Talk:Betz's law

message: 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)
thread_title:  Request for clarity 

message: This article should be renamed.  —Preceding unsigned comment added by 128.40.54.199 (talk) 15:12, 22 February 2008 (UTC)
thread_title:  The Lanchester-Betz Limit 

message: 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)
thread_title:  Questionable Title 

message: 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)
thread_title:  Experimental demonstration? 

message: 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)
thread_title:  The Economic Relevance 

message: 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)
replies: text:  the last para summarises the whole article very succintly chami 15:05, 20 January 2013 (UTC)
thread_title:  Notes on Revised Article 

message: 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)
thread_title:  Suggested Addition to Article 

message: 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)
thread_title:  Request for clarification of observation in "Points of Interest" 

message: There is inconsistency within this page about calling it "Betz' law" versus "Betz's law". Which is correct? I believe "Betz's" would be preferable grammatically, but "Betz'" might be the common spelling for this scientific term (I don't actually know). Thoughts? —Will(B) 15:39, 22 October 2013 (UTC)
thread_title:  That darn apostrophe-S 

message: I just joined Wikipedia. I was under the impression that this a community where anyone can edit. Furthermore, why is Bet'z law 'protected'? It shouldn't be that popular. Do you just lock every mildly popular article for only the 'registered' users to edit? I find that very displeasing. Please consider changing this disgrace to the "community" of Wikipedia. -Larry
Larryswindcatcher (talk) 18:34, 23 March 2015 (UTC)
replies: text: per no request specified. - DVdm (talk) 18:37, 23 March 2015 (UTC)
thread_title:  Semi-protected edit request on 23 March 2015 

message: Unfortunately this article has some serious errors and misrepresentations. The biggest fault is with a claim that Betz limit applies to all turbines, or was calculated for turbines at all. Betz law applies ONLY to PROPELLERS! This is evident to anyone who actually bothered to read his original publication. Albert Betz was studying an ideal propeller, which he describes as "thin air-permeable DISK with infinite blades" That's a verifiable fact, just open his work in any engineering library. The limit does not apply to VERTICAL TURBINES, which use aerodynamics of airplane-like wing, and can take energy from the same airflow multiple times. This is due to the fact that ideal VAWT would be described as a "squirrel cage" and not a "thin disk". This article needs serious corrections, and needs a moderator who actually has serious background in aerodynamics. Please clean up this embarrassment!—Preceding Roger.Tanner comment added by 24.69.137.123 (talk) 08:24, 27 July 2015 (UTC)
replies: text: You can start editing today. All you need is a good source. Ariadacapo (talk) 11:32, 1 August 2015 (UTC)

text: NB: Why am I replying to a 4 year-old comment? I got confused by the history log date vs. autosignage SineBot inconsistency. Apologies for disruption. Ariadacapo (talk) 11:37, 1 August 2015 (UTC)

text: This article is semi-protected. Sign-up to Wikipedia by registering an account; after 4 days and 10 edits you will be able to participate to this article. Ariadacapo (talk) 06:05, 5 August 2015 (UTC)
thread_title:  Serious errors 

message: The second paragraph of the introduction is unsubstantiated.  — Preceding unsigned comment added by 128.8.202.120 (talk) 20:11, 14 January 2016 (UTC)
replies: text: (I suppose you mean unsourced, rather than unsubstantiated). I’m pretty sure it does. Will check in the coming 24 hrs. Ariadacapo (talk) 06:21, 15 January 2016 (UTC)
replies: text:  You were correct: it doesn’t. I removed the reference. A re-write of the lead to reflect the body (WP:LEAD) would be nice. Ariadacapo (talk) 08:32, 16 January 2016 (UTC)
thread_title:  Source 4 does not mention shrouded turbines 

message: Under "concepts", "Betz' Law applies to all Newtonian fluids" suggests that viscosity is relevant. But the law does not explicitly involve a unique constant nonzero viscosity (where for example Stoke's law does). Betz' law should apply to fluids with any stress-strain rate relation -- not just Newtonian.  — Preceding unsigned comment added by 192.0.171.34 (talk) 20:04, 15 June 2016 (UTC)
thread_title:  newtonian fluid is more specific than necessary 

message: Even to propellers it is a nonsensical limit. Propellers exploit only 1% of the whole area through which air passes and obviously dependent not only on the area of blades or their coverage area while rotating, but and on the speed of wind and the speed of rotation. Moreover, stopped air molecules would diffuse due to the density or concentration gradient into the surrounding area, thus this limit can be lifted up till 90% or more for propellers (depending on the air flow speed since it would affect the drag or turbulence during the rotation and other dynamic parameters) and even higher in other designs like vertical axis turbines. Unless they apply this limit to only the area of blades (perpendicular surface of blades to the flow even would not rotate blades at all in perfect conditions) and not to the whole covering area of rotation. — Preceding unsigned comment added by 121.215.23.155 (talk) 01:43, 6 February 2017 (UTC)
thread_title:  the law that is not a law