By Ludwig Prandtl, O. G. Tietjens, Engineering
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We get, by analogy with the previous analysis, d'Yo 'Yo . dp = kop Substituting the expressions found for d'YA/dp and d'Yo/dp in (7), we obtain _dR = V'YA('YA _ 'Yo). 1. 1. 1. 1. L. hon hon (8) As an example of the practical application of this expression we shall find out how high a balloon will ascend in the taut state if a given quantity of ballast is thrown out. For equilibrium we have R = 0. If a quantity of ballast of weight ~Q is thrown out, then, at that instant, ~R = -~Q. By expanding h as a function of R in a Taylor's series and using only the first term, we find dh ~h = dR~R, or, since at the moment of releasing the ballast and since from (8) we can write dh dR ~R = -~Q, nh« = -Q' we obtain M = h~t~Q.
Equation (4) states: V1PI h = VI(PI - p) = - . 100 1 The assumption that v = constant is to a large extent true with liquids, so that in agreement with reality we can write h2 - hi = PI "Y P2 (equation of hydrostatic pressure). APPLICATION OF THE PRESSURE EQUATION 31 With the Eqs. (5) and (6) this becomes ho = 100 = 260 ft. h Thus in a uniform atmosphere at a surface temperature of O°C the pressure decreases every 260 ft by 1 per cent. 12. -We shall now suppose that the temperature is constant throughout the whole atmosphere: T const.
Function of the time, a curve of the kind shown in Fig. 32 will be obtained. The thick parts of the curve correspond to the taut states while for the rest of the time the balloon was in the limp state. 23. -We shall first investigate the stability under adiabatic conditions (no change of temperature by radiation) and afterward determine the influence exerted by changes of temperature, caused by emission or absorption of heat. In both cases we consider first the taut, then the limp state. Equation (5) in combination with (2) gives R = V('YA - 'Yo) - Q, (6) where Q is the total weight (Q = QI + Q2).