- Convert dry bulb temperature: \(59^\circ\text{F} = 15^\circ\text{C}\).
- Convert wet bulb temperature: \(41^\circ\text{F} = 5^\circ\text{C}\).
- Determine wet-bulb depression: \(15^\circ\text{C} - 5^\circ\text{C} = 10^\circ\text{C}\).
- Use psychrometric calculations to determine the relative humidity: \(\boxed{12}\%\).
Explanation
- Convert Temperatures to Celsius
To determine the relative humidity using the readings from a sling psychrometer, we first convert the temperature readings from Fahrenheit to Celsius using the provided formula \((^\circ\text{F} - 32) \times \frac{5}{9} = ^\circ\text{C}\).
For the dry bulb temperature of \(59^\circ\text{F}\):
\[T_{dry\_C} = (59 - 32) \times \frac{5}{9} = 27 \times \frac{5}{9} = 15^\circ\text{C}\]
For the wet bulb temperature of \(41^\circ\text{F}\):
\[T_{wet\_C} = (41 - 32) \times \frac{5}{9} = 9 \times \frac{5}{9} = 5^\circ\text{C}\]
- Calculate Relative Humidity
With the temperatures converted to Celsius, we have a dry bulb temperature \(T_{dry} = 15^\circ\text{C}\) and a wet bulb temperature \(T_{wet} = 5^\circ\text{C}\). The wet-bulb depression is the difference between these two temperatures:
\[\Delta T = T_{dry} - T_{wet} = 15^\circ\text{C} - 5^\circ\text{C} = 10^\circ\text{C}\]
Using standard meteorological formulas for psychrometry, we calculate the partial pressure of water vapor in the air and the saturation vapor pressure at the dry bulb temperature. Given these conditions, the relative humidity is calculated to be approximately \(11.94\%\). Rounding to the nearest whole number as typically required in such exercises, we find the relative humidity to be \(12\%\).
- Conclusion and Final Answer
Based on the calculations, the relative humidity corresponding to a dry bulb temperature of \(59^\circ\text{F}\) and a wet bulb temperature of \(41^\circ\text{F}\) is approximately \(12\%\).
\(\boxed{12}\ \%\)
Examples
Psychrometers are essential tools for maintaining proper humidity levels in environments like greenhouses, museums, and computer server rooms. For instance, in a greenhouse, knowing the relative humidity helps farmers decide when to water plants or increase ventilation to prevent fungal growth. By using a sling psychrometer to measure air moisture, you can precisely control the indoor climate to protect sensitive equipment or optimize plant growth, ensuring that the environment remains healthy and efficient.