1 THEORY OF DUCTED FAN by Giacomo Sacchi august, 2009 Theory of duct fan, by Giacomo Sacchi 1/102 DIFFERENCE BETWEEN FREE FAN END DUCTED FAN The air f...

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by Giacomo

DUCTED FAN

Sacchi

[email protected] august, 2009

Theory of duct fan, by Giacomo Sacchi

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DIFFERENCE BETWEEN FREE FAN END DUCTED FAN The air flow of fan is "ideal" if:

is delimited by a clean boundary

with

the

air

with

surrounding

"quadratic" form; the air speeds up continuously along the way; the air has got a constant speed in every section of the flow; the density of the air is constant and equal to the surrounding one; For the principle of the conservation of the mass (with hypothesis of constant density) if the section halves, the speed doubles,

since

the

area

is

function of the square of the radius as consequence the form of

the

border

must

be

"quadratic". The real flow is less regular, since

the

depression

that

swallows the air in the propeller is attracted from every direction and this clearly does not allow

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clear border between flow and surrounding air, therefore the accelerations and the speeds of the flow are disomogenee between center and peripheral zone. The air flow which comes form the propeller is pushed everywhere by the pressure. In particularly degenerated conditions, the same air tip comes newly inhaled by the propeller generating whirling rings that reduce the push drastically. The reason of this is in the sir properties such as: viscosity, frictions, molecular bonds, etc. The worsening of the flow is given above all for lowlands speed and small propellers in proportion to the applied power. In these cases it is useful to use a duct to address the flow in the propeller, as a consequence to reduce these aerodynamic inefficiencies. This brings to more weight and more construction complexity, moreover the dimensioning of the tube is optimized for a specific condition and far from this ideal condition, the wastes increase remarkably. Generally the employment of ducted propellers is necessary due by the lack of space for larger propellers or by the will to maximize the static thrust or for safety reasons because people must be protected form the propeller 's contact.

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ANALYSIS OF DUCTED FAN FLOW The most important parameters are: ρ0 V0 Ain Av Aout Vin

= density of the air = speed of the surrounding air = in aera = fan area = out area = average speed of the air that enters Vv = air fan speed Vout = average speed of the air that exits Vmax = flow max speed Δp = variation of pressure between before and after the propeller

HUPOTHESIS CONSTANT DENSITY OF THE AIR (ρ) The first hypothesis to fix is that the density remains constant and equal to that of the surrounding air. This is a more legitimate approximation as well as the flow has a much minor speed of that of the sound (1200 KM/H). Moreover the pressure generated by the propeller does not have to reach the highest values, because it would cause accelerations of the air such to induce it to compress itself in order to win its inertia. MASS FLOW (m') Air amount that passes through the propeller (Av) in the unit of time (Kg/sec). TRUST (T) The push is exclusively given by the quantity of air which is accelerated T = mass x acceleration = m' (Vmax - Vo) = m' ΔV This formula refers to the push of the whole system propeller-tube, while the propeller generates a slightly advanced push that is lost in frictions between air and duct.

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TRUST POWER (Pt) Pt = ½ m' (Vmax² - Vo²) = T Vo + ½ T (Vmax - Vo) Obviously the power increases with the push that has to be generated but it "uselessly" increases also with ΔV and Vo. The following considerations are drawn: 1. with constant ΔV and increasing m' I obtain same push with less power, this is obtained with large Av, this is always to maximize in the plan limits 2. with the increase of Vo we need more power; Air that goes into the tube must to be slow but if the air is slowed down by us, there is some wasted energy that is not completely regained, instead if the air is however slowed down (es. boundary layer on wings), it gives a greater output. AIR SPEED INTO DUCT The air speed in the conduct is only given by the principle of conservation of the mass:

ρin Ain Vin = ρv Av Vv = ρout Aout Vout

but with ρ constant: Vin = (m'/ρ)/Ain;

Ain Vin = Av Vv = Aout Vout Vv = (m'/ρ)/Av; Vout = (m'/ρ)/Aout;

Please notice that Vin is generally different from Vo and Vout does not have ties with Vmax; The Vin, Vv, Vout speed does not depend directly from the tube sections and from Vo and Vmax; Moreover please remember that they are average speeds, for example “the ideal” flow could pass only through the central part of the section and the remaining air to be almost firm. In case of real flow are the resistances of sliding to give a bell distribution to the speed in the section. The speed in the section of propeller (Vv) is assumed to be exactly the same hardly before and hardly after and this is an acceptable approximation. For a good efficiency the speeds of the flow (in particular Vmax) are always more minor of the sound speed (approximately 1200 km/h).

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PRESSURES IN THE CONDUCT The action of the propeller is to generate a depression hardly before (p.v1) that attracts the air in the conduct and a pressure hardly after (p.v2) such to accelerate the air in the getting out. Δp = p.v2 - p.v1, this difference of pressure generates the push of the propeller F (>T in order to satisfy the wastes for friction in the tube) F = Δp Av With high Δp more inefficiencies are generated and the air can be compressed, but we have to remember that we want to realize a ducted propeller not a compressor. The pressure in income to the tube (p.in) and the one in escape (p.out) can vary with the sections and the speeds of the air on the basis of Bernulli's equation that with constant ρ and with horizontal conduct (therefore without variations of potential energy) we have: ½ ρVin² + p.in = ½ ρVv² + p.v1 ½ ρVv² + p.v2 = ½ ρVout² + p.out The values of pressure in income and escape stretch to the external pressure (Po) but only if the sections of the conduct are correctly dimensioned to follow the flow and not to hinder it. SWIRL The spin of the propeller generates axial push (f) with a power (Pf), but also some vortices that are an undesired effect as they waste power (Ps). Total power applied to the propeller: Pt = Pf + Ps For the planning is supposed a “distribution of the free vortices”, that is to say that the speed of the vortices increases linearly moving away from the center of the propeller:

Vs = K / r

ωs = K / r²

τs = m' K

K = costant of swirl; it can be experimentally obtained by measuring the power Theory of duct fan, by Giacomo Sacchi

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applied to the propeller and the effective obtained push; the values go from 1 to 6; The power of the vortices is obtained making the integral calculus on Av of (τs x ωs) and it obtains: Ps = 2 π (m' K² /Av) ln (Rv / Rhub) Rv

= fan radius;

Rhub = ogive propeller radius; CHARACTERISTICS OF THE DUCT The income culvert has the only aim to convey the air to the propeller in a regular way introducing less possibly frictions and maintaining the laminar flow. The income must be aerodynamically clean, streamlined. The surfaces must be smooth, regular and with course of the beam of the sections to be reduced in quadratic way. The length of the income culvert is determined by the optimization degree that we wanted to reach to a determined speed. That is to say that if the culvert is lenghtened until the income speed income and the movement speed are the same, to which we want to optimize the push, we obtain the maximum of the yield in that condition but the fast worsening of the efficiency in the others speed. The escape culvert has instead the task to transform the pressure in speed, as it happens in weapons where the pressure of the outbreak is transformed thanks to the cane in speed of the ogive. The escape culvert must be sufficiently long (some decimeters are enough) but not too long in order not to introduce resistances, moreover the section does not have to be increased. In order to have some advantages in the extreme circumstance of ducted propellers with very short ducts it is necessary to hinder the return of the air from the escape to the income (es. tail posts of some helicopters).

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THE PLANNING OF THE DUCT Various approaches can be undertaken, beginning from various ties, but in any case it will be probably necessary to begin with estimated values and to proceed by trials and error in the direction of the aspects to optimize. The procedure chosen by me expects to decide: the propeller diameter: as large as possible; the Vo speed of the external air to which to optimize the push, it could be the cruise speed but it would penalize the takeoff; This is a choice to do estimating the aircraft and the installed power; to estimate the intentional push to Vo; Now the mass flow must be estimated (m') optimal, that is that it diminishes the total power which is necessary. The First method gives better results with low Vo or Ae small: Ptot = power for the push + swirl power (the other wastes as the resistance of the culvert can not be still estimated as the air flow speeds are still not known ) β = variable convenient; Ps = 2 π (m' K² /Av) ln (Rv / Rhub) = β m' T = m' ΔV Ptot = TVo + ½ TΔV + Ps replacing: Ptot = TVo + ½ T²/m' + β m' this formula links the necessary power to the mass flow, but it must diminish the power, therefore we derive in m' and we place equal to zero. m' = T / sqrt(2 β)

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The second method gives better results with high Vo or Ae large m' = ρ Ae Vo Between the two methods it has to be used the mass flow which requires less power. Once m' is found, we can desume the several speeds : Vin = (m'/ρ)/Ain;

Vv = (m'/ρ)/Av; Vout = (m'/ρ)/Aout; Vmax = T/m' + Vo

These speeds must be much lower of the sound speed (approximately 1200 km/h). Moreover it is possible to estimate the power wastes for the resistence (Pr) due to the friction of the air with the conduct that are depending on the speeds. Power to apply to the propeller: Pf = TVo + ½ T²/m' + Ps + Pr If it is excessive it is possible to increase the area of the impeller and/or diminishing the cruise speed (Vo). The fixed push (t) is the wished one, but the propeller must produce a greater push (f) in order to satisfy the wastes for friction (if considered). F = T ((Pf-Ps)/(Pf-Ps-Pr)) ((Vmax+Vo)/2)/Vv The swirl power has to be deducted as it does not contribute to the push. A parameter that will be useful for the planning of the propeller is: Δp = F / Av

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THE PLANNING OF THE FAN Once the conduct is established, the speed of the air to impeller (Vv) is obtained, for the prefixed speed (Vo) and the push (r) that we want to obtained. We have to fix the speed of spin of the impeller (ω) according to the diagram of power/RPM of the engine to the necessary power (Pf), remember that the terminal velocity (Vt = ω R) must be lower of the sound speed (1200 km/h). The shovel must be divided in sections in order to gain for each one the values of chord and twisting angle. The relative speeds are obtained (Vi) and the angles of incidence (αi) of the air flow in the several blade sections. Now it can be decided to apply various choices, like for example to maintain a blade with constant chord (simplifying the propeller but not optimizing the result), or increase the number of the blades in order not to have too much large chord. Angles of twist and optimal chord From the diagram of the chosen profile for the shovels the angle of incidence is found (α') such for which 75% of (the Cp/Cr) maximum are had, so that the profile works as much as possible in good efficiency, but reducing the risk of stalls. We immediately desume the angle of twisting in the several sections = αi α' The width of the chord (ci) is desumed from Δp, Cp to α' and the formula of the lifting capacity P = ½ ρ Si Cp Vi². Si = surface of the shovel section

Good job.

Giacomo Sacchi www.autogiro.it

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