ON THE LIMITS OF THE BETZ S EFFICIENCY COEFFICIENT OF WIND TURBINES WITH THE HORIZONTALLY SHAFT

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1 ON THE LIMITS OF THE BETZ S EFFICIENCY COEFFICIENT OF WIND TURBINES WITH THE HORIZONTALLY SHAFT Prof.dr. ing. Petre Terzi SOCIETATEA PENTRU PROMOVAREA ENERGIILOR REGENERABILE, INEPUIZABILE SI NOI SPERIN Abstract Beginning with the flowing modeling by means of the moing components of the horizontal shaft wind trbines applying the trblent flow theoretical concept and considering the papers pblished by Betz and Sabinin, there are new theoretical concepts presented regarding the limits of incident crrent energy tilization factor on the horizontal shaft wind trbines, by considering the effect of pstream and downstream flow elocities indced by the wind trbine rotor. Key words: aerodynamics, wind trbine, horizontal shaft wind rotor Rezmat Pornind-se de la modelarea crgerii prin] elementele rotorli de trbina eoliana c arbore orizontal, prin aplicarea conceptli de crgere trblrblenta neintbata, pe baza pblicatiilor li Betz si Sabinin, se prezinta n no concept teoretic priind limitele coeficientli de tilizare a energiei antli incident pe sectinea actia a rotorli trbinei eoliene c arbore orizontal c larea in considerare a itezelor indse in amonte si in aal de rotor. Cinte chee: aerodinamica, trbine de ant, rotor eolian c arbore orizontal..introduction Following by means of the moing components of the wind trbines hae been approached in the specialized technical literatre since the latter half of the past centry, in stdies condcted by RANKINE [] AND FROUDE [], sbseqently resmed by BETZ [3] and SABINI [4] dring the period in witch there was laid the basis of the aerodynamic stdy of general flowing in horizontal shaft wind trbines. The present paper, centered on the theories deeloped by BETZ [3] AND SABINI [4], deals with the theoretic aspects of the stdy of flowing in the horizontal shaft wind rotor. Pointing ot the continity between these theories, the paper frther presents some aspects related to the deelopment of the trblent flow theory as proposed by SABINI [4]. Possible soltions of the flowing eqations are presented, by considering the effect of the rotor indced pstream and downstream flow elocities, as well as the effect of the ariation of the rotor indced downstream elocities according to the horizontal shaft trbine rotor operating regime, in order to determine the correlation of the maximm efficiency regime erss the ariation of the elocities indced behind the wind trbine..elementary THEORY OF THE HORIZONTAL SHAFT WIND TURBINES The aerodynamic model proposed by BETZ [3] in order to stdy wind energy catching by means of propeller type deices, considers the fact domain, made p by the moing air crrent. NOMENCLATURE VV V V V V, mm/s mm/s mm/s Far incident elocity incident elocity Aail incident elocity Indced peripheral elocity in amonte or aal section

2 w [m/s Relatie wind crrent elocity incident on the profile P PPa P Air pressre at - section P PPa P Air pressre at - section dfr N Elementary bearing force dfa N Elementary axial force A m m Propeller area P P kw Mechanical power Ns m 4 Volmetric density Special characters E - C P Betz s efficiency ( power factor) - Velocity factor Aerodynamic profile finenesses i i Velocity modle amonte or aal section The aerodynamic model proposed by BETZ [3] in order to stdy wind energy catching by means of propeller type deices, considers the fact domain, made p by the moing air crrent. This aerodynamic model considers the total air pressre discontinity accring in the pstream and downstream sections in the immediate icinity of the rotor. By sing the concept of air crrent tilization factor, also called wind trbine power factor, marked C p then the initial power may be written as: P C b AV 3 P (.)

3 V P V P V Fig.. Aerodynamically model proposed by BETZ Which means that air elocity after crossing the catching system still maintains abot 33% of the initial elocity V? The reslts a maximm ales of the dimensionless power factor of C p =.593. There reslts, therefore that an ideal rotor with an infinite nmber of blades placed in niform crrent can catch at the most 6% of the incident air crrent energy. The elementary theory of the wind catching system points ot the following aspects: the power of catching from an air crrent of incident elocity V is directly proportional to the rotor incidence area A, the cbic initial elocity V downstream far off the section where the catching system is located, the power factor C p and the air braking capacity of the incident air crrent energy Also called air crrent braking factor, V a m V defined by Betz [4]. V im =,5(V + V ) Maximm ales when the braking factor is a 3. A more throgh analysis of the wind energy catching mechanism is to be fond in the following chapter by sing the trblent flow theoretical concept and applying the implse theory to the aerodynamic stdy of the horizontal shaft wind trbine. 3. THE IMPULSE THEORY AS APPLIED TO THE HORIZONTAL SHAFT WIND TURBINE. The application of the implse theory to the horizontal shaft wind trbine dimensioning has been taken oer from the airplane propeller elastic theory, which, together with the classical whirlwind theory has led to highly efficient aerodynamic soltions.

4 One of the forernners of this calclation and dimensioning method was G. K. Sabinin [4] who had his stdies in this field pblished starting from Hypotheses: The application of the implse theory to the calclation and dimensioning of the propeller type horizontal shaft trbines is based on the following main hypotheses: () the air jet crosses the rotor at an een elocity throghot the axial cross section; (3) the rotor lets the air pass throgh the blades withot determining a local elocity discontinity and has an infinite nmber of blades; (3) the presence of the rotor brings abot a pressre ariation between pstream and downstream, in a flid domain delimited downstream the catching system by a cylindrical srface on which an infinite nmber of whose winding is the cylindrical srface corresponding to section A-A, fig., downstream, in its immediate icinity; (4) the crrent tbe delimited by the solenoid srface does not allow for the air exchange between its inside and otside, the air crrent passing throgh the rotor being considered isolated from the ambient; (5) the crrent non-niformity increases at the rotor otlet, which leads to trblent energy losses and therefore to a lower efficiency of catching the wind energy, while the crrent twisting dissipated in alternating whirlpools cased by the instability of the flow downstream the rotor; (6) air pressre in the cross-section in fig.. is assmed to be eqal to the atmospheric one; (7) in cross-section -, pressre rises to ales p <p, yet frther increasing while aiming asymptotically towards ales p, far-off downstream; (8) in the sections downstream the wind trbine rotor, pressre ariation is neglected, considering that the centrifgal forces determined by the refined crrent twisting after rotor crossing are small as compared to the forces cased by the axial implse in the same section; (9) the pressre difference downstream and pstream the catching system leads to the occrrence of an axial force pon the rotor Fa ; as a reslt of the adeqately bilt rotor geometry there appears a tangential component Fr in the rotation plan, leading to the occrrence of the catching system sefl moment. Obiosly, wind-catching systems shold be bilder so that the rotor implse tangential component shold be as big as possible. Fig. 3.. schematically shows the air crrent shape pstream and downstream the rotor according to the aboe hypotheses, as well as the diagram of the modality the whirlwind solenoid srface is formed, delimiting the crrent tbe downstream the rotor; pstream the rotor, namely in the - section in fig.., crrent elocity V is eqal to the far-off pstream elocity (infinite pstream); in the catching system rotor section -, and as getting nearer and nearer to this section, the crrent axial elocity drops to ales V = V V, where V and V are elocities indced by the cylindrical trblent layer generated by the whirlwinds on top of the rotor blades forming p a crrent tbe. As in the wind trbine rotor there appears a rotating moment in the section -, that leads to the occrrence of the indced rotating implse, conterclockwise to the catching system rotation, as a reslt of the whirlwinds indced by the blades haing as winding the crrent tbe cylindrical srface as per the diagram in figre 3. there reslts therefore that downstream the rotor, the air crrent rotates at a elocity eqal to the elocity V in a close enogh section, where the whirlwind tbes do not icinity, the crrent peripheral elocity is considered not ery mch different from that, so that V ~ V, where V is the rotating elocity at the rotor otlet. Fig 3. the diagram of the formation of whirlwind solenoid srface downstream the wind trbine

5 3. Determining of the peripheral force component and of the axial force component acting pon the catching system blade Isolating an annlar area of r radis, dr thick, off the air crrent, the axial component dfa as well as the rotation peripheral component dfr (see fig..3) can be determined in the corresponding annlar section in the rotor plan, fig. 3.. These components lead to the occrrence of the interactions in the rotor constrction elements (blades). By applying the implse theorem and taking into accont that, in keeping with second principle of mechanics, action is eqal to reaction, there can be determined the reactions occrring in the rotor blades, along the axial and tangential directions, respectiely, Where: dfa elementary axial force, and dfr the elementary tangential force. dm π r pvdr Where dm is the nitary moss air flow crossing the annlar section in the rotor, fig. 3. V = V V ; V is the indced elocity before the rotor, cased by the blades By applying to the air mass delimited by the two concentric cylindrical areas of radis r and r + dr, as shown in figre., the Eler theorem for the motional qantity by taking into the expression of the elementary force cased by the action of the elementary air mass enclosed between the two cylindrical areas, pon the blades. dfa = V dm; (3.) Where, In a similar way, in keeping with theorem of the motional qantity moment, the elementary torqe created by the tangential rotational force occrring on the elementary srface of the blades enclosed between the two cylindrical areas of radis r and r+dr, there reslts the relation: dfr = dm (3.3) V,, w, as well the corresponding elocities in a section downstream the rotor are witten with the following relations: V = V, of the blades enclosed between the two cylindrical areas of radis r and r+dr, there reslts the relation: (3.4).

6 ( ) ( r ) ( ) ( r ) (3.) The relation (3.) represents the ratio of the axial indced elocity to the peripheral elocity component s. This ratio is ariable along the blade from hb to apex. ( ) ( ) (3.5) Relation (3.3) is flfilled proided what is in between the brackets is eqal to zero. Ot of this condition, there reslt: And (3.6) NEW SOLUTION The general case corresponds to the condition when relation (3.3) parameters are eqal to one another, yet different from zero, That is condcing to two frther conditions: (3.7) (3.8) In the case of indced elocities as well, there can be admitted a modle elocity defined as the ratio between the indced peripheral elocity and the axial elocity i as i follows: i and i (3.9) By sing the definition of the indced elocity after the rotor, relation.3 can be written: d d i i i i (3.33) d Cancelling of deriate d Leads to: i i (3.) i min By soling ot the eqation in relation with there reslts : the expressions for : max and ( ) i max i (3.)

7 There reslts accordingly: From which the following expression for may be inferred: i i (3.). We specify that the soltion gien by Sabinin (4) namely modle elocity i is alid in the case of a The axial indced elocity drops asymptotically at the same time with the increase of aiming to ales the wind catching system indced downstream flow elocity modle i By explaining the rotor indced downstream flow axial elocity in relation (3.9), there reslts: i i i (3.3) Ot of the calclation of the asymptotic ales there reslts, considering the limit of the fnction gien by relation. (3.39) for : i lim i i i i lim lim i i i i (3.4) Therefore, there reslts that the minimm ale towards which elocities indced after the rotor aim when i are that is the doble of the flow elocities indced pstream the wind trbine rotor. In order to determine the extremes of the ariation fnction of the flow elocity indced after the rotor, as defined by relation (3.3) the first deriate shold be cancelled. There reslts in this case: i i i (3.5) i i (3.6) x i (3.7) xx i ' (3.8)

8 There reslts therefore max i i min i i (3.9) (3.) It is considered that the dependency relation of the indced elocity gien by relation (3.39) erifies the condition at the limit. For, the cre in figres 3. passes throgh the origin. For ales of the modle elocity indced after the rotor, in the range i i the indced rotational elocity after the rotor.ths, the soltion gien by Sabinin [4] is corrected throgh relation (3.9.) 5. CONCLUSIONS The deeloped theory regarding the flowing in the horizontal shaft wind rotor leads to ales that are considerably higher than the maximm tilization efficiency becase of the factor Cpmax =.593 determined by means of the elementary theory of the mechanical offered by Sabinin. This specify limit can hae a mch higher ale, that is Cpmax= REFERENCES: []Rankine, W.J. Mechanical principles of the action of propellers. Trans. Inst. Naal Architects 865, ol.6, pp3-4 []Frode, W. An elementary relation between pitch slip and proplsie efficiency. Trans. Inst. Naal Architects 865, ol 4,pp.69-7, [3]Etz, A. Die Windmhlen am Lichte nerer Fernschng Natrwiesenschaften. 97, XV, pp [4]Dascheici, K.P. Implsnaia theory etreannih digatelei Sabinina Promiselnaia Aerodinamika, ipsc 3, Oboronghiz 959, pp8-.

9 Fig. 3.. Variation of peripheral indced elocity erss the modle of indces elocities i

Efficiency Increase and Input Power Decrease of Converted Prototype Pump Performance

Efficiency Increase and Input Power Decrease of Converted Prototype Pump Performance International Jornal of Flid Machinery and Systems DOI: http://dx.doi.org/10.593/ijfms.016.9.3.05 Vol. 9, No. 3, Jly-September 016 ISSN (Online): 188-9554 Original Paper Efficiency Increase and Inpt Power