Talk:Betz's law

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Request for clarity

It is not clear exactly where the 16/27 comes from. I think an explanatory step is needed. —Preceding unsigned comment added by 24.7.178.142 (talk) 02:28, 25 June 2008 (UTC)

The Lanchester-Betz Limit

This article should be renamed. —Preceding unsigned comment added by 128.40.54.199 (talk) 15:12, 22 February 2008 (UTC)

Betz was an eminent German engineer, Professer Emeritus at the University of Gottingen, and principal assistant to Ludwig Prandtl before succeeding him. Betz published his limit in 1920 however the limit was originally derived by Frederick Lanchester and published in 1915. Interestingly, Lanchester, who was an eminent English mathematician and engineer, developed the idea of circulation around an aerofoil as the mechanism of lift, which was in fact validated mathematically by Prandtl.

Because Betz's name is so commonly associated with the limit, 'Lanchester-Betz Limit' seems appropriate.

Lanchester's limit is published in the following article: "A Contribution to the Theory of Propulsion and the Screw Propeller" Proceedings from the 56th session of the Institution of Naval Architects (now the Royal Institution of Naval Architects), March 25th 1915.

Further discussion on this topic, including noting an independent derivation of the limit by Russian aerodynamicist Joukowski (Zhukovski), in 1920, can be found in the following papers:

"The Lanchester-Betz limit (energy conversion efficiency factor for windmills)" BERGEY, K.H. (Oklahoma, Univ., Norman), Journal of Energy 1979, 0146-0412 vol.3 no.6 (382-384)

"The Lanchester-Betz-Joukowsky limit" Gijs A.M. van Kuik, Delft University Wind Energy Research Institute, Wind Energ. 2007; 10:289–291

--- Peter Johnson

I think Betz Law is not appropriate, there are examples of turbines (with diffusers) that break this law!! I would leave make the title Betz Limit since that is the common name. As far as changing it formally to "The Lanchester-Betz-Joukowsky limit", I think that is arbitrary and pretentious. There are so many things in engineering concepts that are named after people, then different writers will use different combinations of names to refer to the same thing. From the point of view of communication it is stupid. At the end of the day it is all Jargon meant for people to sound smart. —Preceding unsigned comment added by 24.69.137.123 (talk) 08:24, 16 February 2011 (UTC)

Questionable Title

Shouldn't the article's title be "Betz's Law", with an apostrophe and s after his name? I'm reluctant to edit this without discussion, however.

Steve P

This is an interesting and probably a questionable issue.

The rules for possessive forms have been simplified by many grammar references over the years. Past rules contained exceptions for the "'s" added to both singular common and singular proper nouns that ended in s, z, and x. Not all references adopted changes in the same order, so there were many different versions of these rules.

Some of these past exceptions included adding only the apostrophy when the noun or proper noun ended in s, z, or x like in "Chris' exam" or when the the noun or proper noun was followed by a word beginning with an s like in "the boss' sister." Today most references have recognized that these exceptions were unnessary, simply led to confusion, and so changed the rules.

Regarding "Betz' Law," although this form follows a rule of the past, its usage is no longer simply a possessive form of two arbitrary words, but is now the title or proper name of a law of physics. Accordingly, changing it to be currently grammatically correct would also change the spelling of the name of the law. Accordingly, I suspect few reference book editors are willing to make that change. However, if you are Albert Betz's grandson, Ralph, and establish a new law, it would probably be spelled "Betz's Second Law" or "Ralph Betz's Law."

BillinSanDiego (talk) 11:10, 22 February 2008 (UTC)

This Article Is Not a Complete Discussion or Proof

The Wikipedia article contains only the more commonly presented mathematical proof of Betz’ Law. However, it is not a complete proof of Betz’ Law because it does not prove the assumption used in the mathematical proof that the velocity through the turbine (referred to as “a rotor” in the text of the Wikipedia article and as “a disk-shaped actuator” in the accompanying diagram) is equal to the average of the incoming and outgoing flow velocities. Below are my related comments presented in a step by step fashion.

1. No proof of Betz’ Law that I have been able to find including the proof in the Wikipedia article contains a discussion or establishes a rationale for using the average velocity of V1 and V2 as the velocity representing the velocity through the turbine (again, referred to as “a rotor” in the text of the Wikipedia article and as “a disk-shaped actuator” in the accompanying diagram).

2. Although the schematic flow diagram in the Wikipedia article is depicted to be a bottle-shaped tube, flow machines may or may not be shrouded and/or contained within a tube that directs flow. Accordingly, there is no mention of whether Betz’ Law is applicable to specific configurations and whether applicable configurations include both shrouded and open (un-shrouded) types. It is also widely known that fluid couplings can have conversion ratios of 98 percent or better. And although the design of fluid couplings typically incorporates rotational flows using a rotational or radial pump and turbine, there is no discussion clarifying what type or types of flow machines or turbines are limited by Betz’ Law.

3. Regarding the question of shrouded versus open turbines, this distinction raises at least two questions. One question is whether the flow through an open turbine (not surrounded by a tube) actually takes the shape depicted in the diagram (and to what extent). The second question relates to a turbine that is inserted into such a shroud or tube. In this case it should be noted that the incoming velocity, annotated as V1 in the diagram, may or may not remain equal to the approaching flow velocity. Accordingly, if V1 is representative of the approaching velocity, then the shape of shroud must conform to that which does not alter the incoming flow velocity at the entrance to the shroud. According, this can be true for only one set of flow conditions. Additional discussion clarifying this appears necessary.

4. Basic flow and energy equations appear to indicate that the bottle shape affects the efficiency of the turbine. For instance, a constant diameter tube appears to have a maximum conversion efficiency of 0.3849 at a velocity ratio of 0.5774. This compares to the Betz’ Law maximum conversion efficiency of 0.5926 at a velocity ratio of one third (0.3333). In the case of the cylinder, the inlet flow velocity at the entrance to the cylinder would be less than the approach velocity, and the inlet and outlet flow velocities would be the same. Accordingly, the constant diameter of the cylinder prevents flow velocity changes within the cylinder, but not upstream of the entrance.

5. The concept or specific definition of energy conversion efficiency is also somewhat ambiguous. This is because not all of the approaching flow may pass through the turbine and therefore there is one conversion efficiency based on the approaching flow and another based on flow passing through the turbine. For example, if the turbine is contained in a cylinder and if the flow velocity through the cylinder is any velocity less than the approaching velocity, then some of the approaching flow will be diverted around the inlet to the cylinder. Accordingly, the flow energy of the column of flow entering the cylinder is less than the flow volume of the same-diameter column of flow approaching the inlet to the cylinder. Therefore, the ratio of the energy extracted from the incoming energy by the turbine will be different depending on whether that ratio is based on the energy of the flow column approaching the entrance to the cylinder or is based on the energy of the flow column actually entering the entrance of the cylinder.

6. The upstream-downstream placement of a turbine along the centerline within a cylinder-shaped or bottle-shaped tubular flow pattern does not affect the theoretical or ideal conversion efficiency when the reference for this efficiency conversion is the flow column passing through the turbine (as opposed to the same-diameter flow column approaching the turbine). And, although the upstream-downstream location of the turbine in a bottle-shape flow pattern may alter (a) the turbine diameter, (b) the flow velocity through the turbine, and (c) the fluid pressure upstream and downstream of the turbine, the conversion will calculate to the same value. However, if the reference for the efficiency conversion is the energy of the approaching flow column where the diameter of that column is matched to the diameter of the turbine, then the calculation of the energy conversion efficiency will decrease for turbine placements requiring larger diameter turbines even though the extracted power is the same.

7. Related to which column of incoming flow is used as a basis for the theoretical conversion efficiency of the extracted energy, these calculations can have the appearance of violating Betz’ Law. If in the example depicted in the Wikipedia article, we simply calculate the incoming and outgoing energy (actually power), we should expect that the difference should be the extracted energy. This is true because there are no loss terms in the equations used to prove Betz’ Law. Accordingly, if we use V2 to be one third of V1, then we should calculate a theoretical conversion efficiency of 0.5926. To perform the calculations we need variables to represent the cross-sectional areas where the velocities V1 and V2 are measured. We will simply call these areas A1 and A2. Accordingly, the energy rate (power) into the bottle-shaped flow pattern is ½ rho times A1 times V1 cubed (½ p A1 V1^3). Likewise, the energy exiting the bottle-shaped flow pattern is ½ rho times A2 times V2 cubed (½ p A2 V2^3). Therefore the conversion efficiency for this flow condition is A1 times V1 cubed minus A2 times V2 cubed all over A1 times V1 cubed ((A1 V1^3 – A2 V2^3) / A1 V1^3). This easily reduces to 1 - A2/A1 (V2/V1)^3. Substituting A2/A1 equal to 3 and V2/V1 equal 1/3, the efficiency calculates to 1- 1/9 or 0.8889. This exceeds the limit of 0.5926 established by Betz’ Law. Accordingly, although it may appear that these calculations violate Betz’ Law, they only differ from Betz’ Law because they are based on the incoming energy that passes through the turbine rather than on the incoming energy approaching the turbine. In fact, if the calculated efficiency of 0.8889 is multiplied by the ratio of area A1 to area S (used in the diagram), then the efficiency agrees with that calculated by Betz’ Law. Accordingly, because A1 V1 equals A2 V2 equals S Vavg equals S (V1 + V2)/2, then S equals 2 A1 V1 /(V1 + V2). Therefore, in our example, where V1/V2 equals 3, S equals 3/2 A1 and 1/(3/2) times 0.8889 equals 0.5926. And, by the way, note that although Vavg equals (V1 + V2)/2, S does not equal (A1 + A2)/2.

8. To understand turbine flow it may also be helpful to analyze the counterpart to energy extraction, that is, the flow patterns and conditions created by a fan instead of a turbine. Accordingly, although the suction side of a fan may develop a flow pattern that simulates an increasing diameter tube (a portion of the bottle-shaped flow pattern), the exhaust side does not appear to simulate flow in an increasing diameter tube. This approach to analyzing the flow patterns also raises the same questions discussed above regarding the rationale and justification for establishing and/or assuming that it is appropriate to use the average velocity in the proof of Betz’ Law.

9. It appears that Betz’ Law applies to the maximum theoretical efficiency for a turbine if and only if the turbine has a flow velocity equal to the average of the upstream and downstream velocities. Accordingly, if the turbine has a flow velocity equal to the average of the upstream and downstream velocities, then it has a maximum efficiency of 0.5926 at an upstream to downstream flow velocity ratio equal to one third (0.3333). However, if the flow velocity through the turbine is not the average of the upstream and downstream flow velocities, then 0.5926 is not the maximum theoretical efficiency for that turbine and the maximum will not necessarily be at an upstream to downstream flow velocity ratio equal to one third.

10. Because the proof of Betz’ Law in the Wikipedia article does not contain a proof that the flow velocity through the turbine is equal to the average of the upstream and downstream velocities, then the presentation is not actually a proof of Betz’ Law, but rather a proof based on that assumption.

BillinSanDiego (talk) 10:27, 1 February 2008 (UTC)

I don't have answers to all your questions, but I believe I can adress at least some of them. Firstly the proof that the average wind speed inside the turbine equals the average of the speeds at its respective ends can be found using calculus and the continuity equation. Simply use the continuity equation to write the velocity through a thin slice as a function of the cross section, and then do an integral. As for why some turbines (i.e hydroelectric plants and brayton cycles ) can get higher efficiencies, these are fundamentally different situations. While a wind turbine gains its energy by extracting kinetic energy from already moving wind, a hydroelectric plant allows almost stationary water to fall up to hundreds of meters through the earth's gravitational field, extracting the potential energy that is released in the process. In contrast the air that flows through a wind turbine only changes height by a small amount, and since it also has a low density, not much potential energy is released. When it comes to turbine based heat engines , such as the rankine or brayton cycle, then the answer is that the fluid in those turbines is at substantially different temperatures before and after the turbine. The energy extracted by a steam turbine comes from the heat stored in the hot water as it enters, not the velocity by which it travels. Effectively what happens is that the steam continiously expands throughout the turbine, causing it to accelerate and cool down. In a wind turbine the air maintains pretty much the same temperature as it passes through the turbine, and the energy is taken completely from the kinetic energy of the flow, without cooling the fluid. I guess the best way I can explain it is that in a wind turbine the energy extracted comes from the fluids change in velocity, while in a hydroelectric plant it is determined by change in gravitational potential energy, and in the steam turbine it comes from change in temperature. Indeed, in most hydroelectric and steam turbines the velocity of the fluid is almost zero as it enters the turbine, it is then accelerated by the pressure gradient, and finally deccelerated back down to the initial velocity. In contrast the air in a wind turbien has its highest speed as it enters the turbine, and then loses speed as the turbine extracts the energy. 193.216.221.180 (talk) 13:54, 27 July 2008 (UTC)

The same thing struck me when I read this proof: the assumption that the volume of air passing through should be based on the "average" velocity times the swept area seems wrong. This "average" velocity seems meaningless. The volume in must be the same as volume out, which is obviously v1 times the swept area. The lower velocity on the lee side must result in higher pressure and vertical / horizontal deflection. This means you cannot approach 100% efficiency, but it's a more complex model than this "average velocity" idea to determine the volume.

Regarding hydro plants, it's not the release of gravitational p.e. after it's gone through the turbines that produces energy. This energy accelerates the water as it falls and turns into wasted heat and sound as it crashes into the river below. It's the gravitational p.e. of the water above the that creates pressure on the water at the bottom of the dam, and this force is what turns the turbines. So the maximum energy there will be proportional to the height of the dam, not how far the water falls as it comes out.

The above comment was anonymous.

The turbine knows nothing about the gravitational potential energy: it is aware of the kinetic energy at that point (where the blade meets water). Most pneumetic tools work with a compressor and an air turbine. This is also the same principle as most air compressors (e.g., vacuum cleaners). Again, the water leaving the turbine carries away some of the kinetic energy.

Steam turbines are different because they are mostly working as a heat engine and are not isothermal devices like air or water turbines. As the kinetic energy is extracted by the moving blade, the temperature drops and the pressure too. Therefore there is a difference of pressure on the two sides of the blade, unlike in a wind or water turbine.

Constant diameter flow tube with a leakless turbine at the middle is more like a heat engine and will not work unless you include viscosity.

Your comment about using the average velocity is correct. What is important is the velocity at that point. If you extract kinetic energy from air, the temperature must drop. The proof is valid only for very low velocities. chami 14:34, 20 January 2013 (UTC) — Preceding unsigned comment added by Ck.mitra (talk • contribs)

Some Clarifications

1. I have the proof that you may use the average velocity as the velocity at the rotor, it is correct, and I will edit the main page to reflect that.

2. The schematic you choose is irrelevant provided you choose the correct control volume. If your fluid power device has a nozzle for an inlet, your control volume must contain the nozzle, i.e. you must draw your control volume such that all of the fluid affected by your device is included, otherwise you are violating the original assumptions. Rotational flows are of course included, Betz' law is derived by using conservation of mass and energy, and if you induce vorticity into the flow, you still must obey those laws.

3. Same argument as above, you must include the fluid shroud, so your control volume (and corresponding area) must use the bigger opening on the nozzle. Those who have used this equation on devices while neglecting a nozzle often say that they have found a device which exceeds Betz' limit, that is incorrect, they simply violated the assumptions, and if they recalculated using the proper analysis, they would find that they do not exceed the limit.

4. It's not about the geometry you choose, if you choose to restrict the flow (say in a cylinder), you will indeed restrict the performance of the device, but you must always choose a volume for your analysis such that the inlet velocity is constant across your cross section, the same with the outlet. This analysis neglects friction completely, this is known as an 'ideal' fluid power machine, and it theoretically (and correctly) the maximum amount of power available for a given density, cross section, and flow speed.

5. You must choose this carefully as stated above, if you choose your control volume such that some mass is coming in the entrance, but not making it out the exit (say by leaking out the sides), you are violating the initial assumptions, and your number will be wrong. In that case you would be violating conservation of mass, momentum, and energy. You must account for all leaks and potential energy losses if you want to get this right.

8. This will be clarified as well, when you see the resulting analysis with the complete proof, try and work it out for the fan. Examine the assumptions and the power equations and it will make sense then.

9. This is a fact that stems from the power equations, it's not an appearance or an assumption. When the main page is updated, it will be clear.

DrAero (talk) 13:37, 23 September 2008 (UTC)

If it be considered that the distance made good by an air particle before hitting the turbine blades is equal to the distance made good after hitting the blades, it would be necessary to calculate the average before/after speed by the harmonic average [since the distances are same]. The arithmetical mean of 3 and 1 is 2 , but the harmonic is 2:[1/3 h/mi + 1/1 h/mi] = 1.5 [e.g. in mph]. With some overlap it is to be considered that the mean speed operates in the vertical field of the propeller. Not an easy opponent this Betz theory. 84.80.66.78 (talk) 18:34, 13 October 2008 (UTC)Desertfax (talk) 18:36, 13 October 2008 (UTC)desertfaxDesertfax (talk) 07:39, 14 October 2008 (UTC)

--203.91.193.5 (talk) 08:01, 12 July 2011 (UTC) What if I put two rotors, one behind the other? Shouldnt I get 59.3% plus the efficency of the 2nd one (lower than 59.3%)?

Your point #4: Consider a cylinder of infinte length. Fluid is flowing at a constant velocity. This is same as constant flow. You can now put any number of turbines in the tube that will run? What happens to the energy?

The assumption of constant pressure and temperature cannot be satisfied chami 14:56, 20 January 2013 (UTC)

Experimental demonstration?

Without having a copy of Introduction to the Theory of Flow Machines available to me, is there an experimental demonstration of Betz's law that could be cited here? Navuoy (talk) 00:15, 6 January 2009 (UTC)

The Economic Relevance

How do losses in transmission and energy storage affect the output of a turbine? These are losses downstream of generation. —Preceding unsigned comment added by 86.211.246.143 (talk) 07:41, 11 February 2009 (UTC)

Notes on Revised Article

The following discussion is related to the version of the article dated December 10th, 2008. Although the article was improved in the last revision, I still view the article as being deficient in several respects. These specifically include (1) still not providing an actual proof for using the subject average velocity through the turbine and (2) not discussing the applications and limitations of the law. Basically, the ten points that I stated on February 1st, 2008 in the above discussion (“This Article is Not a Complete Discussion or Proof”) still remain true despite the latest revision to the article.

First of all, the reason that I view the complete proof and a discussion of the applications and limitations as being important to the content of article is because without it, it would appear that the reader will be prone to confusion and misunderstanding, as originally happened to me. In my search for clarification I have read numerous comments in discussions where I found Betz’s Law had been applied to all sorts of turbine machines having different shapes and shrouds, where I not only couldn’t see a basis or correlation to Betz’s Law, but where I found actual conflict and contradiction. Accordingly, the complete proof and a discussion of the applications and limitations of Betz’s Law not only appear well within the scope of an encyclopedia article, but appear to be a necessity for providing an accurate understanding, which should be the objective of the article.

Regarding the newly added mathematical proof, this proof is strictly a manipulation of mathematical terms and does not correlate to the theoretical model that is depicted in the diagram. Although this proof uses a form of the momentum equation, the momentum equation as presented does not represent this theoretical model. And, although the equations are dimensionally correct, simply maintaining correct dimensional units does not constitute a proof for the validity of the relationship that delta V equals (v1 - v2), which is the basis for later proving that v equals (v1 + v2)/2.

As presented, the first line of the proof begins with the equation F equals m times a. The second line states that F equals m times dv/dt. The third line states F equals m dot times delta V (where “m dot” is the time-rate change in m). And finally, the fourth line states that F equals rho times S times v times (v1 - v2). Accordingly, although each step is logical in dimensional units, there is no proof that delta V equals (v1 - v2) as implied between the third and fourth lines.

Basically, the first three lines represent the development of an equation for the force created by the momentum of a jet of air flow striking a moving plane that is perpendicular to the direction of flow, similar to that of a jet of water from a nozzle against a moving flat plate. This condition would be true in the turbine model if the incoming air flow were moving at velocity v1 were to directly strike the blades of the turbine (like a jet of water) with the blades having an effective velocity (due to pitch and rotation) equal to velocity v2. However, that is not the case. In the model in the diagram, the incoming air velocity is gradually slowed thereby increasing the pressure on the inlet side of the turbine blades. Basically, in the model there is no delta V at the turbine, but rather a change in pressure (a delta P). According, although the subject equations may represent a possible flow condition, they do not represent the flow conditions of the model in the diagram.

Accordingly, if one were to write equations based on the theoretical model, they would be as follows, where P1 and P2 are pressures corresponding to v1 and v2 respectively, v is velocity through the turbine, and P3 and P4 are the respective pressures on the inlet and outlet sides if the turbine blade:

F = S ( P3 – P4 )

F = ½ rho S ( v1^2 – v4^2 ) = ½ m dot 1/v ( v1^2 –v2^2)

delta V = F / m dot = ( v1^2 – v2^2 ) / ( 2 v )

Accordingly, in this model there is no physical representation for the term delta V, and therefore no rationale for equating delta V to (v1 - v2). Also note that although the above equations do not prove that delta V equals (v1 - v2) and consequently that v equals (v1 + v2)/2, it does not exclude that this condition could be true for other reasons not presented. Obviously, given a range of shroud shapes there is bound to be a shape that allows v to be equal to (v1 + v2)/2. Presumably, Betz discovered and proved that the shape required to achieve this was the same shape as the flow profile when no shroud was used at all. Presumably, a three-dimensional flow analysis should confirm this. And, it is also possible that the analysis is so complex that no one wishes to present it.

This could well be similar to the case of gravitational attraction where for spheres the point of attraction is typically and without question simply assumed or understood to be the center of the sphere. However, closer examination reveals that the problem is not as simple as one’s first observation, and only after performing the double integration to all points within the sphere does one understand that this integration proves, that what was easily assumed incorrectly to be obvious, is actually true.

Also, when I searched the Internet for more detailed proofs, I found no proofs addressing the origin of using the average. In fact, the web site of a Danish windmill company at www.windpower.org called using the average a “reasonable assumption.” The web site also stated that Betz offered a proof of this, but, again, I couldn’t find it. Regarding tracking down Betz’s original proof, when I “googled” “‘Betz’s Law’ +‘original proof’” and numerous other combinations, I received zero responses. The original proof was apparently in German. Perhaps no one has translated it into English.

I also found the discussion in the section headed “Points of Interest” confusing on several points. Regarding the statement that “the preceding analysis has no dependence on the geometry,” the statement is not true in cases where there is a shroud that deviates from a shape that forces the velocity through the turbine to be other than the average of the entrance and exit velocities. Regarding the discussion about “claims of exceeding the Betz' limit,” this needs more clarification. For example, it is not overly obvious that one of the reasons the Betz limit is 59.26 percent is because, at that operational point, one-third of the inlet flow is diverted around the turbine. Accordingly, 88.89 percent of energy is removed from the two-thirds of the airflow that passes through the turbine, while zero energy is removed from the remaining one-third of the airflow that passes around the turbine. This is significant to the meaning and understanding of Betz’s Law and should not be muddled or omitted. Accordingly, two points should be perfectly clear. The first is that Betz’s Law is applicable to transverse-flow turbines without shrouds. And second, that the efficiency of 50.26 percent is the result of removing 88.89 percent of the energy from the flow that passes through the turbine and that one-third of the flow passes around the turbine.

Bill Wolf (talk) 19:22, 16 February 2009 (UTC)

Suggested Addition to Article

The following is suggested text for an additional section to the Wikipedia article. I admit that the readability could be improved if the equations that are presently stated in sentence form were replaced by actual equations. I have simply not expended the time required to do that.

Betz’s Law is applicable to non-shrouded, transverse-flow turbines having a flow velocity through the turbine equal to the average of the incoming and exiting flow velocities. The assumption regarding the average flow velocities is part of the proof, so it is incorrect to apply Betz’s Law to applications where this initial assumption is not the case. Also, the theoretical efficiency of 50.26 percent is the result of theoretically removing 88.89 percent of the energy from the two-thirds of the incoming flow that passes through the turbine and removing zero energy from the remaining one-third of the incoming flow that passes around the turbine. According, the theoretical efficiency of the turbine is dependent on what is defined to be the power of the incoming flow. If the power of the incoming flow is defined in terms of the cross-sectional area of the flow that eventually passes through the turbine, then the maximum theoretical efficiency is 88.89 percent. However, if incoming power is defined in terms of a cross-sectional area equal to the cross-sectional area of the turbine, as in the case of Betz’s Law, then the maximum theoretical efficiency is 50.26 percent.

The maximum theoretical efficiency for turbines that do not correlate to either of the above cases must accordingly be calculated for the specific design. For example, the inclusion of shrouding around the turbine can alter the velocity through the turbine to be other than the average of the incoming and exiting flow velocities (the main assumption used in Betz’s Law) and thereby changes the maximum theoretical efficiency. Accordingly, modifying a design to provide a different maximum theoretical efficiency does not violate Betz’s Law, because Betz’s Law is specific to a set of conditions where the velocity through the turbine must be equal to the average of the incoming and exiting flow velocities. Likewise, altering the definition of what is considered to be incoming power also does not violate Betz’s Law, because incoming power in Betz’s Law is predefined in terms of the incoming cross-sectional area that is equal to that of the turbine.

If you have reservations about the above, then perform the following calculations at the point of maximum theoretical efficiency as determined by Betz’s Law. This occurs when the incoming velocity is three times the exiting velocity. Accordingly: Velocity v(in) is equal to 3 v(ex). The velocity through the turbine v(t) is equal to [v(in) + v(ex)]/2 by definition. v(t) is thereby also equal to 2 v(ex). The outlet cross-sectional flow area A(ex) is equal to 3 A(in). And, the turbine area A(t) is equal 1.5 A(in).

Accordingly: The incoming power P(in) is equal to ½ rho A(in) v(in)^3. The exiting power P(ex) is equal to ½ rho A(ex) v(ex)^3. And, the output power (power removed) P(out), as typically stated in proofs of Betz’s Law, is equal to ½ rho A(t) v(t) [v(in)^2 - v(ex)^2], which should also be equal to the difference of the incoming power and the exiting power.

Performing the calculations and normalizing for A(in) equal to 1, v(ex) equal to 1, and rho equal to 1, results in the following: P(in) equal to 13.5, P(ex) equal to 1.5, and P(out) equal to 12. The efficiency P(out)/P(in) is therefore equal to 12/13.5 or 0.8889. If we modify the definition of incoming power to use the cross-sectional area equal to that of the turbine (as defined by Betz’s Law), P(in) becomes equal to rho A(t) V(in)^3, which is equal to 20.25, and the efficiency thereby decreases to 12/20.25, which equals 59.26 percent, in agreement with Betz’s Law.

Bill Wolf (talk) 19:43, 16 February 2009 (UTC)

Further Clarification of Three Flaws in the Existing Proof

The following discussion is related to the version of the article as of May 7th, 2009. Basically, as previously discussed there remain three questionable problems in the proof of Betz’s Law as presented in the article.

1. Inapplicable Force Equations

Basically, the force equations (the second through the fourth equation in the article) are not representative of the model as depicted in the diagram. This has been discussed previously but perhaps not emphasized. The equations are as follows:

$F=m\cdot a$

$F=m\cdot {\begin{matrix}{\frac {dv}{dt}}\end{matrix}}$

$F={\dot {m}}\cdot \Delta V$

$F=\rho \cdot S\cdot v\cdot (v_{1}-v_{2})$

Basically, the first three equations are representative of a force created by the momentum of a jet of air striking a moving plane that is perpendicular to the direction of flow, similar to that of a jet of water from a nozzle against a moving flat plate. This condition would be true in the turbine model if the incoming air flow were moving at velocity v1 and were to directly strike the blades of the turbine (like a jet of water) with the blades having an effective velocity (due to pitch and rotation) equal to velocity v2. However, that is not the case. In the model in the diagram, the incoming air velocity is gradually slowed thereby increasing the pressure on the inlet side of the turbine blades. Basically, in the model there is no delta V at the turbine, but rather a change in pressure (a delta P). According, although the subject equations may represent a possible flow condition, they do not represent the flow conditions of the model in the diagram.

Regarding delta V in the third equation, it is important to not automatically assume that delta V is simply equal to the quantity (v1 - v2). Although (v1 - v2) can also be represented as by a symbol and variable such as delta v (I intentionally made v in lower case to create a distinction), it is unknown whether that delta v is identical to the delta V in equation three. This is a strong point of confusion in the proof as presented and is a key point in defining and proving Betz’ Law.

Accordingly, instead of using force equations that are unrelated to the model, let’s instead create an equation that corresponds directly to the model. To do this we will first require a few more variables. Appropriately, let p1 and p2 represent the air pressures corresponding to the locations of v1 and v2 respectively: Let v represent the velocity through the turbine, let p3 and p4 represent the respective pressures on the inlet and outlet sides if the turbine blades, and note also that p1 and p2 are equal.

$p_{3}+{\tfrac {1}{2}}\cdot \rho \cdot v^{2}=p_{1}+{\tfrac {1}{2}}\cdot \rho \cdot v_{1}^{2}$

$p_{3}=p_{1}+{\tfrac {1}{2}}\cdot \rho \cdot v_{1}^{2}-{\tfrac {1}{2}}\cdot \rho \cdot v^{2}$

$p_{4}+{\tfrac {1}{2}}\cdot \rho \cdot v^{2}=p_{2}+{\tfrac {1}{2}}\cdot \rho \cdot v_{2}^{2}$

$p_{4}=p_{2}+{\tfrac {1}{2}}\cdot \rho \cdot v_{2}^{2}-{\tfrac {1}{2}}\cdot \rho \cdot v^{2}$

$p_{3}-p_{4}=p_{1}-p_{2}+{\tfrac {1}{2}}\cdot \rho \cdot (v_{1}^{2}-v^{2}-v_{2}^{2}+v^{2})$

$p_{3}-p_{4}={\tfrac {1}{2}}\cdot \rho \cdot (v_{1}^{2}-v_{2}^{2})$

The above equation was derived to be used in following subsequent substitution. These equations represent the force on the turbine blades and the power extracted by the turbine.

$F=S\cdot (p_{3}-p_{4})$

$F={\tfrac {1}{2}}\cdot \rho \cdot S\cdot (v_{1}^{2}-v_{2}^{2})$

$P={\tfrac {1}{2}}\cdot \rho \cdot S\cdot v\cdot (v_{1}^{2}-v_{2}^{2})$

Note that although this equation is consistent with the eleventh equation in the article, in this case it was derived directly from a representative model as opposed to simply unrelated equations. It thereby has a basis for used in the following discussion and is lees like to be misinterpreted or misrepresented.

2. Questionable Derivation of v from v1 and v2

As discussed in previous discussions, the proof of Betz’s Law in the article is based on the assumption that that the flow through the turbine (designated v) is equal to the average of the upstream and downstream flow velocities (v1 and v2). Regarding the added mathematical proof, this proof is strictly a manipulation of mathematical terms and does not correlate to the theoretical model that is depicted in the diagram. Although this proof uses a form of the momentum equation, the momentum equation as presented does not represent the theoretical model for a wind turbine as depicted in the diagram. And, although the equations are dimensionally correct, simply maintaining correct dimensional units does not constitute a proof for the validity of the relationship that delta V equals (v1 - v2), which is the basis for later proving that v equals (v1 + v2)/2.

Also note that although the above equations do not prove that delta V equals (v1 - v2) as defined and consequently that v equals (v1 + v2)/2, it does not exclude that this condition could be true for other reasons not presented. Obviously, given a range of shroud shapes there is bound to be a shape that allows v to be equal to (v1 + v2)/2. Presumably, Betz discovered and proved that the shape required to achieve this was the same shape as the flow profile when no shroud was used at all. Presumably, a three-dimensional flow analysis should confirm this. However, it is not proven by the equations presented or by the proof provided in the article.

To provide credibility to my point that delta V from the force equations may or may not equal (v1 - v2), below I will simply show an example when it does not. Therefore if a case exists when this relationship is not true, it cannot be simply assumed that the relationship is universally true. Accordingly, the relationship, therefore, should require further proof.

Below, I restated the four subject force equations from the existing article. Recall that the questionable step in substitution was in going from the third equation to the fourth. There should be no question that rho S v is universally equal to m dot. However, there is a question whether delta V as represented in this equation is universally equal to (v1 - v2) where v1 and v2 are the velocities as defined.

$F=m\cdot a$

$F=m\cdot {\begin{matrix}{\frac {dv}{dt}}\end{matrix}}$

$F={\dot {m}}\cdot \Delta V$

$F=\rho \cdot S\cdot v\cdot (v_{1}-v_{2})$

Next, let’s recall the force equation that we previously derived from the representative model based on the pressure drop across the flow through the blades or, as noted previously, delta p.

$F={\tfrac {1}{2}}\cdot \rho \cdot S\cdot (v_{1}^{2}-v_{2}^{2})$

$F={\tfrac {1}{2}}\cdot \rho \cdot S\cdot (v_{1}-v_{2})\cdot (v_{1}+v_{2})$

Next, let’s first assume the general case where v, v1, and v2 are not constrained from being independent of each other. Therefore, if:

$(v_{1}-v_{2})=\Delta v\,\!$, and

${\tfrac {1}{2}}(v_{1}+v_{2})=v_{avg}$, then

$F={\tfrac {1}{2}}\cdot \rho \cdot S\cdot (\Delta v)\cdot (2\,v_{avg})$, and

$F=\rho \cdot S\cdot (\Delta v)\cdot (v_{avg})$, and

$P=\rho \cdot S\cdot v\cdot (\Delta v)\cdot (v_{avg})$

Note that although the power equation is now in terms of delta v and v average, it still contains the velocity term v, which represents the velocity through the turbine blades. Furthermore, at this point there is no reason to assume that v is equal to v average.

Next, we will apply a special case where v, v1, and v2 are no longer unconstrained form being completely independent. A simple case for doing this is to change the shape of the flow path from the bottle shape in the diagram to that of a cylinder. Accordingly, for flow through the cylinder v is constrained to be equal to v2 by the walls of the cylinder. If there is a pressure drop through the turbine, then v1 must be greater than v and v2. Upstream of the mouth of the cylinder the velocity is v1. However, as the air enters the cylinder it is slowed to v thereby increasing the pressure on the inlet side of the turbine.

$P=\rho \cdot S\cdot v_{2}\cdot (\Delta v)\cdot (v_{avg})$

Therefore, if:

$\Delta v=(v_{1}-v_{2})\$, and

$v_{avg}={\tfrac {1}{2}}\cdot (v_{1}+v_{2})$, then

$v_{2}=2\cdot v_{avg}-v_{1}$

Also, because of the flow through the cylinder, where v is equal to v2:

$v=v_{2}=2\cdot v_{avg}-v_{1}$

Next, if we then make the assumption that v equals v average as what was deduced in the subject proof, we find that v1 must also be equal to v and v2. So, in equation form, if

$v=v_{avg}\,\!$, and

$v=2\cdot v_{avg}-v_{1}$, then

$v=2\cdot v-v_{1}$, then

$v_{1}=2\cdot v-v=v$

Accordingly, setting v to v average causes a contradiction. It is also fairly intuitive. If v1 is greater than v2 and v is equal to v2, v certainly cannot be equal to the average of v1 and v2. Accordingly, for the case of turbine flow being within a cylinder where v equals v2 and not the average of v1 and v2, the subject proof in the article becomes a contradiction. And, accordingly, one cannot assume from the force equations used in the article that because delta V can be assumed equal to (v1 - v2) for that case, that this relation ship universally holds true for the case of a wind turbine.

3. Power and Work Equations Based on Unproven Derivation

If the derivation of delta V being equal to (v1 - v2) and v being equal to (v1 + v2)/2, then the equations in the section headed “Power and Work” are not universally valid as well.

Again, it should be noted that although the subject equations do not prove that delta V equals (v1 - v2) and consequently that v equals (v1 + v2)/2, it does not exclude that this condition could be true for other reasons not presented in the proof.

Revised: BillinSanDiego (talk) 00:08, 18 February 2010 (UTC)

This is an awfully long treatment so forgive me for not addressing all of it, but there is at least something wrong with some of your reasoning. You are correct that there is no $\Delta V$ at the turbine, but the force equations as you quote them are applied to the control volume, not the turbine. The jet hitting a moving plate you describe is a special case to which the equations apply, for which in control volume analysis the control volume moves with the plate, and extends from its surface to a parallel plane where the jet is uniform and infinitely radially away from the jet (where it can be shown that the radial velocity tends to zero). In the equations you quote, $v$ at the turbine does appear, but only to express ${\dot {m}}$ which, according to the definition of the system and/or its assumptions, is constant.

I'm no expert on proofs and make no assertion on the quality of the one in the article - but it's much easier to pick holes in assertions of specific problems with the proof! Bigbluefish (talk) 02:12, 7 December 2010 (UTC)

Request for clarification of observation in "Points of Interest"

First off, this observation doesn't match the formatting of the rest of the article. It's in italics and was not formatted with LaTeX, although I just fixed the later problem.

More importantly, does anyone know what this observation means? It states that the calculation gives a 50% output, does that mean that the new upper bound on $C_{p}$ is 50%? What motivation do we have to use the middle-following velocity here? The example given doesn't even seem to be physical--how can the average and downstream velocities be zero given the assumptions that go into the Betz limit? This section makes almost no sense to me, and although that doesn't necessarily imply it's wrong, it certainly appears to need more context and introduction.

BaoDehui (talk) 18:57, 29 May 2009 (UTC)