Question

Consider two identical fans, one at sea level and the other on top of a high mountain, running at identical speeds. How would you compare \((a)\) the volume flow rates and (b) the mass flow rates of these two fans?

Short answer

Question: Compare the volume flow rates and mass flow rates of two identical fans operating at the same speed, but at different altitudes (one at sea level and the other on a high mountain). Answer: The volume flow rates of both fans will be the same, while the mass flow rate of the fan at sea level will be greater than that of the fan on the high mountain.

Step-by-step solution

(a) Comparing the Volume Flow Rates

Since both fans are identical, have the same cross-sectional area (A), and are running at the same speed, they will have the same air velocity (v) passing through them. Therefore, their volume flow rates (Q) will be equal. The volume flow rates of both fans can be determined using the formula: $$Q = A \times v$$ Since A and v are the same for both fans, their volume flow rates will be the same.

(b) Comparing the Mass Flow Rates

To compare the mass flow rates, we will need to consider the air density at sea level and on the high mountain. The air density decreases with an increase in altitude, i.e., the air density at sea level ($$\rho_1$$) is higher than that on the high mountain ($$\rho_2$$). The mass flow rates of each fan can be determined using the formula: $$m = Q \times \rho$$ For the fan at sea level, the mass flow rate ($$m_1$$) is: $$m_1 = Q \times \rho_1$$ For the fan on the high mountain, the mass flow rate ($$m_2$$) is: $$m_2 = Q \times \rho_2$$ Since we know $$\rho_1 > \rho_2$$ and their volume flow rates Q are the same, we can conclude that $$m_1 > m_2$$, i.e., the mass flow rate of the fan at sea level is greater than that of the fan on the high mountain.

Key concepts

Volume Flow Rate

When we talk about volume flow rate in fluid mechanics, we are referring to the volume of fluid that passes through a cross-section per unit time. It's often denoted by the symbol 'Q' and is crucial for understanding how fluids like air and water move through systems such as pipes, ducts, or even across fan blades.

In the provided exercise, we've seen that the volume flow rates of two identical fans are the same because the fans have the same dimensions and operate at the same speed. This means the fans are moving the same volume of air per unit time, regardless of the location.

It's essential to distinguish volume flow rate from mass flow rate, which we will discuss in the next section. This is because volume flow rate does not take into account the density of the fluid which can change with temperature, pressure, and specifically altitude as shown in this exercise.

Air Density

Air density, symbolized as \( \rho \), is a measure of how much mass of air is present in a given volume. This value is crucial when transitioning from discussing volume flow rate to mass flow rate of a gas like air. At sea level, air density is at its highest due to the weight of the air above compressing it.

The higher the altitude, the less air there is above and, consequently, the lower the air density. This decrease in air density as altitude increases is an important factor in a variety of applications, from the performance of aircraft to the mass flow rate of air through our two fans in different locations.

The relationship between volume flow rate and air density directly affects the mass flow rate, as seen in the equation \( m = Q \times \rho \), which plays a pivotal role in understanding the differences in fan operation at sea level and at high altitudes.

Altitude Effects on Air Properties

Altitude significantly influences the properties of air, including pressure, temperature, and density. As one ascends to higher altitudes, the atmospheric pressure decreases due to a thinner layer of air above. This change in pressure also brings a drop in temperature, following the principle of adiabatic cooling.

The reduction in air density at higher altitudes is a direct result of the lower atmospheric pressure and temperature. This has a wide range of implications, such as lower oxygen levels for breathing and implications for the design and operation of machinery, including fans and aircraft engines.

Understanding how altitude affects air properties is crucial for predicting and calculating the performance of a device. In the context of the exercise, it explains why the mass flow rate of the mountain-top fan is lower than that of the sea-level fan—demonstrating how environmental conditions play a significant role in the operation of systems involving the flow of fluids.