Dimensionless numbers are an indispensable part of physical chemistry and engineering thermodynamics. But when you are left alone with dozens of conflicting variables on an exam paper or in an industrial project, calculating the Lewis number correctly might not be as easy as it seems.
In this guide, we will simplify the basic formulas behind the Lewis number, perform a manual calculation using a real case, and discuss the common mistakes (especially unit mismatches) frequently made by students or engineers during this process. If you want to set manual calculations aside and guarantee the job, we recommend visiting our Lewis Sayisi Hesaplama tool, which you can use with a single click.
Basic Formula and Parameters
The Lewis number ($Le$), by its basic definition, is found by dividing the rate of thermal diffusivity ($\alpha$) by the rate of mass diffusivity ($D_{AB}$) in a medium.
Main Formula:
$$ Le = \frac{\alpha}{D_{AB}} $$
Here, the SI (International System of Units) unit for both parameters is $m^2/s$. The result is a dimensionless (unitless) value.
However, in real life (or homework questions), the $\alpha$ value is not always given directly. Instead, the thermophysical properties of the fluid are provided. In this case, you need to refer to the second (Extended) formula:
$$ \alpha = \frac{k}{\rho \cdot c_p} $$
If we substitute these values into the main formula, we obtain the Extended Formula:
$$ Le = \frac{k}{\rho \cdot c_p \cdot D_{AB}} $$
Parameter List and Units:
- $k$ (Thermal Conductivity): The ability of the fluid to conduct heat. Unit: $W/(m \cdot K)$
- $\rho$ (Density): Mass per unit volume. Unit: $kg/m^3$
- $c_p$ (Specific Heat): The energy required to raise the temperature of 1 kg of substance by 1 K. Unit: $J/(kg \cdot K)$
- $D_{AB}$ (Mass Diffusion Coefficient): The propagation rate of substance A within substance B. Unit: $m^2/s$
Step-by-Step Example Calculation
Let's say in a heat exchanger design, we are working with a Helium-Air mixture under very low pressure. The data given to you are as follows:
- Thermal Conductivity of the Helium mixture ($k$): $0.14 \ W/(m \cdot K)$
- Density of the Mixture ($\rho$): $0.4 \ kg/m^3$
- Specific Heat ($c_p$): $5190 \ J/(kg \cdot K)$
- Mass Diffusion Coefficient in the mixture ($D_{AB}$): $7.2 \times 10^{-5} \ m^2/s$
Step 1: Finding the Thermal Diffusivity ($\alpha$) Value
First, we must find the thermal diffusivity rate by combining the thermal data:
$$ \alpha = \frac{k}{\rho \cdot c_p} $$
$$ \alpha = \frac{0.14}{0.4 \times 5190} $$
$$ \alpha = \frac{0.14}{2076} \approx 6.743 \times 10^{-5} \ m^2/s $$
Step 2: Calculating the Lewis Number (Le)
Now, we will proportion the $\alpha$ value we found with the mass diffusion coefficient ($D_{AB}$):
$$ Le = \frac{\alpha}{D_{AB}} $$
$$ Le = \frac{6.743 \times 10^{-5}}{7.2 \times 10^{-5}} \approx 0.936 $$
Comment: The Lewis number is less than 1 ($Le = 0.936 < 1$). This result indicates that in the Helium mixture, mass diffusivity moves faster than thermal diffusivity (heat conduction), meaning the concentration boundary layer will be thicker.
Common Mistakes and Unit Traps
Especially during midterm and final exam periods or in laboratory analyses, the common mistakes that completely ruin the calculation are the following:
1. Using kiloJoule (kJ) Instead of Joule (J)
Specific heat ($c_p$) values read from tables are usually given in $kJ/(kg \cdot K)$ (e.g., $\approx 1.005$ for air). Students write this number directly into the formula.
- The Correct Way: When entering the formula, the $kJ$ value must be multiplied by $1000$ to be converted to $J/(kg \cdot K)$. For air, $1005$ should be written instead of $1.005$! Otherwise, the thermal diffusivity $\alpha$ will turn out to be $1000$ times larger than it should be.
2. The Confusion Between Square Centimeters ($cm^2$) and Square Meters ($m^2$)
The mass diffusion coefficient ($D_{AB}$) is mostly given as $cm^2/s$ in chemistry laboratory data.
- The Correct Way: In order for all values to cancel each other out in the SI system, the $cm^2/s$ data must be multiplied by $10^{-4}$ to convert it to $m^2/s$. For example, $0.2 \ cm^2/s = 2 \times 10^{-5} \ m^2/s$.
3. Using the Inverse Ratio (Le = D_AB / alpha)
Some very old sources in academic literature (especially some Russian school fluid mechanics books) might define the Lewis number inversely as the ratio of mass diffusivity to thermal diffusivity ($Le = D_{AB} / \alpha$).
- The Correct Way: In the modern IUPAC Gold Book and NASA terminology, the up-to-date and standard accepted definition is $Le = \alpha / D_{AB}$. You should always follow this rule, and if you are working with a different source, you must read its references carefully.
Conclusion
Calculating the Lewis number correctly is not just about solving a mathematical formula. It is the ability to bring together the physical properties of the material and fluid to read the harmony of energy and mass. As long as you pay attention to basic unit conversions, you can easily turn this hidden dance in boundary layers into numerical data.
If you don't want to waste time or are afraid of making mistakes in complex power values (exponents), you can safely use our Lewis Sayisi Hesaplama module to get the most accurate result for your system quickly!
