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The humidity ratio is a key concept in understanding how much water vapor is present in our moist air. It tells us the mass of water vapor per unit mass of dry air, and is an essential part of thermodynamics, especially when dealing with air conditioning and heating processes. At saturation, this value represents the maximum amount of moisture the air can hold at a given temperature and pressure.
To find the humidity ratio \(W_s\), we need the saturation vapor pressure \(p_{ws}\) at the given temperature \(T_2 = 10^{\circ} \text{C}\). We use the formula:
\[ W_s = 0.622 \frac{p_{ws}}{p - p_{ws}} \]
where \(p\) is the total pressure, here \(100 \text{kPa}\). The factor 0.622 is derived from the molecular weight ratio of water vapor to dry air. This formula helps us determine the humidity ratio when the air is at precisely \(10^{\circ} \text{C}\), showing how the air absorbs moisture during the cooling process.

Saturation properties are crucial when dealing with moist air, as they define the conditions under which air contains the maximum water vapor possible at a given temperature. In this problem, we cool the air to become saturated at \(10^{\circ} \text{C}\).
We use psychrometric tables or charts to look up the saturation vapor pressure \(p_{ws}\). This value represents the pressure at which water vapor is in equilibrium with liquid water at the specified temperature. Knowing this, alongside the saturation humidity ratio, enables us to perform further calculations.

As air is saturated, the dew point, where condensation begins, matches the air temperature. This means our cooled air at \(10^{\circ} \text{C}\) has reached its full capacity for water vapor, an important detail for completing further thermodynamic calculations.

Energy balance is a fundamental principle used to evaluate the flow of energy in and out of a system, ensuring the overall energy remains conserved. In our exercise, we apply this concept to calculate the initial air temperature \(T_1\).
The equation \(m_a h_{a_1} + m_w h_w = m_a h_{a_2} + m_g h_g\) helps us conserve energy throughout the process. Here, \(m_a\) is the dry air flow rate, \(h_{a_1}\) and \(h_{a_2}\) are the specific enthalpies of air at \(T_1\) and \(10^{\circ} \text{C}\), \(m_w\) is the mass flow rate of water, and \(m_g\) (moist air flow rate) includes the added water vapor.

• Specify each substance's enthalpy from tables or calculate it using specific heat capacities.
• Balance the equation to find unknowns like the initial air temperature \(T_1\).

This balance ensures the sprayed water brings the air to saturation without an external heat source.

In thermodynamics, a mass balance equation is used to ensure that mass is conserved in any given system. For our system, we use this principle to calculate the amount of liquid water necessary to saturate the air.
The mass balance at a steady state for our process involves two key equations:
\[ m_g = m_a (1 + W_s) \] and \[ m_w = m_a W_s \]
where \(m_g\) is moist air flow rate, \(m_w\) is the mass flow rate of water, and \(m_a = 2 \text{ kg/s}\) is the dry air flow rate.

• The first equation helps us understand how the addition of water affects total flow.
• With the second equation, calculate \(m_w\) using the known \(m_a\) and the humidity ratio \(W_s\).

By applying these equations, we can determine how much water is sprayed to achieve the required humidity, ensuring the air exiting the system is saturated at the desired conditions.