When it comes to air conditioning (HVAC), cooling systems, and meteorology, the fluid mixture studied the most is undoubtedly the air and water vapor mixture (moist air). From adjusting the temperature of our office to massive cooling towers in power plants, many processes rely on the thermodynamic dance of this duo. The most important factor making this dance manageable from an engineering perspective is the unique, almost magical ratio the Lewis number provides for this mixture: Le ≈ 1.
In this article, we will examine how heat and mass transfer work in perfect harmony (analogy) in air-water vapor systems, and how the $Le = 1$ assumption reduces design costs and complexity. To examine the air properties in your air conditioning systems and test the ratios yourself, you can use our Lewis Sayisi Hesaplama tool.
A Brief Reminder of the Lewis Number
In thermodynamics and fluid mechanics, the dimensionless Lewis number ($Le$) is the ratio of the thermal diffusivity coefficient ($\alpha$) to the mass diffusion coefficient ($D_{AB}$).
- If the system's ability to propagate temperature overcomes its ability to propagate mass, $Le > 1$,
- If the diffusing mass spreads faster than temperature, $Le < 1$.
Under normal conditions, in a randomly selected liquid or gas mixture, this number can take on very different values. However, at standard atmospheric pressure and normal living temperatures, this value for the air-water vapor mixture is approximately between 0.85 and 1.05. This proximity has paved the way for many engineering assumptions by being accepted as $Le = 1$ by engineers.
Psychrometry and the Le=1 Miracle
Psychrometry is the branch of science studying the thermodynamic properties of moist air. On psychrometric charts, "Dry Bulb" (normal temperature of the air) and "Wet Bulb" temperatures are used together to determine the humidity and temperature of the air.
The wet bulb temperature (the lowest temperature measured by passing air over a thermometer wrapped in a wet cloth) is the equilibrium point formed by water evaporating into the air resulting in mass transfer (increasing humidity) and at the same time drawing heat from the thermometer (heat transfer).
If the Lewis number were not 1, the ratio between the heat and mass transfer coefficients of the air would be completely different, and the wet bulb temperature would deviate significantly from the thermodynamically defined "Adiabatic Saturation Temperature" (the state of the system being completely saturated with moisture without any external heat exchange). Only because $Le \approx 1$, can we consider the wet bulb temperature equivalent to the adiabatic saturation temperature and fit psychrometric tables onto a single standard graph!
Design Simplicity with the Chilton-Colburn Analogy
In engineering, it is quite difficult to model situations where both heat and mass are transferred simultaneously in a system. Consider cooling tower design: Hot water sprayed from the top into the tower evaporates into the dry air pulled upward from the bottom (mass transfer) and, in doing so, both cools the water and heats the air (heat transfer).
If you want to measure the mass transfer coefficient ($h_m$) in a laboratory environment, you must conduct very rigorous tests. However, according to the Chilton-Colburn analogy, these two mechanisms progress with the same equations.
$$ \frac{h}{h_m \rho c_p} = Le^{2/3} $$
If we assume $Le = 1$, the formula is reduced to this brilliant simplicity:
$$ h = h_m \cdot \rho \cdot c_p $$
This equation (sometimes known as the Lewis Relation) means: The heat transfer coefficient ($h$) is in direct proportion to the mass transfer coefficient ($h_m$) and the specific heat capacity of the air. Engineers only measure the heat loss of air passing over a pipe in a wind tunnel, find $h$, and thanks to this formula, can instantly and very cheaply determine the rate of water vapor mixing into the air ($h_m$) and build cooling towers.
Situations Where the Analogy is Invalid
Although the harmony of air and water vapor is very successful, there are places where this analogy should not be used blindly:
- Different Gas Mixtures: Instead of air, in Helium-Water vapor, Argon-Water vapor, or industrial flue gases, $Le \neq 1$.
- Extreme Pressures and Temperatures: Psychrometric charts are generally for 1 atm pressure. When the air pressure is very different, such as in aircraft air conditioning or deep mining systems, diffusion and thermal diffusivity ratios change.
- Liquid-Phase Dominated Systems: This analogy is fundamentally based on diffusion in the gas phase.
If you are stepping outside the classic air-water system, you should definitely calculate your Lewis number with current values instead of relying on the analogy.
Conclusion
The fact that the Lewis number of the air and water vapor mixture is very close to 1 is a beautiful coincidence presented to mechanical engineering by nature. This feature has incredibly accelerated, standardized, and enabled the industrialization of all HVAC and cooling tower designs worldwide.
If you want to see how this ratio changes when working with unusual fluid mixtures, without drowning in formulas, you can quickly interpret your results by using our Lewis Sayisi Hesaplama tool.
