(Helium) Equations

(Helium) Equations

Equations, Nomenclature & Drag

Spheres

Surface = (4) pi (radius squared)

Volume = (4/3) pi (radius cubed)

-- Example --

4 foot diameter balloon; surface = (4) pi (4) = 50.3 square feet

Volume = (4/3) pi (8) = 33.5 cubic feet

Gross Lift = 2.1 pounds [63.8 pounds lift per 1,000 cu ft of helium at sea level]

(Travel) Nomenclature

Unlike a ship or a plane, the balloon units have no right or left sides, no “forward” (bow or nose) or “rear” (stern or tail). Therefore new and appropriate nomenclature is needed. The following standard nomenclature is to be used when under way.

1. The unit of length is the width of a balloon (30 ft) which is called one “Bag”. 60 feet is 2 bags (etc.).

2. Distances between units are given in Bags.

3. Angles are given off of the northern axis (north pole) clockwise (12 at the top, 6 at the bottom 3 to the right, 9 to the left).

4. One main unit is called “A” and the other is called “U”.

5. The two T-balloons are named “T1” (for the T1 T-balloon) or “T2” (for the T2 T-balloon).

6. The standard distances are: LALU (bearing from A to U) or LULA (bearing from U to A). LA1 (bearing from A to T1), LA2 (bearing from A to T2), LU1 (bearing from U to T1), LU2 (bearing from U to T2), T12 (bearing from T1 to T2), T21 (bearing from T2 to T1).

7. Other designations are Q1, Q2, Q3...(anchorage locations). The observer is at the location which starts the description.

8. Examples of measurements (estimates) are provided as follows:

“I have a LALU of 6 at 3” (means that measuring from the A unit, the distance to the U unit is 180 feet due East.) [Remember, the observer may only be able to see a small part of another unit due to fog or other obstructions]

”LA1 and LA2 are 20 at 6” (means that both T balloons are 600 feet to the south of unit A.)

“LA1 is 20 at 6 PLUS” and “LA2 is 20 at 6 MINUS” (Both T balloons are 600 feet south of the A unit, but T1 is slightly west of T2.)

“The wind is 10 at 11” (means the wind is 10 knots and is coming from the north north west.)

"AQ3 is 25 at 7" (means Anchorage Q3 is 750 feet south south west of Unit A.)

"UQ1 is 10 at 11" (means Anchorage Q1 is 300 feet north north west of Unit U.)

"AQ2 is zero" (means Anchorage Q2 is under Unit A.)

"T1U is 40 at 6" (means the person on (under) the T1 T-balloon is 1200 feet north of Unit U.)

"T2Q3 is 5 at 9" (means the person on (under) the T2 T-balloon is 150 feet east of Anchorage Q3.)

"Q4A is 10 at 3" (means the person at Anchorage (site) Q4 is 300 feet west of Unit A.)

"Suggest an LU shot of 40 at 10" (proposes a grapple shot of 1200 feet thrown west north west from unit U.)

Rough measurement

Put your hand at arm’s length. A fingernail is about 1/2 inch wide and your fist is about 3 inches wide. A balloon as big as your fist is about 300 feet (10 Bags) away. If it is as big as your fingernail, it is about 1800 feet (60 Bags) away. A half mile is about 88 Bags away (5280/30 = 176 Bags or one mile).

Drag coefficient and Reynolds number

The drag force, FD, depends on the velocity of the sphere, v, the diameter of the sphere, D, the density of the fluid, rho, and the viscosity of the fluid, mu. The powerful technique of dimensional analysis shows that these five variables can be combined into two variables without any loss of ability to describe the drag force.

These two variables, are the drag coefficient, CD = FD/(1/2 rho v^2 A), and the Reynolds number, defined by Re = rho v D/mu, where A is the cross-sectional area of the sphere. Both CD and Re are dimensionless. It can be shown that any model for the drag force can be expressed as a relationship involving only these two dimensionless parameters.

[Adapted from UMAP Module 712, The Drag Force on a Sphere (UMAP Journal, Volume 12, no. 1, Spring 1991, pp. 47-80.), by H. Edward Donley, Copyright 1991, Consortium for Mathematics and its Applications.]

Conclusion: for most situations, the Reynolds number for a single balloon (30 feet in diameter) will be less than 0.5 so the drag will be proportional to the wind velocity (up to about 10 knots; for silicon-dacron fabric). For a steady wind velocity of 10 knots, the tension on a rope restraining it (the drag) will be about 42 pounds.

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