Question

Air at \(300 \mathrm{K}\) and \(100 \mathrm{kPa}\) steadily flows into a hair dryer having electrical work input of \(1500 \mathrm{W}\). Because of the size of the air intake, the inlet velocity of the air is negligible. The air temperature and velocity at the hair dryer exit are \(80^{\circ} \mathrm{C}\) and \(21 \mathrm{m} / \mathrm{s},\) respectively. The flow process is both constant pressure and adiabatic. Assume air has constant specific heats evaluated at \(300 \mathrm{K}\). (a) Determine the air mass flow rate into the hair dryer, in \(\mathrm{kg} / \mathrm{s}\). ( \(b\) ) Determine the air volume flow rate at the hair dryer exit, in \(\mathrm{m}^{3} / \mathrm{s}\).

Short answer

a) Mass flow rate: 0.0277 kg/s b) Volume flow rate at the hair dryer exit: 0.0277 m³/s

Step-by-step solution

Find the specific heat capacity

First, we need to find the specific heat capacity of air at constant pressure (\(c_p\)) which is a property of the fluid. The problem states that we should assume air has constant specific heat capacities evaluated at \(300 \mathrm{K}\). The specific heat capacity of air at constant pressure at this temperature is approximately \(c_p = 1006 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\).

Use the First Law of Thermodynamics to find the mass flow rate

The problem states that the process is adiabatic and has constant pressure. Therefore, we can use the first law of thermodynamics to find the mass flow rate: $$Q - W_s = m \cdot c_p \cdot (T_{out} - T_{in})$$ Since the process is adiabatic, there is no heat transfer \(Q\). Therefore, we have: $$-W_s = m \cdot c_p \cdot (T_{out} - T_{in})$$ The electrical work input is given as \(1500 \mathrm{W}\). We have to convert the temperature in Celsius to Kelvin (\(T_{out} = 80^{\circ}\mathrm{C} + 273.15 = 353.15\mathrm{K}\), \(T_{in} = 300 \mathrm{K}\)): $$-1500 \mathrm{W} = m \cdot (1006 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}) \cdot (353.15\mathrm{K} - 300\mathrm{K})$$ Solving for mass flow rate (m) gives: $$m = \frac{-1500 \mathrm{W}}{(1006 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}})(53.15 \mathrm{K})} \approx -0.0277 \frac{\mathrm{kg}}{\mathrm{s}}$$ Since the mass flow rate is negative, it means the air is flowing into the system which is physically realistic for this problem. Taking the absolute value, we get the mass flow rate into the hair dryer: $$m \approx 0.0277 \frac{\mathrm{kg}}{\mathrm{s}}$$

Use the Ideal Gas Law to find the volume flow rate

Now we need to find the volume flow rate at the hair dryer exit. We can use the ideal gas law: $$PV = nRT$$ First, we have to calculate the number of moles (n) of air. We can use the relationship \(n = \frac{m}{M}\), where M is the molar mass of air (approximately \(29\frac{\mathrm{g}}{\mathrm{mol}}\)). So the number of moles (n) is: $$n = \frac{0.0277\frac{\mathrm{kg}}{\mathrm{s}}}{0.029\frac{\mathrm{kg}}{\mathrm{mol}}} \approx 0.9548 \frac{\mathrm{mol}}{\mathrm{s}}$$ Now we can plug in the values into the ideal gas law to find the volume flow rate at the hair dryer exit (V): $$PV = nRT$$ Using the exit temperature and the given pressure, we have: $$ (100 \times 10^3\mathrm{Pa})(V) = (0.9548 \frac{\mathrm{mol}}{\mathrm{s}})(8.314 \frac{\mathrm{J}}{\mathrm{mol} \cdot \mathrm{K}})(353.15\mathrm{K})$$ Now we can solve for the volume flow rate (V): $$V = \frac{(0.9548 \frac{\mathrm{mol}}{\mathrm{s}})(8.314 \frac{\mathrm{J}}{\mathrm{mol} \cdot \mathrm{K}})(353.15\mathrm{K})}{(100 \times 10^3\mathrm{Pa})} \approx 0.0277 \frac{\mathrm{m}^3}{\mathrm{s}}$$ Final results: a) Mass flow rate: \(0.0277 \frac{\mathrm{kg}}{\mathrm{s}}\) b) Volume flow rate at the hair dryer exit: \(0.0277 \frac{\mathrm{m}^3}{\mathrm{s}}\)

Key concepts

Understanding Specific Heat Capacity

Specific heat capacity, represented by the symbol \(c_p\) for constant pressure or \(c_v\) for constant volume, refers to the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). It is a crucial property in thermodynamics as it helps determine the heat added to or removed from a substance during a temperature change.

For example, in our hair dryer scenario, the specific heat capacity of air at constant pressure is given as \(1006 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\). This value is used to calculate how much heat is involved in changing the air's temperature within the device, a key factor which directly affects the mass flow rate. It's important to note that the specific heat capacity can vary with temperature, but in this case, it is assumed constant at the reference temperature of \(300 \mathrm{K}\), simplifying our calculations.

Applying the First Law of Thermodynamics

The first law of thermodynamics, often referred to as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. The change in internal energy of a system is equal to the heat added to the system minus the work done by the system on its surroundings.

Mathematically, it is expressed as \(\Delta U = Q - W_s\), where \(\Delta U\) is the change in internal energy, \(Q\) is the heat added, and \(W_s\) is the work done by the system. In the context of the hair dryer, this law enables us to determine the mass flow rate of air through the device. Since the process is adiabatic, meaning there's no heat transfer \(Q\), the work done by the electrical input \(W_s\) directly equates to the change in thermal energy of the air, allowing us to solve for the mass flow rate.

Utilizing the Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure \(P\), volume \(V\), and temperature \(T\) of a quantity of ideal gas with its number of moles \(n\). The law is typically formulated as \(PV = nRT\), where \(R\) is the universal gas constant. This equation is essential for predicting the behavior of gases under different conditions.

In our hair dryer exercise, we employed the ideal gas law to calculate the air volume flow rate at the exit. By knowing the number of moles, temperature, and pressure of the air, we can determine the volume it occupies at the hair dryer's exit. It's important to remember that the ideal gas law assumes gases behave ideally, which means the particles are in constant random motion without intermolecular forces and occupy no volume, an approximation that is reasonable under standard conditions.

Determining Mass Flow Rate

Mass flow rate is a measure of the mass of a substance that passes through a given surface per unit time. It is a critical concept in engineering, particularly for systems involving fluid flow, such as our hair dryer example. The mass flow rate tells us how much air is moving into the hair dryer per second, which directly correlates to the performance of the device.

The mass flow rate can be calculated using the first law of thermodynamics when no heat is transferred, as is the case in our adiabatic process. By understanding the specific heat capacity and temperature change of the air, along with the electrical work input, we can find the mass flow rate needed to achieve the desired temperature increase. This value is vital for ensuring the proper functioning of thermal systems.

Analyzing an Adiabatic Process

An adiabatic process is one in which no heat is exchanged between a system and its surroundings. This means that the thermal energy of a system can change only as a result of work being done. In many practical applications, such as our hair dryer example, an assumption of adiabatic conditions greatly simplifies the analysis of a thermal system because it eliminates the variable of heat transfer, focusing solely on the work performed.

In this scenario, the hair dryer does not gain or lose heat to the surroundings but instead relies on the electrical work to increase the internal energy of the air. This results in the temperature rise of the air as it passes through the dryer, under constant pressure, which is a hallmark of an adiabatic process at work.