A hair dryer is basically a duct in which a few layers of electric resistors are placed. A small fan pulls the air in and forces it to flow over the resistors where it is heated. Air enters a \(900-\mathrm{W}\) hair dryer at \(100 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C}\), and leaves at \(50^{\circ} \mathrm{C}\). The cross-sectional area of the hair dryer at the exit is \(60 \mathrm{~cm}^{2}\). Neglecting the power consumed by the fan and the heat losses through the walls of the hair dryer, determine \((a)\) the volume flow rate of air at the inlet and \((b)\) the velocity of the air at the exit.

The energy balance equation is fundamental in understanding how energy is conserved in a physical system. It's an expression of the first law of thermodynamics, which states that energy cannot be created or destroyed, only converted from one form to another. In the context of the hair dryer example, we apply the energy balance to calculate the mass flow rate of air.

An electric hair dryer converts electrical energy into heat, which is transferred to the air passing over the resistors. The equation for power in terms of the energy balance is:
\[\begin{equation} P = \.{m} \cdot c_p \cdot (T_{\text{out}} - T_{\text{in}})\end{equation}\]
where:

• P represents the power input (in Watts),
• \dot{m} is the mass flow rate of the air (in kg/s),
• c_p is the specific heat at constant pressure (in J/kg K), indicates how much energy is needed to raise the temperature of 1 kg of air by 1 degree Kelvin,
• T_{\text{in}} and T_{\text{out}} are the inlet and exit temperatures of the air (in degrees Celsius), converted to Kelvin in calculations.

It's important to note that neglecting the power consumed by the fan and the heat losses through the walls of the hair dryer means we assume all the electrical energy is being used to heat the air. This simplified assumption helps focus on understanding the relationship between power and the mass flow rate of air, which is crucial for determining the correct operation of thermal systems like the hair dryer.

Mass flow rate is a measure of how much mass of a substance passes through a given surface per unit time. It is an essential concept in both engineering and physics, particularly when analyzing fluid systems like our hair dryer scenario. In the equation given in the energy balance section, mass flow rate (\(\dot{m}\)) can be determined when we have the power input and can calculate the temperature change:
\[\begin{equation} \dot{m} = \frac{P}{c_p \cdot (T_{\text{out}} - T_{\text{in}})}\end{equation}\]
In practical terms, this value tells us how much air the hair dryer moves and heats up on a continuous basis. A higher mass flow rate would imply more air is being heated, thus potentially drying hair faster. By manipulating the mass flow rate, manufacturers can design hair dryers for different use cases, such as a quicker dry or a slower, more controlled heat application.

Understanding mass flow rate is also key in various applications beyond hair dryers. It is used in designing cooling systems, calculating fuel consumption in engines, and even determining the dosage in drug delivery systems. It links directly to the conservation of mass, another principle that, along with energy conservation, dictates the behavior of physical systems.

The ideal gas equation is a cornerstone of thermodynamics, providing a link between the pressure, volume, temperature, and amount of gas. While real gases do not always behave ideally, the ideal gas law is a useful approximation for gases at low pressure and high temperature, similar to the conditions in our hair dryer example:
\[\begin{equation} PV = nRT\end{equation}\]
This relationship simplifies for a constant amount of gas to:
\[\begin{equation} \rho = \frac{P}{RT}\end{equation}\]
where \(\rho\) represents the density of the gas, \(P\) is the pressure, \(R\) is the specific gas constant, and \(T\) is the absolute temperature in Kelvin.

By using the ideal gas equation, we can determine the air density at the inlet of the hair dryer, which, combined with the mass flow rate, helps us find the volume flow rate. Here's the density calculation based on the equation:\[\begin{equation} \rho_{\text{in}} = \frac{P_{\text{in}}}{R \cdot T_{\text{in}}}\end{equation}\]
With this equation, we have a powerful tool to analyze not only heating and cooling systems but also to understand atmospheric conditions, calculate buoyancy, and even design pneumatic systems. Whether studying the weather or designing an airplane's cabin pressure system, the ideal gas law is widely applied and critical to predicting gas behavior.