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.
Understand the Problem
The question is asking for the temperature to which air must be heated and the mass of spray water needed per cubic meter of air at room conditions, given specific humidity requirements.
Answer
The temperature must be heated to 24 °C and the mass of spray water required per cubic meter of air is $m_w = \omega_{24} - \omega_s$.
Answer for screen readers
The air must be heated to approximately 24 °C (the same as room temperature) to achieve the required specific humidity, and the mass of spray water required per cubic meter of air is determined by the difference in specific humidities: $$ m_w = \omega_{24} - \omega_s $$
Steps to Solve
- 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).
- 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%: $$ \omega_{24} = 0.622 \frac{0.5 P_{s,24}}{P - 0.5 P_{s,24}} $$
- 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 ($\omega_{24}$): $$ m_w = \omega_{24} - \omega_s $$
- 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.
The air must be heated to approximately 24 °C (the same as room temperature) to achieve the required specific humidity, and the mass of spray water required per cubic meter of air is determined by the difference in specific humidities: $$ m_w = \omega_{24} - \omega_s $$
More Information
In this case, the process illustrates how heating and humidification work together to maintain comfort in indoor environments. The specific humidity is crucial in understanding the air's moisture content, impacting comfort levels.
Tips
- Failing to use the correct saturation pressures for the specific temperatures.
- Miscalculating the atmospheric pressure, which can affect the specific humidity results.
- Not accounting for relative humidity correctly when applying calculations.
