1. A Bates motor is a solid rocket motor with grains cast into multiple separate segments,...

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1. A Bates motor is a solid rocket motor with grains cast into multiple separate segments, each a cylindrical shape as shown. (Actually the sides taper in slightly, so the cross section is really trapezoidal instead of rectangular, but we will ignore this aspect here.) Wall -W. segment. (~ Side View End View (a) Find an expression for the burning surface area A of one motor segment, in terms of 11, 12 and w. Note that both ends burn as well as the inner bore of the cylinder. (b) As a motor segment burns, the burning surface recedes at a rate . Find an expression for the rate A at which the burning surface area changes, in terms of . (c) For a neutral burn (constant thrust), the burning surface area must be constant. What is the relation among 1, 2 and w such that A is zero? (d) Another desirable characteristic of a Bates motor is that the segments should burn to the wall at the same time as, or before, the width w goes to zero. What is the relation among 1, 2 and w such that the burn is neutral and r becomes equal to 12 (i.e., the segment burns to the wall) at the same time as w goes to zero? The mass flow of a solid motor is proportional to the burning surface area: m~ Ab = LWP, (4.35) where A is the burning surface area, L is the grain length and WP is the wetted perimeter, the perimeter of the burning surface, i.e., of the perforation. L is constant, but WP can change as the propellant burns. (Eq. (4.35) assumes the perforation to be the same all along the grain. This is often but not always the case. In the strap-on solid boosters on the Shuttle, the perforation increases from the head end to the nozzle.) Fig. 4.8 shows a grain with a circular perforation in part (a), and the corresponding thrust profile in part (b). As the propellant burns, the hole gets bigger, like the whistling candies in Chitty Chitty Bang Bang. The perforation stays circular but increases in radius. So the wetted perimeter WP = 2R increases with time and the thrust profile is increasing, as shown in fig. 4.8(b). If the burn rate is constant, the thrust increase is linear. A motor with a circular grain has progressive thrust, meaning that the thrust increases with time. 1. A Bates motor is a solid rocket motor with grains cast into multiple separate segments, each a cylindrical shape as shown. (Actually the sides taper in slightly, so the cross section is really trapezoidal instead of rectangular, but we will ignore this aspect here.) Wall -W. segment. (~ Side View End View (a) Find an expression for the burning surface area A of one motor segment, in terms of 11, 12 and w. Note that both ends burn as well as the inner bore of the cylinder. (b) As a motor segment burns, the burning surface recedes at a rate . Find an expression for the rate A at which the burning surface area changes, in terms of . (c) For a neutral burn (constant thrust), the burning surface area must be constant. What is the relation among 1, 2 and w such that A is zero? (d) Another desirable characteristic of a Bates motor is that the segments should burn to the wall at the same time as, or before, the width w goes to zero. What is the relation among 1, 2 and w such that the burn is neutral and r becomes equal to 12 (i.e., the segment burns to the wall) at the same time as w goes to zero? The mass flow of a solid motor is proportional to the burning surface area: m~ Ab = LWP, (4.35) where A is the burning surface area, L is the grain length and WP is the wetted perimeter, the perimeter of the burning surface, i.e., of the perforation. L is constant, but WP can change as the propellant burns. (Eq. (4.35) assumes the perforation to be the same all along the grain. This is often but not always the case. In the strap-on solid boosters on the Shuttle, the perforation increases from the head end to the nozzle.) Fig. 4.8 shows a grain with a circular perforation in part (a), and the corresponding thrust profile in part (b). As the propellant burns, the hole gets bigger, like the whistling candies in Chitty Chitty Bang Bang. The perforation stays circular but increases in radius. So the wetted perimeter WP = 2R increases with time and the thrust profile is increasing, as shown in fig. 4.8(b). If the burn rate is constant, the thrust increase is linear. A motor with a circular grain has progressive thrust, meaning that the thrust increases with time.