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Prandtl Number
Dimensionless ratio comparing momentum diffusivity to thermal diffusivity of a fluid, used in convective heat transfer correlations.
Interpretation
Prandtl number: Pr = c_p·μ/k, ratio of momentum diffusivity to thermal diffusivity. Example: Water at 20°C: Pr ≈ 6.97, indicating stronger momentum transfer.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| Pr | Prandtl number | |
| cp | Specific heat capacity | J/kg.K |
| mu | Dynamic viscosity | Pa.s |
| k | Thermal conductivity | W/m.K |
What it means
The Prandtl number is a dimensionless fluid property defined as Pr = c_p μ / k, where c_p is specific heat at constant pressure, μ is dynamic viscosity, and k is thermal conductivity. It compares the rate of momentum diffusion (viscous effects) to thermal diffusion (heat conduction). A low Pr (e.g., liquid metals, Pr << 1) means heat diffuses faster than momentum; a high Pr (e.g., oils, Pr > 100) means momentum diffuses faster. For air and water, Pr is around 0.7 and 7, respectively. This number is important in heat transfer correlations: it appears in the Nusselt number equations for forced and natural convection. It also helps classify fluids and choose appropriate empirical models. The Prandtl number is essential for heat exchanger design and for understanding thermal boundary layers. It is named after Ludwig Prandtl, the father of modern fluid mechanics.
Worked example
Prandtl Number – Two Examples
Real‑World| Parameter | Value |
|---|---|
| cp | 4180 J/kgK |
| μ | 0.001 Pa·s |
| k | 0.6 W/mK |
| Parameter | Value |
|---|---|
| cp | 1005 |
| μ | 1.8×10⁻⁵ |
| k | 0.026 |
Common mistakes
- Specific heat c_p: At constant pressure, in J/(kg·K).
- Dynamic viscosity μ: In Pa·s (kg/(m·s)).
- Thermal conductivity k: In W/(m·K).
- Units: All SI units give Pr dimensionless.
- Fluid properties: All taken at the film temperature for convection problems.
Applications
The Prandtl number, Pr = c_p·μ/k, is a dimensionless fluid property that represents the ratio of momentum diffusivity to thermal diffusivity. It indicates the relative thickness of the velocity and thermal boundary layers. Pr is used in convective heat transfer correlations, as it influences the Nusselt number. For example, in pipe flow, the Dittus‑Boelter equation includes Pr^(0.3 or 0.4). Engineers use Pr to select working fluids for heat transfer applications: low Pr fluids (like liquid metals) have high thermal conductivity, while high Pr fluids (like oils) have high viscosity. Understanding Pr helps in choosing coolants, designing heat exchangers, and interpreting heat transfer data. It is also essential in scaling laboratory results to industrial equipment.
- Selection of heat transfer fluids (water, oils, refrigerants, liquid metals)
- Interpretation of convective heat transfer correlations
- Thermal design of heat exchangers and cooling loops
- Analysis of boundary layer development in ducts and plates
- Educational tool for understanding transport analogy (heat and momentum)
Frequently Asked Questions
The Prandtl number is the ratio of momentum diffusivity to thermal diffusivity: Pr = c_p·μ / k = ν / α. It indicates the relative thickness of the hydrodynamic and thermal boundary layers.
- Gases: Pr ≈ 0.7 (air, helium).
- Liquids: Pr varies widely – water at 20°C: ≈ 7; oils: 100–100,000; liquid metals: 0.01–0.03.
Most convective heat transfer correlations include Pr as a parameter. For example, the Dittus‑Boelter equation for turbulent pipe flow has Pr^n (n = 0.4 or 0.3). For laminar flow, Nu often depends on (Re·Pr·D/L) or a constant for fully developed flow.
- Evaluating fluid properties at the wrong reference temperature (e.g., bulk vs film temperature).
- Using the wrong definition for c_p (specific heat at constant pressure).
- Confusing Pr with the Schmidt or Lewis numbers.
When Pr = 1, the momentum and thermal boundary layers have the same thickness. This is approximately true for some gases.
In natural convection, Nu is a function of Ra (Rayleigh number) and Pr. Correlations often have the form Nu = C·Ra^n·Pr^m, where m is about 0.25 for many geometries.
The thermal entry length is about L_th ≈ 0.05·Re·Pr·D for laminar flow. The hydrodynamic entry length is L_h ≈ 0.05·Re·D. For Pr > 1, the thermal boundary layer develops more slowly.
Use properties at the film temperature T_f = (T_wall + T_bulk)/2 for external flow, or at the bulk temperature for internal flow, depending on the correlation.