St

which then defines ap in terms of the changes in the temperature factor a at constant chamber pressure.

It is not simple to predict the motor performance, because of changes in grain temperature and manufacturing tolerances. Reference 11-4 analyses the prediction of burning time.

Example 11-2. For a given propellant with a neutrally burning grain the value of the temperature sensitivity at constant burning area is ttk = 0.005/°F or 0.5%/°F; the value of the pressure exponent n is 0.50. The burning rate r is 0.30 in./sec at 70°F at a chamber pressure of px — 1500 psia and an effective nominal burning time of 50 sec. Determine the variation in px and tb for a change of ±50°F or from +20°F to + 120°F assuming that the variation is linear.

SOLUTION. First Eq. 11-5 is modified:

Solving, Ap = ±375 psi or a total excursion of about 750 psi or 50% of nominal chamber pressure.

The total impulse or the chemical energy released in combustion stays essentially constant as the grain ambient temperature is changed; only the rate at which it is released is changed. The thrust at high altitude is approximately proportional to the chamber pressure (with A, and CF assumed to be essentially constant in the equation F = CFpiAt) and the thrust will change also, about in proportion to the chamber pressure. Then the burning time is approximately ti = 50 x 1500/(1500 - 375) = 66.7 sec t2 = 50 x 1500/(1500 + 375) = 40.0 sec

The time change 66.7 - 40.0 = 26.7 sec is more than 50% of the nominal burning time. The result would be somewhat similar to what is described in Fig. 11-8.

In this example the variation of chamber pressure affects the thrust and burning time of the rocket motor. The thrust can easily vary by a factor of 2, and this can cause significant changes in the vehicle's flight path when operating with a warm or a cold grain. The thrust and chamber pressure increases are more dramatic if the value of n is increased. The least variation in thrust or chamber pressure occurs when n is small (0.2 or less) and the temperature sensitivity is low.

Burning Enhancement by Erosion

Erosive burning refers to the increase in the propellant burning rate caused by the high-velocity flow of combustion gases over the burning propellant surface. It can seriously affect the performance of solid propellant rocket motors. It occurs primarily in the port passages or perforations of the grain as the combustion gases flow toward the nozzle; it is more likely to occur when the port passage cross-sectional area A is small relative to the throat area A, with a port-to-throat area ratio of 4 or less. An analysis of erosive burning is given in Ref. 11-5. The high velocity near the burning surface and the turbulent mixing in the boundary layers increase the heat transfer to the solid propellant and thus increase the burning rate. Chapter 10 of Ref. 11-3 surveys about 29 different theoretical analytical treatments and a variety of experimental techniques aimed at a better understanding of erosive burning.

Erosive burning increases the mass flow and thus also the chamber pressure and thrust during the early portion of the burning, as shown in Fig. 11-9 for a particular motor. As soon as the burning enlarges the flow passage (without a major increase in burning area), the port area flow velocity is reduced and erosive burning diminishes until normal burning will again occur. Since propellant is consumed more rapidly during the early erosive burning, there usually is also a reduction of flow and thrust at the end of burning. Erosive burning also causes early burnout of the web, usually at the nozzle end, and exposes the insulation and aft closure to hot combustion gas for a longer period of time; this usually requires more insulation layer thickness (and

FIGURE 11-9. Typical pressure-time curve with and without erosive burning.

more inert mass) to prevent local thermal failure. In designing motors, erosive burning is either avoided or controlled to be reproducible from one motor to the next.

A relatively simple model for erosive burning, based on heat transfer, was first developed in 1956 by Lenoir and Robillard (Refs. 11-3 and 11-6) and has since been improved and used widely in motor performance calculations. It is based on adding together two burn rates: r0, which is primarily a function of pressure and ambient grain temperature (basically Eq. 11-3) without erosion, and re, the increase in burn rate due to gas velocity or erosion effects.

Here G is the mass flow velocity per unit area in kg/m -sec, D is a characteristic dimension of the port passage (usually, D = 4Ap/S, where Ap is the port area and S is its perimeter), p is the density of the unburned propellant (kg/m3), and a and ft are empirically constants. Apparently, ft is independent of propellant formulation and has a value of about 53 when r is in m/sec, /?, is in pascals, and G is in kg/m2-sec. The expression of a was determined from heat transfer considerations to be

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