Question

Design a \(1200-\mathrm{W}\) electric hair dryer such that the air temperature and velocity in the dryer will not exceed \(50^{\circ} \mathrm{C}\) and \(3 \mathrm{m} / \mathrm{s}\), respectively.

Short answer

Answer: The maximum air volume flow rate for the hair dryer design is 5.88 x 10⁻³ m³/s. The power allocated for the heating element is 840 W, and for the fan, it is 360 W.

Step-by-step solution

Determine the maximum air volume flow rate

To ensure the air velocity does not exceed 3 m/s, we need to find the maximum volume flow rate (Q) of air that can pass through the dryer. We can use the following equation: \(Q = A \times V\) Where \(Q\) is the volume flow rate in cubic meters per second (m³/s), \(A\) is the cross-sectional area of the hair dryer's nozzle, and \(V\) is the air velocity in meters per second (m/s). Since we are not given the cross-sectional area of the nozzle, let's first assume a reasonable size and shape for the nozzle. In this case, let's assume a circular nozzle with a diameter of 5 cm (0.05 m). The area of the nozzle can be calculated using the formula for the area of a circle: \(A = \pi r^2\) Where \(r\) is the radius of the nozzle. In our case, the radius is half the diameter, which is equal to 0.025 m. Plugging this value into the formula, we get: \(A = \pi (0.025)^2 = 1.96 \times 10^{-3} m^2\) Now, we can calculate the maximum air volume flow rate of the hair dryer: \(Q = 1.96 \times 10^{-3} m^2 \times 3 m/s = 5.88 \times 10^{-3} m^3/s\) #Step 2: Determine the power required to heat the air#

Determine the power required to heat the air

We need to find the power required to heat the air to the desired temperature without exceeding the maximum power of the hair dryer (1200 W). We can use the following equation for the heating power: \(P = m \times C_p \times \Delta T\) Where \(P\) is the required power in watts, \(m\) is the mass flow rate of the air in kilograms per second (kg/s), \(\Delta T\) is the increase in temperature in Kelvin (K), and \(C_p\) is the specific heat capacity of air at constant pressure, which is approximately \(1006 \, \mathrm{J/(kg \cdot K)}\). From the volume flow rate, we can calculate the mass flow rate using the density of air, \(\rho\), which is about \(1.2\, \mathrm{kg/m^3}\), and the following equation: \(m =\rho \times Q\) Which in our case gives us: \(m = 1.2 \, \mathrm{kg/m^3} \times 5.88 \times 10^{-3} \, \mathrm{m^3/s} = 7.06 \times 10^{-3}\, \mathrm{kg/s}\) We are given that the maximum temperature rise is 50°C. Since we are working in Kelvin, we need to convert this to a temperature rise, so: \(\Delta T = 50^{\circ} \mathrm{C} = 50\, \mathrm{K}\) Now we can calculate the power required to heat the air: \(P = 7.06 \times 10^{-3}\, \mathrm{kg/s} \times 1006 \, \mathrm{J/(kg\cdot K)} \times 50\, K = 354.21\, W\) #Step 3: Design the heating element size#

Design the heating element size

We have found that the required power to heat the air is 354.21 W which is less than the maximum power of the hair dryer (1200 W). Thus, the design is possible. However, the hair dryer needs to also contain a fan to move the air, which would also consume some power. To allocate power for the fan and heating element in the ratio of 70% for the heating element and 30% for the fan: \(P_{heating element} = 0.7 \times 1200\, W = 840\, W\) \(P_{fan} = 0.3 \times 1200\, W = 360\, W\) Since the required power (354.21 W) is less than the power allocated for the heating element (840 W), this design is feasible. The exact dimensions of the heating element will depend on the specific technology used, such as resistive wire or ceramic heating elements. However, with an 840 W heating element and the given constraints, the hair dryer can be designed to ensure that the air temperature and velocity do not exceed the given limits.

Key concepts

Electric Hair Dryer Engineering

The engineering behind an electric hair dryer involves several critical components such as the electric motor, the fan, the heating element, and the nozzle design. The motor and fan work together to draw air into the dryer and push it out through the nozzle. The heated element, commonly a coil of resistive wire or a ceramic component, raises the temperature of the air as it passes by. It is essential to strike a balance between the motor's power, the heating element's capacity, and the airflow to ensure efficient drying without overheating the device or the hair.

The nozzle's cross-sectional area plays a pivotal role in determining the air velocity as it leaves the dryer. An appropriately sized nozzle not only can affect drying time but also ensures a comfortable experience by keeping the air speed at a manageable level, as seen in this design problem where the air velocity must be capped at 3 m/s. By combining these components with electronic controls for heat and speed settings, engineers create a portable and convenient tool for hair drying.

Thermal Heating Power Calculation

The concept of thermal heating power calculation is paramount in accurately determining the energy required to raise the temperature of the air flowing through a hair dryer to a specific level.

The primary formula involved is:
\( P = m \times C_p \times \Delta T \) where \( P \) is the power in watts, \( m \) is the mass flow rate of the air in kg/s, \( C_p \) represents the specific heat capacity of air (usually around \( 1006 \, \mathrm{J/(kg \cdot K)} \)), and \( \Delta T \) is the change in temperature in Kelvin.

This equation calculates the energy necessary to heat a given mass of air by a certain temperature rise. The goal is to design a heating element that can sufficiently warm the air within the power constraints, as in this exercise where we're limited to a 1200-Watt hair dryer.

Understanding this heating power calculation is essential for engineers to verify whether the hair dryer's design conforms to desired temperature specifications without exceeding the device's maximum power capability.

Air Volume Flow Rate

Air volume flow rate is a measure of how much air moves through a particular area over time and is usually expressed in cubic meters per second (m³/s). It is a critical aspect in designing systems that involve airflow, such as hair dryers, ventilation systems, and HVAC units.

The equation to find the air volume flow rate is straightforward:
\( Q = A \times V \) where \( Q \) is the flow rate, \( A \) is the cross-sectional area through which the air flows, and \( V \) is the velocity of the air. A higher flow rate indicates more air is being moved, which typically translates to faster drying times in the context of hair dryers.

To avoid exceeding the maximum desired air velocity of 3 m/s in the hair dryer's design, the cross-sectional area of the nozzle can be adjusted accordingly, resulting in a design that complies with the specified constraints. The flow rate also links to the thermal heating power, as the mass flow rate calculated from the volume flow rate is used to determine the power needed to achieve the desired temperature increase in the hair dryer design.