I need you to use the simulator equations PDF to solve number 10 on the "Extra" PDF. I have already found the equations required, I just need you to Matlab model them. These problems are for extra credit. Each 25 points can be used to replace a quiz or homework, or add 10% to an exam score (you can't go over 100%). You can do as many or as few problems as you want. Work on these problems by yourself. Feel free to ask me questions if anything is unclear. The deadline is 2014-11-19 in the discussion section. Projectile chamber valve barrel Fb Fatm x, projectile location in barrel Pneumatic projectile launchers are used in a variety of different applications, including the launching of radio antenna, hunting, launching of safety flares, and recreation (Paintball, Punkin' Chunkin' championships). These launchers illustrate many different thermodynamic concepts. Assuming ideal gas, constant specific heats, no heat transfer, and isentropic flow, let's see how complex such a simple system is. The launcher works as follows: a projectile (say, a potato, pumpkin, or tennis ball) is loaded at the base of the barrel. There's a pressurized gas chamber separated from the barrel by a valve. As the valve is opened, compressed gas flows from the gas chamber through the valve into the barrel, where it causes the projectile to accelerate. Note that the barrel volume V– b = AbLb, where Ab is the barrel cross sectional area, and Lb is the barrel length. Based on the coordinate system in the drawing, x = 0 when the projectile is at the base of the valve, and x = Lb when it's leaving the barrel. So, the barrel volume at any instant in time is V– b(t) = Abx(t). 1. (20 pts) Starting with the following equations and the mentioned assumptions, derive P V– k = constant: ds = du T + p T dv (1) ds = dh T − v T dp (2) (Hint: An vague outline of the latter part of the derivation is available in section 6.11.2, p. 328, of your textbook.) 2. (5 pts) If the flow rate of the valve is high enough, you can assume that the pressure in the gas chamber and barrel are equal, and model the launcher as a closed system. Using P V– k = constant, derive an equation for the variation of the pressure in the launcher as a function of overall volume (chamber plus barrel)\ 3. (5 pts) As the projectile accelerates down the barrel, what happens to the temperature of the gas? 4. (5 pts) Using the pressure-volume relationship found in the previous part, derive the work done on the projectile by the time it reaches the end of the barrel. 5. (5 pts) Derive an equation for the muzzle velocity, that is, the velocity the projectiletakes as it exits the barrel, based on the work done on the projectile from the previouspart. 6. (5 pts) Derive an equation for the energy efficiency of the launcher. Remember that energy efficiency basically is what you get divided by what you put in. What you put in is the work of the compressor. What you get is the muzzle energy (kinetic energy) of the projectile. Assume the compressor is perfectly efficient (no friction, noise, heat transfer, etc.). (Hints: You really shouldn't have to calculate the compressor work; it's directly related to something you've already calculated. Also, the muzzle velocity is a variable here. In the previous problem, it was calculated under the assumption that the gas chamber and barrel pressures are equal, but that's necessarily true in reality.) 7. (5 pts) The muzzle velocity of a projectile launcher was measured at different pressures. The test had the following characteristics: • chamber pressure: 70 psi (gage) • barrel length: 12 in • gas chamber volume: 1.1 in3 • barrel diameter: 0.527 in • projectile mass: 0.98 g • atmospheric pressure: 101 350 Pa • atmospheric temperature: 85 F The average muzzle velocity over 15 tests was 190.99 ft/s. What is the energy efficiency of this launcher? 8. (10 pts) Plastic pipe is rated for water pressure only for safety reasons. Projectile launchers like this often are made from plastic pipe despite warnings that compressed gas in these pipes poses a safety risk. For a chamber with a volume of 2850 cm3 , calculate the amount of energy that would be released as water and air go from 700 kPa (gage, at atmospheric temperature) to atmospheric pressure. Does this suggest why plastic pipe is only for compressed water? 9. (Harder, 20 pts) In reality, the flow rate of the valve isn't so fast such that the pressures of the gas chamber and barrel are equal. You can model this as a transient open system. Write equations for conservation of mass and energy in the gas chamber and barrel before the projectile leaves the barrel. The valve mass flow rate is m˙ , varies in time, and is assumed to be known. These equations should be like dUchamber/dt = . . .. Note that you don't have to solve these equations, just write them out. 10. (Thermo master level, 100 pts) Write a computer simulation to solve the equations in the previous problem. If you are good with programming, this could take a few hours. If not, but you have a fair bit of free time and you want to learn, try Matlab or Python. You can approximate the time derivative of a function u as follows (forward Euler method): du dt = u n+1 − u n ∆t (3) where ∆t is the time step, and n and n + 1 refer to particular times. Note that t = n∆t here.You will need three extra equations. The first is the ideal gas law. The second is an equation of motion for the projectile location. You can use Newton's first law to find the following (assuming no friction): mprojectile d 2x dt2 = (Pbarrel − Patm)Abarrel (4) You will also need an equation for the mass flow rate of the valve as a function of the pressure differential. Use the ISO valve model:   PuCρrefr Tref Tu s 1 −  Pd/Pu − b 1 − b 2 if Pd Pu > b (subsonic) PuCρrefr Tref Tu if b ≥ Pd Pu (sonic)The subscripts u and d refer to upsteam and downstream. The valve flow could be going either from the gas chamber to the barrel or from the barrel to the gas chamber. Downstream is on the side that things are flowing to. You can assume that b = 0.2, ρref = 1.185 kg/m3 , and Tref = 293.15 K. Play around with this model to get a feel for the dynamics of the system.