THERMODYNAMIC WIND TURBINE

Abstract

The invention discloses improved versions of a horizontal axis wind turbine and new fundamental methodologies for the design of wind turbines, which are capable of extracting both kinetic and thermal energy from the wind. The wind turbines use a large diameter forward inlet fairing to accelerate the airflow to the more effective outer radii of the turbine rotor where the airflow is constrained by an airfoil-shaped flow control ring, which also serves to prevent rotor tip losses, to inhibit wake expansion, and to accelerate the airflow through the turbine. A similarly large diameter aft pressure recovery fairing promotes rotation and contraction of the wake downstream of the turbine. Further methodologies for optimization and an algorithm for detail design are disclosed.

Claims

1. A horizontal axis wind turbine comprising of: a forward central portion consisting of a streamlined inlet fairing of substantial diameter for the purpose of reducing the flow area by an amount sufficient to cause a significant acceleration of the airflow velocity and furthermore to redirect the airflow to the more effective outer regions of the turbine radii; an aft central portion consisting of a streamlined fairing of a diameter approximately equal to said forward inlet fairing for the purpose of a smooth aerodynamic pressure recovery of the airflow aft of turbine; a plurality of conventional airfoil-shaped rotor blades attached to and extending out from either the forward or aft fairing whichever rotates and drives a conventional power generation unit depending on the embodiment; and an outer airfoil-shaped flow control ring with the positive pressure surface of the airfoil of said ring oriented towards the center of the turbine and which is attached to the tips of said rotor blades, rotating with said blades; whereby the airflow entering the wind turbine is accelerated and therefore of higher dynamic pressure to react with the rotor blades and said airflow is constrained at the more effective outer radii of the wind turbine rotor thereby increasing the overall efficiency and power extraction of the wind turbine.

2. The wind turbine according to claim 1, wherein the forward inlet fairing houses the power generation unit and is fixed with the turbine's horizontal axis and at some area attached to a tower but free to rotate about the vertical axis of said tower allowing for the alignment of said turbine with the wind, while the aft fairing is free to spin with its means of internal support and attached rotor blades about the horizontal axis of said wind turbine thereby driving the power generation unit.

3. The wind turbine according to claim 1, wherein the aft fairing houses the power generation unit and is fixed with the turbine's horizontal axis and at some area attached to a tower but free to rotate about the vertical axis of said tower allowing for alignment of said turbine with the wind, while the forward inlet fairing is free to spin with its means of internal support and attached rotor blades about the horizontal axis of said wind turbine thereby driving the power generation unit.

4. The wind turbine according to claim 1, wherein the performance analysis and detailed design of the configuration are carried out using the following algorithm: (a) choose an applicable wind velocity V.sub.1 and initialize an assumed value for inflow velocity ratio a.sub.i=V.sub.2/V.sub.1, the optimum value of a.sub.i should be approximately 0.54; (b) calculate V.sub.2, the minimum velocity that the airflow is slowed to as it approaches the turbine using the formula
V.sub.2=a.sub.iV.sub.1 and calculate V.sub.2.5, the maximum velocity the airflow reaches as it passes through the turbine based on the size of the forward inlet fairing using the formula $V_{2.5}=a_s.Math.a_i.Math.V_1=\frac{a_i.Math.V_1}{1-{\left(\frac{r_s}{R}\right)}^2};$ (c) discretize the flow field into annular elements and layout for each annular element: the number of rotor blades, planform, airfoil chord, pitch angle θ, and rotor velocity Ωr; compute the relative velocity,
V.sub.rel=√{square root over ((V.sub.2.5.sup.2+Ω.sup.2r.sup.2))}, the inflow angle
φ=tan.sup.−1(V.sub.2.5/Ωr), and local angle of attack
α=(φ−θ); (d) using calculations from (c) in combination with known airfoil data or airfoil design software determine local lift coefficient C.sub.l and drag coefficient c.sub.d for each blade element; (e) using calculations from (c) and results from (d), calculate the normal forces on each blade element using standard methods of blade element analysis; (f) sum all the elemental normal forces, computing a total turbine normal force F.sub.n, calculate the thrust coefficient C.sub.T=F.sub.n/(qA), estimate D.sub.C, D.sub.O, and D.sub.I, the drag forces of the center area reduction fairings, the outer flow control ring, and intersection drag, respectfully, then calculate C.sub.D for any non-power extracting flow devices from $C_D=\frac{\left(D_C+D_O+D_I\right)}{\mathrm{qA}},$ and from this calculate the corrected value for a.sub.i using a.sub.i=[1−0.5(C.sub.T+C.sub.D)].sup.2/3; (g) compare the result for a.sub.i from (f) to the initialized value and iterate (a) through (f) until the final value of a.sub.i is in reasonable agreement with the initialized value; the optimum value of the inflow velocity ratio a.sub.i is configured through the aforementioned parameters to be equal to approximately 0.54; (h) calculate differential pressure for each annular element by dividing the elemental blade normal forces by the elemental areas and confirm that these values are reasonably close, if not, adjust chord or pitch angle to maintain reasonably uniform pressure differential between the elements which implies a uniform flow field; iterate (a) through (h) until a.sub.i is in reasonable agreement with the initialized value and the differential pressure between the annular elements is in reasonable agreement representing uniform flow; (i) calculate the normal and tangential forces on each blade element using standard methods of blade element analysis, then compute the total loads and moments on all the blades, finally calculate the turbine power and thrust using standard methods.

5. A new method for optimizing the design of a horizontal axis wind turbine, wherein the wind turbine is configured to maximize the power extracted from a given set of turbine parameters by maximizing the power coefficient for an open flow rotor using the new formula
C.sub.P=2η.sub.Tsa.sub.i(1−a.sub.i.sup.1.5) or for a constrained flow rotor using the new formula
C.sub.P=η.sub.Tsa.sub.sa.sub.i[2(1−a.sub.i.sup.1.5)−C.sub.D], where C.sub.P is the power coefficient defined as $C_P=\frac{{\stackrel{.}{W}}_{\mathrm{out}}}{{\mathrm{qV}}_1.Math.A_2},$ where {dot over (W)}.sub.out is the power extracted, ρ is the density of the air, V.sub.1 is the free stream velocity of the airflow, A is the area swept by the wind turbine, V.sub.2 is the minimum velocity the airflow is slowed to as it approaches the turbine, a.sub.i is the inflow velocity ratio equal to V.sub.2/V.sub.1, which for an open flow rotor is evaluated by the new formula
a.sub.i=(1−0.5C.sub.T).sup.2/3, and for a constrained flow rotor is evaluated by the new formula
a.sub.i=[1−0.5(C.sub.T+C.sub.D)].sup.2/3, where C.sub.T is the thrust coefficient equal to F.sub.n/(qA), C.sub.D is the drag coefficient equal to (D.sub.C+D.sub.O+D.sub.I)/(qA) F.sub.n is the total of blade normal forces acting on the wind turbine rotor, D.sub.C is the drag force of the center area reduction fairings, D.sub.O is the drag force of the outer flow control ring, D.sub.I is the accumulated intersection drag, q is the dynamic pressure of the airflow equal to ρV.sub.1.sup.2/2, η.sub.Ts is the efficiency of the rotor blades at a given annular element and is evaluated by the new formula ${\eta }_{\mathrm{Ts}}=\frac{\frac{L}{D}-\frac{{\lambda }_r}{a_i.Math.a_s}}{\frac{L}{D}+\frac{a_i.Math.a_s}{{\lambda }_r}},$ where L/D is equal to the local lift to drag ratio of the blade element, λ.sub.r is equal to the local speed ratio of the blade element which is equal to Ωr/V.sub.1, r is the local radius of the blade element, Ω is the angular velocity of the rotor blades, and a.sub.s is the flow acceleration factor attributed to the non-power extracting flow devices evaluated by the new formula $a_s=\frac{1}{1-{\left(\frac{r_s}{R}\right)}^2},$ where r.sub.s is the radius of a circular reduction in inlet flow area and R is the outer radius of the rotor blade assembly.

6. The method according to claim 5, wherein the optimum value of the inflow velocity ratio a.sub.i is configured through available parameters to be equal to approximately 0.54.

7. A new method for the enhanced extraction of both kinetic and thermal energy from a horizontal axis wind turbine comprising: a means of accelerating the freestream airflow through a constrained region of the outer radii of a horizontal axis wind turbine thereby increasing the dynamic pressure reacting with the rotor blades at the more effective outer radii of the turbine and thereby increasing the efficiency with which the energy is transferred into the turbine shaft in the form of torque times angular velocity while simultaneously imparting increased angular velocity into the slipstream, in combination with a means of enhancing the smooth aft flow of the slipstream while inhibiting the expansion of the wake downstream of the wind turbine, while promoting the reacceleration of the slipstream thereby increasing both mass flow through the wind turbine and increasing the rate of rotation of the slipstream, wherein the power extracted by said wind turbine using this new method is shown to be a function of the total energy equation including thermodynamic and rotational parameters equal to
{dot over (W)}.sub.t=τΩ=½{dot over (m)}ω.sub.3ΩR.sup.2={dot over (m)}[c.sub.p(T.sub.1−T.sub.6)+½(V.sub.1.sup.2−V.sub.6.sup.2)−¼ω.sub.6.sup.2r.sub.6.sup.2] where {dot over (W)}.sub.out is the power extracted, τ is the torque reacting with the turbine shaft, ω.sub.3 is the angular velocity of the slipstream immediately downwind of the rotor, ω.sub.6 is the final angular velocity of the slipstream, Ω is the angular velocity of the rotor, R is the outer radius of the rotor blade assembly, r.sub.6 is the final radius of the slipstream, c.sub.p is the specific heat capacity of air at constant pressure, T.sub.1 and T.sub.6 are the initial and final temperatures of the airflow respectfully, V.sub.1 is the velocity of the free stream unaffected by the turbine, V.sub.6 is the final velocity of the airflow far down wind of the turbine, {dot over (m)} is the mass flow rate {dot over (m)}=ρa.sub.iV.sub.1A.sub.2, a.sub.i is the inflow velocity ratio equal to V.sub.2/V.sub.1, b.sub.i is the outflow velocity ratio equal to V.sub.6/V.sub.1, both a.sub.i and b.sub.i are evaluated for an open flow rotor by using the new formula
a.sub.i=b.sub.i.sup.2=(1−0.5C.sub.T).sup.2/3, and for a constrained flow rotor are evaluated by using the new formula
a.sub.i=b.sub.i.sup.2=[1−0.5(C.sub.T+C.sub.D)].sup.2/3, where C.sub.T is the thrust coefficient equal to F.sub.n/(qA), C.sub.D is the drag coefficient equal to D.sub.P/(qA), F.sub.n is the total of blade normal forces acting on the wind turbine rotor, D.sub.P is the parasitic drag force of the system, q is the dynamic pressure of the airflow equal to ρV.sub.1.sup.2/2, whereby this method is shown to have the additional advantage of reducing the temperature of the airflow through the turbine, and wherein the final temperature of the affected airflow after the dissipation of rotational kinetic energy can be found to be equal to $T_f=T_1+\frac{\frac{1}{2}.Math.\left(V_1^2-V_6^2\right)-\frac{\tau .Math..Math.\Omega }{\stackrel{.}{m}}}{c_p}.$

Description

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

[0023] FIG. 1A is a perspective view of one embodiment of the basic thermodynamic wind turbine configuration without a tower depicted.

[0024] FIG. 1B is a detail view of the flow control ring showing an approximate airfoil cross section of the ring.

[0025] FIG. 2 is a perspective view of one embodiment of a version of the wind turbine mounted downwind of a tower.

[0026] FIG. 3 is a perspective view of one embodiment of a version of the wind turbine mounted upwind of a tower.

[0027] FIG. 4 is a two dimensional drawing of the profile of the stream tube created by a wind turbine showing relative station positions.

[0028] FIG. 5 is a two dimension drawing of the force and velocity triangles relative to a cross section of a blade element.

[0029] FIG. 6 is a front and side view projection of a thermodynamic wind turbine showing the necessary relative dimensions, areas and velocities required to calculate the accelerated airflow used in the detail design of the thermodynamic wind turbine.

DETAILED DESCRIPTION OF THE INVENTION

[0030] The invention embodies higher efficiency versions of a horizontal axis wind turbine and new fundamental methodologies for the design of wind turbines, which extract both kinetic and thermal energy from the wind.

[0031] FIG. 1A shows the basic configuration for the thermodynamic wind turbine without a tower depicted. It is comprised of but not limited to the following components: [0032] a forward central portion consists of a streamlined or parabolic shaped inlet fairing 120; the forward inlet fairing 120 is of a substantial diameter for the purpose of reducing the flow area by an amount sufficient to cause a desired significant acceleration of the airflow velocity and furthermore to redirect the airflow to the more effective outer regions of the turbine radii; the forward inlet fairing 120 is held by an internal means of concentric support and without any external radial supports; [0033] an aft central portion consists of a streamlined fairing or streamlined and truncated fairing 130; the diameter of the aft fairing 130 is approximately equal to the diameter of the forward inlet fairing 120; the aft fairing 130 is held by a separate internal means of concentric support and without any external radial supports, which would disturb rotational motion downstream; the purpose of the aft fairing 130 is to provide for a smooth aerodynamic pressure recovery of the airflow aft of the turbine without any hindrance to rotational flow within the wake; [0034] a plurality of conventional airfoil-shaped rotor blades 110 are attached to and extend out from either the forward or the aft fairing's means of internal support, whichever rotates and drives a conventional power generation unit (not shown); the rotor blades 110 may be either fixed or able to be rotated by some means about their longitudinal axis for airfoil pitch control; and [0035] an outer airfoil-shaped flow control ring 140 is attached to the tips of the rotor blades and rotates with the blades; the attachment of the blades to the ring may be either fixed or able to pivot for blade pitch control.

[0036] FIG. 1B specifically shows an approximate cross section of the airfoil 145 of the flow control ring 140, which is oriented with the positive pressure surface of the airfoil directed towards the center of the turbine; this airfoil orientation is opposite of diffuser designs and operates at a relatively low angle of attack with the purpose of accelerating airflow into the turbine slipstream and inhibiting wake expansion which is opposite of the effect of a diffuser.

[0037] Therefore, the airflow entering the wind turbine interacts with both the forward inlet fairing and the outer flow control ring to be accelerated, resulting in a higher dynamic pressure reacting with the rotor blades, while being constrained to the more effective outer radii of the wind turbine. In addition, the airfoil-shaped flow control ring serves to prevent rotor tip losses, to inhibit wake expansion, and to promote acceleration of rotating airflow into the slipstream. The cumulative effect thereby increases the overall efficiency and power extraction of the wind turbine as shown in further details.

[0038] FIG. 2 depicts an embodiment of the wind turbine operating downwind of a tower 150. The forward inlet fairing 120 houses the power generation unit and is fixed with the turbine's horizontal axis and attached to the tower 150 by a means that allows the forward inlet fairing 120 to rotate freely about the vertical axis of the tower 150 allowing for the alignment of the wind turbine with the wind. The aft fairing 130 with its means of internal support, the plurality of attached rotor blades 110, and the flow control ring 140 are all free to spin about the horizontal axis of the wind turbine thereby driving the power generation unit.

[0039] FIG. 3 is an embodiment of the wind turbine operating upwind of a tower 150. The aft fairing 130 houses the power generation unit and is fixed with the turbine's horizontal axis and attached to the tower 150 by a means that allows the aft fairing 130 to rotate freely about the vertical axis of the tower 150 allowing for alignment of the turbine with the wind. The forward inlet fairing 120 with its means of internal support, the plurality of attached rotor blades 110, and the flow control ring 140 are all free to spin about the horizontal axis of the wind turbine thereby driving the power generation unit.

[0040] Alternately in other embodiments of this wind turbine, the forward or aft fairing may spin; however, it is not required that either spin. Referring back to FIG. 1A, the forward fairing 120 and aft fairing 130 could both be held by a stationary means to a tower such that they are free to rotate together about the vertical axis of the tower. The plurality of rotor blades 110 and flow control ring 140 could rotate between the fairings about a fixed center axis, driving the power generation unit though a means of gearing.

[0041] Alternately, both the upwind or downwind embodiments could have stability and passive alignment into the wind, both enhanced with the attachment of a conventional vertical fin or tail extending from the aft fairing.

[0042] Additionally, a methodology for enhanced extraction of thermal energy, performance optimization, and an algorithm for detail design of the wind turbine are disclosed in the following details. In order to understand the further disclosures, it is necessary to be familiar with the new corrected momentum and energy equations used, which are fundamental to the design methodology. The variable subscripts used within these equations refer to the turbine flow field station positions depicted in FIG. 4. Station 1 represents the initial position of turbine influence and Station 6 the final position of influence. Stations 2 and 3 represent positions just forward and aft of the turbine respectively. Stations 4 and 5 are reserved for other discussions.

[0043] For the design and analysis of any wind turbine, the first important set of parameters that must be known are the velocity of airflow through the wind turbine V.sub.2 and the final velocity of the airflow far downwind V.sub.6. Relative to the free stream velocity, these can be defined as inflow velocity ratio, a.sub.i=V.sub.2V.sub.1 and outflow velocity ratio b.sub.i=V.sub.6/V.sub.1. Conventional theories typically refer to the less convenient axial induction factor a=(1−a.sub.i). The relationship between the velocity ratios and the thrust or normal force created by the wind turbine is normally derived through Froude's solution of the momentum equation which is equivalent to

a.sub.i=½(b.sub.i+1)=½(1+√{square root over (1−C.sub.T)})  (Eq. 1)

where C.sub.T=(F.sub.n/qA), F.sub.r, is the thrust force of the turbine, q is the dynamic pressure equal to ρV.sub.1.sup.2/2, and A equals the area of the turbine. Equation 1 is a flawed relationship in conventional theory that diverges from real airflow. As can be seen for C.sub.T>1, Froude's solution returns unreal values containing the square root of a negative number. Since C.sub.T can vary between 0 and 2, Froude's equations fail to offer valid solutions for the flow variables for the majority of the possible conditions. The methods and embodiments of this invention do not use Froude's solution but instead use the inventor's unique solution called the laminar wake momentum equation, which is equal to:

a.sub.i=b.sub.i.sup.2=(1−0.5C.sub.T).sup.2/3.  (Eq. 2)

The newly presented Eq. 2 is fundamentally different from previous solutions and is in agreement with both natural observation and empirical data. This equation is essential to the evaluation of the energy equations that are used in the design of all wind turbines.

[0044] At this point, conventional theories incorrectly make use of Bernoulli's equation to compare the pressure differential across the wind turbine to the loss in kinetic energy of the final airflow while ignoring rotational parameters. The methodology presented here does not use Bernoulli's equation but instead precedes by defining the rotational terms, which are applied in the total energy equation.

[0045] The equation for the power output of a wind turbine is derived from Euler's turbine equation, which can be written as:

{dot over (W)}={dot over (m)}Ω(r.sub.iV.sub.θi−r.sub.eV.sub.θe)={dot over (m)}ΩrV.sub.θe,  (Eq. 3)

where Ω is the angular velocity of the turbine, V.sub.θ represents the tangential velocity of the airflow, which at the turbine inlet is assumed to be zero, and mass flow through the turbine {dot over (m)}=ρV.sub.2A=ρa.sub.iV.sub.1A. For the design of a wind turbine, the rotor area is discretized into annular elements. For an annular element, V.sub.θe is equal to ωr, the angular velocity of the slipstream times the radius of the element. The area of the annular element is equal to 2πrdr. This yields the elemental power equation of

d{dot over (W)}=2πρa.sub.iV.sub.1ωΩr.sup.3dr.  (Eq. 4)

Alternately, the power can be expressed per unit area as

[00001]$\begin{array}{cc}\frac{d.Math.\stackrel{.}{W}}{\mathrm{dA}}=\frac{2.Math.\mathrm{\pi \rho }.Math..Math.a_i.Math.V_1.Math.\mathrm{\omega \Omega }.Math..Math.r^3.Math.\mathrm{dr}}{2.Math.\pi .Math..Math.\mathrm{rdr}}=\rho .Math..Math.a_i.Math.V_1.Math.\mathrm{\omega \Omega }.Math..Math.r^2& \left(\mathrm{Eq}..Math.5\right)\end{array}$

or per unit mass flow as

[00002]$\begin{array}{cc}\frac{d.Math.\stackrel{.}{W}}{d.Math.\stackrel{.}{m}}=\frac{2.Math.\mathrm{\pi \rho }.Math..Math.a_i.Math.V_1.Math.\mathrm{\omega \Omega }.Math..Math.r^3.Math.\mathrm{dr}}{2.Math.\mathrm{\pi \rho }.Math..Math.a_i.Math.V_1.Math.\mathrm{rdr}}=\mathrm{\omega \Omega }.Math..Math.r^2.& \left(\mathrm{Eq}..Math.6\right)\end{array}$

From Eq. 5, it can be seen that for uniform flow across the turbine area the power extracted increases in equal and direct proportion with inflow velocity ratio, slipstream rotation and turbine rotation. The most prominent variable in Eqs. 5 and 6 is the radius of the annular element r at which the power extraction occurs. The power extraction increases with the square of the radius for the annular element. This is one of the fundamental concepts of the new turbine design and the new methodology. In order to increase the output of the turbine, it is preferred to redirect the air flowing through the turbine from the inner radii to the more effective outer radii.

[0046] When the turbine rotor blades react with the airflow, the torque on the blades impart an equal and opposite torque into slipstream causing rotational kinetic energy. The rotational kinetic energy per unit mass of air contained in an annular element of the slipstream is equal to

ke.sub.θr=½ω.sup.2r.sup.2  (Eq. 7)

[0047] We can use the above results to derive the correct total energy equations for the flow through the turbine. The following equations represent the energy balance per unit mass for annular elements between the significant station positions:

c.sub.pT.sub.1+½V.sub.1.sup.2=c.sub.pT.sub.2+½V.sub.2.sup.2  (Eq. 8)

c.sub.pT.sub.2+½V.sub.2.sup.2=c.sub.pT.sub.3+½V.sub.3.sup.2+ω.sub.3Ωr.sub.3.sup.2+½ω.sub.3.sup.2r.sub.3.sup.2  (Eq. 9)

c.sub.pT.sub.3+½V.sub.3.sup.2+½ω.sub.3.sup.2r.sub.3.sup.2=c.sub.pT.sub.6+½V.sub.6.sup.2+½ω.sub.6.sup.2r.sub.6.sup.2  (Eq. 10)

c.sub.pT.sub.1+½V.sub.1.sup.2=c.sub.pT.sub.6+½V.sub.6.sup.2+½ω.sub.6.sup.2r.sub.6.sup.2+ω.sub.3Ωr.sub.3.sup.2  (Eq. 11)

[0048] The importance of these equations starts with the understanding of Eq. 9, which represents the energy exchange across the turbine rotor. The continuity of mass flow requires that the velocity of the flow entering and exiting the rotor are the same, V.sub.2=V.sub.3. Therefore, from Eq. 9 it can be shown that the energy extracted per unit mass flow is equal to

ω.sub.3Ωr.sub.3.sup.2=c.sub.p(T.sub.2−T.sub.3)−½ω.sub.3.sup.2r.sub.3.sup.2.  (Eq. 12)

Equation 12 clearly shows that the energy extracted from a wind turbine must be a function of the enthalpy term, which is a function of pressure and temperature in the following equation:

[00003]$\begin{array}{cc}{c}_p\left(T_2-T_3\right)=\left(\frac{p_2}{{\rho }_2}-\frac{p_3}{{\rho }_3}\right)+c_v\left(T_2-T_3\right)={\omega }_3.Math.\Omega .Math..Math.r_3^2+\frac{1}{2}.Math.{\Omega }_3^2.Math.R_3^2.& \left(\mathrm{Eq}..Math.13\right)\end{array}$

These equations explain the thermodynamic process at work as the air passes through the turbine. The energy extracted is shown to be a function of both temperature and pressure; therefore, Bernoulli's equation cannot be used to evaluate the relationships downstream of the wind turbine because the assumed temperature is not constant.

[0049] Therefore, the correct energy equation that is valid for the design of the horizontal axis wind turbine, which extracts energy through rotation, can be derived from Eq. 11 yielding

ω.sub.3Ωr.sub.3.sup.2=c.sub.p(T.sub.1−T.sub.6)+½(V.sub.1.sup.2−V.sub.6.sup.2)−½ω.sub.6.sup.2r.sub.6.sup.2.  (Eq. 14)

Multiplying Eq. 14 by mass flow and integrating from r=0 to R yields the total power output of

{dot over (W)}.sub.out=τΩ=½{dot over (m)}ω.sub.3ΩR.sup.2={dot over (m)}[c.sub.p(T.sub.1−T.sub.6)+½(V.sub.2.sup.1−.sub.6.sup.2)−¼ω.sub.6.sup.2R.sub.6.sup.2].  (Eq. 15)

From Eq. 15 the final temperature of the flow stream after dissipation of rotational kinetic energy can be found with

[00004]$\begin{array}{cc}{T}_f=T_1+\frac{\frac{1}{2}.Math.\left(V_1^2-V_6^2\right)-\frac{\mathrm{\tau \Omega }}{\stackrel{.}{m}}}{c_p}.& \left(\mathrm{Eq}..Math.16\right)\end{array}$

[0050] The above discussion has introduced the thermodynamic process at work within the wind turbine and has shown that in order to increase the power output, it is necessary to increase the mass flow, the rotational properties of the flow, and furthermore to concentrate the energy extraction process at the outer radii of the turbine. The next step is to analyze the detailed energy transfer that occurs at the rotor blades in order to develop a new blade-element design method.

[0051] Conventional methods incorrectly relate the thrust force times velocity to the power extracted, which fails to account for the condition of a free spinning rotor or propeller. When a rotor or propeller is free spinning with no torque applied to its connecting shaft, it is still creating a very significant negative thrust but there is no power exchange other than to frictional losses. The new methodology presented here accounts for this condition with an efficiency factor.

[0052] To derive this new efficiency factor requires an understanding of conventional blade element analysis and the force and velocity triangles relative to the airfoil section of a wind turbine blade. The relative forces and velocities acting on the airfoil section of a wind turbine blade in the outer annular elements of the wind turbine are depicted in FIG. 5; this method puts forth a new way of considering the relationships of these variables. The power entering the turbine can be considered the power available that is equal to the normal force times the velocity through the turbine F.sub.nV.sub.2 and the power extracted must be equal to the tangential values, F.sub.tV.sub.t and these terms can be calculated by

[00005]$\begin{array}{cc}{F}_n=\left(L.Math..Math.\cos .Math..Math.\phi +D.Math..Math.\sin .Math..Math.\phi \right)& \left(\mathrm{Eq}..Math.17\right)\\ F_t=\left(L.Math..Math.\sin .Math..Math.\phi -D.Math..Math.\cos .Math..Math.\phi \right)& \left(\mathrm{Eq}..Math.18\right)\\ \sin .Math..Math.\phi =\left(\frac{V_2}{V_{\mathrm{rel}}}\right)& \left(\mathrm{Eq}..Math.19\right)\\ \cos .Math..Math.\phi =\left(\frac{V_t}{V_{\mathrm{rel}}}\right).& \left(\mathrm{Eq}..Math.20\right)\end{array}$

Power available and power extracted can now be defined as

[00006]$\begin{array}{cc}\mathrm{Power}.Math..Math.\mathrm{Available}={\stackrel{.}{W}}_{\mathrm{in}}=F_n.Math.V_2=\frac{V_2.Math.\Omega .Math..Math.\mathrm{rL}+V_2^2.Math.D}{V_{\mathrm{rel}}}& \left(\mathrm{Eq}..Math.21\right)\\ \mathrm{Power}.Math..Math.\mathrm{Extracted}={\stackrel{.}{W}}_{\mathrm{out}}=F_t.Math.V_t=\frac{V_2.Math.\Omega .Math..Math.\mathrm{rL}-{\left(\Omega .Math..Math.r\right)}^2.Math.D}{V_{\mathrm{rel}}}.& \left(\mathrm{Eq}..Math.22\right)\end{array}$

The efficiency factor can now be derived as

[00007]$\begin{array}{cc}{\eta }_T=\frac{{\stackrel{.}{W}}_{\mathrm{out}}}{{\stackrel{.}{W}}_{\mathrm{in}}}=\frac{V_2.Math.\Omega .Math..Math.\mathrm{rL}-{\left(\Omega .Math..Math.r\right)}^2.Math.D}{V_2.Math.\Omega .Math..Math.\mathrm{rL}-V_2^2.Math.D}.& \left(\mathrm{Eq}..Math.23\right)\end{array}$

Dividing all terms through by DV.sub.1.sup.2 yields

[00008]$\begin{array}{cc}{\eta }_T=\frac{a_i.Math.{\lambda }_r\left(\frac{L}{D}\right)-{\lambda }_r^2}{a_i.Math.{\lambda }_r\left(\frac{L}{D}\right)+a_i^2}& \left(\mathrm{Eq}..Math.24\right)\end{array}$

Equation 24 leaves the efficiency in terms of all non-dimensional parameters, where λ.sub.r is the local rotational speed ratio equal to (Ωr)/V.sub.1 and L/D is the lift to drag ratio for the local airfoil element. Equation 24 can be further simplified to

[00009]$\begin{array}{cc}{\eta }_T=\frac{\frac{L}{D}-\frac{{\lambda }_r}{a_i}}{\frac{L}{D}+\frac{a_i}{{\lambda }_r}}.& \left(\mathrm{Eq}..Math.25\right)\end{array}$

Important findings that are not covered with conventional methods can be noted from Eqs. 24 and 25. The efficiency is lower for higher values of λ.sub.r; in fact, the maximum efficiency occurs at (λ.sub.r/a.sub.i)˜1, which implies tan φ˜1 or in other words the optimum relative flow angle to the rotor blades would be approximately 45 degrees. Also if (λ.sub.r/a.sub.t)=(L/D), then the efficiency for extracting power goes to zero, which precisely explains the case for the free-spinning turbine.

[0053] Now that a relationship has been derived for the efficiency of the turbine blade elements, it can carry forward into an equation for power coefficient. By convention, power coefficient C.sub.P is defined as the power extracted by the turbine divided by the theoretical power contained in the kinetic energy of the free stream or

[00010]$\begin{array}{cc}{C}_P=\frac{{\stackrel{.}{W}}_{\mathrm{out}}}{{\mathrm{qV}}_1.Math.A_2}.& \left(\mathrm{Eq}..Math.26\right)\end{array}$

From Eqs. 21 and 23,

[0054]
{dot over (W)}.sub.out=η.sub.T{dot over (W)}.sub.in=η.sub.TF.sub.nV.sub.2,  (Eq. 27)

Eq. 27 into Eq. 26 yields

[00011]$\begin{array}{cc}{C}_P=\frac{{\eta }_T.Math.F_n.Math.V_2}{{\mathrm{qV}}_1.Math.A_2}={\eta }_T.Math.a_i.Math.C_T,& \left(\mathrm{Eq}..Math.28\right)\end{array}$

and from Eq. 2, C.sub.T=2(1−a.sub.i.sup.1.5) can be inserted into Eq. 28 yielding a new relationship for the power coefficient for an open rotor turbine of

C.sub.P=2η.sub.Ta.sub.i(1−a.sub.i.sup.1.5).  (Eq. 29)

Although efficiency is also a function of λ.sub.r/a.sub.i, it is independent of power available and approaches unity for high L/D ratios. Therefore, we can maximize the C.sub.P equation by taking the derivative with respect to a.sub.i and setting equal to zero while holding η.sub.T constant yielding

[00012]$\begin{array}{cc}\frac{{\mathrm{dC}}_P}{{\mathrm{da}}_i}={\eta }_T\left(2-5.Math.a_i^{1.5}\right)=0.& \left(\mathrm{Eq}..Math.30\right)\end{array}$

Solving Eq. 30 yields an optimum of a.sub.i≈0.543 and C.sub.Pmax≈0.651η.sub.T. This result is of major importance; that is, the realization that the optimum inflow-velocity ratio should be a.sub.i≈0.543 and not the value of 0.667 that other conventional design methods call out. The implication of this and the new efficiency factor is to promote higher-loaded, slower-turning multi-bladed designs to the forefront.

[0055] It is known from Eq. 27 that the power out is equal to η.sub.TF.sub.nV.sub.2; therefore, in order to extract more power from the wind one must either increase the efficiency or increase the power available F.sub.nV.sub.2. However, F.sub.n cannot be increased beyond the value of C.sub.T=(F.sub.n/qA), which returns the optimum value of a.sub.i≈0.543 determined from Eq. 2 without adversely affecting mass flow. V.sub.2 can be increased and this leads to the accelerated flow concept. When flow is constrained through a low-drag area-reduction between Station 2 and the wind turbine, the flow is naturally accelerated to a higher velocity. This process, as used by the present invention, is shown in FIG. 6 where r.sub.s is the radius of the inlet fairing and V.sub.2.5 is the accelerated velocity. Continuity of mass flow requires that V.sub.2A.sub.2=V.sub.2.5A.sub.2.5. Solving for areas as a function of radius and manipulating terms yields

[00013]$\begin{array}{cc}{V}_{2.5}=\frac{a_i.Math.V_1}{1-{\left(\frac{r_s}{R}\right)}^2}.& \left(\mathrm{Eq}..Math.31\right)\end{array}$

A new acceleration factor can be defined by

[00014]$\begin{array}{cc}{a}_s=\frac{1}{1-{\left(\frac{r_s}{R}\right)}^2}.& \left(\mathrm{Eq}..Math.32\right)\end{array}$

This then allows defining

V.sub.2.5=a.sub.sV.sub.2=a.sub.sa.sub.iV.sub.1.  (Eq. 33)

This velocity increase simultaneously increases the kinetic energy and power available within the flow field, dropping the temperatures and increasing the efficiency of the system due to improved flow angles at the rotor and higher dynamic pressures reacting with the rotor blades. This new velocity term can be used to derive a new efficiency term for constrained flow with an area reduction equal to

[00015]$\begin{array}{cc}{\eta }_{\mathrm{Ts}}=\frac{\frac{L}{D}-\frac{{\lambda }_r}{a_i.Math.a_s}}{\frac{L}{D}+\frac{a_i.Math.a_s}{{\lambda }_r}}.& \left(\mathrm{Eq}..Math.34\right)\end{array}$

From Eq. 34, it can be shown that efficiency goes up with acceleration factor. Simultaneously, this makes the new power available term equal to F.sub.nV.sub.2.5 and we can conclude a new equation for power coefficient of

[00016]$\begin{array}{cc}{C}_P=\frac{2.Math.{\eta }_{\mathrm{Ts}}.Math.a_i\left(1-a_i^{1.5}\right)}{1-{\left(\frac{r_s}{R}\right)}^2}=2.Math.{\eta }_{\mathrm{Ts}}.Math.a_s.Math.a_i\left(1-a_i^{1.5}\right).& \left(\mathrm{Eq}..Math.35\right)\end{array}$

[0056] The maximum value of C.sub.P for the accelerated flow thermodynamic wind turbine does not yield a readily available solution. As the area reduction approaches unity, the acceleration factor goes to infinity. In reality, as the drag of the forward inlet fairing, the flow control ring and intersection drag increase, they must be accounted for as they will eventually overwhelm the flow through the turbine decreasing mass flow to zero. This can be accounted for by including a function for the coefficient of drag C.sub.D equal to

[00017]$\begin{array}{cc}{C}_D=\frac{\left(D_C+D_O+D_I\right)}{\mathrm{qA}},& \left(\mathrm{Eq}..Math.36\right)\end{array}$

where D.sub.C, D.sub.O, and D.sub.I are the drag forces of the center area reduction fairings, the outer flow control ring, and intersection drag respectfully. This allows for a more accurate laminar wake momentum equation for constrained flow of

a.sub.i=b.sub.i.sup.2=[1−0.5(C.sub.T+C.sub.D)].sup.2/3.  (Eq. 37)

Equation 37 should also be valid in the analysis of diffusers and other augmented flow designs. The important factor to note here is that the efficiency of flow devices to extract power from their normal force is zero. Therefore, this must be deducted from the power coefficient leaving the final new equation for power coefficient of a constrained flow turbine as

C.sub.P=η.sub.Tsa.sub.sa.sub.iC.sub.T=η.sub.Tsa.sub.sa.sub.i[2(1−a.sub.i.sup.1.5)−C.sub.D].  (Eq. 38)

[0057] One of the keys to making this wind turbine effective is the performance analysis and detail design of the rotor blades using a corrected algorithm that accounts for the new solution to the momentum equation and for the additional energy, which can be extracted from the accelerated flow. The basic algorithm steps for analyzing the turbine's performance are as follows: [0058] Step 1.) Choose an applicable wind velocity V.sub.1 and initialize an assumed value for inflow velocity ratio a.sub.i=V.sub.2/V.sub.1. The optimum value for a.sub.i should be approximately 0.54. [0059] Step 2.) Calculate V.sub.2, the minimum velocity that the airflow is slowed to as it approaches the turbine using the formula

V.sub.2=a.sub.iV.sub.1 [0060] and calculate V.sub.2.5, the maximum velocity the airflow reaches as it passes through the turbine based on the size of the forward inlet fairing using the formula

[00018]$V_{2.5}=a_s.Math.a_i.Math.V_1=\frac{a_i.Math.V_1}{1-{\left(\frac{r_s}{R}\right)}^2}.$ [0061] Step 3.) Discretize the flow field into annular elements and layout for each annular element: the number of rotor blades, planform, airfoil chord, pitch angle θ, and rotor velocity Ωr. Compute the relative velocity,

V.sub.rel=√{square root over ((V.sub.2.5.sup.2+Ω.sup.2r.sup.2))}, [0062] the inflow angle

φ=tan.sup.−1(V.sub.2.5/Ωr), [0063] and local angle of attack

α=(φ−θ). [0064] Step 4.) Using calculations from Step 3 in combination with known airfoil data or airfoil design software, determine local lift coefficient C.sub.l and drag coefficient c.sub.d for each blade element. [0065] Step 5.) Using calculations from Step 3 and results from Step 4, calculate the normal forces on each blade element using standard methods of blade element analysis. [0066] Step 6.) Sum all the elemental blade normal forces, computing a total rotor normal force F.sub.n and calculate the thrust coefficient C.sub.T=F.sub.n/(qA). Then estimate D.sub.C, D.sub.O, and D.sub.I and calculate C.sub.D for the non-power extracting flow devices from the formula

[00019]$C_D=\frac{\left(D_C+D_O+D_I\right)}{\mathrm{qA}},$ [0067] and from these results, calculate the corrected value for a.sub.i using

a.sub.i=[1−0.5(C.sub.T+C.sub.D)].sup.2/3. [0068] Step 7.) Compare the result for a.sub.i from Step 6 with initialized value and iterate Steps 1 through 6 until the final value of a.sub.i is in reasonable agreement with the initialized value. The optimum value of the inflow velocity ratio a.sub.i is configured through the aforementioned parameters to be equal to approximately 0.54. [0069] Step 8.) Calculate differential pressure for each annular element by dividing the elemental blade normal forces by the elemental areas and confirm that these values are reasonably close; if not, adjust chord or pitch angle to maintain reasonably uniform pressure differential between the elements which implies a uniform flow field. Iterate Steps 1 through 8 until a.sub.i is in reasonable agreement with the initialized value and the differential pressures between the annular elements are in reasonable agreement representing uniform flow. [0070] Step 9.) Calculate the normal and tangential forces on each blade element using standard methods of blade element analysis. From these results, compute the total loads and moments on all the rotor blades. Finally, calculate the turbine power and thrust using standard methods.

[0071] The thermodynamic wind turbine and the design methodologies described above are innovations to the horizontal axis wind turbine, which yield higher efficiencies than those previously achieved or thought to be practical. These higher efficiencies are shown from the corrected momentum and energy equations to be the result of a naturally occurring extraction of thermal energy as well as kinetic energy from the wind, therefore reducing the temperature of the air flowing through the turbine. The design methodologies can be easily implemented by those in the engineering fields. The various configurations have the structural advantage of smaller shorter span rotor blades supported at both ends and therefore requiring less weight in structural materials. The various turbine components can be fabricated from any suitable material by those skilled in the art. The basic configurations described have the added benefits of reducing the expansion of the wake and minimizing the loss in the air velocity downwind of the turbine therefore reducing the impact on other turbines in a wind turbine farm. The outer flow control ring may have the added benefit of reducing noise from the rotor and reducing the probability of bird strikes.

[0072] As described, the thermodynamic wind turbine is commercially applicable for the conversion of wind energy to electrical energy but is not limited to this. Further embodiments of the invention could also be used for the conversion of wind energy to mechanical energy such as for pumping or compressing operations. The scope of the invention to be protected is as defined in the claims and it is expressly intended that all such variations and changes, which fall within the spirit and scope as defined within the claims, are thereby included.