly with θ → 𝑚 _𝑚𝑦 = 1 −_2𝜋Φ +𝑚 𝑅

𝑚 , where 𝑚 𝑅 is the flow

rate recirculating and mixing below the volute tongue (𝑚 𝑅 ≈ 5% at first estimate)

 Stagnation pressure decreases linearly with θ → 𝑃02 = 𝑃01− _2𝜋Φ 𝑃01 − 𝑃02

 The vortex exponent m, which takes into account the modification of the free vortex condition, varies with θ → 𝑚 = 𝑚₀− 𝑘𝛷𝑝_{, where}_𝑚

0 = 1 is the exponent at volute

inlet while k and p are experimental constants.

Figure 10: Left – velocity profile across the centre-line of the volute section. Right – variation of centroid radius at two subsequent azimuth positions (Whitfield, [31], adapted)

The swirling flow is subjected to the free vortex relation (conservation of momentum), corrected by the vortex exponent which models the presence of small tangential forces arising from the non ideal volute design

𝐶_𝜃𝑅𝑚 _{= 𝑐𝑜𝑛𝑠𝑡 4.6}

Eqn.(4.6) allows to express the variation of tangential velocity between any two angular locations X and Y separated by a small angle 𝛥𝛷 (see Fig.10 – left)

𝐶_𝜃𝑦 𝐶_𝜃𝑥 = 1 − 𝛥𝛷 2𝜋 𝐶1𝑅1sin 𝛼1 𝐶_𝑥𝑅_𝑥sin 𝛼_𝑥 1 − 𝑆 𝑆𝑥𝑦 𝑅𝑥 𝑅₂ 𝑚𝑥 _𝑅 2 𝑅_𝑦 𝑚𝑦 (4.7)

(30)

19

The flow angle at Y can now be derived from eqn.(4.7) in terms of the known Mach numbers and 𝛼_𝑥 (known as well): the centroid radius 𝑅_𝑦, unknown at this stage, is derived from geometrical considerations (Fig.10 – right).

Finally the conservation of mass is applied between section 1 and the generic section Y in order to find an expression for the area ratio

sin 𝛼𝑦 = 𝑆𝑥𝑦sin 𝛼𝑥 𝑅_𝑥 𝑅2 𝑚𝑥 _𝑅 2 𝑅𝑦 𝑚𝑦 _{1 + 𝛾 − 1 /2 𝑀} 𝑦2 1 + 𝛾 − 1 /2 𝑀𝑥2 1 2 (4.8) 𝑅_𝑦 = 𝑅_𝑥− 𝛥𝑅_𝑥𝑦 = 𝑅_𝑥 − 𝑅𝑦𝛥𝛷 tan 𝛼_𝑥 → 𝑅𝑦 = 𝑅𝑥 1 +_{tan 𝛼}𝛥𝛷 𝑥 (4.9) 𝐴₁ 𝐴_𝑦 = 𝑃0𝑦 𝑃₀₁ 𝑚 𝑚 _𝑦 𝑀𝑦sin 𝛼𝑦 𝑀₁sin 𝛼₁ 1 + 𝛾 − 1 /2 𝑀𝑦2 1 + 𝛾 − 1 /2 𝑀₁2 −_{2 𝛾−1} 𝛾 +1 (4.10)

5.1.2 Implementation of the theoretical procedure

The procedure described above was implemented in a MATLAB®_{code, with the input parameters}

coming from the meanline design.

In order to compare vaneless and vaned volutes at equal working conditions, both configurations are required, at design point, to deliver to the rotor a flow with the same average velocity vector. Thus the design specifications at volute inlet and outlet must be the same as well. However in the vaned case the nozzle ring behaves as a convergent duct, hence the total acceleration is distributed between the two components.

PARAMETER VALUE

Radius at nozzle inlet 𝑟_{𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡} = 52.3 [mm]

Radius at nozzle outlet 𝑟_{𝑛𝑟 −𝑜𝑢𝑡𝑙𝑒𝑡} = 41.8 → 41 [mm] (*)

Blade height 𝑕 = 13 [mm]

Table 1: meanline design parameters for nozzle ring

(*) The radius at outlet was slightly changed from the meanline value so to have complete accordance between the geometrical dimensions of the two volute configurations.

The acceleration that the flow undergoes inside the nozzle ring is derived from the conservation of mass between inlet and outlet sections. This estimate is approximated, as density and temperature of the flow are supposed not to change during the process.

𝑚 _{𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡} = 𝑚 _{𝑛𝑟 −𝑜𝑢𝑡𝑙𝑒𝑡} → 𝜌 𝑀𝑎 𝑢 2𝜋𝑟𝑕 𝐴 _{𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡} = 𝜌 𝑀𝑎 𝑢 2𝜋𝑟𝑕 𝐴 _{𝑛𝑟 −𝑜𝑢𝑡𝑙𝑒𝑡} (4.11)

(31)

20

𝑀_{𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡} = 𝑀_{𝑛𝑟 −𝑜𝑢𝑡𝑙𝑒𝑡} 𝑟𝑛𝑟 −𝑜𝑢𝑡𝑙𝑒𝑡

𝑟𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡 (4.12)

where 𝑟𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡 and 𝑟𝑛𝑟 −𝑜𝑢𝑡𝑙𝑒𝑡 are reported in Tab.1. The input parameters for volute design are

summarized in Tab.2. The solution is shown in Fig.11.

PARAMETER VANELESS VANED (no nozzle)

Mach at volute inlet 𝑀₁ = 0.21 𝑀₁ = 0.21

Mach at volute outlet 𝑀₂ = 0.56 𝑀₂ = 𝑀_{𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡} = 0.44 Flow angle at volute outlet 𝛼₂ = 68 [°] 𝛼₂ = 68 [°]

Radius at volute outlet 𝑅₂ = 41 [𝑚𝑚] 𝑅₂ = 𝑟_{𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡} = 52.3 [𝑚𝑚]

Outlet passage width 𝑕 = 13 [𝑚𝑚] 𝑕 = 13 [𝑚𝑚]

Flow properties at inlet

𝑃₀₁ = 2.56 [𝑏𝑎𝑟] 𝑇₀₁ = 846 [𝐾] 𝑚 ₁ = 0.3322 [𝑘𝑔 𝑠 ] 𝛾 = 𝛾_{𝑎𝑖𝑟 |𝑇}₀₁ = 1.34 𝜌1 = 1.037 [ 𝑘𝑔 𝑚3] 𝑃₀₁ = 2.56 [𝑏𝑎𝑟] 𝑇₀₁ = 846 [𝐾] 𝑚 ₁ = 0.3322 [𝑘𝑔 𝑠 ] 𝛾 = 𝛾_{𝑎𝑖𝑟 |𝑇}₀₁ = 1.34 𝜌1= 1.037 [ 𝑘𝑔 𝑚3] Additional constraints _𝐴𝑅1 = 82.14 [𝑚𝑚] 1 = 2733 [𝑚𝑚^2] 𝑅1 = 82.14 [𝑚𝑚] 𝐴1 = 2733 [𝑚𝑚^2]

Table 2: meanline design parameters for vaneless and vaned volutes

(32)

21

5.1.3 Implementation under additional constraints

The solution deriving from the application of the theoretical procedure may not be implementable due to the presence of additional constraints on the design. In particular, requirements on the maximum size of the turbine may limit radius or area at volute inlet. The designer may then look for a sub-optimal solution, i.e. a solution which is as close as possible to the ideal one yet fulfilling all the constraints.

For the present case the meanline design reports values of 𝑅1 and 𝐴1 which were interpreted as

additional geometrical constraints (Tab.2): however the theoretical solution does not match the values expected at volute inlet, as seen in Fig.11.

Qualitatively, a constraint on 𝑅1 is reflected into a modification of the radius distribution with

respect to the ideal case. Following a strategy suggested by Abidat [1], the modified centroid radius distribution is modeled with a third order Bezier polynomial, a parametric curve defined by 4 monitoring points, conveniently chosen, which allow high control over the shape of the curve and its derivative. Details about the Bezier approach will be given in Section 5.3 “Design of the Rotor”. The polynomial is numerically derived in order to obtain the flow angle distribution, which is geometrically represented by the local tangent to the centroid locus (see Fig.10- right). Given 𝛼 at the generic azimuth position and provided that all the other parameters remain constant, the distribution of cross section area is computed from eqn.(4.10).

The implementation of the solving procedure under constraints is done in a MATLAB®_{code and}

results are reported in Fig.12 and Fig.13.

(33)

22

Figure 13: distribution of centroid radius, cross section area and flow angle with azimuth location. Comparison between theoretical and implemented solution, vaned case

Observations

 Overall the implemented solution follows the theoretical one in the central range of azimuth positions (from 𝜑 ≈ 100° to 𝜑 ≈ 300° ). Major deviations occur close to the inlet, where the constraints are specified, but local modifications are done after a subsequent CFD analysis as part of the iterative phase of the design process.

 Fig.13 shows that a modified radius distribution may lead to a non negligible variation of the outflow angle with respect to the designed value. One solution is to use a higher degree Bezier polynomial, with more control points in order to model the slope of the curve at the edges. However this example highlights the difficulty to fulfill all constraints at once and the limits of inverse design methods.

The Bezier polynomial is transformed into a cartesian curve in space and imported in SOLIDWORKS®_{. The curve represents the locus of the centroids of all the cross sections.}

Circular cross section is chosen, as it is the simplest shape and leads to an efficient configuration according to literature (Shah, [20]). The area of each section is sized according to the solution of the 3D design procedure: slight modifications are made so that, for a correct matching of the components, the outlet passage width is kept constant at all azimuth positions and equal to the designed blade height of the impeller,

(34)

23

Figure 14: 3D geometrical model of the volute casing. Left: vaneless. Right: vaned (no nozzle)

5.2 Design of the nozzle ring

The vaned stator must guarantee the designed flow angle at rotor inlet, ideally without introducing additional losses.

The constraints on the design of the nozzle ring are below (their values are reported in Tab.1)  Radius at inlet (𝑟𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡) and radius at outlet (𝑟𝑛𝑟 −𝑜𝑢𝑡𝑙𝑒𝑡). These must match the outlet

radius of the volute and the inlet radius of the impeller respectively

 Blade height (𝑕). This requirement is set by the volute outlet passage width

Moreover, other parameters from the meanline design are specified: mass flow rate from the volute outlet section (𝑚 = 0.3322 [𝑘𝑔_𝑠 ]), fluid density (𝜌 = 1.037 [_𝑚𝑘𝑔₃]), speed of sound (𝑎 = 564.3 [𝑚_𝑠]), required flow angle at rotor inlet (𝛼 = 68 [°]).

Given the set of design parameters, a final configuration which fulfills all of them may not be implementable if chocking occurs inside the nozzle vanes. Thus the condition 𝑀 ≤ 1 must be verified for the design to be meaningful.

From Fig.15 𝛼 is expressed in terms of nozzle pitch and nozzle throat length as

𝛼 = cos−1 𝑜 𝑠 = cos−1 𝑚 𝜌𝑀𝑎 𝑕𝑍𝑣 2𝜋𝑟_{𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡} 𝑍𝑣 (4.13) where 𝑠 =2𝜋𝑟𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡_𝑍

𝑣 is derived from geometrical considerations while 𝑜 =

𝑚

𝜌𝑀𝑎 𝑕𝑍𝑣 expresses the

(35)

24

Figure 15: nozzle vane geometry definition (Rajoo & Martinez-Botas, [18])

Eqn.(4.13) allows to compute the Mach number. By substituting the design parameters it is found 𝑀 ≈ 0.36, thus chocking is avoided at design point (notice there is no guarantee that 𝑀 < 1 is also verified at off-design conditions)

Moreover eqn.(4.13) shows that the number of vanes has no influence on the outflow angle: according to Rajoo & Martinez-Botas 𝑍_𝑣 is the value which optimizes the pitch/chord ratio so to achieve “compromise between friction losses and good flow guidance” [18]. Meanline design specifies 𝑍_𝑣 = 14.

Figure 16: nozzle ring. Left: sketch in the frontal plane. Right: shape of the blade

(36)

25

inclination angle (Fig.16, left) is set arbitrarily: as initial guess value the designer chooses the outlet flow angle 𝛼, since the flow is expected to “follow the blade” in ideal design. This value is modified iteratively until no separation is detected.

3D CAD model of the nozzle ring is generated in SOLIDWORKS®_{and is shown in Fig.17.}

Figure 17: 3D geometrical model of the nozzle ring

5.3 Design of the rotor

The rotor is the main component of a radial turbine, being the only one which extracts work from the flow. Apart from the requirement of matching with the volute, which sets the dimensions of the diameter and the height of the blade at inlet, there are no constraints on the design of the rotor: the goal is to minimize all sources of losses so to achieve the maximum power output under designed flow conditions.

5.3.1 Preliminary design

As illustrated in the introductory chapter, nominal design condition is associated with a minimum in incidence losses. Therefore the rotor blade must be designed so that for all spanwise locations (from hub to shroud, see Fig.18) the flow enters radially (𝛽₂ = 0) and leaves the rotor axially (𝛼₃ = 0).

In literature [21] it is reported that the condition 𝛽₂ = 0 actually leads to flow recirculation at the suction surface of the blade, and the optimal inlet flow angle is identified within the range 𝛽2 = −10° and 𝛽2 = −40°. Given the flow parameters at volute outlet (𝑀2 = 0.56, 𝛼2 = 68°)

Filippo Valentini

Numerical Modelling of a Radial Inflow Turbine with and without

Nozzle Ring at Design and Off-Design Conditions