Prandtl number
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| Prandtl number | |
|---|---|
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| Name | Prandtl number |
| Named after | Ludwig Prandtl |
| Field | Fluid dynamics, Heat transfer |
| Formula | ν/α |
Prandtl number The Prandtl number is a dimensionless parameter in Fluid dynamics and Heat transfer that relates momentum diffusivity to thermal diffusivity. Developed in the early 20th century by Ludwig Prandtl, it plays a central role in analyses of boundary layers, convection, and transport phenomena in engineering problems encountered by institutions like Siemens and General Electric. Its value distinguishes behaviors in flows studied at facilities such as Imperial College London and Massachusetts Institute of Technology.
Definition and Physical Interpretation
The Prandtl number quantifies the ratio of kinematic viscosity to thermal diffusivity, establishing a link between viscous momentum transport described by Navier–Stokes equations and heat conduction governed by Fourier's law. In physical terms it indicates whether the velocity field or the temperature field has thicker diffusion-dominated layers, a concept central to experiments at Cavendish Laboratory and modeling efforts at NASA and ESA. In contexts from St. Anthony Falls Laboratory measurements to industrial designs at Boeing and Airbus, Pr determines whether momentum or thermal boundary layers control heat transfer analogous to comparisons in classic studies like the Kármán vortex street and observations by Otto von Guericke-era vacuum experiments.
Mathematical Formulation
Mathematically the Prandtl number is defined as the ratio Pr = ν/α where ν denotes kinematic viscosity (μ/ρ) from constitutive relations used in Navier–Stokes equations and α denotes thermal diffusivity from Fourier's law. In derivations of similarity solutions like those by Ludwig Prandtl and Theodore von Kármán, Pr appears alongside the Reynolds number and Nusselt number in nondimensional forms of the energy and momentum equations used in textbooks from Cambridge University Press and McGraw‑Hill. It enters boundary-layer equations employed in classical analyses such as the Blasius solution and extensions by Erwin Hahn and practitioners at Daimler and Shell.
Typical Values and Fluid Dependence
Typical Prandtl numbers vary widely: gases such as air near standard conditions have Pr ≈ 0.7, while liquid metals like mercury or sodium show very low Pr (≈10^-2 to 10^-3), and viscous oils or glycerol exhibit high Pr (≫1). These values are reported in compilations by organizations like NIST and in handbooks used at MIT and ETH Zurich. Environmental fluids studied in projects at Woods Hole Oceanographic Institution and Scripps Institution of Oceanography show Pr values influenced by temperature and composition, as do geothermal fluids investigated by USGS and BP.
Role in Heat Transfer and Boundary Layers
Pr governs the relative thicknesses of thermal and velocity boundary layers in forced and natural convection problems encountered in designs by Siemens Gamesa and ABB. For low Pr (liquid metals) thermal diffusion dominates, producing thicker thermal boundary layers than velocity ones—an effect relevant to research at Argonne National Laboratory and Oak Ridge National Laboratory. For high Pr (oils, polymers), momentum diffuses faster than heat, making thermal gradients confined near walls, a behavior exploited in heat exchanger designs by Alfa Laval and analyzed in standards by ASME. In convection studies such as those by Rayleigh and in investigations of Rayleigh–Bénard convection, Pr combines with the Rayleigh number to determine onset and pattern selection, a topic pursued at Princeton University and California Institute of Technology.
Dimensionless Correlations and Applications
Pr appears in empirical and theoretical correlations relating heat transfer coefficients and flow parameters, including the Dittus–Boelter equation, Sieder–Tate correlation, and forms of the Colburn j-factor. These correlations are used in design and optimization workflows at GE Aviation, Siemens Energy, and process engineering groups like BASF and Dow Chemical Company. In turbulence modeling frameworks like k–ε model and large-eddy simulations developed at Sandia National Laboratories and Los Alamos National Laboratory, Prandtl number (often as turbulent Prandtl number) adjusts scalar transport, affecting predictions in climate models at NOAA and IPCC assessments.
Experimental Measurement and Estimation Methods
Measuring Pr requires characterization of ν and α through experiments such as capillary viscometry for viscosity (methods used at Bureau of Standards laboratories) and laser-based thermal diffusivity measurements like photothermal radiometry employed at Rutherford Appleton Laboratory and industrial labs at Siemens. Estimation approaches leverage tables and correlations from compilations by NIST and reference texts from Elsevier; computational estimation uses property models embedded in software by ANSYS and COMSOL. In-situ measurements in oceanographic campaigns by NOAA and Woods Hole Oceanographic Institution employ microstructure profilers and thermistors to infer Pr through combined ν and α datasets.