Dimensionless

Overview

Dimensionless numbers are fundamental tools in fluid mechanics and engineering that characterize the relative importance of different physical forces and phenomena. By combining variables such as velocity, length scale, fluid properties, and forces into ratios, dimensionless numbers eliminate units and reveal the underlying physics governing fluid behavior. This enables engineers to compare vastly different systems, scale experimental results, predict flow regimes, and simplify complex partial differential equations into universal forms.

The power of dimensionless analysis lies in the Buckingham π theorem, which demonstrates that any physical relationship can be expressed in terms of dimensionless groups. This foundation enables dimensional analysis, scaling laws, and similarity theory—cornerstones of experimental fluid mechanics, process design, and computational validation.

ax.text(1e4, 10, r’Nu = 0.023 Re^{0.8} Pr^{0.4}‘, fontsize=12, bbox=dict(facecolor=’white’, alpha=0.8))

plt.tight_layout() plt.show() ```

Flow Characterization Numbers

Several dimensionless numbers characterize the fundamental nature of fluid flow by comparing different forces or effects.

The Reynolds number (Re = \frac{\rho V L}{\mu}) quantifies the ratio of inertial forces to viscous forces, determining whether flow is laminar or turbulent. This is perhaps the most fundamental dimensionless number in fluid mechanics, governing transition phenomena in pipes, boundary layers, and around objects.

The Froude number (Fr = \frac{V}{\sqrt{gL}}) represents the ratio of inertial forces to gravitational forces, critical in open channel flow, ship design, and hydraulic jumps. When Fr < 1 (subcritical flow), gravity dominates; when Fr > 1 (supercritical flow), inertia dominates. Our FROUDE function calculates this parameter.

The Euler number (Eu = \frac{\Delta P}{\rho V^2}) relates pressure forces to inertial forces, commonly used in analyzing pressure drops and flow resistance. The EULER function computes this ratio for various flow systems.

The Dean number (De = Re\sqrt{\frac{R}{r}}) is a modified Reynolds number for flow in curved pipes, where centrifugal effects create secondary flows. The DEAN function handles this geometry-specific parameter.

Buoyancy and Gravity-Driven Phenomena

When density differences and gravitational forces drive fluid motion, specialized dimensionless numbers characterize the behavior.

The Archimedes number (Ar = \frac{gL^3\rho_f(\rho_p - \rho_f)}{\mu^2}) relates gravitational and viscous forces in particle-laden flows, essential for understanding sedimentation, fluidization, and particle settling. Our ARCHIMEDES function calculates this parameter.

The Bond number (also Eötvös number, Bo = \frac{\Delta\rho g L^2}{\sigma}) compares gravitational forces to surface tension forces, determining bubble and droplet shapes. The BOND function is used in multiphase flow, microfluidics, and surface phenomena.

Interfacial and Multiphase Flow Numbers

When multiple phases interact or surface effects dominate, specialized dimensionless numbers become critical.

The Capillary number (Ca = \frac{\mu V}{\sigma}) represents the ratio of viscous forces to surface tension forces, crucial in microfluidics, porous media flow, and droplet dynamics. The CAPILLARY function computes this parameter.

The Weber number (We = \frac{\rho V^2 L}{\sigma}) compares inertial forces to surface tension, determining droplet breakup, spray formation, and wave breaking.

The Cavitation number (Ca = \frac{P - P_v}{\frac{1}{2}\rho V^2}) predicts the onset of cavitation in flowing liquids, where local pressure drops below vapor pressure. The CAVITATION function is essential for pump and turbine design.

The Confinement number (Co = \frac{1}{L}\sqrt{\frac{\sigma}{g\Delta\rho}}) characterizes two-phase flow in confined geometries like microchannels, calculated by CONFINEMENT.

The Boiling number (Bg = \frac{q''}{G h_{fg}}) is critical in boiling heat transfer, relating heat flux to mass flux and latent heat. The BOILING function supports thermal systems design.

Heat and Mass Transfer Numbers

Dimensionless numbers also characterize transport phenomena beyond momentum.

The Fourier number for heat transfer (Fo = \frac{\alpha t}{L^2}) represents the ratio of heat conduction rate to thermal storage rate, governing transient heat conduction. Our FOURIER_HEAT function calculates this time-dependent parameter.

For mass transfer, the analogous Fourier number for mass (Fo_m = \frac{D t}{L^2}) characterizes diffusion processes, computed by FOURIER_MASS.

The Eckert number (Ec = \frac{V^2}{c_p \Delta T}) relates kinetic energy to enthalpy change, important in high-speed flows where viscous dissipation and compression heating matter. The ECKERT function handles this compressible flow parameter.

The Bejan number has multiple definitions: in heat transfer, it relates pressure drop to heat transfer; in porous media, it compares viscous forces to permeability effects. Our BEJAN function supports both formulations.

Drag and Resistance

The drag coefficient (C_D = \frac{F_D}{\frac{1}{2}\rho V^2 A}) is a dimensionless measure of resistance force, fundamental to aerodynamics, vehicle design, and particle motion. The DRAG function calculates this coefficient for various geometries and flow regimes.

Native Excel capabilities

Excel provides no built-in functions for calculating dimensionless numbers. Engineers typically create manual formulas combining fluid properties and flow parameters, which is error-prone and lacks standardization. The Analysis ToolPak and Solver Add-in do not address fluid mechanics calculations.

The Python functions in this library leverage the fluids package, which provides validated implementations of dimensionless number calculations used throughout industry and academia. These functions ensure consistent application of standard formulas and eliminate unit conversion errors.

Third-party Excel add-ins

• ChemStations CHEMCAD: Process simulation software with Excel integration offering fluid property and dimensionless number calculations, primarily for chemical engineering applications.
• REFPROP (NIST): Provides fluid properties that can be used to manually calculate dimensionless numbers, though it doesn’t compute them directly.
• Engineering-specific add-ins: Various niche add-ins exist for specific industries (e.g., hydraulic engineering, HVAC), but comprehensive dimensionless number libraries are rare in the Excel ecosystem.

The Python-based approach offers superior coverage, validation, and integration with modern computational workflows compared to fragmented Excel-only solutions.

Tools

Tool	Description
ARCHIMEDES	Calculate the Archimedes number (Ar) for a fluid and particle.
BEJAN	Compute the Bejan number (length-based or permeability-based).
BOILING	Calculate the Boiling number (Bg), a dimensionless number for boiling heat transfer.
BOND	Calculate the Bond number using fluids.core.Bond.
CAPILLARY	Calculate the Capillary number (Ca) for a fluid system using fluids.core.Capillary.
CAVITATION	Calculate the Cavitation number (Ca) for a flowing fluid.
CONFINEMENT	Calculate the Confinement number (Co) for two-phase flow in a channel.
DEAN	Calculate the Dean number (De) for flow in a curved pipe or channel.
DRAG	Calculate the drag coefficient (dimensionless) for an object in a fluid.
ECKERT	Calculate the Eckert number using fluids.core.Eckert.
EULER	Calculate the Euler number (Eu) for a fluid flow.
FOURIER_HEAT	Calculate the Fourier number for heat transfer.
FOURIER_MASS	Calculate the Fourier number for mass transfer (Fo).
FROUDE	Calculate the Froude number (Fr) for a given velocity, length, and gravity.