Psychrometric Calculation: Absolute Humidity

This solution details the calculation of absolute humidity ($Y'$) using psychrometric principles, the Antoine equation for saturation vapor pressure, and an energy balance, given the dry-bulb temperature ($T_{db}$), wet-bulb depression ($\Delta T_{wb}$), humid heat ($C_{ph}$), and latent heat of vaporization ($h_{fg}$).

Key Psychrometric Concepts and Equations

• Absolute Humidity ($Y'$): Represents the mass of water vapor present per unit mass of dry air.
• Saturation Vapor Pressure ($P^{sat}$): The maximum partial pressure water vapor can exert at a given temperature. It's calculated using the Antoine equation: $ \ln P^{sat} = 12.00 - \frac{4000}{T-40} $, where $T$ is in $K$ and $P^{sat}$ is in $bar$.
• Wet-Bulb Temperature ($T_{wb}$): The temperature indicated by a thermometer measuring the temperature of air cooled by evaporation of water into it at constant pressure. It is found by $ T_{wb} = T_{db} - \Delta T_{wb} $.
• Humidity Ratio ($Y'$): This is related to the partial pressure of water vapor ($P_v$) and atmospheric pressure ($P_{atm}$) by the formula: $ Y' = \frac{M_{water}/M_{air} \times P_v}{P_{atm} - P_v} $.
• Energy Balance for Adiabatic Saturation: The energy balance around an adiabatic saturation process is crucial. It relates the change in enthalpy to the change in humidity ratio and can be expressed as: $ C_{ph} (T_{db} - T_{wb}) = h_{fg} (Y'_{wb}^{sat} - Y') $. Here, $Y'_{wb}^{sat}$ is the saturation humidity ratio at the wet-bulb temperature.

Step-by-Step Calculation

1. Identify Given Values and Convert Temperatures:

   Dry-bulb temperature, $T_{db} = 30 \ \degree C$. Convert to Kelvin: $ T_{db\_K} = 30 + 273.15 = 303.15 \ K $. Wet-bulb depression, $\Delta T_{wb} = 9 \ \degree C$. Humid heat, $C_{ph} = 1.0 \ kJ \ kg^{-1} \ K^{-1}$. Latent heat of vaporization, $h_{fg} = 2000 \ kJ \ kg^{-1}$. Molecular weight ratio, $M_{water}/M_{air} = 18/28$. Standard atmospheric pressure is assumed: $P_{atm} = 1.01325 \ bar$.

2. Calculate the Wet-Bulb Temperature ($T_{wb}$):

   $ T_{wb} = T_{db} - \Delta T_{wb} = 30 \ \degree C - 9 \ \degree C = 21 \ \degree C $. Convert to Kelvin: $ T_{wb\_K} = 21 + 273.15 = 294.15 \ K $.

3. Determine Saturation Vapor Pressure at $T_{wb}$ ($P_{wb}^{sat}$):

   Using the provided Antoine equation: $ P_{wb}^{sat} = \exp\left(12.00 - \frac{4000}{T_{wb\_K}-40}\right) $ $ P_{wb}^{sat} = \exp\left(12.00 - \frac{4000}{294.15-40}\right) $ $ P_{wb}^{sat} = \exp\left(12.00 - \frac{4000}{254.15}\right) $ $ P_{wb}^{sat} = \exp\left(12.00 - 15.7389\right) = \exp(-3.7389) \approx 0.02387 \ bar $.

4. Calculate the Saturation Humidity Ratio at $T_{wb}$ ($Y'_{wb}^{sat}$):

   $ Y'_{wb}^{sat} = \frac{M_{water}/M_{air} \times P_{wb}^{sat}}{P_{atm} - P_{wb}^{sat}} $ $ Y'_{wb}^{sat} = \frac{(18/28) \times 0.02387}{1.01325 - 0.02387} $ $ Y'_{wb}^{sat} = \frac{0.642857 \times 0.02387}{1.009383} \approx \frac{0.015329}{1.009383} \approx 0.015186 \ kg \ moisture/kg \ dry \ air $.

5. Calculate the Actual Humidity Ratio ($Y'$) using the Energy Balance:

   From the energy balance equation $ C_{ph} (T_{db} - T_{wb}) = h_{fg} (Y'_{wb}^{sat} - Y') $, we rearrange to solve for $Y'$: $ Y' = Y'_{wb}^{sat} - \frac{C_{ph} (T_{db} - T_{wb})}{h_{fg}} $ Substitute the known values: $ Y' = 0.015186 - \frac{1.0 \ kJ \ kg^{-1} \ K^{-1} \times (30 - 21) \ K}{2000 \ kJ \ kg^{-1}} $ $ Y' = 0.015186 - \frac{9}{2000} $ $ Y' = 0.015186 - 0.0045 $ $ Y' = 0.010686 \ kg \ moisture/kg \ dry \ air $.

6. Compute the Final Value and Round:

   The question asks for the value of $1000 \times Y'$. $ 1000 \times Y' = 1000 \times 0.010686 = 10.686 $. Rounding to one decimal place gives $10.7$.