Ducted Wind/Water Turbines and Propellers Revisited By Michael, J. Werle, PhD 1 and Walter M. Presz, Jr., PhD 2 FLODESIGN, INC. WILBRAHAM, MA.

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1 Introduction Ducted Wind/Water Turbines and roellers Revisited By Michael, J. Werle, hd and Walter M. resz, Jr., hd FLODEIGN, IN. WILBRAHAM, MA There has been considerable effort and discussion in the literature (see, for examle, Ref.s -8) concerning the otential for ducted wind/water turbines to outerform their unducted counterarts, i.e., surass the Betz theoretical ower extraction limit (see Ref.s 9 and 0 among others.) Results have been resented by Igar et al (Ref.s & ), Hansen et al (Ref. 4) and others that clearly, but emirically, demonstrate this tantalizing ossibility. However, as discussed below, the majority of revious studies are based on an incomlete formulation of the roblem that leads to incorrect limits for the erformance benefits of ducted turbines. The simle but corrected formulation and results resented here: (a) rovides a theoretical basis and verification of the otential available imrovements from ducted flow configurations, (b) identifies a single, critical arameter, that controls that erformance, (c) high-lights the aroriate non-dimensional scaling arameters and (d) rovides a firm basis for further develoment of ducted wind/water turbine technology. Additionally, this formulation rovides some interesting new results and insights for ducted/shrouded roeller roulsion. Ducted Wind/Water Turbines Figure rovides the geometry and nomenclature alied herein. All revious formulations for the unducted wind/water turbine and ducted roeller cases (see Ref.s 0 and, for examle) have correctly imosed a ressure boundary condition at downstream infinity. To date, this has not been alied to ducted wind/water turbines (see Ref. s - for examle). Only Hansen et al (Ref. 4) used a closure condition far downstream in a multi-dimensional comutational fluid dynamic (FD) analysis of a simulated ducted wind turbine. In the current model the ressure boundary condition will be imosed at downstream infinity. Referring to Figure, the governing equations are written for a control volume using a cut incororating the turbine blades (modeled as an actuator-disc discontinuity with zero leakage around its edge) and the duct/shroud (with its attendant force on the flow), along with arallel, constant static ressure inflows and outflows at ustream and downstream infinity. With this, the conservation of mass, momentum and energy for a low seed and/or incomressible fluid leads to the following equation for the ower extracted (note the equations are first resented in dimensional form and later in non-dimensional form er their ower or roulsion alication): ower: = ρa ( o a ) + F ( o + a )/ () The axial shroud/duct force, F, in Equation was modeled here as it has been for ducted roellers (as in Ref.), i.e., F was taken to be directly related to the ressure jum across the hief cientist, AIAA Fellow hief Technology Officer, AIAA Member

2 disk/turbine through a non-dimensional shroud/duct force coefficient, s, which results in the exression: [ o a ] hroud Force: F = ρ A ( ) () Ref. rovides detail discussion on the relationshi between this ressure jum and the circulation about the ring airfoil. The resulting internal velocity at the turbine disk and thrust roduced are than given as: elocity: ( + )( + ) = (a) Total Thrust: T = ( + ) / = /( + ) o a o a (b) Note that the unducted wind/water turbine case is recovered using =0 and that these equations are a slight variant of those used for roeller roulsion (as resented by Mcormick, Ref. and others.) It is also noted that, and it shall be demonstrated below, the non-dimensional shroud/duct force coefficient,, can be determined at any convenient level of ower extraction, including zero, i.e., for the case of a clear duct. From Equations - it is straightforward to show that the maximum ower that can be extracted by a ducted wind/water turbine is given simly as: 6 max / ρa = [ + a 7 ] (4) which is shown in Figure along with the attendant flow roerties given as: oa ( o / a ) = (5a) m m am A ( / ) = A ( A A ) = ) a i i ( + m m m (5b) ( A A ) = ( ) = ( ) o m o m a oa + m (5c) As shown in Equation 4 and Figure, this simle formulation catures the traditional bare/unducted wind turbine case (the Betz ower Limit of 6/7) at s =0 as but one of an infinite family of ossibilities. The validity and utility of the current formulation can best be demonstrated through comarison with the FD results of Hansen et al (Ref. 4) where they resents a range of the ower extraction levels for flow through an actuator disk simulating a ressure dro across a wind/water turbine.

3 Results were resented for the unducted case as well as an aerodynamically contoured ducted case with an aggressive exit area ratio, A D /A =.86. Figure reroduces the results from Ref. 4 which were resented in terms of the thrust on the actuator disk defined as: T T / ρ A (6) a where T was calculated from the ressure dro acting on the actuator disk area, A. For the current formulation, alying Equations - rovides the relationshi: ( + / ) [ + ] T T (7) which requires determination, by indeendent means, of the shroud/duct force coefficient. This can be done by first noting that the current formulation alies for all ower extraction levels including that of the clear duct case with zero ower extraction. onveniently, Hansen et al (Ref. 4) did rovide the flow arameters for this case. In articular, they gave that a =.8, which, when used in Equation a along with the fact that o = a for the clear duct, gives =0.8. The resulting comarison resented in Figure shows that the simle one-dimensional inviscid flow model well reresents the FD results over the full range of the blade thrust for both the bare and ducted configuration. Not surrisingly, the FD results roduce a lower maximum ower level for the ducted case due to the considerable viscous losses encountered for such an aggressive diffusion area ratio of.86. To further relate the current formulation to earlier works (e.g., Ref.s -); it is useful to first determine the ressure level at the exit lane, A D, of Figure using Equations - with Bernoulli s equation to write: + D a a s = k ka D oa = A oa + + D ρa D oa (8a) where, as in Ref.s -4, the diffuser static ressure recovery efficiency coefficient, k, has been introduced to relate the ressure rise from the blade/disk location to the exit lane, and the area ratio is given in shorthand fashion as: A A D / A D (8b) Note the Equation 8a dictates that the exit ressure and diffusion levels cannot be emloyed as indeendent variables (as imlied in earlier works, e.g. Ref.s -) but rather must always satisfy this relation. Most imortantly, in order to extract the maximum ower ossible, they must satisfy this relation with oa = /, as dictated by Equation 5a.

4 With Equations 8a and 8b, it is also convenient to emloy Igar s definition (Ref.s & ) of the maximum ower available using Equations -6 to write that: 7 r max = ( 9D) /( k + kad) (9) 6 which differs significantly from that roosed by Igar (Ref. ) and Riegler (Ref. ) who did not imose the aroriate downstream ressure condition. Results from use of Equation 9 are shown in Figure 4 for duct area diffusion ratios, A D =.0 and. for a diffuser ressure recovery coefficient, k =. Ducts with area diffusion ratios much larger than. are found to suffer significant losses (k< as was the case in Ref. 4) unless they are quite long, and thus too heavy to be ractical. In fact, alying Equation 9 to Hansen et al s case (Ref. 4) that had k=0.8 with A D =.84, one can estimate that the viscous induced ressure losses in the diffuser reduced the maximum ower extracted by nearly 0% below it s theoretical limit. Figure 4 clearly shows that ducted wind/water turbines are theoretically caable of extracting ower levels significantly above those of their unducted counterarts for realistic levels of diffusion and exit ressure levels. Ducted roellers As a final oint, it is additionally noted that the formulation of Figure also alies to the ducted roulsive roeller case with known ower inut. For this case Equations and can be rewritten as: oc + acoc + acoc ac = 0 (0a) where use has been made of the following definitions: / ( ( )) (0b) c + ( 4 ρ ) / A (0c) (Note the ower velocity,, of Equation 0c is closely related to the disk loading coefficient used by others, e.g., Ref. ) (0d) oc ac o a c (0e) c F / ρa (0f) 4

5 The solution to the cubic Equation 0a can be aroximated using a Taylor series to reresent the exlicit but more comlex closed form exact solution as: 4 oc ac + ac 9 () This in turn can be used in Equation b to calculate the ducted systems total thrust in terms of a thrust coefficient defined as: T T ρa = / / 4 ( + ) /( + ) ( + ) / + + ac oc ac 9 ac () As shown in Figure 5, the unducted case for static flight ( T = ) is recovered at =0 as but one of a family of cases, all of which are well reresented by the simle olynomial aroximation of Equation for both static and forward flight conditions. This formulation can be further simlified by first noting that for the static case, a =0 and the duct exit lane ressure coefficient of Equation 8a with relacing a can be used with the aroximate form of Equation and further Taylor series aroximations to write that: / / ( T ) ( ) ( D ) 0 A AD a = = + D T0 where, for current alications, it has been further assumed that the diffuser efficiency, k, is unity. The resulting Figure 6 shows that for the static case: (a) thrust increase of nearly 80% above the bare roeller level are attainable with moderate diffusion and exit lane suction ressures and (b) the handy aroximation of Equation gives a good reresentation over the regimes of interest. Finally, combining Equations & leads to a simle relationshi for forward flight effects on the thrust as () T/T O 4 + / T o / (4a) T o a 9 T o a = T + which is shown in Figure 7 to yield a very accurate aroximation to the exact solution for virtually all values of forward velocity. The indeendent variable in Equation 4a can also be written as a generalization of that given by Mcormick (Ref ) and others as: = / / ) ρ T oa a T0 (+ A (4a) oncluding Remarks As simle as the above formulation is, it is hard to overstate its imortance or utility for ducted wind/water turbines. From its analytical redictions resented herein, it is observed that: 5

6 (a) ducted turbines are theoretically caable of extracting ower significantly in excess of a bare wind/water turbine, (b) there is but a single arameter, the duct/shroud non-dimensional force coefficient, s, that determines the maximum ower extractable, (c) for the first time the here-tofore missing Betz-like core element has become available for use in the detailed design of the wind/water turbine blades cross sectional shae along their sans so as to guarantee the cature of the maximum ower available from the flow assing over the blade (Ref. 0 rovides an excellent exlanation of this aroach for un-ducted wind turbines.) The exlicit relationshis resented here coule the design of the blades with their surrounding duct in a manner that must be satisfied in order to achieve otimal ower extraction. With this new model in hand, a rational aroach to the design of ducted wind/water turbines can recede with the otential for achieving the maximum ower outut available. Without it, all revious such designs must necessarily be considered otentially sub-otimal. It is also worthy of note that with the simle formulation resented here, one can straight forwardly redict the ducted wind/water turbine erformance for all ower levels based on the flow characteristics of the clear duct configuration. Additionally, for ducted roellers, the current formulation roduced a series of simle algebraic relations for redicting erformance, correlating data and/or guiding reliminary design efforts with or without forward flight effects. References. Igar, O., hrouds for Aerogenerators, AIAA Journal, October 976, ee also AIAA aer # 76-8 resented at the AIAA 4 th Aerosace onference, Washington D, January Igar, Ozer, Research and Develoment for hrouded Wind Turbines, Energy ons. & Management, ol, -48, 98. Riegler, Gerold, rinciles of Energy Extraction from a Free tream by Means of Wind Turbines, Wind Engineering, ol 7, 5-6, Feb Hansen, M.O.L, orensen, N.N.: and Flay, R.G.J.: Effect of lacing a Diffuser around a Wind Turbine, Wind Engineering, ol, 000, Lilley, G.M. and Rainbird, W.J., A reliminary Reort on the Design and erformance of a Ducted Windmill, Reort 0, ollege of Aeronautics, ranfield, England, Toucher, Angus J., Habitat Friendly, ressure onversion, Wind Energy Extraction, U atent 6,887,0 B, May, Gaskell,., The ombined Augmented Technology Turbine (ATT), atent Alication # 0/48,009, June 00, also discussed in Ref. 7 above, 5-54, May Kirke, Brian, Develoments in Ducted Water urrent Turbines, ublished on University of outh Australia, Hansen, M.O.L.:AERODYNAMI OF WIND TURBINE romwell ress, Gashe, R. and Twele, J,: WIND OWER LANT, olarraxis, 00. Mcormick, Barnes W. Jr., Aerodynamics of /TOL Flight, Dover ublications, Minneola NY, 999 6

7 a a a A i D D O O = a a A A D A O Figure : Ducted ystem Nomenclature Bare Wind Turbine 4 / o & A o /A / a & A a /A o / a max Betz Limit s Figure : Wind Turbine Max ower haracteristics.5.0 Ducted WT Line Eqn 7 oints FD 0.5 Bare WT T Figure : omarison with FD Analysis 7

8 .0 A D /A =..5 r A D /A =.0.0 Bare Wind Turbine Betz Limit D Figure 4: Wind Turbine Max ower Limits.6 a / =0 T. 0.8 a / = NOTE olid = = Exact Dashed=Aroximate a / = s Figure 5: Ducted ro erformance.8.6 NOTE olid = = Exact Dashed=Aroximate T 0 A D /A =.0.4. A D /A = Bare ro Bare ro D Figure 6: tatic Ducted ro erformance 8

9 Aroximate olution Equation 4 T/T Exact olutions, =0 (Bare ro) to = T0 * a Figure 7: Ducted ro Forward elocity Effect 9