Answer:
even though the wet-bulb temperature is lower than the dry-bulb temperature, indicating a potential measurement issue, if we proceed with the calculation, we find that the relative humidity at this location is approximately[tex]\(93.04\%\).[/tex]
Explanation:
To determine the relative humidity given the air temperature and wet-bulb temperature, we can use a psychrometric chart or equations derived from psychrometry. One such equation is the Magnus formula for calculating the saturation vapor pressure over water. Once we have the saturation vapor pressure at both the dry-bulb and wet-bulb temperatures, we can calculate the relative humidity.
However, it's important to note that the wet-bulb temperature being lower than the dry-bulb temperature can indicate potential issues with the measurement or conditions. Typically, the wet-bulb temperature should be equal to or higher than the dry-bulb temperature. In this case, with a wet-bulb temperature lower than the dry-bulb temperature, the relative humidity would be 100%.
But let's proceed with the calculation:
Using the Magnus formula, we can find the saturation vapor pressure (\(e_s\)) at both the dry-bulb (\(T\)) and wet-bulb (\(T_w\)) temperatures:
[tex]\[ e_s(T) = 6.112 \times e^{\left(\frac{17.67 \times T}{T + 243.5}\right)} \][/tex]
For \(T = 2°C\), we have:
[tex]\[ e_s(2°C) = 6.112 \times e^{\left(\frac{17.67 \times 2}{2 + 243.5}\right)} \][/tex]
[tex]\[ e_s(2°C) = 6.112 \times e^{\left(\frac{35.34}{245.5}\right)} \][/tex]
[tex]\[ e_s(2°C) = 6.112 \times e^{0.1438} \][/tex]
[tex]\[ e_s(2°C) = 6.112 \times 1.1546 \][/tex]
[tex]\[ e_s(2°C) = 7.0578 \, \text{kPa} \][/tex]
Similarly, for \(T_w = 1°C\), we have:
[tex]\[ e_s(1°C) = 6.112 \times e^{\left(\frac{17.67 \times 1}{1 + 243.5}\right)} \][/tex]
[tex]\[ e_s(1°C) ≈ 6.112 \times e^{\left(\frac{17.67}{244.5}\right)} \][/tex]
[tex]\[ e_s(1°C) = 6.112 \times e^{0.0722} \][/tex]
[tex]\[ e_s(1°C) = 6.112 \times 1.0745 \][/tex]
[tex]\[ e_s(1°C) = 6.5737 \, \text{kPa} \][/tex]
Now, we can calculate the relative humidity (\(RH\)) using the formula:
[tex]\[ RH = \left( \frac{e_s(T_w)}{e_s(T)} \right) \times 100\% \][/tex]
[tex]\[ RH = \left( \frac{6.5737}{7.0578} \right) \times 100\% \][/tex]
[tex]\[ RH = 0.9304 \times 100\% \][/tex]
[tex]\[ RH = 93.04\% \][/tex]
So, even though the wet-bulb temperature is lower than the dry-bulb temperature, indicating a potential measurement issue, if we proceed with the calculation, we find that the relative humidity at this location is approximately[tex]\(93.04\%\).[/tex]
