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OPTIMAL BLADE DESIGN FOR WINDMILL AND

B. L. Blackford Physics Department Dalhousie University Halifax, Canada B3H 3J5

Abstract

This paper discusses the theoretical problem of designing the optimal windmill blade for use on windmill boats and vehicles. The analysis shows that the optimal design for this application is considerably different from that for a conventional stationary windmill. A practical procedure for blade design is given, and experimental results from a 4 m , using a A m diameter turbine, and from a wheeled , using a 1.2 m diameter wind turbine, are presented. The boat achieves an upwind speed of -503! of the wind speed and the vehicle achieves "1002, in agreement with theory.

1. INTBODÖCriON propulsion was revived by the "oil crisis" and the subsequent search for altemative energy sources. (2),(3),(4) During the past several years we A windmill boat is a wind-driven boat in which a have worked on the theory of windmill boats and windmill type air turbine is mechanically coupled vehicles and have carried out niogerous to an underwater propeller. Kinetic energy ex­ experiment8^5),(6), (7)_ objective was to tracted from the wind by the windmill blades is develop a fundamental understanding of these used by the underwater propeller to push the boat devices and, thereby, to identify the parameters directly against the wind. Fig. 1. The boat moves which are iii{>ortant for their efficient, fast forward because the moraentum added to the water can operation. In the present paper we extend this be greater than that removed from the air, despite work to the problem of designing the optimal unavoidable energy losses in the system. In a windmill blade for this application. A detailed similar manner, a wheeled vehicle can move against account of the general theory of wiodmill boats and the wind by coupling the windmill blades to a driv­ vehicles is given in references (5) and (6). ing wheel. In addition to the capability of ­ ing directly upwind, windmill boats and vehicles can also sail at any other direction relative to 2. BLADE ELEMEOT THEOKÏ the wind, by orienting the windmill blade to face the , Fig. 1. 2.1 WIND BLADE

Historically, the concept of windmill boats and vehicles can be traced back tu the early 1900*8, by The modem theory of propeller design, and also of numerous patents and published articles in windmill blade design, was developed during the and America.^'-) More recently, interest in wind early 1900's by Betz, Prandtl and Goldstein.

103 A thorough review of the subject was given where C is the overall efficiency factor **o£ the by Glauert^l''^ and we follow his approach and driving mechanism, 0<4<1. notation where possible. More recently, the propeller theory was revived by Larrabee^11), tailored to computer use and applied to several The driving mechanism consists of the drive shaft, cases, the most notable success being an improved gears, bearings and underwater propeller, or drive propeller for the Gossamer Albatross raan-pouered wheel for the case of the vehicle. The underwater monoplane. propeller is the major source of inefficiency fur the case of che boat.

Our objective here is to use and extend this pro­ peller theory to design the optimum windmill blade Associated with the extraction of power from the and underwater propeller for a windmill boat. Let wind, there is a backwards force, dFy, on the air W be the windspeed and assume that the craft turbine which is given by the longitudinal travels straight upwind* at a speed u, expressed as component of the aerodynamic force acting on the u - f W. For given u/W, we seek to optimize the blade elements, i.e. blade design so as to maximize the net forward force. Referring to Fig, 2, the apparent windspeed experienced by the rotating blade element at radius r is given by dF„ = -P^CNV^CCL COS*+CD sin^) dr . (4) 2

V - C[(u+W)(l-a)]^ + ((l+a')nrf )l/2 (i) The net forward force (dF-dFy) is given by combining Eqns. (2), (3) and (4): where the quantity a expresses the fractional decrease in the wind by the time it reaches the propeller and quantity a* accounts for the induced dF-dFy = -pycGLNV^[c_5. (sini(i-e cos4>) - rotation of the windfield at the propeller tlOJ. The quantity a satisfies 0

The power extracted from the wind by the blade elements of length dr is given by the product of s = nr/(u+W) (6) the tangential component of the aerodynamic force acting on the blade element times the tangential velocity component of the blade element, f!r. That and noting that, from momentum considerations, the is. backwards force can be expressed in the alternative form tlO^,

dP = ^ycm^iCi sin4i-CDC0s«)rir dr (2) dFy = 2p(^(u+W)^a(l-a)2iir dr , (7)

where N is the number of blades, ^ is the apparent The net forward force, Eqn. (5), can then be wind angle, py is the air density, c is the chord written as and CL and Cp are the lift and drag coeffici­ ents of the blade element, respectivelyj Fig. 2. Gj^ and Cp depend upon the particular aerofoil dF-dFw = 4TrpHR^(u+W)^ lÜ^^H(s)s ds (8) section used and upon its with S respect to the velocity V.

The forward force, dF, derived from the power dP where R is the propeller tip radius, S = fiR/(u+W) can be expressed as is Che tip speed ratio, f = u/W and the function H(s) is given by

dF - CdP/u (3)

(a) * For simplicity, we limit the discussion here to the upwind direction, with the intention of publishing the complete theory at a later date.

(b) ** 5 is fundamentally different from the efficiency factor n used in our previous publications '•^'°>'K

104 = 0.13 would give the nwximom T^^j. This H(.) - C(Xn/f). [Cl-')-^"*"')'] - I . (9) result is somewhat counter intuitive because it [(l»«').+c(l-a)] says that at higher boat speeds the propeller must be designed to extract even less energy from the wind, to obtain the maximum possible normalized, Integrating Eqn, (8) over the blade length from F„gj. Of , the value of the maximum F^g^j centre to tip, we get the total net forward force, decreases as the boat speed increases See normaliced for a given propeller radius and Fig. (6). apparent windspeed, i.e.

It is interesting to compare the above result to FNET 1 s the case of a stationary windmill where the " - ƒ a(l-a)H(a)a ds . (10) optimization condition is simply to obtain the 4»Pw?(u+W)* o maximum possible power output, regardless of the backwards force on the windmill blade. The well-known result for this case is that the Our objective is to design the blade so as to optimal a(s) is given by a(8) = 1/3, except for a maximize Fjjgj with respect to the various reduction of a few percent for 8<1. This is a very parameters. Firstly, we see from Eqn. (9) that the much different result from the case of a windmill efficiency C must be large if high boat speeds, f ^ boat or vehicle, which we have presented here. 0,5, are to be obtained. Small values of the drag/ lift ratio, e, are also helpful. Secondly, we keep C, f and E fixed and address the problem of maxi­ The dependence of F^gx °" other parameters in mizing fffg^j. with respect to a(s), a'Cs) and S, Eqn. (1) is shown in Figs. (4), (5) and (6). The Fortunately, a'(a) depends on a(a) and it can be efficiency factor C has the greatest effect on shown, after a considerable number of steps which FuET, as can be seen in Fig. (4). Reduction of 5 are omitted here, that from 1,0 to 0.6 reduces Fj^gf by a factor of seven, for the particular choice of the other parameters given in the figure- For the case of the boat, the underwater propeller is the major «'(s)- - i(l+e/s)+ i/(l+e/s)2+4a[(l-a)/s2-E/8] (U) source of inefficiency and we therefore see the need for good underwater propeller design. This is discussed further in sections (2.2) and (3.2). which is an exact relation, in contrast to the approximate relation used by Larrabee ^ We The effect of the drag/lift ratio, E, is shown in are then left with the problem of optimizing a(s). Fig. (5), where it can be seen that low values of e It is very difficult, if not impossible, to find an are very important, particularly at the higher tip exact analytical function for the optimal a(a), so speed ratios. Small values of E are obtained by we chose instead to use an empirical technique. using good aerofoil sections, combined with high aCs) was chosen to maximize FjigT» Eqns, (9, 10), aspect ratio blades and high quality construction for a series of discrete values of s, ranging from methods. a"0 to s"S. This empirical form for aia) was then approximated by a convenient analytical form. In this way we found that The effect of the boat speed fraction, f, is shown in Fig. (6), The general result is that FJJET decreases rather rapidly as boat speed increases. •Cs) - a„(l-exp(-28)) (12) Of course, the optimal value of is different for each value of f, as discussed previously and as shown in Fig, (6). approximates the optimal a(a) within lOZ. Note that a(s) deviates significantly from the constant value aj, only for s < 2, decreasing to zero at The dependence of F(JET ^^P *peed ratio, S, •-0. • of the windmill blade can be seen in all of the Figs, (3), (4), (5) and (6), For example, in Fig, (5), which shows F^ET ^ several values of e, it can be seen that the best value of S is S ~ 5 Fig. (3) shows plots of the normalized F^ET VB S for e - 0.01 and S = 2.5 for E = 0.05. From a from Eqn. (10), for which a(s) was approximated by practical point of view, E = 0.01 would be very Eqn, (12) and the other pertinent parameters were difficult to achieve, whereas one can easily E"0,02, f"0,5 and C"0,8. Curves are shown for achieve E = 0,05, or less. We have experimented several values of a^ and it can be seen that ag with windmill blades having designed tip speed • 0,21 gives the maximum Fjfjx values of ratios of 2.8 and 6, respectively, and the results the tip speed ratio, S, However, it should be are discussed below in section (4.2). emphasized here that the optimal value of a^ depends upon the other parameters, particularly the boat speed fraction, f. For example, if f=l then

105 2.2 HATER BLADE where a(s) is determined by Eqn. (12) and a'(B) by Eqn. (11). The apparent wind angle as a function of blade radius, iti(r), can be found by making use It is of the utmost importance that the underwater of Eqn. (6), where is given by propeller blades be designed to operate at a high efficiency, as can be seen from Fig. (4). Fortunately, the theory of minimum induced loss S (u+W) fi = propellers does exist in the literature ^'0) gnd (lA) has recently been put into a convenient computa­ tional form by Larrabee which we use here. In practice, it is possible to obtain t ~ 0.8 to For our example, fi " 27,5 rad/sec under the assumed 0.9. The resulting blades are rather long and operating conditions. For purposes of blade narrow, aspect ratio "8, and are approximately 25X of the length of the corresponding windmill construction, it is convenient to calculate ^ for a blades which they are designed co match. The series of discrete values of radii from r • 0 to r propeller has the appearance of an airplane = R, as shown in Table 1, The actual angle between propeller, as opposed to a conventional marine pro­ the chord line of the aerofoil section and the peller. It would be desirable to have C > 0.9 but rotor plane is given by (i(i-a), where a is the angle the resulting blades become so long and narrow that of attack, see Fig. (2), structural problems are encountered, at least when normal propeller materials, such as bronze or aluminium, are used. A practical procedure for Next, the size of the chord as a function of radius designing the underwater blades is given in section can be determined from the expression. (3.2).

C -ilLsin^* cos*[a/(l-a)+a'/(l+a')] (15) 3. PRACTICAL PROCEDURE FOR BLADE DESIGN NCL

3.1 WIND BLADES which can be derived from the basic theory O-"). The value of the , CL, is determined by the choice of aerofoil section and by Firstly, choose a value for the relative boat the angle of attack, a. We have used a NACA-AA12 speed, u/W = f. For example, suppose we choose u/W aerofoil, Table I, and also a Wortraan FX63-137 " 0,5 and assume that we wish to operate in a Of course, the size of the chord is windspeed of W = 10 m/s i~ 20 knots). The boat inversely proportional to the number of blades, N, would then be moving through the water at a speed u which one chooses to use; N=A in Table 1. The = 5 m/s, which would require a certain net force, output power given in the caption of Table 1 was depending on the type and size of boat. This force obtained by numerical integration of Eqn. (2). can be determined experimentally by towing the boat Similarly, the backwards force on the blades and and measuring the tension in the Cow line. the net forward force was obtained by numerical integration of Eqns. (A) and (5), respectively. The optimal values of windspeed reduction, a^,, and tip speed ratio, S, are then determined from Eqns, (10) and (12), as plotted in Fig. (3). For It is worth noting here that this blade design will this example, ao = 0.21 and S = 3.5 are the best continue to be optimal at other values of the choices, where the best value of S depends somewhat windspeed provided that the drag force on the boat on the value of drag/lift ratio, see Fig. (5). (or vehicle) depends on the square of the boat (or vehicle) speed, u^ (5,6)_ This means that the Having determined and S in this manner, the boat (or vehicle) should operate at u/W " 0.5 for a blade tip radius can be determined from Fig, (5) wide range of windspeeds, W. Performance at very and a knowledge of the required net force. For the low windspeeds would be limited by friction in the present example, suppose that a net force of 600 gears, etc., and at high windspeeds by mechanical Newtons is required to move the boat at u = 5 m/s. failure or capsize! The drag force is reasonably If we assume that e = 0.03, which is easily approximated by a u^ dependence for catamaran achieved, then one finds from Fig, (5) that R = type hulls (and vehicles) but not for heavy dis­ 1,90 m, where Py = 1,12 kg/ra^ was used for the placement hulls, whose drag force increases very air density. rapidly near speed.

The apparent wind angle, if, is determined by the It is also worth noting that, although the air relation turbine of Table 1 gives the maximum possible net forward force for a given R, it may not be the best choice for some applications. To be specific, _lrl,l-a suppose that we choose to operate the air turbine *(3) tan 1(-'J:1-)] (13) at a value of 8^=0.10 instead of ao-0.21, while s 1+a' maintaining the same boat speed u/W"0.5 and keeping the other parameters the same. One then finds.

106 from the theory, that a larger turbine radius, good quality, aerodynamically speaking, and were R=2.25 ra, ia required to achieve the same net not designed according to the theory given in the forward force. However, the backwards force, present paper. Fr^eOO Newtons, is now smaller than before (900 N), which is an advantage since it reduces the backwards pitching moment when upwind and In Fig. (8) we present new data from the same wind reduces the heeling moment when sailing off-wind. vehicle using a new turbine designed according to The disadvantage is that one has a longer propeller the present theory and using NACA-4412 aerofoil and a taller structure. sections. As can be seen from the figure, the vehicle achieves u/W - 0.85 ± 0,07 in a windspeed range of 4 to 8 m/sec. an improvement by nearly a . 3.2 WATER BLADES factor of two. The air turbine used was a 2-bladed one having a designed tip speed ratio of 3, Three different drive wheel sizes were tried (20, 24 and To complete the system it is necessary to design an 28 cm diameter) bpt caused no significant change in underwater propeller to properly match the air performance. The turbine operates nearest to its turbine. To do this, we use the computational designed tip speed ratio when the 20 cn wheel is procedure worked out by Larrabee The used. Stalling occurs with a 38 cm wheel. required input parameters are; shaft power, shaft speed, boat speed, propeller radius and drag/lift ratio of the aerofoil (hydrofoil in this case) The experiments were done in natural wind section used. As an example, we give, in Table 2, conditions on a flat asphalt surface having an the detailed parameters of an underwater propeller unobstructed windfield of about 1,5 km. The speeds designed to match the example wind turbine of Table were measured by timing the vehicle at 6 m 1. The theoretical efficiency of this underwater intervals over a 30 m run upwind. The windspeeds propeller is "0.85. Allowing for losses of a few were measured by a cup-type anemometer, which was percent in the right angle drive gears and later calibrated in a wind tunnel against an NRC* bearings, we obtain C"0.82 for the overall Standard, Slippage of the drive wheel was not efficiency. encountered in relative windspeeds ia*V) of up to 13 m/sec, the maxinum encountered.

In Table 2, the shaft power of 9000 watts was obtained from Table 1, allowing for a 5Z loss in It should be noted. Fig. (8), that the vehicle the gears and bearings. The shaft speed of 55 achieved u/H ^ 1 in some of the runs. In our rad/sec takes into account a gear ratio of 2,0 in opinion, this is probably more representative of the right angle drive units, i.e. the underwater the speed capability when all conditions are propeller rotates 2.0 times faster than the air optimal, i.e. when the vehicle is tracking exactly propeller, in this example. The propeller radius upwind in a steady wind field. of R=0.35 m was chosen somewhat arbitrarily to ensure that C>0.8. i; decreases rapidly as R is decreased. For example, C-O.66 for R=0.2 m, which 4.2 WINDMILL BOAT would drastically reduce the net forward force. Fig. 4. In a previous paper we reported results from a windmill boat consisting of a 4 m Aquacat The actual angle between Che chord line of the catamaran using a 2-bladed (3.05 m diameter) air aerofoil section and the rotor plane is given by turbine with a tip speed ratio of 5. The (^+a) in this case, where a is again the angle of underwater propeller was 0.9 m and was also of the attack. Fig. (7). For a NACA-4412 aerofoil 2-bladed type with a tip speed ratio of 5. In that section, it is reasonable to use CL = 0.8 and system, in contrast to Fig. (1), the air turbine a = 4*. and propeller were mounted on opposite ends of a long, straight drive shaft, inclined ac 25° to the l^orizontal '^i^). Since the orientation of 4, EXPERIMENTAL RESULTS AND DISCUSSION the air turbine was fixed, it could sail only upwind or downwind, but not at other angles to the wind. Also, the efficiency was somewhat reduced 4.1 WINDMILL VEHICLE because the turbine was not perpendicular to the wind, even when going upwind. The system did have one advantage, in that the backward pitching moment In two previous publications ^6,/; „g reported encountered with the arrangement of Fig, (l) was experimental results from a windmill vehicle essentially absent. In the upwind direction it consisting of a 1.2 m diameter air turbine mounted gave u/W - 0,34 ± 0,05 and in the downwind on an aluminum tube framework in a ti-icycle direction, u/W - 0.25 ± .05. configuration. Two different turbine types were We now present experimental results from a windmill tested: one 4-bladed with a tip speed ratio of boat using the same catamaran, but with the about 2.5 and one 2-bladed with a tip speed ratio of about 5. The experimental resulcs gave u/W " 0.45 ± 0.05 in a windspeed range of 3 to 8 ra/sec. Those turbine blades were not of a particularly * Nacional Research Council of Canada, Ottawa, configuration shown in Fig. (l). A similar design very well, easily passing racing raonohulls of 8 - was also used by J. Bates with this 10 ra length. Going downwind under the same arrangement the boat can sail in any direction by conditions, the windmill boat appears to be orienting the air turbine to face the apparent somewhat slower than a conventional . In wind. When sailing upwind, in high , there is winds of W < 2 m/s it performs rather poorly, which a considerable backwards pitching moment tending to is presumably due to friction in the drive train. lift the boat at the and to dig in at the . This greatly increases the drag of the boat and, to correct this, a hydrofoil was installed at The 4-bladed air turbine had a designed tip speed the rear of the boat, as shown in Fig. (1). Since ratio, S=2.8 and a designed value of the windspeed the pitching moment is proportional to (u+W)2 and reduction factor a^, = 0.14. The latter means the hydrofoil lift force is proportional to u^, that this turbine would be optimal for a system we see that the proper balance is maintained over a operating at u/W = 0.9, whereas the boat achieved wide range of wind speeds, if u is linearly a/W = 0.5, for which = 0.21 is optimal, see proportional to W. The later is a fairly good Figs. (3) and (6). The 2-bladed turbine did have approximation for the catamaran, see Figs. (9,10). ao = 0.21.

We have tested two different air turbine systems, a The air turbine rotational speeds, fi, were also slow speed 4-bladed one and a high speed 2-bladed measurable from the movie films and these results type. The relevant parameters are given in Table 3 were used to make plots of 0 vs (u+W). The and the experimental results are given in Figs. (9) relationship was a linear one, as expected, and and (10), respectively. The wind speeds were from the slope of the graph one can get the measured by a cup anemometer and the boat speeds observed tip speed ratio. The results (upwind) were measured from movie films taken with a high were 2.6 ± 0.2 for the 4-bladed system and 6.0 ± quality movie camera. The experiments were done in 0.4 for the 2-bladed system, respectively. These relatively sheltered waters, having an unobstructed results compare very favourably with the designed windfield of about 1 km. values of 2.8 and 6.0, respectively. This means that the air turbines were operating very close to their designed point. On the downwind runs the The 4-bladed system, Fig, (9), achieved an average observed tip speed ratios were somewhat higher, value of u/w = 0.48 ± .03 upwind, in windspeeds which is expected because of the reduced loading un from W = 3.5 to 7.5 m/s, with a downward trend at the underwater propeller. the higher windspeeds. In the downwind direction, it achieved u/W = 0,32, The advantages and disadvantages of low tip speed ratio versus high tip speed ratio are of interest. The 2-bladed system, Fig. (10), achieved an average Firstly, consider low tip speed ratios, 8^4. value of u/W = 0.53 ± ,03 upwind, in windspeeds According to the theory presented here (Figs. from W - 3.0 to 7.3 m/s, again with a downward 3,4,5,6), a tip speed ratio of ~ 3.5 should give trend at higher windspeeds. In the downwind the maximum Fj^gf, which is one advantage. Other direction, it achieved u/W = 0.45. The downward advantages are the safer operation associated with trend of u/W at higher speeds suggests that the lower rotational speeds and the lack of aerodynamic drag force of the catamaran hull is increasing noise. However, to operate efficiently in this faster then u^ in this speed range. range, the propeller must have 3 or 4 blades, otherwise the detrimental effects ^^^^ of a smaller number of blades becomes too important. We do not have detailed test results for sailing in The disadvantages of this are the excess weight and other directions with respect to the wind, but blade area aloft, as well as additional cost. qualitative observations suggest Chat the upwind Secondly, consider high tip speed ratios, S ^ 5, performance is maintained for 0 < X < 60°, where In this range only 2-blades are required, with the \ is the angle between the true wind and the boat advantage of less weight, less area and less cost. course. For 60* < X < 180*, the performance The disadvantages are the very high centrifugal gradually diminishes to the downwind value as the forces and aerodynamic noise associated with the boat goes farther off-wind. It is relevant to higher rotational speeds. In addition, the net emphasize that the air turbine used here is optimal forward force is expected to be somewhat less, only for upwind performance. The optimal turbine Figs, (3, 4, 5, 6), for offwind and downwind performance would be different. The details are not given here, but one Fig, (5) shows that tip speed ratios less than 2 finds that Che optimal value of a^ increases as should definitely not be used, which rules out the one goes further off-wind, reaching " 0.4 in multi-bladed American-farm type windmill. At the the downwind direction for our example where f = high speed end, it is probably not useful to exceed 0.5. A controllable pitch system would be needed a tip speed ratio of 7, since E < 0.02 is not to achieve this in practice. easily achieved in practice. The aspect ratio of the blades is also very relevant to this discussion, since low aspect ratios lead to higher We have also made qualitative comparisons with effective values of e. conventional sail boats. Going straight upwind in wind» of W « 4 to 8 m/s, the windmill boat performs

108 TABLE 2 5. CONCLUDING REMARKS

The main points to be drawn from the present study r • <^*CL can be summarized as follows: (1) We give an [cm] [deg] [cm] analysis of the theoretical problem of designing the optimal windmill blade for use on windmill boats and vehicles sailing straight upwind. The 3.5 71.4 1.93 analysis shows that the optimal blade design is 7.0 56.0 4.68 considerably different from that for a stationary 10.5 44.7 5.86 power windmill; (2) We give a practical procedure 14.0 36.6 5.88 for designing the optimal windmill blade for use on 17.5 30.7 5,40 a boat moving upwind at a certain fraction of the 21.0 26.3 4,74 windspeed, provided only that the boat's drag 24,5 23.0 4.01 characteristics are known; (3) We also outline 28.0 20.4 3,20 the procedure for designing an underwater propeller 31,5 18.3 2,21 blade to match che windmill blade; (4) Using the 35,0 16.5 2.04 above theory, we have designed and built two windmill/propeller systems; (5) These propulsion systems have been tested (upwind) on a 4 m Table 2. Relevant parameters for a 2-bladed catamaran hull and the experimental boat speeds underwater propeller designed to match the wind agreed closely with the theoretical predictions, turbine of Table 1. The drag/lift ratio used was which supports the theory; (6) We are continuing € = 0.03 and the input shaft power was assumed to this study with a view to establishing the optimal be 9000 watts at a rotation rate of 55 rad/sec. propeller design for the off-wind and down-wind See text for further discussion. sailing directions.

TABLE 3

TABLE 1 4-Bladed System 2-Bladed System

Air Water Air Water Im] [deg]

0,102 0,1 69,8 0.276 # of 0,3 50.4 0.318 Blades, N 4 2 2 2 0.5 38,1 0.301 0,7 30.0 0.269 0,9 24,6 Blade 0.238 1.8 0.35 1.9 0.40 1,1 20.7 radius, R(ra) 0.210 1.3 17.9 0.188 1.5 15.7 Tip speed 0.169 2.8 4.5 6.0 3.8 1,7 14.0 ratio, S 0,153 1,9 12,6 RPM 210 610 450 450

Table 1, Relevant parameters for a 4-bladed wind Prop 7 2 turbine designed to operate at a boat speed u/W " mass (kg) 12 2 0.5 in a windspeed of W =- 10 m/s. The rotation FX63- FX67K rate ia « - 27.5 rad/sec. The tip speed ratio is NACA- FX67K- 150 137 150 3.5, the assumed efficiency is C = 0.82, and the sect ion 4412 drag/lift ratio is e - 0.03. r = radius, • - apparent wide angle, c = chord and C^ = lift coefficient. The output power ia 9400 watts, the backward force on the blades is 900 Newtons and the Table 3. Relevant parameters of the propeller net forward force is 600 Newtons, If a NACA-4412 systems tested on the windmill boat. The RPM aerofoil section is used then it is reasonable to numbers are for w = 10 m/s and u - 5 m/s. The assume that CL " 0.8 at an attack angle of a = theoretical net forward force (F-Fy) is 400 4*. The actual chord sizes can then be calculated Newtons for the 4-bladed system and 600 Newtons for from the last column in the table. See text for the 2-bladed system. The theoretical backwards further discussion. force Fy is 550 Newtons for the 4-bladed system and 900 Newtons for the 2-bladed system. • Fig. I. Sketch of a windmill boat (catarmaran) sailing straight upwind. The wind turbine can be rotated about the vertical mast so as to face the apparent wind, allowing the boat to be sailed in any direction without . F„ is the backward force on the wind turbine, F is the forward force produced hy the underwater propeller and Fp is the drag force of the water on the boat. The net force (F-F„) produces the forward speed, u, of the boat, at which F-F„-Fn = 0. The underwater hydrofoil at the rear of the boat counteracts the rearward pitching moment due to the forces F„, F and Fp. -2 X10

Fig. 3. Normalized net forward force versus tip speed ratio, showing the effect of the windspeed reduction parameter a^,. Both axes are dimensionless. To obtain FUET in newtons one must multiply by 4IIP„R2(U+W)2. See Eqn. (10).

xte -2 ^0 =. 0.21 E - 0.03 f = 0.5

Fig. 4. Normalized net forward force versus tip speed ratio, showing the effect of the efficiency factor

{u+W)(l*a)

Fig. 7. Sketch of a cross section of the underwater propeller blade element, showing the water veloc components impinging on the blade element.

1.5^ Q - 20 Cill wheel ^ - 24 on wheel • - 28 an wheel

1.0

u/W • AD

0.5

6

W (m/s)

Pigs. 8. Experimental re.sults (u/H vs W) for a windmill vehicle using a turbine designed according to theory presented here.

113 J 1 1 1 1 —I 3 4 5 6 7 8 W (m/s)

Fig. 9. Experimental results (u/W vs W) for a windmill boat using a 4-bIaded air turbine system. Each point on the graph represents an average of several runs.

Fig. 10. Experimental results (u/H vs W) for a windmill boat using a 2-bladed air turbine system. Each point represents an average of several runs. REFERENCES

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ACKNOWLEDGEMENTS

1 wish to acknowledge with thanks the advice and assistance of R. H. March, B. Fullerton, C. Purcell, J. Chevary, A. Feargrieve, R. Jane and M. Blackford. This research was supported by grants fra« Dalhouaie University and the Natural Sciences and Engineering Research Council of Canada.