Calculating Virtual Temperature for Moist Air

The virtual temperature ($T_v$) represents the temperature at which dry air of the same pressure and density would exist. It is calculated using the actual air temperature ($T$) and the mixing ratio ($r$).

Step 1: Convert Temperature to Kelvin

The given ambient air temperature is $T = 27 \text{ °C}$. Convert this to Kelvin (K):

$ T (\text{K}) = T (\text{°C}) + 273.15 $

$ T = 27 + 273.15 = 300.15 \text{ K} $

Step 2: Convert Mixing Ratio to kg/kg

The given mixing ratio is $r = 50 \text{ gm/kg}$. Convert this to kilograms per kilogram (kg/kg):

$ r (\text{kg/kg}) = \frac{r (\text{gm/kg})}{1000} $

$ r = \frac{50}{1000} = 0.050 \text{ kg/kg} $

Step 3: Calculate Virtual Temperature

Use the approximate formula for virtual temperature:

$ T_v \approx T \times (1 + 0.61 \times r) $

Substitute the values in Kelvin and kg/kg:

$ T_v \approx 300.15 \text{ K} \times (1 + 0.61 \times 0.050) $

$ T_v \approx 300.15 \text{ K} \times (1 + 0.0305) $

$ T_v \approx 300.15 \text{ K} \times 1.0305 $

$ T_v \approx 309.30 \text{ K} $

Step 4: Convert Virtual Temperature back to Celsius

Convert the calculated virtual temperature from Kelvin back to Celsius (°C):

$ T_v (\text{°C}) = T_v (\text{K}) - 273.15 $

$ T_v (\text{°C}) \approx 309.30 - 273.15 $

$ T_v (\text{°C}) \approx 36.15 \text{ °C} $

The virtual temperature of the moist air is approximately 36.15 °C.