Prandtl number
The Prandtl number, denoted as Pr, is a dimensionless quantity in fluid mechanics and heat transfer that represents the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity.[1] It is mathematically expressed as Pr = ν / α, where ν is the kinematic viscosity (in m²/s) and α is the thermal diffusivity (in m²/s), with α = k / (ρ c_p), k being the thermal conductivity (in W/m·K), ρ the fluid density (in kg/m³), and c_p the specific heat capacity at constant pressure (in J/kg·K).[2] First formulated by Wilhelm Nusselt in 1909 and named after the German physicist and fluid dynamicist Ludwig Prandtl, this number characterizes the relative rates of momentum and heat diffusion in a fluid, providing insight into boundary layer behaviors during convection.[1] The physical significance of the Prandtl number lies in its indication of how thermal and momentum boundary layers develop relative to each other in convection flows.[1] For Pr > 1, typical of viscous liquids, the thermal boundary layer is thinner than the velocity boundary layer; for Pr < 1, common in liquid metals, the thermal boundary layer is thicker; and for Pr ≈ 1, as in most gases, the boundary layers are comparable.[2] In engineering applications, the Prandtl number is used in dimensionless analysis for convective heat transfer correlations.[1]Definition and Formulation
Mathematical Expression
The Prandtl number, denoted as \Pr, is mathematically defined as the ratio of the kinematic viscosity \nu to the thermal diffusivity \alpha, expressed as \Pr = \frac{\nu}{\alpha}, where \nu has units of \mathrm{m}^2/\mathrm{s} and \alpha has units of \mathrm{m}^2/\mathrm{s}.[3] This formulation emerges naturally from the non-dimensionalization of the governing equations for fluid flow and heat transfer. Specifically, the incompressible Navier-Stokes equations for momentum conservation include a diffusive term \nu \nabla^2 \mathbf{u}, while the energy equation for temperature includes a diffusive term \alpha \nabla^2 T. Upon scaling lengths by a characteristic length L, velocities by a characteristic velocity U, time by L/U, pressure by \rho U^2, and temperature by a characteristic temperature difference \Delta T, the non-dimensional momentum equation features a viscous diffusion coefficient of $1/\mathrm{Re} (where \mathrm{Re} = UL/\nu is the Reynolds number), and the non-dimensional energy equation features a thermal diffusion coefficient of $1/(\mathrm{Re} \Pr). Thus, \Pr quantifies the relative scaling of momentum diffusivity to thermal diffusivity in these transport processes.[4][3] An equivalent expression for the Prandtl number is \Pr = \frac{\mu c_p}{k}, where \mu is the dynamic viscosity (in \mathrm{kg}/(\mathrm{m \cdot s})), c_p is the specific heat capacity at constant pressure (in \mathrm{J}/(\mathrm{kg \cdot K})), and k is the thermal conductivity (in \mathrm{W}/(\mathrm{m \cdot K})).[3] This form derives from substituting \nu = \mu / \rho and \alpha = k / (\rho c_p) into the primary definition, where \rho is the fluid density, yielding a combination of transport properties that inherently cancels dimensions to produce a dimensionless scalar.[4]Dimensionless Characteristics
The Prandtl number is a dimensionless quantity because the kinematic viscosity, denoted as ν, and the thermal diffusivity, denoted as α, both share the same units of length squared per time (m²/s), rendering their ratio unitless. This intrinsic dimensionlessness allows the Prandtl number to serve as a universal parameter that characterizes flow similarity across different physical scales, fluids, and conditions without dependence on specific units. As a result, it facilitates the scaling of transport phenomena in fluid mechanics, ensuring that solutions remain consistent for geometrically and dynamically similar systems.[5][6] Within the framework of the Buckingham π theorem, the Prandtl number arises as a fundamental dimensionless π group in convection problems that incorporate variables related to viscosity (momentum transport), thermal conductivity (heat conduction), and specific heat capacity (energy storage). The theorem, which reduces the number of governing parameters by identifying independent dimensionless combinations from the dimensional variables, highlights the Prandtl number's role alongside other groups like the Reynolds and Nusselt numbers in describing heat and mass transfer processes. This systematic derivation underscores its necessity for capturing the interplay of diffusive mechanisms in forced and natural convection scenarios.[7][8] Typical Prandtl number values range from about 0.01 to 10⁵, reflecting the broad spectrum of relative transport rates across fluids; low Prandtl numbers indicate that thermal diffusion dominates over momentum diffusion, promoting rapid heat spreading relative to velocity gradients, whereas high values signify the reverse, where viscous effects prevail and slow thermal propagation. This range encapsulates the parameter's sensitivity to molecular properties, influencing the structure of boundary layers and overall flow behavior in thermal-fluid systems.[9][10] The Prandtl number is essential for non-dimensionalizing the governing equations in fluid dynamics, such as the Navier-Stokes momentum equations and the energy equation, where it appears as a coefficient governing the balance between viscous and thermal diffusion terms. By scaling velocities, lengths, and temperatures with characteristic values, the equations reduce to a form dependent on Prandtl number (along with other dimensionless groups), enabling similarity analyses and computational simulations that predict flow regimes efficiently without resolving every dimensional detail. This non-dimensional approach is particularly valuable in numerical methods, as it allows results from one set of conditions to generalize to others with matching Prandtl numbers, reducing computational complexity while maintaining physical fidelity.[4][11]Physical Interpretation
Ratio of Diffusivities
The Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity, encapsulating the relative rates at which these two transport mechanisms operate within a fluid. Momentum diffusivity, known as kinematic viscosity and denoted by ν, describes the diffusion of momentum that governs the relaxation of velocity gradients through intermolecular frictional forces. In contrast, thermal diffusivity, denoted by α, characterizes the diffusion of heat that smooths temperature gradients via molecular energy transfer.[1][12] This ratio provides a physical analogy for the competing influences of viscous and thermal transport: a high Prandtl number indicates that momentum diffuses more rapidly than heat, such that viscous shearing dominates the fluid's behavior over thermal conduction. Conversely, a low Prandtl number implies that heat diffuses faster than momentum, allowing thermal gradients to propagate more readily than velocity variations.[1] Qualitatively, in a fluid flow with localized sources of shear or heating, the diffusion layers illustrate this interplay; the layer over which momentum spreads—driven by viscous momentum exchange—extends farther relative to the thermal diffusion layer when Pr > 1, reflecting slower heat propagation, and vice versa for Pr < 1. This conceptual view highlights how the Prandtl number scales the comparative extents of these diffusive zones over time.[1] At the molecular level, momentum diffusivity stems from the transport of momentum during collisions between molecules, where faster-moving particles from high-velocity regions impart their momentum to slower ones, establishing viscosity. Thermal diffusivity, similarly rooted in molecular collisions, involves the transfer of kinetic or internal energy; in gases, this occurs through the exchange of translational kinetic energy in binary collisions, yielding a near-unity Prandtl number in simple kinetic theory models like the BGK approximation, though actual values around 2/3 for monatomic gases arise from detailed collision dynamics, and ~0.7 for diatomic gases like air. In liquids, both viscosity and thermal conductivity are dominated by short-range intermolecular forces and molecular collisions, but the involvement of specific heat capacity often leads to higher Prandtl numbers compared to gases.[12][13]Effects on Flow Regimes
The Prandtl number plays a crucial role in determining the relative thicknesses of the momentum (velocity) and thermal boundary layers in convective flows. When Pr > 1, the thermal boundary layer is thinner than the momentum boundary layer, as momentum diffusivity exceeds thermal diffusivity, leading to steeper temperature gradients near the wall. In contrast, for Pr < 1, the thermal boundary layer becomes thicker than the momentum boundary layer, resulting in more gradual temperature variations and broader heat distribution across the flow field.[9][10] In high-Prandtl-number flows, such as those involving oils where Pr can exceed 10^4, the elevated kinematic viscosity relative to thermal diffusivity enhances viscous damping, suppressing velocity fluctuations and promoting more stable, slower flow motion. This configuration also impedes heat transfer due to the reduced thermal diffusivity, confining thermal effects to a narrow layer near the surface and limiting overall convective efficiency.[14][15] Conversely, low-Prandtl-number flows, typical of liquid metals like sodium or mercury with Pr ≈ 0.01, exhibit rapid thermal diffusion that outpaces momentum diffusion, creating nearly isothermal conditions throughout much of the flow domain. The dominant thermal boundary layer facilitates efficient heat spreading, often rendering molecular conduction significant even in turbulent regimes and altering the structure of near-wall heat transfer.[10][16] The Prandtl number also affects flow stability and the transition from laminar to turbulent regimes in boundary layers, with effects varying by flow type. In some configurations like compressible plane Couette flow, lower Pr values can promote instability by increasing disturbance growth rates through enhanced thermal perturbations, while higher Pr stabilizes by damping these effects. However, in compressible boundary layers at high Mach numbers, the influence may differ.[17]Historical Development
Ludwig Prandtl's Role
Ludwig Prandtl (1875–1953), a pioneering German physicist and fluid dynamicist, is widely recognized as the founder of boundary layer theory, which revolutionized the understanding of viscous flows near solid surfaces.[18] His seminal 1904 lecture at the International Mathematics Congress in Heidelberg introduced the boundary layer concept, laying the groundwork for analyzing momentum and heat transfer in fluids with small viscosity effects.[18] In his 1910 paper titled "Eine Beziehung zwischen Wärmeaustausch und Strömungswiderstand der Flüssigkeiten," Prandtl examined heat transfer in turbulent pipe flows and highlighted the significance of the ratio of kinematic viscosity to thermal diffusivity, now known as the Prandtl number, in relating heat exchange to flow resistance.[19] Working as director of the Institute for Applied Mechanics at the University of Göttingen since 1904, Prandtl integrated this dimensionless ratio into his broader framework for turbulent transport, emphasizing its role in distinguishing between momentum and thermal diffusion processes.[18] Prandtl's explicit adoption of the symbol "Pr" for this number appeared in his subsequent works on convection and turbulence, gaining widespread acceptance in the fluid mechanics community by the post-1920s era.[20] This notation facilitated its integration with his 1925 mixing-length theory, where the Prandtl number helped model eddy diffusivities for both momentum and heat in turbulent flows, influencing heat transfer predictions in engineering applications.[21]Evolution in Fluid Dynamics
Following Ludwig Prandtl's foundational contributions to boundary layer theory, the Prandtl number was integrated into subsequent developments by Theodore von Kármán during the 1930s, particularly in analyses of turbulent boundary layers where it quantified the relative diffusion of momentum and heat.[22] Von Kármán's momentum integral approach and similarity hypotheses extended Prandtl's framework by incorporating the Prandtl number to predict the ratio of velocity to thermal boundary layer thicknesses, enabling more accurate modeling of heat transfer in high-speed and compressible flows.[23] This adoption marked an early refinement, shifting focus from laminar to turbulent regimes while emphasizing the parameter's role in scaling transport processes. By the 1960s, the Prandtl number achieved standardization in engineering practice through comprehensive heat transfer handbooks and technical reports. The inaugural edition of the Handbook of Heat Transfer (1973) by Warren M. Rohsenow and James P. Hartnett systematically included the Prandtl number in empirical correlations for forced and natural convection across diverse flow geometries and fluid properties.[24] Concurrently, NASA technical reports from the era, such as those evaluating Prandtl numbers for gases in aerospace applications, reinforced its routine use in design and analysis, promoting consistent application in propulsion and reentry simulations.[25] These resources solidified the parameter as a cornerstone for predictive modeling in thermal engineering. A key milestone in its evolution occurred with the 1981 publication of Fundamentals of Heat and Mass Transfer by Frank P. Incropera and David P. DeWitt, which presented the Prandtl number as a fundamental dimensionless group in convection chapters, influencing generations of textbooks and curricula.[26] In contemporary fluid dynamics, the Prandtl number has been extended to computational fluid dynamics (CFD) simulations of complex systems, including non-Newtonian fluids where variable viscosity modifies its effective value to capture shear-thinning behaviors in polymer processing.[27] Similarly, in multiphase flows, such as slurries or bubbly mixtures, CFD models employ the Prandtl number to resolve interfacial heat transfer and phase interactions, enhancing predictions for energy systems like nuclear reactors.[28]Values for Fluids
Experimental Data for Gases and Liquids
Experimental determination of the Prandtl number for gases and liquids involves measuring the underlying thermophysical properties: dynamic viscosity (μ), specific heat capacity at constant pressure (Cp), and thermal conductivity (k), since Pr = μ Cp / k. Viscosity is typically measured using capillary viscometers, where the flow rate through a tube under gravity or pressure provides the value via Poiseuille's law, with accuracies around ±1-2% for Newtonian fluids. Thermal conductivity is assessed through steady-state methods, such as the guarded hot plate apparatus for liquids or the hot-wire technique for both gases and liquids, achieving uncertainties of ±3-5%. Specific heat is obtained via differential scanning calorimetry or adiabatic calorimetry, with precisions better than ±1%. Direct Prandtl number measurements can also employ aerodynamic techniques, like the adiabatic recovery factor in boundary layer flows over flat plates, particularly for gases, or acoustic resonators for precise gas property evaluations.[25][29][30] The Prandtl number exhibits temperature dependence due to variations in these properties. For gases like air, Pr remains nearly constant (around 0.7) over a wide temperature range because both viscosity and thermal conductivity increase proportionally with temperature, leading to minimal change; typical variations are less than 5% from 200 K to 1000 K. In liquids, the behavior differs: for water, Pr decreases from about 5.9 at 300 K to around 1.8 at 373 K as thermal conductivity rises and viscosity falls. For mercury, a liquid metal, Pr is low (∼0.025) and shows weak temperature dependence, decreasing slightly to ∼0.02 at higher temperatures. Engine oils, with high initial Pr (>1000), experience a sharp decline (e.g., by orders of magnitude) with rising temperature due to the strong inverse temperature dependence of viscosity. These trends are derived from experimental data in engineering handbooks and NIST-referenced correlations, with overall measurement accuracies of ±5% for compiled Pr values.[31][32][10] Representative experimental values at 300 K (27°C) for common fluids are tabulated below, drawn from thermophysical property compilations and direct measurements. These illustrate the range from low-Pr liquid metals to high-Pr viscous oils, with data uncertainties noted.| Fluid | Prandtl Number (Pr) | Temperature (K) | Notes/Accuracy |
|---|---|---|---|
| Air (dry) | 0.707 | 300 | ±2%; constant over wide T range[31] |
| Water | 5.86 | 300 | ±5%; decreases with T[32] |
| Mercury | 0.0248 | 300 | ±3%; from μ=1.523×10^{-3} Pa·s, Cp=139.3 J/kg·K, k=8.54 W/m·K[33] |
| Engine Oil | 6400 | 300 | ±10%; high viscosity dominates, drops sharply with T[34] |