Conversions between units in different measurement systems can seem challenging, especially when dealing with two entirely different dimensions—such as mass flow rate (pound per minute) and volume flow rate (cubic meters per hour). To solve this problem, we must break it down step by step, making sure to carefully account for unit conversions and the properties of the substance involved.
Let’s dive into the detailed process of converting 0.45 pounds per minute (lb/min) into cubic meters per hour (m³/h).
Step 1: Relationship Between Mass and Volume
To convert from mass flow rate (pounds per minute) to volume flow rate (cubic meters per hour), we need to know the density of the substance being measured. This is because the relationship between mass and volume is governed by the formula:Volume=MassDensity\text{Volume} = \frac{\text{Mass}}{\text{Density}}Volume=DensityMass
In this case, we’ll assume the substance is water, as it is one of the most common fluids for such conversions. The density of water is approximately 62.4 pounds per cubic foot (lb/ft³) or 1,000 kilograms per cubic meter (kg/m³) at standard temperature and pressure.
Step 2: Convert Pounds to Kilograms
Since 1 pound is equal to 0.453592 kilograms, we can convert the mass flow rate from pounds per minute to kilograms per minute:0.45 lb/min×0.453592 kglb=0.2041164 kg/min0.45 \, \text{lb/min} \times 0.453592 \, \frac{\text{kg}}{\text{lb}} = 0.2041164 \, \text{kg/min}0.45lb/min×0.453592lbkg=0.2041164kg/min
Step 3: Convert Minutes to Hours
Next, we need to express the flow rate in terms of kilograms per hour. Since there are 60 minutes in an hour, we multiply by 60:0.2041164 kg/min×60 minhour=12.246984 kg/h0.2041164 \, \text{kg/min} \times 60 \, \frac{\text{min}}{\text{hour}} = 12.246984 \, \text{kg/h}0.2041164kg/min×60hourmin=12.246984kg/h
So, the mass flow rate is now 12.246984 kilograms per hour.
Step 4: Use the Density of Water to Convert Mass to Volume
The density of water in SI units is 1,000 kilograms per cubic meter. Using the formula:Volume (m³)=Mass (kg)Density (kg/m³)\text{Volume (m³)} = \frac{\text{Mass (kg)}}{\text{Density (kg/m³)}}Volume (m³)=Density (kg/m³)Mass (kg)
we calculate the volume flow rate:Volume Flow Rate=12.246984 kg/h1,000 kg/m³=0.012246984 m³/h\text{Volume Flow Rate} = \frac{12.246984 \, \text{kg/h}}{1,000 \, \text{kg/m³}} = 0.012246984 \, \text{m³/h}Volume Flow Rate=1,000kg/m³12.246984kg/h=0.012246984m³/h
Step 5: Final Answer
After completing all the conversions, we find that 0.45 pounds per minute equals approximately 0.0122 cubic meters per hour.
Key Takeaways
- Understand the Properties of the Substance: The density is crucial for converting between mass and volume.
- Break the Conversion into Manageable Steps: Converting units systematically (e.g., pounds to kilograms, minutes to hours) ensures accuracy.
- Know the Common Densities: For water, 1,000 kg/m³ is standard, but for other substances, the density will vary.
This problem highlights the importance of unit consistency and systematic calculation in engineering, science, and everyday applications. Mastering such conversions not only improves accuracy but also enhances your problem-solving skills.