Question

The ventilating fan of the bathroom of a building has a volume flow rate of \(30 \mathrm{L} / \mathrm{s}\) and runs continuously. If the density of air inside is \(1.20 \mathrm{kg} / \mathrm{m}^{3},\) determine the mass of air vented out in one day.

Short answer

Answer: The mass of air vented out in one day is 3,110.4 kg.

Step-by-step solution

Convert the volume flow rate to \(\mathrm{m}^{3}/s\)

First, we need to convert the volume flow rate from \(\mathrm{L/s}\) to \(\mathrm{m}^3/\mathrm{s}\). We can do this using the conversion factor: \(1 \mathrm{L} = 0.001 \mathrm{m}^3\). Volume flow rate = \(30 \mathrm{L/s} \times \frac{0.001 \mathrm{m}^3}{1 \mathrm{L}} = 0.03 \mathrm{m}^3/\mathrm{s}\)

Calculate the mass flow rate

Now we can use the density of the air to find the mass flow rate. We can do this using the formula: Mass flow rate = Volume flow rate × Density Mass flow rate = \(0.03 \mathrm{m}^3/\mathrm{s} \times 1.20 \mathrm{kg}/\mathrm{m}^3 = 0.036 \mathrm{kg/s}\)

Convert one day to seconds

In order to find the total mass of air vented out in one day, we need to convert one day into seconds. 1 day = 24 hours × 60 minutes/hour × 60 seconds/minute = 86,400 seconds

Calculate the mass of air vented out in one day

Finally, we can find the mass of air vented out in one day by multiplying the mass flow rate by the total number of seconds in one day. Mass of air vented out in one day = Mass flow rate × Total seconds in one day Mass of air vented out in one day = \(0.036 \mathrm{kg/s} \times 86,400 \mathrm{s}\) Mass of air vented out in one day = \(3,110.4 \mathrm{kg}\) So, the mass of air vented out in one day is \(3,110.4 \mathrm{kg}\).

Key concepts

Volume Flow Rate

Volume flow rate is a measure of how much fluid (in this case, air) passes through a given surface per unit of time. It's typically measured in liters per second \(\text{L/s}\) or cubic meters per second \(\text{m}^3/s\). In thermodynamics and engineering, it's a fundamental quantity for analyzing ventilation systems like the bathroom fan in our exercise. To understand the movement of air or any fluid within a system, the volume flow rate provides essential information about the system's capacity and performance. It's crucial in determining how long it will take a certain amount of fluid to fill up or empty a space, which is exactly the problem we solved when calculating the mass of air vented out from the bathroom.

In practical terms, for the bathroom fan moving air at 30 L/s, this translates to a continuous replacement of air, ensuring that humidity and odors are expelled efficiently. By knowing the volume flow rate, we can proceed to determine the mass flow rate when the density of the medium is known.

Air Density

Air density, the mass per unit volume of Earth's atmosphere, is another critical factor in calculating mass flow rate. It's usually measured in kilograms per cubic meter \(\text{kg/m}^3\). This property of air is influenced by temperature, pressure, and humidity. Warmer air tends to be less dense than cooler air, and dry air tends to be less dense than humid air.

In our exercise, we used the air density value of 1.20 \(\text{kg/m}^3\), which is approximately the density of air at room temperature and atmospheric pressure. This value is crucial because it links the volume flow rate (how much space the air occupies) with the mass flow rate (how much the air weighs). Understanding the density of air allows us to convert easily from a volume-based measurement to a mass-based measurement, enabling the performance calculations of ventilation systems.

Unit Conversion

Unit conversion is a fundamental aspect of many scientific calculations, including those in thermodynamics. It involves converting a quantity expressed in one set of units to another set of units without altering the quantity's value. In our exercise, we converted the volume flow rate from liters per second to cubic meters per second \(30 \text{L/s} = 0.03 \text{m}^3/s\) using the conversion factor: \(1 \text{L} = 0.001 \text{m}^3\).

Such conversions are essential for consistency in equations and computations, especially when combining measurements from different systems, like the metric system and the imperial system. Mastering unit conversion is critical because inaccurate conversions can lead to erroneous results, thereby affecting the outcomes of engineering projects and scientific research.

Thermodynamics Principles

Thermodynamics is the study of the relationships between heat, work, temperature, and energy. The principles of thermodynamics guide the behavior of substances under various conditions and are fundamental in understanding and designing a plethora of machines and processes, including heating, ventilation, and air conditioning (HVAC) systems as illustrated in our exercise.

The calculation of the mass flow rate of air vented out by the fan over a day involves these thermodynamic principles. The ability of the fan to continuously cycle air at a given volume flow rate is directly tied to the energetic performance of the system. Also, understanding that energy can neither be created nor destroyed but can change forms (first law of thermodynamics) informs us that the work done by the fan is transferred to the movement of the air, which keeps the environment inside the building habitable and comfortable.