The force due to friction or drag $ F_D = C_D \frac{\rho}{2} A D_z $ (This assumes that there is no vertical motion of the air in which the balloon system is floating. We shall later consider the case where an atmospheric force is causing vertical motion of the air.)
Where:
- $ \rho $ = mass density of the air surrounding the balloon system
- $ A $ = projected area of the balloon on a plane perpendicular to the relative velocity
- $ D_z $ = vertical velocity of the balloon system (Velocity in the direction of greater altitude is considered positive.)
- $ C_D $ = a coefficient of drag, dependent on Reynolds number $ N_R = \frac{D_z d \rho}{\mu} $ where:
- $ d $ = diameter of sphere (ft.)
- $ \rho $ = mass density of surrounding fluid ( $ \frac{\text{lb. sec.}^2}{\text{ft.}^4} $ )
- $ \mu $ = viscosity of surrounding fluid ( $ \frac{\text{lb. sec.}}{\text{ft.}^2} $ )
A plot of drag coefficient against Reynolds number for a sphere is shown in Figure 27.
[figure]
Figure 27. Drag coefficient vs. Reynolds Number, for sphere.
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