1

Systems Problem 4

Lecture Notes

payload

2

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

3

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

4

Three state variables

€

˙ h i = Vi

˙ V i =Ti

mi

− g −1

2ρ airVi Vi

CDAbottle

mi

˙ m i = −ρ water u eiAthroat

5

Open items

• Initial conditions– Modeling stage 1

• Thrust (Ti)?

• Exit velocity (uei)?

• Integrating the equations

• Analyzing the results

• Developing a design + rationale

6

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

7

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

8

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

9

My spreadsheet