Talk:Betz's law

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)[reply]

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)[reply]

Betz was an eminent German engineer, Professer Emeritus at the University of Gottingen, and principle 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

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)[reply]

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 V1 A1 equals V1 A2 equals Vavg S equals (V1 + V2)/2 S, then S equals 2 V1 A1 /(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, 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)[reply]

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)[reply]

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.

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)[reply]

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 mean before/after speed by the harmonic mean [since the distances are same]. The arithmetical mean of 3 and 1 is 2 , but the harmonic mean 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[reply]