Lewis Number for Wet-Bulb & Adiabatic Saturation Temperature Equality
The wet-bulb temperature ($T_{wb}$) and adiabatic saturation temperature ($T_{as}$) are closely related concepts in psychrometrics, describing the temperature resulting from evaporative cooling under adiabatic conditions.
Equality Condition Explained
These two temperatures, $T_{wb}$ and $T_{as}$, are theoretically equal when the Lewis number ($Le$) of the mixture is equal to 1.
Defining the Lewis Number
The Lewis number ($Le$) is a dimensionless ratio comparing thermal diffusivity ($ \alpha $) to mass diffusivity ($ D_{AB} $). It quantizes the relative effectiveness of heat and mass transfer by diffusion.
The relationship is defined as:
$ Le = \frac{\alpha}{D_{AB}} = \frac{k}{\rho c_p D_{AB}} $
- $ \alpha $: Thermal diffusivity
- $ D_{AB} $: Mass diffusivity (of vapor in gas)
- $ k $: Thermal conductivity
- $ \rho $: Density
- $ c_p $: Specific heat capacity
Significance of $Le = 1$
A Lewis number of 1 indicates that the rates of heat and mass diffusion are balanced. This balance is the underlying principle for the equality between the wet-bulb and adiabatic saturation temperatures in idealized scenarios. While real mixtures (like air-water) often have $ Le \approx 0.8-0.9 $, the value $ Le = 1 $ represents the theoretical condition for exact equality.
Result
Based on the theoretical relationship, the Lewis number for which the wet-bulb temperature and adiabatic saturation temperature are nearly equal is 1.
Important Questions from Humidification
- Vapor pressure of water at various temperature is given in the table.
Temperature (in K) 284 289 294 299 310 Vapor pressure of water (in kPa) 1.28 1.80 2.50 3.40 6.40
An air and water vapor mixture at $100 \text{ kPa}$ with a relative humidity of $20\%$ has a dry-bulb temperature of $310 \text{ K}$. Assume latent heat of vaporization for water is $44 \text{ kJ mol}^{-1}$ and specific heat capacity of the air and water vapor mixture is $0.035 \text{ kJ mol}^{-1}\text{K}^{-1}$. Which one of the following is the closest to its wet-bulb temperature (in K)? Consider moist air with absolute humidity of 0.02 (kg moisture)/(kg dry air) at 1 bar pressure. The vapour pressure of water is given by the equation
$\ln P^{sat} = 12 - \frac{4000}{T-40}$
where $P^{sat}$ is in bar and $T$ is in K. The molecular weight of water and dry air are 18 kg/kmol and 29 kg/kmol, respectively. The dew temperature of the moist air is ____________ °C (rounded off to the nearest integer).Partially saturated air at 1 bar and $50 ^{\circ}C$ is contacted with water in an adiabatic saturator. The air is cooled and humidified to saturation, and exits at $25^{\circ}C$ with an absolute humidity of 0.02 kg water per kg dry air. Use latent heat of vaporization of water as 2450 kJ. $kg^{-1}$, and average specific heat capacity for dry air and water, respectively as 1.01 kJ.$kg^{-1}K^{-1}$ and 4.18 kJ. $kg^{-1}K^{-1}$. If the absolute humidity of air entering the adiabatic saturator is $H \times 10^{-3}$ kg water per kg dry air, the value of $H$ is ________ (rounded off to two decimal places).
The dry-bulb temperature of air in a room is $30 \ \degree C$. The Antoine equation for water is given as
$ln \ P^{sat} = 12.00 - \frac{4000}{T-40}$
where $T$ is the temperature in $K$ and $P^{sat}$ is the saturation vapor pressure in $bar$. The latent heat of vaporization of water is $2000 \ kJ \ kg^{-1}$, the humid heat is $1.0 \ kJ \ kg^{-1} \ K^{-1}$, and the molecular weights of air and water are $28 \ kg \ kmol^{-1}$ and $18 \ kg \ kmol^{-1}$, respectively. If the absolute humidity of air is $Y' \ kg$ moisture per $kg$ dry air, then for a wet-bulb depression of $9 \ \degree C$, $1000 \times Y' = $ _________ (rounded off to one decimal place).The humidity of air at a dry-bulb temperature of $65^{\circ}C$ is $0.025 \; kg \; water/kg \; dry \; air$. The latent heat of vaporization of water at $0 ^{\circ}C$ is $2500 \; kJ/kg$. The psychrometric ratio of air is $0.95 \; kJ \; (kg \; dry \; air)^{-1} \; K^{-1}$. Considering $0 ^{\circ}C$ as reference temperature, the enthalpy of air (in kJ/kg) at its adiabatic saturation temperature of $35 ^{\circ}C$ is ________ (rounded off to two decimal places).
