The paper introduces a new approach being referred as Direct Multiplication Fractional Changes Method (DMFCM) in the derivation of power coefficient. Wind energy extraction process involves conversion of the kinetic energy of the wind into mechanical energy of the rotor. An actuator disc model is commonly used in the derivation of power coefficient and it is a representation of an ideal horizontal-axis wind turbine. Power from the wind is a product of force and velocity. Force from wind is the change in kinetic pressure multiplied by the effective cross-section area. Therefore, the three variables used in calculation of wind power are given by Equation (2.1).

$P=pAv$ (2.1)

where; P is the power, p is the kinetic pressure and v is the wind velocity.

The new approach considers fractional change of each variable that constitute wind power. These individual fractional changes are directly multiplied to obtain the overall fractional change in power. An assumption in this approach is that a flow sensor attached to the turbine blades can accurately measure instantaneous velocity.

A fractional change in power can, therefore, be expressed as direct multiplication of individual fractional changes (DMFCM) for the variables as shown in Equation (2.2).

$\frac{\Delta P}{P}=\frac{\Delta p}{p}\cdot \frac{\Delta A}{A}\cdot \frac{\Delta v}{v}$ (2.2)

where: ∆P is change in power, ∆p is change in pressure, ∆A is the change in cross-sectional area, and ∆v is the change in velocity.

Kinetic pressure, p at a given velocity is given by Equation (2.3).

$p=\frac{1}{2}\rho v^2$ (2.3)

where; $\rho$ is the density of air and v is the instantaneous velocity.

Kinetic pressure decreases downstream and it is converted into static pressure. Therefore, kinetic pressure at the rotor disc, $p_2$ is less than upstream kinetic pressure $p_1$ as shown in Figure 3 . The fractional change in kinetic pressure at the turbine is given by Equation (2.4).

$\frac{\Delta p}{p}=\frac{v_2^2-v_1^2}{v_1^2}$ (2.4)

where: $v_2$ is the velocity at the turbine and $v_1$ is the velocity upstream.

Slowing down wind causes the flow stream to expand downstream because of conversion of kinetic pressure into static pressure. The affected cross-section area $A_1$ upstream is two thirds of the rotor disc $A_2$ [8] . The fractional change in cross-sectional area is given as Equation (2.5).

$\frac{\Delta A}{A}=\frac{A_2-A_1}{A_1}=\frac{1}{2}$ (2.5)

The fractional change in velocity is given as Equation (2.6).

$\frac{\Delta v}{v}=\frac{v_2-v_1}{v_1}$ (2.6)

Substituting Equations (2.4) - (2.6) into Equation (2.2) gives Equation (2.7).

$\frac{\Delta P}{P}=\frac{1}{2}\cdot \frac{v_2^2-v_1^2}{v_1^2}\cdot \frac{v_2-v_1}{v_1}=\frac{1}{2}\left(b^2-1\right)\left(b-1\right)$ (2.7)

where: b is a ratio of velocity at the disc, $v_2$ divided by undisturbed velocity, $v_1$.

Figure 4 shows power coefficient curve from Equation (2.7). It is a new curve illustrating how power coefficient changes from undisturbed wind upstream, +b up to a fully recovered wind downstream, −b behind the disc.

The optimum power coefficient in Figure 4 is in agreement with the hypothetical velocity curve where the minimum velocity occurs downstream and is one third of the undisturbed wind as shown in Figure 3(b) . Considering the distance to be positive upstream and negative downstream, the Betz’s limit can practically occur behind the disc at an instant velocity of negative a third of the undisturbed wind. It is impossible to extract power behind the disc and therefore the Betz’s limit is an ideal theoretical value. If the hypothetical velocity curve is correctly formulated, it will be possible to predict the distance behind the disc where slowing wind will have its minimum velocity.

Velocity is a vector quantity with both directions and magnitude. Wind velocity reduces downstream changing both in magnitude and direction. Assuming the flow is streamlined and there is no rotating wake, wind velocity is modelled as a vector quantity that is undergoing geometrical transformation at different positions in the direction of flow downstream as shown in Figure 5 .

Some of undisturbed wind $v_1$ temporarily stops to form an air wedge when it is blocked by the rotor disc. The wind direction will be diverted through an angle $\theta$ and the magnitude reduced to $v_2=v_1\cos \theta$. Past the edges of the rotor disc, the wind vector returns to its original direction (horizontal) further reducing the magnitude to $v_3=v_1{\cos }^2\theta$. There is a low-pressure region behind the disc and some wind particles in this region are not in motion. The wind velocity vector past the edges will be diverted again to vector $v_4=v_1{\cos }^3\theta$ to occupy the low-pressure region. The stationary wind particles in the low-pressure region will acquire some kinetic energy after combining with the moving wind particles. Consequently, the velocity is reduced further to $v_5=v_1{\cos }^4\theta$. Velocity vector, $v_5$ will recovering downstream to attain the original magnitude and direction.

Hypothetical velocity curve is mathematically formulated by combination of the vectors $v_1$, $v_2$, $v_3$, and $v_5$ depending on the position of flow downstream. From Figure 5 , $\cos \theta$ can be expressed in terms of distance, x and radius, r of the rotor disc by Equation (2.8).

$\cos \theta =\frac{x}{\sqrt{x^2+r^2}}$ (2.8)

Wind velocity remain undisturbed when radius $r=0$ or the disc is removed and $\cos \theta =1$. An increase in the size of the radius will correspond with an increase in distance x which is the altitude of the air wedge. The maximum size of angle $\theta =45˚$ and therefore $v_2=0.7v_1\approx 0.6\stackrel{\dot{}}{6}{v}_1$. This value is in agreement with the two thirds of the undisturbed wind when passing at the rotor disc.

A number of simulations were done. Equations (2.9) and (2.10) were used to simulate $v_3\left(x\right)$ and $v_5\left(x\right)$ respectively. The two vectors represent velocities that are horizontal along the direction of flow.

$v_3\left(x\right)=v_1{\left(\frac{x+1.225r}{\sqrt{{\left(x+1.225r\right)}^2+r^2}}\right)}^2;-\infty <x<+\infty$ (2.9)

$v_5\left(x\right)=v_1{\left(\frac{x+1.225r}{\sqrt{{\left(x+1.225r\right)}^2+r^2}}\right)}^4;-\infty <x\le 0$ (2.10)

where: 1.225r is a constant distance used to shift the cosine curve to the right to have velocity ratio as $0.6\stackrel{\dot{}}{6}{v}_1$ at the rotor disc position.

Taking the radius $r=2\text{m}$, Figure 6 shows a simulation illustrating upstream velocity $v_3\left(x\right)$ with the velocity at the rotor disc as $0.6\stackrel{\dot{}}{6}{v}_1$ and downstream velocity $v_5\left(x\right)$ starting from zero behind the rotor disc.

Figure 6 shows a special case when the two velocities do not combine. Velocity $v_3\left(x\right)=0$ (minimum) at a distance a distance, $x=-1.225r=-2.45\text{m}$ behind the rotor disc. This is the position where wind is expected to lose all its energy before recovering. The two velocities will equalize around eight meters downstream from the disc after which they continue to recover together. The velocity-distance curve predicts that the two vectors will fully recover at a distance 286 m downstream from the rotor disc at a velocity fraction of 1.0000 (4 decimal places).

In a case where the two velocities are combined behind the rotor disc, Equations (2.9) and (2.10) are added to form Equation (2.11).

${v^{\prime }}_4\left(x\right)=v_3\left(x\right)+v_5\left(x\right);-1.225r\le x\le 0$ (2.11)

where: ${v^{\prime }}_4\left(x\right)$ is resultant velocity behind the rotor disc.

The resultant continuous velocity-distance curve is show in Figure 7 for a distance range from the rotor disc ( $x=0$) to $x=-1.225r$.

The minimum velocity of ${v^{\prime }}_4\left(x\right)=0.3\stackrel{\dot{}}{3}{v}_1$ at a distance 1.87 m behind the rotor disk. This result is in agreement with the prediction in the DMFC derivation of Betz’s upper limit and a confirmation that it occurs at a position behind the rotor disk.

Figure 8 shows a comparison of velocity-distance curves for three different radii.

The three curves had their velocity equal to $0.6\stackrel{\dot{}}{6}{v}_1$ at the rotor disc, but they differed at the position where the minimum velocity occurred. The distance for the minimum velocity is related to the radius by $x_{\min }=0.93r$. The main factor that affects the distance for the minimum velocity occurring behind is the radius of the rotor disc.

An individual wind particle goes past the disc position at different angle of attack. From the flow simulation used to derive velocity-distance curve, the angle of attack for particles within the air wedge, ranges ${45}^{\circ }<\theta \le {90}^{\circ }$. However, the bulk of particles pass outside the air wedge have their angle of attack $\theta \cong {45}^{\circ }$ $\Rightarrow$ $\cos {45}^{\circ }=0.7\approx 2/3$. The velocity fraction b in Equation (1.2) can be substituted with the cosine of the angle of attack, as demonstrated in Equation (2.12).

$b=\frac{v}{v_1}=\cos \theta$ (2.12)

where; $v_1$ is the undisturbed wind velocity upstream, v is the instantaneous velocity after disturbance and $\theta$ is the angle of attack.

Therefore, it is possible to relate the power coefficient curve given in Figure 4 with the angle of attack as shown in Figure 9 .

The angles shown in the model exclusively represent the velocity fractions of wind particles located on the upstream side of the disc. An angle of zero signifies that the wind velocity remains undisturbed upstream, resulting in a power coefficient of zero. In contrast, wind particles that are entirely obstructed within the air wedge of the model exhibit an angle of attack of 90˚. These particles must effectively traverse through a right angle to bypass the disc. Interestingly, it is these fully deflected particles that contribute to the highest power coefficient, achieving a value of 0.5.

Results from DMFCM show that power coefficient varies as wind slows down during energy extraction and also as it regains its energy some distance downstream. From the findings, the range for affected velocity is given as; $-v\le v\le +v$ where +v is the undisturbed velocity upstream and −v is the fully recovered velocity downstream. Figure 4 illustrates the whole spectrum of power coefficient within the entire range of the affected velocities. On the positive side (front side of the rotor disc), the maximum coefficient is 0.5 and can be obtained from those wind particles that are stopped temporally within the air wedge, as shown in Figure 5 . Wind particles flowing past the edges into the rear side of the rotor disk have a maximum coefficient of 0.59 at a distance from disc where instantaneous wind velocity reduces to its minimum value of a third of the undisturbed velocity. This is in agreement with Mahmoud et al., (2023) that the highest actual power coefficient ranges between 0.45 and 0.50.

The power coefficient curve shown in Figure 4 from DMFCM is different from the existing curves that are already published using EVAM in Figure 1(b) [8] - [10] [13] [14] and those using LVM in Figure 2 [11] [12] . The other two existing curves caters for power coefficient in the velocity range $0\le v\le \left|v\right|$ which is difficult to tell whether wind is slowing down or recovering after an interaction with the rotor disc. In comparison with the new curve, the curve derived using LVM in Figure 2 provides a power coefficient on the rear side of the disc and ignores what is happening in the front side. Practically, the maximum power coefficient is 0.5 at the rotor disc. The implication of this finding is Betz’s limit of 0.59 remains a theoretical value because it can only be realized a distance behind the disc.

There are many different approaches used in the analysis of Betz’s model and the derivation of power coefficient factor. From the reviewed literature, there are two common expressions for power coefficient predicting the same optimum value of 59% but generating different graphs for power-coefficient against velocity. The expression for power coefficient used to generate curves in the EVAM method is expressed as a product of two factors by Equation (3.1).

$C_P=4a{\left(1-a\right)}^2=4\left(1-\frac{v}{v_1}\right){\left\{1-\left(1-\frac{v}{v_1}\right)\right\}}^2=\left(1-\frac{v}{v_1}\right){\left(2\frac{v}{v_1}\right)}^2$ (3.1)

where; a is the axial induction factor, v is instantaneous velocity, and $v_1$ is the undisturbed velocity.

To compare $C_P$ in EVAM method with the new approach of fractional change in total power, individual fractional changes in pressure, area, and velocity are given in Equation (3.2).

$\frac{\Delta p}{p}={\left(2\frac{v_4}{v_1}\right)}^2,\frac{\Delta A}{A_1}=1\text{and}\frac{\Delta v}{v_1}=\frac{v_2-v_1}{v_1}=-a$ (3.2)

According to EVAM, cross-sectional area has a changing factor of one, hence causing no effect on power coefficient. The first correction required in EVAM approach is to change the positive axial induction factor, +a into a negative because velocity downstream is smaller than the velocity upstream. Therefore, the optimum $C_P$ will occur at $a=-0.3$ behind the rotor disc and not at $a=\left|0.3\right|$ ahead of the rotor disc. The second correction required is on calculation of the change in static pressure. Pressure $P_4$ at velocity $v_4$ as shown in Figure 3 is at a distance far behind the small effective area $A_1$. Figure 10 shows variation of power coefficient against axial induction factor after the correction in the EVAM approach.

The curve in Figure 10 predicts a negative power coefficient in the front side of the disc which is scientifically not correct. This is because, the cross-sectional area $A_1$ used in the derivation is upstream and the static pressure considered is downstream (on the negative). The graph shows a positive power generation starts behind the disc downstream which is impractical. Therefore, EVAM approach is not appropriate for obtaining a practical power coefficient.

This finding is collaborated by a suggestion that some solution to the momentum equation in actuator disc theory is incorrect giving results that diverge from realistic airflow [15] . Burton et al. (2011) recommended empirical adjustment for values of $a>0.5$ because the wake velocity approaches zero or becomes negative making momentum theory not applicable. A comparable observation was reported by Manwell et al. (2010), indicating that when $a=0.5$, the wind velocity behind the rotor decreases to zero, rendering the basic theory inapplicable.

For the purpose of comparison with the new method, the expression for power coefficient in the LVM is given as product of factor by Equation (3.3) [11] .

$C_p=\left(\frac{1}{2}\right)\left(1+\frac{v}{v_1}\right)\left(1-{\left(\frac{v}{v_1}\right)}^2\right)=\frac{1}{2}\left(1+\left(\frac{v}{v_1}\right)-{\left(\frac{v}{v_1}\right)}^2-{\left(\frac{v}{v_1}\right)}^3\right)$ (3.3)

where, v is the instantaneous velocity and $v_1$ is the undisturbed wind velocity.

Power coefficient using LVM method has three changing variables when compared with the new approach of fractional change in total power. The equation of extracted power in LVM approach is given by Equation (3.4) where the cross-section area, static pressure and instant velocity are all taken at the location of the disc.

$\Delta P=\frac{1}{2}\rho A_2\left(v_1^2-v_2^2\right)\left(v_1-v_2\right)$ (3.4)

where; ∆P is the change in power, $\rho$ density of air, $A_2$ is the cross-section area of the disc, $v_1$ is the undisturbed wind upstream, and $v_2$ is the instantaneous wind at the disc.

The changes in static pressure and the instantaneous velocity are both calculated at the disc. The correction required in this approach is to change $\left(v_1-v_2\right)$ into $\left(v_2-v_1\right)$ because $v_2$ is the affected velocity downstream. This will slightly modify the $C_P$ given in Equation (3.3) into Equation (3.5).

$C_p=\frac{1}{2}\left(\frac{v}{v_0}-1\right)\left(1-{\left(\frac{v}{v_0}\right)}^2\right)$ (3.5)

The power coefficient curve generated by the modified Equation (3.5) is similar to the one shown in Figure 4 for new DMFC approach.

Three different approaches were used in the numerical derivation of C_p in this paper. For the purpose of identification, they were named EVAM, LVM and DMFCM. Table 1 shows a comparison of the evaluated methods.

Continued

The formulated velocity-distance function and the simulations results shown in Figure 8 indicates the position of Betz’s limit is downstream for three different radii. This confirms hypothetical velocity-distance curve given in Figure 3(b) . This formulated velocity-distance curve is not a single continuous function. It consists of three step functions. The first step function $v_3\left(x\right)$ is given in Equation (2.9) for $0\le x\le +\infty$ and it has no turning point. The second step function ${v^{\prime }}_4\left(x\right)=v_3\left(x\right)+v_5\left(x\right)$ is given in Equation (2.11) for $-122r\le x\le 0$ and has a turning point at $x=-0.33$. The third step function $v_5\left(x\right)$ is given in Equation (2.10) for $-\infty \le x\le -122r$ and it has no turning point.

From the results, the rotor disc affects the flow both upstream and downstream. The distance affected can be predicted using this formulated velocity-distance curve. When a rotor radius $r=1\text{m}$ was used, the distance affected upstream (in front) and downstream was approximately 140 m and 143 m, respectively. When a rotor radius $r=2\text{m}$ was used, the distance affected upstream (in front) and downstream was approximately 281 m and 286 m, respectively. For every one meter of the radius, a distance of approximately 142 m is affected both downstream and upstream.