Dealing with complex differential equations in fluid dynamics and thermodynamics is one of the daily challenges for engineers. One of the most elegant ways to overcome this challenge is using "dimensionless numbers". Reducing formulaic clutter to a single number to understand heat, mass, and momentum transfer is key to managing processes.
In this article, we will examine the three fundamental dimensionless numbers (Prandtl, Schmidt, and Lewis numbers) that form the golden triangle in the transport mechanisms of fluids, and the mathematical, as well as physical, relationship between them. If you want to perform your numerical analyses practically, you can use our Lewis Sayisi Hesaplama tool.
The Three Main Pillars of Transport Phenomena
The interactions that liquids and gases (fluids) have with their environment while in motion are defined by three main transport phenomena:
- Momentum Transfer: Friction and movement transfer between fluid layers (Related to viscosity).
- Heat Transfer: Energy conduction arising from temperature differences (Related to thermal diffusivity).
- Mass Transfer: Matter transition arising from concentration differences (Related to mass diffusivity).
As a result of comparing these three transport phenomena in pairs, three fundamental dimensionless ratios, critical in engineering design, emerge.
Prandtl Number (Pr): Momentum and Heat
The Prandtl number gives the ratio of the momentum diffusion rate (kinematic viscosity, $\nu$) to the heat diffusion rate (thermal diffusivity, $\alpha$) in a fluid.
$$ Pr = \frac{\nu}{\alpha} $$
The Prandtl number shows the relationship between the velocity boundary layer thickness and the thermal boundary layer thickness. For example, for air $Pr \approx 0.7$, and for water it is around $Pr \approx 7$. Water conducts heat slower than momentum.
Schmidt Number (Sc): Momentum and Mass
The Schmidt number expresses the ratio of the momentum diffusion rate (kinematic viscosity, $\nu$) to the mass diffusion rate (mass diffusivity, $D_{AB}$).
$$ Sc = \frac{\nu}{D_{AB}} $$
The Schmidt number determines the relative thicknesses of the momentum boundary layer in fluid mechanics and the concentration boundary layer in mass transfer. It is used especially in gas-liquid interphase diffusion and air purification systems.
Lewis Number (Le): Heat and Mass
The Lewis number ($Le$), as we frequently mention, is the ratio of the thermal diffusivity ($\alpha$) rate to the mass diffusion ($D_{AB}$) rate.
$$ Le = \frac{\alpha}{D_{AB}} $$
This number, which is not directly related to momentum transfer (viscosity), is effective in all situations where only thermal and mass phenomena are compared, from combustion chambers to meteorological analyses.
The Triple Relationship: Intersection of Formulas
These three numbers are mathematically inextricably linked to each other. If we look closely at the formulas, we see that the concept of kinematic viscosity ($\nu$) is a common numerator (or denominator) in both Prandtl and Schmidt numbers.
If we divide the Schmidt number ($Sc$) by the Prandtl number ($Pr$):
$$ \frac{Sc}{Pr} = \frac{\nu / D_{AB}}{\nu / \alpha} = \frac{\alpha}{D_{AB}} = Le $$
This extremely elegant mathematical equation ($Le = Sc / Pr$) shows the following: If you know the Schmidt and Prandtl numbers of a gas or liquid experimentally through momentum transfer properties, you can directly calculate the Lewis number without needing an additional mass or heat transfer experiment.
Why is This Relationship Important in Practice?
If a chemical or mechanical engineer is working with a new multi-component mixture (for example, natural gas-hydrogen mixtures), the mass diffusion coefficient ($D_{AB}$) is one of the most difficult parameters to measure precisely in a laboratory environment.
On the other hand:
- The viscosity and density of the fluid are easily measured, hence kinematic viscosity ($\nu$) is found.
- Thermal diffusivity ($\alpha$) is derived from thermal conductivity, density, and specific heat values, from which $Pr$ is calculated.
- If we have certain empirical relationships and the Lewis number is known, the Schmidt number can be easily reached from the equation $Sc = Le \cdot Pr$, and from there, the mysterious mass diffusion coefficient ($D_{AB}$) we are looking for can be obtained.
Thus, difficult and costly experimental measurements turn into simple thermodynamic proportions.
Use in Analogy (Similarity) Approaches
The similarity of heat, mass, and momentum transfer has led engineers to develop approaches like the "Chilton-Colburn j-factor analogy".
According to the Chilton-Colburn analogy, the friction coefficient, heat transfer coefficient, and mass transfer coefficient can be made equivalent to each other. In this equation, Prandtl and Schmidt numbers take place as correction factors to the power of $2/3$.
For example, the ratio of mass to heat transfer coefficients can be expressed as:
$$ \frac{h}{h_m \rho c_p} = \left( \frac{Sc}{Pr} \right)^{2/3} = Le^{2/3} $$
In systems where the Lewis number is $Le=1$, the right side becomes directly $1$, leaving a tremendously simple direct proportion between the heat transfer coefficient ($h$) and the mass transfer coefficient ($h_m$). Air and water vapor systems (HVAC) are among the rare examples where this assumption of 1 works perfectly.
Conclusion
The engineering world produces strong designs to the extent that it can reduce complexity to simplicity. This dimensionless tripod consisting of Prandtl, Schmidt, and Lewis numbers forms the DNA of transport phenomena. Grasping the $Le = Sc / Pr$ relationship among them offers the engineer very powerful insights when faced with difficult thermodynamic problems.
Do not forget to try our Lewis Sayisi Hesaplama tool, where you can use your system properties as input both to convert the coefficients you get from literature and to minimize calculation errors. Keep exploring the limits of mathematics in transport phenomena!
