**2. One-dimensional fluid dynamics models**

In this section, one-dimensional fluid dynamics models are analyzed and formulated based on the extended Bernoulli equation [37], accounting for losses that are assumed to be pro‐ portional to the square of the velocity of the air crossing the rotor blades of the wind turbine. The performance characteristics of the wind turbine are given by the power and thrust coef‐ ficients. The efficiency of the wind turbine is addressed and its relation to the power coeffi‐ cient is discussed.

While developing a model to describe the performance of a wind turbine, common assump‐ tions regarding the fluid flow are as follows [18]:


Other assumptions are given for the specific models.

#### **2.1. Bare wind turbine**

Consider a wind turbine that intercepts the flow of air moving with velocity *V0*. The differ‐ ent quantities involved in the physics of the moving air are the pressure, *p*, the velocity, *V*, the cross section area, *A* (at different locations along the stream lines), and the thrust on the blades of wind turbine, *T*. The upstream condition is identified with a subscript of zero, the downstream condition is identified with a subscript of 3, and the turbine locations are iden‐ tified by subscripts of 1, 2, and t (see Figure 1).

**Figure 1.** Schematics of the bare wind turbine. In the upper part, airstream lines are shown crossing the turbine's ro‐ tor. The velocity of the air at the rotor is the same based on the mass flow rate: *V1*=*V2*=*Vt.*The pressure drop is deter‐ mined by ∆p=p*1*-p*2*.

The steady state mass flow rate is given by:

$$
\dot{m} = \rho V\\\dot{A} = const.\tag{1}
$$

The modified Bernoulli equation (with reference to the turbine head), which describes the energy balance through the wind turbine, is written between the entrance and exit sections, and is given by:

$$\frac{p\_0}{\gamma} + \frac{1}{2} \frac{V\_0^2}{g} = \frac{p\_0}{\gamma} + \frac{1}{2} \frac{V\_3^2}{g} + h\_t + h\_{\text{loss}} \tag{2}$$

In this equation *ht* is the head of the turbine (related to the amount of power extraction), *hloss* represents the head losses, and *γ* is the specific weight. The loss term in the energy equation accounts for friction (mechanical and fluid) and is assumed proportional to the kinetic ener‐ gy of the rotor blades, expressed as follows:

$$h\_{\rm loss} = C\_{\rm loss} \frac{V\_t^2}{2g} \tag{3}$$

Equations (2) and (3) are rearranged and the head of the turbine is given by:

$$h\_t = \frac{1}{2} \frac{V\_0^2}{g} - \frac{1}{2} \frac{V\_3^2}{g} - \mathcal{C}\_{\text{loss}} \frac{V\_t^2}{2g} \tag{4}$$

The power output from the turbine, *P*, is given by:

**t**

**Figure 1.** Schematics of the bare wind turbine. In the upper part, airstream lines are shown crossing the turbine's ro‐ tor. The velocity of the air at the rotor is the same based on the mass flow rate: *V1*=*V2*=*Vt.*The pressure drop is deter‐

The modified Bernoulli equation (with reference to the turbine head), which describes the energy balance through the wind turbine, is written between the entrance and exit sections,

. (1)

+ = + ++ (2)

*m VA const* & = = r

2 2 0 00 3 1 1

> g*g g*

2 2 *t loss p Vp V h h*

In this equation *ht* is the head of the turbine (related to the amount of power extraction), *hloss* represents the head losses, and *γ* is the specific weight. The loss term in the energy equation

mined by ∆p=p*1*-p*2*.

10 Advances in Wind Power

and is given by:

The steady state mass flow rate is given by:

g

$$P = \gamma V\_t A\_t h\_t = \frac{1}{2} \rho V\_0^3 A\_t \frac{V\_t}{V\_0} \left( 1 - \left(\frac{V\_3}{V\_0}\right)^2 - C\_{\text{loss}} \left(\frac{V\_t}{V\_0}\right)^2 \right) \tag{5}$$

From equation (5) we can determine the power coefficient, CP, by:

$$C\_P = \frac{P}{\frac{1}{2}\rho V\_0^3 A\_t} = \frac{V\_t}{V\_0} \left(1 - \left(\frac{V\_3}{V\_0}\right)^2 - C\_{loss} \left(\frac{V\_t}{V\_0}\right)^2\right) \tag{6}$$

The developed thrust, *T*, on the turbine blades is given by the linear moment equation (see [15] for more details), as follows:

$$T = \rho V\_t A\_t (V\_0 - V\_3) \tag{7}$$

The thrust coefficient, *CT,* based on equation (7), is given by:

$$C\_T = \frac{T}{\frac{1}{2}\rho V\_0^2 A\_t} = 2\frac{V\_t}{V\_0} \left(1 - \frac{V\_3}{V\_0}\right) \tag{8}$$

Equations (5) and (7) are governed by the following relationship:

$$P = TV\_t \tag{9}$$

By observation, the velocity of air decreases toward the downstream. Therefore, we can sim‐ plify calculations by introducing the parameter, *a*, to express the velocity at the cross section of the turbine, and the upstream velocity can be determined by:

$$V\_t = (1 - a)V\_0 \qquad \qquad 0 < a < 1\tag{10}$$

Equating equations (5) and (9) gives the velocity of the air, *V3*, at the downstream, and after some algebraic manipulation, it could be shown to be given by:

$$V\_3 = \left(1 - a - \sqrt{\left(a^2 - \mathbb{C}\_{\text{loss}}^\* \left(1 - a\right)^2\right)}\right) V\_0 \tag{11}$$

The normalized loss coefficient is defined by:

$$\mathbf{C}\_{\text{loss}}^{\*} = \mathbf{C}\_{\text{loss}} V\_0^2 \tag{12}$$

Equations (11) and (12) are useful to calculate the power coefficient (equation (6)) and the thrust coefficient (equation (8)). For the case where the losses are negligible, the known re‐ sults in the literature are reproduced and given by the following equations:

The downstream velocity is given by:

$$V\_3 = 1 - 2a \tag{13}$$

The power coefficient is given by:

$$C\_P = 4a \left(1 - a\right)^2\tag{14}$$

The thrust coefficient is given by:

$$C\_T = 4a(1 - a) \tag{15}$$

Inversing the relation given in equation (15), the parameter, *a,* is given as a function of *CT* by:

$$a = \frac{1 - \sqrt{1 - C\_T}}{2} \tag{16}$$

Finally, the power coefficient as a function of the thrust coefficient is given by:

$$\mathbf{C}\_{P} = \frac{\mathbf{C}\_{T} + \mathbf{C}\_{T}\sqrt{1 - \mathbf{C}\_{t}}}{2} \tag{17}$$

#### **2.2. Augmented wind turbine**

By observation, the velocity of air decreases toward the downstream. Therefore, we can sim‐ plify calculations by introducing the parameter, *a*, to express the velocity at the cross section

Equating equations (5) and (9) gives the velocity of the air, *V3*, at the downstream, and after

( ( ) ) <sup>2</sup> 2 \*

Equations (11) and (12) are useful to calculate the power coefficient (equation (6)) and the thrust coefficient (equation (8)). For the case where the losses are negligible, the known re‐

Inversing the relation given in equation (15), the parameter, *a,* is given as a function of *CT* by:

1 1 2

sults in the literature are reproduced and given by the following equations:

3 0 1 1 *V a aC a V loss* æ ö = -- - - ç ÷

<sup>0</sup> (1 ) 0 1 *V aV a <sup>t</sup>* = - << (10)

è ø (11)

\* 2 *C CV loss loss* <sup>0</sup> <sup>=</sup> (12)

<sup>3</sup> *V a* = -1 2 (13)

( )<sup>2</sup> 4 1 *C aa <sup>P</sup>* = - (14)

4 1( ) *C aa <sup>T</sup>* = - (15)

*CT <sup>a</sup>* - - <sup>=</sup> (16)

of the turbine, and the upstream velocity can be determined by:

some algebraic manipulation, it could be shown to be given by:

The normalized loss coefficient is defined by:

12 Advances in Wind Power

The downstream velocity is given by:

The power coefficient is given by:

The thrust coefficient is given by:

In order to exploit wind power as economically as possible, it was suggested that the wind turbine should be enclosed inside a specifically designed shroud [38, 39]. Several models were reported in the literature to analyze wind turbine rotors surrounded by a device (shroud), which was usually a diffuser [18, 25, and 26]. Others suggested different ap‐ proaches [28].

In this section, the extended Bernoulli equation and mass and momentum balance equations are used to analyze the augmented wind turbine. The power coefficient and the thrust coef‐ ficients are derived, accounting for losses in the same manner as was done for the bare tur‐ bine case. The efficiency of the wind turbine could be defined as the ratio of the net power output to the energy input to the system. The efficiency based on this definition agrees with the Betz limit.

The schematics of the shrouded wind turbine are shown in Figure 2.

**Figure 2.** Schematics of the shrouded wind turbine. There is a vertical element at the exit of the wind turbine. This element contributes to reducing the power at the downstream side of the turbine, an effect that extracts more air through the wind turbine. (Idea reproduced similar to the description given by Ohya [40].)

This type of design has been recently reported [40], and it was shown that the power coeffi‐ cient is about 2-5 times greater when compared to the performance of the bare wind turbine. The vertical part at the exit of the shroud reduces the pressure and therefore, the wind tur‐ bine draws more mass.

The balance equations are followed in the same manner as for the bare wind turbine. The modified Bernoulli equation differs by the pressure at the exit and is given by:

$$\frac{p\_0}{\gamma} + \frac{1}{2} \frac{V\_0^2}{g} = \frac{p\_3}{\gamma} + \frac{1}{2} \frac{V\_3^2}{g} + h\_t + h\_{\text{loss}} \tag{18}$$

The pressure drop between inlet and outlet (*p0-p3*) is rewritten as proportional (with *CF* pro‐ portionality coefficient) to the difference in kinetic energies and it is given by:

$$
\Delta p = p\_0 - p\_3 = \frac{1}{2}\rho \left(V\_0^2 - V\_3^2\right) \mathcal{C}\_F \tag{19}
$$

The power coefficient for the shrouded wind turbine is given by:

$$\mathbf{C}\_{P} = \frac{P}{\frac{1}{2}\rho V\_{0}^{3}A\_{t}} = \frac{V\_{t}}{V\_{0}}\left(\mathbf{C}\_{F} + \mathbf{1}\right)\left(\mathbf{1} - \left(\frac{V\_{3}}{V\_{0}}\right)^{2}\right) - \mathbf{C}\_{\text{loss}}\left(\frac{V\_{t}}{V\_{0}}\right)^{2}\tag{20}$$

The thrust coefficient is given by:

$$C\_T = \frac{T}{\frac{1}{2}\rho V\_0^2 A\_t} = 2\left(C\_F + 1\right)\frac{V\_t}{V\_0}\left(1 - \frac{V\_3}{V\_0}\right) \tag{21}$$

Manipulating equations (9)-(12) makes it possible to produce sample plots to consider in the next section.

#### **2.3. Wind turbine efficiency**

Usually, efficiency is defined as the ratio between two terms: the amount of net work, *w,* to the input, *qin,* energy to the device. Efficiency can be alternatively defined as the ratio be‐ tween the derived power, *Pout,* and the rate of energy flowing to the system, *Pin*. Based on these definitions, the efficiency of power generating machine is given by:

$$\eta = \frac{w\_{net}}{q\_{in}} = \frac{P\_{out}}{P\_{in}} \tag{22}$$

As was observed by Betz, the maximal achievable efficiency of the bare wind turbine is giv‐ en by the Betz number *B* = 16/27. In section 2.1, the power coefficient of the bare turbine was considered under the assumption of frictional losses. In this case, the power coefficient can also be identified as the efficiency of the wind turbine in this case. However, the power coef‐ ficient for the shrouded wind turbine as considered in section 2.2 is not efficiency. Based on the definition of efficiency, one could observe that if we divide the power coefficient by the factor (CF + 1) (taking into account the increased mass flow to the system due to pressure drop), a similar expression of the shrouded wind turbine could be given by equation (6). Thus, according to the modeling assumptions and with special care in treatment of the loss coefficient, one could conclude that the efficiency of the wind turbine could not exceed the Betz limit, although the power coefficient in general could exceed the Betz limit, as was ob‐ served previously by others.

#### **2.4. Maximum windmill efficiency in finite time**

2 2 0 03 3 1 1

> g*g g*

portionality coefficient) to the difference in kinetic energies and it is given by:

( )

*P F loss*

*C CC*

0

*C C*

*T F t*

these definitions, the efficiency of power generating machine is given by:

h

1 2

r*V A*

The power coefficient for the shrouded wind turbine is given by:

0

*t*

1 2

The thrust coefficient is given by:

**2.3. Wind turbine efficiency**

next section.

14 Advances in Wind Power

r*V A*

g

2 2 *t loss p Vp V h h*

The pressure drop between inlet and outlet (*p0-p3*) is rewritten as proportional (with *CF* pro‐

03 0 3 1 <sup>2</sup> *<sup>F</sup>* D= - = - *pp p V VC* r

( ) 2 2

3

( ) <sup>3</sup> 2 0 0

*t*

*V V*

æ ö

= = (22)

2 11

*T V V*

Manipulating equations (9)-(12) makes it possible to produce sample plots to consider in the

Usually, efficiency is defined as the ratio between two terms: the amount of net work, *w,* to the input, *qin,* energy to the device. Efficiency can be alternatively defined as the ratio be‐ tween the derived power, *Pout,* and the rate of energy flowing to the system, *Pin*. Based on

> *net out in in w P q P*

As was observed by Betz, the maximal achievable efficiency of the bare wind turbine is giv‐ en by the Betz number *B* = 16/27. In section 2.1, the power coefficient of the bare turbine was considered under the assumption of frictional losses. In this case, the power coefficient can also be identified as the efficiency of the wind turbine in this case. However, the power coef‐ ficient for the shrouded wind turbine as considered in section 2.2 is not efficiency. Based on

= =+ - ç ÷

æ ö æ ö æö æö

*t t*

*V VV*

3 0 00

1 1

= = +- - ç ÷ ç ÷ ç÷ ç÷ ç ÷ èø èø è ø è ø

*P V VV*

+ = + ++ (18)

2 2

(19)

è ø (21)

(20)

In a different approach, a model to estimate the efficiency of a wind turbine was introduced [41] and the efficiency at maximum power output *ηmp*was derived. Although the power de‐ veloped in a wind turbine derives from kinetic energy rather than from heat, it was possible to view the basic model of the wind turbine in a schematic way, which is similar to the heat engine picture. After the wind turbine accepts energy input in its upstream side, it extracts power at the turbine blades and ejects energy at the downstream. Details of this approach are given elsewhere [41].

The derived value for the efficiency at maximum power operation was shown to be a func‐ tion of the Betz number, *B,* and is given by the following formula:

$$
\eta\_{mp} = 1 - \sqrt{1 - B} \tag{23}
$$

This value is 36.2%, which agrees well with those for actually operating wind turbines.

With an algebraic manipulation this expression could be approximated by:

$$
\eta\_{mp} = \frac{B}{2 - \frac{B}{2}} \tag{24}
$$

The expression given in equation (24) differs by only a small percentage when compared to equation (23). With the aid of equation (24), one could estimate the efficiency as 8/23.

As compared to the efficiency of heat engines, the system efficiency could be defined as the mean value between the maximum efficiency (Carnot efficiency) and the efficiency at maxi‐ mum power point (Curzon-Ahlborn efficiency). Thus, the efficiency of the wind turbine sys‐ tem *ηts*could be given by:

$$
\eta\_{\rm ts} = \frac{\left(B + 1 - \sqrt{1 - B}\right)}{2} \tag{25}
$$

If equation (23) is used while neglecting the contribution of the Number *B*/2 compared to 2, the turbine efficiency given by equation (24) would be approximated by:

$$
\eta\_{ts} = \frac{3}{4}B \tag{26}
$$

#### **2.5. Wind turbine efficiency and the golden section**

The golden section has been considered in different disciplines as a measure of beauty [42-49]. The schematics of the golden section are given in Figure 3.

**Figure 3.** Schematics of the golden ratio. In the upper part to the left, the oval shape is divided into two parts, *x* and 1 *x*. in the upper part to the right, the same oval is divided into two parts with the following engine parameters: *W* represents the net work output; *Eout* represents the energy left the machine. *Ein* represents the energy that was put into the machine. In the lower part, a construction of the golden section is depicted by the isosceles triangle with a base angle of 72°. Comparing the different parts, *x* is defined as *W/Ein* or *Eout/Ein*, depending upon parts that have been produced and rejected.

In the upper part of the figure, the oval shape is divided into two parts, x and 1-x (as can be seen in the left side of the figure). In the right side, the same oval shape depicts the relation to quantities considered in engine machines. In the lower part of Figure 3, a golden section construction is depicted using the isosceles triangle with sides of unity and base triangle of 72°. If we apply the result of the golden section ratio (the golden section ratio is related to

the sine of the angle of 18º, thus 2sin(18)= 5−1 <sup>2</sup> ) to wind power efficiency, which is usually defined as the work gained divided by the energy input to the system, we can observe that

the Betz limit agrees very well to the golden section (0.593 compared to 0.618). We can also

see that the practical efficiencies of wind turbines in the proximity of 40% are comparable with the smaller part of the golden section 38.2%.

The golden section beauty can be related to wind turbines if we recall that the kinetic energy of the wind usually splits into two parts: useful and rejected. According to the Betz limit, about 60% of the energy is used to produce electric power and the rest is rejected. On the other hand, the real wind turbines extract about 40% and the rest is wasted. These findings are in match with the golden section division (61.8% and 38.2%). One could conclude by asking: Is this just a fortuitous result or is there something more deep and inherent in the beauty of nature?

#### **2.6. Factors that affect the efficiency of the wind turbine**

If equation (23) is used while neglecting the contribution of the Number *B*/2 compared to 2,

3 4 *ts* h

The golden section has been considered in different disciplines as a measure of beauty

**Figure 3.** Schematics of the golden ratio. In the upper part to the left, the oval shape is divided into two parts, *x* and 1 *x*. in the upper part to the right, the same oval is divided into two parts with the following engine parameters: *W* represents the net work output; *Eout* represents the energy left the machine. *Ein* represents the energy that was put into the machine. In the lower part, a construction of the golden section is depicted by the isosceles triangle with a base angle of 72°. Comparing the different parts, *x* is defined as *W/Ein* or *Eout/Ein*, depending upon parts that have been

In the upper part of the figure, the oval shape is divided into two parts, x and 1-x (as can be seen in the left side of the figure). In the right side, the same oval shape depicts the relation to quantities considered in engine machines. In the lower part of Figure 3, a golden section construction is depicted using the isosceles triangle with sides of unity and base triangle of 72°. If we apply the result of the golden section ratio (the golden section ratio is related to

5−1

defined as the work gained divided by the energy input to the system, we can observe that the Betz limit agrees very well to the golden section (0.593 compared to 0.618). We can also

<sup>2</sup> ) to wind power efficiency, which is usually

= *B* (26)

the turbine efficiency given by equation (24) would be approximated by:

[42-49]. The schematics of the golden section are given in Figure 3.

**2.5. Wind turbine efficiency and the golden section**

produced and rejected.

16 Advances in Wind Power

the sine of the angle of 18º, thus 2sin(18)=

In terms of wind turbine efficiency, it is possible to highlight different parts of the turbine when estimating its value. Most of the studies discussed above considered extracting power from the kinetic energy of the wind. This could be defined as kinetic energy efficiency*ηKE*. Other considerations should be accounted for, such as the mechanical efficiency, *ηme*(when power is decreased due to mechanical friction), the conversion to electricity efficiency,*ηcon*, and the blockage efficiency, *ηbl* (which is defined as the amount of air blocked by the turbine blades as depicted by Figure 4). The overall turbine efficiency is given by:

$$
\eta\_{net} = \eta\_{bl}\eta\_{cm}\eta\_{me}\eta\_{KE} \tag{27}
$$

*cross*

*A A*

Figure 4. Schematic of the cross section of the rotor blades. The cross section illustrates the fact that the physical body of the rotor blades blocks some of the air particles, reducing potential power production from the air. The blocking efficiency can be defined as 1 *rotor bl* . In this **Figure 4.** Schematic of the cross section of the rotor blades. The cross section illustrates the fact that the physical body of the rotor blades blocks some of the air particles, reducing potential power production from the air. The blocking efficiency can be defined asη*bl* =1<sup>−</sup> *Arotor Across* . In this equation, *Arotor* is the projected cross section of the rotor blades and *Across* is the cross section without the rotor blades.

The one-dimensional bare wind turbine model without losses has been treated extensively and is well documented in textbooks [15]. The velocity of the air crossing the wind turbine velocity is assumed to be a fraction of the upstream air velocity. This fraction is introduced as a parameter, *a*, which expresses the ratio between the latter and the former. Parameter *a* is assigned a value in the range of numbers between zero and one. It is important to note that the physical range of this parameter is limited, for example, 0<a<0.5, otherwise the velocity at the downstream becomes negative. To help clarify this point, calculations were performed covering the full range of parameter *a*. Equations (6) and (8) give the power coefficient and the thrust coefficient, respectively. Figure 5 shows these coefficients as functions of parameter *a* (a very well-known result in the professional literature in the field).

> **Power and thrust coefficients for the bare wind turbine with zero loss coefficeint vs. the parameter a**

Figure 5. Power and thrust coefficients for the ideal bare wind turbine as a function of the parameter a (the ratio between the air velocity crossing the turbine blades and the upstream velocity of the air). The plot is reproduced similar to what is known in the literature, but highlighting the

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a**

**0<a <0.5 0.5<a<1**

For explicit presentation, the physical range of parameter *a* (up to the value of 1/2) was drawn in a thick black color, while the rest of the plot was prepared using a thinner red color. It is clear from the figure that the coefficients vanish at the zero and one values of the parameter. In between, the maximum thrust coefficient (with a value of unity) occurs at the value of *a*=1/2. On the other

physical region (0<*a*<0.5) with thicker black color and the non-physical region (0.5<a<1) with thinner red color.

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1**

**Power and thrust coefficients.**

equation, *Arotor* is the projected cross section of the rotor blades and *Across* is the cross section without the rotor blades.

**3. Numerical considerations** 

In this section sample, plots of the results are considered.

**3.1. The ideal bare wind turbine model** 
