# Patent

## Title
Mini notched turbine generator

## Abstract

The present invention is turbine generator capable of integration into a bio-physiological or microfluidic system. The generator can convert biomechanical energy into electrical energy by using electromagnetic subsystems to transform the kinetic energy to electricity. These systems have the potential to convert hydraulic energy (such as flow of body fluid, blood flow, contraction of blood vessel, dynamic fluid in nature) into electric energy that may be sufficient for self-powering nano/micro devices and systems, such as artificial organs, valves, sensors, micro motors, and micro robots. The system incorporates a new turbine model having, notched blades; a rotor in levitation; and a special casing capable of integration into a bio-physiological or microfluidic system.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This nonprovisional application claims priority to provisional application No. 61/883,702, entitled “MINI NOTCHED TURBINE GENERATOR,” filed Sep. 27, 2013 by the same inventors.


BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to turbine generators, and more specifically to a turbine generator capable of integration into a bio-physiological or microfluidic system.
2. Brief Description of the Related Art
Currently, there are many devices, such as a cochlear implant, a cardiac pacemaker, or an artificial heart, that can be integrated into the human body to enhance the quality of life or allow for the continuation of life. While these implantable devices do improve the quality of life, they are limited due to their use of batteries. The problem is not only that the battery is made from toxic chemicals, but also that the battery life is finite. At the end of the battery's life, the patient with the implant must have the battery surgically replaced. Repeated surgeries are expensive and may adversely affect the patient's quality of life.
Accordingly, there is a need for an efficient means of powering or recharging these implantable devices. The development of a turbine generator capable of integration into a bio-physiological or microfluidic system would provide a solution.
However, in view of the art considered as a whole at the time the present invention was made, it was not obvious to those of ordinary skill in the field of this invention how the shortcomings of the prior art could be overcome.
SUMMARY OF THE INVENTION
The long-standing but heretofore unfulfilled need for turbine generator capable of integration into a bio-physiological or microfluidic system is now met by a new, useful, and nonobvious invention.
The novel structure includes a turbine generator having a turbine casing, a rotor, and a magnet generator assembly that is specifically designed to reduce pressure differentials and turbulent flow. The turbine casing includes an inlet section, a nozzle fluidly coupled to the inlet section, a rotor chamber fluidly coupled to the nozzle, and an outlet section fluidly coupled to the rotor chamber. Additionally, the turbine casing incudes hollows adapted to receive inductor coils and the inlet section and the outlet section are separated by one hundred degrees.
The rotor includes a central hub, a plurality of blades, and two support discs, and is adapted for receipt by the rotor chamber. Each blade has a central transversal axis, a distal edge, a proximal edge attached to the central hub, and a notch disposed along the central transversal axis adjacent to the proximal edge. The support discs are located on either sides of the blades and are axially aligned with the rotor's rotational axis.
The magnet generator assembly includes a plurality of PMs disposed on the rotor and a plurality of inductor coils disposed in the hollow of the turbine casing. The inductor coils are arranged in an orientation perpendicular to the centrifugal movement of the PMs in the rotor to induce a current.
In a certain embodiment, the present invention includes a levitation system. The levitation system comprises of a plurality of permanent magnets disposed in a first wall of the turbine casing, a plurality of permanent magnets disposed in a second wall of the turbine casing, and a plurality of permanent magnets disposed in the rotor, wherein the permanent magnets disposed in the rotor and turbine casing are configured to magnetically suspend the rotor structure in the rotor chamber such that the rotor structure is not in contact with the turbine casing.
In a certain embodiment, the plurality of inductor coils include a 3-phase, 9-slot winding layout, where each phase has three circular coils separated by 120 degrees in a star configuration and connected in series.
In a certain embodiment, the hub of the rotor has a cylindrical shape with a radius that is generally equal to half of a total radius of the rotor. In a certain embodiment, the notches in each blade are semicircular in shape and have a radius equal to or less than half of the radius of one of the blades.
In a certain embodiment, the present invention uses a turbine casing designed for a bio-physiological or microfluidic system.
These and other important objects, advantages, and features of the invention will become clear as this disclosure proceeds.
The invention accordingly comprises the features of construction, combination of elements, and arrangement of parts that will be exemplified in the disclosure set forth hereinafter and the scope of the invention will be indicated in the claims.



BRIEF DESCRIPTION OF THE DRAWINGS
The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.
For a fuller understanding of the invention, reference should be made to the following detailed description, taken in connection with the accompanying drawings, in which:
FIG. 1A is a perspective view of the inlet section of the turbine casing.
FIG. 1B is a side view of the inlet section of the turbine casing and a portion of the rotor chamber.
FIG. 1C is provides an illustration of a fluid leaving the nozzle and impacting a blade.
FIG. 1D is an illustration of theoretical velocity triangles at different locations on a blade.
FIG. 1E is an illustration of the theoretical fluid forces on internally notched blades.
FIG. 2 is a perspective view of a certain embodiment of the rotor having PMs located in the support discs to aid in rotor levitation.
FIG. 3A is an illustration of the geometry of a certain embodiment of the rotor.
FIG. 3B is an illustration of the geometry of a certain embodiment of the rotor and the notches.
FIG. 4 is a perspective view of a certain embodiment of the inlet section of the turbine casing.
FIG. 5A is a side cutout view of a certain embodiment of the turbine casing and the rotor.
FIG. 5B is a side cutout view of a certain embodiment of the turbine casing without the rotor.
FIG. 5C is a side cutout view of a certain embodiment of the turbine casing having PMs for rotor levitation.
FIG. 6A is an illustration of the theoretical variable affecting the rotor's rotation.
FIG. 6B is an illustration of the theoretical velocity triangles on the first blade.
FIG. 6C is an illustration of the theoretical velocity triangles on the second blade.
FIG. 7A is a perspective view of a certain embodiment of the rotor having a ring of PMs in the rotor support discs.
FIG. 7B is a top view of the rotor in FIG. 7A.
FIG. 8A is a perspective view of a certain embodiment of the turbine casing illustrating a certain location for the coils.
FIG. 8B is a perspective view of FIG. 8A when the turbine casing is closed.
FIG. 8C is a perspective cutout view of a certain embodiment of the turbine casing and the rotor.
FIG. 9 is a graph of the electromotive force response of a mathematical model of the present invention.
FIG. 10 is graph providing the characterization of a HDD motor used as PM-Generator
FIG. 11 is a turbine flow simulation representing the 3.175× non-notch model with a 90-degree separation between the inlet and outlet sections of the turbine casing.
FIG. 12 is a turbine flow simulation representing the 4× non-notch model with a 90 degree separation between the inlet and outlet sections of the turbine casing.
FIG. 13 is a turbine flow simulation representing the 3.175× notched model.
FIG. 14A is a graph of turbine simulations time response of systems behavior using volume flow rate Q=2.5 gallons per minute. The turbine model used is the same as FIG. 11, 3.175× non-notch model, with a 90-degree separation between the inlet and outlet sections of the turbine casing.
FIG. 14B is a graph of turbine simulations time response of systems behavior using volume flow rate Q=2.5 gallons per minute. The turbine represented is the 4× notch lacking model of FIG. 12, the 4× non-notched model having a 90-degree separation between the inlet and outlet section of the turbine casing.
FIG. 14C is a graph of turbine simulations time response of systems behavior using volume flow rate Q=2.5 gallons per minute. The turbine represented is the 3.175× notched model.
FIG. 15 is a graph showing voltage versus volume flow rate for the three tested models.
FIG. 16 is a graph showing volume flow rate versus angular velocity for the three tested models.
FIG. 17 is a graph showing voltage versus volume flow rate for notched blade model.
FIG. 18 is a side view showing a certain embodiment of the levitation system.



DETAILED DESCRIPTION OF THE INVENTION
In the following detailed description of the preferred embodiments, reference is made to the accompanying drawings, which form a part thereof, and within which are shown by way of illustration specific embodiments by which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the invention.
Glossary of Claim Terms
Cross-Flow Turbine: is an impulse turbine where the water passes through the turbine transversely, or across the turbine blades.
Impulse Turbine: is a turbine that is driven by jets directed against turbine blades.
Inductor Coil: is a passive two-terminal electrical component that resists changes in electric current passing through it.
Internally Notched Blade: is a blade where the notch is on the edge of the blade nearest the axis of rotation (proximal edge).
Magnet Generator: is a system that uses the changes in the magnetic field to induce current in inductor coils.
Permanent Magnet: is a magnet that retains its magnetic properties in the absence of an inducing field or current.
Rotor Chamber: is a housing for the rotor structure in the turbine casing.
Rotor Structure: is a mechanical piece consisting of the hub, notched blades, and two rotor support discs on either side of the blades.
Total Radius: is the radius to the outermost edge of the rotor.
Turbine Casing: is a housing for the turbine consisting of an inlet section, a nozzle, a rotor chamber, an outlet section.
Turbine: is a device that can extract energy from a fluid, such as water, steam, air, or combustion gases.
The present invention provides a new option of green energy conversion; focusing on the use of turbines to extract energy from microfluidics, with diverse application in biomedical, military/aerospace, and home areas. These systems have the potential of converting mechanical movement energy, and hydraulic energy into electric energy that may be sufficient for self-powering micro-sized devices and micro-sized systems. A flow, with constant pressure, a magnetic generator, and a novel impulse turbine design are combined to form a self-contained miniaturized generator system. The turbine consists of two main parts: a bearingless rotor and the enclosure or casing; while the miniaturized magnetic generator is a permanent magnet brushless machine, preferably comprising of permanent magnets in a ring configuration and radial coils. A permanent pressure, from microfluidic pressure system, is the force used to move the blades. This rotational motion of the turbine is transformed into electricity using magnetic induction, formed by permanent magnets on the rotor and nine coils fixed in the holder of the turbine. The electricity is generated when the magnetic field rotates and moves past the conductor, which induces a current according to Faraday's Law. The system has potential uses not only in medical equipment, but also in automotive applications, home appliances, and aquatic and ventilation systems.
The present invention is a turbine generator, that utilizes an impulse turbine design, notched blades, a rotor in levitation, and special turbine casing designed for a bio-physiological or microfluidic system. The preferred embodiment contains a cross-flow turbine design and can be installed and used in different orientations and positions; in contrast to the more standard turbine designs and energy generating systems, which only work in one position and have limited orientations.
Mathematical Background
Equations Governing Flow and Pressure
The principle of fluid machines is centered on the utilization of useful work due to the force applied by a fluid jet, impacting and moving, over a series of blades attached to a rotating wheel that spins about an axis. The equations of flow and pressure in a tube flow system, such as in FIG. 1A, are defined through the equations of continuity of steady flow (mass conservation) and momentum equations [15], [69]. These equations may be written as follows:

ΣρV·A=0  1.1

−ρ1A1ν1+ρ2A2ν2=0

ρ1A1ν1=ρ2A2ν2  1.2

The variables ρ1, A1, and ν1 are density, transversal area, and velocity respectively. The principles of conservation of mass and momentum, which predicts that a fluid of constant volume and flow rate, with density p, and flowing during a period of time through an area, produces simultaneously an increase in the mean velocity and a decrease in pressure at the outlet of the system.
To determine the rate of change of momentum in FIGS. 1B and 1C, it is important to analyze the momentum of fluid entering and leaving in the stream conical pipe in the time Δt. The volume of the fluid in time Δt, is moved a distance νiΔt, and the momentum equations can be written as

Momentum of fluid entering=mass×inlet velocity

{dot over (m)}iνinΔt=ρA1νinνinΔt=ρA1νin2Δt  1.3

Momentum of fluid leaving=mass×oulet velocity

{dot over (m)}oνnΔt=ρAnνnνnΔt=ρAnνn2Δt  1.4

If equations 1.3 and 1.4 are combined with Newton's second law, is possible to find the total force supplied by the fluid. The law states that the force is equal to the rate of change of momentum [69], [70], [73], and can be written as




  
    
      
        F
        =
        
          
            (
            
              
                
                  A
                  n
                
                ⁢
                
                  v
                  n
                  2
                
                ⁢
                Δ
                ⁢
                
                  
                
                ⁢
                t
              
              -
              
                ρ
                ⁢
                
                  
                
                ⁢
                
                  A
                  i
                
                ⁢
                
                  v
                  in
                  2
                
                ⁢
                Δ
                ⁢
                
                  
                
                ⁢
                t
              
            
            )
          
          
            Δ
            ⁢
            
              
            
            ⁢
            t
          
        
      
    
    
      1.5
    
  




The equation 1.5 defines the force inside of the conical pipe, when it is acting in the direction of the flow of the fluid. The volume flow rate (Q) will be considered constant through the system, if the fluid is incompressible, density ρ is constant, and one-dimensional [69], [70], [73]. The expression governing Q will be written as

Q=Aiνin=Anνn  1.6

where nozzle area is defined as





  
    
      
        
          A
          n
        
        =
        
          π
          *
          
            (
            
              
                D
                n
              
              2
            
            )
          
          ⁢
          
            (
            
              m
              2
            
            )
          
        
      
    
    
      1.7
    
  




Therefore, the Q and An equations can be used to rewrite the force equation as

F=Qρ(νn−νin)  1.8

The Bernoulli equation defines the fluid dynamic relationship between fluid velocity (v), fluid pressure (p), and height (h) [69], [70], which describe the behavior of fluids in a tube, but some assumptions such as flow viscosity=0 (internal friction), constant density (ρ), steady flow, and Incompressible flow, must be defined to assure the applicability of this equation. The analysis of forces and velocities in a turbine could be developed using similar shapes and variables, shown in FIG. 1A-1C, and reported by [69], [74]-[77].
Bernoulli's equation [69], [70], is defined as:




  
    
      
        
          
            p
            in
          
          +
          
            
              pv
              in
              2
            
            2
          
          +
          
            ρ
            ⁢
            
              
            
            ⁢
            
              gh
              1
            
          
        
        =
        
          
            p
            n
          
          +
          
            
              ρ
              ⁢
              
                
              
              ⁢
              
                v
                n
                2
              
            
            2
          
          +
          
            ρ
            ⁢
            
              
            
            ⁢
            
              gh
              2
            
          
        
      
    
    
      1.9
    
  




Where

    ρ=fluid density (kg/m3);
    g=acceleration due to gravity on Earth (m/s2);
    h=height from an arbitrary point in the direction of gravity (m).
    pin=Inlet pressure (N/m2, Pa).
    pn=Nozzle pressure (N/m2, Pa).
    νin=mean inlet velocity (m/s).
    νn=mean nozzle velocity (m/s).


If h1=h2, then the equation is reduced to:




  
    
      
        
          
            ρ
            2
          
          ⁢
          
            (
            
              
                v
                n
                2
              
              -
              
                v
                in
                2
              
            
            )
          
        
        =
        
          
            p
            in
          
          -
          
            p
            n
          
        
      
    
    
      1.10
    
  




On the other hand, the potential power or power extracted [69], [70], [73] from the turbine nozzle (Pn) is

pn=pgQh  1.11

But,




  
    
      
        
          v
          n
        
        =
        
          
            2
            ⁢
            gh
          
        
      
    
    
      1.12
    
  




to find an expression to power (Pn), in function of known and defined variables, combine Q, Pn, and νn equations:





  
    
      
        
          p
          n
        
        =
        
          
            ρ
            ⁢
            
              
            
            ⁢
            
              A
              n
            
            ⁢
            
              v
              n
              2
            
          
          2
        
      
    
    
      1.13
    
  




The velocity triangle shown in FIG. 1D is typically used to explain the interaction between the fluid and the blades in a turbine, and determines the components of velocities and the forces acting on the rotor. The rotor is moved by the action of forces from the nozzle jet stream, which impacts the blades. The kinetic energy is transformed to mechanical energy, producing a continuous spin of the rotor. The mass of the liquid striking the blades per second [78]-[81] is given by

{dot over (m)}=ρAnVn  1.14

Where An is the transversal area of the nozzle, Vn is the velocity of the fluid from nozzle to blades, and ρ is the density of the liquid. The product of mass of the fluid striking per second and the component of velocity impacting the blades in tangential direction (Vwi) is known as the momentum of the fluid striking the blades [69], [73] and the general equation is given by

Mi={dot over (m)}Vwi=ρAnVnVwi  1.15

Similar momentums are produced by the fluid from the nozzle, striking the blade, and the fluid striking the outlet of each blade. The new components are Vw1 in the tangential direction of the blade inlet, and Vw2 in the tangential direction of the outlet of each blade. The velocity components can be positive or negative. If it contributes to the movement, it is positive, otherwise it will be negative.
If the momentum of the liquid at the nozzle and the momentum leaving the blade are combined as a product of radial distances, the results will be the angular momentum per second at the inlet and at the outlet of the blade.

MAi=ρAnVnVwiR1  1.16

MA2=ρAnVnVwiR2  1.17

The equations 1.19 and 1.20 are related in the impulse momentum theorem [69], [70], [82], which states that the rate of change of angular momentum is equal to the torque on the wheel or torque on the rotor. The equation is given by

T=ρAnVnVwiR1±ρAnVnVwiR2 

T=ρAnVn(VwiR1±VwiR2)  1.18

In addition, the torque equation could be written as,

T=FtR1  1.19

Where Ft is the total force applied to each blade, in a direction tangential to the rotor, and Ri is the distance or radius from which the forces are applied, measured from rotor axis of rotation.
Power from the turbine (Pr) is a function that relates torque and angular velocity, and is expressed in the Euler turbomachine equation [47], [69], [70]. The power delivered to the fluid is thus,

Pr=Tω  1.20

The angular velocity ω and torque T are variables that characterize the rotor behavior, which were defined in equations 1.18 and 1.21,




  
    
      
        ω
        =
        
          
            U
            i
          
          
            R
            i
          
        
      
    
    
      1.21
    
  




The Ui variables are the blade tangential velocities, measured in different points with radius Ri measured from the axis of rotation to each defined point on the blade. Finally, the power from the rotor is a relationship between rotor torque (T) and angular velocity (ω). The rotor torque is in direct relation with the forces impacting the blades, especially the blades located between the nozzle output and the turbine output zone as shown in FIG. 1E.
Finally, all equations are related in one expression known as efficiency of the system (η) [69], [70], [77], which reflects the system's behavior, showing the percent of losses into the system, when the potential energy is converted to mechanical energy. The efficiency is calculated with the following relation,




  
    
      
        
          Efficiency
          ⁢
          
            
              
            
            ⁢
            
              
            
          
          (
          η
          )
        
        =
        
          
            P
            r
          
          
            P
            n
          
        
      
    
    
      1.22
    
  




Equations Describing Electromagnetic Field
There are constitutive relations in electromagnetic systems which describe how two or more physical parameters are related, depending on the medium's properties involved in the process. The relationship between electric field intensity (E) and the conductive current due to the movements of free charges (J) is given by the microscopic form of the Ohm's law, J=σE and D=∈E, where σ and ∈ are electric conductivity and dielectric permittivity; also, the relationship between the magnetic induction or magnetic flux density (B), the magnetic field strength (H), and the magnetic polarization (M) are given by B=μH and M=XH, where μ and X are the magnetic permeability and magnetic susceptibility [13], [14], [83], [84]. Assuming, σ, ∈, μ, and X describe the electromagnetic properties of a material, then an electromagnetic phenomena can be described by four relationships known as Maxwell's equations [13], [14], [84], which describe how objects can be influenced by the propagation of electric and magnetic fields and how the system response could be in accordance with that kind of electromagnetic input. The simpler form preferred by authors in books [13], [14] and papers [12], [85], [86] are written as,




  
    
      
        
          ∇
          
            ×
            
              H
              _
            
          
        
        =
        
          
            J
            _
          
          +
          
            
              ∂
              D
            
            
              ∂
              t
            
          
        
      
    
    
      1.23
    
  




This equation is known as Ampere's law, and





  
    
      
        
          ∇
          
            ×
            
              E
              _
            
          
        
        =
        
          -
          
            
              ∂
              B
            
            
              ∂
              t
            
          
        
      
    
    
      1.24
    
  




is known as Faraday's law. Now, using a form of the Gaussian theorem and the relationship [87]-[89], the next two equations are derived. The magnetic Gauss's law states that if the magnetic source is contained in a closed surface, the net magnetic flux crossing through this surface is zero,

∇·B=0  1.25

The relationship between magnetic field intensity H and magnetic field density B can be written as

B=μH=(μrμ0)H 

Where μr is the relative permeability of the medium (unit less) and μ0 is the magnetic permeability in vacuum (μ0=4π×10−7 henry/m) [87]-[89].

Finally, the magnetic Gauss's law may be derived from Faraday's and Ampere's laws going from open surface to become a closed surface. This Maxwell equation is defined in terms of electric flux density and electric charge density, to state how the electric field acts around electric charges.

∇·D=ρ  1.27

The total magnetic flux density B inside the magnetic material and the presence of an external magnetic field H, are related through magnetization M, and can be written as

∇×B=μ0(H+M)  1.28

Faraday discovered the phenomenon, known as electromagnetic induction, which provides that an electric current is induced, in a wire in loop shape, by changing the magnetic field. Faraday's law states that an electromotive force (emf) is generated, if for any reason, the magnetic flux changes with the time. The average emf, induced in a coil of N turns is expressed in the following equation:




  
    
      
        ɛ
        =
        
          
            -
            
              N
              ⁡
              
                (
                
                  
                    ∅
                    -
                    
                      ∅
                      0
                    
                  
                  
                    t
                    -
                    
                      t
                      0
                    
                  
                
                )
              
            
          
          =
          
            
              
                -
                N
              
              ⁢
              
                
              
              ⁢
              
                Δ∅
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            =
            
              
                -
                ω
              
              ⁢
              
                
              
              ⁢
              
                Δ∅
                Δθ
              
            
          
        
      
    
    
      1.29
    
  




Where N is the number of turns on each coil, ω is the angular velocity, and Ø is the magnetic flux. The magnetic flux is the total flux through a surface. Ø is found by integrating the flux density over this area

Ø=∫BdA  1.30

If the flux density is constant through this area, the above equation will be expressed as,

Ø={right arrow over (B)}·{right arrow over (A)}=BA×cos θ  1.31

Where θ is the angle between {right arrow over (B)} and the normal vector from a transversal coil area {right arrow over (A)}, so if the spatially uniform magnetic field {right arrow over (B)} is used, this equation will be written as

Ø=−BRsαtlt  1.32

Where Rs is the radius of the stator surface, αt is the coil arc, expressed in radians, and lt is the effective axial length of each coil. The combination of flux density and magnetic field produces the induced voltage equation, which can be written as




  
    
      
        ɛ
        =
        
          
            -
            
              N
              ⁡
              
                [
                
                  
                    
                      d
                      
                        
                      
                    
                    
                      d
                      t
                    
                  
                  ⁢
                  
                    (
                    
                      BA
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    )
                  
                
                ]
              
            
          
          =
          
            -
            
              N
              ⁡
              
                [
                
                  
                    
                      (
                      
                        
                          d
                          B
                        
                        
                          d
                          t
                        
                      
                      )
                    
                    ⁢
                    A
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                  +
                  
                    
                      B
                      ⁡
                      
                        (
                        
                          
                            d
                            A
                          
                          
                            d
                            t
                          
                        
                        )
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                  -
                  
                    BA
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      θ
                      ⁡
                      
                        (
                        
                          
                            d
                            θ
                          
                          
                            d
                            t
                          
                        
                        )
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      1.33
    
  




As is stated in Faraday's law [90], [91], the three components of the induced voltage equation determine that an emf can be induced, by modifying at least one of the three parameters involved in this expression, the magnetic field, the coil transversal area, or the angle between {right arrow over (A)} and {right arrow over (B)}. In this research work, only the case when the magnetic field is rotated around the coils was considered, resulting in the following equation,




  
    
      
        ɛ
        =
        
          
            -
            
              N
              ⁡
              
                [
                
                  
                    
                      d
                      
                        
                      
                    
                    
                      d
                      t
                    
                  
                  ⁢
                  
                    (
                    
                      BA
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    )
                  
                
                ]
              
            
          
          =
          
            
              -
              
                N
                ⁡
                
                  [
                  
                    
                      (
                      
                        
                          d
                          B
                        
                        
                          d
                          t
                        
                      
                      )
                    
                    ⁢
                    A
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                  ]
                
              
            
            =
            
              -
              
                N
                ⁡
                
                  [
                  
                    
                      (
                      
                        
                          d
                          B
                        
                        
                          d
                          t
                        
                      
                      )
                    
                    ⁢
                    
                      R
                      s
                    
                    ⁢
                    
                      α
                      t
                    
                    ⁢
                    
                      l
                      t
                    
                  
                  ]
                
              
            
          
        
      
    
    
      1.34
    
  




This equation represents the variation of the electromagnetic field into the coils, which induces currents into coils and generates emf (voltage) in the coils terminals.
Faraday's law of induction is a basic law of electromagnetism, which supposes that a magnetic field will create an electric current, relating to the operating principles of generators, electrical motors and transformers; also, Lenz's law reaffirms the applicability of Faraday's law, which states that the induced emf in the coil, results from a changing magnetic flux, and has a polarity that leads to an induced current, which in turn tends to oppose the change in magnetic flux that induces such currents.
Permanent Magnets (PMs) Brushless Machines
PMs brushless machines are electromechanical systems that can be operated as generators or motors. There are different topologies of PM brushless machines [90], [92], [96], [97], but these can be grouped in two general types: radial flux permanent magnets (RFPM) and axial flux permanent magnets (AFPM), which in turn can be classified into slotted or slotless, and internal or external rotor. Although various rotor and stator topologies may be employed to develop motor and generator machines, the most common RFPM and AFPM use configurations of permanent magnets attached on the rotor, and the coil or armature winding mounted on the stator. The magnetic distribution and flux flow direction are the principal differences of AFPM and RFPM machines. The magnetic field in RFPM machines flows radially between rotor and stator, while the magnetic field in AFPM flows axially in the direction parallel to the vertical axis of the rotor.
Permanent-magnet brushless turbine generators are machines (PMBLM) that produce electricity through variation magnetic field created by moving PMs, which induce currents within stator windings. The core benefit of a PMBLM is that it does not require any additional or external excitation current.
Example of the Present Invention
The design of the present invention includes three main parts—the turbine system, the motor generator system, and the integration of both in a closed system.
Turbine System
The turbine is divided in two main parts: rotor and holder or casing. The rotor is a new model of impulse turbine to be immersed, which additionally uses reaction characteristics to contribute to the rotor spin. As shown in FIG. 2, a certain embodiment of rotor structure 100 includes central hub 102, two support discs 104 either side of central hub 102, and notched blades 106 arranged concentrically around central hub 102 supported by rotor support discs 104. Support discs 104 assure blade stability, and canalize and redirect the fluid flow to central hub 102 or internal notch 108 of each blade 106. Complementally, the top and bottom of the rotor have a special design, which joins the hub and the blades in a unique piece, making the blade stronger and increasing its resistance. In addition, the top and bottom design helps to support the assembling of the spindle machine, which converts the mechanical energy to electric energy. Further support for the improvements created from the notched design is incorporated in the following sections. FIG. 2 also includes PM receptacles 107, which area adapted to receive PMs that correspond to PMs in the rotor housing to allow for rotor levitation, resulting in frictionless rotation. As provided in FIG. 2, these PM receptacles are preferably located in the outer surface of support discs 104.
As shown in FIG. 3A, the blades have curved forms to improve the capabilities of the design. The intersection point between the reference line and the radius of the internal circle R3, plus the blade thickness, defines the shape and curvature of each blade. FIG. 3A provides an exemplary geometry, where the reference line, denoted by reference element “C”, is 45° degrees, the radius of the inner circle or hub 102 (R3=2.15 mm) is half of the total turbine radius (R1=4.3 mm), the blade thickness (about 0.4 mm) is 10% of the total turbine radius. The internal blade radius R=1.9 mm, and the internal blade arc is 90° degrees, measured between intersection point “A” (intersection point of the vertical axis and the circle of radius R2) and intersection point “B” (intersection point of reference line and the circle of radius R3, which represents the hub). R2 is equivalent to the total turbine radius R1 minus the blade thickness. The design is scalable so that the system can be used for any objective.
As highlighted in FIG. 3B, blades 106 include semicircular notch 108 on the internal proximal edge of blades 106. The notches are centrally located about the tangential axis of the turbine. The radius of the notches arc is equal to or less than half of the radius of a blade. In this exemplary embodiment, blade tangential reference circle RBTR is drawn having a diameter of 6.45 mm, such that the circle intersects the notch along the internal blade arc at point B2. A tangent line “T” is drawn from point B2 and the tangent line intersects the circle of radius R2 at point A2, which marks the location of the next blade. The tangent line provides the angle at which the notch is created in the blade.
The internally notched blades permit a continuous circulation of fluid inside the rotor chamber, which decreases the possibility of clogging, fouling, or stagnation of the flow. As part of the reaction benefits, the notch also assures more interaction between the fluid and the blades because at the time of jet stream impact on the blade, the notch redirects the fluid so that more than one blade is impacted to increase the impulse force and the angular velocity of the blades. When the blades are initially impacted by the liquid, the direction of the liquid is changed, but the notched blades are able to redirect around 40% of the fluid to the outer surface of the next blade of the rotor chamber, adding more force that contributes to the continuous spin of the rotor. Therefore, the notch not only contributes toward increasing rotation, but towards minimizing the vortices between blades, thus, guaranteeing the proper rotation of the fluid inside of the rotor chamber and between blades.
Turbine Casing Design
The holder or casing is a new design, the final product of three years of research work. The CAD designs, simulations, prototypes and testing of pieces, helped find the best model, and were done according to potential applications as well as to satisfy the basic principles of stability, compatibility, adaptability, and simple assembly. As shown in FIG. 5, turbine casing 103 includes main three sections: inlet-nozzle 110, rotor chamber, 118 and outlet 120. The input and output of the turbine were designed using a standard circular shape, with initial inlet and final output diameters of 4 mm, which is an ideal size to connect the micro turbine with microfluidic systems such as the circulatory system, micro pumps, mixers and refrigeration systems found in medical equipment, automotive systems, and home appliances. Similar to the rotor, the casing is scalable to allow the turbine generator to be used in other applications as is known to a person having ordinary skill in the art.
The inlet-nozzle section 110, shown in FIG. 4, illustrates a gradual contraction between the inlet 112 and the nozzle 114, where a conical section 116 is formed with sharp discontinuities at the intersections. The inlet and nozzle dimensions used are a response to the laminar behavior requirement, especially in the blood system. The nozzle geometry has 10° degrees of inclination with respect to the horizontal axis, or in other words, an angle of 100 degrees between the inlet and outlet sections. The inclination concentrates the jet on the maximum area of the first blade and optimizes the impact. Moreover, this inclination contributes to the discharge through the notch and reduces the vortices in the central area of the rotor. The inlet was designed into a gradual contraction pipe or funnel shape to create a venturi effect with the fluid flow, increasing the output velocity of the fluid at the nozzle and focusing the jet on the blades.
As shown in FIG. 5, rotor chamber 118 and outlet section 120 have unique geometries that permit free flow circulation to the output. The rotor chamber was designed with a curvature, which combines parabolas and circular geometries, to ease the flow discharge, and direct the impact of the fluid on more than one blade, and to increase the pressure on the output area of the turbine. The outlet section is designed to reestablish the inlet pressure, which was modified when the fluid passed into the inlet conical section and through the nozzle to the rotor chamber.
In addition, the curved shape of the walls surrounding the rotor chamber contributes to the free flow of the fluid, avoiding any type of accumulation or stenosis, thus reducing stress on walls and assuring a good fluid displacement to the outlet turbine. Finally, and looking to have the best adaptability, the inlet 110 and outlet 120 of the turbine were designed as circular pipes given that it is the most common shape used to transport and connect microfluidic systems. In a certain embodiment, as shown in FIG. 5C, rotor chamber 118 includes PMs 122, which interact with PMs on the rotor, such as the ones shown in FIG. 2, allowing for rotor levitation.
The characteristics of the turbine casing, inlet, rotor chamber, and outlet, were defined according to biomedical and circulatory system specifications. The rotor chamber was designed to keep the turbine rotor aligned in the center and frictionless and to keep a variable gap between rotor and enclosing walls. The gap contributes to the circulation and redirects the fluid to the blades; also, the gap design avoids the friction when the rotor is in motion. The outlet chamber is a free zone to discharge the fluid from rotor chamber with a conical shape and circular end.
Turbine Mathematical Model
In the turbine mathematical model, the mass of fluid per second, entering from nozzle to the rotor chamber, is used to find the momentum of the liquid when the blades are pushed, and establishes the relationship with the forces striking the blades, torque and mechanical power generated by the system.
A vector representation of velocities and flow direction is shown in FIG. 6A, where the dashed arrows represent the velocity trajectories of the fluid applied from nozzle. Curved block arrows represent the permanent internal flow fluid circulation, and block arrows are the forces produced by the injected flow fluid, which create the torque to keep the rotor in constant speed. To concentrate the analysis and develop the mathematical model, three principal points, central, external, and internal, were defined on each blade.
To know the minimum and maximum values of volume flow rate required to satisfy the mechanical power and angular velocities of this turbine, an analysis involving torques, forces, areas, and linear velocities affecting the blades was developed. Starting with the velocity and fluid flow behavior to the first blade, FIG. 6B, where the total velocity is represented as the sum of angular velocity (U) on rotor and the lineal velocity of flow fluid entering from nozzle (G). However, the analysis required the real impact of the fluid entering on each of blade is centered on the velocity vector, which is a subtraction between angular velocity on rotor and the lineal velocity of flow fluid entering from nozzle. FIG. 6B shows the first blade vector analysis, which represents the components of velocities of the working fluid in a turbomachine. This analysis produced as a result, the equation that connects the desired rotor behavior (speed and torque) with the inlet volume flow rate, velocity, and forces of the fluid flow pushing the blades.
According to the shapes and geometries defined for this novel turbine, two velocity triangles on two points (internal A2 and external A1 points) of the first blade are defined, but a blade-by-blade analysis is developed. This is followed by the mechanical power and the efficiency equations of the turbine. The velocity triangles used in this analysis explain the flow direction and state the principal points of pressure, through which torque is applied to spin the rotor.
Through the vector analysis of velocities (velocity triangles), on two points of the first blade, the equations that explain the behavior of the notched blade turbine as an impulse turbine are defined. In FIG. 6B, Vw1 and Vw2 are the tangential velocities at A1 and A2, and Vn is the jet velocity at the output of the nozzle. In addition, an angle α1 is formed between the nozzle and the horizontal plane tangent to the rotor radius. Also, the horizontal plane and the tangential line in A2, create an angle α2, just into the notch of the first blade, in the direction of the second blade.
The system on the first blade is analyzed as an impulse turbine and the equations on the first blade are developed at two positions, A1 and A2 resulting in the tangential velocities Vw1 and Vw2. In both cases, the results of the tangential velocities are functions of the nozzle velocity and the angles involved in the input and output of the fluid on the blade. Additionally, the fluid flow leaving the first blade has a velocity VA. The equations resulting from the analysis on the first blade are written as:
Tangential Inlet Velocity:

Vw1=Vn cos α1−U1 

Tangential Outlet Velocity (Notch Velocity):

Vw2=(VN cos α1−U1)cos ∝2 

The fluid velocity crossing the notch of the first blade is written as

VA=Vw2+U2=(VN cos α1−U1)cos ∝2+U2 

The model of forces on the first blade was found using the geometry shown in FIGS. 6.A-6.B, but applying the momentum balance analysis combined with the velocity triangles. In addition, the torque equation on first blade was found as the product of these forces and the radial distances.
Force on First Blade

FA=ρQ(Vw1+Vw2)=(VN cos α1−U1)(1+cos ∝2)

Torque on First Blade

TA=ρQ(Vw1r1+Vw2r2)=ρQ[VN cos α1(r1+r2 cos ∝2)−U1(r1+r2 cos ∝2)]

TA=ρQ[(VN cos α1−U1)(r1+r2 cos ∝2)]

If A=(r1+r2 cos ∝2) then

TA=ρQ(VNA cos α1−U1A)

The analysis to find the equations on the second blade starts with the flow fluid from the first blade after passing by the notch, and goes until it totally leaves the second blade, but includes the central impact on second blade. As FIG. 6.C shows, the velocity VA from the notch area of the first blade enters to the second blade with an inclination angle α3, measured between the central point of the second blade, in the tangential direction to the rotor.
The equations for the second blade on the B1, B2, and the central point Bm are described as follows.
Second Blade:
VA is the inlet velocity at the second blade, but has an inclination angle α3 casing measured between the flow fluid direction and the center blade on the tangential line of this point. FIG. 6C, shows the geometry,

VA=Vw2+U2=(VN cos α1−U1)cos α2+U2 

Tangential inlet velocity on second blade, denoted as VMB is written as



  
    
      V
      
        M
        B
      
    
    =
    
      
        
          V
          A
        
        ⁢
        cos
        ⁢
        
          
        
        ⁢
        
          α
          3
        
      
      =
      
        
          
            (
            
              
                V
                
                  w
                  2
                
              
              +
              
                U
                2
              
            
            )
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
        =
        
          
            [
            
              
                
                  (
                  
                    
                      
                        V
                        N
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                    
                    -
                    
                      U
                      1
                    
                  
                  )
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  α
                  2
                
              
              +
              
                U
                2
              
            
            ]
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        V
        
          M
          B
        
      
      =
      
        
          [
          
            
              
                (
                
                  
                    
                      V
                      N
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      α
                      1
                    
                  
                  -
                  
                    U
                    1
                  
                
                )
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                α
                2
              
            
            +
            
              
                U
                1
              
              ⁢
              
                
                  r
                  2
                
                
                  r
                  1
                
              
            
          
          ]
        
        ⁢
        cos
        ⁢
        
          
        
        ⁢
        
          α
          3
        
      
    
  




  
    
      
    
    ⁢
    
      
        V
        
          M
          B
        
      
      =
      
        
          
            V
            N
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            1
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            2
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
        -
        
          
            
              U
              1
            
            ⁡
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    α
                    2
                  
                
                -
                
                  
                    r
                    2
                  
                  
                    r
                    1
                  
                
              
              )
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        if
        ⁢
        
          
        
        ⁢
        B
      
      =
      
        
          cos
          ⁢
          
            
          
          ⁢
          
            α
            2
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
          ⁢
          
            
          
          ⁢
          and
          ⁢
          
            
          
          ⁢
          C
        
        =
        
          
            (
            
              B
              -
              
                
                  
                    r
                    2
                  
                  
                    r
                    1
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  α
                  3
                
              
            
            )
          
          ⁢
          
            
          
          ⁢
          then
        
      
    
  




  
    
      
    
    ⁢
    
      
        V
        
          M
          B
        
      
      =
      
        
          
            V
            N
          
          ⁢
          B
          ⁢
          
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            1
          
        
        -
        
          
            U
            1
          
          ⁢
          C
        
      
    
  



Flow fluid velocity from the first blade impacts the second blade on center place, then the fluid flow is divided into two components after pushing the second blade, the first velocity component is directed to the external edge B1, and the second component is directed to the internal edge B2 or notch. The velocities results on B1 and B2 have a tangential direction in the same line of the rotor tangential velocity on these points. The component on B1 is opposite of that of tangential velocity U1, while the component on B2 has the same direction as the tangential velocity on this point U2.



  
    
      V
      
        B
        1
      
    
    =
    
      
        U
        1
      
      -
      
        
          (
          
            
              V
              
                M
                B
              
            
            -
            
              U
              M
            
          
          )
        
        ⁢
        cos
        ⁢
        
          
        
        ⁢
        
          β
          1
        
      
    
  




  
    
      V
      
        B
        2
      
    
    =
    
      
        U
        2
      
      +
      
        
          (
          
            
              V
              
                M
                B
              
            
            -
            
              U
              M
            
          
          )
        
        ⁢
        cos
        ⁢
        
          
        
        ⁢
        
          α
          2
        
      
    
  




  
    
      Δ
      ⁢
      
        
      
      ⁢
      
        V
        
          B
          1
        
      
    
    =
    
      
        V
        
          M
          B
        
      
      -
      
        V
        
          B
          1
        
      
    
  




  
    
      Δ
      ⁢
      
        
      
      ⁢
      
        V
        
          B
          1
        
      
    
    =
    
      
        
          V
          
            M
            B
          
        
        ⁡
        
          (
          
            1
            +
            
              cos
              ⁢
              
                
              
              ⁢
              
                β
                1
              
            
          
          )
        
      
      -
      
        
          U
          1
        
        ⁡
        
          (
          
            1
            +
            
              
                
                  r
                  m
                
                
                  r
                  1
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                β
                1
              
            
          
          )
        
      
    
  




  
    
      Δ
      ⁢
      
        
      
      ⁢
      
        V
        
          B
          2
        
      
    
    =
    
      
        V
        
          M
          B
        
      
      -
      
        V
        
          B
          2
        
      
    
  




  
    
      Δ
      ⁢
      
        
      
      ⁢
      
        V
        
          B
          2
        
      
    
    =
    
      
        
          V
          
            M
            B
          
        
        ⁡
        
          (
          
            1
            -
            
              cos
              ⁢
              
                
              
              ⁢
              
                α
                2
              
            
          
          )
        
      
      -
      
        
          U
          1
        
        ⁡
        
          (
          
            
              
                r
                2
              
              
                r
                1
              
            
            -
            
              
                
                  r
                  m
                
                
                  r
                  1
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                α
                2
              
            
          
          )
        
      
    
  



The force on second blade is represented by



  
    
      
    
    ⁢
    
      
        F
        B
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    B
                    1
                  
                
              
              +
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    B
                    2
                  
                
              
            
            )
          
        
      
    
  




  
    
      F
      B
    
    =
    
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                V
                
                  M
                  B
                
              
              ⁡
              
                [
                
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                  +
                  
                    (
                    
                      1
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          α
                          2
                        
                      
                    
                    )
                  
                
                ]
              
            
            )
          
        
      
      -
      
        
          U
          1
        
        ⁡
        
          [
          
            
              (
              
                1
                +
                
                  
                    
                      r
                      M
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    β
                    1
                  
                
              
              )
            
            +
            
              (
              
                
                  
                    r
                    2
                  
                  
                    r
                    1
                  
                
                -
                
                  
                    
                      r
                      m
                    
                    
                      r
                      2
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    α
                    2
                  
                
              
              )
            
          
          ]
        
      
    
  



By using the above equations plus the mass flow rate ({dot over (m)}=ρQ) then the torque on second blade is found



  
    
      
    
    ⁢
    
      
        T
        B
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    B
                    1
                  
                
                ⁢
                
                  r
                  1
                
              
              +
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    B
                    2
                  
                
                ⁢
                
                  r
                  2
                
              
            
            )
          
        
      
    
  




  
    
      T
      B
    
    =
    
      ρ
      ⁢
      
        
      
      ⁢
      
        Q
        ⁡
        
          [
          
            
              
                V
                
                  M
                  B
                
              
              ⁡
              
                (
                
                  
                    
                      r
                      1
                    
                    ⁡
                    
                      (
                      
                        1
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            β
                            1
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      r
                      2
                    
                    ⁡
                    
                      (
                      
                        1
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            2
                          
                        
                      
                      )
                    
                  
                
                )
              
            
            -
            
              
                U
                1
              
              ⁡
              
                (
                
                  
                    
                      r
                      1
                    
                    ⁡
                    
                      (
                      
                        1
                        +
                        
                          
                            
                              r
                              M
                            
                            
                              r
                              1
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            β
                            1
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      r
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        -
                        
                          
                            
                              r
                              m
                            
                            
                              r
                              1
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            2
                          
                        
                      
                      )
                    
                  
                
                )
              
            
          
          ]
        
      
    
  



If D=r1(1+cos β1)+r2(1−cos α2) and




  E
  =
  
    
      
        r
        1
      
      ⁡
      
        (
        
          1
          +
          
            
              
                r
                M
              
              
                r
                1
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              β
              1
            
          
        
        )
      
    
    +
    
      
        r
        2
      
      ⁡
      
        (
        
          
            
              r
              2
            
            
              r
              1
            
          
          -
          
            
              
                r
                m
              
              
                r
                1
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              α
              2
            
          
        
        )
      
    
  




then torque on second blade is rewritten as

TB=ρQ(VMBD−U1E)=ρQ[VNBD cos α1−U1CD−U1E]

TB=ρQ[VNBD cos α1−U1(CD+E)]

The analysis to find the equations in the next blades, third to sixth, use the same physical principles and the same geometry as the first and second blades, which starts with the fluid flow from the previous blade, passing through the notch in direction of the next blade at the central point of the blade. As shown in FIG. 6, there are components of velocity from the notch areas by each blade, which enter to the next blade with an inclination angle β1, measured between the central point of the blade and in the rotor radial direction. The components of velocity have the same angle on entry and exit on each blade, but the magnitudes of these vectors are different, having the maximum value in the first blade and a minimum in the sixth blade.
Third Blade:

VMC=VB2 cos α3=(U2+(VMB−UM)cos ∝2)cos α3 

VMC=(U2+(VNB cos α1−U1C−UM)cos ∝2)cos α3




  
    
      V
      
        M
        C
      
    
    =
    
      (
      
        
          
            V
            N
          
          ⁢
          B
          ⁢
          
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            1
          
          ⁢
          cos
        
        ⁢
        
          ∝
          2
        
        ⁢
        
          
            cos
            ⁢
            
              
            
            ⁢
            
              α
              3
            
          
          +
          
            
              U
              1
            
            ⁢
            
              
                r
                2
              
              
                r
                1
              
            
            ⁢
            cos
          
        
        ⁢
        
          ∝
          3
        
        ⁢
        
          
            -
            
              U
              1
            
          
          ⁢
          C
          ⁢
          
            
          
          ⁢
          cos
        
        ⁢
        
          ∝
          2
        
        ⁢
        
          
            cos
            ⁢
            
              
            
            ⁢
            
              α
              3
            
          
          -
          
            
              U
              1
            
            ⁢
            
              
                r
                m
              
              
                r
                1
              
            
            ⁢
            cos
          
        
        ⁢
        
          ∝
          2
        
        ⁢
        
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
      
      )
    
  




  
    
      
    
    ⁢
    
      
        V
        
          M
          C
        
      
      =
      
        [
        
          
            
              V
              N
            
            ⁢
            
              B
              2
            
            ⁢
            cos
          
          ⁢
          
            ∝
            1
          
          ⁢
          
            +
            
              
                U
                1
              
              ⁡
              
                (
                
                  
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      
                        -
                        B
                      
                      ⁢
                      
                        
                      
                      ⁢
                      C
                    
                    -
                    
                      B
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          r
                          m
                        
                        
                          r
                          1
                        
                      
                    
                  
                
                )
              
            
          
        
        ]
      
    
  




  
    
      
    
    ⁢
    
      
        V
        
          C
          1
        
      
      =
      
        
          U
          1
        
        -
        
          
            (
            
              
                V
                
                  M
                  C
                
              
              -
              
                U
                M
              
            
            )
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            β
            1
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        V
        
          C
          2
        
      
      =
      
        
          
            
              U
              2
            
            +
            
              
                (
                
                  
                    V
                    
                      M
                      C
                    
                  
                  -
                  
                    U
                    M
                  
                
                )
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            Δ
            ⁢
            
              
            
            ⁢
            
              V
              
                C
                1
              
            
          
        
        =
        
          
            V
            
              M
              C
            
          
          -
          
            V
            
              C
              1
            
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            C
            1
          
        
      
      =
      
        
          
            V
            
              M
              C
            
          
          ⁡
          
            (
            
              1
              +
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  β
                  1
                
              
            
            )
          
        
        -
        
          
            U
            1
          
          ⁡
          
            (
            
              1
              +
              
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  β
                  1
                
              
            
            )
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            C
            2
          
        
      
      =
      
        
          V
          
            M
            C
          
        
        -
        
          V
          
            C
            2
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            C
            2
          
        
      
      =
      
        
          
            V
            
              M
              C
            
          
          ⁡
          
            (
            
              
                1
                -
                cos
              
              ⁢
              
                ∝
                2
              
            
            )
          
        
        -
        
          
            U
            1
          
          ⁡
          
            (
            
              
                
                  
                    r
                    2
                  
                  
                    r
                    1
                  
                
                +
                
                  
                    
                      r
                      m
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
              
              ⁢
              
                ∝
                2
              
            
            )
          
        
      
    
  



Force on Third Blade



  
    
      
    
    ⁢
    
      
        F
        C
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    C
                    1
                  
                
              
              +
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    C
                    2
                  
                
              
            
            )
          
        
      
    
  




  
    
      F
      C
    
    =
    
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                V
                
                  M
                  C
                
              
              ⁡
              
                [
                
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                  +
                  
                    (
                    
                      
                        1
                        -
                        cos
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
                ]
              
            
            )
          
        
      
      -
      
        
          U
          1
        
        ⁡
        
          [
          
            
              (
              
                1
                +
                
                  
                    
                      r
                      M
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    β
                    1
                  
                
              
              )
            
            +
            
              (
              
                
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  -
                  
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                
                ⁢
                
                  ∝
                  2
                
              
              )
            
          
          ]
        
      
    
  



Torque on Third Blade



  
    
      
    
    ⁢
    
      
        T
        C
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    C
                    1
                  
                
                ⁢
                
                  r
                  1
                
              
              +
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    C
                    2
                  
                
                ⁢
                
                  r
                  2
                
              
            
            )
          
        
      
    
  




  
    
      T
      C
    
    =
    
      ρ
      ⁢
      
        
      
      ⁢
      
        Q
        [
        
          
            
              V
              
                M
                C
              
            
            ⁡
            
              (
              
                
                  
                    r
                    1
                  
                  ⁡
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                
                +
                
                  
                    r
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        1
                        -
                        cos
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
              
              )
            
          
          -
          
            
              U
              1
            
            ⁡
            
              (
              
                
                  
                    r
                    1
                  
                  ⁢
                  
                    (
                    
                      1
                      +
                      
                        
                          
                            r
                            M
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                
                +
                
                  
                    r
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        -
                        
                          
                            
                              r
                              m
                            
                            
                              r
                              1
                            
                          
                          ⁢
                          cos
                        
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
              
              )
            
          
        
        ]
      
    
  



If D=r1(1+cos β1)+r2(1−cos ∝2) and




  E
  =
  
    
      
        r
        1
      
      ⁡
      
        (
        
          1
          +
          
            
              
                r
                M
              
              
                r
                1
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              β
              1
            
          
        
        )
      
    
    +
    
      
        r
        2
      
      ⁡
      
        (
        
          
            
              
                r
                2
              
              
                r
                1
              
            
            -
            
              
                
                  r
                  m
                
                
                  r
                  1
                
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
        
        )
      
    
  




then torque on third blade is rewritten as





  
    T
    C
  
  =
  
    
      ρ
      ⁢
      
        
      
      ⁢
      
        Q
        ⁡
        
          (
          
            
              
                V
                
                  M
                  C
                
              
              ⁢
              D
            
            -
            
              
                U
                1
              
              ⁢
              E
            
          
          )
        
      
    
    =
    
      ρ
      ⁢
      
        
      
      ⁢
      
        Q
        ⁡
        
          [
          
            
              
                V
                N
              
              ⁡
              
                (
                
                  
                    
                      B
                      2
                    
                    ⁢
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    1
                  
                
                )
              
            
            +
            
              
                U
                1
              
              ⁡
              
                (
                
                  
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      
                        -
                        C
                      
                      ⁢
                      
                        
                      
                      ⁢
                      B
                      ⁢
                      
                        
                      
                      ⁢
                      D
                    
                    -
                    
                      B
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          r
                          m
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      D
                    
                    -
                    E
                  
                
                )
              
            
          
          ]
        
      
    
  




Fourth Blade:



  
    
      
    
    ⁢
    
      
        V
        
          M
          D
        
      
      =
      
        
          
            V
            
              C
              2
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
        =
        
          
            (
            
              
                
                  U
                  2
                
                +
                
                  
                    (
                    
                      
                        V
                        
                          M
                          C
                        
                      
                      -
                      
                        U
                        M
                      
                    
                    )
                  
                  ⁢
                  cos
                
              
              ⁢
              
                ∝
                2
              
            
            )
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
      
    
  




  
    
      V
      
        M
        D
      
    
    =
    
      
        (
        
          
            
              U
              2
            
            +
            
              
                (
                
                  
                    
                      V
                      N
                    
                    ⁢
                    
                      B
                      2
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    1
                  
                  ⁢
                  
                    
                      +
                      
                        
                          U
                          1
                        
                        ⁡
                        
                          (
                          
                            
                              
                                
                                  r
                                  2
                                
                                
                                  r
                                  1
                                
                              
                              ⁢
                              cos
                            
                            ⁢
                            
                              ∝
                              3
                            
                            ⁢
                            
                              
                                
                                  -
                                  C
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                B
                              
                              -
                              
                                B
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  
                                    r
                                    m
                                  
                                  
                                    r
                                    1
                                  
                                
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      U
                      M
                    
                  
                
                )
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
        
        )
      
      ⁢
      cos
      ⁢
      
        
      
      ⁢
      
        α
        3
      
    
  




  
    
      V
      
        M
        D
      
    
    =
    
      (
      
        
          
            V
            N
          
          ⁢
          
            B
            3
          
          ⁢
          cos
        
        ⁢
        
          ∝
          1
        
        ⁢
        
          
            +
            
              U
              1
            
          
          ⁢
          
            
              r
              2
            
            
              r
              1
            
          
          ⁢
          cos
        
        ⁢
        
          ∝
          3
        
        ⁢
        
          
            +
            
              U
              1
            
          
          ⁢
          B
          ⁢
          
            
          
          ⁢
          
            
              r
              2
            
            
              r
              1
            
          
          ⁢
          cos
        
        ⁢
        
          ∝
          3
        
        ⁢
        
          
            
              -
              
                U
                1
              
            
            ⁢
            C
            ⁢
            
              
            
            ⁢
            
              B
              2
            
          
          -
          
            
              U
              1
            
            ⁢
            
              
                r
                m
              
              
                r
                1
              
            
            ⁢
            
              B
              2
            
          
          -
          
            
              U
              1
            
            ⁢
            
              
                r
                m
              
              
                r
                1
              
            
            ⁢
            B
          
        
      
      )
    
  




  
    
      V
      
        M
        D
      
    
    =
    
      [
      
        
          
            V
            N
          
          ⁢
          
            B
            3
          
          ⁢
          cos
        
        ⁢
        
          ∝
          1
        
        ⁢
        
          +
          
            
              U
              1
            
            ⁡
            
              (
              
                
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  3
                
                ⁢
                
                  
                    +
                    B
                  
                  ⁢
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  3
                
                ⁢
                
                  
                    
                      -
                      C
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      B
                      2
                    
                  
                  -
                  
                    
                      B
                      2
                    
                    ⁢
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                  
                  -
                  
                    B
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                  
                
              
              )
            
          
        
      
      ]
    
  




  
    
      
    
    ⁢
    
      
        V
        
          D
          1
        
      
      =
      
        
          U
          1
        
        -
        
          
            (
            
              
                V
                
                  M
                  D
                
              
              -
              
                U
                M
              
            
            )
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            β
            1
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        V
        
          D
          2
        
      
      =
      
        
          
            
              U
              2
            
            -
            
              
                (
                
                  
                    V
                    
                      M
                      D
                    
                  
                  -
                  
                    U
                    M
                  
                
                )
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            Δ
            ⁢
            
              
            
            ⁢
            
              V
              
                D
                1
              
            
          
        
        =
        
          
            V
            
              M
              D
            
          
          -
          
            V
            
              D
              1
            
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            D
            1
          
        
      
      =
      
        
          
            V
            
              M
              D
            
          
          ⁡
          
            (
            
              1
              +
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  β
                  1
                
              
            
            )
          
        
        -
        
          
            U
            1
          
          ⁡
          
            (
            
              1
              +
              
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  β
                  1
                
              
            
            )
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            D
            2
          
        
      
      =
      
        
          V
          
            M
            D
          
        
        -
        
          V
          
            D
            2
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            D
            2
          
        
      
      =
      
        
          
            V
            
              M
              D
            
          
          ⁡
          
            (
            
              
                1
                -
                cos
              
              ⁢
              
                ∝
                2
              
            
            )
          
        
        -
        
          
            U
            1
          
          ⁡
          
            (
            
              
                
                  
                    r
                    2
                  
                  
                    r
                    1
                  
                
                -
                
                  
                    
                      r
                      m
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
              
              ⁢
              
                ∝
                2
              
            
            )
          
        
      
    
  



The Force on the Fourth Blade



  
    
      
    
    ⁢
    
      
        F
        D
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    D
                    1
                  
                
              
              +
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    D
                    2
                  
                
              
            
            )
          
        
      
    
  




  
    
      F
      D
    
    =
    
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                V
                
                  M
                  D
                
              
              ⁡
              
                [
                
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                  +
                  
                    (
                    
                      
                        1
                        -
                        cos
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
                ]
              
            
            )
          
        
      
      -
      
        
          U
          1
        
        ⁡
        
          [
          
            
              (
              
                1
                +
                
                  
                    
                      r
                      M
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    β
                    1
                  
                
              
              )
            
            +
            
              (
              
                
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  -
                  
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                
                ⁢
                
                  ∝
                  2
                
              
              )
            
          
          ]
        
      
    
  



And the torque on the fourth blade as



  
    
      
    
    ⁢
    
      
        T
        D
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    D
                    1
                  
                
                ⁢
                
                  r
                  1
                
              
              +
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    D
                    2
                  
                
                ⁢
                
                  r
                  2
                
              
            
            )
          
        
      
    
  




  
    
      T
      D
    
    =
    
      ρ
      ⁢
      
        
      
      ⁢
      
        Q
        [
        
          
            
              V
              
                M
                D
              
            
            ⁡
            
              (
              
                
                  
                    r
                    1
                  
                  ⁡
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                
                +
                
                  
                    r
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        1
                        -
                        cos
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
              
              )
            
          
          -
          
            
              U
              1
            
            ⁡
            
              (
              
                
                  
                    r
                    1
                  
                  ⁢
                  
                    (
                    
                      1
                      +
                      
                        
                          
                            r
                            M
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                
                +
                
                  
                    r
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        -
                        
                          
                            
                              r
                              m
                            
                            
                              r
                              1
                            
                          
                          ⁢
                          cos
                        
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
              
              )
            
          
        
        ]
      
    
  



Substituting D=r1(1+cos β1)+r2(1−cos ∝2) and




  E
  =
  
    
      
        r
        1
      
      ⁡
      
        (
        
          1
          +
          
            
              
                r
                M
              
              
                r
                1
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              β
              1
            
          
        
        )
      
    
    +
    
      
        r
        2
      
      ⁡
      
        (
        
          
            
              
                r
                2
              
              
                r
                1
              
            
            -
            
              
                
                  r
                  m
                
                
                  r
                  1
                
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
        
        )
      
    
  




then torque on fourth blade is rewritten as




  
    
      
    
    ⁢
    
      
        T
        D
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                
                  V
                  
                    M
                    D
                  
                
                ⁢
                D
              
              -
              
                
                  U
                  1
                
                ⁢
                E
              
            
            )
          
        
      
    
  




  
    
      T
      D
    
    =
    
      ρ
      ⁢
      
        
      
      ⁢
      
        Q
        ⁡
        
          [
          
            
              D
              ⁡
              
                (
                
                  
                    
                      V
                      N
                    
                    ⁢
                    
                      B
                      3
                    
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    1
                  
                  ⁢
                  
                    +
                    
                      
                        U
                        1
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                r
                                2
                              
                              
                                r
                                1
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            cos
                          
                          ⁢
                          
                            ∝
                            3
                          
                          ⁢
                          
                            
                              +
                              
                                
                              
                              ⁢
                              B
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                r
                                2
                              
                              
                                r
                                1
                              
                            
                            ⁢
                            cos
                          
                          ⁢
                          
                            ∝
                            3
                          
                          ⁢
                          
                            
                              
                                -
                                C
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                B
                                2
                              
                            
                            -
                            
                              
                                B
                                2
                              
                              ⁢
                              
                                
                                  r
                                  m
                                
                                
                                  r
                                  1
                                
                              
                            
                            -
                            
                              B
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  r
                                  m
                                
                                
                                  r
                                  1
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                
                )
              
            
            -
            
              
                U
                1
              
              ⁢
              E
            
          
          ]
        
      
    
  




  
    
      T
      D
    
    =
    
      ρ
      ⁢
      
        
      
      ⁢
      
        Q
        ⁡
        
          [
          
            
              
                V
                N
              
              ⁢
              
                B
                3
              
              ⁢
              
                
              
              ⁢
              cos
            
            ⁢
            
              ∝
              1
            
            ⁢
            
              +
              
                
                  U
                  1
                
                ⁡
                
                  (
                  
                    
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        +
                        
                          
                        
                        ⁢
                        B
                      
                      ⁢
                      
                        
                      
                      ⁢
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        
                          -
                          
                            B
                            2
                          
                        
                        ⁢
                        C
                        ⁢
                        
                          
                        
                        ⁢
                        D
                      
                      -
                      
                        
                          B
                          2
                        
                        ⁢
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            r
                            m
                          
                          
                            r
                            1
                          
                        
                      
                      -
                      
                        B
                        ⁢
                        
                          
                        
                        ⁢
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            r
                            m
                          
                          
                            r
                            1
                          
                        
                      
                      -
                      E
                    
                  
                  )
                
              
            
          
          ]
        
      
    
  



Fifth Blade:



  
    
      
    
    ⁢
    
      
        V
        
          M
          E
        
      
      =
      
        
          
            V
            
              D
              2
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
        =
        
          
            (
            
              
                
                  U
                  2
                
                +
                
                  
                    (
                    
                      
                        V
                        
                          M
                          D
                        
                      
                      -
                      
                        U
                        M
                      
                    
                    )
                  
                  ⁢
                  cos
                
              
              ⁢
              
                ∝
                2
              
            
            )
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            α
            3
          
        
      
    
  




  
    
      V
      
        M
        E
      
    
    =
    
      
        (
        
          
            
              U
              2
            
            +
            
              
                (
                
                  
                    
                      V
                      N
                    
                    ⁢
                    
                      B
                      3
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    1
                  
                  ⁢
                  
                    
                      +
                      
                        
                          U
                          1
                        
                        ⁡
                        
                          (
                          
                            
                              
                                
                                  r
                                  2
                                
                                
                                  r
                                  1
                                
                              
                              ⁢
                              cos
                            
                            ⁢
                            
                              ∝
                              3
                            
                            ⁢
                            
                              
                                +
                                
                                  
                                
                                ⁢
                                B
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  r
                                  2
                                
                                
                                  r
                                  1
                                
                              
                              ⁢
                              cos
                            
                            ⁢
                            
                              ∝
                              3
                            
                            ⁢
                            
                              
                                
                                  -
                                  C
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  B
                                  2
                                
                              
                              -
                              
                                
                                  B
                                  2
                                
                                ⁢
                                
                                  
                                    r
                                    m
                                  
                                  
                                    r
                                    1
                                  
                                
                              
                              -
                              
                                B
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  
                                    r
                                    m
                                  
                                  
                                    r
                                    1
                                  
                                
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      U
                      M
                    
                  
                
                )
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
        
        )
      
      ⁢
      cos
      ⁢
      
        
      
      ⁢
      
        α
        3
      
    
  




  
    
      V
      
        M
        E
      
    
    =
    
      (
      
        
          
            V
            N
          
          ⁢
          
            B
            4
          
          ⁢
          cos
        
        ⁢
        
          ∝
          1
        
        ⁢
        
          
            +
            
              U
              1
            
          
          ⁢
          
            
              r
              2
            
            
              r
              1
            
          
          ⁢
          cos
        
        ⁢
        
          ∝
          3
        
        ⁢
        
          
            +
            
              U
              1
            
          
          ⁢
          B
          ⁢
          
            
          
          ⁢
          
            
              r
              2
            
            
              r
              1
            
          
          ⁢
          cos
        
        ⁢
        
          ∝
          3
        
        ⁢
        
          
            +
            
              U
              1
            
          
          ⁢
          
            
          
          ⁢
          
            B
            2
          
          ⁢
          
            
              r
              2
            
            
              r
              1
            
          
          ⁢
          cos
        
        ⁢
        
          ∝
          3
        
        ⁢
        
          
            
              -
              
                U
                1
              
            
            ⁢
            C
            ⁢
            
              
            
            ⁢
            
              B
              2
            
          
          -
          
            
              B
              3
            
            ⁢
            
              U
              1
            
            ⁢
            
              
                r
                m
              
              
                r
                1
              
            
          
          -
          
            
              B
              2
            
            ⁢
            
              U
              1
            
            ⁢
            
              
                r
                m
              
              
                r
                1
              
            
          
          -
          
            
              U
              1
            
            ⁢
            
              
                r
                m
              
              
                r
                1
              
            
            ⁢
            B
          
        
      
      )
    
  




  
    
      V
      
        M
        E
      
    
    =
    
      [
      
        
          
            V
            N
          
          ⁢
          
            B
            4
          
          ⁢
          cos
        
        ⁢
        
          ∝
          1
        
        ⁢
        
          +
          
            
              U
              1
            
            ⁡
            
              (
              
                
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  3
                
                ⁢
                
                  
                    +
                    B
                  
                  ⁢
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  3
                
                ⁢
                
                  
                    +
                    
                      B
                      2
                    
                  
                  ⁢
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  3
                
                ⁢
                
                  
                    
                      -
                      C
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        B
                        ⁢
                        
                          
                        
                      
                      3
                    
                  
                  -
                  
                    
                      B
                      3
                    
                    ⁢
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                  
                  -
                  
                    
                      B
                      2
                    
                    ⁢
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                  
                  -
                  
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    B
                  
                
              
              )
            
          
        
      
      ]
    
  




  
    
      
    
    ⁢
    
      
        V
        
          E
          1
        
      
      =
      
        
          U
          1
        
        -
        
          
            (
            
              
                V
                
                  M
                  E
                
              
              -
              
                U
                M
              
            
            )
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            β
            1
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        V
        
          E
          2
        
      
      =
      
        
          
            
              U
              2
            
            -
            
              
                (
                
                  
                    V
                    
                      M
                      E
                    
                  
                  -
                  
                    U
                    M
                  
                
                )
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            Δ
            ⁢
            
              
            
            ⁢
            
              V
              
                E
                1
              
            
          
        
        =
        
          
            V
            
              M
              D
            
          
          -
          
            V
            
              D
              1
            
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            E
            1
          
        
      
      =
      
        
          
            V
            
              M
              E
            
          
          ⁡
          
            (
            
              1
              +
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  β
                  1
                
              
            
            )
          
        
        -
        
          
            U
            1
          
          ⁡
          
            (
            
              1
              +
              
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  β
                  1
                
              
            
            )
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            E
            2
          
        
      
      =
      
        
          V
          
            M
            E
          
        
        -
        
          V
          
            E
            2
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          
            E
            2
          
        
      
      =
      
        
          
            V
            
              M
              E
            
          
          ⁡
          
            (
            
              
                1
                -
                cos
              
              ⁢
              
                ∝
                2
              
            
            )
          
        
        -
        
          
            U
            1
          
          ⁡
          
            (
            
              
                
                  
                    r
                    2
                  
                  
                    r
                    1
                  
                
                -
                
                  
                    
                      r
                      m
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
              
              ⁢
              
                ∝
                2
              
            
            )
          
        
      
    
  



Force on Fifth Blade



  
    
      
    
    ⁢
    
      
        F
        E
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    D
                    1
                  
                
              
              +
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    D
                    2
                  
                
              
            
            )
          
        
      
    
  




  
    
      F
      E
    
    =
    
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                V
                
                  M
                  E
                
              
              ⁡
              
                [
                
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                  +
                  
                    (
                    
                      
                        1
                        -
                        cos
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
                ]
              
            
            )
          
        
      
      -
      
        
          U
          1
        
        ⁡
        
          [
          
            
              (
              
                1
                +
                
                  
                    
                      r
                      M
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    β
                    1
                  
                
              
              )
            
            +
            
              (
              
                
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  -
                  
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                
                ⁢
                
                  ∝
                  2
                
              
              )
            
          
          ]
        
      
    
  



Torque on Fifth Blade



  
    
      
    
    ⁢
    
      
        T
        E
      
      =
      
        ρ
        ⁢
        
          
        
        ⁢
        
          Q
          ⁡
          
            (
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    E
                    1
                  
                
                ⁢
                
                  r
                  1
                
              
              +
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    E
                    2
                  
                
                ⁢
                
                  r
                  2
                
              
            
            )
          
        
      
    
  




  
    
      T
      E
    
    =
    
      ρ
      ⁢
      
        
      
      ⁢
      
        Q
        [
        
          
            
              V
              
                M
                E
              
            
            ⁡
            
              (
              
                
                  
                    r
                    1
                  
                  ⁡
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                
                +
                
                  
                    r
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        1
                        -
                        cos
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
              
              )
            
          
          -
          
            
              U
              1
            
            ⁡
            
              (
              
                
                  
                    r
                    1
                  
                  ⁡
                  
                    (
                    
                      1
                      +
                      
                        
                          
                            r
                            M
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          β
                          1
                        
                      
                    
                    )
                  
                
                +
                
                  
                    r
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        -
                        
                          
                            
                              r
                              m
                            
                            
                              r
                              1
                            
                          
                          ⁢
                          cos
                        
                      
                      ⁢
                      
                        ∝
                        2
                      
                    
                    )
                  
                
              
              )
            
          
        
        ]
      
    
  



Substituting D=r1(1+cos β1)+r2(1−cos ∝2) and




  E
  =
  
    
      
        r
        1
      
      ⁡
      
        (
        
          1
          +
          
            
              
                r
                M
              
              
                r
                1
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              β
              1
            
          
        
        )
      
    
    +
    
      
        r
        2
      
      ⁡
      
        (
        
          
            
              
                r
                2
              
              
                r
                1
              
            
            -
            
              
                
                  r
                  m
                
                
                  r
                  1
                
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
        
        )
      
    
  




then torque on fifth blade is rewritten as





  
    
      
        
          T
          E
        
        =
        
        ⁢
        
          ρ
          ⁢
          
            
          
          ⁢
          
            Q
            ⁡
            
              (
              
                
                  
                    V
                    
                      M
                      E
                    
                  
                  ⁢
                  D
                
                -
                
                  
                    U
                    1
                  
                  ⁢
                  E
                
              
              )
            
          
        
      
    
  
  
    
      
        
          T
          E
        
        =
        
        ⁢
        
          ρ
          ⁢
          
            
          
          ⁢
          
            Q
            [
            
              D
              (
              
                
                  
                    V
                    N
                  
                  ⁢
                  
                    B
                    4
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  1
                
                ⁢
                
                  +
                  
                    
                      U
                      1
                    
                    (
                    
                      
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      ⁢
                      
                        
                          +
                          B
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      ⁢
                      
                        
                          +
                          
                            B
                            2
                          
                        
                        ⁢
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      -
                    
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
              
              ⁢
              
                
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  
                    B
                    3
                  
                
                -
                
                  
                    B
                    3
                  
                  ⁢
                  
                    
                      r
                      m
                    
                    
                      r
                      1
                    
                  
                
                -
                
                  
                    B
                    2
                  
                  ⁢
                  
                    
                      r
                      m
                    
                    
                      r
                      1
                    
                  
                
                -
                
                  
                    
                      r
                      m
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  B
                
              
              )
            
            )
          
          -
          
            
              U
              1
            
            ⁢
            E
          
        
        ]
      
    
  
  
    
      
        
          T
          E
        
        =
        
        ⁢
        
          ρ
          ⁢
          
            
          
          ⁢
          
            Q
            [
            
              
                
                  V
                  N
                
                ⁢
                
                  B
                  4
                
                ⁢
                D
                ⁢
                
                  
                
                ⁢
                cos
              
              ⁢
              
                ∝
                1
              
              ⁢
              
                +
                
                  
                    U
                    1
                  
                  (
                  
                    
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        +
                        B
                      
                      ⁢
                      
                        
                      
                      ⁢
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        +
                        
                          B
                          2
                        
                      
                      ⁢
                      D
                      ⁢
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              
                B
                3
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              D
            
            -
            
              
                B
                3
              
              ⁢
              D
              ⁢
              
                
                  r
                  m
                
                
                  r
                  1
                
              
            
            -
            
              
                B
                2
              
              ⁢
              D
              ⁢
              
                
                  r
                  m
                
                
                  r
                  1
                
              
            
            -
            
              
                
                  r
                  m
                
                
                  r
                  1
                
              
              ⁢
              B
              ⁢
              
                
              
              ⁢
              D
            
            -
            E
          
          )
        
        ]
      
    
  




Sixth Blade:




  
    
      
        
          V
          
            M
            F
          
        
        =
        
        ⁢
        
          
            
              V
              
                E
                2
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              α
              3
            
          
          =
          
            
              (
              
                
                  
                    U
                    2
                  
                  +
                  
                    
                      (
                      
                        
                          V
                          
                            M
                            E
                          
                        
                        -
                        
                          U
                          M
                        
                      
                      )
                    
                    ⁢
                    cos
                  
                
                ⁢
                
                  ∝
                  2
                
              
              )
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              α
              3
            
          
        
      
    
  
  
    
      
        
          V
          
            M
            F
          
        
        =
        
        ⁢
        
          (
          
            
              U
              2
            
            +
            
              (
              
                
                  
                    V
                    N
                  
                  ⁢
                  
                    B
                    4
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  1
                
                ⁢
                
                  +
                  
                    
                      U
                      1
                    
                    (
                    
                      
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      ⁢
                      
                        
                          +
                          B
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      ⁢
                      
                        
                          +
                          
                            B
                            2
                          
                        
                        ⁢
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      -
                    
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        C
                        ⁢
                        
                          
                        
                        ⁢
                        
                          B
                          3
                        
                      
                      -
                      
                        
                          B
                          3
                        
                        ⁢
                        
                          
                            r
                            m
                          
                          
                            r
                            1
                          
                        
                      
                      -
                      
                        
                          B
                          2
                        
                        ⁢
                        
                          
                            r
                            m
                          
                          
                            r
                            1
                          
                        
                      
                      -
                      
                        
                          
                            r
                            m
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        B
                      
                    
                    )
                  
                  -
                  
                    U
                    M
                  
                
                )
              
              ⁢
              cos
            
            ⁢
            
              ∝
              2
            
          
          )
        
        ⁢
        cos
        ⁢
        
          
        
        ⁢
        
          α
          3
        
      
    
  








  
    
      
        
          V
          
            M
            F
          
        
        =
        
        ⁢
        
          (
          
            
              
                V
                N
              
              ⁢
              
                B
                5
              
              ⁢
              cos
            
            ⁢
            
              ∝
              1
            
            ⁢
            
              
                +
                
                  U
                  1
                
              
              ⁢
              
                
                  r
                  2
                
                
                  r
                  1
                
              
              ⁢
              cos
            
            ⁢
            
              ∝
              3
            
            ⁢
            
              
                +
                
                  U
                  1
                
              
              ⁢
              B
              ⁢
              
                
              
              ⁢
              
                
                  r
                  2
                
                
                  r
                  1
                
              
              ⁢
              cos
            
            ⁢
            
              ∝
              3
            
            ⁢
            
              
                +
                
                  U
                  1
                
              
              ⁢
              
                B
                2
              
              ⁢
              
                
              
              ⁢
              
                
                  r
                  2
                
                
                  r
                  1
                
              
              ⁢
              cos
            
            ⁢
            
              ∝
              3
            
            +
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              U
              1
            
            ⁢
            
              B
              3
            
            ⁢
            
              
                r
                2
              
              
                r
                1
              
            
            ⁢
            cos
          
          ⁢
          
            ∝
            3
          
          ⁢
          
            
              
                -
                
                  U
                  1
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              
                B
                4
              
            
            -
            
              
                B
                4
              
              ⁢
              
                U
                1
              
              ⁢
              
                
                  r
                  m
                
                
                  r
                  1
                
              
            
            -
            
              
                B
                3
              
              ⁢
              
                U
                1
              
              ⁢
              
                
                  r
                  m
                
                
                  r
                  1
                
              
            
            -
            
              
                B
                2
              
              ⁢
              
                U
                1
              
              ⁢
              
                
                  r
                  m
                
                
                  r
                  1
                
              
            
            -
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            U
            1
          
          ⁢
          
            
              r
              m
            
            
              r
              1
            
          
          ⁢
          B
        
        )
      
    
  
  
    
      
        
          V
          
            M
            F
          
        
        =
        
        ⁢
        
          [
          
            
              
                V
                N
              
              ⁢
              
                B
                5
              
              ⁢
              cos
            
            ⁢
            
              ∝
              1
            
            ⁢
            
              +
              
                
                  U
                  1
                
                (
                
                  
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      +
                      B
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      +
                      
                        B
                        2
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  +
                
              
            
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              
                B
                3
              
              ⁢
              
                
                  r
                  2
                
                
                  r
                  1
                
              
              ⁢
              cos
            
            ⁢
            
              ∝
              3
            
            ⁢
            
              
                
                  -
                  C
                
                ⁢
                
                  
                
                ⁢
                
                  B
                  4
                
              
              -
              
                
                  B
                  4
                
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
                ⁢
                
                  B
                  3
                
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
              
              -
              
                
                  B
                  2
                
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
              
              -
              
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
                ⁢
                B
              
            
          
          )
        
        ]
      
    
  
  
    
      
        
          V
          
            F
            1
          
        
        =
        
        ⁢
        
          
            U
            1
          
          -
          
            
              (
              
                
                  V
                  
                    M
                    F
                  
                
                -
                
                  U
                  M
                
              
              )
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              β
              1
            
          
        
      
    
  
  
    
      
        
          V
          
            F
            2
          
        
        =
        
        ⁢
        
          
            
              U
              2
            
            +
            
              
                (
                
                  
                    V
                    
                      M
                      F
                    
                  
                  -
                  
                    U
                    M
                  
                
                )
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
        
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            V
            
              F
              1
            
          
        
        =
        
        ⁢
        
          
            V
            
              M
              F
            
          
          -
          
            V
            
              F
              1
            
          
        
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            V
            
              F
              1
            
          
        
        =
        
        ⁢
        
          
            
              V
              
                M
                F
              
            
            ⁡
            
              (
              
                1
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    β
                    1
                  
                
              
              )
            
          
          -
          
            
              U
              1
            
            ⁡
            
              (
              
                1
                +
                
                  
                    
                      r
                      m
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    β
                    1
                  
                
              
              )
            
          
        
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            V
            
              F
              2
            
          
        
        =
        
        ⁢
        
          
            V
            
              M
              F
            
          
          -
          
            V
            
              F
              2
            
          
        
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            V
            
              F
              2
            
          
        
        =
        
        ⁢
        
          
            
              V
              
                M
                F
              
            
            ⁡
            
              (
              
                
                  1
                  -
                  cos
                
                ⁢
                
                  ∝
                  2
                
              
              )
            
          
          -
          
            
              U
              1
            
            ⁡
            
              (
              
                
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  -
                  
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                
                ⁢
                
                  ∝
                  2
                
              
              )
            
          
        
      
    
  




Force on Sixth Blade




  
    
      
        
          F
          F
        
        =
        
        ⁢
        
          ρ
          ⁢
          
            
          
          ⁢
          
            Q
            ⁡
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      1
                    
                  
                
                +
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      2
                    
                  
                
              
              )
            
          
        
      
    
  
  
    
      
        
          F
          F
        
        =
        
        ⁢
        
          
            ρ
            ⁢
            
              
            
            ⁢
            
              Q
              ⁡
              
                (
                
                  
                    V
                    
                      M
                      F
                    
                  
                  ⁡
                  
                    [
                    
                      
                        (
                        
                          1
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              β
                              1
                            
                          
                        
                        )
                      
                      +
                      
                        (
                        
                          
                            1
                            -
                            cos
                          
                          ⁢
                          
                            ∝
                            2
                          
                        
                        )
                      
                    
                    ]
                  
                
                )
              
            
          
          -
          
            
              U
              1
            
            [
            
              
                (
                
                  1
                  +
                  
                    
                      
                        r
                        M
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      β
                      1
                    
                  
                
                )
              
              +
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          (
          
            
              
                
                  r
                  2
                
                
                  r
                  1
                
              
              -
              
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
                ⁢
                cos
              
            
            ⁢
            
              ∝
              2
            
          
          )
        
        ]
      
    
  




Torque on Sixth Blade




  
    
      
        
          T
          F
        
        =
        
        ⁢
        
          ρ
          ⁢
          
            
          
          ⁢
          
            Q
            ⁡
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    V
                    
                      F
                      1
                    
                  
                  ⁢
                  
                    r
                    1
                  
                
                +
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    V
                    
                      F
                      2
                    
                  
                  ⁢
                  
                    r
                    2
                  
                
              
              )
            
          
        
      
    
  
  
    
      
        
          T
          F
        
        =
        
        ⁢
        
          ρ
          ⁢
          
            
          
          ⁢
          
            Q
            [
            
              
                
                  V
                  
                    M
                    F
                  
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        1
                      
                      ⁡
                      
                        (
                        
                          1
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              β
                              1
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        r
                        2
                      
                      ⁡
                      
                        (
                        
                          
                            1
                            -
                            cos
                          
                          ⁢
                          
                            ∝
                            2
                          
                        
                        )
                      
                    
                  
                  )
                
              
              -
              
                
                  U
                  1
                
                (
                
                  
                    
                      r
                      1
                    
                    ⁡
                    
                      (
                      
                        1
                        +
                        
                          
                            
                              r
                              M
                            
                            
                              r
                              1
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            β
                            1
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              r
              2
            
            ⁡
            
              (
              
                
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  -
                  
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                
                ⁢
                
                  ∝
                  2
                
              
              )
            
          
          )
        
        ]
      
    
  




Substituting D=r1(1+cos β1)+r2(1−cos ∝2) and




  E
  =
  
    
      
        r
        1
      
      ⁡
      
        (
        
          1
          +
          
            
              
                r
                M
              
              
                r
                1
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              β
              1
            
          
        
        )
      
    
    +
    
      
        r
        2
      
      ⁡
      
        (
        
          
            
              
                r
                2
              
              
                r
                1
              
            
            -
            
              
                
                  r
                  m
                
                
                  r
                  1
                
              
              ⁢
              cos
            
          
          ⁢
          
            ∝
            2
          
        
        )
      
    
  




then torque on sixth blade is rewritten as

TF=ρQ(VMFD−U1E)





  
    
      
        
          T
          E
        
        =
        
        ⁢
        
          ρ
          ⁢
          
            
          
          ⁢
          
            Q
            [
            
              D
              (
              
                
                  
                    V
                    N
                  
                  ⁢
                  
                    B
                    5
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  1
                
                ⁢
                
                  +
                  
                    
                      U
                      1
                    
                    (
                    
                      
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      ⁢
                      
                        
                          +
                          B
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      ⁢
                      
                        
                          +
                          
                            B
                            2
                          
                        
                        ⁢
                        
                          
                            r
                            2
                          
                          
                            r
                            1
                          
                        
                        ⁢
                        cos
                      
                      ⁢
                      
                        ∝
                        3
                      
                      +
                    
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
              
              ⁢
              
                
                  
                    B
                    3
                  
                  ⁢
                  
                    
                      r
                      2
                    
                    
                      r
                      1
                    
                  
                  ⁢
                  cos
                
                ⁢
                
                  ∝
                  3
                
                ⁢
                
                  
                    
                      -
                      C
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      B
                      4
                    
                  
                  -
                  
                    
                      B
                      4
                    
                    ⁢
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                  
                  -
                  
                    
                      B
                      3
                    
                    ⁢
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                  
                  -
                  
                    
                      B
                      2
                    
                    ⁢
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                  
                  -
                  
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    B
                  
                
              
              )
            
            )
          
          -
          
            
              U
              1
            
            ⁢
            E
          
        
        ]
      
    
  
  
    
      
        
          T
          F
        
        =
        
        ⁢
        
          ρ
          ⁢
          
            
          
          ⁢
          
            Q
            [
            
              
                
                  V
                  N
                
                ⁢
                
                  B
                  5
                
                ⁢
                D
                ⁢
                
                  
                
                ⁢
                cos
              
              ⁢
              
                ∝
                1
              
              ⁢
              
                +
                
                  
                    U
                    1
                  
                  (
                  
                    
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        +
                        B
                      
                      ⁢
                      
                        
                      
                      ⁢
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        +
                        
                          B
                          2
                        
                      
                      ⁢
                      D
                      ⁢
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              
                B
                3
              
              ⁢
              
                
              
              ⁢
              D
              ⁢
              
                
                  r
                  2
                
                
                  r
                  1
                
              
              ⁢
              cos
            
            ⁢
            
              ∝
              3
            
            ⁢
            
              
                
                  -
                  
                    B
                    4
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                D
              
              -
              
                
                  B
                  4
                
                ⁢
                D
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
              
              -
              
                
                  B
                  3
                
                ⁢
                D
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
              
              -
              
                
                  B
                  2
                
                ⁢
                
                  
                
                ⁢
                D
                ⁢
                
                  
                
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
              
              -
              
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
                ⁢
                B
                ⁢
                
                  
                
                ⁢
                D
              
              -
              E
            
          
          )
        
        ]
      
    
  




Rotor Mechanical Power




  
    
      
        
          
            α
            1
          
          =
          
          ⁢
          10
        
        ,
      
    
  
  
    
      
        
          
            α
            2
          
          =
          
          ⁢
          
            30.01
            ⁢
            °
          
        
        ,
      
    
  
  
    
      
        
          
            α
            3
          
          =
          
          ⁢
          
            50.35
            ⁢
            °
          
        
        ,
      
    
  
  
    
      
        
          
            β
            1
          
          =
          
          ⁢
          
            9.6
            ⁢
            °
          
        
        ,
      
    
  
  
    
      
        
          
            r
            1
          
          =
          
          ⁢
          
            3.95
            ⁢
            
              
            
            ⁢
            mm
          
        
        ,
      
    
  
  
    
      
        
          
            r
            2
          
          =
          
          ⁢
          
            3.1
            ⁢
            
              
            
            ⁢
            m
          
        
        ,
      
    
  
  
    
      
        
        ⁢
        and
      
    
  
  
    
      
        
          r
          m
        
        =
        
        ⁢
        
          3.5
          ⁢
          
            
          
          ⁢
          mm
        
      
    
  
  
    
      
        ρ
        =
        
        ⁢
        
          1000
          ⁡
          
            [
            
              
                k
                ⁢
                
                  
                
                ⁢
                g
              
              
                m
                3
              
            
            ]
          
        
      
    
  
  
    
      
        Q
        =
        
        ⁢
        
          
            V
            N
          
          ⁢
          
            
              A
              N
            
            ⁡
            
              [
              
                
                  m
                  3
                
                s
              
              ]
            
          
        
      
    
  
  
    
      
        
          A
          N
        
        =
        
        ⁢
        
          
            
              π
              ⁡
              
                (
                0.00065
                )
              
            
            2
          
          ⁡
          
            [
            
              m
              2
            
            ]
          
        
      
    
  
  
    
      
        A
        =
        
        ⁢
        
          
            (
            
              
                
                  r
                  1
                
                +
                
                  
                    r
                    2
                  
                  ⁢
                  cos
                
              
              ⁢
              
                ∝
                2
              
            
            )
          
          =
          0.0066
        
      
    
  
  
    
      
        B
        =
        
        ⁢
        
          
            cos
            ⁢
            
              ∝
              2
            
            ⁢
            
              cos
              ⁢
              
                
              
              ⁢
              
                α
                3
              
            
          
          =
          0.8528
        
      
    
  
  
    
      
        C
        =
        
        ⁢
        
          
            (
            
              cos
              ⁢
              
                ∝
                2
              
              ⁢
              
                -
                
                  
                    r
                    2
                  
                  
                    r
                    1
                  
                
              
            
            )
          
          =
          0.0582
        
      
    
  
  
    
      
        D
        =
        
        ⁢
        
          
            
              
                r
                1
              
              ⁡
              
                (
                
                  1
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      β
                      1
                    
                  
                
                )
              
            
            +
            
              
                r
                2
              
              ⁡
              
                (
                
                  
                    1
                    -
                    cos
                  
                  ⁢
                  
                    ∝
                    2
                  
                
                )
              
            
          
          =
          0.0082
        
      
    
  
  
    
      
        E
        =
        
        ⁢
        
          
            
              
                r
                1
              
              ⁡
              
                (
                
                  1
                  +
                  
                    
                      
                        r
                        M
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      β
                      1
                    
                  
                
                )
              
            
            +
            
              
                r
                2
              
              ⁡
              
                (
                
                  
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    -
                    
                      
                        
                          r
                          m
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                  
                  ⁢
                  
                    ∝
                    2
                  
                
                )
              
            
          
          =
          0.0074
        
      
    
  




Torque on first blade:TA=ρQ(VNA cos α1−U1A)

Torque on second blade:TB=ρQ(VMBD−U1E)

Torque on third blade:TC=ρQ(VM9di CD−U1E)

Torque on fourth blade:TD=ρQ(VMDD−U1E)

Torque on fifth blade:TE=ρQ(VMED−U1E)

Torque on sixth blade:TF=ρQ(VMFD−U1E)

Turbine Total Torque:

TT=ρQ(VNA cos α1+D(VMB+VMC+VMD+VME+VMF)−U1(5E+A))

The tangential inlet velocities on each blade can be analyzed as follows:




  
    
      
        
          V
          
            M
            A
          
        
        =
        
        ⁢
        
          
            
              V
              N
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              α
              1
            
          
          -
          
            U
            1
          
        
      
    
  
  
    
      
        
          V
          
            M
            B
          
        
        =
        
        ⁢
        
          
            
              V
              N
            
            ⁢
            B
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              α
              1
            
          
          -
          
            
              U
              1
            
            ⁢
            C
          
        
      
    
  
  
    
      
        
          V
          
            M
            C
          
        
        =
        
        ⁢
        
          [
          
            
              
                V
                N
              
              ⁢
              
                B
                2
              
              ⁢
              cos
            
            ⁢
            
              ∝
              1
            
            ⁢
            
              +
              
                
                  U
                  1
                
                ⁡
                
                  (
                  
                    
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        
                          -
                          B
                        
                        ⁢
                        
                          
                        
                        ⁢
                        C
                      
                      -
                      
                        B
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            r
                            m
                          
                          
                            r
                            1
                          
                        
                      
                    
                  
                  )
                
              
            
          
          ]
        
      
    
  
  
    
      
        
          V
          
            M
            D
          
        
        =
        
        ⁢
        
          [
          
            
              
                V
                N
              
              ⁢
              
                B
                3
              
              ⁢
              cos
            
            ⁢
            
              ∝
              1
            
            ⁢
            
              +
              
                
                  U
                  1
                
                ⁡
                
                  (
                  
                    
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        +
                        B
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                    ⁢
                    
                      ∝
                      3
                    
                    ⁢
                    
                      
                        
                          -
                          C
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          B
                          2
                        
                      
                      -
                      
                        
                          B
                          2
                        
                        ⁢
                        
                          
                            r
                            m
                          
                          
                            r
                            1
                          
                        
                      
                      -
                      
                        B
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            r
                            m
                          
                          
                            r
                            1
                          
                        
                      
                    
                  
                  )
                
              
            
          
          ]
        
      
    
  
  
    
      
        
          V
          
            M
            E
          
        
        =
        
        ⁢
        
          [
          
            
              
                V
                N
              
              ⁢
              
                B
                4
              
              ⁢
              cos
            
            ⁢
            
              ∝
              1
            
            ⁢
            
              +
              
                
                  U
                  1
                
                (
                
                  
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      +
                      B
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      +
                      
                        B
                        2
                      
                    
                    ⁢
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      
                        -
                        C
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        B
                        3
                      
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              
                B
                3
              
              ⁢
              
                
                  r
                  m
                
                
                  r
                  1
                
              
            
            -
            
              
                B
                2
              
              ⁢
              
                
                  r
                  m
                
                
                  r
                  1
                
              
            
            -
            
              
                
                  r
                  m
                
                
                  r
                  1
                
              
              ⁢
              B
            
          
          )
        
        ]
      
    
  
  
    
      
        
          V
          
            M
            F
          
        
        =
        
        ⁢
        
          [
          
            
              
                V
                N
              
              ⁢
              
                B
                5
              
              ⁢
              cos
            
            ⁢
            
              ∝
              1
            
            ⁢
            
              +
              
                
                  U
                  1
                
                (
                
                  
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      +
                      B
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  ⁢
                  
                    
                      +
                      
                        B
                        2
                      
                    
                    ⁢
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                  ⁢
                  
                    ∝
                    3
                  
                  +
                
              
            
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              
                B
                3
              
              ⁢
              
                
                  r
                  2
                
                
                  r
                  1
                
              
              ⁢
              cos
            
            ⁢
            
              ∝
              3
            
            ⁢
            
              
                
                  -
                  C
                
                ⁢
                
                  
                
                ⁢
                
                  B
                  4
                
              
              -
              
                
                  B
                  4
                
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
              
              -
              
                
                  B
                  3
                
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
              
              -
              
                
                  B
                  2
                
                ⁢
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
              
              -
              
                
                  
                    r
                    m
                  
                  
                    r
                    1
                  
                
                ⁢
                B
              
            
          
          )
        
        ]
      
    
  




Substituting VMB, VMC, VMD, VME, and VMF on total torque equation, TT is rewritten as,




  
    T
    T
  
  =
  
    ρ
    ⁢
    
      
    
    ⁢
    Q
    ⁢
    
      {
      
        
          
            V
            N
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            
              α
              1
            
            ⁡
            
              [
              
                A
                +
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  
                    B
                    ⁡
                    
                      (
                      
                        1
                        +
                        B
                        +
                        
                          B
                          2
                        
                        +
                        
                          B
                          3
                        
                        +
                        
                          B
                          4
                        
                      
                      )
                    
                  
                
              
              ]
            
          
        
        -
        
          
            U
            1
          
          ⁡
          
            [
            
              A
              +
              
                5
                ⁢
                
                  
                
                ⁢
                E
              
              +
              
                D
                ⁡
                
                  (
                  
                    
                      
                        (
                        
                          
                            
                              B
                              ⁢
                              
                                
                                  r
                                  m
                                
                                
                                  r
                                  1
                                
                              
                            
                            -
                            
                              
                                
                                  r
                                  2
                                
                                
                                  r
                                  1
                                
                              
                              ⁢
                              cos
                            
                          
                          ⁢
                          
                            ∝
                            3
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          4
                          +
                          
                            3
                            ⁢
                            
                              
                            
                            ⁢
                            B
                          
                          +
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            
                              B
                              2
                            
                          
                          +
                          
                            B
                            3
                          
                        
                        )
                      
                    
                    +
                    
                      B
                      ⁢
                      
                        
                      
                      ⁢
                      
                        C
                        ⁡
                        
                          (
                          
                            1
                            +
                            B
                            +
                            
                              B
                              2
                            
                            +
                            
                              B
                              3
                            
                          
                          )
                        
                      
                    
                    +
                    C
                  
                  )
                
              
            
            ]
          
        
      
      }
    
  




The objective of this mathematical analysis was to find a general equation modeling the behavior of the turbine to find the minimum and maximum values of volume flow rate required to satisfy the mechanical power and angular velocities of the model. The sum of torques on each blade or total torque, multiplied by the rotor angular velocity, produces the new mechanical power general equation of this turbine, which is a function involving tangential velocity on the rotor and the velocity from the nozzle. However, as the principles of conservation of mass and momentum relate the volume flow rate and the velocity, then the mechanical power equation developed on the rotor (PR=TTω), satisfies the objective.




  
    
      
        
          P
          T
        
        =
        
        ⁢
        
          
            
              T
              T
            
            ⁢
            ω
          
          =
          
            
              T
              T
            
            ⁢
            
              
                U
                1
              
              
                r
                1
              
            
          
        
      
    
  
  
    
      
        
          P
          T
        
        =
        
        ⁢
        
          
            
              ρ
              ⁢
              
                
              
              ⁢
              Q
            
            
              r
              1
            
          
          ⁢
          
            {
            
              
                
                  V
                  N
                
                ⁢
                
                  U
                  1
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  
                    α
                    1
                  
                  ⁡
                  
                    [
                    
                      A
                      +
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        
                          B
                          ⁡
                          
                            (
                            
                              1
                              +
                              B
                              +
                              
                                B
                                2
                              
                              +
                              
                                B
                                3
                              
                              +
                              
                                B
                                4
                              
                            
                            )
                          
                        
                      
                    
                    ]
                  
                
              
              -
              
                
                  U
                  1
                  2
                
                [
                
                  A
                  +
                  
                    5
                    ⁢
                    
                      
                    
                    ⁢
                    E
                  
                  +
                
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          D
          (
          
            
              
                (
                
                  
                    
                      B
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          r
                          m
                        
                        
                          r
                          1
                        
                      
                    
                    -
                    
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                  
                  ⁢
                  
                    ∝
                    3
                  
                
                )
              
              ⁢
              
                (
                
                  4
                  +
                  
                    3
                    ⁢
                    
                      
                    
                    ⁢
                    B
                  
                  +
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      B
                      2
                    
                  
                  +
                  
                    B
                    3
                  
                
                )
              
            
            +
          
        
      
    
  
  
    
      
        
          
            
            ⁢
            
              
                B
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  (
                  
                    1
                    +
                    B
                    +
                    
                      B
                      2
                    
                    +
                    
                      B
                      3
                    
                  
                  )
                
              
              +
              C
            
            )
          
          ]
        
        }
      
    
  




The maximum mechanical power developed on the rotor is found by differentiating the mechanical power equation and is found where the derivative is equal to zero.




  
    
      
        
          
            d
            
              P
              T
            
          
          
            d
            
              U
              1
            
          
        
        =
        
        ⁢
        
          
            
              ρ
              ⁢
              
                
              
              ⁢
              Q
            
            
              r
              1
            
          
          ⁢
          
            {
            
              
                d
                
                  d
                  
                    U
                    1
                  
                
              
              ⁢
              
                (
                
                  
                    
                      V
                      N
                    
                    ⁢
                    
                      U
                      1
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        α
                        1
                      
                      ⁡
                      
                        [
                        
                          A
                          +
                          
                            D
                            ⁢
                            
                              
                            
                            ⁢
                            
                              B
                              ⁡
                              
                                (
                                
                                  1
                                  +
                                  B
                                  +
                                  
                                    B
                                    2
                                  
                                  +
                                  
                                    B
                                    3
                                  
                                  +
                                  
                                    B
                                    4
                                  
                                
                                )
                              
                            
                          
                        
                        ]
                      
                    
                  
                  -
                
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            U
            1
            2
          
          [
          
            A
            +
            
              5
              ⁢
              
                
              
              ⁢
              E
            
            +
            
              D
              (
              
                
                  
                    (
                    
                      
                        
                          B
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              r
                              m
                            
                            
                              r
                              1
                            
                          
                        
                        -
                        
                          
                            
                              r
                              2
                            
                            
                              r
                              1
                            
                          
                          ⁢
                          cos
                        
                      
                      ⁢
                      
                        ∝
                        3
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      4
                      +
                      
                        3
                        ⁢
                        
                          
                        
                        ⁢
                        B
                      
                      +
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          B
                          2
                        
                      
                      +
                      
                        B
                        3
                      
                    
                    )
                  
                
                +
              
            
          
        
      
    
  
  
    
      
        
          
            
              
              ⁢
              
                
                  B
                  ⁢
                  
                    
                  
                  ⁢
                  
                    C
                    ⁡
                    
                      (
                      
                        1
                        +
                        B
                        +
                        
                          B
                          2
                        
                        +
                        
                          B
                          3
                        
                      
                      )
                    
                  
                
                +
                C
              
              )
            
            ]
          
          )
        
        }
      
    
  
  
    
      
        
          
            d
            
              P
              T
            
          
          
            d
            
              U
              1
            
          
        
        =
        
        ⁢
        
          0
          =
          
            
              
                V
                N
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                
                  α
                  1
                
                ⁡
                
                  [
                  
                    A
                    +
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      
                        B
                        ⁡
                        
                          (
                          
                            1
                            +
                            B
                            +
                            
                              B
                              2
                            
                            +
                            
                              B
                              3
                            
                            +
                            
                              B
                              4
                            
                          
                          )
                        
                      
                    
                  
                  ]
                
              
            
            -
            
              2
              ⁢
              
                U
                1
              
              ⁢
              
                {
                
                  A
                  +
                  
                    5
                    ⁢
                    
                      
                    
                    ⁢
                    E
                  
                  +
                
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          D
          [
          
            
              
                (
                
                  
                    
                      B
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          r
                          m
                        
                        
                          r
                          1
                        
                      
                    
                    -
                    
                      
                        
                          r
                          2
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                  
                  ⁢
                  
                    ∝
                    3
                  
                
                )
              
              ⁢
              
                (
                
                  4
                  +
                  
                    3
                    ⁢
                    
                      
                    
                    ⁢
                    B
                  
                  +
                  
                    2
                    ⁢
                    
                      B
                      2
                    
                  
                  +
                  
                    B
                    3
                  
                
                )
              
            
            +
          
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              B
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                (
                
                  1
                  +
                  B
                  +
                  
                    B
                    2
                  
                  +
                  
                    B
                    3
                  
                
                )
              
            
            +
            C
          
          ]
        
        }
      
    
  
  
    
      
        
          V
          N
        
        =
        
        ⁢
        
          2
          ⁢
          
            
          
          ⁢
          
            U
            1
          
          ⁢
          
            
              
                
                  
                    {
                    
                      A
                      +
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        E
                      
                      +
                      
                        D
                        [
                        
                          
                            
                              (
                              
                                
                                  
                                    B
                                    ⁢
                                    
                                      
                                        r
                                        m
                                      
                                      
                                        r
                                        1
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        r
                                        2
                                      
                                      
                                        r
                                        1
                                      
                                    
                                    ⁢
                                    cos
                                  
                                
                                ⁢
                                
                                  ∝
                                  3
                                
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                4
                                +
                                
                                  3
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  B
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    B
                                    2
                                  
                                
                                +
                                
                                  B
                                  3
                                
                              
                              )
                            
                          
                          +
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        
                          B
                          ⁢
                          
                            
                          
                          ⁢
                          
                            C
                            ⁡
                            
                              (
                              
                                1
                                +
                                B
                                +
                                
                                  B
                                  2
                                
                                +
                                
                                  B
                                  3
                                
                              
                              )
                            
                          
                        
                        +
                        C
                      
                      ]
                    
                    }
                  
                
              
            
            
              cos
              ⁢
              
                
              
              ⁢
              
                
                  α
                  1
                
                ⁡
                
                  [
                  
                    A
                    +
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      
                        B
                        ⁡
                        
                          (
                          
                            1
                            +
                            B
                            +
                            
                              B
                              2
                            
                            +
                            
                              B
                              3
                            
                            +
                            
                              B
                              4
                            
                          
                          )
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
  




The relationship between VN and U1 is found by substituting the following constants values on the above equation:




  
    
      
        
          
            α
            1
          
          =
          
          ⁢
          10
        
        ,
      
    
  
  
    
      
        
          
            α
            2
          
          =
          
          ⁢
          
            30.01
            ⁢
            °
          
        
        ,
      
    
  
  
    
      
        
          
            α
            3
          
          =
          
          ⁢
          
            50.35
            ⁢
            °
          
        
        ,
      
    
  
  
    
      
        
          
            β
            1
          
          =
          
          ⁢
          
            9.6
            ⁢
            °
          
        
        ,
      
    
  
  
    
      
        
          
            r
            1
          
          =
          
          ⁢
          
            3.95
            ⁢
            
              
            
            ⁢
            mm
          
        
        ,
      
    
  
  
    
      
        
          
            r
            2
          
          =
          
          ⁢
          
            3.1
            ⁢
            
              
            
            ⁢
            m
          
        
        ,
      
    
  
  
    
      
        
        ⁢
        and
      
    
  
  
    
      
        
          r
          m
        
        =
        
        ⁢
        
          3.5
          ⁢
          
            
          
          ⁢
          mm
        
      
    
  
  
    
      
        A
        =
        
        ⁢
        
          
            (
            
              
                
                  r
                  1
                
                +
                
                  
                    r
                    2
                  
                  ⁢
                  cos
                
              
              ⁢
              
                ∝
                2
              
            
            )
          
          =
          0.0066
        
      
    
  
  
    
      
        B
        =
        
        ⁢
        
          
            cos
            ⁢
            
              ∝
              2
            
            ⁢
            
              cos
              ⁢
              
                
              
              ⁢
              
                α
                3
              
            
          
          =
          0.8528
        
      
    
  
  
    
      
        C
        =
        
        ⁢
        
          
            (
            
              cos
              ⁢
              
                ∝
                2
              
              ⁢
              
                -
                
                  
                    r
                    2
                  
                  
                    r
                    1
                  
                
              
            
            )
          
          =
          0.0582
        
      
    
  
  
    
      
        D
        =
        
        ⁢
        
          
            
              
                r
                1
              
              ⁡
              
                (
                
                  1
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      β
                      1
                    
                  
                
                )
              
            
            +
            
              
                r
                2
              
              ⁡
              
                (
                
                  
                    1
                    -
                    cos
                  
                  ⁢
                  
                    ∝
                    2
                  
                
                )
              
            
          
          =
          0.0082
        
      
    
  
  
    
      
        E
        =
        
        ⁢
        
          
            
              
                r
                1
              
              ⁡
              
                (
                
                  1
                  +
                  
                    
                      
                        r
                        M
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      β
                      1
                    
                  
                
                )
              
            
            +
            
              
                r
                2
              
              ⁡
              
                (
                
                  
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    -
                    
                      
                        
                          r
                          m
                        
                        
                          r
                          1
                        
                      
                      ⁢
                      cos
                    
                  
                  ⁢
                  
                    ∝
                    2
                  
                
                )
              
            
          
          =
          0.0074
        
      
    
  








  
    If
    ⁢
    
      
    
    ⁢
    F
  
  =
  
    {
    
      A
      +
      
        5
        ⁢
        
          
        
        ⁢
        E
      
      +
      
        D
        [
        
          
        
        ⁢
        
          
            
              (
              
                
                  
                    B
                    ⁢
                    
                      
                        r
                        m
                      
                      
                        r
                        1
                      
                    
                  
                  -
                  
                    
                      
                        r
                        2
                      
                      
                        r
                        1
                      
                    
                    ⁢
                    cos
                  
                
                ⁢
                
                  ∝
                  3
                
              
              )
            
            ⁢
            
              (
              
                4
                +
                
                  3
                  ⁢
                  
                    
                  
                  ⁢
                  B
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    B
                    2
                  
                
                +
                
                  B
                  3
                
              
              )
            
          
          +
          
            B
            ⁢
            
              
            
            ⁢
            
              C
              ⁡
              
                (
                
                  1
                  +
                  B
                  +
                  
                    B
                    2
                  
                  +
                  
                    B
                    3
                  
                
                )
              
            
          
          +
          C
        
        ]
      
    
    }
  




And G=cos α1[A+DB(1+B+B2+B3+B4)]

    
    
        then F=0.0435 and G=0.0509 and VN is rewritten as
    
    






  
    V
    N
  
  =
  
    2
    ⁢
    
      
    
    ⁢
    
      U
      1
    
    ⁢
    
      F
      G
    
  




The maximum power is reached when VN=5.5106 U1.
On the other hand, the potential power or power extracted from the nozzle turbine is Pn=pgQh, but, since




  
    V
    n
  
  =
  
    
      2
      ⁢
      
        
      
      ⁢
      gh
    
  




then a new equation to potential power is written as,





  
    P
    n
  
  =
  
    
      ρ
      ⁢
      
        
      
      ⁢
      
        QV
        n
        2
      
    
    2
  




Finally, the equations of mechanical power on the rotor and the potential power from nozzle equations are related in one expression known as hydraulic efficiency of the system (η) [67], [68], [75], which reflects the system behavior, showing the percent of losses into the system, when the potential energy is converted to mechanical energy. Hydraulic efficiency is the ratio of the power developed by the turbine rotor (rotor power) to the power supplied at the nozzle by the fluid (fluid power). The efficiency is calculated with the following relation,



  
    
      η
      max
    
    =
    
      
        
          P
          T
        
        
          P
          N
        
      
      =
      
        
          
            
              T
              T
            
            ⁢
            ω
          
          
            
              1
              2
            
            ⁢
            
              V
              N
              2
            
            ⁢
            ρ
            ⁢
            
              
            
            ⁢
            Q
          
        
        =
        
          
            
              
                
                  ρ
                  ⁢
                  
                    
                  
                  ⁢
                  Q
                
                
                  r
                  1
                
              
              ⁢
              
                (
                
                  
                    
                      V
                      n
                    
                    ⁢
                    
                      U
                      1
                    
                    ⁢
                    G
                  
                  -
                  
                    
                      U
                      1
                      2
                    
                    ⁢
                    F
                  
                
                )
              
            
            
              
                1
                2
              
              ⁢
              
                V
                N
                2
              
              ⁢
              ρ
              ⁢
              
                
              
              ⁢
              Q
            
          
          =
          
            
              2
              ⁢
              
                (
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      U
                      1
                    
                    ⁢
                    
                      F
                      G
                    
                    ⁢
                    
                      U
                      1
                    
                    ⁢
                    G
                  
                  -
                  
                    
                      U
                      1
                      2
                    
                    ⁢
                    F
                  
                
                )
              
            
            
              
                
                  r
                  1
                
                ⁡
                
                  (
                  
                    2
                    ⁢
                    
                      U
                      1
                    
                    ⁢
                    
                      F
                      G
                    
                  
                  )
                
              
              2
            
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        η
        max
      
      =
      
        
          
            G
            2
          
          
            2
            ⁢
            
              r
              1
            
            ⁢
            F
          
        
        =
        
          0.7499
          =
          
            74.97
            ⁢
            %
          
        
      
    
  



Magnetic Generator System Design

The present invention is a combination of two systems to harvest the conversion of mechanical energy to electrical energy. The first system (previously analyzed) is the turbine, and the second is an alternating current generator, which uses electromagnetic principles to induce alternating current into coils, when permanent magnets are rotated around of the wires, creating on them a periodical change of magnetic flux.
PMs in a ring configuration, with alternated polarity and magnetic fields in the radial direction are combined with coils to form the PM generator. The PMs around the static central axis are rotated, and the magnetic flux from them crosses the coils to induce an electromotive force (emf). The coils are distributed in a central and stationary way, inside of the circle formed by the PM ring, with the PMs.
PM Generator Mathematical Model
The designs of magnets and coils used here have the same shape and distribution as most of the commercial HDD motor designs, which use a three-phase system with 9 coils and 12 magnets, distributed in a ring around the coils, with radial alternated polarity. Also, to select the number of armature coils and the number of pair poles the equation from [96], which supports the currents micro motor designs used in commercial hard drive disk machines, was used.




  
    n
    coil
  
  =
  
    
      3
      4
    
    ⁢
    
      n
      p
    
  




Where np is the number of permanent magnets used in the ring configuration. The present invention preferably uses a 3-phase winding with S=9 slots and a ring of np=12 PMs, creating a permanent magnet system of (S/3)/P= 3/12=1⁄4. In this relation, the numerator 1 in 1⁄4 is the number of slots for each phase in each pole phase and the denominator 4 is the number of poles required for a complete pattern. However, some laws and principles must be used to find the relationship between parameters of the permanent magnet machine. For instance, Ampere's Law relates the net magnetic field along a closed loop to the electric current passing through the loop, Biot-Savart's law describes the magnetic fields induced by a current, and Lenz's Law states the direction of the induced electromotive force (emf) resulting from a changing magnetic flux which has a polarity that leads to an induced current in the coils. Finally, as defined in the second chapter, Faraday's Law states that an emf is generated, if the magnetic flux changes for any reason A, B or Ø change.
As mentioned before, the simplest practical generator consists of circular coils and permanent magnets rotating. The coils are stationary and the ring of permanent magnets rotates with constant angular velocity co, and a uniform magnetic field of strength B. The flux per pole area Ø is defined as the integral of the flux density over the pole area Apm=




  
    l
    ⁢
    
      
        P
        r
      
      
        n
        p
      
    
  
  ,




(Where np is the number of poles, l is the pole height, and pr is the ring of magnets perimeter (pr=2πrPM), also, rPM is the radius of the ring of PMs), if the flux density is considered sinusoidally distributed. The flux per pole can be calculated using the winding function method [97]-[99], which is given by:




  
    Ø
    =
    
      
        
          ∫
          
            
              -
              π
            
            /
            2
          
          
            π
            /
            2
          
        
        ⁢
        
          
            B
            p
          
          ⁢
          
            A
            pm
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          θ
          ⁢
          
            d
            θ
          
        
      
      =
      
        
          (
          
            2
            ⁢
            
              B
              p
            
            ⁢
            
              A
              pm
            
          
          )
        
        =
        
          
            2
            ⁢
            
              B
              p
            
            ⁢
            l
            ⁢
            
              
                p
                r
              
              
                n
                p
              
            
          
          =
          
            2
            ⁢
            
              B
              p
            
            ⁢
            l
            ⁢
            
              
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  r
                  PM
                
              
              
                n
                p
              
            
          
        
      
    
  




  
    Ø
    =
    
      
        4
        ⁢
        π
        ⁢
        
          
        
        ⁢
        
          lr
          PM
        
        ⁢
        
          B
          p
        
      
      
        n
        p
      
    
  



When the rotor is moving and the magnets spin around the coils at constant electrical angular velocity ωe, the flux linked by coil is given by:

λ(t)=Nφ cos(ωet)

The induced voltage (emf) in the full-pitch stator coil is,



  
    emf
    =
    
      
        -
        
          
            d
            λ
          
          
            d
            t
          
        
      
      =
      
        
          -
          
            (
            
              
                N
                ⁢
                
                  
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        e
                      
                      ⁢
                      t
                    
                    )
                  
                
                ⁢
                
                  
                    d
                    φ
                  
                  
                    d
                    t
                  
                
              
              -
              
                N
                ⁢
                
                  
                
                ⁢
                
                  φω
                  e
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  ω
                  e
                
                ⁢
                t
              
            
            )
          
        
        =
        
          N
          ⁢
          
            
          
          ⁢
          
            φω
            e
          
          ⁢
          sin
          ⁢
          
            
          
          ⁢
          
            ω
            e
          
          ⁢
          t
        
      
    
  




  
    emf
    =
    
      
        
          4
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            lr
            PM
          
          ⁢
          
            B
            p
          
        
        
          n
          p
        
      
      ⁢
      
        ω
        e
      
      ⁢
      N
      ⁢
      
        
      
      ⁢
      sin
      ⁢
      
        
      
      ⁢
      
        ω
        e
      
      ⁢
      t
    
  



The mechanical angular velocity and the electrical angular velocity are related in this relationship as,




  
    ω
    e
  
  =
  
    
      
        n
        p
      
      2
    
    ⁢
    
      ω
      m
    
  




(np is the number of machine poles, and ωm is the mechanical angular velocity).




  
    emf
    =
    
      
        
          4
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            lr
            PM
          
          ⁢
          
            B
            p
          
        
        
          n
          p
        
      
      ⁢
      
        
          n
          p
        
        2
      
      ⁢
      
        ω
        m
      
      ⁢
      N
      ⁢
      
        
      
      ⁢
      sin
      ⁢
      
        
      
      ⁢
      
        (
        
          
            
              n
              p
            
            2
          
          ⁢
          
            ω
            m
          
          ⁢
          t
        
        )
      
    
  




  
    emf
    =
    
      2
      ⁢
      π
      ⁢
      
        
      
      ⁢
      
        lr
        PM
      
      ⁢
      
        B
        p
      
      ⁢
      
        ω
        m
      
      ⁢
      N
      ⁢
      
        
      
      ⁢
      
        sin
        ⁡
        
          (
          
            
              
                n
                p
              
              2
            
            ⁢
            
              
            
            ⁢
            
              ω
              m
            
            ⁢
            t
          
          )
        
      
    
  



The maximum value is reached when sin




  
    
      (
      
        
          
            n
            p
          
          2
        
        ⁢
        
          ω
          m
        
        ⁢
        t
      
      )
    
    =
    1
  
  ,




then

efmmax=2πlrPMBpωmN 

The maximum value of induced RMS value of speed voltage is written as,




  
    E
    RMS
  
  =
  
    
      
        φ
        ⁢
        
          
        
        ⁢
        
          ω
          e
        
        ⁢
        N
      
      
        2
      
    
    =
    
      
        
          2
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            lr
            PM
          
          ⁢
          
            B
            p
          
          ⁢
          
            ω
            m
          
          ⁢
          N
        
        
          2
        
      
      ⁡
      
        [
        Volt
        ]
      
    
  




Permanent magnets machines or high power AC machines may have distributed or short-pitch windings and a winding factor must be included in the ERMS equation, but in this research, this factor will be included in the general losses of the system.
Complete Energy Generation System Theoretical Calculations
A new mathematical analysis is developed to find a general equation that establishes the relationship between the mechanical system (turbine) and the permanent magnet generator system. The initial conditions and some predefined geometry parameters are summarized in the next table,







TABLE 1







Energy Generation System Design Dimensions.








Turbine
Permanent Magnet Machine (PMM)












Parameter
Dimension
Units
Parameter
Dimension
Units















α1
10
degree
np
12



α2
30.01
degree
Slots (S)
9



α3
50.35
degree
l
1
mm


β1
9.6
degree
rPM
2.6
mm


r1
3.95
mm
pr
16.34
mm


r2
3.1
mm
N
60
Turns


rm
3.5
mm
Bp
1
Tesla


rN
0.65
mm
Apm
1.361 E−6
m2


rr
4
mm
ωm
1000-8000
RPM


AN
1.327 E−6
m2
Electrical
 70-700
mW





Power









The coil and magnet shapes were inspired by the motors used in computer hard disc drive (HDD), combined with the fundamental principles of hydro turbines. The HDD motor is a kind of brushless machine, which uses a conjunct of coils oriented in the radial direction inside a stator chamber, surrounded by a ring of frictionless permanent magnets, assembled on a rotor. In this invention, the permanent magnetic configuration is used as a magnet generator to produce current in the inductor coils, transforming the rotational motion of the turbine into electricity.
As shown in FIG. 7, PMs 124 are preferably arranged in a ring configuration, with alternating polarity and radial direction. Additionally, PMs 124 are set into rotor support discs 104 and the ring has a central axis aligned with the rotational axis of the rotor. As shown in FIG. 8, coils 126 are preferably integrated into the top and bottom of turbine casing 103. In this embodiment, there are nine coils 126 configured in three phases on both the top and bottom of turbine casing 103. The first phase is identified by coils 126a, the second phase is identified by coils 126b, and the third phase is identified by coils 126c. The rings of PMs 124 are assembled on the top and bottom of rotor 100, around coils 126, but with a micro gap to avoid any friction when the rotor is in movement. A double combination of coils and permanent magnets assures a higher efficiency that finally translates into more energy being transformed.
The relationship between turbine and PM generator starts in the novel design of the turbine, which is integrated with the PMs and coils in a unique compact system. The final system includes the PMs and coils attached inside of the rotor and walls of the turbine casing to maintain the preferred shape of the turbine casing and rotor while using the benefits of the rotor spin to create a variable magnetic field on the fixed coils and to generate electricity. A system of three phases (3Φ) of electrical energy is the result of the electromagnetic induction. The induction of electric currents into the coils fixed into the casing is produced by the magnetic fields from the ring of PMs, which spin when the rotor is moving. The PMs are assembled in alternating polarity on the rotor, and distributed in a ring configuration to produce radial lines of magnetic field that cross the coils in a perpendicular direction to generate electric current, induced when the movement of the rotor changes in position with respect to the PMs.
The mathematical models and the equations developed in the previous sections are related through the angular velocities, mechanical angular velocity (ωm) produced on the rotor and the electrical angular velocity (ωe) which is created on the ring of PMs. As previously defined, the mechanical angular velocity and the electrical angular velocity are related as




  
    ω
    e
  
  =
  
    
      
        n
        p
      
      2
    
    ⁢
    
      
        ω
        m
      
      .
    
  




So, if the maximum mechanical power is reached when VN=5.4528 U1 (VN is the fluid velocity after leave the nozzle in the rotor direction, that is known as nozzle velocity), and U1=ωmr1, then a new equation emerges as a result of combining them,





  emf
  =
  
    
      2
      ⁢
      π
      ⁢
      
        
      
      ⁢
      
        lr
        PM
      
      ⁢
      
        B
        p
      
      ⁢
      
        ω
        m
      
      ⁢
      N
      ⁢
      
        
      
      ⁢
      
        sin
        ⁡
        
          (
          
            
              
                n
                p
              
              2
            
            ⁢
            
              ω
              m
            
            ⁢
            t
          
          )
        
      
    
    =
    
      2
      ⁢
      π
      ⁢
      
        
      
      ⁢
      l
      ⁢
      
        
      
      ⁢
      
        r
        PM
      
      ⁢
      
        B
        p
      
      ⁢
      
        
          U
          1
        
        
          r
          1
        
      
      ⁢
      N
      ⁢
      
        
      
      ⁢
      
        sin
        ⁡
        
          (
          
            
              
                n
                p
              
              2
            
            ⁢
            
              ω
              m
            
            ⁢
            t
          
          )
        
      
    
  




Substituting VN and the constants values (l, rPM, r1, np) then the equation is rewritten as,



  
    emf
    =
    
      2
      ⁢
      π
      ⁢
      
        
      
      ⁢
      l
      ⁢
      
        
      
      ⁢
      
        r
        PM
      
      ⁢
      
        B
        p
      
      ⁢
      
        
          U
          1
        
        
          r
          1
        
      
      ⁢
      N
      ⁢
      
        
      
      ⁢
      
        sin
        ⁡
        
          (
          
            
              
                n
                p
              
              2
            
            ⁢
            
              ω
              m
            
            ⁢
            t
          
          )
        
      
    
  




  
    emf
    =
    
      
        
          2
          ⁢
          π
          ⁢
          
            
          
          ⁢
          l
          ⁢
          
            
          
          ⁢
          
            r
            PM
          
          ⁢
          
            B
            p
          
        
        5.4528
      
      ⁢
      
        
          V
          N
        
        
          r
          1
        
      
      ⁢
      N
      ⁢
      
        
      
      ⁢
      
        sin
        ⁡
        
          (
          
            
              
                n
                p
              
              2
            
            ⁢
            
              ω
              m
            
            ⁢
            t
          
          )
        
      
    
  




  
    emf
    =
    
      0.1313
      ⁢
      
        V
        N
      
      ⁢
      
        
      
      ⁢
      
        sin
        ⁡
        
          (
          
            282.14
            *
            
              V
              N
            
            *
            t
          
          )
        
      
    
  



The theoretical emf response when a volumetric flow rate




  Q
  =
  
    3000
    ⁢
    
      ml
      min
    
  




goes through the system, with angular velocity ωe=1550.4 RPMs and produces a mechanical power from the rotor, Pr=102.3 mW can be seen in FIG. 9.

Experimentation

To determine the functionality of the assembled prototype and to develop the final tests, the turbine inlet system was connected to a domestic water system. The measurements of volume flow rate were taken on the turbine inlet, through a volume flow rate digital meter, connected in series with the water system and a gradual flow controller. The measurements of induced voltage were done on the terminals of the 3-phase system through a voltmeter and an oscilloscope connected in parallel with the motor-generator terminals. Three prototypes were tested as shown in Table 2 below:







TABLE 2







Comparative analysis of three prototypes.











Prototype


Geometry and
Additional


Model
Shape
Size
curvature
Characteristics





Model 1
8 blades
3.175X,
90 degree
Impulse turbine




Small
between inlet
characteristics,




rotor
and outlet,
rotor spin is require





innovative
to free flow





casing design



Model 2
6 blades
4X,
90 degree
Impulse turbine




Big rotor
between inlet
characteristics,





and outlet, 
rotor spin is require





innovative 
to free flow





casing design



Model 3
6 notched
3.175,
100 degree
Impulse and



blades
Medium
between inlet
reaction turbine




rotor
and outlet,
characteristics, 





innovative
cross-flow





casing design
design, free flow









The first two generation of prototypes were printed in the Virtual Manufacturing and Design Laboratory for Medical Devices (VirtualMDLab) at USF, by members of Dr. Lai-Yuen's team, and the last two generations were printed by USF's Engineering Technical Support Services (ENG TSS). Most of the prototypes developed through this dissertation work were made using a 3D prototyping printer which uses Acrylonitrile Butadiene Styrene (ABS) plastic to build parts (Dimension sst-768), while only two sets were made using a different 3D prototyping printer which uses a proprietary powder and binder system to build the parts (3D Systems' ZPrinter-450).
Model 3 is the preferred embodiment. The turbine casing has an angle of 100 degrees between the inlet and outlet sections, this inclination results in the fluid being directed more towards the distal end of each blade, which has greater surface area since the notch is located on the internal, proximal edge. In other words, the first is a nozzle inclination of 10 degrees with respect to the radial and tangential axes. The behavior simulations of the fluid through the turbine are shown in FIG. 14C. FIG. 14C illustrates how the contribution of the nozzle inclination aids in reducing the vortices between the blades and on the central area of the rotor.
Motor Magnet Generator
There are different models of electromagnetic systems, which could be used as electromagnetic generators, but after an extensive cost benefit analysis, it was decided to use the shape and configuration of coils and permanent magnets used in typical hard disk drive (HDD) motors. These motors generally consist of a ring of permanent magnets and a winding in nine slots, which interact to create a three-phase electric power system.
In the assembly process of the three prototypes, 4 HDD motors were used, two of them from 3.5 inch HDD motors (3.5 inch is the physical dimensions of the drive), and two from 2.5 inch HDD motors (2.5 inch is the physical dimensions of the drive). These kinds of motors have the same shapes, but the sizes of all elements involved in the models are different, hence the magnetic induction response is different. As was previously defined, the induced voltage (emf) will be determined by the number of PMs, the surface area of each PM, the intensity and density of the magnetic field, and the rotation speed of the magnets around the coils.
Energy Generation System







TABLE 3







Turbine and permanent magnets dimensions and configuration.








Turbine
Permanent Magnet Machine (PMM)












Parameters
Dimensions
Units
Parameters
Dimensions
Units















α1
10
degree
nP
12



α2
30.01
degree
Slots (S)
9



α3
50.35
degree
l
1
mm


β1
9.6
degree
rPM
2.6
mm


r1
3.95
mm
pr
16.34
mm


r2
3.1
mm
N
60
Turns


rm
3.5
mm
Bp
1
Tesla


rN
0.65
mm
Apm
1.361 E−6
m2


rr
4
mm
ωm
1000-8000
RPM


AN
1.327 E−6
m2
Electrical 
 70-700
mW





Power









Energy Generation System Testing Results

The three selected models have been simulated by using ANSYS Workbench V. 13.0, and the associated software to include geometry, mesh, pre-processing and post-processing. The simulation processes used the same solver control, output control, and data expressions, to maintain the same reference point for the analysis. The velocity and pressure behavior of the three models are shown in FIGS. 11-13.
FIG. 10 shows the characterization data and linearization of a small HDD motor used as PM-generator. The data of FIG. 10 was collected at USF's AMBIR lab and was to find the relationships between the voltage and angular velocity, and between the volume flow rate and angular velocity, when the HDD motor is assembled into the turbines and used as a motor-generator machine.
The characteristics of the three models are summarized in Table 3; and the testing results are shown in Table 4 below. The data points of voltage and volume flow rate in Table 4 were taken using the Extech 430 (part number EX430) True RMS Autoranging Multimeter and the OMEGA Digital Paddlewheel Meter FP2001-RE respectively. The volume flow meter was connected between the domestic water system and the turbine inlet to measure how many gallons per minute were entering the Turbine-generator system, and to use this meter as a flow regulator. The volume flow rate measured was between 0 and 4 gallons per minute. The maximum Q of the turbine was sufficient to move the motor/generator in a safe range of revolutions per minute, without causing any damage to the internal structure of the generator's motor. The HDD motors used and assembled into the prototypes were designed to work at angular velocity of 5400 RPM [100]-[103].







TABLE 4







Test results: Volume flow rate vs. voltage produced.









Prototype 1
Prototype 2
Prototype 3 


(Old-3.175X)
(Old-4X)
(Final-3.175)












Volume

Volume

Volume



Flow Rate

Flow Rate

Flow Rate



Q (Gallons/
Voltage
Q (Gallons/
Voltage
Q (Gallons/
Voltage


Minute)
(V)
Minute)
(V)
Minute)
(V)















0.92
0.07
0.52
0.044
0.63
0.142


1.1
0.168
0.62
0.092
0.71
0.173


1.14
0.19
0.68
0.12
0.81
0.205


1.2
0.22
0.72
0.141
0.9
0.26


1.21
0.232
0.82
0.184
1.02
0.32


1.34
0.271
0.9
0.19
1.11
0.362


1.42
0.3
1
0.276
1.21
0.421


1.48
0.323
1.1
0.322
1.31
0.471


1.5
0.334
1.2
0.371
1.4
0.51


1.55
0.34
1.3
0.409
1.51
0.56


1.62
0.372
1.4
0.452
1.61
0.6


1.7
0.401
1.51
0.47
1.71
0.64


1.78
0.435
1.6
0.535
1.8
0.666


1.8
0.45
1.71
0.58
1.85
0.684


1.83
0.44
1.8
0.6
1.9
0.7


1.92
0.47
1.83
0.615
1.95
0.737


2
0.518
1.91
0.635
2
0.76


2.3
0.62
2
0.684
2.11
0.794


2.4
0.625
2.1
0.716
2.2
0.83


2.5
0.684
2.23
0.79
2.3
0.87


2.66
0.706
2.3
0.807
2.4
0.91


2.75
0.738
2.4
0.86
2.5
0.95


2.8
0.76
2.5
0.905
2.6
0.985


2.9
0.81
2.71
0.98
2.71
1.025


3
0.83
2.8
1
2.8
1.05


3.2
0.875
2.91
1.043
2.9
1.06


3.46
0.943
3.02
1.09
3
1.13


3.79
1.04
3.08
1.11
3.12
1.16




3.13
1.14
3.2
1.185




3.29
1.19
3.25
1.195




3.3
1.195
3.3
1.125




3.35
1.21
3.47
1.26




3.43
1.24









Analysis of the Results

The internal behavior of the turbines, when a fluid flows through the system, is shown in FIGS. 11, 12, and 13. These simulations show the internal differences between the three turbine models and confirm the adaptability problems with the non-notched models, when operating in a closed circulatory system. In contrast to the non-notched models (FIGS. 11 and 12), the notched blade turbine system, shown in FIG. 13, contains several innovations, the most important being the inclusion of a notch on each blade. As shown in FIG. 13, the notches in the internal central edge of each blade allow for continuous circulation of liquid inside of the rotor-chamber, and ensure more interaction between the fluid and the blades. In this novel design, the fluid strikes the first blade and the notch redirects the fluid, to impact more than one blade.
The simulations in FIGS. 11-13 show the impulse jet behavior on the first blade in the three prototyped models. Only in the notched system is most of the fluid directed clockwise, resulting in more than one blade being impacted and contributing to an increased rotor spin. The simulations of the notched turbine in FIG. 13, illustrates how the notched design results in more interaction between the fluid and the blades than that of non-notched blades. FIGS. 11 and 12 show the fluid impacting the first non-notched blade, producing only a single force impulse on the rotor, while the notched blade model in FIG. 13 shows a direct jet impact on the first blade, and an indirect impact on second and third blades. While this benefit is mostly due to the notches in the blades, the blade curvature and casing shape also aid in the directed flow.
Other differences between models are in the transient response (see FIGS. 14A-C), which is immediate in the notched turbine (FIG. 14C). In addition, flow circulation problems were detected in the non-notched bladed turbines because, when the flow was interrupted, the turbine inlet pressure and the pressure in the first blade increased dramatically, causing vortices inside of the rotor blades. These vortices could reduce the lifetime of the turbine system.
The testing results of Voltage (V) vs. Volume Flow Rate (Q) and Voltage (V) vs. Angular Velocity (ω) of the three prototypes are shown in FIGS. 15 and 16, respectively. Using the same volume flow rate of the three models, an increase in voltage can be seen in the notched model when compared to the models without the notch. Moreover, to spin the rotor in the three prototypes, a higher volume flow rate is needed to spin the non-notched models, than in the notched model. In summary, the test results validate the novel notched blade design of the present invention. Furthermore, these analyses support the conclusion that the notched system is more efficient than the non-notched models, and satisfies the minimal efficiency general standard of impulse turbines, which reach an average of 70% efficiency.
FIG. 17 was developed to understand the difference between the theoretical calculations and the experimental data. The mathematical model was used to find the relationship between volume flow rate and the ideal voltage that must be produced by the notched blade energy generation system. The parameters of a 3.175× size model were used and the final relationship is shown FIG. 17. The experimental data of the notched blade energy generation prototype was used to complete FIG. 17. Under ideal conditions, the experimental and theoretical results of FIG. 17 should be close to being equal. The mathematical relationship of experimental to theoretical produces a result of 53%. This result has multiple causes, which are listed as follows:

    
    
        a. The theoretical calculations were developed using a perfect model, which did not include the losses, by structure, frictions, material, viscosity, and gravity.
        b. The materials used to develop the prototypes, and the limitations of the 3D printer machines at USF labs created assembly problems.
        c. The prototype quality was impacted by structural problems in the casing walls, and many tiny holes increased wall porosity and changed the internal behavior of the system.
        d. The relationship between the inlet and outlet volume flow rate was affected by the porosity of the material, which became a considerable factor that adversely affected the efficiency of the system.
        e. The materials, facilities, and tools used presented limitations, but were sufficient to show proof of concept.
        f. Finally, the HDD motors used as motor-generators, which added friction, weight, forces, and torques, were not taken into account in the theoretical model resulting in an efficiency of 53% when compared with the experimental results.
    
    


In a certain embodiment, the present invention may utilize any number of notched blades known to a person having ordinary skill in the art, however, there are preferably between six and twelve blades.
In a certain embodiment, a levitation system may be used on the rotor to create rotor levitation as previously mentioned. A permanent-magnet configuration in the casing and the rotor creates the rotor levitation effect. This special characteristic could minimize friction inside of the turbine, reducing pressure through the turbine, as well as contribute to increased rotor spin and efficiency. In a certain embodiment, as shown in FIG. 18, the present invention utilizes twenty four PMs 130 attached in the lateral side of each rotor disk, and twenty four PMs 132 inside of the rotor chamber to create rotor levitation. In a certain embodiment, PMs 130 and 132 are arranged in the rotor and rotor chamber so that they are in the same plane and can perfectly aligned with each other in the longitudinal direction, see FIG. 18. These PMs have 30 degree of inclination with respect to the radial direction and all of them have the same external polarity, to produce repulsion when the PM's frontal cross section are longitudinally aligned and attraction when there are 15 degree or more between them. The orientation of the PMs allows for rotor levitation (bearingless, brushless and without spindles), resulting in frictionless rotation.
In a certain embodiment, the present invention is made of certain materials that allow for integration into a bio-physiological system, including a titanium turbine, permanent magnets coated with biocompatible material, and copper inductor coils coated with biocompatible material. Additionally, the circuit elements and connectors exposed to an in vivo environment will be constructed of silicon carbide, and any surface in contact with blood or tissue will be biocompatible.
The advantages set forth above, and those made apparent from the foregoing description, are efficiently attained. Since certain changes may be made in the above construction without departing from the scope of the invention, it is intended that all matters contained in the foregoing description or shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense.
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## Claims

1. A rotor, comprising:
a plurality of blades, wherein each blade has a proximal edge, a distal edge, and a length running between the proximal edge and the distal edge, the proximal edges of the blades are disposed at a predetermined concentric orientation about a rotational axis of the rotor; and
a notch centrally and adjacently disposed on the proximal edge of each blade.

2. The rotor of claim 1, further including a central hub axially aligned with the rotational axis of the rotor, and the axial hub having a lateral surface in communication with the proximal edge of each blade.

3. The rotor of claim 2, wherein the hub has a cylindrical shape with a radius that is equal to half of a total radius of the rotor.

4. The rotor of claim 1, further including a first support disc and a second support disc, wherein each support disc is axially aligned with the rotational axis of the rotor and in communication with a side of each blade in the plurality of blades.

5. The rotor of claim 1, wherein the notch in each blade is semicircular in shape and has a radius equal to or less than half of a radius of one of the blades.

6. A turbine generator, comprising:
a turbine casing, wherein the turbine casing further includes an inlet section, a nozzle fluidly coupled to the inlet section, a rotor chamber fluidly coupled to the nozzle, an outlet section fluidly coupled to the rotor chamber, and hollows adapted to receive inductor coils;
a rotor disposed in the rotor chamber, the rotor including a plurality of blades, wherein each blade in the plurality of blades has a central transversal axis, a distal edge, a proximal edge disposed at a predetermined concentric orientation about a rotational axis of the rotor and a notch disposed along the central transversal axis;
a magnet generator assembly further comprising:
at least one permanent magnet disposed on the rotor; and
at least one inductor coil disposed in the hollows of the turbine casing in an orientation perpendicular to the centrifugal movement of the at least one permanent magnet.

7. The turbine generator according to claim 6, further comprising a levitation system including a plurality of permanent magnets disposed in a first wall of the turbine casing, a plurality of permanent magnets disposed in a second wall of the turbine casing, and a plurality of permanent magnets disposed in the rotor, wherein the permanent magnets disposed in the rotor and turbine casing are configured to magnetically suspend the rotor in the rotor chamber such that the rotor is not in contact with the turbine casing.

8. The turbine generator according to claim 6, wherein the rotor further includes a cross-flow design such that the notch on each blade is adjacent to the proximal edge of each blade.

9. The turbine generator according to claim 6, wherein the rotor further includes a cylindrical hub axially aligned with the rotor's axis of rotation, wherein the hub includes a lateral surface in communication with the proximal edge of each blade.

10. The turbine generator according to claim 9, wherein the hub has a radius that is generally equal to half of a total radius of the rotor.

11. The turbine generator according to claim 6, wherein the at least one inductor coil includes a plurality of inductor coils having a 3-phase, 9-slot winding layout, where each phase has three circular coils separated by 120 degrees in a star configuration and connected in series.

12. The turbine generator according to claim 6, further comprising each part of the turbine generator made of or coated in a biocompatible material.

13. The turbine generator according to claim 6, wherein the inlet section and outlet section of the turbine casing are separated by an angle of one hundred degrees with respect to the rotational axis of the rotor.

14. The turbine generator according to claim 6, further including a first support disc and a second support disc, wherein each support disc is axially aligned with the rotational axis of the rotor and in communication with a side of each blade.

15. The turbine generator according to claim 6, wherein the notch in each blade is semicircular in shape and has a radius equal to or less than half of the radius of one of the blades.

16. A turbine generator, comprising:
a turbine casing, wherein the turbine casing further includes:
an inlet section;
a nozzle fluidly coupled to the inlet section;
a rotor chamber fluidly coupled to the nozzle;
an outlet section fluidly coupled to the rotor chamber;
hollows adapted to receive inductor coils; and
one hundred degrees of separation between the inlet section and the outlet section;

a rotor disposed in the rotor chamber, wherein the rotor further includes:
a central hub;
a plurality of blades, wherein each blade has a central transversal axis, a distal edge, a proximal edge attached to the central hub, and a notch disposed along the central transversal axis adjacent to the proximal edge;
a first support disc and a second support disc, wherein each support disc is axially aligned with the rotational axis of the rotor and in communication with a side of each blade;

a magnet generator assembly further comprising:
at least one permanent magnet disposed in the rotor;
at least one inductor coil disposed in the hollows of the turbine casing in an orientation perpendicular to the centrifugal movement of the at least one permanent magnet; and

a levitation system including a plurality of permanent magnets disposed in a first wall of the turbine casing, a plurality of permanent magnets disposed in a second wall of the turbine casing, and a plurality of permanent magnets disposed in the rotor, wherein the permanent magnets disposed in the rotor and turbine casing are configured to magnetically suspend the rotor structure in the rotor chamber such that the rotor structure is not in contact with the turbine casing.

17. The turbine generator according to claim 16, wherein the at least one inductor coil includes a plurality of inductor coils having a 3-phase, 9-slot winding layout, where each phase has three circular coils separated by 120 degrees in a star configuration and connected in series.

18. The turbine generator according to claim 16, wherein the hub of the rotor has a cylindrical shape with a radius that is generally equal to half of a total radius of the rotor.

19. The turbine generator according to claim 16, wherein the notch in each blade is semicircular in shape and has a radius equal to or less than half of the radius of one of the blades.