91Ó°ÊÓ

Understanding how to calculate the volume of a cylinder is vital in many thermodynamic problems. To start, we need to know the formula, which is given by the base area of the cylinder multiplied by its height. In this exercise, the base is a circle, so its area is calculated using \(\text{Base Area} = \pi r^{2}\) where \( r \) is the radius.

Given the diameter is 4 meters, we can find the radius by dividing the diameter by 2, resulting in \(\text{radius} = 2\text{ meters}\). Next, we can calculate the base area: \( \text{Base Area} = \pi \times (2)^{2} = 4\pi \ text {square meters}\).

Finally, the volume of the cylinder is computed by multiplying this base area by the height (or length) of the tank, which is given as 25 meters: \(V = 4\pi \times 25 = 100 \ pi \ text {cubic meters}\). Thus, the total volume of the tank is \(100\ \ pi \text{ cubic meters})\).

The mass flow rate is a crucial concept because it tells us how much mass is moving through a given point per unit time. In our example, the mass flow rate is given as 10 kg/s. This means that each second, 10 kilograms of propane enter the tank.

Using the mass flow rate allows us to easily determine the total amount of mass that has entered the tank over a period. Since 10 kg of propane flows each second, knowing the total mass required will help us calculate the time needed to fill the tank, which we discuss in later sections.

Density is the mass per unit volume of a substance. For propane at given conditions (20°C and 9 bar), the density is approximately 493 kg/m³. To calculate the mass of propane required to fill a tank, we use the formula: \( \text{Mass} = \text{Density} \times \text{Volume} = 493 \times 100 \pi = 49300\ \ pi \ text{ kg}\).

This means that the propane needed to fill the tank is about 49300\ \ pi \ kilograms. Density thus serves as a bridge between volume and mass, helping us understand material properties and behavior under specified conditions.

Calculating time in thermodynamics problems often means determining how long a process will take. Here, we use the mass flow rate and total mass to find the time required to fill the tank. The formula is: \(\text{Time} = \frac{\text{Mass}}{\text{Flow Rate}}\)

Substituting the values, we get: \( \frac{49300 \ pi}{10 } \ = 4930\text{pi}\ \text{ seconds}\). To convert seconds into minutes, since there are 60 seconds in a minute, we divide: \( \text{Time in minutes} = \frac { 4930 \ pi}{60} = 82.17 \pi ≈ 258.19 \ \text{minutes}\).

Thus, it will take approximately 258.19 minutes to fill the tank with propane. This step-by-step approach helps ensure a comprehensive understanding of the time calculation process.