Hello. The standard Betz limit of 16/27 indicates the maximum power that can be extracted by a turbine from the upstream kinetic power of an open flow. However, the theoretical limit was first devised for wind turbines. But if we consider a trivial or a tidal turbine that stream flow could then be confined in a shallow free layer. We will see that the confinement induced by the free surface may significantly impact the theoretical Betz limit. This image gives an example of how a single turbine may affect the free surface in a small scale about the channel. Conversely, the presence of this free surface will impact the pressure distribution around the turbine. According to the standard Betz calculation, the power extraction will be maximized when the downstream velocity, V3, behind the turbine is one third of extreme velocity, V0. This implies that the cross section of the flow passing through the turbine expand by a factor two in the turbine wake. One of the main assumption of the Betz calculation is that the surrounding flow, which do not exchange energy with a turbine, remains unperturbed. If the downstream area, 2A, becomes larger than the general cross-section, this assumption will hike down. Hence, we introduce the confinement parameter, epsilon, which is equal to the ratio of the turbine swept area over the whole cross-channel section. For the Betz calculation, the area of the flow surrounding the turbine is infinitely large. Therefore, the Betz low will be recovered when epsilon tends to zero. As for the Betz calculation, we consider a control surface which bound the volume of the fluid passing through the turbine. Inside this control surface, the three parcels interact with the turbine and transfer both, momentum and kinetic energy. Due to the expansion of this control surface, the unperturbed fluid will be deflected and constrained to pass through a smaller cross-section area behind the turbine. Hence, due to the mass conservation, the velocity around the turbine will be accelerated. If we assume for each layer, a uniform velocity and pressure in the transverse section, A3 and A4, the velocity shear will then appear in the turbine wake. V3, the velocity inside the surface control will be smaller than the upstream velocity, V0, and the velocity, V4, outside the surface control will be higher than V0. The extreme flux is separated in two, the flow, A3V3, will interact with the turbine while the outside flux, A4V4, will keep its energy and momentum. If now we consider the stream line of a free parcel outside the surface control. The difference between the upstream pressure, P0, and the downstream pressure, P4, will be fixed by the binary equation. We can consider that on average for the whole cross-section. The height variation is zero and therefore, due to the velocity acceleration, the downstream pressure, P4, will be smaller than the upstream pressure, P0. This phenomena is similar to a venturi tube where a constricted section of a pipe leads to an increase of velocity and the pressure drop. Then, if we take into account the pressure continuity on the surface control, the inside pressure, P3, should be equal to the outside pressure, P4. Hence, and lags the stunned out Betz calculation, the downstream pressure, P3, is now smaller than the upstream pressure, P0. To estimate this impact on the thrust exerted by the flow on the turbine, we consider a streamline in the downstream wake. That rely the pressure peak to just behind the turbine to the downstream pressure, P3. According to the boundary relation, this pressure difference depends only on the velocity difference inside the surface control. If we assume that the surface control is weakly impacted by the confinement, this pressure difference, P2 minus P3, will not be affected by the confinement and the pressure, P2, behind the turbine will then follow the same evolution as the downstream pressure, P3, and will be reduced by the confinement. Therefore, the thrust exerted by the flow on the turbine will increase with the confinement parameter, epsilon, and be larger than the one obtain in the stand down Betz calculation. However, the full calculation of the power coefficient for confined turbine was done by Garrett and Cummins in 2007. And they get a complex relation that depends on the free velocities, V0, V3, and V4. We can then introduce two dimensions variables, A and B, which are the ratio of the two downstream velocities over the upstream velocity, V0. Then, we can rewrite the power coefficient Cp with these two parameters only. However, the confinement parameter does not appear in this complex formula. We then need to use the mass conservation and the momentum balance to obtain a single relation that rely the parameter, B to A, in epsilon. If now, we replace B in the first calculation of the power coefficient, we see that Cp depends only on the confinement parameter epsilon and the velocity ratio, A. This complex formula leads to simple graphs. And we plot here, in solid lines, the case without confinement, why are plotted in dash and dotted lines. The two curves corresponding respectively to a confinement parameter epsilons of 10 and 20 percent. If we focus on the maximal value of the power coefficient, we notice that the optimal efficiency of the turbine is always obtained when the ratio of the downstream velocity over the upstream velocity is equal to one third. And surprisingly, this optimal ratio do not depend on the confinement parameter. Besides, this maximal Cp value increases when epsilon increases. Hence, the confinement will increase the optimal efficiency of the turbine. A typical example where the confinement should be taken into account is the laboratory test of turbine efficiency. The finite size of a wind flume or an ionic channel may impact the results in comparison with a larger open flow condition. Another case where the flow confinement should be taken into account is when a larger A of turbine is displayed in a river or a tidal channel. For this following example using long roll off helical turbines, the confinement parameter is directly proportional to the ratio of the turbine diameter over the water height. And typical values up to 20 or 30 percent of confinement could be reached. If we plot the power curves for such values, we get the maximal power coefficient is significantly larger than the standard Betz limits. And for a confinement of 30 percent, we get a value above unity which means that the turbine have more than 100 percent efficiency. Is it possible? Is it a violation of the energy budget? In fact, when the power coefficient is equal to unity, it means that the power extracted from the flow is exactly equal to the amount of the upstream kinetic power. But, for a river or a tidal flow, the upstream flow also contains potential energy, and the available power is kinetic and potential. Hence, for a free surface flow, the power extracted by a turbine could exceed extreme kinetic power. And the available power is then given by the hydraulic power formula where the extracted power is equal to the density row times the gravity, times the flow rate Q, times the hydraulic head variation. In other words, the available power depends on the variation of the kinetic power and the potential energy flux. But when the flow number is low, which is always a case for large rivers or tidal flows, the hydraulic head variation is almost equal to the free surface variation. Hence, the energy extraction will have an impact on the free surface. On this image, we can see a small drop in the water height just after the turbine. The free surface contains a large amount of available potential energy, and the drop of the water height isn't proportional to the potential energy loss. To sum up, the height of the atmospheric layer is so large that we can neglect confinement effect for wind turbines. However, if we consider river or marine flows, we should introduce a confinement parameter and check his value. When the confinement parameter is larger than 10 percent, we should take into account its impact on the power coefficient. Confined turbines could have a power coefficient that exceed the standard Betz limit. Hence, if the 16/27 low is a theoretical limit for wind turbine efficiency, this is not the case for river or marine turbines. Indeed, for my layer the free surface is the source of potential energy, and the turbine could extract both kinetic and also, potential energy from the flow. Thank you.
