39. Injectors

An injector is a specially designed tube or nozzle which atomises the discharged liquid, i.e., the jet into the atmosphere (or a space filled with gas under pressure) is broken up into a fine spray.

The most common type of injector used in gas-turbine aircraft and liquid-propellant rocket engines is the so-called swirl injector for spraying fuel in the combustion chamber.

In such an injector whirl is imparted to the stream, which is then contracted (Fig. 95). The angular momentum created by the tangen­tial approach of the liquid remains approximately constant during flow through the injector. Consequently, as the stream converges the whirl velocity component increases considerably and large cen­trifugal forces develop which throw the liquid to the walls, forming a thin film which issues from the injector in a fine spray. Along the centre line of the injector there is formed an air (gas) vortex in which the pressure at the surface is close to atmospheric (if the discharge is into the atmosphere). This vortex is completely analogous to that formed when liquid flows out of a hole in the bottom of a vessel (Fig. 96), though in the injector it is much more intensive.

Thus, the jet issuing from a swirl injector does not fill the whole cross-section of the injector nozzle; it is annular, with an air vortex along the centre (Fig. 97). Therefore, the contraction coefficient ε of a swirl injector is usually much less than unity.

Because of this, and also because the resultant velocity of dis­charge V is directed at an angle to the cross-section of the injector (Fig. 95), the tangent of which is equal to the ratio of the whirl component и to the axial component v the coefficient of discharge is always much less than unity, varying within broad limits depending upon the geometry and relative dimensions of the injector.

In order to determine the rate of discharge of an injector according to the basic equation (9.6)

the precise value of the coefficient of discharge ц must be known.

The injector theory developed by Professor G. N. Abramovich enables |i to be determined from the dimensions and geometry of the injector. We shall consider this theory very briefly for the case of an ideal fluid. The following are the three initial equations:

(i) Bernoulli's equation between sections 1-1 and 2-2 (see Fig. 95):

or

where is the rated head;

v and и are the axial and whirl components of the velocity V at the boundary of the air vortex at section 2-2. (ii) The equation for the conservation of angular momentum of the fluid with respect to the centre line at the given cross-sections:

or

where r₀ = radius of the air vortex at section 2-2.

(iii) The continuity equation between the sections*:

or

where

From the last expression we have

which gives, after substitution into the second equation,

Using the third equation instead of the foregoing one, we obtain

where is a parameter characterising the geometry of theinjector.

Substitution of the expression for и into Bernoulli's equation at (i) yields

whence

* Assuming a uniform distribution of the axial velocities about the annulaj cross-section at the injector outlet.

Now we can express the rate of discharge as the product of the ve­locity and the annular area of the stream at the injector outlet, i.e.,

Thus, the discharge coefficient of the injector is

(9.20)

The coefficient e, however, is unknown, as we do not know the di­mensions of the air vortex (the radius r_v for a given r₀ and A). To determine it some additional condition must be introduced. Such a condition was put forward as a hypothesis by Prof. Abramovich, and later it was confirmed experimentally. He postulated that a steady vortex would be of such size as to ensure a maximum rate of dis­charge Q for a given head H or, in other words, the flow regime is such that a given discharge is produced by the least possible head.

Let us find the value of e corresponding to the maximum coefficient of discharge μ. For this differentiate the expression under the radical in Eq. (9.20) with respect to e and equate the derivative to zero:

whence

(9.21)

This formula can be used to plot 8 as a function olA. The curve can be used together with Eq. (9.20) to compute values of μ for a series of values of A and to plot a graph of μ as a function of A (Fig. 98).

The diagram shows that with A increasing the coefficient μ de­creases. Physically this is explained by the fact that A increasing means a higher whirl velocity u at the injector outlet as compared with the entrance velocity u_l and, consequently, a more intensive whirl of the fluid. This leads to a vortex of larger diameter and a smaller cross-sectional area of the stream, which means that more and more of the available energy H is dissipated on building up the whirl velocity. At A = 0 (R =0), μ = 1, i.e., there is no whirl and the injector works as an ordinary nozzle.

The foregoing equations can be used to determine the angle of di­vergence of the spray or flame of the injector (see Fig. 98). With A increasing angle a increases and the discharge coefficient drops. Therefore in injector design A must be so chosen as to ensure a suf­ficiently wide angle a (up to 60°) without reducing μ too much.

The injector theory set forth here is developed for the case of an ideal fluid. The effect of viscosity on injector performance is mani­fest in that the angular momentum is not uniform, diminishing to­wards the outlet.

Accordingly, the whirl velocity component at the outlet is usually smaller, and the discharge greater than in the case of an ideal fluid. At first glance this might seem paradoxical.

The effects of viscosity can be taken into account by introducing an equivalent parameter A_eg, which is determined from the formula

where X_t = friction factor of the injector, which can be taken from Table 5, depending on the Reynolds number computed for the diameter and velocity at the injector inlet.

Table 5

	1.5x103	3 x103	5 x103	l x104	2 x104	5x104
	0.22	0.11	0.077	0.055	0.04	0.03

The equivalent injector parameter A_eq thus computed is used to determine the discharge coefficient μ and the angle a taking into account viscosity according to Abramovich's diagram in Fig. 98, where A_eg now stands for A. Since as a rule A_e < A, the coeffi­cient μ taking into account viscosity is somewhat greater, and the angle a is smaller, than in the case of an ideal fluid.

In many liquid-propellant rocket injectors whirl is imparted to the fluid by means of a two- or three-step helix built into the nozzle (Fig. 99).

The injector theory is valid in this case, the factor A being comput­ed according to the formula

(9.22)

where r_m = mean radius of helical groove;

S_n = area of flow cross-section normal to groove; n = number of coils; φ = helix angle.

In modern gas-turbine engines there are more commonly usedadjustable swirl injectors in which the coefficient of discharge (or the area of the outlet) varies automatically with the changing fuel pressure. Such nozzles enable the range of fuel supply to be in­creased for a given pressure range while maintaining the required quality of spraying.

Widely employed are twin-nozzle and two-stage injectors and injectors with by-pass arrangement. Common to them all is a valve device which opens (or closes) an auxiliary conduit when tjie pressure rises, thereby increasing the coefficient of dis­charge or the outlet area.

The twin-nozzle injector (Fig. 100) actually comprises two

injectors, one inside theother. At low pressures the valve is closed and the inner injector operates;when the pressure rises the valve opens and the second injec­tor becomes operative. The supply of fuel increases sharply.

The two-stage injector (Fig. 101) has one nozzle and one swirl chamber with two inlet pipes. At low pressures the fuel is supplied through one pipe, at high pressures, through both. As a result, the pa­rameter A decreases and the coefficient μ increases.

The by-pass injector (Fig. 102) is provided with a valve-controlleddrain. The lower the pressure the wider opens the valve; at peak pressure the valve is closed completely. Thus, at low pressure the intake velocity is higher, which is equivalent to a smaller inlet area, and this means an increase in the equivalent parameter A_eq and a reduction of μ. This is just what is needed to widen the discharge range.

These adjustable injec­tors can be calculated on the basis of Abramovich's theory, with due consid­eration of specific features (see examples).

Example 1. Plot the flow characteristics of a twin-nozzle injector (Fig. 100) showing the relationship between the rate of discharge and the pressure drop in the injector, if the second nozzle (the second channel) be­comes operative at a pressure drop equal to Apn- The geome­try of both channels (dimensions and parameters A_x and 42) and the flow characteristics pf the regulating valve are given. Solve the problem in general form.

Solution, (i) According to the given values of A_x and A% determine μ_J and JX2 from the graph in Fig. 98. (ii) Applying the equations

plot the characteristics of the first and second conduits of the injector (Fig. 103).

(iii) Sum the characteristics of the control valve and the second conduit. In performing this operation remember that the pressure drop Ap in the second conduit is expended in the injector proper and in the valve. Therefore the ord i-nates of Ap must be summed for the given rates of discharge.

(iv) To obtain the total flow characteristic of the injector, sum the character­istics of the first and second conduits and the valve, and the rates of discharge (Fig. 103).

Example 2. Find the expression for the parameter A_x of a swirl injector with by-pass if the total input of fuel is , where Q, - is the dis­charge through the injector nozzle and Q_bp is the by-pass aischarge (see Fig. 102).

Solution, (i) Introduce a by-pass coefficient k_bp:

(ii) The supply of fuel through the input pipes is

(iii) The quantity of fuel passing through the injector nozzle is

Consequently

whence

(iv) Substituting the expression for v_x into the angular momentum equa­tion, and then into Bernoulli's equation, determine the velocity v in the same way as before:

(v) The fuel supply through the injector is

(vi) Comparing this equation with the expression (9.20)_t we find that

C H A P T E R X