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Edits to SIM_balloon README. #1210

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@ -5,7 +5,7 @@ A hot-air balloon is a lighter than air aircraft consisting of
2. a burner system to heat the air in the envelope, and
3. a basket to carry fuel and passengers.
The motion of the balloon is controlled by changing the temperature of the air within the envelope. As the temperature of the air within the envelope increases, its density, and therefore its mass decreases. When the total mass of the balloon becomes less the mass of the cooler outside air that it displaces, the balloon will ascend.
The motion of the balloon is controlled by changing the temperature of the air within the envelope. As the temperature of the air within the envelope increases, its density, and therefore its mass decreases. When the total mass of the balloon becomes less the mass of the cooler outside air that it displaces, the balloon ascends.
![Balloon Parts](Images/Picture1.png)
@ -57,11 +57,12 @@ The forces acting on our balloon will be those of gravity, buoyancy and aerodyna
To determine the balloon’s motion, we first need to know it’s acceleration. We can determine this by calculating and summing the forces acting on the balloon, and then dividing that sum by the balloons total mass.
<a id=Equation-1></a>
![Equation 1](Images/Equation1.png)
To help us calculate these forces we use the US Standard Atmosphere model described above.
To find the state of the balloon, we integrate acceleration over time to get velocity. We then integrate velocity over time to get position.
To find the state of the balloon, we integrate acceleration and velocity, over time, to get velocity and position.
---
@ -69,15 +70,17 @@ To find the state of the balloon, we integrate acceleration over time to get vel
Applying the acceleration of gravity to Newton’s 2nd Law:
![Equation 1](Images/Equation2.png)
<a id=Equation-2></a>
![Equation 2](Images/Equation2.png)
At sea-level, **g** is around 9.81 m/s2. But as altitude increases **g** decreases. To determine **g** for a given altitude we’ll use the ```US_STD_gravity(altitude)``` function from the atmosphere model library.
At sea-level, **g** is around 9.81 m/s2. But as altitude increases **g** decreases. To determine **g** for a given altitude we use the ```US_STD_gravity(altitude)``` function from the atmosphere model library.
#### Balloon Mass
The total balloon mass is the sum of the fixed mass and the mass of the heated air inside the balloon envelope.
![Equation 1](Images/Equation3.png)
<a id=Equation-3></a>
![Equation 3](Images/Equation3.png)
##### Balloon Fixed Mass
@ -96,31 +99,43 @@ The fixed mass is simply the sum of the balloon component masses:
The mass of the air in the balloon envelope is the product of the volume of the balloon envelope and the density of the heated air within the envelope.
![Equation 1](Images/Equation4.png)
<a id=Equation-4></a>
![Equation 4](Images/Equation4.png)
#### Balloon Volume
To calculate the volume the the balloon, we separate the balloon into two parts: 1) a spherical dome and 2) a cone.
To calculate the volume the the balloon, we separate the balloon into two parts:
1. a spherical dome and
2. a cone.
![Balloon Volume](Images/Picture3.png)
**R** represents the radius of the spherical part. **&#x3B8;** represents angle at which the spherical part of the balloon transitions to the cone. The height (h) for each of the spherical and conical parts is given shown in the diagram. They are each used in their respective volume calculations below.
**R** represents the radius of the spherical dome. **&#x3B8;** represents angle at which the spherical dome transitions to the cone. The height (h) for each of the spherical and conical parts is given shown in the diagram. They are each used in their respective volume calculations below.
The volume of the spherical dome is:
![Equation 1](Images/Equation5.png)
<a id=Equation-5**></a>
![Equation 5](Images/Equation5.png)
The volume of the cone is:
![Equation 1](Images/Equation6.png)
<a id=Equation-6></a>
![Equation 6](Images/Equation6.png)
The total volume of the envelope is the sum of the two component volumes.
![Equation 1](Images/Equation7.png)
<a id=Equation-7></a>
![Equation 7](Images/Equation7.png)
#### Air Density
To calculate air density (𝝆) we use the following form of the Ideal Gas Law.
![Equation 1](Images/Equation8.png)
<a id=Equation-8></a>
![Equation 8](Images/Equation8.png)
* p is just the standard pressure (in pascals) at the balloon's current altitude. Use ```US_STD_pressure(altitude)```.
* R<sub>air</sub> is the specific gas constant for dry air. R<sub>air</sub> = 287.055 J/kg K.
* T is the temperature of the gas in kelvin. The conversion from Celsius to kelvin is T<sub>kelvin</sub> = T<sub>celsius</sub> + 273.15.
* T is the temperature of the gas in kelvin. The conversion from celsius to kelvin is T<sub>kelvin</sub> = T<sub>celsius</sub> + 273.15.
---
@ -128,26 +143,28 @@ To calculate air density (𝝆) we use the following form of the Ideal Gas Law.
Buoyancy is a force on an object, that opposes gravity, by a fluid within which it’s immersed. This force is equal to the mass of the displaced fluid times the acceleration of gravity.
![Equation 1](Images/Equation9.png)
<a id=Equation-9></a>
![Equation 9](Images/Equation9.png)
Here, we can do that same calculation for air mass as before, using
Eq#4, but this time with 𝝆 = US_STD_density(altitude), which assumes U.S Standard temperature at the given altitude.
[Equation #4](#Equation-4), but this time with 𝝆 = ```US_STD_density(altitude)```, which assumes U.S Standard temperature at the given altitude.
---
### Force of Drag
As the balloon moves upward, or downward, it will be subjected to an atmospheric drag force. Drag is a function of the balloon’s shape [
presented by the coefficient of drag (Cd)], the density of the air (𝝆), and the cross-sectional area (A) perpendicular to the velocity (v). The drag force points in the opposite direction as the velocity.
As the balloon moves upward, or downward, it will be subjected to an atmospheric drag force. Drag is a function of the balloon’s shape [ represented by the coefficient of drag (Cd) ], the density of the air (𝝆), and the cross-sectional area (A) perpendicular to the velocity (v). The drag force points in the opposite direction as the velocity.
![Equation 1](Images/Equation10.png)
<a id=Equation-10></a>
![Equation 10](Images/Equation10.png)
For our balloon, we’ll assume Cd = 0.5, the coefficient of drag for a sphere.
Since the balloon will be moving through our “standard atmosphere” 𝝆 = ```US_STD_density(altitude)```.
We’ll calculate the cross-sectional area (A) from the radius of the spherical portion of our balloon.
![Equation 1](Images/Equation11.png)
<a id=Equation-11></a>
![Equation 11](Images/Equation11.png)
## Graphics