systems problem 4 lecture notes
DESCRIPTION
payload. Systems Problem 4 Lecture Notes. Learning objectives. After completing this SP4 and SP5 you will have: Applied material from 8.01, (Unified thermodynamics) and Unified fluid mechanics to develop a model for a single stage water rocket - PowerPoint PPT PresentationTRANSCRIPT
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Systems Problem 4
Lecture Notes
payload
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Learning objectives
• After completing this SP4 and SP5 you will have:
– Applied material from 8.01, (Unified thermodynamics) and Unified fluid mechanics to develop a model for a single stage water rocket
– Demonstrated an ability to integrate a system of ordinary differential equations using a spreadsheet
– Explored how external aerodynamics, structural weight, propellant mass fraction, payload mass, internal fluid mechanics and thermodynamics jointly determine the dynamic behavior of a single stage water rocket.
– Demonstrated an ability to describe conceptually how the performance of the water rocket changes as a function of important design parameters
– Developed a preliminary design for a water rocket that you and a partner may build and test for SP6
SP4
SP5
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3 stages
Stage 1Quasi-static adiabatic expansionas rocket lifts off launch rodConstant mass
Stage 2Quasi-static adiabatic expansionWater ejected from rocketGravity, drag, thrust forces
Stage 3Ballistic Gravity and drag forces
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Three state variables
€
˙ h i = Vi
˙ V i =Ti
mi
− g −1
2ρ airVi Vi
CDAbottle
mi
˙ m i = −ρ water u eiAthroat
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Open items
• Initial conditions– Modeling stage 1
• Thrust (Ti)?
• Exit velocity (uei)?
• Integrating the equations
• Analyzing the results
• Developing a design + rationale
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Modeling the gas expansion
• Work = change in energy
• Adiabatic, quasi-static
• = 1.4 for air
pressure
volume€
p final = pinitialVol initial
Vol final
⎛
⎝ ⎜
⎞
⎠ ⎟γ
€
W = pdVolVol initial
Vol final
∫
€
W =p finalVol final − pinitialVol initial
γ −1
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Thrust from the momentum equation
• Force = time rate of change of momentum
€
Ti = ρwateruei uei Athroat
€
Thrust =mass
time×momentum
mass
€
Thrust = mass flow rate × velocity
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Forward Euler finite difference equations
€
hi+1 = hi + Vi[ ]Δt
Vi+1 = Vi +Timi
− g−1
2ρ airVi Vi
CDAbottlemi
⎡
⎣ ⎢
⎤
⎦ ⎥Δt
mi+1 = mi − ρwaterueiAthroat[ ]Δt
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My spreadsheet