The pressure difference across the fan selected to ventilate the bathroom is approximately 1.3861 Pascals.
To determine the pressure difference across the fan in Pascals, we can use the following steps:
Step 1: Calculate the volume of air that needs to be replaced:
Given that the bathroom dimensions are 2 m x 3 m x 3 m, the volume of air in the bathroom is:
Volume = Length x Width x Height
Volume = 2 m x 3 m x 3 m
Volume = 18 m³
Step 2: Calculate the mass of air to be replaced in the bathroom:
The density of air is given as 1.25 kg/m³. Therefore, the mass (m) of air to be replaced is:
Mass (m) = Density x Volume
m = 1.25 kg/m³ x 18 m³
m = 22.5 kg
Step 3: Determine the time required to replace the air:
The fan replaces the entire volume of air in 15 minutes, which is 15/60 hours:
Time (t) = 15 minutes = 15/60 hours
t = 0.25 hours
Step 4: Calculate the airflow rate (Q) in cubic meters per second (m³/s):
Q = Volume / Time
Q = 18 m³ / 0.25 hours
Q = 72 m³/hour
Since 1 hour = 3600 seconds, we can convert the airflow rate to m³/s:
Q = 72 m³/hour x (1 hour / 3600 seconds)
Q ≈ 0.02 m³/s
Step 5: Calculate the velocity of the air (V) through the fan:
Velocity (V) = Airflow rate / Cross-sectional area
The cross-sectional area is the area of the bathroom, which is 2 m x 3 m = 6 m².
V = 0.02 m³/s / 6 m²
V ≈ 0.00333 m/s
Step 6: Calculate the pressure difference (ΔP) across the fan:
ΔP = (ρ * V²) / (2 * η)
Where:
ρ = Density of air = 1.25 kg/m³
V = Velocity of air = 0.00333 m/s
η = Combined efficiency of the fan-motor unit = 50% or 0.5
ΔP = (1.25 kg/m³ * (0.00333 m/s)²) / (2 * 0.5)
ΔP ≈ (1.25 * 1.1089) / 1
ΔP ≈ 1.3861 Pa
So, the pressure difference across the fan selected to ventilate the bathroom is approximately 1.3861 Pascals.