Massachusetts Institute of Technology Department of Aeronautics and Astronautics Cambridge, MA 02139 16.01/16.02 Unified Engineering I, II Fall 2003 Systems Problem 4 Name: Due Date: Parts 1, 2, 3 - 10/2/2003 Parts 4, 5, 6, 7 - 10/9/2003 Time Spent (min) Part 1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7 Study Time Announcements: Please turn in this cover sheet on 10/9/2003MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Aeronautics and Astronautics 16.010 / 16.020 Unified Engineering Systems Problems #4 Issued: September 25, 2003 Part 1, 2, and 3 Due: 5:00p October 2, 2003 Parts 4, 5, 6, and 7 Due: 5:00p October 9 2003 WATER BOTTLE ROCKET PERFORMANCE ANALYSIS Objectives After completing this systems problem, you should be able to: • Apply rocket performance equations and thermodynamics such that you can call yourself a Novice Rocket Scientist. • Apply concepts from freshman year (especially 8.01) and from Unified to model a system of modest complexity. • Use simple computational tools such as a spreadsheet to compute the performance of a system based on an engineering model. • Explain an engineering model, its underlying concepts, its assumptions, and its limitations. • Use an engineering model for parameter design to improve performance. Discussion This is the first of three consecutive systems problems devoted to the design of a small rocket propelled by compressed air and water. In this systems problem, you will estimate rocket performance and make a preliminary analysis of the impact of two design parameters on the maximum altitude achieved by the rocket. This includes developing a computer spreadsheet to aid in the analysis. The second systems problem (next week) will focus on your design of a new, higher-performance rocket, while the third systems problem will involve construction of your rocket, flight testing, and final performance analysis. 1 of 8We will begin with a baseline rocket system that uses a standard 2 liter soda bottle for the rocket structure and fuel storage. The bottle will be partially filled with water and mounted on a rocket launch mechanism (Figure 1). The air in the bottle will then be pressurized, and the launch will be executed. We will focus on this baseline rocket in this systems problem and predict its performance. In the following systems problem, you will design your own rocket and will perform a performance analysis of your design. So, your work here lays the groundwork for the next several weeks. WaterLaunch Rod Release pin Two-liter soda bottle Air at 60 psig 7 in Figure 1. Water Bottle Rocket and Launching Mechanism 2 of 8The same launching mechanism will be used for the baseline rocket we examine here and for the rocket you design later. The mechanism uses a 7-inch-long rod to provide stability and to ensure that the rocket proceeds in a straight line. The bottle is inverted onto the rod, and the launcher is staked to the ground. A metal launch restraining pin is inserted over the lip of the bottle neck to keep the bottle on the launcher until it is ready for blastoff. Air is then pumped through the rod to pressurize the bottle; a small rubber o-ring on the launch rod provides an airtight seal at the mouth of the bottle. The rocket launch consists of three stages. The first stage covers the period between the start of the launch until the rocket nozzle (bottle mouth) just reaches the top end of the launch rod. The second stage involves the compressed air forcing the water out of the bottle at high speed as the bottle rises into the sky. We will assume that any compressed air that remains after the water is expelled produces negligible additional thrust. The third stage is a ballistic stage, in which the rocket continues upward under only the influence of gravity and drag, reaching some maximum altitude before falling back to the ground. First Second Third Stage Stage Stage Figure 2. Rocket Launch Stages 3 of 8Relevant parameters Rocket Volume of bottle Outer diameter Nozzle diameter Empty mass Coefficient of drag Launcher Launch rod length Launch rod diameter Environmental Conditions Air pressure Air temperature Air density Water density Baseline Launch Procedure 2.3x10-3 m3 1.118x10-1 m 2.159x10-2 m 4.7x10-2 kg 0.5 (estimated)1.778x10-1 m 2.159x10-2 m 1.013x105 Pa 288 K 1.225 kg/m3 1x103 kg/m3 1. Bottle filled 40% of volume with water2. Bottle inserted onto launch stand3. Launch stand is securely staked to the ground and launch retainer pin is inserted4. Bottle is slowly pressurized to 60 psig (60 pounds per square inch gauge pressure = 60psi above ambient air pressure = 4.137x105 Pa above ambient air pressure) 1. Thermodynamic Analysis The rocket launch process actually forms a complete thermodynamic cycle, composed of the following states: State 0: air at ambient pressure and temperature State 1: air compressed in the bottle, at ambient temperature, just before launch State 2: air compressed in the bottle, at the start of the second stage State 3: air at atmospheric pressure immediately following the second stage The transitions between states are: State 0 to State 1: air is compressed into the bottle isothermally 4 of 8State 1 to State 2: air expands adiabatically as the launch rod leaves the bottle (first stage) State 2 to State 3: air expands adiabatically to atmospheric pressure (second stage) State 3 to State 0: air warms to 288K at atmospheric pressure, completing the cycle (a) Determine the complete state of the air (p, V, T) at each state in the cycle (0, 1, 2, and 3). (b) Compute the work W, heat flow Q, and change in internal energy ∆E in each of the four transitions between states. (c) Verify that the work performed over the entire cycle equals the heat flow, and that the change in internal energy after a complete cycle is zero. (note: you may not find that these hold true exactly, due to roundoff error and to the assumption of constant Cp [specific heat actually varies slightly with temperature] -- getting answers to within a few Joules is fine). (d) Draw a P-V plot and a T-V plot for this cycle (to scale, not just sketched), allowing one to clearly read off the values of p, V, and T at each state. 2. First Stage During the first stage, you may neglect friction forces due to the launch rod and you may neglect aerodynamic drag forces (the velocity of the bottle is still relatively small at this point). Do not neglect the weight of the bottle and water, however. You may assume
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