On the Ascensional Rate of a Balloon

Abstract

Recently, it is frequently observed that the ascensional rate of a balloon for aerological observation gradually diminishes with increasing altitude. However, the cause of this effect is not clear as yet. From data of double theodolite observations made by Dines, Horiguti, Oishi, and Ishimaru, the number of ascents being some 1, 800 in total, the author computed the drag coefficient C_D and the corresponding Reynolds' number R_e by means of the following equations:
Here ρ=density of the surrounding air; μ=viscosity of air; α=1-σ/ρ;σ=density of inner gas (hydrogen); υ=ascensional rate; L=free lift; Q=total lift; and g=acceleration of gravity.
Because the available data of v from the referred materials were the average values for the layer of 0_??_3km altogether, the values of ρ and μ for the height of 1.5km were employed in this computation. The results are plotted in Fig. 1, in which C_D gradually decreases with increasing Re, and no indication of steep change in C_D can be seen for Reynolds' numbers up to 6×10⁵. The obtained drag curve seems to join rather smoothly with that by Bacon and Reid (1922), who made experiments dropping various spheres from an airplane.
When a balloon ascends, the Reynolds' number decreases with increasing altitude, so the drag coefficient increases. Whether the ascensional rate increases or decreases with height depends on the relation between C_D and Re. Differentiating (1) and (2), and replacing μ∝T^5/6, we obtain: where g/R=autoconvection gradient, and γ=-dT/dh.
If we put n=dlnC_D/dlnRe, and eliminate Re and C_D in (3) and (4), we have:
This means that the rate of increase of v with height is determined by the slope of C_D curve on double log-paper, n, and the temperature lapse rate γ. The loci of 1/v dv/dh in % per km as a function of n and γ are shown in Fig. 2, where T=300°K is assumed. The area surrounded by hatched border indicates the region of decreasing ascensional rate. It is shown by this figure that in usual conditions the decrease of ascensional rate is rather small. Observed large values of decrease appear to be due to other causes, say, the effect of radiation on balloons, the excess pressure in balloons, the vertical currents in the atmosphere, etc.
It is also shown theoretically, that the steep part of C_D curve where n<-2 can not be realized in free ascension or fall of a sphere, because any point on such a part of C_D curve is unstable, when the resistance of the sphere should be kept constant.