Dimensionless Group
A dimensionless group is a combination of physical variables and constants that yields a pure number with no units, enabling the comparison and scaling of physical phenomena independent of the system of measurement. In chemical engineering, dimensionless groups like the Reynolds, Nusselt, and Prandtl numbers are used to correlate experimental data, predict transport phenomena, and scale laboratory results to industrial equipment. They arise from dimensional analysis (Buckingham Pi theorem) and are fundamental to similarity theory and the development of engineering correlations.
Worked Example
Problem
Water (ρ = 998 kg/m³, μ = 0.001 Pa·s) flows at v = 2 m/s through a pipe of diameter D = 0.05 m. Calculate the Reynolds number and determine the flow regime.
Solution
Step 1: Reynolds number formula: Re = ρ × v × D / μ. Step 2: Re = 998 × 2 × 0.05 / 0.001 = 998 × 0.1 / 0.001 = 99.8 / 0.001 = 99,800. Step 3: Compare with critical values: Re < 2,300 → laminar; 2,300 < Re < 4,000 → transitional; Re > 4,000 → turbulent. Step 4: Re = 99,800 >> 4,000, so flow is fully turbulent.
Answer
Re = 99,800; flow is turbulent
Key Dimensionless Groups in Chemical Engineering
| Name | Symbol | Definition | Physical Meaning | Application |
|---|---|---|---|---|
| Reynolds | Re | ρvL/μ | Inertia / viscous forces | Flow regime prediction |
| Nusselt | Nu | hL/k | Convective / conductive heat transfer | Heat transfer correlation |
| Prandtl | Pr | μCp/k | Momentum / thermal diffusivity | Fluid heat transfer |
| Schmidt | Sc | μ/(ρD_AB) | Momentum / mass diffusivity | Mass transfer in fluids |
| Damköhler | Da | reaction rate / transport rate | Reaction vs transport | Reactor design |
| Biot | Bi | hL/k_solid | External / internal resistance | Transient conduction |
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Related Terms
Heat Transfer Coefficient
The heat transfer coefficient (h) is a proportionality constant that quantifies the rate of heat transfer per unit area per unit temperature difference between a surface and a fluid in contact with it. It combines the effects of conduction through the fluid boundary layer and convection driven by fluid motion, making it central to the design of heat exchangers, reactors, and process equipment. Higher values indicate more efficient heat transfer, and the coefficient depends strongly on fluid properties, flow velocity, geometry, and surface roughness.
Pressure Drop
Pressure drop (ΔP) is the reduction in fluid pressure between two points in a flow system due to frictional resistance from pipe walls, fittings, valves, packed beds, or other flow restrictions. It determines the pumping or compression power required to maintain flow and is a critical factor in the economic design of pipelines, heat exchangers, distillation columns, and catalytic reactors. For incompressible flow in pipes, the Darcy-Weisbach equation relates pressure drop to fluid velocity, pipe geometry, and friction factor.
Chemical Reactor Design
Chemical reactor design is the discipline of selecting and sizing reactor vessels that achieve a desired chemical conversion at specified conditions of temperature, pressure, and flow rate. It integrates reaction kinetics, thermodynamics, and transport phenomena to predict concentration and temperature profiles within the reactor. Industrial applications range from petroleum refining and polymer synthesis to pharmaceutical manufacturing and environmental remediation.
The term "dimensionless" refers to having no physical dimensions (length, mass, time). The systematic framework was developed by Lord Rayleigh (1877) and formalised by Edgar Buckingham in his Pi theorem (1914). Named groups honour their originators: Osborne Reynolds (1883), Wilhelm Nusselt (1910s), Ludwig Prandtl (1900s), and Ernst Damköhler (1930s).