Formula and Method for the Prandtl Number
The Prandtl number (Pr) is a dimensionless quantity that compares a fluid's ability to diffuse momentum (through viscosity) to its ability to diffuse heat (through thermal conduction). It is defined as Pr = cp·μ / k, where cp is specific heat capacity, μ is dynamic viscosity, and k is thermal conductivity. Equivalently, Pr = ν/α, the ratio of kinematic viscosity ν = μ/ρ to thermal diffusivity α = k/(ρ·cp). This calculator computes Pr along with ν and α from your fluid's properties.
How the calculation works
Enter the fluid's dynamic viscosity (in centipoise or pascal-seconds), specific heat capacity, thermal conductivity, and density. The calculator first converts viscosity to SI units (Pa·s) if you entered centipoise, then multiplies specific heat capacity by dynamic viscosity and divides by thermal conductivity to get Pr = cp·μ/k. It also computes kinematic viscosity (ν = μ/ρ) and thermal diffusivity (α = k/(ρ·cp)) so you can confirm Pr = ν/α independently.
Common mistakes
- Confusing dynamic and kinematic viscosity: dynamic viscosity μ (Pa·s) and kinematic viscosity ν (m²/s) differ by a factor of density — mixing them up throws Pr off by orders of magnitude.
- Using molar heat capacity: Pr requires specific (per-mass) heat capacity in J/(kg·K), not molar heat capacity in J/(mol·K).
- Ignoring temperature dependence: Pr changes significantly with temperature for most fluids, especially liquids — use property values at the actual operating temperature rather than a generic table value.
Real-world applications
- Convective heat transfer correlations, such as Nu = 0.023·Re^0.8·Pr^n for turbulent pipe flow, use Pr to relate fluid friction and heat transfer.
- In boundary layer theory, Pr indicates whether the thermal boundary layer is thicker or thinner than the velocity boundary layer — low-Pr liquid metals have thick thermal boundary layers relative to velocity, while high-Pr oils have the opposite.
- Heat exchanger and cooling system design uses Pr to choose correlations appropriate for the working fluid, whether air, water, oil, or a liquid-metal coolant.
Frequently Asked Questions
What does the Prandtl number measure?
The Prandtl number is the ratio of momentum diffusivity (kinematic viscosity, ν) to thermal diffusivity (α) in a fluid: Pr = ν/α = cp·μ/k. A low Pr means heat diffuses much faster than momentum, as in liquid metals; a high Pr means momentum diffuses much faster than heat, as in oils.
What are typical Prandtl numbers for common fluids?
Air at room temperature has Pr ≈ 0.71, water at 20°C has Pr ≈ 7.0, engine oil ranges from about 100 to over 10,000 depending on temperature, and liquid metals such as mercury or sodium have Pr around 0.01-0.03.
How does the Prandtl number relate to boundary layer thickness?
For Pr > 1, the thermal boundary layer is thinner than the velocity (momentum) boundary layer, roughly by a factor of Pr^(1/3) in laminar flow over a flat plate. For Pr < 1, the thermal boundary layer extends further from the surface than the velocity boundary layer.
Can I calculate the Prandtl number directly from kinematic viscosity and thermal diffusivity?
Yes. If you already know kinematic viscosity (ν) and thermal diffusivity (α) for your fluid, Pr = ν/α directly, without needing density, specific heat, or thermal conductivity separately. This calculator uses the equivalent form Pr = cp·μ/k because those fluid properties are more commonly published in reference tables.
Common Tip Percentages
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|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
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| Needs | 50% | Rent, groceries, utilities, insurance |
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |