Modern Layout and Design Strategy for Axial Fans

Home , Stagnation point, Turbomachinery

Maria Teodora Pascu

Moderne Auslegungs- und Entwurfsstrategie für Axialventilatoren

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Modern Layout and Design Strategy for Axial Fans

Moderne Auslegungs- und Entwurfsstrategie für Axialventilatoren

Der Technischen Fakultät der Friedrich–Alexander–Universität Erlangen–Nürnberg

zur Erlangung des Grades

DOKTOR–INGENIEUR

vorgelegt von

Maria Teodora Pascu

Erlangen, 2009

Als Disseration genehmigt von der Technischen Fakultät der Universität Erlangen-Nürnberg

Tag der Einreichung: 11.12.2008 Tag der Promotion: 24.04.2009

Dekan: Prof. Dr. J. Huber Berichterstatter: Prof. Dr. Dr. h.c. F. Durst Prof. Dr. M. Wensing

Acknowledgements

This work received financial support from the Bavarian Science Foundation in the form of an individual grant, which is gratefully acknowledged.

I would like especially to thank my supervisor, Prof. Dr. Dr. h.c. F. Durst, who from the first day we met, during the Summer Academy Kuşadasi 2004, captured my attention for fluid mechanics research, and by supporting my diploma thesis at the Institute of Fluid

Mechanics LSTM Erlangen, brought me in close contact with the topic and stirred my interest in further academic education. He has enabled me to work in the field of turbomachines and constantly supported the present work.

I would also like to thank Prof. Dr. M. Wensing for his kind acceptance to review the present work.

Furthermore, I would like to thank Prof. Dr. A. Delgado, head of the Institute of Fluid

Mechanics LSTM Erlangen, who supported and encouraged the present work, always taking a genuine interest in the outcome of the investigations.

My deepest acknowledgements go to Dr. J. Jovanović, head of the turbulence research at

LSTM, for his sustained moral support, to whom I dedicate the successful turnout of the experimental investigations included in the thesis.

I would also like to express my gratitude to Alu Automotive GmbH for the kind support offered to the present work, and especially to the company manager, Mr. Felix Hellmuth.

I would especially like to thank Dr. Ph. Epple, head of the Turbomachinery Optimization research group at LSTM Erlangen, for his constant supervision and strong commitment to the present work.

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My warm acknowledgement also goes to Dipl.-Ing. M. Miclea-Bleiziffer for his support and numerous brainstorming sessions that led to the present layout of the work.

I warmly acknowledge the technical department of the institute for their continual collaboration, and especially to Mr. F. Kaschak and Mr. C. Bakeberg.

Finally, I would like to thank all my colleagues at LSTM for the wonderful and friendly working atmosphere, to the workshop and administration, who all contributed to the achievements in the present work.

Şi cel mai important, doresc să îi mulţumesc familiei mele, pentru sprijinul necondiţionat si încrederea deplină pe care mi le-au acordat in permanenţă, si cărora le datorez în

întregime tot ceea ce sunt azi.

Erlangen, December 2008 Maria Teodora Pascu

List of Contents

15H 6H Introduction and aim of work 117H

1.18H General9H introduction 1110H

1.21H Classification12H of turbomachines 1113H

1.314H Aim15H of work 1116H

217H 18H Basic equations of fluid mechanics as applied in turbomachines 1119H

2.120H 21H Navier–Stokes equations in rotating systems 112H

2.223H Energy24H transfer in turbomachines 1125H

326H 27H Survey of the available design methods for axial impellers 1128H

3.129H Basic30H features of turbomachinery design 1131H

3.232H Two3H – dimensional cascade theory 1134H

3.2.135H Aerodynamics36H forces and governing equations 1137H

3.338H Design39H methods based on the airfoil theory 1140H

3.3.141H Airfoil42H families. Mean-line and thickness distribution 1143H

3.3.24H Design45H parameters 1146H

3.3.347H Cascade48H losses. Diffusion factor 1149H

3.450H Three-dimensional51H character of the flow in axial turbomachines 1152H

453H 54H Proposed design strategy for axial fans 115H

4.156H Mean-line57H calculation 1158H

4.259H Outlet60H conditions 1161H

4.362H Meridional63H flow analysis for axial fans 1164H

4.465H The6H indirect design problem 1167H

4.568H Parameterization69H of the total pressure in the span-wise direction for an

axial fan blade 1170H

4.671H Blade72H shape computation 1173H

4.774H Further75H design assumptions based on profile analysis 1176H

4.7.17H Static-to-static78H cascade efficiency 1179H

4.7.280H Total-to-total81H cascade efficiency 1182H

4.7.383H Profiling84H the camber line 1185H

4.886H Design87H Solver (DS) 118H

4.989H DS90H output 1191H

592H 93H Numerical flow analysis 1194H

5.195H Mathematical96H model 1197H

5.298H Mesh9H generation 1110H

5.310H Numerical102H models and boundary conditions 11103H

5.4104H Appropriate105H performance indicators 11106H

5.5107H Optimum108H span-wise pressure distribution 11109H

5.610H Profile1H analysis 1112H

5.6.113H Flow14H domain around the profiles 1115H

5.6.216H Mesh17H generation 1118H

5.6.319H Numerical120H results 1112H

612H 123H Experimental validation of the proposed design strategy 11124H

6.1125H Investigated126H impellers 11127H

6.2128H Experimental129H facility 11130H

6.313H Measured132H parameters 1113H

6.4134H Measuring135H equipment 11136H

6.5137H Experimental138H results 11139H

6.6140H Validation14H of the results 11142H

7143H 14H Integrated ideal efficiency for axial fans 11145H

7.1146H The147H Cordier diagram 11148H

7.2149H Ideal150H efficiency for axial fans 1115H

8152H 153H Conclusions and outlook 11154H

Index of Symbols

A [m2] area

M [N m] torque

P [W] power

Q [m3/s] flow rate

R [–] degree of reaction b [m/s2] acceleration c [m/s] absolute velocity l [m] blade chord n [rpm] rotational speed r [m] radius u [m/s] peripheral velocity w [m/s] relative flow velocity wm [m/s] meridional component of the relative velocity wu [m/s] tangential (peripheral) component of the relative velocity

P [Pa] pressure difference

Greek symbols

 [-] diameter coefficient

 [o] gliding angle

 [o] blade angle

 [–] efficiency

 [–] flow coefficient (dimensionless flow)

 [–] pressure ratio between the fan outlet and the inlet static pressures

 [kg/m3] density

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 [o] camber turning angle

  [–] temperature ratio between the fan outlet and inlet temperatures

 [–] speed coefficient

 [rad/s] angular velocity

 [–] pressure coefficient (loading)

 [–] circulation

 [–] vorticity

Subscripts

1 blade inlet

2 blade outlet h hub t tip, total s static d dynamic

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Abstract

The present work addresses the application of computational fluid dynamics in the research and development process of axial fans of the kind used in numerous fields of engineering and in daily life. In this sense, a modern layout and design strategy for axial impellers are proposed, as basis for optimization in this engineering field. Essentially, the strategy is a combined inverse-direct method, based on a design solver which computes the optimum blade profile according to the flow conditions in the fan, and does not make use of any predefined profiles. When applied in a rigorous manner, the proposed design strategy delivers high- performance design solutions for axial fans, and this is thoroughly confirmed by both numerical and experimental results. The design calculation scheme starts with the one – dimensional hypothesis of the mean streamline, based on which the blade inlet (at all sections) and outlet (at the hub) conditions are determined. Then, by computing the blade as a succession of several cascades, the two-dimensional nature of the flow is considered. Finally, the blade profile is fully resolved by implementing a three-dimensional (meridional) analysis into the design process. By assuming an arbitrary vortex flow, the optimum pressure distribution in the span-wise direction is determined and the parameterization of the outlet blade angle is achieved, as a function of one of the most important constructive characteristics of an axial fan, i.e. the hub ratio. The advantages of employing the suggested design strategy as an optimization tool are first emphasized by fully converged CFD solutions, which show the substantial improvements in efficiency achieved by the new designs over the reference model, i.e. an engine cooling fan currently used in the automotive industry. Moreover, the employment of the non-free vortex assumption at the design stage is proved to be beneficial for the fan performance, since the design obtained accordingly performs efficiently through a wider flow range. Even though modern CFD nowadays achieves excellent flow predictions, phenomena with impact on the performance are neglected, hence the motivation for the experimental confirmation of the proposed designs. The performance curves of the non-free vortex

flow design against the reference impeller show an absolute increase in the measured (total-to-static) efficiency of 10% for the proposed design. Finally, the present work proposes an analytical computation of the integrated ideal efficiency for axial fans, a concept which was derived as a response to the incapacity of the classical Cordier diagram to predict the actual performance of axial impellers operating in the low-pressure regimes, due to proven inconsistencies for this type of turbomachine with regard to the definitions of the parameters employed. It is shown that the proposed design strategy delivers an axial fan whose performance comes very close to that of the ideal machine.

Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit der Anwendung von numerischen Strömungssimulationen in der Forschungs- und Entwicklungsarbeit axialer Gebläse, die im Ingenieursbereich und täglichen Alltag vielfältig zu finden sind. In diesem Sinne wird für diesen Bereich des Ingenieurwesens eine moderne Auslegungs- und Designstrategie für axiale Gebläse vorgeschlagen. Die entwickelte Auslegungsstrategie ist hauptsächlich eine kombinierte inverse–direkte Methode, die sich auf einen Design-Löser stützt, der in Abhängigkeit der Strömungsbedingungen im Laufrad, ein optimales Schaufelprofil berechnet, ohne andere vorgegebene Profile zu nutzen. Die vorgeschlagene Auslegungsstrategie liefert, wenn sie richtig angewendet wird, Hochleistungslösungen für das Design axialer Gebläse, die von numerischen und experimentellen Ergebnissen bestätigt wurden. Das Auslegungsschema beginnt mit der ein-dimensionalen Hypothese der Hauptstromlinie, auf der die Bedingungen am Schaufeleintritt (auf allen Querschnitten) und Schaufelaustritt (an der Nabe) bestimmt werden. Die Zwei-dimensionalität der Strömung wird mithilfe der Berechnung zwei-dimensionaler Kaskaden berücksichtigt. Das Schaufelprofil wird im Designverlauf über die Einführung einer drei-dimensionalen (Merdidionalen) Analyse vollständig berechnet. Nach der Einstellung einer freien Wirbelströmung werden in der Spannweitenrichtung die optimale Druckverteilung und die Parametrierung des Austrittwinkels als Funktion der wichtigsten Eigenschaft eines Axialgebläses - dem Nabe-Gehäuse-Verhältnis - berechnet. Der Vorteil der vorgeschlagenen Auslegungsstrategie als Optimierungswerkzeug wird erst durch vollständig konvergierte CFD-Ergebnisse unterstützt. Die neuen Gebläse, zum Beispiel Lüftungsgebläse der Automobilindustrie, zeigen im Vergleich zum Referenzmodell erhebliche Verbesserungen des Wirkungsgrads. Außerdem erweist sich die Anwendung der „Nicht-freien Wirbel“ Annahmen vorteilhaft für das Leistungsverhalten des Gebläses, da das neue Design über einem weiten Bereich der Durchfluss-Kennlinie effizienter arbeitet. Obwohl moderne CFD heutzutage ausgezeichnete Strömungsvorhersagen erreicht und somit erlauben, aus einer Serie von Auslegungen die beste auszuwählen, wurde die beste

Auslegung am Prüfstand verifiziert. Damit konnte das Auslegungsverfahren experimentell verifiziert werden. Nach der Auslegung und Nachrechnung mit CFD, wurde für das beste Laufrad ein Prototyp gebaut und am Prüfstand vermessen und somit die Auslegung verifiziert. Ein Vergleich der Kennlinien der mithilfe der „Nicht-freien Wirbel“-Auslegungsmethode ausgelegten Gebläse mit denen von Referenzgebläsen zeigen in der gemessenen Effizienz (total -zu- statisch) eine 10 %-ige absolute Erhöhung. Die vorliegende Arbeit schlägt abschließend eine analytische Berechnung der integrierten-idealen Effizienz von Axialgebläsen vor. Dieses Konzept wurde als eine Ergänzung zum klassischen Cordier-Diagramm abgeleitet, da dieses wegen schon bekannter Widersprüche in den eingesetzten Parametern für diesen Typ von Turbomaschinen nicht in der Lage ist, die Leistungen von axialen Gebläsen vorherzusagen. Es wird gezeigt, dass die vorgeschlagene Auslegungsstrategie axiale Gebläse liefert, deren Wirkungsgrad einer idealen Maschine sehr nahe kommen.

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1. Introduction0B and aim of work

1 Introduction0B and aim of work

1.1 General introduction Fluid mechanics is a subject that has developed over centuries to become an independent field of science, based on laws that are generally accepted, such as conservation of mass, momentum, and energy, and the basic laws that describe the thermodynamic properties of the fluids. These laws are best and most condensed written down in terms of tensor notation:  Conservation of mass (continuity equation) for Newtonian fluids   U i 0 (1.1) txi  Momentum equation

UUjjP  ij  Ugij (1.2) txxxiji  Molecular transport of momentum

U UU2 j ik   (1.3) ij ij xijxx3  k  Thermal energy equation UU eeqi j j UPiij (1.4) txiiji  xx   x eCT v (1.5)  Molecular transport of heat T qi  (1.6) xi

1. Introduction0B and aim of work

 Equations of state for gas flows PRT  (1.7) The above equations describe the ideal gas as a medium, and the different types of flows are given only by different initial and boundary conditions. These equations are therefore used nowadays to solve flow problems which occur in different areas of engineering,

science and medicine, and also in nature and daily life, Durst [31]:15H  Heat exchanger, cooling and drying technology  Reaction technology and reactor layout  Aerodynamics of vehicles and airplanes  Semiconductor-crystal production, thin-film technology, vapor-phase deposition processes  Layout and optimization of pumps, valves and nozzles  Development of measuring instruments and production of sensors  Ventilation, heating and air-conditioning techniques, layout and tests, laboratory vents  Problem solutions for roof ventilation and flows around buildings  Production of electronic components, micro-systems analysis engineering  Layout of stirrer systems, propellers, and turbines  Sub-domains of biomedicine and medical engineering  Layout of combustion units Among the numerous fields, the above equations are used to solve fluid flow problems in the field of flow machinery in general and turbomachines, in particular. It is the latter field to which the present thesis attempts to make a contribution. The flow in turbomachines is turbulent, and there is a crucial difference when modeling the physical phenomena between laminar and turbulent flow. For the latter, the appearance of turbulence eddies occurs on a wide range of length scales and typical flow domains in this case would require computing meshes of 109–1012 grid points, Tu et al.

[103].156H With the present-day computing power, the computing requirements for a direct numerical solution (DNS) of the time-dependent Navier–Stokes equations of the fully

1. Introduction0B and aim of work turbulent flows at high Reynolds numbers are truly phenomenal. Engineers, however, require computational procedures that can supply adequate information associated with the turbulent processes, but wish to avoid the need to predict all the effects caused by each individual eddy in the flow. This category of CFD users is almost always satisfied with information about the time-averaged properties of the flow. Therefore, in order to study the dynamics of turbulence in common engineering applications, Reynolds turbulence decomposition, for the instantaneous velocity and pressure, and time averaging of the Navier–Stokes equations are required, yielding in a system of equations

for the mean flow, Jovanović [55]:157H

UUP  Uuugjj      (1.8) iijj xxxxijii In Eq.(1.8) the Reynolds equations for turbulent flows are written (for the incompressible case), including the “Reynolds stresses” uuij, which are caused by the turbulent motion. In this way, new unknowns are introduced, which are in fact correlations of velocity fluctuations. This yields a system of differential equations which is unclosed, and in order to close this system, the application of turbulence models is required. Here, the standard

two-equation k- model should be mentioned, Launder and Spalding [64],158H which closes the above system of equations with two additional transport equations: one for the turbulent kinetic energy k, and the other for the rate of dissipation of the turbulent energy  . This model is probably the most widely used and validated turbulence model, and its performance has been assessed against a considerable number of practical flows. However, despite the many successful applications in handling industrial problems, the standard k- model demonstrates only moderate agreement when predicting unconfined flows, hence numerous turbulence models have been developed recently in order to obtain accurate solutions of the more complex flow situations, such as the flow in turbomachines. Solving such complex flow behaviors is addressed by most computational fluid dynamics (CFD) commercial codes available today, and due to the proven robustness of the flow solver for turbomachines, ANSYS CFX was intensively employed in the present work, and excellent convergence, in terms of both computational time and resources, was obtained

by applying the Shear Stress Turbulence (SST) model, Menter [77].159H A detailed

1. Introduction0B and aim of work explanation of how this turbulence model works and why it is particularly applicable in the case of turbomachines will be given in Chapter 5 of the thesis.

1.2 Classification of turbomachines Essentially, the turbomachine is an energy conversion device, converting mechanical energy to thermal/pressure energy or vice versa. The conversion is done through the dynamic interaction between a continuously flowing fluid and a rotating machine component, and both momentum and energy transfer are involved. In turbomachines, the fluid flows freely between the inlet and outlet of the machine, without any intermittency. All turbomachines have a freely and continuously rotation part known as a runner, impeller or rotor, which allows uninterrupted flow through it. Therefore, the energy transfer between the rotor and the fluid is continuous, as a result of

the rate of change in angular momentum, Aksel [3].160H

Much has been written on classifying turbomachinery, Wright [112],16H and without repeating the entire classification procedure, available in most introductory chapters of books addressing this field, the author feels that a short summary of the different types will be helpful for a clear statement of the aim of the present work within the large class of flow devices to which the terminology refers. In most of the available literature, a major subdivision can be achieved based on the power criterion, identifying whether power is added (power absorbing) or extracted from the fluid (power producing). In this respect, pumps can be assigned to the category of the power-absorbing turbomachines, and they include liquid pumps, fans, blowers and compressors. Turbines are power- producing machines and they include windmills, water wheels, modern hydroelectric turbines, the exhaust side of automotive engine turbochargers, and the power extraction end of an aviation gas turbine engine. They also operate with various types of fluids, including gases, liquids, and mixtures of the two. Another possibility to classify the turbomachines is from the perspective of the fluid

medium handled, either compressible or incompressible, Peng [84].162H

The above subdivisions can be summarized as indicated in Figure163H 1-1:

1. Introduction0B and aim of work

Figure 1-1 Classification of turbomachines, adapted from Aksel [3]164H According to the nature of the flow path through the passages of the rotor, the device is termed an axial-flow turbomachine when the path of the through-flow is wholly or mainly parallel to the axis of rotation, radial when the path is in a plane perpendicular to the rotation axis, and mixed-flow turbomachines, when at the rotor outlet both radial and

axial velocity components are present in significant amounts, Dixon [29].165H With regard to the first criterion cited, i.e. that of whether power is being extracted from or added to the working fluid, the present work addresses the category of power- absorbing turbomachinery, namely fans. Since, as will be shown in the later sections, the flow in the investigated fans, i.e. air, is characterized by Mach numbers below the compressibility limit, a further placement in the general classification can be made, according to the working fluid, i.e. fans operating with incompressible flow. When considering the nature of the flow path with respect to the axis of rotation, the present work addresses the axial flow machines, and thus, axial fans. As a class, axial flow fans include high-capacity, low-head (pressure), single-stage machines. In small sizes, motor-driven axial fans are sometimes built in two stages owing

to speed limitations but are classed essentially as low head machines, Thwaites [102].16H

1. Introduction0B and aim of work

1.3 Aim of work The advent of the gas turbine engine during the Second World War demanded rapid developments in aerodynamic design and analysis techniques linked to wind tunnel and model testing, and in response, the field of “Internal Aerodynamics” was born and has expanded with remarkable speed and complexity over the entire class of turbomachines,

Lewis [68],167H including axial fans. However, in the area of axial fans, a lack of test and design data in the early stages of their development is responsible for the attempt of most

designers in this field to make extensive use of airfoil profiles, Stepanoff [100].168H The main scope of the design process of axial fans, employing either airfoil theory or other more direct design methods, is to deliver high-efficiency blades. The same area is addressed in the present work, but with strong emphasis on developing a new design strategy for axial fans, which does not make use of any predefined profiles, such as airfoils, but instead computes high-efficiency profiles, according to the operational requirements of the investigated impellers. Of course, designing an efficient profile implies that the required shape has to be aerodynamically efficient and therefore, the classical design considerations based on the airfoil theory will not be neglected. However, the method presented is not limited only to such considerations, and even though the airfoil theory method of axial impeller design is invariably associated with the free-vortex energy distribution along the radius, variations from this well-known design assumption are investigated and found more to be appropriate for fan design purposes. Hence the proposed design method, addressing the field of low-pressure axial fans, suggests an innovative blend of one-, two-, and three-dimensional flow considerations, with the aim of delivering the best performing blade profiles, and thus fan models approaching the ideal flow machine. The concept of the ideal machine is quantified differently in the literature, and it seems that there is really no standardized method to determine what the maximum achievable performance of a specified class of flow machines, might be. Often, most designers refer

to the Cordier diagram in this sense, as presented in Figure169H 1-2. Essentially, this diagram delivers, for an optimum pair of rotor dimensions and operating conditions (quantified by the so-called “diameter number” , and the “speed number” ,

respectively), the highest efficiency for the investigated impeller, Eck [33].170H  and  are

1. Introduction0B and aim of work derived based on the flow and pressure (head) coefficients, i.e.  and  , respectively.  and  are essential for the dimensionless analysis in the field of turbomachinery, and one can interpret them as the equivalent of the Nusselt number (Nu) and Euler number (Eu) from the Navier–Stokes equations. Based on a given pair ,  , the flow type of the investigated rotor is determined on the Cordier diagram: from radial (higher  and smaller ), to diagonal and finally, to axial (smaller  and higher ).

Figure 1-2 Original Cordier diagram, adapted from Cordier [21]17H As stated in the above paragraphs, the present work addresses axial flow fans operating in low-pressure ranges. When identifying the points corresponding to this type of impellers on the diagram, it seems that they are mostly concentrated in the upper half of the chart,

1. Introduction0B and aim of work

i.e.  opt  0.3, and when associating a trend-line to cover most of these points, a

“probable” curve of efficiency for low-pressure fans is obtained, as marked in Figure172H 1-2. Hence any axial fan, characterized by optimum dimensions and operating conditions, should deliver its highest efficiency if placed on this curve. However, this concept of ideal/maximum efficiency, as quantified by the Cordier diagram, is to some extent inconsistent from the axial impeller point of view. As will be presented in later sections, axial turbomachines are characterized by integral properties, and all the parameters influencing the efficiency of the impeller ( and  ) need to be integrated values of the local ones. Hence the ideal efficiency of axial impellers should represent the result of the integration of this local efficiency over the entire flow area. This matter is referred to in the present work and an analytical, integrated expression of the ideal efficiency of axial fans is proposed.

2. Basic1B equations of fluid mechanics as applied in turbomachines

2 Basic1B equations of fluid mechanics as applied in turbomachines

In the previous chapter, a general statement of the essential laws which govern the flow in turbomachines was made and it was emphasized that full solutions of these equations are obtained based on the correct treatment of the turbulent Reynolds stresses. However, before even considering the turbulent aspect of the flow in such machines, a thorough understanding of these basic equations is required, as they are central for the Computational Fluid Dynamics (CFD) technique. CFD is fundamentally based on the governing equations of fluid dynamics, which essentially represent statements of the conservation laws of physics. The purpose of the present chapter is to introduce the derivation of these fundamental equations and their employment for the CFD analysis of turbomachines, where the following physical laws are adopted:  Mass is conserved for the fluid  Newton’s second law, the rate of change of momentum equals the sum of all forces acting on the fluid  First law of thermodynamics, the rate of change of energy equals the sum of rate of heat addition to the fluid and the rate of work done on the fluid In this sense, the derivation of the continuity, momentum, and energy equations in rotating reference systems will be addressed, as they are applicable to turbomachines, as

in the case of the axial fan depicted in Figure173H 2-1, where the rotation is about the x3-axis.

2. Basic1B equations of fluid mechanics as applied in turbomachines

Figure 2-1 Axial flow fan within the cartesian coordinate system

2.1 Navier–Stokes equations in rotating systems Let us consider, for the upcoming derivations, a control volume defined around the fluid

element of known mass m, as shown in Figure174H 2-2:

Figure 2-2 Control volume around a fluid element The flow velocity (denoted U in the previous section) will be referred to as c, as it is the common notation in the theory of turbomachines. The rate of change in the mass is given by: dm    cdA (2.1) dt A The continuity equation states that there is no change in mass with respect to time, hence

2. Basic1B equations of fluid mechanics as applied in turbomachines

dm  0 (2.2) dt

In the case of the an incompressible fluid, i.e. the fluid density is constant, (2.2)175H becomes   cdA  0 (2.3) A

In tensorial form (employing Einstein’s summation), Eq. (2.3)176H can be written as c i  0 ,i 1, 2, 3 (2.4) xi

Equation (2.4)17H expresses the continuity equation for an incompressible, steady flow. The momentum equation (Newton’s second law) states that the rate of change in the momentum equals the sum of all forces acting on the fluid element:  dM d    ccdA  F (2.5) dt dt A A

In (2.5),178H the sum of all forces acting on the fluid element includes: pressure forces, friction forces and gravity forces: d    ccdAF  F F (2.6)   pfrg dt A Written in tensorial form, for the cartesian coordinate system, the momentum equation becomes

ccj jijGP11  ci (2.7) txxij  x j  x i    gravity pressure friction force force force

On multiplying by the density, then Eq. (2.7)179H becomes

ccj jijGP  ci (2.8) txijji  xxx

In Eq. (2.8),180H the following terms can be identified: G g j  x j where G is the gravitational potential and    

2. Basic1B equations of fluid mechanics as applied in turbomachines

where  is the kinematic viscosity and  is the dynamic viscosity.

Equation (2.8)18H becomes

ccjjP  ij  cgij (2.9) txiji  xx

For the fluid mechanically ideal case of inviscid flow, i.e. ij  0 , the momentum equation

in (2.9)182H becomes

ccjjP  cgij (Euler equations) (2.10) txij  x

The continuity equation in (2.4)183H and momentum balance equation, for the viscous flow

case, in (2.9),184H form a system of four equations with ten unknowns: the flow velocityccccj 123,, , the viscous term  ij 11,,,,, 12 13 22 23 33  and the pressure P. In order to close this system, additional equations have to be solved.

According to Durst [31],185H the viscous term due to the momentum transport  ij is a c function of j and is described by the following relation: xi

c cc2 j ik   (2.11) ij ij xijxx3  k Considering   const and   const , then

22cc c iii (2.12) x j xxxxxiijji    The continuity equation for the incompressible flow states that c i  0 (2.13) xi

Equation (2.12)186H becomes 22cc ii0 (2.14) xxji  xx i j

On inserting (2.14)187H into (2.11),18H then the viscous term in the momentum equation can be written as

2. Basic1B equations of fluid mechanics as applied in turbomachines

2  ij  ci  2 (2.15) xiix

Equation (2.9)189H becomes  2 ccjjP  ci  cgij2  txiji  xx   (2.16) c  i  0  xi

Equation (2.16),190H coupled with the continuity equation (2.13),19H form the Navier–Stokes system of equations for incompressible flow, which is a system of four equations with four unknowns, and hence a closed system. This system, however, in order to describe accurately the flow in a rotating system, such as the rotor of a turbomachine, needs to be solved with respect to a rotating (relative) system. Let us consider a point P and its position with respect to two systems: the absolute

reference system (AS) and the relative reference system (RS), as depicted in Figure192H 2-3.  Its origin is at point OR, it rotates with angular velocity  and its translation from the AS  is given by the vector a .

Figure 2-3 Absolute and relative reference systems at the same time step t. Adapted from Fister [40]193H  At the same time step t, the position of the point P is given by the vector q with respect  to AS and by r with respect to RS. The rate of change in the position of P with respect to AS represents the absolute velocity and is given by   dq c  OA (2.17) dt The rate of change in the position of P with respect to RS is the relative velocity:

2. Basic1B equations of fluid mechanics as applied in turbomachines

  dr w  OR (2.18) dt

The rotation of RS with respect to AS creates the circumferential velocity, Fister [40],194H which is defined as  ur (2.19)

On writing the Navier–Stokes system in (2.16)195H with respect to RS, then two additional forces in the momentum balance appear: one due to the centrifugal force, and the other due to the Coriolis force:   ww  P     wg r2 w (2.20) tr  rr   centrifugal Coriolis force force

In tensorial form, (2.20)196H can be written for all three directions of the cartesian system, i.e.  as indicated in Figure197H 2-1, whereas for the RS, the angular velocity vector  indicates the

positive x3-direction, Epple [37]:198H

wwjjP  ij 2  wgxxwijiijijkijki2  (2.21) txxxiji  The system is then completed with the continuity equation: w i  0 (2.22) xi

In (2.21)19H and (2.22),20H the Navier–Stokes equations with respect to a relative system are written for the case of the incompressible flow. These are the equations which describe the rotating frames of reference and, in the case of the turbulent flow in turbomachines, Reynolds decomposition is applied to this system. The energy equation for the same relative reference system is obtained by multiplying

(2.21)201H with wj under the following assumptions: stationary flow, neglecting the molecular moment transport  ij and considering a constant angular velocity:

Dpuw2 2  j G 0  Dt 22

 w2 pu2  wGj  0 (2.23) i  xi 22

2. Basic1B equations of fluid mechanics as applied in turbomachines

On integrating (2.23)20H according to Figure203H 2-2, then the following expression for the energy equation for a rotating frame is obtained:

 w2 pu2  wGwdAj   x iii22 A i  which finally leads to 1 Wccuuww22 22 2 2 (2.24) 2 12 12 2 1

Equation (2.24)204H is the energy equation for a relative system and in the theory of turbomachines this is referred to as the Euler equation for pumps and turbines (the difference is made by the sign convention: positive work for turbines, negative for pumps). This equation relates to the energy transfer in such flow machines and, since it is essential for the analysis of turbomachines and central to the design process, it will be treated separately in the following section.

2.2 Energy transfer in turbomachines

1 22 The first term in (2.24),205H cc12 , represents the energy transfer due to the change in 2 the absolute kinetic energy of the fluid during its passage between the entrance and exit 1 sections. In a pump or compressor, the discharge kinetic energy from the rotor, c2 , may 2 2 be considerable. Normally, it is the static head or pressure that is required as useful energy. Usually, the kinetic energy at the rotor outlet is converted into a static pressure head by passing the fluid through a diffuser. In a turbine, the change in absolute kinetic energy represents the power transmitted from the fluid to the rotor due to an impulse effect. As this absolute kinetic energy change can be used to accomplish a rise in pressure, it can be called a “virtual pressure rise” or a “pressure rise” which it is possible to attain. The amount of the pressure rise in the diffuser depends, of course, on the efficiency of the diffuser. Since this pressure rise comes from the diffuser, which is 1 external to the rotor, cc22 is sometimes called the “external effect”. 2 12

The other two terms in Eq. (2.24)206H are factors that produce a pressure rise within the rotor itself, and hence they are called “internal diffusion”. The centrifugal effect, given by the

2. Basic1B equations of fluid mechanics as applied in turbomachines

1 term uu22 , is due to the centrifugal forces that are developed as the fluid particles 2 12 move outwards towards the rim of the machine, and this effect is produced if the fluid changes the radius from the entrance to the exit of the impeller. This is not the case with the axial-flow machines, where the flow particles enter and leave the rotor at the same radius, and hence uu12 .

1 22 The last term, ww21 , represents the energy transfer due to the change in the relative 2 kinetic energy of the fluid. If ww21 , the passage acts like a nozzle, and if instead ww21 , then it acts like a diffuser.

Figure 2-4 Meridional flow through a turbomachine and flow through an elementary stream tube: a) meridional flow through a pump or a fan rotor; b) stream tube along the surface of revolution mapped out by the meridional streamline  0 . Adapted from Lewis [68]207H The Euler pump and turbine equation as derived previously is a one-dimensional equation in the sense that it is applicable to a unit mass of fluid flowing along the line mapped out

by the elementary stream tube illustrated in Figure208H 2-4b. The circumferential projection of such infinitely thin stream tubes on to the (x,r) plane leads to the definition of a family

of so-called meridional streamlines illustrated in Figure209H 2-4a, of which the hub and casing form the boundary streamlines. It is clear that one Euler equation must be derived for each meridional streamline during the design phase of a turbomachine and these equations will lead to a precise specification of the swirl velocity change from cu1 to cu2 required for a specified change in the work of the rotor, W. The Euler equation is thus central to the design process.

3. Survey2B of the available design methods for axial impellers

3 Survey2B of the available design methods for axial impellers

In the field of turbomachinery, multiple design methods and concepts can be identified, and often important design choices are made based on the designer’s experience. Even though such design choices appear to be numerous and developed for the particular class of flow machines under consideration, the design process itself, no matter what its content, can be laid out in simple steps that need to be followed if the resulting design is to be a successful one.

3.1 Basic features of turbomachinery design Much of the established literature on designing flow machines makes valuable references

to the existent design methods, Balje [9],210H and accordingly three basic steps in the design analysis are important to designers and analysts of turbomachinery:  The one-dimensional design or the so-called “mean-line design” method (or critical path line) usually provides the first step in the design procedure  Two-dimensional and three-dimensional methods, including blade and vane definition, supported by methods based on the potential flow analysis  Advanced viscous 3D calculations on the basis of the Reynolds equations, including also complex structural calculations The role of each of these steps needs to be clearly understood for a successful design and the value of each must not be underrated compared with the others. The first step includes the preliminary designs, and also some preliminary design optimization studies, using a variety of one-dimensional design and analysis tools. These tools attempt to model the basic flow physics at distinct stations through the

3. Survey2B of the available design methods for axial impellers turbomachine where important events occur or where important thermodynamic and fluid

dynamic book-keeping must be carried out, Japikse and Baines [53].21H Usually these analyses utilize the basic conservation laws (mass, momentum and energy) applied either along a mean streamline through the machine or along a critical streamline. Even though the description of this initial step sounds essentially simple, in reality, at this level, most of the important design decisions must be made, and it is normally the case that such decisions are entirely based on empirical data and the designer’s experience. It is difficult, however, to predict, at this level, whether one or the other design choice is better suited for the application of the machine, and therefore this process involves much trial and error before a final decision is made. The design solutions for a basic concept are normally derived for the design point (see

Figure21H 3-1), and several trial geometries are usually generated. After the basic one-dimensional configuration has been laid out (i.e. the typical passage height and mean-line velocity triangles, flow rates, speeds, and power levels are calculated through the one-dimensional analysis), the designer can proceed to the specification of the two-dimensional problem, i.e. to specify the actual blade shapes. Circular arcs, multiple circular arcs, arcs plus straight lines, and polynomials are the most commonly employed design choices. Traditionally, the first and often most appropriate tool is the two-dimensional inviscid flow-field analysis. Such calculations enable one to confront the most essential elements of the flow process, from a physical standpoint, with reasonably practical numerical calculations. Equally important at this stage of the design is the choice of important constructive parameters of the full rotor. In this respect, an excellent summary was presented by

Stepanoff [100],213H where empirical formulations for critical design parameters, such as the hub ratio and number of blades, were presented. Finally, the design process is normally concluded with the performance evaluation of the rotor, and usually one analyzes the so-called “system characteristic”, which is the map of the system’s behavior at various operating points. All important parameters reflecting the performance, i.e. head and efficiency, are observed before one or the other design is

chosen. A typical system characteristic of an axial fan is depicted in Figure214H 3-1.

3. Survey2B of the available design methods for axial impellers

Figure 3-1 The “design point” (point of maximum efficiency) on a typical system characteristic for an axial fan The design point is the point of maximum efficiency on the efficiency curve depicted in

Figure215H 3-1, and the corresponding flow rate is the design flow, Qdesign . Most designs are normally evaluated in this manner and the best performing design is usually a blend between good efficiency rates and pressure (or head) which can be achieved by the impeller. Most designs today require optimization to ensure good performance under diverse operating conditions. Therefore, to carry out an optimization requires repeated analysis, such as the ones mentioned above, under many different conditions, and the cost of the two- and three-dimensional analysis flow field calculation is excessive for this approach. Hence two- and three-dimensional tools are used for detailed refinement of a basic concept that has been previously optimized with effective mean-line calculations.

3.2 Two – dimensional cascade theory The operation of any turbomachine is directly dependent upon changes in the working fluid’s angular momentum as it crosses individual blade rows. A deeper insight into turbomachinery mechanics may be gained from consideration of the flow changes and forces exerted within these individual blade rows. The complex three-dimensional flow can be treated as the superposition of a number of two-dimensional flows. This leads to a more manageable blade design and profile selection techniques. For an axial impeller with hub and casing, it is reasonable to assume that the stream surfaces at the entry to the annulus remain cylindrical as they progress through the

3. Survey2B of the available design methods for axial impellers machine. Each cylindrical meridional stream surface will then intersect the blade row to

form a circumferential array of blade shapes known as cascade, Lewis [68].216H If one such cylindrical surface were to be unwrapped from the cylindrical coordinate system x,r

to a cartesian one x, y , then it would have the aspect depicted in Figure217H 3-2.

unwraping of the cylindrical surface

from cylindrical to cartesian coordinate system through y=rθ

Figure 3-2 Development of a cylindrical blade-to-blade section into an infinite rectilinear cascade in

the cartesian coordinate system. Adapted from Lewis [68]218H The full three-dimensional flow could then be modeled by a series of such plane two- dimensional cascades, one for each of the cylindrical meridional surfaces equally spaced between hub and casing. To obtain truly two-dimensional flow, one would require a cascade of infinite extent. However, cascades must be limited in size, and careful design is needed to ensure that at least the central regions (where the flow analyses and measurements are made) operate with approximately two-dimensional flow. In particular for axial-flow machines of high hub-to-tip ratio, radial velocities are negligible and, to a close approximation, the flow may be described as two-dimensional,

Dixon [29].219H The flow in a cascade is then a reasonable model of the flow in the machine. For impellers with lower hub-to-tip ratios, the blades will normally have an appreciable amount of twist along their length, the amount depending upon the chosen design. Even so, data obtained from two-dimensional cascades can still be of value to a designer requiring the performance at discrete blade sections of such blade rows.

3. Survey2B of the available design methods for axial impellers

3.2.1 Aerodynamics forces and governing equations

Considering the two-dimensional airfoil cascade depicted in Figure20H 3-2, the fluid approaches the cascade from far upstream with a velocity w1 at an angle 1 and leaves far

downstream the cascade with a velocity w2 at  2 . The following analysis assumes that the fluid is incompressible and the flow is steady. The first assumption is valid since most of the cascade tests and measurements, even for axial compressors, are made for low Mach numbers, i.e. below 0.3, when the compressibility effects can be neglected. However, a correlation between the compressible and incompressible cascades is possible and the

available literature offers detailed techniques on doing so, Csnady [22].21H With regard to the steady flow assumption, this is valid for an isolated cascade row, but at the rotor level, relative motions between successive blade rows appear, causing unsteadiness.

An explanatory scheme of a portion of an isolated cascade is depicted in Figure2H 3-3.

Figure 3-3 Forces acting upon an airfoil in a cascade It is common in turbomachinery analysis to refer to the airfoil aerodynamic properties

with respect to w , which is the velocity of the undisturbed flow, far in front and far behind the profile. In a cascade, the action of the fluid on the profile can be considered similar to that taking place on an airfoil in a wind tunnel, provided that the velocity of the

undisturbed flow w is an average of the inlet and outlet relative velocities, Bohl [12]:23H

3. Survey2B of the available design methods for axial impellers

2 2 ww12uu wm ww m and tan   (3.1) 2 ww12uu 2

In this case, the velocity vectors diagram has the aspect depicted in Figure24H 3-4.

Figure 3-4 Velocity vectors diagram

Two forces can be recognized as acting upon the profile: one in the axial direction ( Fax ) and the other in the tangential direction ( Ft ). These forces are exerted by unit depth of blade upon the fluid, exactly equal and opposite to the forces exerted by the fluid upon the unit depth of blade.

If p1 is the pressure upstream the cascade and p2 is the pressure downstream, then

Fppbtax 21 (3.2) where b represents the radial length of the blade profile.

When analyzing the velocity diagrams in Figure25H 3-4, then the following relations can be developed between the characteristic velocities of the fluid: www222 111um (3.3) 22 2 ww222um w Because of the incompressibility assumption made in the previous paragraph, the meridional component of the flow velocity is conserved along the profile, and thus Q ww (3.4) mm12bt The energy balance between the inlet and the outlet of the cascade results in

3. Survey2B of the available design methods for axial impellers

p wpw22 11 2 2 (3.5) 22

With Eqs. (3.3)26H and (3.4)27H inserted into Eq. (3.5),28H the following expression for the axial force acting upon the profile can be established

 2 2 22 22 Fbtwwbtwwax u12 u  1122sin  sin  (3.6) 22 An expression for the tangential force can be obtained by means of the impulse equation:

Fmwwtuu 12 (3.7) where mQ   is the mass flow. The volume flow rate is determined from Eq.(3.4):29H

Qwbt m

Further manipulation of Eq. (3.7)230H yields

Fwbtwwtmuu 12 wbtw m 1122sin w sin  (3.8)

The resulting force acting on the profile, in the absence of friction (see Figure231H 3-3), is the

lift, and after replacing expressions (3.6)23H and (3.8),23H this force can be written as:

22 LFFax t btwww  u12  u (3.9)

Equation (3.9)234H defines the lift force acting upon a profile in an isolated cascade, in the ideal case of the frictionless flow. If one considers instead the real case of the flow with friction, then also a drag force, D, can be included in the force balance. Considering the unit depth of a cascade blade, the lift force acts perpendicular to the flow

direction, while the drag is parallel to the flow, as indicated in Figure235H 3-3, and both forces

can be expressed in terms of Fax and Ft , Anderson [5]:236H LFsin F cos tax (3.10) DFtaxcos  F sin 

In Eq. (3.10),237H the angle of the undisturbed flow   is given by (3.1).238H The aerodynamic properties of the airfoil are usually presented in terms of dimensionless coefficients, i.e. the lift and drag coefficients, defined as

3. Survey2B of the available design methods for axial impellers

L CL   2 wl 2 (3.11) D C  D  wl2 2  The calculation of the lift force acting on an airfoil can be achieved also by considering the circulation in a large circuit enclosing the airfoil, and hence, the velocity far in front the cascade:

Lw  (3.12)

Equation (3.12)239H expresses the Kutta–Joukowski theorem for a single isolated airfoil for the ideal case of the frictionless flow, when D  0 . The circulation is the contour integral of velocity around a closed curve and inserting Eq.

(3.9)240H into Eq. (3.12)241H yields:

bt wuu12  w (3.13) The Kutta–Joukowski theorem is directly related to the flow at the trailing edge of the

airfoil, Hughes and Brighton [52],24H and it can be reformulated as follows: for a given airfoil, at a specified angle of attack, the value of the circulation about the airfoil is such that the flow leaves the trailing edge smoothly. Apart from the Kutta–Joukowski theorem, the literature includes numerous methods and further approximations for the calculation of the lift and drag, for particular families of

airfoils, and an excellent review was given by Thwaites, [102],243H in his chapter 5.

3.3 Design methods based on the airfoil theory There are two approaches to blade profile selection which are often referred to as the

direct (analysis) method and the inverse (synthesis) method, Lewis [68].24H The direct method assumes the profile generation through systematic geometrical techniques, and then series containing geometries generated as such are analyzed by means of measurements or theoretical investigation, resulting in the determination of the most efficient profiles and their detailed aerodynamic performance. In engineering practice, the systematic procedures of the direct method offer a special attraction for building up experimental and theoretical data for closely related families of cascades.

3. Survey2B of the available design methods for axial impellers

Several direct design methods for generating two-dimensional blade shapes, applicable

for the design of axial turbomachinery cascades, were developed by Korakianitis [60].245H The first method specifies the airfoil shapes with analytical polynomials and it shows that continuous curvature and continuous slope of curvature are necessary conditions to minimize the possibility of flow separation and to lead to improved blade designs. The second method specifies airfoil shapes with parametric fourth-order polynomials, which result in continuous slope-of-curvature airfoils, with smooth Mach numbers and pressure distributions. Finally, a third method is presented, in which the airfoil shapes are specified by using a mixture of analytical polynomials and mapping the airfoil surfaces from a desirable curvature distribution. This method provides blade surfaces with desirable performance in very few direct-design iterations. The inverse method allows the designer to specify the velocity or pressure distribution along the blade surface and to calculate the profile accordingly. When prescribing velocity distributions (PVD), two options are available for airfoil or cascade design by the inverse design method. The first method permits the designer to specify a prescribed velocity distribution (PVD), and therefore pressure distribution, on both the upper and lower surfaces, resulting in automatic synthesis of the entire profile to meet this specification. Although this sounds attractive, the procedure has its drawbacks. At worst, the designer may choose an impossible PVD for which there is no corresponding blade profile. At best, the chosen PVD may lead to an unsuitable profile thickness distribution.

In view of the latter problems, Wilkinson [111]246H proposed another option for airfoil design whereby the PVD is limited to the more aerodynamically sensitive upper surface only, but a profile thickness is also prescribed. In effect, the inverse method then involves designing the camber line shape required to achieve the desired PVD on the upper surface. The velocity distribution on the lower surface is simply accepted to adjust freely

to whatever it will. Such design techniques were developed by Ackert [2],247H Railly [89],248H

Cheng [19],249H and Lewis [67].250H The inverse design methods, by defining the blade geometry corresponding to a prescribed pressure distribution, have been the subject of many theoretical and computational studies and have greatly matured in recent years. They are nowadays

3. Survey2B of the available design methods for axial impellers reckoned to be a valuable alternative to the iterative use of direct methods, de Vito et. al.

[25].251H Although the latter allow an easier control of various geometric parameters to meet the design requirements, they demand a lot of physical insight by the designer to predict the changes required to reach a target pressure distribution. Hence inverse methods allow a more direct interaction with the blade pressure distribution and require much less

computational effort. In this respect, Leonard and Van den Braembussche [66]25H and

Demeulenaere and Van den Braembussche [26],253H made remarkable efforts to develop an efficient inverse design method based on Euler solvers. Their method allows the definition of the 3D geometry required to obtain a prescribed pressure distribution on the whole blade surface. Also, by prescribing the blade loading distribution (i.e. the difference between the blade suction and pressure surfaces), the profile geometry can be

inversely computed, Dang et. al., [23]254H [105].25H

3.3.1 Airfoil families. Mean-line and thickness distribution

Since most of the parameters that are critical for the design process of axial fans rely entirely on the airfoil theory and wind-tunnel measurements of such profiles, the author feels that a brief explanation of the main characteristics of these airfoils is helpful. An airfoil can be conceived of as a curved camber line upon which a profile thickness

distribution is symmetrically superimposed, Figure256H 3-5.

Figure 3-5 Geometrical description of a cascade profile The traditional approach to the aerodynamical design of axial-flow compressors and fans is to use various families of airfoils as the basis for blade design, and probably the most popular ones belong to the NACA family and the British C-series. The development of

3. Survey2B of the available design methods for axial impellers the NACA profiles started in 1929 in the Langley variable-density wind tunnel. There exists a huge data base on these profiles and their measured characteristics, and probably the most comprehensive material on the topic is the summary of Abbot and Von

Doenhoff 257H[1]. Several series of NACA profiles can be identified (commonly used in the design of axial compressors and fans are the four and five digit series) and each of these series is characterized by a different thickness distribution along the mean camber line. All airfoils in the NACA 4-digit family were designed for the same basic thickness distribution and the amount of camber (curvature) was systematically varied to produce the family of related airfoils. By extending the investigation to airfoils having the same thickness distribution, but moving the positions of the maximum camber far forward on the airfoil, the 5-digit series was obtained. The process of combining the mean-line and thickness distribution to obtain the NACA

airfoils is described in Figure258H 3-6.

Figure 3-6 Sample calculations for the derivation of the NACA 65 series. Adapted from Abbot and

Von Doenhoff [1]259H Ordinates of the cambered airfoil are obtained by laying off the thickness distribution perpendicular to the mean-line. If xU and yU represent the abscissa and ordinate, respectively, of a typical point of the upper surface of the airfoil and yt is the ordinate of the symmetrical thickness distribution at the chordwise position x , the upper-surface coordinates are given by the following relations: xxy sin Ut (3.14) yyyUt cos Analogous relations for the lower surface can be derived.

In Eq. (3.14),260H yt is the thickness distribution and detailed formulations of the applied thicknesses to all NACA families are presented in Abbot’s summary of airfoil data.

3. Survey2B of the available design methods for axial impellers

3.3.2 Design parameters

In the process of designing axial impellers, the initial analyses of the flow through the blade rows assume that the velocity vectors along a blade are constrained to lie along the mean camber line of the profile. Although this is true exactly on the surface of the profile, the flow is not generally constrained completely by the blade shape through the entire

flow channel, Wright [112].261H Since the flow pattern is highly dependent on the geometric characteristics of the channel, i.e. blade-to-blade distance, it becomes obvious that whatever camber angle value is prescribed through the design method, it will not coincide with the actual flow angle, especially in the blade passage. To increase the accuracy of the angle prediction and the design calculations, one needs to formulate a quantitative model of this flow behavior. The most important geometric parameters, critical for profile design, are depicted in

Figure26H 3-5 and the differences between the designed blade angles and the actual flow angles can be underlined as follows:  The difference between the prescribed blade angle and the flow angle achieved at

the inlet section is called the incidence, i  11  .  The difference between the blade and flow angles, at the outlet section, is called

the deviation,  22. The deviation reflects the failure to achieve the expected level of turning of the flow vector from the ideal angle and it is a function of the geometric and velocity properties of the blade cascade. There are many empirical correlations available in the literature, which are focused on the prediction of the flow pattern in the channel by applying various values for the incidence and deviation angles. As far as the incidence angle is considered, it frequently refers to the particular inlet angle for which the leading edge (LE) stagnation point is situated precisely on the end of the profile camber line, i.e. i  0 , and this is the particular case of the “shock- free” inlet (see

Figure263H 3-7b).

3. Survey2B of the available design methods for axial impellers

Figure 3-7 Stagnation streamline and the “shock-free” inflow For greater or smaller inlet angles, the stagnation point will move instead on to the lower

or upper surface respectively, as illustrated in Figure264H 3-7a and c. The so-called “shock- free” inlet flow condition ensures the smoothest entry conditions into the cascade and is

thus likely to be close to the “minimum loss” situation, Lewis [68].265H In the present work, a zero incidence assumption is included in the design process. Including in the design process correlations for the deviation angles accounts for a good

prediction, from the design stage, of the actual flow angles. In this respect, Lieblein [70]26H proposed corrections for the NACA 65 series and the British C4 profiles, in terms of empirical formulations for the incidence and deviation.

Howell [51]267H developed an analytical method that allowed the modeling of the flow angles with excellent results: by fixing a “shock-free” entrance of the flow in the cascade, the vector field can be resolved by expressing the trailing edge (TE) deviation as a simple function of the channel proportions. Howell’s correlation can be written as follows: m   1 (3.15)  2 where  represents the cascade solidity and is defined as the ratio between the chord l length of the profile and the blade spacing (pitch):   .  is the camber turning angle: t

 12 The coefficient m is calculated according to the following expression:

3. Survey2B of the available design methods for axial impellers

1 m 0.41 0.2 100 Another empirical correlation for the deviation, derived also for the zero incidence

condition, was proposed by Carter and Hughes, [17]:268H     12 (3.16) 4  The outlet flow angle predicted by Carter’s rule is then

22 (3.17)

Finally, McKenzie [76]269H also suggested an empirical formulation of the deviation angle:

1  1.1 0.313 (3.18) It can be concluded that, with respect to the prediction of the flow angles at critical stations on the profile, such as the trailing edge, the literature offers numerous methods (such as the ones presented above) to correct the design angles, so that the desired flow– blade congruency is achieved. However, most of these methods are empirical and were determined for a specified class of axial impellers, i.e. axial compressors, and it is the choice of the designer which of the formulations should be included in the design process. In the present work, the value of the deviation was assumed to be zero, since it was found more appropriate for thin profiles; it will be shown in the following section that, for the fan application of interest for the present work, thin profiles are better suited rather than airfoils. However, the possibility to assume other values for the deviation is incorporated in the mathematical routine used for the profile calculation and any of the

presented formulations in Eqs. (3.15),270H (3.16)271H and (3.18)27H can be employed.

3.3.3 Cascade losses. Diffusion factor

The total pressure loss of a cascade depends on many factors. Under normal operating conditions, the boundary layer on the suction (upper) surface will be much thicker than that on the pressure (lower) surface of the airfoil, and hence, to a first approximation, the latter can be neglected. Then, the thickness of the wake (and therefore the total pressure loss) will be primarily determined by that fraction of the suction surface over which the velocity difference is negative, since that is where the majority of the boundary layer

growth occurs, Brennen [13]273H and Lakshminarayana [63].274H In other words, the suction

3. Survey2B of the available design methods for axial impellers surface has a large pressure rise which can cause the viscous boundary layer to separate, and this is not desirable due to the associated higher losses of the total pressure.

One should visualize the deceleration (diffusion) of the fluid from wmax to w2, where wmax is the maximum velocity on the suction surface. Hence, a direct correlation between the total pressure loss and the diffusion (caused by the deceleration of the fluid) on the upper

side is useful. According to Lieblein et al. [72],275H the amount of diffusion is given by the diffusion factor, which is defined as w  w DF  max 2 (3.19) wmax Lieblein argued the momentum thickness of the wake, θ, should be correlated with the diffusion factor. In this respect, several measurements were performed for the NACA65 and British C4 series profiles and it was found out that a value of DF = 0.6 imposes an

upper limit for the allowable diffusion factors, as indicated in Figure276H 3-8. Above this value, a dramatic increase in the diffusion in the boundary layer will occur.

Figure 3-8 Wake momentum thickness versus overall diffusion factor DF for NACA 65 and C4

airfoils at minimum loss incidence. Adapted from Lewis [68]27H

The diffusion factor, as defined in Eq.(3.19),278H represents an important design parameter and offers valuable information, from the design stage, on the performance of the prescribed profile, since a good design should be characterized by limited deceleration in the flow velocity. Moreover, even though at the design stage a frictionless flow is

3. Survey2B of the available design methods for axial impellers assumed, by calculating the DF and imposing limitations to its value one can assume that the real flow with friction will cause fewer losses due to these preventive measurements taken from the early design stage.

3.4 Three-dimensional character of the flow in axial turbomachines In the previous section, the most important design considerations in the field of axial impellers were reviewed and it can be concluded that such methods have the benefit of the simple two-dimensional flow modeling. However, the designer must not forget that in truth, the flow in turbomachines is really three-dimensional and any design considerations based on the 2D cascade theory result in a simplified flow prediction, which overlooks the 3D character of the flow and its side effects, i.e. secondary flows.

Probably the most notable effort in this area was made by Wu [113],279H who recognized the truly three-dimensional nature of the flow in turbomachines and proposed the remarkably

sophisticated computational scheme illustrated in Figure280H 3-9.

Figure 3-9 S-1 and S-2 stream surface according to Wu. Adapted from Lewis [68]281H Wu proposed a general theory of steady three-dimensional flow of a non-viscous fluid in subsonic and supersonic turbomachines having arbitrary hub and casing shapes and a finite number of blades. The solution of the three-dimensional direct and inverse problem was obtained by investigating appropriate combinations of flows on relative stream

3. Survey2B of the available design methods for axial impellers surfaces whose intersections with a z-plane, either upstream of or somewhere inside the blade row, form a circular arc or a radial line. The fully 3D flow was treated by the superposition of a number of 2D flows, but in this case located on the S-1 and S-2 stream surfaces. S-2 surfaces follow the primary fluid deflection caused by the blade profile curvature and its associated aerodynamic loading. Due to the variation of static pressure between the convex surface of the 1st blade and the concave surface of the 2nd blade, the curvature of each S-2 stream surface will vary, thus

calling for the introduction of several surfaces for adequate modeling, Lewis [68].28H The S-1 surfaces are equivalent to the meridional surfaces of revolution which are allowed to twist in order to accommodate the fluid movements caused by the variations of the S-2 surfaces. The S-1 and S-2 surfaces represent, in fact, a selection of the true stream surfaces passing through the blade row. By solving equations of motion for the flows on this adaptable model, successively improved estimates of the S-1 and S-2 surfaces may be obtained, allowing also the fluid dynamic coupling between them. The iterative approach to achieve a good estimate of the fully three-dimensional flow was very comprehensively

laid out by Wu [113].283H The first major computational scheme based on Wu’s work dealt with axisymmetric

meridional flow located on an averaged S-2 surface, Marsh [74].284H

Also based on Wu’s treatment of the flow in turbomachines, Denton [27]285H developed a time-marching method, which practically initiated the path for design codes for

compressible three-dimensional flow analysis. Similar efforts were made by Potts [86],286H who also developed a time-marching method to study the twisting of the S-1 surfaces within highly swept turbine cascades. A second aspect of the three-dimensional character of the flow in turbomachines is the appearance of the so-called “secondary flows”, which are of great importance in axial turbomachinery aerodynamics, where boundary layer growth that occurred on the casing and hub walls of the machine are deflected by the blade rows. Many studies have been entirely dedicated to the investigation of the boundary layer on the annulus walls of axial flow machines and it has been concluded that axial impellers operate at nominal

conditions with significant boundary layer separation, Lieblein [71]287H and Schlichting [94].28H

3. Survey2B of the available design methods for axial impellers

The separation in the boundary layer is an important issue for the design of axial turbomachinery blading and major attempts at understanding and solving the complicated three-dimensional flows which appear due to this separation have been made, Horlock

and Lakshminarayana [49]289H and Hawthorne and Novak [46].290H However, it is generally agreed that classical boundary layer considerations alone are insufficient for capturing the full 3D nature of the secondary flows, and full CFD calculations of the whole flow are

generally required, Horlock [50].291H The presence of the 3D secondary flows impacts directly on the performance of the impeller and therefore methods of controlling the separation phenomenon have to be included in the design process. One such method is to predict, from the early design stage, the location where the separation might occur. In this respect, Ramirez Camacho

and Manzanares Filho [90]29H developed a model for the cascade computation of axial impellers, able to predict the separation point near the TE and reattaching the flow by introducing fictitious velocities to achieve the viscous effect of the attached flow. Other possible methods to control the separation flows are either to introduce a tip

clearance, Gbadebo et al. [41],293H or through the suction (aspiration) of the boundary layer

on the suction surface of the profile and at the end walls, Gbadebo et al. [42]294H and

Kerrebrock et al. [59].295H Even though both methods yield in immediate increase in the impeller performance, the first method appears more attractive, especially in the case of axial fans, since the tip clearance is a design parameter that is fairly easy to control, due to the general constructive simplicity of the fan casing, which dictates the size of the clearance. However, an optimum tip gap has to be found and it is normally in the range of a few percent of the chord length. Finally, also accounting for the three-dimensional character of the flow, is the meridional analysis, which is particularly focused on the flow prediction in axial machines. Since the meridional analysis of axial fans is of major importance for the design strategy proposed in the present work, a detailed explanation of the concept will be presented in the following chapter.

4. Proposed3B design strategy for axial fans

4 Proposed3B design strategy for axial fans

The layout and design strategy proposed in the present work is focused on the blade design of axial fans that operate under low-pressure regimes, delivering high flow rates, and thus deals with fans normally employed for cooling purposes. This method involves a preliminary design stage, based on mean-line performance calculations, and also a detailed design stage, which includes three-dimensional flow considerations to generate the initial blade profiles. Essentially, the design procedure presented below is a design-point method, meaning that the runner blade is derived for a single point on the system characteristic depicted in

Figure296H 3-1, i.e. the point of maximum efficiency corresponding to the design flow rate,Qdesign . According to the available literature, the design-point technique appears to be an inverse method. However, the presented design strategy is a combined inverse–direct method used in an iterative process to generate an actual geometry, where one can specify the fluid dynamic boundary conditions and the governing equations, and then, effect solutions for the geometry required to establish these conditions. The approach is very appealing from the design point of view. The goal of the design strategy is to deliver high-performance designs for direct industrial applications. For this, the constructive dimensions and operating regime of an impeller, currently manufactured in industry and used for engine cooling purposes, are employed as a reference solution. These details will be referred to later.

4.1 Mean-line calculation

As it was referred in the introduction to section 3.1,297H the first step in the proposed design strategy is the one-dimensional flow analysis, i.e. the mean-line calculation. The advantage of this approach is that Euler’s equation for turbomachinery can be resolved

4. Proposed3B design strategy for axial fans

for each cascade independently. Consequently, the first design consideration is that the impeller is a succession of such cascades, at different distances from the hub center, and the mean-line calculations will be performed for each cascade individually. In the case of axial impellers, the flow particles enter and leave the blade at the same section, and therefore uuu12. In this case, Euler’s equation states that the total pressure difference accomplished by a profile assuming inviscid, incompressible flow has a static and a dynamic component:    22 22 (4.1) pwwcct 12   21  2  static dynamic pressure pressure The first design assumption (confirmed by both CFD and experiments) is that the fluid has an axial entry in the cascade. This assumption remains valid as long as the entrance in the impeller is not disturbed, and hence there are no stator vanes in front of the fan blades which dictate the flow inlet in the rotor.

A simplified scheme of the mean-line calculation is presented in Figure298H 4-1.

4. Proposed3B design strategy for axial fans

Figure 4-1 Mean-line calculation of the velocity diagram with axial entry Assuming axial entry of the fluid simplifies the inlet velocity diagram (point 1 in the above sketch). The tangential component of the absolute flow velocity is zero, i.e. cu1  0 , hence there is no pre-whirl induced in the cascade. This implies that wuu1  (4.2) where urn 2 and n represents the rotational speed of the rotor cascade.

With this assumption, Eq.(4.1)29H becomes

 222  22 2 ptuuuwwc122    uw  u 2  uw  u 2  uuw   u 2  uc u 2 (4.3) 22

On analyzing the velocity diagrams in Figure30H 4-1, the following expressions can be immediately stated for the inlet and outlet blade angles in the cascade:

4. Proposed3B design strategy for axial fans

wwmm tan1  (4.4) wuu1

wm tan 2  (4.5) wu2 Basically, provided that the flow rate and rotational speed are known, by assuming axial entry, the inlet blade angle can be calculated.

The meridional component of the relative velocity wm is conserved along the blade profile, and therefore through out the cascade, and is equal to the ratio of the flow rate to the flow area. The flow area of the cascade is in fact the flow area corresponding to the entire impeller, and therefore the area of the ring between the hub and tip sections of the rotor:

22 A  rrth (4.6) The expression of the inlet blade angle becomes Q 1 tan  d (4.7) 1 222 rn  rrth For to the derivation of the outlet blade angles, more assumptions are required to solve the velocity diagram at the cascade exit. This matter will be addressed in the following section.

4.2 Outlet conditions In the previous section, the expression for the inlet blade angle was determined according to the axial entry assumption. This angle is a function of the design point, i.e. flow rate and rotational speed of the cascade, and, of course, the radius.

Equation (4.1)301H expresses the total pressure difference achieved by the axial cascade, according to Euler. It can be observed that it has two components: a static pressure difference, given by the relative flow velocity gradient, and a dynamic component, given by the absolute flow velocity gradient. What normally is of interest for the system characteristic of an axial rotor is the static pressure component:

 22 22 22 psuuuww12   ww 12   uw  2 (4.8) 222

4. Proposed3B design strategy for axial fans

If the tangential component of the relative flow velocity at the cascade exit is rewritten

according to Eq. (4.5),302H then

wwmm tan 22wu (4.9) wu22tan

Inserting Eq. (4.9)30H into Eq. (4.8),304H the following expression of the static pressure difference is established:

2  2 wm Pus   2 (4.10) 2tan 2

According to Figure305H 3-1, the static pressure difference is maximum when the flow rate is

zero and, conversely, the flow rate is maxim at zero pressure difference. If Eq. (4.10)306H is written for the highest possible flow rate, i.e. zero pressure difference, then it yields

22 2222wQm max 1 urn2224  (4.11) tan 22A tan  Hence the maximum flow rate is

QrnAmax 2tan  2 (4.12)

From Eq. (4.12)307H it can immediately be observed that the maximum flow rate, at a specified blade section, is “maximum” when the outlet flow angle

 2 (r ) 90deg (Qmax ). Of course, this value is radius dependent, hence it can be implemented at one blade section only. The question is which section is best suited for this angle. Applying this angle means, in terms of velocity diagrams, that wu2  0 and cuu2  , i.e. cu2 is maximum. The tangential component of the absolute velocity cu2 represents the swirl at the exit of the cascade and the product cru2 is a direct measure of

the blade loading. For an axial cascade, the total pressure difference in Eq. (4.3)308H can be written in terms of the loading and hence

ptuuc222 2 rn c uu  Kc r (4.13) where Kn 2  is a constant and after further equating, it can be concluded that the swirl velocity downstream of the cascade is inversely proportional to the rotor radius: K c  (4.14) u2 r

4. Proposed3B design strategy for axial fans

Since the swirl velocity cu2 expresses losses and, according to Eq. (4.13)309H high swirl means high blade loading, a good design should limit this component. Moreover, through deceleration of cu2 , a recovery in the static pressure can be achieved. Since the hub section has the highest expected loading and here the profile should not turn more than axial, this is where a blade angle of 90o is best suited. For the other blade radii, the outlet blade angles will be determined recurrently, using this startup value at the hub.

4.3 Meridional flow analysis for axial fans The previous paragraphs were mainly concerned with the mean-line calculations leading to the determination of the inlet blade angles, and, by assuming the “maximum flow rate” condition at the hub section, the derivation of the outlet blade angle at this section was achieved. The present section deals with the three-dimensional treatment of the flow in an axial fan and further design assumptions will be made according to this complex flow treatment.

The starting point of the three-dimensional treatment was made in Figure310H 2-4, where the notions of “cascade” and “meridional flow” were introduced. It was shown that fully three-dimensional flow can be treated, for a more manageable framework, as an axisymmetric or circumferentially averaged meridional flow, and a series of superimposed cascade flows to define blade profiles at selected sections from hub to tip. Previously, the fluid motion through the blade rows was assumed to be two-dimensional in the sense that radial (span-wise) velocities do not exist. This is not an unreasonable assumption for axial rotors with a high hub-to-tip ratio. However, for ratios less than 0.8, the radial velocities through a blade row may become appreciable, the consequent redistribution of mass flow (with respect to radius) seriously affecting the outlet velocity

profile and, thus, the flow angle distribution, Dixon [29].31H Such radial flows are mainly caused by the imbalance between the strong centrifugal forces exerted on the fluid and the radial pressures. The flow in an annular passage, in which there is no radial component of the velocity, with circular streamlines and cylindrical surfaces and which is axisymmetric, is commonly know as “radial equilibrium flow”. The treatment of the three-dimensional flow in an axial turbomachine based on the assumption that only radial

4. Proposed3B design strategy for axial fans

flow which may occur is completed within a blade row, the flow outside the row being in

equilibrium, is the radial equilibrium method, Dixon [29].312H Since the correct understanding of this method is central for the proposed design strategy, the author feels that a brief explanation of the theoretical treatment behind is necessary.

Figure 4-2 Radial equilibrium through: a) a rotor; b) a fluid element (cr = 0) For a small fluid element of mass dm , of unit depth and at an angle d from the axis, rotating about the axis, with a tangential velocity cu at a radius r , the radial equilibrium

(centrifugal force is balanced by the pressure force), as indicated in Figure31H 4-2 b, can be stated as follows: 1 dp c2  u (4.15)  dr r For incompressible flow, which is normally the case for axial fans, the so – called stagnation pressure can be defined as p p cc22c2 oxr u (4.16) 222 where cx represents the axial velocity, cr is the radial component of the velocity, and cu represents the tangential velocity.

Since the radial equilibrium condition, as stated previously, imposes cr  0 , on

differentiating Eq. (4.16)314H with respect to r one obtains 11dpdp dc dc 0 ccx u (4.17) dr drxu dr dr dp

Introducing from Eq. (4.15),315H then dr

4. Proposed3B design strategy for axial fans

1 dp dc c drc 0 c xu u (4.18)  drx dr r dr

Equation (4.18)316H states the equation for the radial equilibrium condition for incompressible flows. The key element for the present work in this equation was the loading given by the product cru and the most important design choices were made with respect to this term.

Equation (4.18)317H can be applied to two sets of problems: 1) the indirect problem (design method), in which the tangential (swirl) velocity is specified and the axial velocity is calculated accordingly. 2) the direct problem, in which the swirl angle is given and the axial and tangential velocities are calculated. The proposed design strategy is focused on the first method.

4.4 The indirect design problem The indirect design problem requires specification of the tangential velocity, calculating the resulting axial velocity. There are several approaches to this problem and they are

generally focused on the product cru  in Eq. (4.18).318H

Probably the most popular design assumption is that the product cru  is constant, also known as the “free-vortex flow” assumption. Essentially, this assumption implies that

cru  k (4.19) When considering an element of the ideal inviscid flow rotating about a fixed axis, as

indicated in Figure319H 4-2b, then the circulation  (the vortex strength) is involved:  c  (4.20) u 2 r The vorticity at a point is defined as the ratio between the limiting value of the circulation   and the elementary area  A, as  A becomes vanishing small:    lim (4.21)  A0  A

Differentiating Eq. (4.20):320H dcdcrdrdcrduu    u

4. Proposed3B design strategy for axial fans

dcuu c drddr   (4.22) dr r

Equation (4.21)321H then becomes d 1 dcr   u (4.23) dA r dr

dcr u  If the vorticity expressed in Eq.(4.23)32H is zero, then  0 , with the result that dr cu r const (4.24)

Equation (4.24)32H embodies the absolute condition to be satisfied by a free-vortex flow. Returning to the expression for the total pressure difference achieved by an axial cascade

in Eq. (4.3),324H the free-vortex flow assumption can then immediately be translated into a “constant total pressure” assumption, since

ptuuc22  k c u r (4.25) As already mentioned, this is probably the most popular design choice for axial fans, especially when the two-dimensional airfoil theory is employed in the design process. However, the use of the three-dimensional analysis with axial fans is not limited only to this assumption, and the available literature makes references to several other possibilities, often referred to as “non-free” vortex flow (forced vortex) or “solid body”,

Dixon [29],325H Lewis [68],326H Augnier [6],327H and Carolus [16].328H Essentially, these inverse methods assume some variation of the product cru  , deriving the axial velocity according to this assumption. One of the preferred hypotheses is that of constant swirl velocity, cconstu  .Even though such methods are mentioned, little use has been of them and the effects of employing a non-free vortex design assumption are not well known. At this point in the analysis, it becomes of great interest to establish which design choice, whether free- or non-free vortex flow, delivers, for a specified fan configuration, the best- performing design. This leads to the motivation for the present work: to determine, for a given class of fans, the optimum vortex design specification.

4. Proposed3B design strategy for axial fans

4.5 Parameterization of the total pressure in the span-wise direction for an axial fan blade Typical design methods for axial fans make intensive use of the free-vortex flow assumption. Basically, this implies that along the blade, a constant total pressure

difference, according to Euler’s equation, is applied, as indicated in Eq. (4.25).329H For the area of interest in the present work, i.e. axial fans used for cooling purposes, the

work of Wallis [108]30H is relevant, in which an inlet guide vane–rotor–stator installation was investigated. The system considered was of the free-vortex flow type and several important parameters, e.g. lift-to-drag ratio, were fixed. This resulted in explicit expressions for efficiency and total pressure rise as a function of tip speed ratio, hub ratio, and downstream losses.

Focusing on the same class of fan application, Dugao et al. [30]31H considered the numerical design optimization of a rotor–stator configuration for mine ventilation. Employing a free-vortex design method, a considerable improvement in efficiency was achieved compared with an existing installation. As an additional advantage, it was found that the noise emission from the fan installation was reduced. It thus seems that the free-vortex solution is preferred by most designers in the field. There is little information available on design techniques which make use of an arbitrary- vortex assumption, and the differences that such a design consideration might bring about when compared with the classical zero-vorticity condition.

One of the few references in this respect relates to the work of Sørensen et al. [98]32H and

[99],3H which deals with combining an optimization algorithm with the arbitrary vortex flow, so that a wider range of design alternatives are investigated in an efficient manner. The fixed design parameters were the tip radius, the number of blades and the tip clearance. The design variables included the hub radius, the chord distribution, the camber angle, the total pressure rise and the velocity diagrams, and a series of constraints were applied to these parameters. The objective function was the efficiency of the rotor, considered over a design interval of flow rates, not only at the design point, and the design flow rate was fixed in the centre of the interval. The goal of the optimization was to maximize the mean value of the aerodynamic fan efficiency.

4. Proposed3B design strategy for axial fans

Since good prospects of employing a non-free vortex flow condition are created by such results, it seems desirable to undertake the development of a design strategy which, for a specified fan application, determines whether improvements in the performance can be achieved by employing the free-vortex flow or whether some variation is better suited. Hence the motivation of the present analysis was to determine the optimum pressure distribution along an axial fan blade, by considering both free and non-free vortex flow assumptions, and parameterizing the solution according to the fan specifications.

Figure 4-3 Sketch of an axial impeller Considering various sections of the blade (hub, tip and an intermediate section), as

indicated in Figure34H 4-3, the free-vortex assumption can be written in terms of the total pressure difference, at different blade sections, as follows: P tr, 1 (4.26) Pth,

The total pressure differences mentioned in Eq. (4.26)35H are defined as stated in Eq. (4.13).36H

According to Carolus [16],37H employing such an assumption introduces, from the design stage, a correct blade loading (pressure distribution) at the designated section. A proper blade loading at the tip section (where the radius is larger) demands considerably smaller values of cu2 (responsible for the swirl) than at the hub section (where the radius is smaller). This means that close to the hub the absolute flow, characterized by higher cu2, has to be redirected to the tip section, thus keeping the product cru2 constant. The desired effect of such a consideration is that the flow detachments will occur earlier and hence the hub section will be more stressed than the tip.

4. Proposed3B design strategy for axial fans

The free-vortex assumption in Eq. (4.26)38H is directly connected to the swirl component at the exit of the cascade and it is employed whenever airfoil profiles are used in designing an axial flow machine. However, there is nothing in the airfoil theory to prevent the desired head distribution along the blade radius. By changing the pressure distribution in the span-wise direction, designs providing higher overall performances may be achieved with a small impact on the dimensions of the swirl. Hence it is the designer’s task to investigate, for the specified design parameters and operating conditions, the optimum pressure distribution which balances the efficiency and the swirl component at the same time. In the present study, it was assumed that starting from the hub and advancing to the tip, the Eulerian pressure difference variation can be expressed as a function of the radius: P tr,  f ()r (4.27) Pth,

By employing the expression for the total pressure difference in Eq. (4.3)39H and making use

of the outlet velocity diagram in Figure340H 4-1, a recurrent relationship between the outlet blade angles, at different blade sections, can be determined:

wwmm f()2 r rhh n 2 r n 2  rn 2 rn  tan 22hr tan ww mm22 rnrfrrfrr2()() hh tan 22rh tan 12 n r 1 22 h rfrrfr()h () (4.28) tan 22rmrw r tan h

Equation (4.28)341H can be further simplified with another assumption made in the previous section, namely that at the hub, the outlet blade angle is 90o: 12 n 22 rfrr()h (4.29) tan 2rmrw

It can be seen in Eq. (4.29)342H that the expression for the outlet blade angle can be parameterized according to the prescribed variation of the total pressure difference in the span-wise direction. The use of such a formulation can be further extended if the equation can be expressed in terms of relevant geometric parameters for the design process, such as the hub ratio.

4. Proposed3B design strategy for axial fans

As already mentioned in the introduction to the present chapter, the major focus of the present work is the design of axial fans used for cooling purposes (engine cooling fans). This is an important factor for the parameterization of the pressure variation, since such devices are normally low-pressure high-flow delivering machines. Hence the pressure should not be increased to unrealistic values in the range where the impeller will never operate anyway. For example, for the reference model and its constructional details, according to the Cordier diagram, a realistic operational value of the pressure difference achieved by the impeller is below 4000 Pa. Also, rapid variations of the pressure from one section to the next, which might result from employing exponential or higher order polynomial laws for the function f r , should be avoided. Another major constraint in the parameterization process of the function f ()r is the immediate impact that this expression has on the value of the outlet blade angle. The angle, at the specified blade radius, delivered by f ()r should be a realistic value (no negative values) and should be incorporated in the geometry of the full blade, i.e. to respect the trend imposed by the neighboring sections and not induce sudden twists in the blade aspect. Furthermore, the aim of the present formulation is to derive an expression in which, by changing one factor, the desired pressure variation along the blade can be achieved. An iterative routine, in which all of the above constraints were incorporated and solved simultaneously, gave the following exact expression for the function f ()r :

x rtip 1.35 fr() x r rh 1 (4.30) rhub The parameter to be changed during the design process is x . It can be immediately seen that for x  0 the classical assumption of constant pressure (free-vortex flow) is

employed, since fr() 1 and Eq. (4.26)34H comes about. For x 1 the pressure variation is linear with the inverse of the hub ratio, for x  2 parabolic, and so on. The exponential

1.35 in Eq. (4.30)34H yielded from pressure variation considerations. This exponential was iterated until a smooth pressure increase from one section to the next was obtained; with higher degree exponentials the increase was to steep.

Equation (4.30)345H then becomes:

4. Proposed3B design strategy for axial fans

x 12 n 22rtip 1.35 rx rr hh1 r (4.31) tan rw r 2rm hub

Equation (4.31)346H allows the determination of the outlet blade angle, at the specified radius, based on the pressure variation from hub to tip. Essentially, this variation has a major influence on the resulting blade shape and as a consequence, an optimum profile demands proper prescription of the pressure. Therefore, several test cases of axial fan designs will be investigated, including the classical constant pressure assumption, in order to determine, for specified dimensions and operating conditions, the optimum pressure variation in the span-wise direction, which delivers high efficiency and small losses due to the swirl component. Three designs, corresponding to x  0DesignI  , x 1DesignII  and x  2DesignIII respectively, are derived and their performance is assessed against the reference model.

4.6 Blade shape computation The computation of the blade shape is carried out considering the cartesian system

depicted in Figure347H 4-4. The inlet profile angle is1 , at an intermediate point along the profile  m , and the outlet of the profile is described by  2 . Each of the three stations on the profile is characterized by corresponding coordinates  x, y .

Figure 4-4 View of the calculated blade shape in (x,y) coordinates Basically, the blade shape at a specified section can be readily computed by imputing the y coordinates and angle distribution  (y) , since

4. Proposed3B design strategy for axial fans

y tan (y )  (4.32) x

To apply effectively any driving action on the fluid, the blade angle is increased from 1 to  2 . The difference between the two,  21  , is a measure of the blade curvature along any given blade section. The increase from 1 to  2 can be estimated by any type of variation, either linear or higher degree polynomial. At this point, the presented design strategy requires another assumption, connected to the blade angle distribution along the calculated profile. It will be assumed that this

distribution is parabolic since, according to Pascu and Epple [82],348H this assumption is appropriate for ducted axial fans. Pascu and Epple showed by means of streamline analysis performed on the designed blades, that very good agreement between the actual flow angles and the prescribed blade angles is achieved by employing such a design consideration. Considering such a distribution  (y)  Ay 2  By  C :

yy12 

yymm  (4.33)

yy21  Since both the inlet and outlet conditions are fixed, the blade shape computation is carried out with only one degree of freedom given by ( mm ,y ) . The geometry is iterated until the axial chord constraints are matched and also corrected for abrupt flow parameters.

4.7 Further design assumptions based on profile analysis Considering the flow over a profile, the fluid approaches the profile from upstream with a velocity w1 at an angle 1 and leaves the profile with a velocity w2 at an angle  2 , as

indicated in Figure349H 3-3.

4. Proposed3B design strategy for axial fans

Figure 4-5 Forces acting upon the profile for small gliding angles

Figure350H 4-5 repeats the analysis on the forces acting upon a profile from Figure351H 3-3, except that it underlines the presence of  , often referred to as the “gliding” angle of the profile, which is defined as the drag-to-lift ratio: D C  D (4.34) LCL In the literature, several optimum values are proposed based on extensive experimental

results, Eckert [34],352H and it is normally the case that the values of the gliding angle are very small. The value of  is an important design choice since it denotes the impeller losses by friction, and thus influences the cascade efficiency and hence the overall performance of the rotor, as it will be shown in the following paragraphs. The design value of  will be chosen based on this analysis.

4.7.1 Static-to-static cascade efficiency

Compared with the ideal case of the friction-less flow, when   0 , the drag force

introduced by  in the direction of w (Figure35H 4-5) calls for a decrease in the pressure difference.

According to the analysis of the forces acting upon a profile presented in section 3.2.1,354H the lift force, acting perpendicular to the flow direction, has an axial and a tangential

component. The axial component is given by Eq.(3.2):35H

FppbtPbtax21  s (4.35)

4. Proposed3B design strategy for axial fans

Considering the case of the flow with friction (the real flow), then the tangential force can be written as a function of the gliding angle:

Fax tan 90    Ft F 1 ax  (4.36) Ft tan    

The definition of the tangential force is given by Eq.(3.8):356H

Fwbtwwtmuu 12

Combining (4.36)357H and (3.8),358H the following expression for the axial force acting upon the profile yields

wbtwmuu12 w Fax  (4.37) tan  

Substituting Eq. (4.37)359H into Eq. (4.35),360H the pressure difference caused by considering the drag force caused by in the flow direction, i.e. friction, becomes

Fax wwmu12 w u Psfr, (4.38) bt tan  

The trigonometric expression tan      can be expanded to

tan   tan tan   (4.39) 1tantan   

For very small gliding angles, which is normally the case, Bohl [12],361H the above expression can be simplified to

  tan  tan   (4.40) 1tan   To facilitate correspondence with the classical cascade evaluation techniques, the

following annotations will be employed, according to Vavra [106]:362H w   m (4.41) u ww   uu12 (4.42) u

4. Proposed3B design strategy for axial fans

The theoretical degree of reaction is given by the ratio of the static pressure difference to the total pressure gradient, achieved by the cascade, according to Euler’s equation, for the

case of frictionless flow, as defined by (4.8)36H and (4.3),364H respectively: P R  s (4.43) Pt Writing these pressure differences for the case of axial entry, the reaction becomes  ww22 uu121 ww R 2 uu12 (4.44) ucu2 2 u

Further more, the expression for the angle   , as given in Eq. (3.1),365H can also be reformulated in terms of the dimensionless coefficients defined above:

wm  tan   (4.45) wwuu12 R 2

Further equating (4.38),36H the static pressure difference for the case of flow with friction can be rewritten as  1 Pu2 R (4.46) sfr,    R

By putting   0 in Eq. (4.46),367H the static pressure difference, for the ideal, frictionless flow, can be written as 1 Pu 2 (4.47) s  R At this point, a first evaluation of the cascade performance is possible, since expressions for both the ideal (for the case of frictionless flow) and the real (flow with friction) static pressure differences are derived. The ratio between the two terms is referred as the static- to-static cascade efficiency and it has the following form:  P 1  sfr, R (4.48) ss P R s 1  

4. Proposed3B design strategy for axial fans

Equation (4.48)368H expresses the losses in the static-to-static profile efficiency caused by considering the drag force acting upon the profile. For the case of ideal frictionless flow, this efficiency is maximum, i.e. 1, as can be readily observed by putting   0 in the above formulation.

4.7.2 Total-to-total cascade efficiency

Compared with the ideal flow, in the case of flow with friction, the drag force arises in the direction of the fluid and, due to this force, work is dissipated, and according to Eck

[32]369H this can be formulated as

Dw  Ploss w m bt (4.49)

From Eq. (4.49),370H the expression for the decrease in the total pressure caused by the drag force can be derived:

Dw Ploss (4.50) wbtm

The definition of the drag force was stated in Eq.(3.11):371H wlb2 DC  (4.51) D 2

Equation (4.50)372H then becomes

2 ww l PCloss D  (4.52) wtm 2 Returning to the definition of the tangential component of the lift force acting upon the

profile, given by Eq. (3.10),37H and including the definitions of the lift and drag forces, as

indicated in Eq. (3.11),374H then the following expression is obtained:   FLsin D cos  C wlb22 sin  C wlb cos  (4.53) tLD22 

However, the tangential force is also given by Eq. (3.8):375H

Ftmuuwbtw12 w From the identity of the equations, it is found that   wbtw w C wlb22sin  C wlb cos muu12 L22 D   Further manipulations of the above equation yield

4. Proposed3B design strategy for axial fans

2 wtmu w12 w u wlC Lsin  cos  (4.54)

From the velocity triangle depicted in Figure376H 3-4, it can be written that wwm  sin (4.55) The peripheral relative velocity gradient is

wwwuu12  u

Equation (4.54)37H then becomes

2wu l 1 CL 1  (4.56) wttan

The left-hand term in Eq. (4.56)378H can be rewritten in terms of the total pressure difference,

as defined in Eq. (4.3):379H

2Pt l  CL 1 uw t tan

l  PuwCtL 1  (4.57) 2tant  

Equation (4.57)380H expresses the total pressure difference achieved by an axial profile cascade when considering the real case of flow with friction and the drag force acting accordingly upon the profile. This equation is essential for the design process of the cascade, since it contains one of the most important design parameters, i.e. the cascade l solidity , and it allows the calculation of the optimum solidity, according to the t prescribed pressure. On considering the ideal flow without friction, when the gilding angle is zero, then  l PuwC (4.58) tL2  t In the present work, the cascade solidities were fixed to the values of the reference

impeller (presented below, in Figure381H 4-7) and with respect to this parameter a design

choice could not be made. However, in the absence of such constraints, Eq. (4.58)382H can be used for the dimensioning of an ideal cascade. At this stage, it is of interest to see how much of the total pressure gradient of the ideal

cascade is reflected in the losses induced by the drag force, expressed in Eq.(4.52):38H

4. Proposed3B design strategy for axial fans

P w2 loss    (4.59) Puwtm

Equation (4.59)384H can be written in terms of the reaction in Eq. (4.44)385H and the

dimensionless flow in Eq. (4.41)386H as follows:

R2 PPloss t (4.60)  At this point, a second performance parameter of the cascade can be defined as the ratio between the total pressure difference of a real cascade and the total pressure difference of an ideal cascade, i.e. total-to-total efficiency:

2 PPtloss R tt 1  (4.61) Pt 

Similarly to Eq. (4.48),387H the efficiency formulated above is maximum for   0 . The nature of the fan applications of interest for the present work imposes low-pressure regimes, and at low-pressure, attributing values to the gliding angle, others than zero, will only cause additional losses. Hence the present study assumes   0 . However, the possibility of including in the design process other values of  is incorporated in the mathematical routine used for the profile computation, and blade shapes according to such an assumption can be computed.

4.7.3 Profiling the camber line

Another parameter central for the design process is the camber angle,  . Basically, the

resulting shape indicated in Figure38H 4-4 is the camber line and by attributing some thickness around it, a profile is obtained. The thickness distribution is again an important design choice and one has to carefully weigh whether profiling really pays. Most design methods for axial fans use a variable thickness for the profile (profiling the blade shape), according to the thickness distribution functions existing in the literature, i.e. NACA and

British-C4 series, Wallis [109].389H However, such methods were originally proposed for axial flow water turbines, where the operating regime requires high pressures. For axial fans, where the pressures are in the range of thousands of Pa, a variable thickness along

the camber is unnecessary since according to Eckert [34],390H when measuring two impellers

4. Proposed3B design strategy for axial fans

operating at low-pressure, one with airfoil profiles and the other with constant thickness

(thin profiles), their performances are identical, as shown in Figure391H 4-6.

Figure 4-6 Performance curves for constant thickness blade and an airfoil, Eckert [34]392H Another restriction for profiling results from the narrow pitching induced, i.e. the distance between two consecutive profiles in a cascade is considerably reduced, and thus, the cross-section becomes fairly restricted due to profiling, and the high speeds generated, with the corresponding high friction, often offset the advantages of profiling. Hence, for the calculated profiles, constant thickness will be applied, thus introducing the last assumption made at the design stage: due to constant thickness distribution, there will be no point of maximum thickness, and hence the camber angles, at all points on the profile, will coincide with the computed blade angles.

4.8 Design Solver (DS) In the previous sections, all the design assumptions required to compute the blade shape were made. For summary purposes and also to help with a better understanding of all the steps of the proposed design strategy, a simplified flow chart of the design steps is

presented in Figure39H 4-7.

4. Proposed3B design strategy for axial fans

Step 1: Mean-line calculations  Velocity triangles

Axial + Shockless entry  Inlet Maximum flow rate condition + blade

blade angle1 loading considerations  Outlet blade angle at the hub 2,h

Step 2: Meridional (3D) flow analysis

- indirect design problem - arbitrary vortex-flow assumption - parameterization of the total pressure difference in span-wise direction

 Outlet blade angle at specified section  2 r

Step 3: Blade angle distribution (parabolic)

Step 4: Profile cascade dimensioning

- initial design value for the gliding angle   0 - constant thickness  camber angles = blade angles l - initial values for the cascade solidity correspond to the reference t impeller

No. of blades z [–] Characteristic Pitch t [mm] Chord l [mm] section [mm] 8 147 (hub) 2 r 121 187 t  130 227 z 137 280 (tip) 144

Step 5: Design Solver  Blade Shape

Figure 4-7 Simplified flow chart of the design steps All of these steps have to be solved for each individual cascade. The DS incorporates all the design parameters and their derivation into a mathematical routine, which delivers the

blade profile at the specified section, as indicated in Figure394H 4-8.

4. Proposed3B design strategy for axial fans

START

Cascade Parameters

air,rmmrmm hub 147 , tip  280 3 nrpmQms3000 ,design 4 /

zbl,,1  r

Velocity triangles urn 2 Q w  design m 22  rrth

wm wuaxialentryu11tan wu1

NO YES Hub section x 12 n 22rtip 1.35 rx--1 rrhh r   90deg tan rw r 2,h 2rm hub

wm tan 22wu wu2

(, mmy )

(continued)

4. Proposed3B design strategy for axial fans

 ()y  Ay2  By C

yy 12 

yymm 

yy21 

 2  y    i  1

x yytan  

xii xx1 

yx  y

22 lxxyy22121

(continued)

4. Proposed3B design strategy for axial fans

(, mmy )

ll12

w wy m sin y max wy w DF  2 max wy

 22 PwwPucsuutu12 ,  2 2

Fax P s bt, F t  w m bt w u12  w u

  0   0

D  0

www1  P  0  muu, R 12 loss uu2 2 ss 1 R Pwbtloss m PPloss t, D  tt 1  w  1 2 Psfr, R R ss,1 tt  P R  s 1   

Figure 4-8 Mathematical structure of the DS

4. Proposed3B design strategy for axial fans

As already stated in the previous section, the proper blade shapes for the fan application presently investigated are the thin profiles, since it was shown that, at low-pressures, assuming a variable thickness around the camber line does not bring any improvements in the performance of the impellers. For such thin profiles, all design assumptions are made for the case of the frictionless flow   0 and accordingly, the DS completes its

routine on the right branch of the mathematical structure depicted in Figure395H 4-8. However, for different impeller applications, such as the axial compressors, which operate at considerably higher pressures, airfoils are more appropriate according to the

available literature, Aungier [6].396H In this case, assuming at the design stage that   0 might deliver profiles with higher aerodynamical performance and an optimum value of  can be determined by switching the profile calculations to the left branch of the DS.

4.9 DS output

According to the cascade dimensions mentioned in Step 4 in Figure397H 4-7, the resulting profiles, in x, y coordinates, for the investigated models, i.e. x  0 , x 1 and x  2 , and

also the reference profiles, are presented in Figure398H 4-9.

4. Proposed3B design strategy for axial fans

inlet

Figure 4-9 Computed camber lines It can be observed that at the hub section, the camber lines of the three models are identical, since the inlet angle is always given by the axial entry condition and the outlet angle is always fixed at 90o. The differences in the computed profiles become obvious at some distance from the hub, and half-way through the span the characteristic shape of the camber lines is noticeable, i.e. “S” shape (see sections r227 and r280).

The solidity (the ratio l/t) of the reference varies with the radius, as indicated in Figure39H 4-10, and it is kept constant for all the models.

4. Proposed3B design strategy for axial fans

Figure 4-10 Variation of the solidity with the blade radius The computed blade angles, for all three designs, and also the angle distributions of the

reference profiles, are presented in Figure40H 4-11.

inlet

Figure 4-11 Blade angle distribution

4. Proposed3B design strategy for axial fans

Again, the remark regarding the identical angle distribution at the hub sections applies. Moreover, it can be observed that the reference cascades are characterized by large angle gradients, from the inlet to the exit from the cascade, especially from mid-span on, and the immediate result of such a design assumption can be observed in the relative flow

velocity distributions, as indicated below in Figure401H 4-12.

Figure 4-12 Relative flow velocity distribution The reference model is characterized by velocity distributions with huge gradients, very unlikely to be achieved by the actual flow, according to Carnot’s basic principles for hydraulic machinery to achieve its maximum efficiency. One of these principles states that: “the fluid flow should be such that there is no decay, no turbulence, and no sudden velocity reduction”, Epple [38]. The proposed designs, independent of the corresponding pressure variation, have the calculated velocities in a fairly uniform range. This is a first indication of the

4. Proposed3B design strategy for axial fans

improvements which can be achieved with the proposed design strategy, right from the design stage. The same conclusion can also be drawn by analyzing the values of the

diffusion factor, as expressed in Eq. (3.19),402H shown in Figure403H 4-13.

Figure 4-13 Variation of the diffusion factor DF with the radius All of the above results underline the advantage, at least from a design point of view, of employing the proposed design strategy for the derivation of the blade profiles according to the operational requirements of the impeller, rather than to use predefined profiles, which may have been derived for completely different flow conditions, with pressure and/or velocity values other than those required during normal operation. However, this statement is made, as already mentioned, strictly from the design point of view, and it remains to be seen whether the proposed models perform better according to the flow analysis, and this will be discussed in the following chapter.

5. Numerical4B flow analysis

5 Numerical4B flow analysis

Previously, the first key element of the design strategy was presented, i.e. a design solver which computes the optimum blade profile for axial fans, based entirely on the impeller specifications, and does not make use of predefined profiles, such as those in airfoil databases. The next important step in the design process is numerical flow analysis, and the present work makes intensive use of CFD for the investigation of the proposed designs. There are many advantages in considering CFD as an integrated part of the design and optimization process. First, CFD presents the perfect opportunity to study specific terms in the governing flow equations in a more detailed fashion. Second, CFD complements experimental and analytical approaches by providing an alternative cost-effective mean of simulating real fluid flows and substantially reduces lead times and costs in designs and production compared with an experimental-based approach. With the technological improvements and competition requiring a higher degree of optimal designs and as new high-technology applications demand the precise prediction of flow behaviors, CFD is becoming an integral part of the engineering design and analysis environment and it is intensively used to predict the performance of new designs before they are manufactured.

5. Numerical4B flow analysis

Figure 5-1 CFD analysis frame work. Adapted from Tu and Liu [103]40H

An outline of the most important steps involved in full CFD analysis is depicted in Figure405H 5-1. This is also the sequence that will be followed next.

5.1 Mathematical model

In the previous chapter, a design system for axial fans was presented, which basically consisted of a quasi-three-dimensional blade design module to generate the initial blading geometries (detailed design stage), after a mean-line performance calculation was completed (preliminary design stage). This design system was employed to derive an optimum solution for a reference impeller (RI), i.e. a baseline model of an axial fan currently used in the automotive industry for engine cooling purposes. This impeller was made available by the manufacturer, and for flow investigation purposes, the CAD model of the blade was also provided. For the present investigation, four mathematical models of axial fans were derived, corresponding to x  0 , x 1 and x  2 , and also the reference blade. All four models are characterized by the same constructive dimension: Dmmhub  294 , Dmmtip  560 (corresponding hub ratio 0.5), and all impellers have eight blades. Since the reference blade was characterized by a thickness of approximately 3mm, when building the CAD model of the new blades, a constant thickness of 3.2mm was applied symmetrically to the

camber lines presented in Figure406H 4-9.

5. Numerical4B flow analysis

Exemplary depictions of the mathematical models for the reference impeller and x  2DesignIII are presented bellow:

a) b)

Figure 5-2 CAD models: a) reference impeller; b) x=2(Design III) The characteristic “S” shape of the new designs can be readily observed, especially towards the tip section of the rotor, while the reference profile is characterized by circular profiles. The CAD models of the three new designs were built with Pro/Engineer

Wildfire 3.0 [121].407H All four impellers were placed inside a pipe with a tip clearance of approximately 3.5% from the chord length at the tip section, i.e. 5mm, and hence Dmmpipe  570 . To bring the simulation results closer to real-life operating conditions, two extra components were added to the CAD model: at the inlet of the impeller a rig section long enough that the flow entering the impeller domain can be considered fully developed (the length of the rig section was chosen as 3Dpipe ), and at the outlet (ambient) another pipe with a length

2Dpipe was included. An exemplary depiction of the CAD models is shown in Figure408H 5-3.

5. Numerical4B flow analysis

Figure 5-3 CAD model of the full system

5.2 Mesh generation

Following the flow chart depicted in Figure409H 5-1, the next step in the numerical flow analysis is the mesh generation of the mathematical model, and probably the most important matter to be addressed at this level is the mesh-independent results. Theoretically, the errors in the solution related to the grid must disappear for an increasing number of grid points. Hence, also from a theoretical point of view, the finer the mesh, the more accurate are the solutions, and often it is very difficult to determine the level of fine enough grids, owing to computational time and resources issues. Hence the motivation of the following investigation was the determination of the optimum grid structure and size so that mesh-independent results are obtained, with no limitation from the available computational resources.

All grids were generated using ANSYS ICEM 11.0 [118],410H applying the Generalized Grid Interface (GGI) capability of the software. GGI makes it possible to independently generate a set of meshes for different sections of a problem using any tools or mesh structure within each component. The individual grids can then be combined in a single model. For the present models, due to the complexity of the impeller models, tetrahedral

5. Numerical4B flow analysis elements were chosen. However, tetrahedral cells are not desirable near walls if the boundary layer needs to be resolved because the first grid point must be very close to the wall while relatively large grid sizes can be used in the directions parallel to the wall. These requirements lead to long thin tetrahedra, creating problems in the approximation

of diffusive fluxes, Ferziger and Perić [39].41H For this reason, it is preferable that, during the meshing process, first a layer of prisms or hexahedra near solid boundaries is generated, starting with a triangular or quadrilateral discretization of the surface, and on top of this layer, a tetrahedral mesh is generated automatically in the remaining part of the domain. This, however, depending on the number of the prismatic layers generated, relates again to the mesh size issue.

Several meshes of the CAD model presented in Figure412H 5-3 were generated, with and without the prismatic layers, and the essential parameters for the fan performance, i.e. torque in the shaft and the static pressure at the inlet of the impeller, were monitored

closely. The results are presented in Figure413H 5-4.

Figure 5-4 Grid-independent results study (monitored parameters for Qdesign) For greater transparency, the investigated grid characteristics are summarized in the table below.

5. Numerical4B flow analysis

Characteristics Number of grid points Prismatic Layers Mesh 1 approx. 895,000 none Mesh 2 approx. 1,455,000 none Mesh 3 approx. 1,700,000 5 layers, initial height 0.09, exponential growth from the wall Mesh 4 approx. 2,000,000 10 layers, initial height 0.02, exponential growth from the wall Mesh 5 approx. 2,200,000 10 layers, initial height 0.09, linear growth from the wall Mesh 6 approx. 2,250,000 none Mesh 7 approx. 3,000,000 10 layers, initial height 0.09, exponential growth from the wall Mesh 8 approx. 3,500,000 none Mesh 9 approx. 5,500,000 7 layers, initial height 0.09, exponential growth from the wall Table 5-1 Mesh characteristics It can be observed that both the pressure and the torque, for the meshes which do not include prismatic layers, have strong peaks, and a reasonable trend for their variation as a function of the grid size can not really be established. This aspect is avoided by introducing the prismatic layers into the mesh generation process, and the variation of the monitored parameters becomes fairly stable with increasing number of grid points.

Another parameter essential for the near-wall treatment isY , which basically represents the dimensionless distance of the first node away from the wall, and for correct assessment this value should be as small as possible. Considering all these delicate matters of the near-flow wall treatment, Mesh 9 is found appropriate and hence the grid

characteristics mentioned in Table41H 5-1 will be employed in the following investigation. A detail of the mesh around the essential parts of the flow domain, i.e. blade and hub, is

depicted in Figure415H 5-5 and Figure416H 5-6.

5. Numerical4B flow analysis

Detail A

Figure 5-5 Final mesh

Detail A

Figure 5-6 Mesh detail indicating the smooth transition from structured to unstructured grid

5.3 Numerical models and boundary conditions

The numerical simulations were carried out with the commercial code ANSYS CFX 11.0. The simulation type was set initially to be steady. The fluid flow was set to be viscous (air properties as an ideal gas were considered). The flow was solved with the Navier– Stokes equations, assuming conservation of mass, momentum and energy.

5.3.1 Flow physics

wD Typical values for the Reynolds number, calculated as Re  m tip were around  800,000. One of the main problems in turbulence modelling is the accurate prediction of

5. Numerical4B flow analysis flow separation from a smooth surface. Standard two-equation turbulence models often fail to predict the onset and the amount of flow separation under adverse pressure gradient conditions. To avoid this problem, the model used in the present computations of

the air flow is the Shear Stress Transport model, Menter [77].417H The model works by solving a turbulence/frequency-based model (k-ω) at the wall and k-ε model in the bulk flow. A blending function ensures a smooth transition between the two models. The interface between the different frames of reference is taken to be a Frozen Rotor. The Frozen Rotor model has the advantage of being robust, using less computer resources than the other frame change models, e.g. the stage interface model, which is not suitable for applications with tight coupling of components and/or significant wake interaction effects and may not accurately predict loading. The Frozen Rotor model treats the flow from one component to the next by changing the frame of reference while maintaining the relative position of the components. This model must be used for non-axisymmetric flow domains, such as impeller/volute or classifier/casing.

5.3.2 Boundary conditions

An inlet boundary with specified mass flow was applied to the rig domain, flow direction normal to the boundary condition and medium turbulence intensity (5%). A rotational speed of 3000 rpm was applied to the fan domain. The outlet boundary was applied to the ambient domain by specifying a 25oC temperature (air density 1.05 kg/ m3 ). Typical Mach numbers were around 0.25 (for Mach numbers below 0.3, the fluid can be

considered incompressible, Anderson [5]).418H Several simulations were carried out for all models by varying of the flow rate at 2, 3, 4, 5, 6 and 8 ms3 / .

5.3.3 Convergence history

The simulations were completely converged; all runs reached the convergence criterion (residual type RMS and residual target to default value 10–4). The convergence criterion value can be increased, but this means also a considerable increase in the computational time required and not at the cost of significant differences in results. Further, during the simulations, the most important physical quantities, such as flow rate, pressure and efficiency, were monitored and it was observed that all these relevant quantities

5. Numerical4B flow analysis converged properly with the 10–4 residual target. All simulations were solved by parallel running on 16 CPUs and the typical computation time required for full convergence is in

3 wall clock seconds of around 510 [122].419H A detail of the convergence history is depicted

exemplarily in Figure420H 5-7.

Figure 5-7 Convergence history

5.4 Appropriate performance indicators

A large number of definitions of the efficiency of turbomachines have been given in the literature, and the correct assessment of turbomachines, especially when it comes to comparing the performance of more rotors, depends on defining appropriate performance indicators. For an axial fan with casing, where the power input to the rotor is in fact the power to the shaft, the efficiency can be defined as total() hydrodynamic energy input to fluid in unit time Efficiency  power input to coupling of shaft In other words, the efficiency of the impeller can be written as the ratio of the hydraulic power to the shaft power:

5. Numerical4B flow analysis

hydraulic power Efficiency  (5.1) shaft power Ideally, the two parameters should be equal so that the efficiency of the machine is 1. The shaft power developed by the impeller can be written as the product of the angular speed and the required torque:

PMshaft   (5.2) If the total-to-static pressure difference between the inlet and the outlet of the impeller is measured, then the hydraulic power is

PPQhydraulic s When the total pressure difference across the impeller is considered, then the hydraulic power becomes

PPQhydraulic t Accordingly, two efficiencies, conveniently named total – to static and total – to – total efficiencies, can be defined: PQ   s ts M (5.3) PQ   t tt M Also of high relevance for practical use is the polytropic efficiency (small stage efficiency), defined as  1ln   (5.4) poly  ln p T where the air ratio is   1.4 ,  2 and   2 p1 T1

5.5 Optimum span-wise pressure distribution

After post – processing the results of the converged numerical simulations, the following system characteristics for the investigated models were obtained:

5. Numerical4B flow analysis

Figure 5-8 a) Variation of the static pressure difference with the flow rate; b) variation of the torque with the flow rate

In Figure421H 5-8a, the system characteristic curves for all four models are plotted. It can be observed that, while the proposed designs are characterized by similar values of the static pressure difference between the inlet and the outlet of the impellers, the reference model

has higher pressure values. Since, according to Eq. (5.3),42H the second parameter, which

essentially influences the efficiency of the impeller, is the torque, in Figure423H 5-8b the resulting torque along the rotation axis is plotted. Again, similar values are obtained for the proposed designs, and much higher values for the reference. There is one point, however, on both curves, in the behavior of the reference where a sudden decrease in pressure and torque occurs. Even so, this does not influence the efficiency trends, as

indicated in Figure42H 5-9.

Figure 5-9 Total-to-static efficiency

5. Numerical4B flow analysis

In Figure425H 5-9 the total-to-static efficiency, as expressed by Eq. (5.3),426H is plotted. As expected, no substantial difference can be observed in the performance of the proposed designs, at least for the first interval of the investigated flow range. For higher flow rates however, this is no longer the case, since the differences in the calculated efficiencies become substantial, i.e. Design III has an absolute increase in efficiency of 16% for 8 m3/s, compared with Design I. The same can be concluded also from the total – to – total

and polytropic efficiency curves, as shown in Figure427H 5-10, and absolute increases of 13% and 18%, respectively, for Design III are obtained.

Figure 5-10 a) Total-to-total efficiency; b) polytropic efficiency At this point, the advantage of considering at the design stage the possibility of a non-free vortex flow (i.e. assuming that in the span-wise direction the total pressure is not constant) becomes obvious. Employing such a design assumption impacts directly on the extension of the flow range under which the fan can effectively operate, since, according

to Figure428H 5-9 and Figure429H 5-10, for higher flow rates, Design I – characterized by the free- vortex flow assumption – shows a rapid decrease in efficiency, whereas Design III, corresponding to x = 2 non-free vortex flow assumption, performs substantially better. Since in practice fans often operate far from the design point (and often with low

efficiency), Bolton [14],430H it therefore seems desirable to choose the design that, for an interval of flow rates, performs better not only at the design point but also away from this value. Hence it can be concluded that for the investigated fan specifications, i.e. with a hub ratio of 0.5, the best performing design is delivered by the x=2 design assumption, and

5. Numerical4B flow analysis therefore the optimum span-wise pressure distribution corresponds to a parabolic increase with the hub ratio, from hub to tip:

2 Ptr, rtip 1.35 21rrh (5.5) Prth,  hub Further more, on comparing the performances of the suggested design and the reference impeller, significant absolute increases can be observed for Design III: up to 18% in total-to-static efficiency (which is the relevant performance indicator), up to 4% in the total-to-total efficiency, and up to 9% in the polytropic efficiency. Hence, as a solution to the optimization problem of the reference impeller, Design III is suggested.

Note: since the obvious trend in the variation of the x parameter in Eq. (4.30)431H is to increase it, a fourth design, corresponding to x  2.5 , was investigated and it was observed that a further increase in the pressure distribution (more than parabolic with rrtip/ hub ) is not recommended, since decreases in all performance indicators, compared with x  2 , were obtained.

5.6 Profile analysis

In the previous section, the best performing solution to the optimization problem of a reference impeller of known dimensions was found, and it was shown that, for the indicated constructive dimensions (hub ratio 0.5), the optimum variation of the total pressure difference along the blade can be parameterized according to the equation

2 Ptr, rtip 1.35 21rrh (5.6) Prth,  hub

The efficiency curves presented in Figure432H 5-9 and Figure43H 5-10 show substantial increases in the performance of Design III, and represent the integrated values of the efficiency at the cascade level. They can therefore be referred to as a quantitative analysis. A qualitative analysis may be the flow aspect around the investigated blades, and this

matter is addressed in Figure43H 5-11. Even though, from the aspect point of view, the flow around the proposed designs is very similar, and compared with the reference case appears much more attached to the blades, it is very difficult to draw conclusions about the performance of the computed profiles from this cascade perspective. Hence the motivation of the following section is to perform a thorough flow analysis, directly on the

5. Numerical4B flow analysis single profiles, for both Design III and the reference case, so that the best performing profile, from the aerodynamic point of view is determined.

reference

Design I (x=0)

Design II (x=1)

Design III (x=2)

Figure 5-11 Velocity streamlines around the investigated models for Qdesign at r = 227 mm

5. Numerical4B flow analysis

5.6.1 Flow domain around the profiles

In order to carry out the flow analysis over the investigated blade shapes, isolated profiles were simulated so that the aerodynamic performance of the single profile was captured. For all characteristic sections (hub, tip, and the two intermediate sections) the profiles of both blades (Design III and the reference) were investigated, and the flow domain around the profiles was large enough that the theoretical considerations of the undisturbed flow far in front of and far behind the profile were satisfied. Before choosing the specific lengths of the CAD model of the full flow domain around the profiles, several simulations were carried out so that the dimensions of the domain at which the cascade influence becomes negligible were chosen. The thickness of the profile was approximately 10 mm and the flow was investigated on the center-line of the profile, where no influence from the margins of the side walls was observed. The length of the

chord was kept to the values indicated above in Figure435H 4-7. A typical CAD of the profiles

and the flow domain around them is depicted in Figure436H 5-12.

Figure 5-12 Model of the flow domain around the considered profiles Special attention was paid to the setting of the boundary conditions. Excellent convergence residuals were obtained by setting the following boundary conditions. At the inlet section, the specified flow velocity wm is

5. Numerical4B flow analysis

Q wmsdesign 22.4 / (5.7) m 22  rrth At the outlet section of the domain, opening boundary to the ambient pressure was applied; on the lateral surfaces of the domain, symmetry boundary was set; for the top and bottom surfaces again symmetry was applied since it was observed that, after a height of the domain of 20 times the chord length from top to bottom, the difference in results between applying symmetry or opening boundary was negligible. The simulations were carried out for viscous flow with air as ideal gas.

5.6.2 Mesh generation

In order to calculate a qualitative mesh able to solve accurately the flow around the profile, and especially the delicate problem of the flow in the near-wall regions, a grid study was carried out. For this, three different grids were initially generated on the CAD

model depicted in Figure437H 5-12: a fine tetrahedral mesh (approx. 120,000 cells), a course hexagonal mesh (approx. 65,000 cells), and a fine hexagonal mesh (approx. 200,000 cells). The hexagonal grids were obtained by dividing the flow domain into blocks, which were then subdivided into grids with good properties, and O-grid structured topology was applied for the block corresponding to the profile. The two different hexagonal grids were calculated by varying the height of the elements along the resulting edges (smaller heights for the fine mesh).

5. Numerical4B flow analysis

Figure 5-13 Velocity streamlines for the reference profile at r = 280 mm: a) tetrahedral grid; b) course hexagonal grid; c) fine hexagonal grid On comparing the flow images, it can be immediately seen that whereas the streamlines for the tetrahedral grid appear smooth and completely attached to the profile, for the course hexagonal grid the trajectory deviates slightly and the streamlines are distancing from the profile, and finally, in the case of the fine hexagonal grid, an obvious flow

detachment from the contour is plotted. As already mentioned in section 5.2,438H in the near- wall regions, the structured hexahedra or prismatic layers are preferred, since they resolve the boundary layer more accurately, and for this reason, the fine hexahedra was used for

further calculations. A detail of the grid generated around the profile is depicted in Figure439H 5-14.

5. Numerical4B flow analysis

Figure 5-14 Fine hexagonal grid for the reference profile at r=280 mm

5.6.3 Numerical results

A first analysis of the numerical results is the qualitative appreciation of the pressure

contours, as depicted in Figure40H 5-15. Pressure plots on the blade are helpful in identifying the different sources of losses in the system, such as the ones caused by sound sources, i.e. the leading edge of the blade and areas close to tip section, due to the tip clearance,

Carolus et al. [91].41H

r147 r147

TE LE

r187 r187

5. Numerical4B flow analysis

r227 r227

r280 r280

Figure 5-15 Pressure contours around the reference (left) and Design III (right)

It can be concluded from Figure42H 5-15 that at all characteristic sections, the suction on the reference profile is much more pronounced than for Design III. Additionally, the size of the suction on the reference profile increases with increase in the radius, indicating an increase in the boundary layer thickness, causing an associated increase of the diffusion on the profile. At the same time, the suction on the Design III profile decreases with increase in the radius and the contour plots indicate fairly uniform distributions on the

upper surfaces, causing the small diffusion predicted at the design stage (see Figure43H

4-13). Moreover, the “stagnation point” (see Figure4H 3-7) is much more pronounced on the reference profiles than on the Design III profiles, indicating that the latter are closer to

“the minimum loss situation”, Lewis [68].45H Such pressure contours can be numerically quantified with the help of the so-called pressure coefficient, defined as p  p C  1 (5.8) P 1 w2 2 1 where p represents the static pressure along the profile, and p1 and w1 are the static pressure and velocity, respectively, upstream of the cascade.

5. Numerical4B flow analysis

LE Pressure (lower) surface

Suction (upper) surface

Figure 5-16 CP distributions along the investigated profiles

CP, as defined in Eq. (5.8),46H is normally plotted against the dimensionless profile length, defined as the ratio between the position of the camber in the flow direction and the chord, x / l .

The values of CP are always positive on the pressure (lower) side of the profiles and negative on the suction side. As expected, the CP distributions agree with the results

depicted by contour plots in Figure47H 5-15: at all sections, the Design III profiles are characterized by smaller CP values compared with the reference profiles, indicating smaller losses through diffusion in the flow around the profiles and, implicitly, higher performance for the x = 2 shapes. There is one exception, however: at the hub section, CP for Design III is higher in the region corresponding to the trailing edge, due to the forced outlet angle, i.e. the maximum flow rate condition and, thus, 2  90deg .

5. Numerical4B flow analysis

Figure 5-17 Dimensionless velocity distribution Finally, the performance of the investigated profiles can be analyzed also from the

velocity distribution point of view, as shown in 48HFigure 5-17. To maintain the consistency of the dimensionless analysis, a velocity coefficient is defined as the ratio of the flow velocity around the profile to the velocity upstream of the cascade, ww/ 1 . At all sections, the velocity coefficients corresponding to Design III indicate smooth variations along the profile length, with no sudden gradients, compared with the reference shapes. It can be therefore concluded that, from the profile analysis point of view, the design strategy delivered aerodynamically superior shapes compared with the reference– classical circular profiles with a symmetrical thickness distribution.

6. Experimental5B validation of the proposed design strategy

6 Experimental5B validation of the proposed design strategy

The most important outcome of the previous chapters is the substantial increase in the efficiency of the newly designed impellers compared with the reference solution. After a

careful numerical flow analysis, it was concluded in chapter 549H that, for the investigated hub ratio, Design III, corresponding to the x  2 non-free vortex flow assumption, had the best performance according to all performance indicators analyzed. Therefore, for the final comparison with the reference impeller, this design was suggested. However, the efficiency curves presented were purely numeric, and even though modern CFD tools achieve excellent flow predictions, some idealization of the flow phenomena with a significant impact on the performance is nevertheless being made. Hence the motivation for the following investigation was to capture experimentally the efficiency increase of the proposed design.

6.1 Investigated impellers

As already mentioned above, two impellers were experimentally investigated, i.e. the reference impeller, made available by the manufacturer, and a prototype of the proposed design.

In Figure450H 6-1a, the reference impeller is depicted, and in Figure451H 6-1b, a detail of the blade shape at the tip is shown.

6. Experimental5B validation of the proposed design strategy

Figure 6-1 Detail of the reference impeller: a) the full rotor; b) detail of the blade shape at the tip The impeller corresponding to Design III (x = 2) was built through the rapid prototyping

method [119].452H Eight identical blades were manufactured according to the CAD model of

the blade, shown in Figure453H 5-2, and captured in the hub. A detail of the prototype blades

is shown in Figure45H 6-2.

6. Experimental5B validation of the proposed design strategy

Figure 6-2 Prototype blades

Both impellers are characterized by the same hub and tip diameters, i.e. Dmmhub  294 and Dmmtip  558 , and of course, the same number of blades, z  8 . The experimental investigation was focused on the comparison of the system characteristic and efficiency curves of the two models, and hence the pressure and the torque (required at the shaft) were measured for each of the investigated flow rate.

6.2 Experimental facility

The experimental analysis was carried out according to the standard norm DIN 24 163

Part 2 for measurements on axial fans [28]:45H

6. Experimental5B validation of the proposed design strategy

Figure 6-3 Standard norm for measurements on axial impellers, DIN 24 163 Part 2 During the measurements, one of the two wind tunnel facilities available at LSTM

Erlangen was used, characterized by a diameter Dmmtunnel  2000 and a total length

Lmmtunnel  6000 , as shown in Figure456H 6-4.

Figure 6-4 Simplified scheme of the experimental setup The wind tunnel is equipped with an inlet orifice where Venturi nozzles with different diameters can be fitted, so that the required inlet flow rate is achieved. For the investigated impellers and their corresponding system characteristics, a nozzle with

Dmmnozzle  400 was chosen. The flow was driven by a centrifugal blower, which was controlled by a frequency converter so that the rotation of the fan could be adjusted. For flow rate control purposes, the wind tunnel is equipped with a jalousies system, and by opening/closing the jalousies, the different flow rates on the system characteristic were achieved.

6. Experimental5B validation of the proposed design strategy

The air stream was guided through a settling chamber consisting of alternate perforated plates and honeycombs to ensure uniform flow conditions at the outlet of the tunnel, which was smoothly connected to the impeller inlet.

At the outlet of the wind tunnel, a pipe with a diameter Dmmpipe  560 and a total length

Lmmpipe 1200 was installed, and inside this pipe, with a tip gap of approximately 1mm in radius, the two impellers were mounted.

In Figure457H 6-5, a detail of the inlet of the impeller is depicted, as it is mounted on the outlet of the wind tunnel.

Figure 6-5 Inlet of the impeller

In Figure458H 6-6, the entire measuring facility is presented, with the pipe attached to the wind tunnel and the impeller discharging into the ambient. The impeller is belt driven by a motor, which can be observed below the pipe, in the measuring scheme.

6. Experimental5B validation of the proposed design strategy

Figure 6-6 Wind tunnel with pipe and impeller mounted

6.3 Measured parameters

As mentioned previously, the measurements were focused on obtaining the experimental

efficiencies of the two impellers, for comparison purposes. As defined in chapter 5,459H the efficiency is a function of achieved pressure difference (static or total), flow rate, rotational speed and required torque to the shaft:  f PQnM,,, (6.1) At the nozzle inlet, both the static pressure at the wall and the ambient temperature were measured, allowing the calculation of the inlet flow rate according to Bernoulli’s

6. Experimental5B validation of the proposed design strategy equation. The variation on the investigated flow range was achieved with the jalousies system. At a length of approximately 300 mm from the outlet of the tunnel and inlet in the impellers, a second pressure tap was placed, which measured the pressure difference between the static pressure at the wall and the ambient, Pinlet . Very close to the impeller inlet and outlet, two additional thermocouples were placed, so that the temperature difference achieved was calculated. Additionally, torque and rotational speed measurements were carried out for each of the investigated flow rates.

A scheme of the measurement points is depicted in Figure460H 6-7.

Figure 6-7 Scheme of the measuring points on the experimental setup

6.4 Measuring equipment

Each pressure tap was connected to an OEI pressure scanner with pressure tubes. The pressure measurements were made with a Höntzch pressure transducer and the pressure scanner was used to switch from the first pressure channel (at the nozzle) to the second (at the impeller inlet). The torque measurements were done with an HBM T4WA-S3 torque sensor for nominal values between 5Nm and 1kNm. The output was registered with an HBM SCOUT 55 amplifier. Both the pressure transducer and torque amplifier were connected to integrating DVs type DISA 55D31, with voltage output.

6. Experimental5B validation of the proposed design strategy

The impellers were belt driven by an AC motor with a maxim rotation of 3500 rpm and a nominal power of 18 kW, operating at 3000 rpm, and its control was achieved with a frequency regulator. The value of the rotational speed was always maintained at 3000 rpm and for these readings a Hall-effect sensor, with maximum frequency of 10 MHz and TTL signal output, was used. The temperature was measured with K- Type thermocouples.

6.5 Experimental results

After the measurements on both impellers had been performed, the characteristic pressure

and torque values, depicted in Figure461H 6-8 were obtained. It can be observed that, for the investigated flow range, the pressure differences between the ambient and the static pressure at the inlet of the impellers are similar, and in this respect, the behavior of the two fans does not differ very much.

Figure 6-8 Measured static pressure difference However, the same cannot be stated when analyzing the torque measured at the shaft,

presented in Figure462H 6-9.

6. Experimental5B validation of the proposed design strategy

Figure 6-9 Measured torque It can be readily observed that, throughout the entire flow range, the reference impeller is characterized by substantially higher torque values than Design III. In fact, starting from the value of 3/ms3 the absolute increase in the measured torque of the reference is 6%, whereas for 5/ms3 and 6/ms3 , the difference increases to 11%. Having completed the pressure difference and torque measurements, the efficiency of the two impellers can be calculated as the total-to-static efficiency: PQ   s (6.2) total to static M

On plotting the calculated efficiency for the two fans as expressed in Eq. (6.2),463H then the

chart depicted in Figure46H 6-10 is obtained.

6. Experimental5B validation of the proposed design strategy

Figure 6-10 Total-to-static measured efficiency Design III is characterized throughout the entire flow range by a considerable increase in the total-to-static efficiency, and for the interval 46 ms3 / this increase is maintained to 11% absolute difference compared with the efficiency curve of the reference. These differences in the efficiency curves of the two impellers were, of course, expected, since according to the previous two plots, with no significant difference in the achieved pressure, Design III is characterized by much smaller torque values for all flow rates and,

according to Eq. (6.2),465H these are the two most important parameters which influence the efficiency. Essentially, these results can be generalized to the conclusion that the proposed design strategy delivered a “torque-optimized” axial fan.

In Figure46H 6-11, the measured temperature differences for both fans are plotted.

6. Experimental5B validation of the proposed design strategy

Figure 6-11 Measured temperature differences It can be observed that the cooling capabilities of the impellers are very similar and the temperature gradients achieved differ by a less than 0.5o for all flow rates investigated.

This result, coupled with the efficiency graphs depicted in Figure467H 6-10, confirms that the same cooling effect can be achieved by both impellers, but Design III requires much less energy, thus resulting in direct cost savings.

6.6 Validation of the results

In chapter 5468H of the thesis, based on extensive numerical analysis, the optimum pressure distribution in the span-wise direction, and hence the optimum vortex-flow design assumption, were proposed, and it was concluded that Design III was, according to the numerical prediction, the best performing of the three investigated. These were the premises for the experimental procedures presented in this chapter, where the reference impeller and the suggested design were measured according to the standard DIN norm. The experimental curves for the total-to-static efficiency showed a considerable increase for Design III throughout the investigated flow range, thus confirming the numerical advantage presented earlier. However, the numerical investigations of the four models were carried out for boundary conditions completely different from the actual measuring stand. Hence the motivation of the following analysis was to compare the numerical

6. Experimental5B validation of the proposed design strategy prediction of the behavior of the rotors, for the exact conditions of the experimental setup, with the actual experimental data.

For this, a new CAD model, corresponding to the dimensions in Figure469H 6-4, was built and the flow domain of the impellers was adjusted to the same gap, i.e. 1 mm. The mesh

characteristics were kept at the values indicated in section 5.2.470H Data planes were placed in the exact locations as the measuring pressure taps and thermocouples. All of the relevant parameters that influence the performance of the fan were compared with the measured values, for both the reference impeller and Design III, and excellent _ agreement was found, as shown in Figure471H 6-12 Figure472H 6-14.

Figure 6-12 Numerical and experimental static pressure curves

Figure 6-13 Numerical and experimental torque curves

6. Experimental5B validation of the proposed design strategy

Figure 6-14 Numerical and experimental total-to-static efficiency curves

7. Integrated6B ideal efficiency for axial fans

7 Integrated6B ideal efficiency for axial fans

7.1 The Cordier diagram As mentioned in the introduction to this thesis, probably the most often addressed method in the literature with respect to the concept of maximum efficiency is the Cordier diagram. Essentially, this diagram is based on the head ( ) and flow ( ) coefficients,

which are defined, according to Strub and Eck [32],473H as c   m utip (7.1) 22Yctu2  2 uutip tip where Yt represents the total work achieved by the rotor, and cm is the meridional flow velocity.

100

7. Integrated6B ideal efficiency for axial fans

axial rotor radial rotor

Figure 7-1 Different types of rotors as a function of the flow direction. Adapted from Bohl [12]47H For the following analysis, first the case of a radial rotor described by an outlet diameter

DD2  , as depicted in Figure475H 7-1, will be considered. In this case, the representative flow area is given by the area of a circle with diameter D:  D2 A  (7.2) 4

where utip represents the peripheral velocity at the exit of the rotor; in the case of radial impellers, this is given by the peripheral velocity calculated at the tip section:

uDntip   (7.3) The meridional flow velocity is given by the ratio between the flow rate and the flow area: QQ4 c  (7.4) m A  D2

On inserting Eqs. (7.3)476H and (7.4)47H into Eq. (7.1),478H the expression for the flow coefficient becomes 4Q   (7.5)  23Dn

From Eq. (7.5),479H a first formulation of the outlet diameter of a radial impeller can be derived:

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7. Integrated6B ideal efficiency for axial fans

1 4Q 3 D  21 1 (7.6)  33n  3

The head coefficient in Eq. (7.1)480H can be written as 2Y   t (7.7)  222Dn

From Eq. (7.7),481H a second expression for the outlet diameter of the impeller can be derived:

1 2 2Yt D  1 (7.8)  n 2

Of course, Eqs. (7.6)482H and (7.8)483H have to be equal, and hence

11 32 42QYt 21 1 1 (7.9) 333n  n 2

From Eq. (7.9),48H the following formulation for the rotational speed n can be extracted:

3 1 4 2 2Yt  1 n  131 (7.10) 4Q 242 

Cordier [21]485H defined the second ratio in Eq. (7.10)486H as the “speed” number,  :

1  2   3 (7.11)  4

Inserting Eq. (7.11)487H into Eq. (7.10):48H

3 4 2Yt 1 2nQ n 11 3 (7.12) 22 4 42QY t

Returning to the definitions of the pressure and flow coefficients as expressed in Eq. (7.1)489H , and switching the parameter of interest from the diameter D to the rotational speed n, then the following identity is obtained:

1 4Q 2Y 2  (7.13) 23D 1  D 2

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7. Integrated6B ideal efficiency for axial fans

From Eq. (7.13),490H the following expression for the diameter is derived:

11 21Q 24 D  111 (7.14) 422 2Yt   Similarly to the speed coefficient, Cordier defined a “diameter” number as

1  4   1 (7.15)  2

Inserting Eq. (7.15)491H into Eq. (7.14),492H the following expression for the diameter number can be established:

2Yt    4 D (7.16) Q2 2

The expressions of  and  in Eqs. (7.11)493H and (7.16),49H respectively, are employed in the

Cordier diagram, as shown in Figure495H 7-2. Cordier found that for the optimum pair of speed and diameter numbers  opt, opt , any impeller is characterized by its highest efficiency. On this diagram, the points corresponding to the low-pressure fans were identified as being located mostly in the upper-half, i.e.  opt  0.3. By fitting a trend line

to cover all these points, the curve marked in Figure496H 7-2 was obtained, and this was termed the “probable” curve of maximum efficiency for axial fans. At this point, it becomes very interesting to see how close to this curve, indicated by Cordier’s results for the low-pressure range, the performance of Design III (x = 2) is at the design point. The

corresponding values of  and  for Design III are calculated according to Eqs. (7.11)497H

and (7.16),498H respectively, using the characteristic fan dimensions and measured static

pressure difference, as presented in Table49H 7-1.

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7. Integrated6B ideal efficiency for axial fans

Design III characteristics   0.19   0.3  1.71  1.07

Dmmtip  558 Dmm 294 hub nrpm 3000 3 Qmsdesign  4/

Measured head: PPas 1150

(see Figure50H 6-8) Table 7-1 Calculated speed and diameter numbers for Design III

Figure 7-2 The Cordier diagram

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7. Integrated6B ideal efficiency for axial fans

On placing Design III on the Cordier diagram, according to the calculated values of the

speed and diameter numbers indicated in Table501H 7-1, it can be immediately observed that its efficiency lies exactly on the estimated curve of maximum efficiency, between the curves for 80 and 85 % efficiency. However, due to the very similar pressure differences measured for both impellers, i.e.

reference and Design III, as indicated in Figure502H 6-8, the position of the first impeller in the Cordier diagram coincides with that of the latter:

reference 1.73   reference 1.04 Hence, at least from the Cordier diagram point of view, there is no noticeable difference in the efficiencies of the two rotors, since they are both characterized by the same values of the speed and diameter numbers, and they are already located on the curve of maximum efficiency. Therefore, just from this perspective, there would have been no need to optimize the reference, and this is definitely contradicted by the results presented so far. This diagram uses only the rotor characteristic dimensions and operating parameters, and there is no reference to crucial parameters for the performance of the impeller, such as the torque. For this reason, for two axial fans with identical dimensions, which deliver similar pressures at the considered flow rate, no difference in their performance is obtained as far as the Cordier diagram is concerned. However, by calculating an appropriate performance indicator, i.e. the total-to-static efficiency in

Figure503H 6-10, the advantage of one design over the other can be observed, and thus the need for optimization of the reference. At this stage of the analysis, some inconsistencies of the Cordier diagram, at least from the point of view of axial impellers, should be discussed. First, in the original publication of Cordier, there is no specification of which type of pressure was being measured, either static or total. However, since it is common for fan measurements, it can be assumed that it is either the total pressure or the total-to-static pressure. The total pressure is usually obtained by measuring the total-to-static pressure and by adding then the dynamic pressure at the exit of the volute. This dynamic component is calculated as the ratio of the flow rate to the cross section area. In doing so, the swirl, which is the dominant component of the flow velocity, is neglected. Therefore

105

7. Integrated6B ideal efficiency for axial fans the total pressure calculated such will not differ much from the measured total-to-static pressure. From the above considerations it is clear that any performance indicator should take into account the actual measured parameters and defined accordingly. Second, the definition of the flow coefficient for axial impellers employed in this diagram does not take into account the integral properties of these flow machines. As suggested

by Bohl [12],504H rather different expression of  should be employed, which takes into consideration the fact that the characteristic flow area in this case is given by the ring

formed between the hub and the tip sections (see Figure50H 7-1), and not by the circle with the tip diameter, as it is in the case of the radial rotors:

22 4DDth A  (7.17) axial 4 The flow coefficient should be then calculated as 4Q   (7.18) axial 22 2  DDDntht Due to such inconsistencies, the concept of maximum efficiency for axial impellers cannot be properly quantified by the Cordier diagram. Hence the motivation of the following analysis is to derive a valid formulation of the ideal efficiency of axial fans, considering all crucial parameters that influence the performance of such devices.

7.2 Ideal efficiency for axial fans

In the previous section, it was mentioned that the Cordier diagram, as presented in Figure506H 7-2, links the optimum operating conditions, i.e. flow rate and specific head, with the optimum diameter and rotational speed, and the intersection reveals the point of maximum efficiency. Theoretically, any single-stage turbomachine can be plotted on the Cordier diagram with the help of the dimensionless speed and diameter numbers, as

introduced in Eqs. (7.11)507H and (7.15):508H

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7. Integrated6B ideal efficiency for axial fans

Pt 2 1  4   4  D  opt2 tip Q2 1  2 1 Q  2 opt 2 n 33 44 Pt  2  These numbers are formulated based on the flow ( ) and head ( ) coefficients, defined

at the impeller outlet. In the case of the radial rotor, as depicted exemplarily in Figure509H 7-3, the impeller outlet coincides with the tip section, and thus c   m utip (7.19) 22Yctu2  2 uutip tip

Figure 7-3 Typical velocity diagrams for a radial impeller. Adapted from Epple [37]510H However, this is not the case for the axial impellers, where the flow particles enter and leave the blade at the same radius, and hence the peripheral velocity is constant for a given blade section: uu122 rn (7.20)

In fact, for axial impellers (Figure51H 7-1b), considering only the peripheral velocity at the tip is section is an incorrect evaluation, and instead one should calculate the dimensionless flow and pressure with an integrated value of u over the entire flow area of the impeller, which corresponds to the ring area between the hub and the tip diameters:

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7. Integrated6B ideal efficiency for axial fans

1 rt uurdr 2 (7.21) axial  rr22  thrh

As a result, Eq. (7.19)512H will be rewritten for the particular case of the axial impeller, at one blade section: c   m (7.22) section u 2c   u2 (7.23) section u The expression for the head coefficient can be further developed according to the velocity

diagrams presented in Figure513H 4-1:

2uw u2 wwum2 1  section 21 21 (7.24) uuutan 2

Inserting Eq. (7.22)514H into Eq. (7.24),51H the following expression is obtained:

1  section21 sec tion (7.25) tan 2 Let us consider the maximum reaction, at the cascade level, according to Euler’s equation, defined as the ratio of the static pressure difference that can be achieved to the total pressure difference:

Ps Rcascade  (7.26) Pt

Considering the expressions in Eqs. (4.3)516H and (4.8),517H then the reaction expressed in Eq.

(7.26)518H can be rewritten as

22 111uwuuu222 uw w Rcascade 1 (7.27) 222uu wu2 u u

According to the outlet velocity triangle in Figure519H 4-1:

wm wu2  (7.28) tan 2 This means that

1111wm Rcascade11 sec tion (7.29) 2u tan 22 2 tan

108

7. Integrated6B ideal efficiency for axial fans

According to Eq. (7.29),520H the reaction of an axial cascade is always higher than 1/2, and for very small outlet blade angles, its value approaches infinity. It can be immediately concluded that the reaction is not really a precise performance indicator, and it cannot be employed for the determination of a theoretical limit of performance, at least from the point of view of axial impellers. Instead, different performance indicators have to be considered, and for this it is very useful to analyze the problem from a practical perspective. In the previous chapter, detailed information about the standard measuring techniques for axial fans was given. It was shown that, in reality, the measured pressure was the difference between the static pressure at the wall and the ambient pressure, i.e. a static pressure difference. This statement can be confirmed also from the numerical simulations point of view, and for exemplification, the velocity streamlines behind the impeller are

depicted in Figure521H 7-4. It can be readily observed that even at the outlet of the considered flow domain (approximately 2 m length after the impeller) the swirl is still not decaying, c2 this swirl being the dominant component in the dynamic pressure  2 , where 2

22 2 cc22um c 2.

Figure 7-4 Swirl development after the impeller

109

7. Integrated6B ideal efficiency for axial fans

If one wishes to measure the total pressure difference, as it is included in the various definitions of efficiency, then also the dynamic component (and hence the swirl) would have to be measured, and this, from the practical point of view, would be very difficult. In this sense, a reconsideration of the expression for the static pressure difference is useful, according to the rig measurements conditions, by subtracting the dynamic

component from the total pressure, Epple [36]:52H  PPc2 (7.30) srig,2 t 2 The ideal efficiency, following this consideration, can then be expressed as: P 2 srig, 1 c2 cascade 1 (7.31) Puctu2 2

On further equating Eq. (7.31)523H according to the velocity diagrams, then

2 11wcmu2 cascade 1  (7.32) 22ucu2 u

Equation (7.32)524H can be written in terms of the dimensionless flow and pressure

coefficients, as defined in Eqs. (7.22)52H and(7.23):526H

2 section sec tion cascade 1  (7.33)  section 4 The next step is to express the pressure coefficient in terms of the flow coefficient:

cuwu22 u w u 2 w m1  sec tion  section 2  2  21  21  21 (7.34) uu u utan 22 tan

Equation (7.33)527H then becomes

2 11section sec tion cascade 11   (7.35) 22tansection  2 1 tan 2

Equation (7.35)528H reveals the theoretical efficiency that can be achieved by an axial cascade and it is a direct function of the flow coefficient (flow area and flow rate implicitly) and the value of the outlet blade angle, at the specified cascade radius. To extrapolate from such a formulation, valid only for a blade section, to the full impeller, an integration

similar to Eq. (7.21),529H over the entire flow area, is required:

110

7. Integrated6B ideal efficiency for axial fans

1 rt  2 rdr (7.36) axial rr22  cascade thrh However, this is associated with the difficulty of the variation of the outlet blade angle

with the radius. For the proposed design strategy, this variation is given by Eq. (4.31):530H x 12 n 22rtip 1.35 rx rr hh1 r tan rw r 2rm hub On substituting this into the expression for the cascade efficiency, as defined in Eq.

(7.35),531H the analytical integration with the radius to the overall efficiency of the impeller becomes impossible, due to the complicated formulation. The solution to this problem is to analyze, for known impeller specifications, the variation of the efficiency with the flow rate, for averaged angle values, between the hub and the tip sections. For the prototype considered, an average value between the calculated outlet blade

angles, according to Eq. (4.31),532H where x = 2, was considered:

radius [mm] 147 187 227 280 o 2 [ ] 90 49.68 36.1 29.63 o 2, average [ ] 51 Table 7-2 Averaged angle for Design III

Equation (7.36)53H then becomes



r  11t 2 1 112section  sec tion rdr (7.37) axial  rr22  22tan thrh section 2,average 1 tan 2,average

The corresponding curve for the integrated efficiency according to Eq. (7.37),534H for the

investigated flow range, is depicted in Figure53H 7-5.

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7. Integrated6B ideal efficiency for axial fans

Figure 7-5 Design III ideal efficiency Before commenting on the performance of Design III with respect to the integrated efficiency curve, first some remarks should be made. For this, it is very useful to identify on the performance plot three intervals, covering the range of small flow rates, the design

flow rate, and the higher flows range, respectively, as indicated in Figure536H 7-5. In the first interval, it can be seen that the integrated ideal efficiency curve starts from 0.5, whereas the measured performance of Design III indicates that for zero flow rate the efficiency would be zero. Similarly, in the third interval, the slope of the ideal curve decreases very slowly, while the measured efficiency drops very suddenly, shortly after the design interval. Such large differences, which characterize both the first and third intervals, are due to a series of losses which inevitably appear during normal fan operation, and are not

accounted for in the case of the ideal machine. According to Epple [37],537H such losses include:  shock losses, which appear as soon as the system operates away from the design point  friction losses, which account for the energy dissipation due to the contact of the fluid with solid walls  mechanical losses, caused by disc friction or bearing losses.

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7. Integrated6B ideal efficiency for axial fans

The mechanical losses do not influence the pressure of the system, and hence the hydraulic power. However, they impact heavily on the power input, i.e. the power to the

shaft. Accordingly, the measured efficiency of the system, as expressed in Eq. (5.1),538H will considerably differ from the ideal value. Because of such losses, differences between the ideal curve and the measured efficiency, as in the first and the third intervals, cannot be avoided, no matter what the fan system. Focusing on the middle interval and thus on the design point and the regions around it, it

3 can observed that, for the design flow rate, i.e. Qmsdesign  4/, the efficiency of Design III is less than 9% (absolute percent) below the ideal value, indicating that the considered design calculations and assumptions delivered a design solution in close proximity to the ideal machine, characterized by an infinite number of blades, infinitely small thickness, and inviscid flow (these are the assumptions under which the Euler equation of turbomachinery is formulated). By considering the integral properties of axial impellers, it was shown that the efficiency expectations suggested by the Cordier diagram for the investigated class of impellers are not realistic, and cannot be achieved, not even analytically, due to several inconsistencies with regard to the nature of the flow in such machines. A correct formulation of the maximum/ideal efficiency concept for axial fans should be the result of the integration, over the entire flow area, of the performance of each individual cascade, and the present analysis suggests one analytical method to do so.

It can be concluded, based on the performance plot depicted in Figure539H 7-5, that the proposed layout and design strategy delivered an efficient design model for axial fans, as initial solution to the existing optimization problem, based on the correct theoretical treatment of the flow in such flow machines, on the one hand, and also incorporates a series of innovative design assumptions delivering high-efficiency blades, on the other. Without any further improvements (related to geometry or by employing genetic algorithms), the outcome is a fan already operating close to its theoretical maximum achievable performance.

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8. Conclusions7B and outlook

8 Conclusions7B and outlook

In the present work, a layout and design strategy was developed and applied to axial ventilators/blowers, focused on delivering high-performance design solutions. This strategy may be employed as an optimization tool in this engineering field. Essentially, this procedure is a combined inverse–direct design method, which computes the optimum blade profiles, according the fan specifications and operational requirements, and does not make use of any predefined profiles, such as airfoils. Central to the proposed design strategy was the so-called “Design Solver”, in which a mathematical routine was implemented to solve the flow in three consecutive steps: one- dimensional (mean-line calculations), two-dimensional (cascade calculation of the blade), and three-dimensional (meridional analysis). Most design strategies in this field rely entirely on the extensive data on airfoil profiles and make use almost exclusively of the free-vortex flow assumption. The present work was concerned with finding, for given fan dimensions, the optimum vortex-flow consideration, and the possibility of a non-free vortex flow was also included in the design process. Moreover, according to the vortex assumption, parameterization of the pressure distribution in the span-wise direction was possible, which later dictated the outlet blade angle variation with the radius. All new designs were derived on the basis of the so-called “reference impeller”, an axial fan with a hub ratio of 0.5, currently used in the automobile industry for cooling purposes. Thorough numerical investigation showed that for such fan application (high-flow and low-pressure), the best performance during operation was achieved by the design corresponding to the non-free vortex flow assumption, i.e. Design III. Moreover, compared with the efficiency curves for the reference, substantial increases, of up to

114

8. Conclusions7B and outlook

18%, were obtained, thus proving the optimizing potential of the suggested layout and design method. The advantage of the proposed strategy came about also from aerodynamic analysis performed on the reference profiles and Design III profiles, and it was shown, by means of dimensionless pressure and velocity plots, that the latter cause less diffusion in the flow around them, and thus smaller losses. Modern Computational Fluid Dynamics tools achieve excellent flow predictions, and due to time- and cost-reduction considerations, experiments are nowadays being replaced more often by such numerical methods. However, for validation purposes, the reference impeller, which was made available by the manufacturer, Alu Automotive GmbH, and a prototype of Design III were investigated experimentally according to the standard DIN norm. The relevant parameters for the efficiency were measured, i.e. the static pressure and the torque required at the shaft to drive the system. It was concluded that although the two impellers produce similar pressures, the torque required to drive the Design III impeller was substantially reduced compared with the reference. Of course, this was immediately observed in the total-to-static efficiency curves, where an absolute increase of 11% for Design III was obtained. After analyzing the cooling capabilities of the two impellers, it was observed that similar cooling effects were achieved by both fans, but with considerable less consumption by Design III, resulting in immediate decreases in energy and costs. Another aspect central to this work was the analytical formulation of the integrated ideal efficiency for axial fans. This concept was derived as a response to the incapacity of the classical Cordier diagram to predict the actual performance of axial impellers operating in the low-pressure regimes, due to inconsistencies in the definitions employed. The suggested analytical formulation can be extended to any type of axial impeller of known dimensions and design characteristics, thus offering a reliable method to determine the maximum performance achievable by any fan. It was shown that, at the design point, the proposed layout and design strategy delivered a fan operating already in the close proximity to the ideal machine, quantified by the integrated ideal efficiency curve. This result represents a fully optimized impeller and underlines the validity of the proposed layout and design method. The method is

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8. Conclusions7B and outlook

essentially based on the correct theoretical treatment of the flow in axial fans, and further improvements in the efficiency of the proposed design can be achieved by employing

genetic algorithms [104],540H or by improving the geometry of the system, i.e. hub or shroud

[78],541H thus bringing the impeller closer to the ideal machine. It has to be kept in mind, however, that these latter methods are more costly and require a good starting solution. The latter can be given by the present method. The presented layout and design strategy, although applied to axial fans, has potential to be employed for other types of axial impellers operating under different pressure regimes, such as the axial compressors. The proposed design solver handles flow calculations under both frictionless flow and flow with friction hypotheses. In the present work, the frictionless flow assumption was employed since it was found appropriate for the case of low-pressure axial fans. However, for different impeller application, the second assumption, i.e. design for flow with friction, might be found more effective. In this respect, a new research venue could determine optimum values for the gliding angle  , as function of the different parameters influencing the performance of the impeller, on the one hand, but also as function of the impeller application, on the other.

116

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