/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 100 A helium balloon is to lift a pa... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 100

A helium balloon is to lift a payload to an altitude of \(40 \mathrm{km},\) where the atmospheric pressure and temperature are 3.0 mbar and \(-25^{\circ} \mathrm{C},\) respectively. The balloon skin is polyester with specific gravity of 1.28 and thickness of 0.015 \(\mathrm{mm} .\) To maintain a spherical shape, the balloon is pressurized to a gage pressure of 0.45 mbar. Determine the maximum balloon diameter if the allowable tensile stress in the skin is limited to \(62 \mathrm{MN} / \mathrm{m}^{2}\). What payload can be carried?

Short Answer

Expert verified
The maximum diameter of the balloon is obtained from step 1. The helium balloon can carry a payload of the amount calculated in step 3.

Step by step solution

01

Calculation of Maximum Diameter

The balloon skin is subject to tensile stress, which is given by \( \sigma = Pr/2t \), where P is the pressure difference across the skin, r is the radius of the balloon, and t is the thickness of the skin. This is rearranged to \( r = 2t\sigma / P \). The pressure difference is the gage pressure of the balloon, which is 0.45 mbar. Given that the allowable tensile stress in the skin (\( \sigma \)) is 62 MN/m虏, the thickness of the skin (t) is 0.015 mm, and the pressure difference across the balloon skin (P) is 0.45 mbar, the maximum radius of the balloon is determined. Then, the diameter is found by multiplying the radius by 2.
02

Calculation of Volume and Lift Force

The volume of the balloon is found using the formula \( V= 4/3\pi r^3 \) where r is the radius determined in the previous step. Then, the number of moles of helium inside the balloon is found using the ideal gas law \( PV = nRT \). This is rearranged to find n, which is the number of moles, using \( n = PV/RT \), where P is the pressure inside the balloon (the atmospheric pressure at 40 km altitude plus the gage pressure of the balloon), V is the volume calculated earlier, R is the ideal gas constant, and T is the temperature of the air at 40 km. Since the lift force is given by the weight of the displaced air, and the weight is the product of the mass and gravity, we find the mass of the air displaced by the balloon using the ideal gas law, given that the air is at 40 km altitude. The lift force is then calculated by multiplying the mass of the displaced air by the acceleration due to gravity.
03

Calculation of Payload

The weight of the balloon skin is calculated using the formula \( m = \rho V \), where \( \rho \) is the density of the balloon skin material and V is the volume of the balloon skin. The volume of the balloon skin is found using the formula \(V = 4\pi r^2t\), where r is the radius and t is the thickness of the skin. With this weight and the weight of the helium gas inside the balloon, the payload that can be carried by the balloon is found by subtracting these weights from the lift force.

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