Question

The air in a \(6-m \times 5-m \times 4-m\) hospital room is to be completely replaced by conditioned air every 15 min. If the average air velocity in the circular air duct leading to the room is not to exceed \(5 \mathrm{m} / \mathrm{s}\), determine the minimum diameter of the duct.

Short answer

Answer: The minimum diameter of the duct required to replace the air in the room completely within 15 minutes is approximately 0.289 meters.

Step-by-step solution

Calculate the volume of the room

The volume of the room can be found using the formula for the volume of a rectangular prism: Volume = Length × Width × Height. The dimensions of the room are given as 6 meters, 5 meters, and 4 meters. Volume = 6 m × 5 m × 4 m = 120 m^3 The volume of the room is 120 m3.

Calculate the volume of air to be replaced per second

Since the air in the room needs to be replaced completely every 15 minutes, we can find the volume of air to be replaced per second by dividing the total volume by 15 min (converted to seconds). Volume per second = (Volume of room) / (15 min * 60 sec/min) Volume per second = 120 m^3 / (15*60) s = 120 m^3 / 900 s = 2/15 m^3/s The volume of air to be replaced per second is 2/15 m^3/s.

Calculate the minimum cross-sectional area of the duct

The average air velocity (v) in the duct is given as 5 m/s. We will use the formula for velocity: Velocity = (Volume flow rate) / (Cross-sectional area). Rearranging the formula to find the cross-sectional area (A), we get: Area = (Volume flow rate) / (Velocity) A = (2/15 m^3/s) / (5 m/s) = 2/(15*5)m^2 A = 2/75 m^2 The minimum cross-sectional area of the duct is 2/75 m^2.

Calculate the minimum diameter of the duct

Since the air duct is circular, the cross-sectional area can be defined as Area = π * (Diameter/2)^2. Rearranging the formula to find the diameter (D) and plugging in the value of the area we found earlier, we get: D = 2 * sqrt(A/π) = 2 * sqrt((2/75)/π) After calculating, we get: D ≈ 0.289 m The minimum diameter of the duct required to replace the air in the room completely within 15 minutes is approximately 0.289 meters.

Key concepts

Understanding Volume Flow Rate

Volume flow rate is a critical concept in thermodynamics and fluid mechanics, representing the volume of fluid that passes through a specified cross-sectional area per unit time. In practical applications such as air conditioning systems, volume flow rate corresponds to how quickly air is being moved or cycled through ductwork.

Considering our exercise, where the goal is to replenish the air in a hospital room, we first identify the amount of space we're dealing with. The room volume is a prerequisite for calculating the volume flow rate, as it provides a baseline for how much air needs to be replaced. Once we've ascertained the volume of the room ( 120 m^3), we determine the volume flow rate necessary to achieve complete air replacement within the target time frame, in this case, every 15 minutes.

Translating this situation into a formula, we get the volume flow rate by dividing the room's volume by the replacement period, resulting in a need to move air at a rate of 2/15 m^3/s. This calculation is vital for designing HVAC systems to ensure efficient and effective air circulation.

Calculating Cross-Sectional Area of Ducts

The cross-sectional area calculation is essential for duct sizing, as it directly relates to the volume flow rate that can pass through the duct and the speed at which the fluid (or air) flows. This area determines how much space is available for the air to move through and thus impacts the system's efficiency.

In our scenario, the duct's cross-sectional area is calculated to accommodate the derived volume flow rate of air, while not exceeding the maximum allowed velocity of air through the duct, which is 5 m/s. By using the formula Area = (Volume flow rate) / (Velocity), we established that the duct must have a minimum area of 2/75 m^2 to satisfy the airflow requirements.

The relevance of proper cross-sectional area sizing cannot be overstated. Being too generous might result in bulky and costly installations, while skimping on size could lead to insufficient air movement and potential system strain.

Air Replacement Rate for Ventilation

The concept of air replacement rate is linked to indoor air quality and the efficiency of ventilation systems. It measures how frequently the air within a space is completely replaced with fresh air and is often expressed in terms of air changes per hour (ACH).

In the example provided, the air replacement rate is necessary to prevent stagnation and ensure adequate ventilation in the hospital room. It is directly tied to occupant comfort and health regulations. With a 15-minute target for full room air replacement, we're looking at 4 air changes per hour (since there are 60 minutes in an hour).

Ensuring a proper air replacement rate is implemented in buildings like hospitals is critical, not just for comfort but also for controlling contaminants, maintaining temperature, and reducing humidity levels. This concept is crucial in designing HVAC systems to meet health and safety standards while providing a comfortable environment for occupants.