Question

The dry-bulb temperature of air in a room is 30 °C. The Antoine equation for water is \[ \ln P^{sat} = 12.00 - \frac{4000}{T - 40} \] where $T$ is in K and $P^{sat}$ is in bar. The latent heat of vaporization is 2000 kJ kg$^{-1}$, humid heat is 1.0 kJ kg$^{-1}$ K$^{-1}$, and the molecular weights of air and water are 28 and 18 kg kmol$^{-1}$. If absolute humidity is $Y'$ kg moisture per kg dry air, then for a wet-bulb depression of 9 °C, $1000\,Y' =$ \(\underline{\hspace{2cm}}\) (rounded to one decimal place).

Hint:

Wet-bulb depression strongly influences humidity; use psychrometric relations with Antoine equations for accurate vapor pressure estimates.

Answer 1

Correct Answer: 10

Step 1: Calculate Saturation Vapor Pressure at Wet-Bulb Temperature

Using the Antoine equation at $T_{wb} = 294 K$:

$$\ln P^{sat}_{wb} = 12.00 - \frac{4000}{294 - 40}$$

$$\ln P^{sat}_{wb} = 12.00 - \frac{4000}{254} = 12.00 - 15.748 = -3.748$$

$$P^{sat}_{wb} = e^{-3.748} = 0.0236 \text{ bar}$$

Step 2: Calculate Saturation Vapor Pressure at Dry-Bulb Temperature

Using the Antoine equation at $T_{db} = 303 K$:

$$\ln P^{sat}_{db} = 12.00 - \frac{4000}{303 - 40}$$

$$\ln P^{sat}_{db} = 12.00 - \frac{4000}{263} = 12.00 - 15.209 = -3.209$$

$$P^{sat}_{db} = e^{-3.209} = 0.0406 \text{ bar}$$

Step 3: Apply the Psychrometric Equation

The psychrometric equation relates humidity to wet-bulb depression:

$$Y' = Y'{wb} - \frac{c_s}{\lambda}(T{db} - T_{wb})$$

Where $Y'_{wb}$ is the saturation humidity at wet-bulb temperature.

The saturation humidity is given by:

$$Y'{sat} = \frac{M{water}}{M_{air}} \times \frac{P^{sat}}{P - P^{sat}}$$

Assuming total pressure $P = 1$ bar:

$$Y'_{wb} = \frac{18}{28} \times \frac{0.0236}{1 - 0.0236} = 0.643 \times \frac{0.0236}{0.9764} = 0.643 \times 0.02417 = 0.01554$$

Step 4: Calculate Absolute Humidity Y'

$$Y' = Y'{wb} - \frac{c_s}{\lambda}(T{db} - T_{wb})$$

$$Y' = 0.01554 - \frac{1.0}{2000}(30 - 21)$$

$$Y' = 0.01554 - \frac{1.0}{2000} \times 9$$

$$Y' = 0.01554 - 0.0045 = 0.01104$$

Step 5: Calculate $1000 \times Y'$

$$1000 \times Y' = 1000 \times 0.01104 = 11.04 \approx 11.0$$

Answer

The value of $1000 \times Y'$ is 11.0 (rounded to one decimal place).