AAE 450 Spring 2008

William Yeong Liang Ling2/27/2008Propulsion

Analysis of Balloon Rise Time

Propulsion

AAE 450 Spring 2008Propulsion

Lift

Mass

Drag 0 1000 2000 3000 4000 5000 60004.5

5

5.5

6

6.5

7Change in Reynolds Number over time

Time (s)

log1

0(Re

)

0 1000 2000 3000 4000 5000 60000

1000

2000

3000

4000Change in balloon drag over time

Time (s)

Drag

(N)

0 1000 2000 3000 4000 5000 60000

0.01

0.02

0.03

0.04

0.05Change in balloon acceleration over time

Time (s)

Acce

lera

tion

(m/s

2 )

Determination of rise timeAssumptions• Constant sphere• Constant CD = 0.2• Barometric formula• Kinematic viscosity

variation with temperature• Constant acceleration

over timesteps of 1 second

AAE 450 Spring 2008Propulsion

1 hour 36 minutes to reach 30km.

• Compares well with high altitude balloon rise times

• Final velocity of 19.7m/s upwards

• In reality, balloon will either rupture or oscillate about 30km

Determination of rise time

0 1000 2000 3000 4000 5000 60000

5

10

15

20

25

30

35X: 5741Y: 30.01

Change in balloon altitude over time

Time (s)

Altit

ude

(km

)

Future work

• Determine maximum drift radius due to wind gusts

AAE 450 Spring 2008Propulsion

0 0.5 1 1.5 2 2.5 3

x 104

2.95

3

3.05

3.1

3.15

3.2

3.25

3.3

3.35x 10

4

Altitude (meter)

Lift

ing

Forc

e (N

ewto

ns)

Lifting Force of the Balloon

Thanks to Jerald Balta for modifying the balloon code to output this.

AAE 450 Spring 2008Propulsion

0 0.5 1 1.5 2 2.5 3

x 104

10

20

30

40

50

60

70

80

90

Altitude (meter)

Diam

eter

of B

allo

on (m

)Change in diameter with altitude

Thanks to Jerald Balta for modifying the balloon code to output this.

AAE 450 Spring 2008Propulsion

0 1000 2000 3000 4000 5000 60000

2

4

6

8

10

12

14

16

18

20Change in balloon velocity over time

Time (s)

Velo

city

(m/s

)

X: 5741Y: 19.7

AAE 450 Spring 2008Propulsion

function Output = Balloon_Rise(GLOW) close all%Timestep = 1 second % This is fixed within the code, i.e. dt = 1Altitude = 0;g = 9.80665; %% SUMMARY % This function determines the rise time of the balloon to an altitude of% 30,000m. As a bonus, it also determines the drag, Reynolds Number,% acceleration and velocity experienced by the balloon over the rise time.%% x = 0;v = 0;i = 1;t = 0; while x < 30000 Variables = Balloon_Model(GLOW, x);Force = Variables(1);Force = Force - GLOW.*g;Volume = Variables(2);Diameter = Variables(3); [Density_Air Pressure_Air Temperature_Air] = Barometric_Formula(x); Drag = 0.2.*0.5.*Density_Air.*v.^2.*(pi./4).*Diameter^2; if Drag > Force Drag = Force;end

AAE 450 Spring 2008Propulsion

Acceleration = (Force - Drag)./GLOW; x = x + v + 0.5.*Acceleration;v = v + Acceleration; Altitude(i) = x;Velocity(i) = v;Acceleration_Grid(i) = Acceleration;Drag_Grid(i) = Drag;t = t + 1;Time(i) = t; % Dynamic Viscosity determined by a best fit curve by Ierardi, James.Dynamic_Viscosity = (-1.1555.*10^-14).*Temperature_Air^3 + (9.5728.*10^-11).*Temperature_Air^2 + (3.7604.*10^-8).*Temperature_Air - (3.4484.*10^-6);Re(i) = (Density_Air.*v.*Diameter)./(Dynamic_Viscosity); i = i + 1; end figure(1)plot(Time,Altitude./1000);title('Change in balloon altitude over time')xlabel('Time (s)')ylabel('Altitude (km)')

AAE 450 Spring 2008Propulsion

function [Density_Air Pressure_Air Temperature_Air] = Barometric_Formula(Altitude) %% Fixed Constraints g = 9.80665; % Gravitational acceleration, assumed to be constant [m/s^2]Molar_Air = 0.0289644; % Molar mass of Earth's air [kg/mol]R = 8.31432; % Universal gas constant [N·m/(mol·K)] %% Density of Air using the Barometric Formula if Altitude < 11000 Density_b = 1.2250; Temperature_b = 288.15; Lapse_b = -0.0065; Height_b = 0; Pressure_b = 101325; elseif Altitude < 20000 Density_b = 0.36391; Temperature_b = 216.65; Lapse_b = 0; Height_b = 11000; Pressure_b = 22632.1; elseif Altitude < 32000 Density_b = 0.08803; Temperature_b = 216.65; Lapse_b = 0.001; Height_b = 20000; Pressure_b = 5474.89; elseif Altitude < 47000 Density_b = 0.01322; Temperature_b = 228.65; Lapse_b = 0.0028; Height_b = 32000; Pressure_b = 868.019;

AAE 450 Spring 2008Propulsion

elseif Altitude < 51000 Density_b = 0.00143; Temperature_b = 270.65; Lapse_b = 0; Height_b = 47000; Pressure_b = 110.906; elseif Altitude < 71000 Density_b = 0.00086; Temperature_b = 270.65; Lapse_b = -0.0028; Height_b = 51000; Pressure_b = 66.9389; else Density_b = 0.000064; Temperature_b = 214.65; Lapse_b = -0.002; Height_b = 71000; Pressure_b = 3.95642;end if Lapse_b == 0 Density_Air = Density_b.*exp((-1.*g.*Molar_Air.*(Altitude - Height_b))/(R.*Temperature_b)); %[kg/m^3]else Density_Air = Density_b.*((Temperature_b./(Temperature_b + (Lapse_b.*(Altitude - Height_b))))^(((g.*Molar_Air)./(R.*Lapse_b)) + 1)); %[kg/m^3]end %% Pressure of air using barometric formula if Lapse_b == 0 Pressure_Air = Pressure_b.*exp((-1.*g.*Molar_Air.*(Altitude - Height_b))/(R.*Temperature_b)); %[Pa]else Pressure_Air = Pressure_b.*((Temperature_b./(Temperature_b + (Lapse_b.*(Altitude - Height_b))))^(((g.*Molar_Air)./(R.*Lapse_b)))); %[Pa]end %% Temperature of air using ideal gas law Temperature_Air = (Molar_Air.*Pressure_Air)./(Density_Air.*R); %[K]