Saturated air at 5 °C is required to be supplied to a room where the temperature must be held at 24 °C with a relative humidity of 50 %. The air is heated and then water at 24 °C i... Saturated air at 5 °C is required to be supplied to a room where the temperature must be held at 24 °C with a relative humidity of 50 %. The air is heated and then water at 24 °C is sprayed in to give the required humidity. Determine the temperature to which the air must be heated and the mass of spray water required per meter cube of air at room conditions. Neglect the fan power. Read more

Steps to Solve

1. Identify the properties of saturated air at 5 °C

At 5 °C, find the saturation pressure ($P_s$) and the specific humidity ($\omega_s$) using psychrometric charts or equations. For this example, using standard values, we find:

• Saturation pressure at 5 °C: $P_s \approx 0.872$ kPa
• Specific humidity ($\omega_s$) can be calculated using the formula: $$ \omega_s = 0.622 \frac{P_s}{P - P_s} $$ Here, $P$ is the atmospheric pressure (around 101.3 kPa).

2. Calculate specific humidity at 24 °C with 50% relative humidity

At 24 °C, determine the saturation pressure ($P_{s,24}$) and then calculate the specific humidity at 50% RH:

• Saturation pressure at 24 °C: $P_{s,24} \approx 2.979$ kPa
• The specific humidity ($\omega_{24}$) at 50%: $$ \omega_{24} = 0.622 \frac{0.5 P_{s,24}}{P - 0.5 P_{s,24}} $$

3. Determine the mass of water to be added

To find the mass of spray water needed ($m_w$) per cubic meter of air, subtract the specific humidity at 5 °C ($\omega_s$) from that at 24 °C ($\omega_{24}$): $$ m_w = \omega_{24} - \omega_s $$

4. Calculate the necessary heating temperature

Using the specific humilities and the fact that the air is heated before water is added, find the final temperature by analyzing the conditions until the desired humidity is achieved. The relationship between these properties involves the psychrometric relationships to adjust for the different conditions before and after. The relationship will ultimately require finding an equilibrium point where the specific humidity is achieved at a higher temperature.