i

Politecnico di Milano SCHOOL OF INDUSTRIAL AND INFORMATION ENGINEERING

Master of Science – Energy Engineering

Preliminary design of a centrifugal

compressor for an industrial blower

application

Supervisor Prof. Mister TOMMASO LUCCHINI

Candidate Miss MICHELLE WONG – 894419

Academic Year 2019 – 2020

iii

Acknowledgements

The writer would like to acknowledge the people that kept her sane during the development of

this project. Half my sanity is attributed to my friends, those who live in the old, the new and the

far too old continent. Those who live in this country have my deep gratitude; with a special

mention to Syed Ali Zaryab, Lucia Villalba, Giulia Botti and Fabio Gardella. The other half of

my sanity I attribute to my table tennis team, with a deep thank you to Silvio Carnevale and

Paolo Milza for welcoming me into the team.

The writer would like to acknowledge the people that made this project possible. My advisor

Prof. Tommaso Lucchini and Prof. Gianluca Montenegro, who had the arduous job to guide me

through the math contained in this volume. A special mention goes to Prof. Paolo Gaetani, for

his patience and assistance. Ing. Marco Riso and Ing. Mauro De Falco, from Nilfisk , who

proposed this project. Ing. Joel Lopez and Ing. Agustin Lozano, for their tangible contribution,

This document is dedicated to my parents, for the good genes and to Claire Saffitz, for the lesson

in perseverance.

v

Abstract

The aim of this thesis is to design a compressor to replace the side channel blower employed by

Nilfisk in the assembly of a vacuum cleaner. The blowers used by Nilfisk are produced by Elmo

Rietschle. Nilfisk provided the technical data of the G series side channel blowers by Elmo

Rietschle. The blowers had a very low efficiency and Nilfisk intended to start producing their

own compressors, instead of blowers. The aim of the design is to design a more efficiency

machine, that can replace the blower. The efficiency of the most efficient point had to exceed

80% in the impeller and 50% in the overall compressor, including the diffuser. The current

blower employed by Nilfisk is a side channel blower called 2BH1 930. The blower has two types

of operation. It operates as a vacuum and as a compressor. The blowers all seemed to be working

in very similar points in the Balje diagram. This way they all had very similar expected

efficiencies. Nevertheless, the expected efficiencies were quite low. The goal was to provide a

much better efficiency considering as a starting point that the rotating speed had to increase. The

method used to calculate the compressor is one dimensional. The calculation comprises: the inlet

flange, the impeller and the vaneless diffuser. The method is based on the methodology from the

book, but it was adjusted to avoid doing too many iterations and to have the least amount of

iterative equations. The Balje diagram was used to determine the speed and size of the

compressor, the rest of the parameters were calculated to maximize the efficiency. The expected

efficiency was determined based on the calculation of the loss correlations in the impeller and

the diffuser. Since the size and speed of the machine were already determined based on the Balje

diagram, the variable that changed was the volumetric flow entering the compressor. After the

compressor was designed to reach the target pressure ratio, the other working points were

calculated varying the inlet volumetric flow rate. The performance of the compressor in the

design point has an efficiency of 84% along the impeller and 72% along the whole compressor.

The lowest efficiency in the compressor is 57%, which is higher than the highest efficiency in

the side channel blower.

vii

Table of Contents

Acknowledgements ........................................................................................................................ iii

Abstract ........................................................................................................................................... v

Table of Contents .......................................................................................................................... vii

List of figures ................................................................................................................................. ix

List of tables ................................................................................................................................... xi

Chapter 1 Introduction ............................................................................................................... 1

1.1. Analysis of the blower configuration ............................................................................... 1

Chapter 2 State of the art ........................................................................................................... 6

2.1. Stage of the compressor ................................................................................................... 6

2.2. Rotating coordinate system .............................................................................................. 8

2.3. Basic Equations .............................................................................................................. 10

2.3.1. The continuity equation .......................................................................................... 10

2.3.2. Momentum equation ............................................................................................... 10

2.3.3. Energy equation ...................................................................................................... 11

2.3.4. Second law of thermodynamics .............................................................................. 12

2.4. Dimensionless parameters .............................................................................................. 13

2.5. Fluid dynamics ............................................................................................................... 15

2.6. Velocity triangles ........................................................................................................... 17

2.7. Slip factor ....................................................................................................................... 17

2.8. Performance parameters ................................................................................................. 19

2.8.1. Definition of efficiency ........................................................................................... 19

2.8.2. The isentropic process............................................................................................. 20

2.8.3. Diffuser performance parameters ........................................................................... 20

2.9. Losses in centrifugal compressors ................................................................................. 21

2.9.1. Incidence losses ...................................................................................................... 21

viii

2.9.2. Blade loading losses ................................................................................................ 23

2.9.3. Skin friction losses .................................................................................................. 23

2.9.4. Disk friction losses .................................................................................................. 24

2.9.5. Recirculation losses ................................................................................................ 24

2.9.6. Clearance losses ...................................................................................................... 25

2.9.7. Shock wave losses................................................................................................... 25

Chapter 3 Stage design ............................................................................................................ 26

3.1. Assumptions made to make the calculation: .................................................................. 26

3.2. Calculation of the compressor inlet (stage 0-1): ............................................................ 26

3.3. Calculation of the impeller discharge (stage 1-2) .......................................................... 29

3.4. Entropy calculation in the compressor ........................................................................... 32

3.4.1. Incidence enthalpy loss ........................................................................................... 32

3.4.2. Skin friction losses .................................................................................................. 33

3.4.3. Loading losses ......................................................................................................... 34

3.4.4. Disk friction losses .................................................................................................. 34

3.4.5. Recirculation and clearance losses.......................................................................... 35

3.5. Design of vaneless diffuser (stage 2-3) .......................................................................... 35

Chapter 4 Working points of the compressor .......................................................................... 38

Chapter 5 Results of the design ............................................................................................... 41

5.1. Results of the impeller design ........................................................................................ 41

5.2. Working points of the compressors ................................................................................ 45

5.2.1. Compressor 1 .......................................................................................................... 45

5.2.2. Compressor 2 .......................................................................................................... 48

5.3. Vaneless diffusor ............................................................................................................ 57

Chapter 6 Conclusions ............................................................................................................. 59

First Appendix: Calculation of the design of the compressor....................................................... 60

Second Appendix: Calculation of the other working points ......................................................... 66

Third Appendix: Flow diagram for the impeller design ............................................................... 71

Bibliography ................................................................................................................................. 72

ix

List of figures

Figure 1.1: Compressor 930 in the Balje diagram

Figure 2.1: Side view of single stage compressor

Figure 2.2: Multistage compressor stage

Figure 2.3: Enthalpy-entropy diagram of a compressor

Figure 2.4: Curvilinear coordinate system

Figure 2.5: Coordinates in a stream surface

Figure 2.6: Control volume

Figure 2.7: Temperature-entropy diagram

Figure 2.8: Typical impeller styles

Figure 2.9: Balje diagram

Figure 2.10: Cordier line

Figure 2.11: Velocity triangles

Figure 2.12: Concept of slip factor

Figure 2.13: Velocity triangle with the slip

Figure 2.14: Triangles before and after incidence

Figure 5.1: Balje diagram with the design points of both compressors

Figure 5.2: Balje diagram with the three working points of the first compressor

Figure 5.3: Balje diagram with the four working points of the second compressor

Figure 5.4: Volumetric rate - pressure difference graph of first compressor

Figure 5.5: Power - pressure difference graph of first compressor

Figure 5.6: Temperature difference rate - pressure difference graph of first compressor

Figure 5.7: Volumetric rate - pressure difference graph of second compressor

Figure 5.8: Power - pressure difference graph of second compressor

Figure 5.9: Temperature difference rate - pressure difference graph of second compressor

Figure 6.1: Volumetric rate - pressure difference graph of final compressor

Figure 6.2: Power - pressure difference graph of final compressor

Figure 6.3: Temperature difference rate - pressure difference graph of final compressor

xi

List of tables

Table 1.1: Main parameters taken from the blower technical sheet

Table 5.1: Starting properties of the compressor

Table 5.2: Geometrical parameters of the compressor

Table 5.3: Velocity triangle magnitude at the inlet

Table 5.4: Velocity triangle magnitude at the outlet

Table 5.5: Velocity triangles at the inlet and outlet of the compressors

Table 5.6: Stagnation properties in the impeller

Table 5.7: Dimensionless losses in the impeller

Table 5.8: Performance parameters in the impeller

Table 5.9: Velocity triangle magnitude in the inlet of the first impeller

Table 5.10: Velocity triangle magnitude in the outlet of the first impeller

Table 5.11: Velocity triangles in the working points in the first compressor

Table 5.12: Stagnation properties in the first impeller

Table 5.13: Dimensionless losses in the first impeller

Table 5.14: Performance parameters in the first impeller

Table 5.15: Velocity triangle magnitude in the inlet of the second impeller

Table 5.16: Velocity triangle magnitude in the outlet of the second impeller

Table 5.17: Stagnation properties in the second impeller

Table 5.18: Dimensionless losses in the second impeller

Table 5.19: Performance parameters in the second impeller

Table 5.20: Velocity triangles in the working points in the second compressor

Table 5.21: All working points of the first and second compressor

Table 5.22: Geometrical parameters of the diffuser

Table 5.23: Velocity components of the diffuser

Table 5.24: Stagnation properties of the diffuser

Table 5.25: Performance parameters of the diffuser

Table 6.1: Geometrical parameters of the final compressor

Table 6.2: Velocity triangle magnitude in the inlet of the final impeller

Table 6.3: Velocity triangle magnitude in the outlet of the final impeller

Table 6.4: Stagnation properties in the first impeller

Table 6.5: Dimensionless losses in the final impeller

Table 6.6: Performance parameters in the first impeller

Table 6.7: Velocity triangles in the final compressor

1

Chapter 1

Introduction

The aim of this thesis is to design a compressor to replace the side channel blower employed by

Nilfisk in the assembly of a vacuum cleaner. The Nilfisk Group is headquartered in Denmark on a

vision of producing and selling products of the highest quality worldwide. Their main product lines

are floorcare equipment, vacuum cleaners and high pressure washers. This project started as an

initiative from Nilfisk to start fabricating their own compressors instead of purchasing them for

their vacuum cleaners.

The blowers used by Nilfisk are produced by Elmo Rietschle. This company was awarded the

Imperial Patent for the world's first liquid ring vacuum pump in 1903. Three years later a smaller

Elmo vacuum pump was used in Germany's first vacuum cleaner. The first side channel blower

was introduced in 1963.

Nilfisk provided the technical data of the G series side channel blowers by Elmo Rietschle. The

blowers had a very low efficiency and Nilfisk intended to start producing their own compressors,

instead of blowers. The aim of the design is to design a more efficiency machine, that can replace

the blower. The efficiency of the most efficient point had to exceed 80% in the impeller and 50%

in the overall compressor, including the diffuser. The scope of the design considers a flange before

the impeller, the impeller and a vaneless diffuser right after the impeller.

The technical data provided by Nilfisk was comprised of a technical sheet and the drawing of the

side channel to be used as a model. Since the compressor and the blower are two different types

of machines, many considerations were taken from the literature based on empirical data to provide

an optimal design. The working points of the blower are the same points that the compressor must

achieve at the outlet of the diffuser. No volute or collector was considered in the design, because

it is not inside the scope of this thesis.

1.1. Analysis of the blower configuration

The current blower employed by Nilfisk is a side channel blower called 2BH1 930. Nilfisk

provided a set of blowers, all of them were analyzed. Nilfisk provided the technical sheet of every

blower. Every blower was placed in the Balje diagram to determine the expected efficiency for

each blower. By the end of the analysis, 11 blowers were placed on the Balje diagram. The model

chosen as a starting point for the design was chosen because of its size, since it was the biggest

2

and most efficient. The performance curves of the machine are shown in Figure 1.1, Figure 1.2 and

Figure 1.3.

Figure 1.1: Suction capacity vs total pressure difference of the side channel blower

Figure 1.2: Power consumption at the shaft vs total pressure difference

Figure 1.3: Temperature rise in the air vs total pressure difference

3

The same curves will be provided for the resulting designed compressor to be able to perform a

comparison. The blower has two types of operation. It operates as a vacuum and as a compressor.

Both configurations are featured in the performance curves. The data presents 4 working points

for each configuration. The data provided by the curves is:

a) Suction capacity

b) Total pressure difference in the blower

c) Power consumption at the pump shaft

d) Temperature rise of the conveyed air

These parameters are used to estimate the efficiency of the blower and the dimensionless numbers

of the machine. The specific diameter and the specific speed are calculated to place the compressor

configuration points in the Balje diagram. The following table gives a summary of the calculations

made to get the aforementioned parameters.

Vacuum Compressor

4 3 2 1 1 2 3 4

V (m3/h) 450 620 880 1050 1075 950 880 700

P (kW) 17 15 13 8 8 13 15 18

Temperature

difference 113 75 40 25 20 35 45 70

Pressure

difference 320 280 220 120 110 200 260 340

Pressure

ratio 1.47 1.38 1.28 1.13 1.12 1.25 1.35 1.51

Ideal work

(m2/s2) 33683 28453 21272 10756 9789 19035 25977 36448

Specific

speed 0.044 0.059 0.08 0.160 0.174 0.099 0.07 0.05

Specific

diameter 18.77 15.33 11.9 9.2 8.91 11.20 12.58 15.35

Efficiency 0.296 0.377 0.529 0.428 0.487 0.541 0.574 0.518

Mass flow

(kg/s) 0.144 0.213 0.320 0.367 0.379 0.345 0.325 0.256

Table 1.1: Main parameters taken from the blower technical sheet

The positioning of the working points in the Balje diagram determined that the blower was working

as a rotating compressor. The main goal of the design was to shift the working points to the

Turbocompressor zone. The compressor had to be smaller and rotate faster. The main point of

using the Balje diagram was to get an expected efficiency and to determine if the appropriate size

4

and speed of the compressor. This allowed for the design of a compressor with a significant amount

of freedom in terms of size and power output. The compressor still had to be smaller than the side

channel blower. As the guide for the size of the compressor, the drawing of the side channel was

used to determine the size of the compressor. The drawing was also used to determine the specific

diameter of the machine to be placed in the Balje diagram.

Figure 1.4: Front view of the side channel blower

The blowers all seemed to be working in very similar points in the Balje diagram. This way they

all had very similar expected efficiencies. Nevertheless, the expected efficiencies were quite low.

The goal was to provide a much better efficiency considering as a starting point that the rotating

speed had to increase. The maximum efficiency found was bordering the 55% in the calculus but

was around 50% in the Balje diagram. Since there is no way to calculate the losses with the limited

information and the Balje diagram can only give an estimate based on empirical information, the

calculated efficiency was taken as the more accurate calculation. The new compressor must be an

improvement on the blower. The delivery of a machine with a significant improvement in

efficiency had to justify the use of the faster motor to move the shaft of the compressor.

Both configurations were placed in the Balje diagram, but only the compressor configuration was

to be designed. The pressure ratio in the working point was taken as the aim output pressure in the

5

impeller and the volumetric flow rate was used to give an idea of the inlet volumetric flow rate in

the working points.

Figure 1.5: Compressor 930 in the Balje diagram

6

Chapter 2

State of the art

2.1. Stage of the compressor

The centrifugal compressor can be a single or multi-stage compressor. Figure 2.1 and Figure 2.2

feature both configurations respectively. The stage consists of a rotating impeller that transfers

energy into the fluid and a diffuser, which recovers some of the fluids kinetic energy before the

flow enters in the volute. The diffuser can be vaned or vaneless. In the latter case it is just a simple

annular passage. The volute is used to collect the flow from the diffuser into the discharge pipe. A

multistage compressor does not have a volute, instead it has a crossover and a return channel. They

are both used to guide the fluid into the next stage of the compressor.

Figure 2.1: Side view of single stage compressor (Aungier, 2000)

7

Figure 2.2: Multistage compressor stage (Aungier, 2000)

Figure 2.1 features an unshrouded impeller, which means the outer wall of the impeller is

stationary. Figure 2.2 features a closed or shrouded impeller, in this case the outer wall rotates with

the impeller. This configuration is usually used in multistage industrial compressors where it can

be difficult to hold acceptable tight clearances between the impeller blades and a stationary shroud

for several stages. It should also be noted that the Figure 2.1 features a full inducer impeller where

the inducer starts at the eye of the impeller. Figure 2.2 features a semi inducer impeller. This

inducer does not start in the eye of the impeller. The choice of inducer depends on the dimensional

parameters of the impeller.

8

Figure 2.3: Enthalpy-entropy diagram of a compressor (Whitfield, 1990)

2.2. Rotating coordinate system

The analysis of the flow in the impeller is best calculated based on a rotating frame of reference.

The coordinate system rotates with the impeller. A curvilinear coordinate system has component

m, which is measured along a stream surface; θ, which is the polar angle of the cylindrical

coordinates; and n, which is the normal to the stream surface. The velocities in the absolute frame

of reference are designated as C and the velocities in the relative frame of reference are designated

as W. Figure 2.5 features the conversion between the curvilinear coordinate system and the

absolute and relative frame of references. In this figure, the subscripts m and U designate the

meridional and tangential velocity components, respectively.

9

Figure 2.4: Curvilinear coordinate system (Aungier, 2000)

Figure 2.5: Coordinates in a stream surface (Aungier, 2000)

10

2.3. Basic Equations

The thermodynamic state of the fluid will be determined using two properties. In this case,

temperature and pressure will be used. The typical design methodology used for centrifugal

compressors is a one dimensional design. Therefore, the fluid moves in streamlines which follow

the geometry of the blades. This model is based on four equations:

a. The continuity equation

b. The momentum equation

c. The energy equation or first law of thermodynamics

d. The second law of thermodynamics

Figure 2.6: Control volume (Aungier, 2000)

2.3.1. The continuity equation

This equation states that in steady flow the mass flow rate of the fluid entering a control volume

is the same as the rate of fluid exiting the control volume.

�̇� = 𝜌1𝐶1𝐴1 = 𝜌2𝐶2𝐴2

Equation 2.1: Continuity equation

Where C is the velocity, A is the area normal to C and ρ is the density. The velocity has a tangential

and meridional component. Based on the coordinate system explained previously the continuity

equation can be adjusted to the desired coordinate system.

2.3.2. Momentum equation

The rate of change of momentum of a fluid is equal to the net applied force on the fluid in a control

volume in the direction of the flow. In the case of the steady flow, the formula can be written in

this way.

11

𝐹𝑥 = �̇�(𝐶𝑥2 − 𝐶𝑥1)

Equation 2.2: External force

A useful control volume analyzed in turbomachines is the impeller or rotor, because the external

work done in this section is due to the torque in the shaft. Therefore, the previous equation can be

used to calculate the torque and the work of the fluid passing through the impeller.

𝜏 = �̇�(𝑟2𝐶𝑥2 − 𝑟1𝐶𝑥1)

Equation 2.3: Torque in the shaft

�̇�

�̇�=

𝜏𝜔

�̇�= 𝑈2𝐶𝑡2 − 𝑈1𝐶𝑡1

Equation 2.4: Momentum equation

2.3.3. Energy equation

The net change in the energy of a fluid undergoing any process is equal to the net transfer of work

and heat between the fluid and its surroundings. The energy of a steady flow of fluid going through

a duct can be considered as the sum of the internal energy, the work done by the fluid on the

surroundings, the kinetic energy and the potential energy. The potential energy is negligibly small,

and we consider the turbomachine to be adiabatic. An appropriate equation is yielded from the

general form.

�̇�

�̇�−

�̇�

�̇�= (ℎ2 − ℎ1) +

1

2(𝐶2

2 − 𝐶12) + 𝑔(𝑧2 − 𝑧1)

Equation 2.5: General energy equation

�̇�

�̇�= (ℎ2 +

1

2𝐶2

2) − (ℎ1 +1

2𝐶1

2)

Equation 2.6: Energy equation in the compressor

12

2.3.4. Second law of thermodynamics

A full description of the second law is of no use here. Instead we present the thermodynamically

ideal standard against which processes may be compared. Since this is a non-cyclic process, we

can define entropy to describe the change of state brought about by a reversible process.

𝑠2 − 𝑠1 = (𝑑�̇�

𝑇)

𝑟𝑒𝑣

Equation 2.7: Entropy change

The entropy change can be associated as the deviation of the process from the ideal reversible

process. A way to illustrate these processes is the enthalpy-entropy diagram or the temperature-

entropy diagram.

Figure 2.7: Temperature-entropy diagram (Whitfield, 1990)

13

2.4. Dimensionless parameters

These parameters can provide information regarding the achievable performance and the most

effective design. The main parameters used are the following.

𝜙 = �̇�/(𝜌0𝑡𝜋𝑟22𝑈2) = 𝑄0/(𝜋𝑟2

2𝑈2)

Equation 2.8: Stage flow coefficient

Where m is the mass flow, the subscript 2 refers to the conditions in the exit of the impeller and

the subscript t refers to the stagnation condition. U2 refers to the impeller tip speed.

𝜇 = 𝐻𝑟𝑒𝑣/𝑈22

Equation 2.9: Head coefficient

Where Hrev refers to the reversible head provided to the compressor.

𝑀𝑈 = 𝑈2/𝛼0𝑡

Equation 2.10: Rotational Mach number

Where α refers to the speed of sound. These numbers are used in the present calculus to

characterize the current machine and the designed machines. An alternative to these numbers are

the specific diameter and specific speed.

𝑛𝑆 = 1.773√𝜙/𝜇𝑖𝑠0.75

Equation 2.11: Specific speed

𝑑𝑆 = 1.128𝜇𝑖𝑠0.25/√𝜙

Equation 2.12: Specific diameter

The subscript “is” refers to a reversible thermodynamic process or isentropic process. The stage

flow coefficient characterizes the stage type and the efficiency level that can be expected. Figure

2.8 features the typical impeller styles used in multistage compressors according to the stage flow

coefficient. At very high values of the stage flow coefficient, the wide passages lead to large

“curvature losses” to limit the achievable efficiency levels. This can be alleviated using a mixed

14

flow impeller. On intermediate values, more conventional impellers with rather good efficiencies

are expected.

Figure 2.8: Typical impeller styles (Aungier, 2000)

The specific diameter and specific speed are used in the Balje diagram to give an approximation

of the type of machine we are analyzing, and the expected efficiency provided by that machine.

The concept of best efficiency point comes from the use of this diagram. The BEP intersects the

Cordier line which passes through the Balje diagram and provides the best efficiency point for a

given specific speed or specific diameter and the corresponding specific diameter and specific

speed, respectively. The selection and then design of a machine can be done using the Cordier line

using the specific speed as an input to get the corresponding specific diameter and from there get

a design of the machine. The rest of the design is performed using basic Thermofluidic dynamics.

Figure 2.9: Balje diagram (Balje, 1981)

15

Figure 2.10: Cordier line (Whitfield, 1990)

2.5. Fluid dynamics

The stagnation state is defined as the state at which the fluid is brought isentropically to rest. The

definition of the stagnation enthalpy is the sum of the static enthalpy and the kinetic energy.

ℎ0 = ℎ +1

2𝐶2

Equation 2.13: Stagnation enthalpy

By replacing the values of enthalpy with the values of temperature the relationship between the

stagnation temperature and the static temperature can be obtained.

𝑇0 = 𝑇 +1

2𝐶2/𝐶𝑝

Equation 2.14: Stagnation temperature

This equation can be rearranged to get the relationship between the total and static temperature

using the Mach number. This formula can again be rearranged using the equation of state of the

gases to yield the relationship between the temperature and pressure of the total and static state.

16

𝑇0

𝑇= 1 +

𝛾 − 1

2𝑀2

Equation 2.15: Relationship between the total enthalpy and the Mach number

𝑃0

𝑃= (

𝑇0

𝑇)

𝐶𝑝/𝑅

= (𝑇0

𝑇)

𝛾/(𝛾−1)

Equation 2.16: Relationship between the pressure and temperature

Adiabatic flow in which no work transfer occurs characterizes all the stationary blade rows and

passages of a turbomachine. This type of flow has constant stagnation enthalpy or temperature.

The impeller has work, in which work transfer occurs. The property that remains constant in the

impeller is called rothalpy. Starting from the equation of Euler work, we can derive an equation

for rothalpy that is constant in a turbomachinery with adiabatic flow, regardless of the work

transfer or the radius changes.

ℎ02 − ℎ01 = 𝑈2𝐶𝑡2 − 𝑈1𝐶𝑡1

Equation 2.17: Euler equation of work

𝐼 = ℎ +1

2𝑊2 = ℎ0

′ −1

2𝑈2

Equation 2.18: Definition of rothalpy

ℎ02′ − ℎ01

′ =1

2(𝑈2

2 − 𝑈12)

Equation 2.19: Relative stagnation enthalpy

This definition is significant specially for the calculation of the losses later on in the impeller. The

equation for relative stagnation enthalpy can be used in any passage of the machine and proofs that

for a passage with a same inlet and outlet radius the relative stagnation enthalpy is conserved along

the passage. It is a general expression that can be used in stationary and rotating components.

17

2.6. Velocity triangles

The velocity triangles determine the relation between the blade speed and the gas velocity, absolute

and relative to the moving blades of the impeller. The large difference between the hub and the

shroud in a centrifugal compressor means that different velocity triangles need to be taken into

account for the hub, at the tip and in a mean radius. Figure 2.11 features the sign convention used

in the following calculus. The angles are measured from the meridional direction.

Figure 2.11: Velocity triangles (Whitfield, 1990)

2.7. Slip factor

The flow entering the impeller can be considered irrotational. In the rotating reference frame, a

relative eddy is present to maintain the flow irrotational in the absolute frame of reference. This

eddy will prevent the flow from being perfectly guided by the blades. This unperfect guiding is

referred to as a slip. The effect of slip is to reduce the magnitude of the tangential component of

18

the velocity from that which is ideally attainable. This can also affect the delivered pressure ratio,

by reducing it. This slip is present even in an ideal impeller. As a consequence, a bigger and faster

impeller is needed to get the required pressure ratio. This increases the relative velocity and the

friction losses. The quantification of the slip can be illustrated in the velocity triangle. A simple

model for the slip factor is provided by Stodola.

Figure 2.12: Concept of slip factor (Aungier, 2000)

𝜇 = 1 −𝐶𝑠𝑙𝑖𝑝

𝑈2= 1 −

𝐶𝑡2∞ − 𝐶𝑡2

𝑈2

Equation 2.20: Slip factor by Stodola

19

Figure 2.13: Velocity triangle with the slip (Whitfield, 1990)

This equation is purely kinematic. This is the definition of slip factor used in this calculation. There

are also correlations for the slip factor provided by Busemann and Stanitz, which are based upon

theoretical analyses of the flow of an ideal fluid. It has also been argued that the value of the slip

factor varies with the flow rate and it is not a single value for the whole impeller. This happens

due to the different degrees of separation which occur within the impeller passage as the flow rate

varies. At the best efficiency point, many impellers with different designs but the same number of

blades with present a difference in degrees of separation.

2.8. Performance parameters

2.8.1. Definition of efficiency

The efficiency of the machine is one of the most important performance parameters. This

parameter relates the actual work transfer to that which would occur if the working fluid followed

an ideal flow process.

20

𝜂𝐶 =𝑤𝑜𝑟𝑘 𝑖𝑛𝑝𝑢𝑡 𝑖𝑛 𝑎𝑛 𝑖𝑑𝑒𝑎𝑙 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑠𝑡𝑎𝑡𝑒𝑠

𝑎𝑐𝑡𝑢𝑎𝑙 𝑤𝑜𝑟𝑘 𝑖𝑛𝑝𝑢𝑡

Equation 2.21: Efficiency of the compressor

This definition depends on the definition of the inlet and outlet states and the definition of the ideal

process.

2.8.2. The isentropic process

It is an adiabatic process, where the efficiencies are known as adiabatic efficiencies. The work

transfer in adiabatic processes can be expressed ad changes in the total enthalpy. The total to total

efficiency is defined.

𝜂𝑡𝑡 =ℎ02𝑠 − ℎ01

ℎ02 − ℎ01

Equation 2.22: Total to total efficiency

2.8.3. Diffuser performance parameters

The efficiency in the diffuser can be defined in the following equation. Nevertheless, this defined

efficiency is not defined as the previous efficiencies. Since there no external work transfer in the

diffuser, those definitions are inapplicable.

𝜂𝑡𝑡 =ℎ3𝑠 − ℎ2

ℎ3 − ℎ2=

(𝑃3

𝑃2⁄ )

(𝛾−1)/𝛾

− 1

𝑇3𝑇2

⁄ − 1

Equation 2.23: Efficiency in the diffuser

𝐶𝑃 =𝑃3 − 𝑃2

𝑃02 − 𝑃2

Equation 2.24: Diffuser pressure recovery coefficient

21

𝐶𝑃𝑖 =𝑃3𝐸 − 𝑃2

𝑃02 − 𝑃2

Equation 2.25: Ideal diffuser pressure recovery coefficient

2.9. Losses in centrifugal compressors

For the calculation of the losses in the impeller there are many empirical methods that have been

developed over the years. This method are condensed into one optimization study to calculate the

losses in a one dimensional analysis. The losses in the impeller are:

a. Incidence losses

b. Shock wave losses

c. Internal losses

i. Blade loading losses

ii. Skin friction losses

iii. Clearance losses

d. External losses

i. Disk friction losses

ii. Recirculation losses

The external losses are the losses which increase the impeller discharge stagnation enthalpy

without any corresponding increase in pressure. They also include any heat transfer from an

external source.

All these losses each have an array of correlations used to calculate them. The cited paper used

three different combinations by three authors to calculate the losses on a designed centrifugal

compressor. The correlations used in this calculation are based on the optimal correlation

suggested by this paper.

2.9.1. Incidence losses

Incidence loss is caused by the direction of the gas flow diffusing from the blade angle, which

greatly affects the compressor performance characteristics at off-design conditions. The

calculation of the optimum angle of incidence leads to zero blade loading at the leading edge. This

produce no change in whirl as the flow enters the impeller.

𝜖 = 𝛽𝑥 𝑜𝑝𝑡 − 𝛽𝑦

Equation 2.26: Incidence angle developed by (Stanitz, 1953)

22

Figure 2.14: Triangles before and after incidence (Whitfield, 1990)

According to Figure 2.14, we can calculate the optimum angle. If we consider x to be the pre

incidence condition and y the post incidence condition.

𝛽𝑥 𝑜𝑝𝑡 = 𝑈𝑥/𝐶𝑚𝑥 𝛽𝑦 = 𝑈𝑦/𝐶𝑚𝑦

Equation 2.27: Relationship between pre and post incidence

Galvas calculated the relative velocity component normal to the optimum flow direction. This

velocity is used to calculate the relative stagnation enthalpy loss.

𝑊𝐿 = 𝑊𝑥 𝑠𝑖𝑛(|𝛽𝑥 − 𝛽𝑥 𝑜𝑝𝑡|)

Equation 2.28: Relative velocity component normal to the optimum flow direction

∆ℎ0𝑖 = 𝑘𝑊𝐿/2

Equation 2.29: Stagnation enthalpy loss by incidence

The value of k is between 0.5 and 0.7. This value takes into account the change in kinetic energy

associated with the tangential component of velocity. This model can be used for rotating and

stationary blades.

23

2.9.2. Blade loading losses

Boundary layer growth in the impeller is highly dependent on the diffusion of the working fluid

internal to the impeller itself. The correlation developed by (Lieblein, Schwenk, & Broderick,

1953) is intended to be used in axial compressors.

∆𝑞𝑏𝑙 = 0.05𝐷2

Equation 2.30: Blade loading correlation

𝐷 = 1 −𝑊𝑦

𝑊𝑥𝑠+ 0.75∆𝑞𝑡ℎ

𝑊𝑦

𝑊𝑥𝑠[𝑍𝐵

𝜋(1 −

𝑟𝑥𝑠

𝑟𝑦) +

2𝑟𝑥𝑠

𝑟𝑦]

−1

Equation 2.31: Diffusion factor

∆𝑞𝑡ℎ = (𝐶𝑡𝑦𝑈𝑦 − 𝐶𝑡𝑥𝑈𝑥)/𝑈𝑦2

Equation 2.32: Specific enthalpy jump

The diffusion factor has been used without justification for axial impellers. Rogers used

experimental data to get a correlation better suited for centrifugal impellers.

𝐷 = 1 −𝑊𝑦

𝑊𝑥 𝑟𝑚𝑠+

𝜋𝑟𝑦∆𝑞𝑡ℎ𝑈𝑇

𝑍𝐵𝐿𝑊𝑥 𝑟𝑚𝑠+ 0.1

𝑏

𝑟𝑠(1 +

𝑊𝑦

𝑊𝑥 𝑟𝑚𝑠)

Equation 2.33: Diffusion factor by (Rodgers & Sapiro, 1972)

2.9.3. Skin friction losses

Skin friction losses are due to shear forces in the boundary layer. The correlation is calculated

using the hydraulic diameter an length and the average relative velocity in the impeller. The

calculation of the friction factor can be taken from the Moody diagram or using the Colebrook-

White equation.

∆𝑞𝑡ℎ = 4𝐶𝑓𝐿�̅�2/2𝐷𝑈𝑇2

Equation 2.34: Skin friction correlation by (Jansen & Qvale, 1967)

24

If the friction factor is taken from the Moody diagram for straight ducts, then the friction factor

can be modified using the (Musgrave, 1980) method. The modification takes into account the

Reynolds number and mean radius of curvature of the flow path.

4𝐶𝑓′ = 4𝐶𝑓[𝑅𝑒(𝑑/2𝑅𝑐)2]0.05

Equation 2.35: Modified friction factor

2.9.4. Disk friction losses

This specific loss is due to the shear forces between the impeller back face and the stationary

surface. The disk friction depends on the torque coefficient. The torque coefficient calculation

varies based on the Reynolds number.

∆𝑞𝑑𝑓 = 0.25�̅�𝑈𝑇𝑟𝑇2𝐾𝑓/�̇�

Equation 2.36: Correlation for disk friction losses by (Daily & Nece, 1960)

𝐾𝑓 =

3.7 (𝜖

𝑟𝑇)

0.1

𝑅𝑒𝑦0.5⁄ 𝑓𝑜𝑟 𝑅𝑒𝑦 < 3 × 105

0.102 (𝜖

𝑟𝑇)

0.1

𝑅𝑒𝑦0.2⁄ 𝑓𝑜𝑟 𝑅𝑒𝑦 > 3 × 105

Equation 2.37: Torque coefficient by (Daily & Nece, 1960)

2.9.5. Recirculation losses

Recirculation loss results from the working fluid backflow into the impeller. As the discharge flow

angle increases and it approaches the tangential direction, the external losses increase significantly.

The aforementioned diffusion coefficient is used to calculate the recirculation losses.

∆𝑞𝑟𝑒 = 0.02𝐷2√𝑡𝑎𝑛(𝛼𝑦)

Equation 2.38: Recirculation losses correlation by (Coppage, et al., 1956)

25

2.9.6. Clearance losses

Significant flow leakage occurs through the gap between the impeller and the casing resulting from

the pressure difference between the pressure side and the suction side of the compressor. The

correlation developed by Jansen considers that the flow through the clearance gap undergoes a

sudden contraction followed by a sudden expansion.

∆𝑞𝑐𝑙 = 0.6𝜖

𝑏𝑦

𝐶𝑡𝑦

𝑈𝑇{

4𝜋

𝑏𝑦𝑍𝐵[

𝑟𝑥𝑠2 − 𝑟𝑥ℎ

2

(𝑟𝑦 − 𝑟𝑥𝑠)(1 + 𝜌𝑦/𝜌𝑥)]

𝐶𝑡𝑦

𝑈𝑇

𝐶𝑚𝑥

𝑈𝑇}

1/2

Equation 2.39: Clearance loss correlation by (Jansen & Qvale, 1967)

The aforementioned correlation is only suitable for clearance ratios under 0.03. If this ratio is

exceeded there is another correlation to be used. This correlation gives an overall efficiency

decrement due to clearance.

∆𝜂𝑐𝑙 =0.35𝜖

𝑏𝑦− 0.01

Equation 2.40: Clearance loss correlation by (Musgrave, 1980)

Clearance loss decreases with a reduced mass flow and a low shaft speed for high pressure ratio

compressors. They also then to be greater for low pressure ratio compressors, considering two

compressors with the same tip clearance ratio. The loss in efficiency due to the increase of

clearance is nonlinear. The efficiency will fall more rapidly at small clearances.

2.9.7. Shock wave losses

These are the losses from the shock waves form in supersonic flow, including direct total pressure

losses through the wave and boundary layer separation caused by shock interaction. These losses

are mostly present in high pressure ratio compressors because it may be necessary to operate with

transonic inlet flow conditions. In this case it will be necessary to account for the shock wave

losses. Since there is a lack of correlations to account for this loss, there is some empirical research

that can be used to estimate the impact of the shock wave losses. It seems to not be very significant

for Mach numbers arriving up to 1.2.

26

Chapter 3

Stage design

The aim is to get a machine using the best efficiency point in the Balje diagram and analyze the

obtained geometry to achieve the desired pressure ratio with the best efficiency possible. The

previous machine had four working points. The following calculation is used for the two middle

working points to see which one can achieve all the other working points with the best efficiency.

The method used to calculate the compressor is one dimensional.

3.1. Assumptions made to make the calculation:

The total conditions in the compressor inlet are 288 K and 101300 Pa, for temperature and pressure

respectively. The aim is to achieve a pressure ratio of 1.45 at the exit of the impeller, the calculation

of the machine’s dimensions will be controlled to assure getting to this pressure ratio. The inlet

volumetric flow rate is 1000 m3/h. This volumetric flow rate is slightly higher than the one used

on the previous machine; this is done to improve efficiency. This new flow rate helps to get a

higher efficiency .The ratio between the hub and the shroud in the inducer is a value assumed based

on the literature and helps get a more cohesive design. The subscript 0 is for the inlet of the

compressor, the subscript 1 is for the inlet of the impeller, the subscript 2 is for the exit of the

impeller and subscript 3 is for the exit of the diffuser.

Based on the example provided we can expect a low stage flow coefficient, according to , we may

not be able to choose a full inducer impeller. This is the reason also there are a difference between

the impeller eye condition and the blade inlet impeller. In this calculation we calculate a flange at

the inlet of the compressor before the inducer. This is a single stage compressor, since the aimed

pressure ratio is in the common range.

According to the literature, some parameters must be available to optimize a design.

a. The inlet stagnation pressure and temperature; the conditions given by the manufacturer

b. The degree of prewhirl; here the inducer has zero prewhirl in the entrance

c. The mass flow rate of the working fluid

3.2. Calculation of the compressor inlet (stage 0-1):

The compressor inlet is calculated as a stationary duct. First the relative Mach number at the inlet

must be minimized. This working point minimizes the blade incidence losses in the impeller

27

passage to be able to work on the best efficiency point. A β angle similar to -55 degrees is assumed

to start, this value will be varied based on the resulting geometry of the impeller. Based on

(Whitfield, 1990), the minimized Mach number will always be related to a value similar to -60

degrees in the inlet for a no prewhirl inducer. From there the first guess of the relative Mach

number in the inlet is obtained. This first guess is good to make preliminary calculations.

cos2(𝛽1𝑠) =3 + 𝐺 × 𝐴𝑀1𝑅

2

2 × 𝐴𝑀1𝑅2 {1 − [1 −

4 × 𝐴𝑀1𝑅2

(3 + 𝐺 × 𝐴𝑀1𝑅2)

2]

1/2

}

Equation 3.1: Equation to minimize the relative Mach number

The equation to get the β angle in the inlet of the inducer comes from the velocity triangle with

zero prewhirl. Using the definition of non-dimensional mass and the inlet triangle, two equations

are yielded that can be rearranged to get an equation with the relative Mach number as an unknown.

This equation is aimed to maximize the mass flow for any relative Mach number. This equation

can be rearranged to be in terms of the relative Mach number and the flow angle. We can

differentiate this rearranged equation to yield the maximum mass flow. Once the relative Mach

number is obtained and therefore, the β angle in the inlet, the properties of the inlet flow and the

velocity triangle can be obtained. Since the properties in the flange are calculated, the

transformation from stage 0 to stage 1 is assumed to be isentropic and the velocity in 0 and 1 based

on the continuity equation is obtained.

𝐴𝑀1 = 𝐴𝑀1𝑅 × cos(𝛽1𝑠)

Equation 3.2: Relationship between relative and absolute Mach number

𝑇01/𝑇1 = 1 + (𝐺 − 1)/2 × 𝐴𝑀12

Equation 3.3: Total to static temperature

𝑃01/𝑃1 = (𝑇01/𝑇1)𝐺

𝐺−1

Equation 3.4: Total to static pressure

(𝑇00𝑅

𝑇0)

𝐺𝐺−1

=𝑃00𝑅

𝑃0

Equation 3.5: Total relative to static pressure

28

Using the relationship between the relative and absolute Mach number the value of the absolute

Mach number can be obtained and use it to get the relationship between the total and static

conditions in the inlet. Then a value for the diameter of the shroud inlet needs to be assumed. The

diameter of the shroud is equal to the value of the inlet diameter of the compressor, from there the

inlet area can be obtained and the speed at which the air enters the compressor from the continuity

equation.

Using the value of the speed in the flange, the static pressure and temperature at the flange can be

obtained. Since this is a stationary duct, all the relative fluid properties are conserved, and the

value of the total temperature on the inducer can be obtained. From the value of temperature and

pressure the density and speed of sound can be obtained.

𝑇0 = 𝑇 +𝐶2

2 × 𝐶𝑝

Equation 3.6: Stagnation temperature

𝑇01𝑅 = 𝑇00𝑅 = 𝑇00

Equation 3.7: Assumption for stationary ducts

𝛼01 = √𝐺 × 𝑅 × 𝑇01

Equation 3.8: Speed to sound

𝜌1 =𝑃01

𝑅 × 𝑇01

Equation 3.9: Density of the flow

From the velocity in the inlet, the relative conditions in the inlet can be obtained and since we are

working with an isentropic transformation, the relative temperature and pressure in the inlet are

conserved. From the conditions in the inducer and from there we can confirm if the value of

diameter of shroud that we assumed is correct and we can adjust the value of the relative Mach

number and the β angle. A larger diameter corresponds to a lower Mach number so we can adjust

the value to get an appropriately sized inducer.

𝑟1ℎ = 𝑟1𝑠 × 𝐴𝑁𝑈

Equation 3.10: Ratio between shroud and hub

29

𝐶1 = 𝑊1 × 𝑐𝑜𝑠(𝛽1𝑠)

Equation 3.11: Inlet triangle

�̇� = 𝑄0 × 𝐶0 = 𝐶1 × 𝜌1 × 𝐴1

Equation 3.12: Continuity equation

Using the ratio between the shroud and the hub the value of the diameter of the hub can be obtained.

Based on the value of the diameter in the hub and shroud, the area in the inlet of the inducer can

be obtained. We can complete the value of the properties in the inlet of the inducer and with that

get the value of the velocity at the inlet of the inducer. Following this calculation, we can move on

to the calculation of the impeller.

3.3. Calculation of the impeller discharge (stage 1-2)

The Balje diagram is used to get the best radial machine possible. From the Balje diagram we can

get the specific diameter and the specific speed. The value of the volumetric flow rate have to be

adjusted based on the value of the specific speed and the inlet of the impeller must be recalculated.

Since there is the β1 angle the value of the tangential velocity in the inlet can be obtained and from

there the rotational speed of the machine can be obtained. The rotational speed allows to calculate

the specific speed that will be placed in the Balje diagram. The value of the inlet flow rate can be

arranged to change the resulting shroud diameter until the specific speed point is in the desired

location in the diagram. Now because we want the best machine possible, the value of the

corresponding specific diameter can be obtained from the graph. The choice in outlet diameter

comes from the Balje diagram. The ideal work comes from the pressure ratio that we want to

achieve. Here the calculations made in the previous section must adjusted so that the inlet triangle

corresponds to the outlet chosen based on the best efficiency point. Using the value of the diameter

in the outlet, from the specific diameter equation, and the rotational speed, from the tangential

speed, the tangential speed in the outlet of the impeller can be obtained. From the efficiency in the

Balje diagram the real work can be obtained that will be needed by the compressor as a first guess.

In this calculation the geometric β angle of the discharge to about -40 degrees are assumed. This

value is based on the researched literature as a fitting angle for optimal compressors, and from

there the discharge velocity triangle can calculated. The negative sign represents backward swept

blades. The advantages of the backward swept blades are documented by (Came, McKenzie, &

Dadson, 1979). They produce a reduction of the impeller discharge absolute Mach number, thereby

reducing the diffusion requirements of the vaneless diffuser. They have a broader stable operating

range because of an increased negative gradient of the work input and pressure ratio. They reduce

the secondary flow and the losses associated with them, due to the increase streamline curvature

30

in the blade to blade plane. the tangential component of the absolute velocity can be obtained from

the ideal work of the compressor and the rest of the components of the absolute velocity from the

assumption of β. the value of β at the inlet can be calculated as well considering the velocity

components of the triangle in the outlet. Α must have a value between 60 and 80, which is an

optimal range according to (Rodgers & Sapiro, 1972).

𝑙𝑖𝑑𝑒𝑎𝑙 = 𝐶𝑝 × 𝑇01 × (𝑃𝑅𝐺−1

𝐺 − 1)

Equation 3.13: Ideal work of the compressor

𝑙𝑟𝑒𝑎𝑙 = 𝑈2 × 𝐶𝑡2

Equation 3.14: Real work of the compressor

𝑆𝐹 = 1 −𝜋 × 𝑐𝑜𝑠(𝛽2𝑏)

𝑁𝑏

Equation 3.15: Slip factor

𝐶𝑚2 = (𝐶𝑡2 − 𝑈2 × 𝑆𝐹)/ 𝑡𝑎𝑛(𝛽2𝑏)

Equation 3.16: Meridional outlet velocity

𝛼2 = 𝑡𝑎𝑛−1𝐶𝑡2

𝐶𝑚2

Equation 3.17: Α at discharge

Since there are all the geometrical parameters chosen, the slip factor in the impeller can be

calculated. The number of blades is imposed based on the chosen geometrical angles in the inlet

and discharge of the impeller. The slip factor equation used is the Stodola equation because of the

range of β angles in which it works. The slip factor affects the real work performed by the machine.

There is the geometrical velocity triangle, as well as the flow velocity triangle. The flow velocity

triangle has a new value of β.

𝐶𝑠𝑙𝑖𝑝 = 𝑈2 × (1 − 𝑆𝐹)

Equation 3.18: Slip effect

31

𝐶𝑡2𝑔𝑒𝑜 = 𝐶𝑡2 + 𝐶𝑠𝑙𝑖𝑝

Equation 3.19: Geometrical tangential component of the absolute velocity

From the outlet triangle, the properties of the fluid in the discharge can be obtained. The equations

for relative total conditions can be used to get the relative stagnation temperature at the outlet and

the relative stagnation enthalpy at the outlet. The ideal work of the compressor allows to get the

isentropic difference of enthalpy in the compressor. There are both the stagnation enthalpy at the

inlet and outlet from the real work. The real work comes from the efficiency which will be

recalculated using the losses. There are the static properties in the discharge from the relative total

conditions. All the real conditions at the outlet and the isentropic ones from the ideal work can be

obtained. The corresponding losses to the assumed efficiency can be calculated. These losses will

be compared to the calculated losses to converge the efficiency. The value of the total relative

outlet pressure comes from the value of the entropy jump. The entropy jump can be calculated

from the losses. So far there is one variable: efficiency and one equation, which is the equation

comparing the losses based on the assumed efficiency and the losses calculated with the loss’s

correlations.

𝑇02𝑅 = 𝑇01𝑅 + (𝑈22 − 𝑈1

2)/(2 × 𝐶𝑝)

Equation 3.20: Rothalpy conservation

∆ℎ = ℎ02𝑅 − ℎ02𝑅𝑠

Equation 3.21: Enthalpy loss in the impeller

𝐴𝑀2 = 𝐶2/√𝑅 × 𝐺 × 𝑇2

Equation 3.22: Absolute outlet Mach number

The total outlet pressure can be calculated from the relative total pressure. We must check that the

real pressure ratio in the impeller is not that far off from the ideal design pressure ratio. The static

properties in the outlet are obtained based on the current velocity triangle. This velocity triangle

will change according to the efficiency. After the value of the outlet is obtained total temperature

rise in the impeller can be obtained. The value of the relative and absolute Mach number can be

calculated to ensure that the impeller is not working with a supersonic flow. The blade width comes

from the use of the continuity equation to make sure the mass flow is consistent. Now that all the

geometry of the impeller has been calculated the entropy jump in the impeller must be calculated

to confirm the previous assumption.

32

𝑃02𝑅 = 𝑃01𝑅 × 𝜎/ ((𝑇01𝑅/𝑇02𝑅)𝐺

𝐺−1)

Equation 3.23: Effect of entropy on P02R

𝑏2 = �̇�/(𝜌2 × 𝑑2 × 𝜋 × 𝐶𝑚2)

Equation 3.24: Blade width in the outlet

3.4. Entropy calculation in the compressor

The entropy gain comes from the following formula (Balje, 1981) and will be used in the end to

make sure the calculation of the losses converges with the efficiency assumed. Once the new

enthalpy is calculated and therefore, the new efficiency the losses can be calculated again and from

there make sure they both converge. This process is iterative and takes into account balancing all

the equations that have been previously used; except the calculation of the flange, which is

independent.

𝜎 = (1 − (𝐺 − 1)/(𝑅 × 𝐺 × 𝑇02𝑅) × 𝑈22 × ∆ℎ)

𝐺𝐺−1

Equation 3.25: Entropy jump in the impeller

3.4.1. Incidence enthalpy loss

Due to the incidence angle on the impeller we experience some losses which must be taken into

account to calculate the entropy gain. To start the optimum incidence angle must be calculated.

There will be a difference between the real flow angle in the inlet and the optimum angle

calculated. The optimum angle is calculated based on the geometry of the inlet of the impeller.

This incidence angle represents zero incidence loss. The method used to calculate the incidence

losses is provided by (Galvas, 1973) and has the objective to calculate the difference between the

real and the optimum angle of incidence. This optimum difference is defined as the conditions that

lead to zero blade loading at the leading edge, that is those which produce no change in whirl as

the flow enters the impeller. From the number of blades, the thickness of the blade can be

calculated also and make sure it is a realizable compressor. (Conrad, Raif, & Wessels, 1980)

considers the loss to be a proportion of the change in kinetic energy associated with the tangential

33

component of velocity. Thus, there is a parameter k, which is considered between 0.5 and 0.7. In

this case, the chosen factor was 0.6.

𝜀 = 𝛽1𝑜𝑝𝑡 − 𝛽1𝑠

Equation 3.26: Optimum angle difference

𝛽1𝑜𝑝𝑡 = 𝑡𝑎𝑛−1(𝐴1/𝐴1𝑝 × 𝑡𝑎𝑛(𝛽1𝑏))

Equation 3.27: Calculation of the optimal angle (Stanitz, 1953)

∆ℎ𝑖𝑛𝑐 = 𝑊12 × 𝑠𝑖𝑛(𝛽1𝑠 − 𝛽1𝑜𝑝𝑡)

2× 0.6/2

Equation 3.28: Calculation of the incidence losses (Galvas, 1973)

3.4.2. Skin friction losses

The (Jansen & Qvale, 1967) method is used to get the losses. There is an equation for the viscosity

of the fluid to be able to calculate the Reynolds number of the flow. The Reynolds number is

calculated based on the average values on the compressor. These are losses due to shear forces

exerted on the fluid in the boundary layer, similar to pipe friction losses. The formula for the

friction coefficient is adapted to work for the impeller of the compressor. The calculation of the

friction losses uses several parameters, which formulas are detailed in the following equations.

∆ℎ𝑠𝑓 = 2 × 𝐶𝑓 × 𝐿𝑏/𝐷ℎ𝑏 × (𝑊𝑓𝑙/𝑈2)2

Equation 3.29: Skin friction losses

𝑊𝑓𝑙 = √(𝑊12 + 𝑊2

2)/2

Equation 3.30: Average relative velocity

𝐿𝑏 =𝜋

8× [𝑑2 −

𝑑1𝑠 + 𝑑1ℎ

2− 𝑏2 + 2 × 𝐿𝑧] × (

2

𝑐𝑜𝑠(𝛽1𝑠) + 𝑐𝑜𝑠(𝛽1ℎ)2 + 𝑐𝑜𝑠(𝛽2)

)

Equation 3.31: Hydraulic length

34

𝑑ℎ𝑏 =𝑑2 × cos(𝑏𝑒𝑡𝑎2)

[𝑍𝜋 +

𝑑2 × cos(𝑏𝑒𝑡𝑎2)𝑏2

]+

1/2 × (𝑑1𝑠𝑑2

+𝑑1ℎ𝑑2

) × (cos 𝑏𝑒𝑡𝑎1𝑠 + cos 𝑏𝑒𝑡𝑎1ℎ

2 )

𝑍𝜋 + (

𝑑1𝑠 + 𝑑1ℎ𝑑1𝑠 − 𝑑1ℎ

) × (cos 𝑏𝑒𝑡𝑎1𝑠 + cos 𝑏𝑒𝑡𝑎1ℎ

2 )

Equation 3.32: Hydraulic diameter

3.4.3. Loading losses

Now there are the blade loading losses. As it is shown all the losses are adimmensional. The

method for the blade losses is by (Coppage, et al., 1956). Losses due to boundary layer growth and

separation and secondary flows.

∆ℎ𝑏𝑙 = 0.05 × 𝐷𝑓2

Equation 3.33: Blade loading losses

𝐷𝑓 = 1 −𝑊2

𝑊1𝑠+

0.75 × ∆ℎ𝑡ℎ × 𝑊2

[𝑍𝜋 (1 −

𝑑1𝑠𝑑2

) + 2 ×𝑑1𝑠𝑑2

]

Equation 3.34: Diffusion factor

∆ℎ𝑡ℎ =𝑈2 × 𝐶𝑡2

𝑈22

Equation 3.35: Specific enthalpy jump

3.4.4. Disk friction losses

For the disk friction losses, the (Galvas, 1973) method is used. An enthalpy rise due to the work

done on the fluid by shear between the rear face of the impeller and any adjacent stationary surface.

The calculation of the torque coefficient depends on the Reynolds number.

∆ℎ𝑑𝑓 = 0.25 ×𝜌𝐴𝑉𝐺 × 𝐾𝑓

4 × �̇�× 𝑑𝐴𝑉𝐺

2 × 𝑈2

Equation 3.36: Disk friction losses

35

𝐾𝑓 = 3.7 × (0.02/𝑟2)0.1/𝑅𝑒0.5 for Re < 3x105 𝐾𝑓 = 0.102 × (0.02/𝑟2)0.1/𝑅𝑒0.2 for Re > 3x105

Equation 3.37: Torque coefficient

3.4.5. Recirculation and clearance losses

For the recirculation and clearance losses the (Jansen & Qvale, 1967) method is used.

Recirculation is an additional enthalpy rise due to recirculation of low momentum fluid from the

vaneless space back into the impeller passage. Clearance are losses resulting from the leakage of

fluid from the pressure to the suction side of the blades of the unshrouded impellers.

∆ℎ𝑟𝑒 = 0.02 × 𝐷𝑓2 × √cot(𝛼2)

Equation 3.38: Recirculation losses

∆ℎ𝑐𝑙 = 0.6 ×0.02

𝑏2×

𝐶𝑡2

𝑈2

√4 × 𝜋

𝑏2 × 𝑍[

𝑟1𝑠2 + 𝑟1ℎ

2

(𝑟2 − 𝑟1𝑠) × (1 − 𝜌2 /𝜌1)] (

𝐶𝑡2

𝑈2) (

𝐶𝑚2

𝑈2 )

Equation 3.39: Clearance losses

After there is all the losses they must converge with the assumed efficiency. The parameter that

change is the efficiency.

3.5. Design of vaneless diffuser (stage 2-3)

After the impeller, there is the presence of a diffuser for the purpose of pressure recovery. Since

this is a unguided swirling flow in a stationary passage we can assume, in a frictionless flow that

there is a free vortex relationship. Nevertheless, the formulas developed by (Wallace, Baines, &

Whitfield, 1976) will be used that take into account the friction in the diffuser to be more exact.

As in the stationary duct in the inlet the relative temperature is conserved. We take this as a starting

point to calculate the properties of the flow exiting the diffuser.

Based on literature, a good starting point for the diffuser is to calculate a diffuser with a defined

ratio between the impeller discharge diameter and the diffuser diameter. This parameter will be

changed to search for the iteration with the highest efficiency.

𝑑3 = 𝑑2 × 𝑅𝑅

Equation 3.40: Radius ratio

36

The main assumptions are the value of the outlet flow angle and the Mach number at the outlet.

The two equations used to get the Mach number and the outlet flow angle are the following

equation and the continuity equation.

𝑠𝑖𝑛(𝛼3) =𝐶𝑡3 × [1 + (𝐺 − 1)/2 × 𝐴𝑀3

2]0.5

𝐴𝑀3 × √𝐺 × 𝑅 × 𝑇03

Equation 3.41: Flow angle equation

𝐶𝑡2

𝐶𝑡3=

𝑟3

𝑟2+

2 × 𝜋 × 𝐶𝑓 × 𝜌2 × 𝐶𝑡2 × (𝑟32 − 𝑟2 × 𝑟3)

�̇�

Equation 3.42: Vortex equation including shear effect

�̇� = 𝐴3 × 𝐶𝑚3 × 𝜌3

Equation 3.43: Continuity equation in the diffuser

Because we need the area at the outlet the value of the width of the outlet of the diffuser will be

imposed. In this case to be a bit smaller to the width of the impeller outlet. Since there is the

rotational speed. The value of the tangential velocity at the outlet of the diffuser can be obtained.

The adapted free vortex equation can be used with the friction considered and the assumption of

the flow angle can be used to complete the velocity triangle at the exit of the diffuser.

𝐶𝑓 = 0.02 × (1.8 × 105

𝑅𝑒)0.2

Equation 3.44: Friction coefficient

The following equation is used to get the properties in the discharge of the diffuser. There is three

unknowns: the flow angle, the Mach number and the tangential component of the absolute velocity

at the outlet of the diffuser. The two equations and the continuity equation are used to yield the

correct geometrical values of the diffuser.

𝑇03𝑅 = 𝑇02𝑅 + (𝑈32 − 𝑈2

2)/(2 × 𝐶𝑝)

Equation 3.45: Rothalpy conservation in the diffuser

An important parameter of the design of the diffuser is the ideal pressure recovery coefficient.

The formula for the pressure recovery coefficient depends on the diffuser discharge to inlet area,

37

labeled AR and the radius ratios of the diffuser, labeled RR. This parameter is a better indicator of

the efficiency of the diffuser than the actual calculation of the efficiency, so this value is calculated

to make sure there is an optimal design.

𝐶𝑃𝑖 = 𝑐𝑜𝑠(𝛼2)2 × (1 − 1/𝐴𝑅2) + 𝑠𝑖𝑛(𝛼2)2 × (1 − 1/𝑅𝑅2)

Equation 3.46: Ideal pressure recovery coefficient

𝐴𝑅 = 𝐴3/𝐴2

Equation 3.47: Area ratio

After solving the three equations there is the value of the velocity triangle and the properties at the

exit of the diffuser and the real pressure recovery coefficient can be calculated.

𝐶𝑃 =𝑃3 − 𝑃2

𝑃02 − 𝑃2

Equation 3.48: Pressure recovery coefficient

𝑒𝑓𝑓𝑑 =(

𝑃3𝑃2

)

(𝐺−1)𝐺

− 1

(𝑃3𝑒𝑃2

)

(𝐺−1)𝐺

− 1

Equation 3.49: Efficiency of the diffuser

After we get the geometrical values of the diffuser the losses in the diffuser can be calculated. The

losses are used to calculate the entropy jump in the diffuser. The entropy jump assumed at the

beginning of the calculation of the diffuser must converge with the value of the entropy jump

calculated with the losses in the diffuser. The entropy jump is calculated just as the entropy jump

calculated in the impeller. The only losses calculated for the diffuser are the friction losses. Once

there is convergence the efficiency of the diffuser can be calculated and get the new total efficiency

of the compressor.

∆𝑞 =𝐶𝑓 × 𝑟2 × [1 − (𝑟2/𝑟3)1.5](𝐶2/𝑈3)2

1.5 × 𝑏2 × 𝑐𝑜𝑠(𝛼2)

Equation 3.50: Friction losses in a vaneless diffuser (Coppage, et al., 1956)

38

Chapter 4

Working points of the compressor

Once there is the calculated impeller in the design point, three additional working points can be

calculated to make some performance graphs of the compressor. The geometrical parameters of

the compressor remain the same.

a. Shroud diameter

b. Hub diameter

c. Outlet diameter

d. Geometrical blade angles

e. Blade width

The aim of this calculation is to find the required mass flow to reach a desired pressure. The losses

in the point are calculated accordingly and give the efficiency of the compressor at this point.

The start of this calculation imposes the volumetric flow rate. The volumetric flow rate yields the

mass flow and the properties in the flange. The calculation of the density in the inlet of the flange

allows to calculate the mass flow along the machine. The properties in the inlet of the impeller

come from the conservation of rothalpy in stationary duct. The calculation of the velocity triangle

on the impeller inlet comes from the convergence of the mass flow conservation. The new velocity

triangle has a new β flow angle, which will affect the incidence losses. The optimum incidence

angle was calculated with the blade angle and remains constant.

𝑃01 × √2 × 𝐶𝑝 × (𝑇01 − 𝑇1)

𝑇1 × 𝑅 × (𝑇01𝑇1

)

𝐺𝐺−1⁄

=�̇�

𝐴1

Equation 4.1: Continuity equation in the flange

The calculation of the outlet begins making an assumption meridional outlet velocity in the

working point. This assumption was easier for this calculation, but another variable could have

been chosen. The losses calculated based on this velocity will be compared to the losses calculated

based on the correlations. Both values will converge, and we can calculate the resulting pressure

ratio for the imposed volumetric flow. The result will be evaluated to decide whether or not the

current compressor can reach the pressure ratios required.

39

𝑇02𝑅 = 𝑇01𝑅 + (𝑈22 − 𝑈1

2)/(2 × 𝐶𝑝)

Equation 4.2: Total relative outlet temperature in the impeller

The conservation of rothalpy yields the relative properties in the outlet. The assumed velocity gives

the velocity triangle in the outlet of the impeller. The remaining conditions starting from the

relative stagnation conditions and the real work can be obtained. Calculating the remaining

properties from the velocity triangle is crucial to achieve a cohesive calculation. The values of the

angles in the velocity triangle in the outlet will determine if the compressor will be able to achieve

every working point. If the angles are higher than 80 degrees then perhaps the losses will be too

big, and the pressure will not be achieved.

𝜌2 = �̇�/(𝐴2 × 𝐶𝑚2)

Equation 4.3: Mass conservation in the impeller for every point

The real work comes from the velocity triangle and the ideal work is calculated based on the

efficiency. The isentropic conditions in the impeller is finally calculated. The calculation of the

losses comes from the difference between the real and ideal enthalpy in the relative frame of

reference. This calculated value is the losses of a impeller working with the imposed velocity.

These losses do not reflect the imposed mass flow rate. The real losses are calculated with the

same correlations as the designed impeller. Both calculated losses should be the same. The

imposed velocity must be iterated until this condition is met. The new Mach number in the inlet

and discharge must be checked to avoid supersonic flow.

𝜎 = 𝑃02𝑅/ (𝑃01𝑅 × (𝑇02𝑅/𝑇01𝑅)𝐺

𝐺−1)

Equation 4.4: Calculation of entropy jump based on Rothalpy

∆ℎ =(1 − 𝜎

𝐺−1𝐺⁄ )

(𝐺 − 1) × 𝑈22 × (𝑅 × 𝐺 × 𝑇02𝑅)

Equation 4.5: Enthalpy losses based on entropy jump

40

𝑒𝑓𝑓𝑡𝑡 =((𝑃02/𝑃01)

(𝐺−1)𝐺⁄ ) − 1

𝑇02𝑇01

− 1

Equation 4.6: Efficiency calculation in the impeller

𝐴𝑀2 = 𝐶2/√𝑅 × 𝐺 × 𝑇2

Equation 4.7: Calculation of the Mach number in the outlet of the impeller

The choice in the meridional velocity as a variable was taken based on its effect in the mass

conservation equation and because this velocity remains constant in the calculation of the slip

factor effect in the impeller. The design uses the efficiency as a iterative variable. In this case, the

efficiency was not a possible variable because the real and ideal work were unknown. It is

suggested to use a component of the velocity triangle as an iterative value.

41

Chapter 5

Results of the design

5.1. Results of the impeller design

Following the previously described procedure, there are a resulting two compressors. Each

compressor has a different target pressure ratio. The inlet conditions for both compressors are the

same. The volumetric flow rate differs in order to get a better efficiency on the different target

points. The two target points are the middle working points of the previous machine used.

Starting properties Compressor 1 Compressor 2

Total inlet temperature (K) 288 288

Total inlet pressure (Pa) 101300 101300

Target pressure ratio in impeller 1.28 1.45

Volumetric flow rate (m3/h) 1000 900

Hub to shroud ratio 0.4 0.4 Table 5.1: Starting properties of the compressor

The resulting geometry based on the starting conditions yields two very different compressors.

The similitudes between the two are mostly deliberate, as they were a design choice. In order to

resolve the variables in the calculation, the software SOLVER was used to achieve the converge

of the equality equations. For example, for the equation that compares the imposed losses in the

impeller to the calculated losses in the impeller, an equation subtracting both is created. This

equation must yield the value of zero. The software iterates the value of the imposed losses until

the equation yields the desired value. The tolerance of the solver is of about 103. Therefore, a

number inferior to 0.001 is considered to be zero for all equations. The following table shows the

geometric parameters of the compressor.

Geometrical parameter Compressor 1 Compressor 2

Discharge diameter (cm) 20.57 18.66

Hub diameter (cm) 3.95 3.41

Shroud diameter (cm) 9.87 8.53

Blade width (cm) 1.59 0.69

Inlet blade angle (degrees) -55.1 -55.3

Discharge blade angle (degrees) -40 -40

Number of blades 15 15

42

Blade thickness (mm) 2.17 1.88 Table 5.2: Geometrical parameters of the compressor

The velocity triangles are featured in the following figure, the magnitudes are scaled. There is a

velocity triangle for the inlet one for the discharge. This impeller has backwards swept blades, as

is reflected in the blade angles sign. The following table shows the true magnitudes of the

velocities. The flow angles, which differ from the blade angle, are also depicted in both the

triangles and the table.

Velocity magnitude inlet Compressor 1 Compressor 2

Tangential velocity (m/s) 63.15 77.69

Absolute velocity (m/s) 44.05 53.81

Relative velocity (m/s) 77.00 94.51

Β flow angle (degrees) -55.1 -55.3 Table 5.3: Velocity triangle magnitude at the inlet

Velocity magnitude outlet Compressor 1 Compressor 2

Tangential velocity (m/s) 188.10 242.78

Tangential component of the absolute velocity (m/s) 136.73 157.81

Tangential component of the relative velocity (m/s) 51.37 84.98

Meridional component of the velocity (m/s) 25.25 54.85

Absolute velocity (m/s) 139.04 167.07

Relative velocity (m/s) 57.24 101.14

Β flow angle (degrees) -63.82 -57.16

Α flow angle (degrees) 79.54 70.83 Table 5.4: Velocity triangle magnitude at the outlet

43

Compressor 1 – Inlet triangle Compressor 2 – Inlet triangle

Compressor 1 – Outlet triangle

Compressor 2 – Outlet triangle

Table 5.5: Velocity triangles at the inlet and outlet of the compressors

The resulting stagnation properties are depicted in the following table. Even though the working

points are different, the difference in temperature depending on the pressure ratio can be compared.

In this table it is also featured the real and ideal work produced by the impeller.

Stagnation properties Compressor 1 Compressor 2

Total temperature - inlet of the impeller (K) 286.01 285.00

Total temperature - outlet of the impeller (K) 311.62 315.55

Total pressure - inlet of the impeller (Pa) 98877 97649

Total pressure - outlet of the impeller (Pa) 126448 141554 Table 5.6: Stagnation properties in the impeller

44

The enthalpy losses in the impeller can be an interesting start point to view the magnitude of the

real losses in the impeller. The only way to really know the actual magnitude of the losses in the

impeller is to simulate or experiment with a prototype. The calculation of the losses using the

correlations should give a very similar outcome as the simulation because the correlations are

based on experimental data. The following data gives a comparison of the different losses in the

impeller. All the losses are depicted in a relative frame of reference.

Losses in the impeller Compressor 1 Compressor 2

Incidence losses (m2/s2) 0.1364 0.2120

Skin friction losses 0.0409 0.0448

Blade loading losses 0.0158 0.0063

Disk friction losses 0.0386 0.0470

Recirculation losses 0.0342 0.0015

Clearance losses 0.0026 0.0039

Total losses 0.1322 0.1035

Entropy jump 0.9473 0.9339 Table 5.7: Dimensionless losses in the impeller

Finally, the performance parameters of the impellers can be viewed in the following table. The

chosen impeller is not simply the one with the highest efficiency. Another deciding factor is the

ability for the compressor to reach the desired working points.

Performance parameters in the impeller Compressor 1 Compressor 2

Specific diameter (m) 4.7 5.0

Specific speed (rad/s) 0.55 0.54

Rotational speed (RPM) 17465 24847

Slip factor 0.84 0.84

Efficiency 0.818 0.841 Table 5.8: Performance parameters in the impeller

A 3D model of the compressor 2 is featured in the appendix. We can locate both compressors in

the Balje diagram. Since they are intentionally placed near the best efficiency line, both points

seem to intersect.

45

Figure 5.1: Balje diagram with the design points of both compressors

5.2. Working points of the compressors

The choice of a compressor relays in the ability of the compressor to reach the desired working

points. The calculation of the working points begins with identifying the parameters which remain

constant along the working points. For example, the geometrical parameters and the inlet

parameters in the compressor. The following tables show the velocity triangles, performance

parameters and stagnation properties of the working points in the first and second compressor.

5.2.1. Compressor 1

Velocity magnitude inlet Point 1 Point 2 Point 3

Tangential velocity (m/s) 63.15 63.15 63.15

Absolute velocity (m/s) 84.52 44.25 22.02

Relative velocity (m/s) 105.50 77.11 66.88

Β flow angle (degrees) -36.77 -54.98 -70.77 Table 5.9: Velocity triangle magnitude in the inlet of the first impeller

46

Table 5.10: Velocity triangle magnitude in the outlet of the first impeller

Inlet

Point 1 Point 2 Point 3

Outlet

Point 1

Velocity magnitude outlet Point 1 Point 2 Point 3

Tangential velocity (m/s) 188.09 188.09 188.09

Tangential component of the absolute velocity

(m/s) 117.63 137.00 147.34

Tangential component of the relative velocity

(m/s) 70.46 51.09 40.75

Meridional component of the velocity (m/s) 48.01 24.92 12.60

Absolute velocity (m/s) 127.05 139.25 147.88

Relative velocity (m/s) 85.26 56.85 42.66

Β flow angle (degrees) -55.73 -64.00 -72.81

Α flow angle (degrees) 67.80 79.69 85.11

47

Point 2

Point 3

Table 5.11: Velocity triangles in the working points in the first compressor

Stagnation properties Point 1 Point 2 Point 3

Total temperature - inlet of the impeller (K) 286.01 286.01 286.01

Total temperature - outlet of the impeller (K) 308.04 311.67 313.60

Total pressure - inlet of the impeller (Pa) 98877 98877 98877

Total pressure - outlet of the impeller (Pa) 121648 128293 129518 Table 5.12: Stagnation properties in the first impeller

Losses in the impeller Point 1 Point 2 Point 3

Incidence losses (m2/s2) 313.09 0.08 104.16

Skin friction losses 0.0816 0.0408 0.0279

Blade loading losses 0.0131 0.0160 0.0198

Disk friction losses 0.0202 0.0388 0.0771

Recirculation losses 0.0033 0.0027 0.0023

Clearance losses 0.0029 0.0027 0.0021

Total losses 0.1300 0.1010 0.1322

Entropy jump 0.9482 0.9596 0.9473 Table 5.13: Dimensionless losses in the first impeller

Performance parameters Point 1 Point 2 Point 3

Volumetric flow rate (m3/h) 1900 1000 500

Specific diameter (m) 3.26 4.76 6.80

Specific speed (rad/s) 0.87 0.53 0.36

Power (kW) 14.02 8.72 4.71

Pressure ratio 1.20 1.27 1.28

48

Efficiency 0.792 0.861 0.831 Table 5.14: Performance parameters in the first impeller

5.2.2. Compressor 2

Velocity magnitude inlet Point 1 Point 2 Point 3 Point 4

Tangential velocity (m/s) 77.70 77.70 77.70 77.70

Absolute velocity (m/s) 108.97 83.95 53.81 26.75

Relative velocity (m/s) 133.84 114.39 94.51 82.17

Β flow angle (degrees) -35.49 -42.78 -55.29 -71.00 Table 5.15: Velocity triangle magnitude in the inlet of the second impeller

Velocity magnitude outlet Point 1 Point 2 Point 3 Point 4

Tangential velocity (m/s) 242.78 242.78 242.78 242.78

Tangential component of the absolute velocity

(m/s) 99.67 128.24 157.45 180.53

Tangential component of the relative velocity

(m/s) 143.11 114.54 85.33 62.26

Meridional component of the velocity (m/s) 124.13 90.08 55.27 27.77

Absolute velocity (m/s) 159.20 156.72 166.87 182.65

Relative velocity (m/s) 189.44 145.72 101.67 68.17

Β flow angle (degrees) -49.06 -51.82 -57.07 -65.96

Α flow angle (degrees) 38.76 54.92 70.66 81.25 Table 5.16: Velocity triangle magnitude in the outlet of the second impeller

49

Inlet

Point 1 Point 2

Point 3 Point 4

Outlet

Point 1

50

Point 2

Point 3

Point 4

Table 5.17: Velocity triangles in the working points in the second compressor

Table 5.18: Stagnation properties in the second impeller

Losses in the impeller Point 1 Point 2 Point 3 Point 4

Incidence losses (m2/s2) 586.72 169.66 0.21 157.99

Stagnation properties Point 1 Point 2 Point 3 Point 4

Total temperature - inlet of the impeller (K) 285.00 285.00 285.00 285.00

Total temperature - outlet of the impeller (K) 309.26 316.23 323.14 328.65

Total pressure - inlet of the impeller (Pa) 97649 97649 97649 97649

Total pressure - outlet of the impeller (Pa) 116685 129998 141820 147067

51

Skin friction losses 0.1244 0.0793 0.0448 0.0265

Blade loading losses 0.0005 0.0024 0.0063 0.0131

Disk friction losses 0.0219 0.0295 0.0470 0.0933

Recirculation losses 0.0002 0.0008 0.0015 0.0021

Clearance losses 0.0030 0.0037 0.0039 0.0034

Total losses 0.1600 0.1185 0.1035 0.1411

Entropy jump 0.8993 0.9247 0.9340 0.9108 Table 5.19: Dimensionless losses in the second impeller

Performance parameters Point 1 Point 2 Point 3 Point 4

Volumetric flow rate (m3/h) 1800 1400 900 450

Specific diameter (m) 2.92 3.74 5.00 7.25

Specific speed (rad/s) 1.36 0.83 0.54 0.36

Power (kW) 14.45 14.66 11.64 6.70

Pressure ratio 1.15 1.28 1.40 1.45

Efficiency 0.613 0.777 0.841 0.810 Table 5.20: Performance parameters in the second impeller

The first compressor has three additional working points. The second one only has two additional

ones because the fourth point was unable to be reached. The mass was imposed until the target

pressure was reached. It can be noted that the pressures achieved are not exactly the same as the

target pressures in the previous machine. The iteration of the mass aimed to achieve a similar

pressure with the aim of proving that the machine was able to reach the working point. If both

machines could have reached all points, then a bigger effort would have been made to compare

both machines in the same working points to determine, which one had the best average efficiency

in all the points.

The calculation of the iterative variable in this calculation was done with SOLVER as well. The

criteria was the same as the one used in the design of the impeller. For example, for the equation

that compares the imposed losses in the impeller to the calculated losses in the impeller, an

equation subtracting both is created. This equation must yield the value of zero. The software

iterates the value of the imposed losses until the equation yields the desired value. The tolerance

of the solver is of about 103. Therefore, a number inferior to 0.001 is considered to be zero for all

equations.

The working points were placed in the Balje diagram as depicted in the following picture. The way

the machines are placed in the diagram gives an idea of how the rest of the working points would

function and what would their efficiency be.

52

Figure 5.2: Balje diagram with the three working points of the first compressor

Figure 5.3: Balje diagram with the four working points of the second compressor

53

Inlet

Compressor 1 Compressor 2

Outlet

Compressor 1

Compressor 2

Table 5.21: All working points of the first and second compressor

54

In the following pictures we can compare the performance of the previous machine and the two

new compressors designed in the Balje diagram. As shown in previous chapters, the model

machine came with a serious of graphs, from which the starting parameters for the design where

extracted. Here is a side by side comparison between both compressors using the same graphs as

the previous machine.

Graphs of the first compressor:

Figure 5.4: Volumetric rate - pressure difference graph of first compressor

0

200

400

600

800

1000

1200

1400

1600

1800

2000

- 50.00 100.00 150.00 200.00 250.00 300.00 350.00

Q (

m3

/h)

delta P (mbar)

Volumetric flow rate vs pressure difference

1

2

3

55

Figure 5.5: Power - pressure difference graph of first compressor

Figure 5.6: Temperature difference rate - pressure difference graph of first compressor

-

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

- 50.00 100.00 150.00 200.00 250.00 300.00 350.00

Po

wer

(kW

)

delta P (mbar)

Power vs pressure difference

1

2

3

0

5

10

15

20

25

30

- 50.00 100.00 150.00 200.00 250.00 300.00 350.00

del

ta T

(K

)

delta P (mbar)

Temperature difference vs pressure difference

1

2

3

56

Graphs of the second compressor:

Figure 5.7: Volumetric flow rate - pressure difference graph of second compressor

Figure 5.8: Power - pressure difference graph of second compressor

0

200

400

600

800

1000

1200

1400

1600

1800

2000

- 100.00 200.00 300.00 400.00 500.00 600.00

Q (

m3

/h)

delta P (mbar)

Volumetric flow rate vs pressure difference

1

2

3

4

-

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

- 100.00 200.00 300.00 400.00 500.00 600.00

Po

wer

(kW

)

delta P (mbar)

Power vs pressure difference

1

2

3

4

57

Figure 5.9: Temperature difference rate - pressure difference graph of second compressor

From this point forward, we will no longer discuss the first compressor. The second compressor

was the only one, which was able to reach all the working points. Therefore, the chosen compressor

for this application is the second compressor. If the first compressor were able to reach all four

points, the decision would require extra analysis and calculation. However, we can choose the

second compressor safely knowing that the first compressor fails to reach a goal of the design.

5.3. Vaneless diffusor

The design of the diffusor was only made on the second compressor for reasons previously stated.

The diffusor pressure recovery coefficient is the most important parameter in the calculation. This

calculation uses at least three iterative variables. As well as in the previous design case, the

SOLVER software was used. The criteria was the same as the one used in the design of the

impeller. For example, for the equation that compares the imposed losses in the impeller to the

calculated losses in the impeller, an equation subtracting both is created. This equation must yield

the value of zero. The software iterates the value of the imposed losses until the equation yields

the desired value. The tolerance of the solver is of about 103. Therefore, a number inferior to 0.001

is considered to be zero for all equations.

The main concern with the diffusor was achieving the best pressure recovery possible without

making the width of the diffusor unrealizable. Since the width of the blade in the discharge of the

impeller was already very narrow, the width of the diffusor is slightly narrower and does not

change along the iterations. The diameter is set to be 1.8 times bigger than the discharge diameter.

-

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

- 100.00 200.00 300.00 400.00 500.00 600.00

del

ta T

(K

)

delta P (mbar)

Temperature differerence vs pressure difference

1

2

3

4

58

Later on, it is changed to improve efficiency and not avoid having an odd shape in the compressor.

The first choice was based on the literature. The following table shows the geometrical parameters

of the diffusor.

Geometrical parameter Diffuser Outlet

Discharge diameter (cm) 33.59

Diffuser width (cm) 0.621 Table 5.22: Geometrical parameters of the diffuser

Velocity magnitude outlet Point 1 Point 2 Point 3 Point 4

Tangential component of the absolute

velocity (m/s) 50.44 60.46 64.19 55.01

Meridional component of the velocity

(m/s) 71.24 51.45 31.49 15.89

Absolute velocity (m/s) 87.29 79.39 71.50 57.26

Α flow angle (degrees) 35.30 49.60 63.87 73.89 Table 5.23: Velocity components of the diffuser

Stagnation properties Point 1 Point 2 Point 3 Point 4

Total temperature - outlet of the

impeller (K) 309.26 316.23 323.14 328.65

Total pressure - outlet of the impeller

(Pa) 116053 128732 139083 144859

Table 5.24: Stagnation properties of the diffuser

The losses and the performance parameters of the diffuser are shown in the following table. The

efficiency is calculated to be able to calculate the overall efficiency of the compressor. The

efficiency seems to increase with a narrower passage but since we must take into account the

construction viability of the duct, the design was left as it was.

Performance parameters Point 1 Point 2 Point 3 Point 4

Pressure recovery coefficient 0.65 0.66 0.68 0.62

Losses 0.0108 0.0143 0.0276 0.0712

Entropy jump 0.9928 0.9907 0.9825 0.9562

Efficiency 0.929 0.909 0.858 0.723

Overall efficiency of the compressor 0.569 0.706 0.722 0.586 Table 5.25: Performance parameters of the diffuser

59

Chapter 6

Conclusions

- The values of the specific speed and specific diameter are 0.54 and 5, respectively. These

dimensionless parameters situate the compressor in an optimal point in the Balje diagram.

This point should have a very high efficiency, based on empirical results. The use of

dimensionless parameters determines the expected shape of the compressor as well. The

stage flow coefficient is 0.37, this value confirms our desire to have a thin compressor, in

the axial direction.

- The design of a compressor must be done using a middle working point as a design point.

In this case, between the four working points, the third larger pressure ratio point was

chosen. The choice depends on the ability of the resulting compressor to reach all the

desired working points. If the compressor does not reach the points in the impeller then it

is impossible for the compressor to reach the desired pressure after the diffuser. The

pressure ratio between the inlet of the flange and the outlet of the diffuser is 1.35. The

pressure difference in the impeller is usually 10% higher than between the flange and the

diffuser. Therefore, the design starting from the flange was adapted to be able reach the

desired pressure ratio. This process is iterative and depends on your design choices.

- The working points of the blower are situated in the left side of the Balje diagram. The

desired outcome is to move the working points from the rotating machine section to the

turbo compressor section. The desired points are reached increasing the rotating velocity

and the diameter of the outlet of the impeller. The rotating velocity of the machine has to

be at least doubled to be able to place the working points in the high efficiency section of

the Balje diagram. The dimensions of the blower cannot be used as a guide to dimension

the compressor because there both different machines.

- The incidence losses in the compressor depend on the optimum incidence angle. The losses

increase when difference between the incidence angle of the working point and the

optimum incidence angle increases. In this case, the second working point is the one with

the highest incidence losses. This is due to the fact that the angle differs from the optimum

angle but also the relative velocity is higher as you aim for a lower pressure difference. The

working points of the compressor have an increase loss in skin friction for lower pressures.

This increase has to do with the increase in the average relative velocity in the impeller.

The working points of the compressor have an increase loss in blade loading, disk friction

and recirculation for higher pressures.

60

First Appendix: Calculation of the design of the

compressor

Property/Equation Variable Equation

Relative Mach number inlet AM1R Imposed

Inlet β shroud angle Β1s cos2(𝑏𝑒𝑡𝑎1𝑠) =

3 + 𝐺 × 𝐴𝑀1𝑅2

2 × 𝐴𝑀1𝑅2{1

− [1 −4 × 𝐴𝑀1𝑅2

(3 + 𝐺 × 𝐴𝑀1𝑅2)2]

1/2

}

Absolute Mach number AM1 𝐴𝑀1 = 𝐴𝑀1𝑅 × cos(𝑏𝑒𝑡𝑎1𝑠)

Total to static inlet

temperature

T01T1 𝑇01𝑇1 = 1 + (𝐺 − 1)/2 × 𝐴𝑀12

Total to static inlet pressure P01P1 𝑃01𝑃1 = (𝑇01𝑇1)𝐺

𝐺−1

Diameter of the shroud D1s Imposed

Volumetric flow rate Q0 Imposed

Flange inlet area A0 𝐴0 = 𝜋𝑑1𝑠2/4

Flange velocity C0 𝐶0 = 𝑄0/𝐴0

Total relative inlet flange

temperature

T00R 𝑇00𝑅 = 𝑇00

Total relative inlet flange

pressure

P00R 𝑃00𝑅 = 𝑃00

Total inlet impeller

temperature

T01 𝑇01 = 𝑇1 +𝐶12

2 × 𝐶𝑝

Total inlet speed of sound A01 𝐴01 = √𝑅 × 𝐺 × 𝑇01

Total inlet impeller pressure P01 𝑃01 = 𝑃1 × (𝑇01/𝑇1)𝐺

𝐺−1

Total inlet density Rho01 𝑟ℎ𝑜01 = 𝑃01/(𝑅 × 𝑇01)

Shroud to hub ratio ANU Assumed

Hub diameter D1h 𝑑1ℎ = 𝐴𝑁𝑈 × 𝑑1𝑠

Area at inlet of the impeller A1 𝐴1 = 𝜋(𝑑1𝑠2 + 𝑑1ℎ2)/4

Inlet of impeller temperature T1 𝑇1 = 𝑇00𝑅/(1 + (𝐺 − 1)/2 × 𝐴𝑀1𝑅2)

Inlet of impeller pressure P1 𝑃1 =𝑃00𝑅

(𝑇00𝑅

𝑇1)

𝐺𝐺−1

Inlet flange static temperature T0 𝑇0 = 𝑇00 +𝐶02

2 × 𝐶𝑝

61

Inlet flange static pressure P0 𝑃0 =𝑃00𝑅

(𝑇00𝑅

𝑇0)

𝐺𝐺−1

Inlet flange density Rho0 𝑟ℎ𝑜0 = 𝑃0/(𝑅 × 𝑇0)

Inlet impeller density Rho1 𝑟ℎ𝑜1 = 𝑃1/(𝑅 × 𝑇1)

Inlet impeller relative velocity W1 𝑊1 = 𝐴𝑀1𝑅 × √𝑇1 × 𝑅 × 𝐺

Inlet impeller velocity C1 𝐶1 = W1 × cos(𝑏𝑒𝑡𝑎1𝑠)

Equation for shroud diameter - 𝑄0 × 𝐶0 = 𝐶1 × 𝑟ℎ𝑜1 × 𝐴1

Mass flow rate M 𝑚 = 𝐶0 × 𝑄0

Specific diameter Ds Imposed from Balje

Specific speed Omegas 𝑜𝑚𝑒𝑔𝑎𝑠 = 𝑜𝑚𝑒𝑔𝑎 × √𝑄0/𝑙𝑖𝑑𝑒𝑎𝑙3/4

Ideal specific work Lideal 𝑙𝑖𝑑𝑒𝑎𝑙 = 𝐶𝑝 × 𝑇01 × (𝑃𝑅𝐺−1

𝐺 − 1)

Efficiency Efftt Iterative variable

Real specific work Lreal 𝑙𝑟𝑒𝑎𝑙 = 𝑙𝑖𝑑𝑒𝑎𝑙/𝑒𝑓𝑓𝑡𝑡

Rotational speed Omega 𝑜𝑚𝑒𝑔𝑎 = 𝑈1/(𝑑1/2)

Outlet impeller diameter D2 𝑑2 = 𝐷𝑠/𝑙𝑖𝑑𝑒𝑎𝑙1/4 × √𝑄0

Peripheral outlet velocity U2 𝑈2 = 𝑑22 × 𝜋/4

Peripheral inlet velocity U1 𝑈1 = 𝐶1 × tan(𝑏𝑒𝑡𝑎1)

Number of blades Nb Imposed

Outlet blade angle Β2b Imposed

Slip factor SF 𝑆𝐹 = 1 −𝜋 × cos 𝑐𝑜𝑠𝑏1𝑠𝑏

𝑁𝑏

Effect of the slip Cslip 𝐶𝑠𝑙𝑖𝑝 = 𝑈2 × (1 − 𝑆𝐹)

Tangential component of the

absolute velocity

Ct2 𝐶𝑡2 = 𝑙𝑟𝑒𝑎𝑙/𝑈2

Geometrical tangential

component of the absolute

velocity

C2geo 𝐶𝑡2𝑔𝑒𝑜 = 𝐶𝑡2 + 𝐶𝑠𝑙𝑖𝑝

Meridional component of the

absolute velocity

Cm2 𝐶𝑚2 = (𝐶𝑡2 − 𝑈2 × 𝑆𝐹)/ tan(𝑏𝑒𝑡𝑎2𝑏)

Tangential component of the

relative velocity

Wt2 𝑊𝑡2 = 𝑈2 − 𝐶𝑡2

Absolute velocity C2 𝐶2 = √𝐶𝑡22 + 𝐶𝑚22

Outlet flow angle α2 𝑎𝑙𝑝ℎ𝑎2 = tan−1𝐶𝑡2

𝐶𝑚2

Outlet blade flow angle Β2 𝑏𝑒𝑡𝑎2 = tan−1𝑊𝑡2

𝐶𝑚2

Total relative inlet

temperature

T01R 𝑇01𝑅 = 𝑇00𝑅

Total relative outlet

temperature

T02R 𝑇02𝑅 = 𝑇01𝑅 + (𝑈22 − 𝑈12)/(2 × 𝐶𝑝)

62

Relative outlet velocity W2 𝑊2 = √𝑊𝑡22 + 𝑊𝑚22

Total relative outlet enthalpy H02R ℎ02𝑅 = 𝐶𝑝 × 𝑇02𝑅

Outlet static enthalpy H2 ℎ2 = ℎ02𝑅 − 𝑊22/2

Outlet static temperature T2 𝑇2 = ℎ2/𝐶𝑝

Total inlet enthalpy H01 ℎ01 = 𝑇01 × 𝐶𝑝

Outlet total enthalpy H02 ℎ02 = ℎ2 + 𝐶22/2

Isentropic total outlet

enthalpy

H02s ℎ02𝑠 = ℎ01 + 𝑙𝑖𝑑𝑒𝑎𝑙

Isentropic static outlet

enthalpy

H2s ℎ2𝑠 = ℎ02𝑠 − 𝐶22/2

Isentropic total relative outlet

enthalpy

H02Rs ℎ02𝑅𝑠 = ℎ2𝑠 + 𝑊22/2

Enthalpy difference deltah 𝑑𝑒𝑙𝑡𝑎ℎ = ℎ02𝑅 − ℎ02𝑅𝑠

Entropy jump Sigma 𝑠𝑖𝑔𝑚𝑎 = (1 − (𝐺 − 1)/(𝑅 × 𝐺 × 𝑇02𝑅) × 𝑈22 × 𝑑𝑒𝑙𝑡𝑎ℎ)𝐺

𝐺−1

Total relative inlet pressure P01R 𝑃01𝑅 = 𝑃1 × (𝑇01𝑅/𝑇1)𝐺

𝐺−1

Total relative outlet pressure P02R 𝑃02𝑅 = 𝑃01𝑅 × 𝑠𝑖𝑔𝑚𝑎/ ((𝑇01𝑅/𝑇02𝑅)𝐺

𝐺−1)

Absolute outlet Mach number AM2 𝐴𝑀2 = 𝐶2/√𝑅 × 𝐺 × 𝑇2

Relative outlet Mach number AM2R 𝐴𝑀2𝑅 = 𝑊2/√𝑅 × 𝐺 × 𝑇2

Outlet static pressure P2 𝑃2 = 𝑃02𝑅/(𝑇02𝑅/𝑇2)𝐺

𝐺−1

Total outlet temperature T02 𝑇02 = 𝑇2 + 𝐶22/(2 × 𝐶𝑝)

Total outlet pressure P02 𝑃02 = 𝑃2 × (𝑇02/𝑇2)𝐺

𝐺−1

Static outlet density Rho2 𝑟ℎ𝑜2 = 𝑃2/(𝑅 × 𝑇2)

Blade width in the outlet B2 𝑏2 = 𝑚/(𝑟ℎ𝑜2 × 𝑑2 × 𝜋 × 𝐶𝑚2)

Shroud peripheral velocity U1s 𝑈1𝑠 = 𝑜𝑚𝑒𝑔𝑎 × 𝑑1𝑠/2

Hub peripheral velocity U1h 𝑈1ℎ = 𝑜𝑚𝑒𝑔𝑎 × 𝑑1ℎ/2

Absolute shroud velocity C1s 𝐶1𝑠 = U1s × cot(𝑏𝑒𝑡𝑎1𝑠)

Relative shroud velocity W1s 𝑊1𝑠 = √𝐶1𝑠2 + 𝑈1𝑠2

Average relative velocity in

the inlet

Wfl 𝑊𝑓𝑙 = √(𝑊12 + 𝑊22)/2

Average density rho_AVG 𝑟ℎ𝑜_𝐴𝑉𝐺 = √(𝑟ℎ𝑜12 + 𝑟ℎ𝑜22)/2

Average velocity C_AVG 𝐶_𝐴𝑉𝐺 = √(𝐶12 + 𝐶22)/2

Reynolds number Re 𝑅𝑒 = 𝑑ℎ𝑏 × 𝐶𝐴𝑉𝐺 × 𝑟ℎ𝑜𝐴𝑉𝐺/𝜇2

Blockage factor BF Imposed

Blade passage inlet angle A1p 𝐴1𝑝 = 𝐵𝐹 × 𝐴1

Blade inlet angle Ap 𝐴𝑝 = 𝐴1 − 𝐴1𝑝

Optimum inlet β angle Β1opt 𝑏𝑒𝑡𝑎1𝑜𝑝𝑡 = tan−1(𝐴1/𝐴1𝑝 × tan(𝑏𝑒𝑡𝑎1𝑏))

Incidence losses Deltahinc 𝑑𝑒𝑙𝑡𝑎ℎ𝑖𝑛𝑐 = 𝑊12 × sin(𝑏𝑒𝑡𝑎1𝑠 − 𝑏𝑒𝑡𝑎1𝑜𝑝𝑡)2 × 0.6/2

Skin friction losses deltahsf 𝑑𝑒𝑙𝑡𝑎ℎ𝑠𝑓 = 2 × 𝐶𝑓 × 𝐿𝑏/𝐷ℎ𝑏 × (𝑊𝑓𝑙/𝑈2)2

63

Hydraulic length Lb 𝐿𝑏

=𝜋

8× [𝑑2 −

𝑑1𝑠 + 𝑑1ℎ

2− 𝑏2 + 2 × 𝐿𝑧]

× (2

cos(𝑏𝑒𝑡𝑎1𝑠) + cos(𝑏𝑒𝑡𝑎1ℎ)2

+ cos(𝑏𝑒𝑡𝑎2))

Dhb Dhb 𝑑ℎ𝑏

=𝑑2 × cos(𝑏𝑒𝑡𝑎2)

[𝑍𝜋

+𝑑2 × cos(𝑏𝑒𝑡𝑎2)

𝑏2]

+1/2 × (

𝑑1𝑠𝑑2

+𝑑1ℎ𝑑2

) × (cos 𝑏𝑒𝑡𝑎1𝑠 + cos 𝑏𝑒𝑡𝑎1ℎ

2)

𝑍𝜋

+ (𝑑1𝑠 + 𝑑1ℎ𝑑1𝑠 − 𝑑1ℎ

) × (cos 𝑏𝑒𝑡𝑎1𝑠 + cos 𝑏𝑒𝑡𝑎1ℎ

2)

Blade loading losses deltahbl 𝑑𝑒𝑙𝑡𝑎ℎ𝑏𝑙 = 0.05 × 𝐷𝑓2

Diffusion factor Df 𝐷𝑓 = 1 −𝑊2

𝑊1𝑠+

0.75 × 𝑑𝑒𝑙𝑡𝑎ℎ𝑡ℎ × 𝑊2

[𝑍𝜋

(1 −𝑑1𝑠𝑑2

) + 2 ×𝑑1𝑠𝑑2

]

Specific enthalpy jump deltahth 𝑑𝑒𝑙𝑡𝑎ℎ𝑡ℎ =𝑈2 × 𝐶𝑡2

𝑈22

Torque coefficient Kf 𝐾𝑓 = 3.7 × (0.02/𝑟2)0.1/𝑅𝑒0.5 for Re < 3x105

𝐾𝑓 = 0.102 × (0.02/𝑟2)0.1/𝑅𝑒0.2 for Re > 3x105

Disk friction losses deltahdf 𝑑𝑒𝑙𝑡𝑎ℎ𝑑𝑓 = 0.25 ×𝑟ℎ𝑜𝐴𝑉𝐺 × 𝐾𝑓

4 × 𝑚× 𝑑𝐴𝑉𝐺

2 × 𝑈2

Recirculation losses Deltahre 𝑑𝑒𝑙𝑡𝑎ℎ𝑟𝑒 = 0.02 × 𝐷𝑓2 × √cot(𝑎𝑙𝑝ℎ𝑎2)

Clearance losses deltahcl 𝑑𝑒𝑙𝑡𝑎ℎ𝑐𝑙

= 0.6 ×0.02

𝑏2

×𝐶𝑡2

𝑈2√

4 × 𝜋

𝑏2 × 𝑍[

𝑟1𝑠2 − 𝑟1ℎ2

(𝑟2 − 𝑟1𝑠) × (1 + 𝑟ℎ𝑜2/𝑟ℎ𝑜1)] (

𝐶𝑡2

𝑈2) (

𝐶𝑚2

𝑈2)

Diffuser outlet diameter D3 𝑑3 = 𝑑2 × 𝑅𝑅

Outlet flow angle Α3 Iterative variable

Diffuser outlet Mach number AM3 Iterative variable

Diffuser outlet width B3 Imposed

Outlet diffuser area A3 𝐴3 = 𝑏3 × 𝑑3 × 𝜋

Tangential diffuser outlet

velocity

U3 𝑈3 = 𝑑3 × 𝑜𝑚𝑒𝑔𝑎/2

Area ratio in the diffuser AR 𝐴𝑅 = 𝐴3/𝐴2

Tangential component of the

absolute velocity in the outlet

of the diffuser

Ct3 Iterative variable

Absolute velocity in the outlet

of the diffuser

C3 𝐶3 = 𝐶𝑡3/ sin(𝑎𝑙𝑝ℎ𝑎3)

Density average in the

diffuser

rho_DAVG 𝑟ℎ𝑜_𝐷𝐴𝑉𝐺 = √(𝑟ℎ𝑜22 + 𝑟ℎ𝑜32)/2

64

Velocity average in the

diffuser

C_DAVG 𝐶_𝐷𝐴𝑉𝐺 = √(𝐶22 + 𝐶32)/2

Diameter average in the

diffuser

D_DAVG 𝐷_𝐷𝐴𝑉𝐺 = √(𝐷22 + 𝐷32)/2

Friction coefficient Cf 𝐶𝑓 = 0.02 × (1.8 × 105

𝑅𝑒)0.2

Tangential component of the

relative velocity in the outlet

of the diffuser

Wt3 𝑊𝑡3 = 𝑈3 − 𝐶𝑡3

Meridional component of the

absolute velocity in the outlet

of the diffuser

Cm3 𝐶𝑚3 = √(𝐶32 − 𝐶𝑡32)

Relative velocity in the outlet

of the diffuser

W3 𝑊3 = √(𝑊𝑡32 − 𝐶𝑚32)

Blade angle at the outlet of the

diffuser

Β3 tan−1(𝑊𝑡3/𝐶𝑚3)

Total relative outlet

temperature in the diffuser

T03R 𝑇03𝑅 = 𝑇02𝑅 + (𝑈32 − 𝑈22)/(2 × 𝐶𝑝)

Enthalpy loss in the diffuser Deltah_D Iterative variable

Total isentropic relative outlet

temperature in the diffuser

T03Re 𝑇03𝑅𝑒 = 𝑇03𝑅 − 𝑑𝑒𝑙𝑡𝑎ℎ𝐷 × 𝑈32/𝐶𝑝

Total absolute outlet

temperature in the diffuser

T03 𝑇03 = 𝑇03𝑅 −𝑊32

2 × 𝐶𝑝+

𝐶32

2 × 𝐶𝑝

Static absolute outlet

temperature in the diffuser

T3 𝑇3 = 𝑇03𝑅 −𝑊32

2 × 𝐶𝑝

Equation of Ct3 - 𝐶𝑡2

𝐶𝑡3=

𝑟3

𝑟2+

2 × 𝜋 × 𝐶𝑓 × 𝑟ℎ𝑜2 × 𝐶𝑡2 × (𝑟32 − 𝑟2 × 𝑟3)

𝑚

Equation of α3 - sin(𝑎𝑙𝑝ℎ𝑎3) =𝐶𝑡3 × [1 + (𝐺 − 1)/2 × 𝐴𝑀32]0.5

𝐴𝑀3 × √𝐺 × 𝑅 × 𝑇03

Reynolds number Re 𝑅𝑒 = 𝑑𝐷𝐴𝑉𝐺 × 𝐶𝐷𝐴𝑉𝐺 × 𝑟ℎ𝑜𝐷𝐴𝑉𝐺 /𝜇3

Entropy jump in the diffuser SigmaD 𝑠𝑖𝑔𝑚𝑎𝐷 = (1 − (𝐺 − 1)/(𝑅 × 𝐺 × 𝑇03𝑅) × 𝑈32

× 𝑑𝑒𝑙𝑡𝑎ℎ𝐷)𝐺

𝐺−1

Total relative outlet pressure

in the diffuser

P03R 𝑃03𝑅 = 𝑃02𝑅 × 𝑠𝑖𝑔𝑚𝑎𝐷/ ((𝑇02𝑅/𝑇03𝑅)𝐺

𝐺−1)

Outlet static pressure in the

diffuser

P3 𝑃3 = 𝑃03𝑅/(𝑇03𝑅/𝑇3)𝐺

𝐺−1

Equation of continuity - 𝑚 = 𝐴3 × 𝐶𝑚3 × 𝑟ℎ𝑜3

Density in the outlet of the

diffuser

Rho3 𝑟ℎ𝑜3 =𝑃3

𝑅 × 𝑇3

Ideal pressure recovery

coefficient

CPi 𝐶𝑃𝑖 = cos(𝑎𝑙𝑝ℎ𝑎2)2 × (1 − 1/𝐴𝑅2)

+ sin(𝑎𝑙𝑝ℎ𝑎2)2 × (1 − 1/𝑅𝑅2)

65

Pressure recovery coefficient CP 𝐶𝑃 =𝑃3 − 𝑃2

𝑃02 − 𝑃2

Losses in the diffuser deltaq 𝑑𝑒𝑙𝑡𝑎𝑞 =𝐶𝑓 × 𝑟2 × [1 − (𝑟2/𝑟3)1.5](𝐶2/𝑈3)2

1.5 × 𝑏2 × cos(𝑎𝑙𝑝ℎ𝑎2)

66

Second Appendix: Calculation of the other

working points

Property/Equation Variable Equation

Pressure ratio PR Design parameter

Outlet diameter D2 Geometry parameter

Hub diameter D1h Geometry parameter

Shroud diameter D1s Geometry parameter

Blade width B2 Geometry parameter

Outlet blade angle Β2 Geometry parameter

Volume flow rate Q0 Input

Inlet flange velocity C0 𝐶0 = 𝑄0/𝐴0

Static inlet flange temperature T0 𝑇0 = 𝑇00 − 𝐶02/(2 × 𝐶𝑝)

Static inlet flange pressure P0 𝑃0 = 𝑃00/(𝑇00/𝑇0)𝐺

𝐺−1

Inlet flange density Rho0 𝑟ℎ𝑜0 = 𝑃0/(𝑅 × 𝑇0)

Mass flow M 𝑚 = 𝑄0 × 𝑟ℎ𝑜0

Total relative inlet impeller

temperature T01R 𝑇01𝑅 = 𝑇00

Relative inlet velocity W1

𝑃01 × √2 × 𝐶𝑝 × (𝑇01 − 𝑇1)

𝑇1 × 𝑅 × (𝑇01𝑇1

)

𝐺𝐺−1⁄

=𝑚

𝐴1

Inlet of impeller static

temperature T1 𝑇1 = 𝑇01𝑅 − 𝑊12/(2 × 𝐶𝑝)

67

Inlet impeller velocity C1 𝐶1 = √𝑊12 − 𝑈12

Total inlet impeller

temperature T01 𝑇01 = 𝑇1 + 𝐶12/(2 × 𝐶𝑝)

Inlet β angle Β1 𝑏𝑒𝑡𝑎1 = sin−1(𝑈1/𝑊1)

Total inlet enthalpy H01 ℎ01 = 𝑇01 × 𝐶𝑝

Total relative inlet impeller

pressure P01R 𝑃01𝑅 = 𝑃00

Total relative outlet

temperature T02R 𝑇02𝑅 = 𝑇01𝑅 + (𝑈22 − 𝑈12)/(2 × 𝐶𝑝)

Total relative outlet enthalpy H02R ℎ02𝑅 = 𝑇02𝑅 × 𝐶𝑝

Inlet of impeller static pressure P1 𝑃1 = 𝑃01𝑅/(𝑇01𝑅/𝑇1)𝐺

𝐺−1

Inlet impeller density Rho1 𝑟ℎ𝑜1 = 𝑃1/(𝑅 × 𝑇1)

Meridional component of the

absolute velocity Cm2 Iterative value

Tangential component of the

absolute velocity Ct2 𝐶𝑡2 = 𝑈2 × 𝑆𝐹 + 𝐶𝑚2 × tan(𝑏𝑒𝑡𝑎2𝑏)

Real specific work Lreal 𝑙𝑟𝑒𝑎𝑙 = 𝑈2 × 𝐶𝑡2

Total outlet enthalpy H02 ℎ02 = ℎ01 + 𝑙𝑟𝑒𝑎𝑙

Total outlet temperature T02 𝑇02 = ℎ02/𝐶𝑝

Tangential component of the

relative velocity Wt2 𝑊𝑡2 = 𝑈2 − 𝐶𝑡2

Absolute velocity C2 𝐶2 = √𝐶𝑡22 + 𝐶𝑚22

Flow outlet angle Α2 𝑎𝑙𝑝ℎ𝑎2 = tan−1(𝐶𝑡2/𝐶𝑚2)

Relative outlet velocity W2 𝑊2 = √𝑊𝑡22 + 𝐶𝑚22

Flow β outlet angle β2 𝑏𝑒𝑡𝑎2 = tan−1(𝑊𝑡2/𝐶𝑚2)

68

Outlet static enthalpy H2 ℎ2 = ℎ02 − 𝐶22/2

Outlet static temperature T2 𝑇2 = ℎ2/𝐶𝑝

Outlet impeller density Rho2 𝑟ℎ𝑜2 = 𝑚/(𝐴2 × 𝐶𝑚2)

Outlet static pressure P2 𝑃2 = 𝑟ℎ𝑜2 × 𝑅 × 𝑇2

Total outlet pressure P02 𝑃02 = 𝑃2 × (𝑇02/𝑇2)𝐺

𝐺−1

Total relative outlet pressure P02R 𝑃02𝑅 = 𝑃2 × (𝑇02𝑅/𝑇2)𝐺

𝐺−1

Entropy jump Sigma 𝑠𝑖𝑔𝑚𝑎 = 𝑃02𝑅/ (𝑃01𝑅 × (𝑇02𝑅/𝑇01𝑅)𝐺

𝐺−1)

Enthalpy losses deltah 𝑑𝑒𝑙𝑡𝑎ℎ =(1 − 𝑠𝑖𝑔𝑚𝑎

𝐺−1𝐺⁄ )

(𝐺 − 1) × 𝑈22× (𝑅 × 𝐺 × 𝑇02𝑅)

Isentropic total relative outlet

enthalpy H02Rs ℎ02𝑅𝑠 = ℎ02𝑅 − 𝑑𝑒𝑙𝑡𝑎ℎ × 𝑈22

Isentropic total relative outlet

temperature T02Rs 𝑇02𝑅𝑠 = ℎ02𝑅𝑠/𝐶𝑝

Efficiency Efftt 𝑒𝑓𝑓𝑡𝑡 =((𝑃02/𝑃01)

(𝐺−1)𝐺⁄ ) − 1

𝑇02𝑇01

− 1

Ideal specific work Lideal 𝑙𝑖𝑑𝑒𝑎𝑙 = 𝑙𝑟𝑒𝑎𝑙 × 𝑒𝑓𝑓𝑡𝑡

Isentropic total absolute outlet

enthalpy H02s ℎ02𝑠 = ℎ01 + 𝑙𝑖𝑑𝑒𝑎𝑙

Isentropic total outlet

temperature T02s 𝑇02𝑠 = ℎ02𝑠/𝐶𝑝

Relative Mach number inlet AM1R 𝐴𝑀1𝑅 = 𝑊1/√𝑅 × 𝐺 × 𝑇1

Absolute Mach number AM1 𝐴𝑀1 = 𝐴𝑀1𝑅 × cos(𝑏𝑒𝑡𝑎1𝑠)

Shroud peripheral velocity U1s 𝑈1𝑠 = 𝑜𝑚𝑒𝑔𝑎 × 𝑑1𝑠/2

Hub peripheral velocity U1h 𝑈1ℎ = 𝑜𝑚𝑒𝑔𝑎 × 𝑑1ℎ/2

Absolute shroud velocity C1s 𝐶1𝑠 = U1s × cot(𝑏𝑒𝑡𝑎1𝑠)

69

Relative shroud velocity W1s 𝑊1𝑠 = √𝐶1𝑠2 + 𝑈1𝑠2

Average relative velocity in the

inlet Wfl 𝑊𝑓𝑙 = √(𝑊12 + 𝑊22)/2

Average density rho_AVG 𝑟ℎ𝑜_𝐴𝑉𝐺 = √(𝑟ℎ𝑜12 + 𝑟ℎ𝑜22)/2

Average velocity C_AVG 𝐶_𝐴𝑉𝐺 = √(𝐶12 + 𝐶22)/2

Reynolds number Re 𝑅𝑒 = 𝑑ℎ𝑏 × 𝐶𝐴𝑉𝐺 × 𝑟ℎ𝑜𝐴𝑉𝐺 /𝜇2

Blockage factor BF Imposed

Blade passage inlet angle A1p 𝐴1𝑝 = 𝐵𝐹 × 𝐴1

Blade inlet angle Ap 𝐴𝑝 = 𝐴1 − 𝐴1𝑝

Optimum inlet β angle Β1opt 𝑏𝑒𝑡𝑎1𝑜𝑝𝑡 = tan−1(𝐴1/𝐴1𝑝 × tan(𝑏𝑒𝑡𝑎1𝑏))

Incidence losses Deltahinc 𝑑𝑒𝑙𝑡𝑎ℎ𝑖𝑛𝑐 = 𝑊12 × sin(𝑏𝑒𝑡𝑎1𝑠 − 𝑏𝑒𝑡𝑎1𝑜𝑝𝑡)2 × 0.6/2

Skin friction losses deltahsf 𝑑𝑒𝑙𝑡𝑎ℎ𝑠𝑓 = 2 × 𝐶𝑓 × 𝐿𝑏/𝐷ℎ𝑏 × (𝑊𝑓𝑙/𝑈2)2

Hydraulic length Lb

𝐿𝑏

=𝜋

8× [𝑑2 −

𝑑1𝑠 + 𝑑1ℎ

2− 𝑏2 + 2 × 𝐿𝑧]

× (2

cos(𝑏𝑒𝑡𝑎1𝑠) + cos(𝑏𝑒𝑡𝑎1ℎ)2

+ cos(𝑏𝑒𝑡𝑎2))

Dhb Dhb

𝑑ℎ𝑏

=𝑑2 × cos(𝑏𝑒𝑡𝑎2)

[𝑍𝜋

+𝑑2 × cos(𝑏𝑒𝑡𝑎2)

𝑏2]

+1/2 × (

𝑑1𝑠𝑑2

+𝑑1ℎ𝑑2

) × (cos 𝑏𝑒𝑡𝑎1𝑠 + cos 𝑏𝑒𝑡𝑎1ℎ

2)

𝑍𝜋

+ (𝑑1𝑠 + 𝑑1ℎ𝑑1𝑠 − 𝑑1ℎ

) × (cos 𝑏𝑒𝑡𝑎1𝑠 + cos 𝑏𝑒𝑡𝑎1ℎ

2)

Blade loading losses deltahbl 𝑑𝑒𝑙𝑡𝑎ℎ𝑏𝑙 = 0.05 × 𝐷𝑓2

Diffusion factor Df 𝐷𝑓 = 1 −

𝑊2

𝑊1𝑠+

0.75 × 𝑑𝑒𝑙𝑡𝑎ℎ𝑡ℎ × 𝑊2

[𝑍𝜋

(1 −𝑑1𝑠𝑑2

) + 2 ×𝑑1𝑠𝑑2

]

Specific enthalpy jump deltahth 𝑑𝑒𝑙𝑡𝑎ℎ𝑡ℎ =𝑈2 × 𝐶𝑡2

𝑈22

Torque coefficient Kf 𝐾𝑓 = 3.7 × (0.02/𝑟2)0.1/𝑅𝑒0.5 for Re < 3x105

𝐾𝑓 = 0.102 × (0.02/𝑟2)0.1/𝑅𝑒0.2 for Re > 3x105

Disk friction losses deltahdf 𝑑𝑒𝑙𝑡𝑎ℎ𝑑𝑓 = 0.25 ×𝑟ℎ𝑜𝐴𝑉𝐺 × 𝐾𝑓

4 × 𝑚× 𝑑𝐴𝑉𝐺

2 × 𝑈2

Recirculation losses Deltahre 𝑑𝑒𝑙𝑡𝑎ℎ𝑟𝑒 = 0.02 × 𝐷𝑓2 × √cot(𝑎𝑙𝑝ℎ𝑎2)

70

Clearance losses deltahcl

𝑑𝑒𝑙𝑡𝑎ℎ𝑐𝑙

= 0.6 ×0.02

𝑏2

×𝐶𝑡2

𝑈2√

4 × 𝜋

𝑏2 × 𝑍[

𝑟1𝑠2 − 𝑟1ℎ2

(𝑟2 − 𝑟1𝑠) × (1 + 𝑟ℎ𝑜2/𝑟ℎ𝑜1)] (

𝐶𝑡2

𝑈2) (

𝐶𝑚2

𝑈2)

71

Third Appendix: Flow diagram for the impeller

design

Inlet area

•Calculation of the inlet based on the maximization of the mass flow

•Assume a relative inlet Mach number and a shroud inlet blade angle

•Considering a isentropic flange

•Calculation of the diameter of the shroud based chosen mass flow rate

Calculation of the velocity

triangle

•Assume an efficiency, outlet diameter, number of blades and a outlet blade angle

•Calculation of the ideal work based on the pressure ratio and calculation of the velocity triangle.

Calculation of the outlet properties

•Calculation of the relative outlet total temperature based on rothalpy

•Calculation of the isentropic total relative enthalpy based on the enthalpy losses

•Calculation of the relative total pressure based on the losses

Calculation of the losses in

the compressor

•Incidence losses, skin friction losses, blade loading losses, disk friction losses, recirculation losses and clearance losses

Recalculation of the assumed

parameters

•Entropy jump and efficiency of the compressor

72

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