# Model of the Balloon Trajectory

# Constraints

- We need to keep the ascension rate below 5 m/s. Any ascent rate over 5 m/s will increases exponentially the risk of the balloon creating unsteady oscillating drag in its wake as it ascends though the atmosphere. This will cause the payload to swing quite violently and could cause damage to the payload and balloon.
- A simple model to calculate a more precise limit for the ascent rate is this:

$v_{z}=\sqrt{\frac{8Rg}{3c}(1-\frac{3m_{total}}{4 \rho \pi R^3})}$ where $v_{z}$ is the ascent rate, $m_{total}$ is the total mass (balloon envelope + lifting gas + payload), R is the balloon effective radius ( this value increases while the balloon ascend) and $\rho$ is the density of the air (also changes depending on the altitude.

- To calculate an upper boundary of the ascension rate we use “extremes” values for R and $\rho$, a more precise model is gonna be implemented based on Ascent Rate in order to maximizes the efficiency.
- The ascension time.
- The amount of helium.
- Type of balloon (Pawan 1200g)

# Estimated Practical Values

To calculate the amount of Helium that the balloon needs to carry our payload, first we are gonna use the *Balloon Performance Calculator* using as variable the positive lift that we are gonna add, in order to obey the constraints.

For a payload of 1800 grams, keeping an ascent rate of $4.5578 m/s$ with an estimated burst altitude of $30723 \pm 1000 m$ and an ascent time of $110 \pm 5 minutes$, we will need a positive lift of $1000 grams$.

This means we will have to fill the balloon with $3845 \pm 250 liters$ of $Helium$.

The estimations are an average of the the results provided by the *Balloon Performance Calculator* and the *Balloon Burst Calculator*.

## Online Tools

- Balloon Performance Calculator http://tools.highaltitudescience.com/# :
**Inputs:**- Balloon size (grams)
- Payload Weights (grams)
- Positive Lift (grams)

- Balloon Burst Calculator http://habhub.org/calc/
**Inputs**- Payloads Mass (grams)
- Balloon Mass (grams)
- Target Ascent rate or Target burst altitude (it's not recommended to give as input the Target burst altitude, because in this way we're unable to control the ascent rate, thus our payload can get damage or even destroyed during the ascension).

- CUSF Landing Predictorhttp://bitmorse.com/habhubpredict/
**Main Inputs:**- Ascent rate
- Burst altitude

References:

- Modeling the ascent of sounding balloons: derivation of the vertical air motion

A. Gallice1, F. G. Wienhold1, C. R. Hoyle1,*, F. Immler2, and T. Peter1