Question

A spherical hot-air balloon is initially filled with air at \(120 \mathrm{kPa}\) and \(20^{\circ} \mathrm{C}\) with an initial diameter of \(5 \mathrm{m}\). Air enters this balloon at \(120 \mathrm{kPa}\) and \(20^{\circ} \mathrm{C}\) with a velocity of \(3 \mathrm{m} / \mathrm{s}\) through a \(1-\mathrm{m}\) diameter opening. How many minutes will it take to inflate this balloon to a \(15-\mathrm{m}\) diameter when the pressure and temperature of the air in the balloon remain the same as the air entering the balloon?

Short answer

Answer: To determine the time required to inflate the balloon, we first find the initial and final volume of the balloon, which are \(V_{initial}\) and \(V_{final}\), respectively. Next, we calculate the volume flow rate (Q) of the air entering the balloon. Finally, we find the time required by dividing the change in volume (∆V) by the volume flow rate (Q). The calculated time needed to inflate the balloon to the desired diameter is then converted from seconds to minutes.

Step-by-step solution

Calculate the initial and final volume of the balloon

To determine the time required to inflate the balloon to the desired volume, first, we need to find the initial and final volume of the balloon. To do this, we can use the following volume formula for spheres: \(V_{sphere} = \frac{4}{3} \pi r^{3}\) where \(V_{sphere}\) is the volume of the sphere and \(r\) is the radius. The radius of the balloon is half of the diameter. Initially, the diameter of the balloon is \(5 \mathrm{m}\), so the initial radius is \(2.5 \mathrm{m}\). \(V_{initial} = \frac{4}{3} \pi (2.5)^{3}\) And the final diameter is \(15 \mathrm{m}\), so the final radius is \(7.5 \mathrm{m}\). \(V_{final} = \frac{4}{3} \pi (7.5)^{3}\) Calculate the initial and final volume.

Calculate the volume flow rate of air entering the balloon

Now, we need to calculate the volume flow rate (Q) of the air entering the balloon. This can be done using the following formula: \(Q = A \cdot v\) where \(A\) is the cross-sectional area of the opening and \(v\) is the velocity of the air entering the balloon. The diameter of the opening is \(1 \mathrm{m}\), so the radius is \(0.5 \mathrm{m}\). The area can be calculated using the formula: \(A = \pi r^{2}\) \(A = \pi (0.5)^{2}\) Now, we can calculate the volume flow rate: \(Q = A \cdot v\) The velocity of the air entering the balloon is \(3 \mathrm{m} / \mathrm{s}\). \(Q = \pi (0.5)^{2} \cdot 3 \mathrm{m/s}\) Calculate the volume flow rate.

Calculate the time required to inflate the balloon

Now that we have the initial and final volume of the balloon and the volume flow rate of air entering the balloon, we can calculate the time required to inflate the balloon. First, find the change in volume by subtracting the initial volume from the final volume: \(\Delta V = V_{final} - V_{initial}\) Next, use the volume flow rate to find the time required to inflate the balloon: \(time = \frac{\Delta V}{Q}\) Substitute the known values and calculate the time in seconds. Finally, convert the time from seconds to minutes. The time needed to inflate the balloon to the desired diameter in minutes is the answer.