RESIGNIFICACIÓN DE LOS CONCEPTOS: CIENCIA Y TECNOLOGÍA

A propeller uses the torque of a shaft to produce axial thrust. This it does by increasing the momentum of the fluid in which it is submerged: the reaction to the force on the fluid provides a forward force on the propeller itself, and this force is used for propulsion. Besides the momentum and energy equa- tions further information is needed for the complete design of a propeller. Nevertheless, the application of these equations produces some illuminating results, and we shall here make a simple analysis of the problem assuming one-dimensional flow.

Figure 4.9 shows a propeller and its slipstream (i.e. the fluid on which it directly acts) and we assume that it is unconfined (i.e. not in a duct, for example). So that we can consider the flow steady, we shall assume the pro- peller is in a fixed position while fluid flows past it. Far upstream the flow is undisturbed as at section 1 where the pressure is p1and the velocity u1.

Just in front of the propeller, at section 2, the pressure is p2 and the mean

axial velocity u2. Across the propeller the pressure increases to p3. Down-

stream of the propeller the axial velocity of the fluid increases further, and for a constant-density fluid, continuity therefore requires that the cross-section of the slipstream be reduced. At section 4 the streamlines are again straight and parallel; there is thus no variation in piezometric pressure across them and the pressure is again that of the surrounding undisturbed fluid.

Applications of the momentum equation 151

Fig. 4.9

This is a simplified picture of what happens. For one thing, it suggests that the boundary of the slipstream is a surface across which there is a discontinu- ity of pressure and velocity. In reality the pressure and velocity at the edge of the slipstream tail off into the values outside it. In practice too, there is some interaction between the propeller and the craft to which it is attached, but this is not amenable to simple analysis and allowance is usually made for it by empirical corrections.

The fluid in the vicinity of the propeller has rotary motion about the pro- peller axis, in addition to its axial motion. The rotary motion, however, has no contribution to make to the propulsion of the craft and represents a waste of energy. It may be eliminated by the use of guide vanes placed downstream of the propeller or by the use of a pair of contra-rotating propellers. Certain assumptions are made for the purpose of analysis. In place of the real propeller we imagine an ideal one termed an actuator disc. This is assumed to have the same diameter as the actual propeller; it gives the fluid its rearward

Actuator disc

increase of momentum but does so without imparting any rotational motion. Conditions over each side of the disc are assumed uniform. This means, for example, that all elements of fluid passing through the disc undergo an equal increase of pressure. (This assumption could be realized in practice only if the propeller had an infinite number of blades.) It is also assumed that changes of pressure do not significantly alter the density and that the disc has negligible thickness in the axial direction. Consequently the cross-sectional areas of the slipstream on each side of the disc are equal and so u2 = u3 by continuity.

(At the disc the fluid has a small component of velocity radially inwards but this is small enough to be neglected and all fluid velocities are assumed axial.) The fluid is assumed frictionless.

Consider the space enclosed by the slipstream boundary and planes 1 and 4 as a control volume. The pressure all round this volume is the same, and,

152 The momentum equation

for the frictionless fluid assumed, shear forces are absent. Consequently the only net force F on the fluid in the axial direction is that produced by the actuator disc. Therefore, for steady flow,

F= Q (u4− u1) (4.10)

This is equal in magnitude to the net force on the disc. Since there is no change of velocity across the disc this force is given by(p3− p2)A. Equating

this to eqn 4.10 and putting Q= Au2, where A represents the cross-sectional

area of the disc, we obtain

p3− p2= u2(u4− u1) (4.11)

Applying Bernoulli’s equation between sections 1 and 2 gives

p₁₊1_2u2_{1= p2+}1_2u2₂ (4.12) the axis being assumed horizontal for simplicity. Similarly, between sections 3 and 4:

p3+1₂u23= p4+1₂u24 (4.13)

Now u2= u3and also p1= p4= pressure of undisturbed fluid. Therefore,

adding eqns 4.12 and 4.13 and rearranging gives

p3− p2= 1₂

u2₄− u2₁ (4.14) Eliminating p3− p2from eqns 4.11 and 4.14 we obtain

u2=

u1+ u4

2 (4.15)

The velocity through the disc is the arithmetic mean of the upstream and downstream velocities; in other words, half the change of velocity occurs before the disc and half after it (as shown in Fig. 4.9). This result, known as Froude’s theorem after William Froude (1810–79), is one of the principal assumptions in propeller design.

If the undisturbed fluid be considered stationary, the propeller advances through it at velocity u1. The rate at which useful work is done by the

propeller is given by the product of the thrust and the velocity:

Power output_{= Fu1= Q (u4− u1) u1} (4.16) In addition to the useful work, kinetic energy is given to the slipstream which is wasted. Consequently the power input is

Applications of the momentum equation 153

since u4−u1is the velocity of the downstream fluid relative to the earth. The

ratio of the expressions 4.16 and 4.17 is sometimes known as the Froude efficiency: ηFr= Power output Power input = u1 u1+1₂(u4− u1) (4.18) This efficiency, it should be noted, does not account for friction or for the effects of the rotational motion imparted to the fluid. A propulsive force requires a non-zero value of u4−u1(see eqn 4.10) and so even for a friction-

less fluid a Froude efficiency of 100% could not be achieved. Equation 4.18 in fact represents an upper limit to the efficiency. It does, however, show that a higher efficiency may be expected as the velocity increase u4− u1becomes

smaller. The actual efficiency of an aircraft propeller is, under optimum conditions, about 0.85 to 0.9 times the value given by eqn 4.18. At speeds above about 650 km· h−1, however, effects of compressibility of the air (at the tips of the blades where the relative velocity is highest) cause the effi- ciency to decline. Ships’ propellers are usually less efficient, mainly because of restrictions in diameter, and interference from the hull of the ship.

The thrust of a propeller is often expressed in terms of a dimensionless thrust coefficient C_{T= F/}1_2u2

1A. It may readily be shown that ηFr=

2 1+(1 + CT)

Since the derivation of eqn 4.18 depends only on eqns 4.10, 4.16 and 4.17 no assumption about the form of the actuator is involved. Equation 4.18 may therefore be applied to any form of propulsion unit that works by giving momentum to the fluid surrounding it. The general conclusion may be drawn that the best efficiency is obtained by imparting a relatively small increase of velocity to a large quantity of fluid. A large velocity(u4− u1)

given to the fluid by the actuator evidently produces a poor efficiency if u1,

the forward velocity of the craft relative to the undisturbed fluid, is small. This is why jet propulsion for aircraft is inefficient at low speeds.

For a stationary propeller, as on an aircraft before take-off, the approach velocity u1 is zero and the Froude efficiency is therefore zero. This is also

true for helicopter rotors when the machine is hovering. No effective work is being done on the machine and its load, yet there must be a continuous input of energy to maintain the machine at constant height. With u1 = 0,

eqn 4.15 gives u2= 1₂u4and eqn 4.10 becomes F= Qu4= Au2× 2u2.

From eqn 4.17, power input= 1₂Qu2₄= ₂1Fu4= Fu2 =√(F3/2A). This

result shows that, for a helicopter rotor to support a given load, the larger the area of the rotor, the smaller is the power required. The weight of the rotor itself, however, increases rapidly with its area, and so the rotor diameter is, in practice, limited.

The foregoing analysis of the behaviour of propellers is due to W. J. M. Rankine (1820–72) and William Froude. Although it provides a valuable picture of the way in which velocity changes occur in the slip- stream, and indicates an upper limit to the propulsive efficiency, the basic

154 The momentum equation

assumptions – particularly those of lack of rotary motion of the fluid and the uniformity of conditions over the cross-section – lead to inaccuracy when applied to actual propellers. To investigate the performance of an actual propeller having a limited number of blades, a more detailed analysis is needed. This, however, is outside the scope of the present book.