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Result
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What the three fields produce
The calculator evaluates μ × cₚ ÷ k exactly as entered. Take all three values from one row of a property table, for one fluid at one temperature, and the result is the Prandtl number of that state, equal to the ratio ν/α of kinematic viscosity to thermal diffusivity. Pipe diameter and flow speed play no part, so the same properties give the same answer whatever device you are analysing. The heat-transfer coefficient, which does depend on the flow, comes in the next step, when this Pr goes into a correlation together with Re.
Variables and SI units
| Symbol | Meaning | SI unit | Fill example values | IAEA water, 20 °C |
|---|---|---|---|---|
| μ | Dynamic viscosity of the fluid | Pa·s | 0.001 | 1.0000 × 10⁻³ |
| cₚ | Specific heat at constant pressure | J/(kg·K) | 4180 | 4185 |
| k | Thermal conductivity of the fluid | W/(m·K) | 0.6 | 0.5995 |
Common unit mistakes
- If the table gives specific heat in kJ/(kg·K), multiply by 1000. Forgetting this makes the result exactly a thousand times too small, and because the leading digits still look right it is easy to miss.
- Viscosity in cP or mPa·s must be multiplied by 0.001 to give Pa·s. If a table heading carries a multiplier such as 10⁴(Pa·s) or 10⁶(Pa·s), divide the tabulated number by that factor.
- If the table lists only kinematic viscosity ν (m²/s), multiply by the density at the same temperature, μ = νρ, before entering it.
- Some sources list conductivity in mW/(m·K); divide those values by 1000.
Reading the size of Pr
A result above 1 means momentum spreads faster than heat, and in laminar flow the velocity boundary layer tends to be the thicker one; below 1 the reverse holds. To judge whether a result is plausible, compare it with these ranges. In the IAEA tables air stays between 0.71 and 0.74 from 20 °C to 1000 °C, and liquid water runs from 13.456 at 0 °C to 1.747 at 100 °C. Liquid metals are often below 0.05 (sodium at 450 °C is 0.0045), and viscous oils sit far above water. If you entered water and got something below 0.01 or above 1000, a unit multiplier is almost certainly wrong.
Examples you can reproduce in the calculator
Example 1. What water do the "Fill example values" inputs describe?
Inputs: the button fills μ = 0.001 Pa·s, cₚ = 4180 J/(kg·K) and k = 0.6 W/(m·K), room-temperature water properties rounded to two or three figures.
Calculation: Pr = 0.001 × 4180 ÷ 0.6 = 6.97.
Comparison: entering saturated water at 20 °C from IAEA Table 3.3 (μ = 1.0000×10⁻³ Pa·s, cₚ = 4185 J/(kg·K), k = 0.5995 W/(m·K)) gives 6.981, the same as the tabulated Pr. The rounded inputs come out 0.2% lower, which is fine for rough work. When you want results from several calculators to agree to the last digit, keep using one property set throughout.
Example 2. Liquid sodium at 450 °C: building μ from a table that lists only ν
Given (IAEA Table 3.10, liquid at 450 °C): kinematic viscosity 3×10⁻⁷ m²/s, density 844 kg/m³, specific heat 1.272 kJ/(kg·K), thermal conductivity 71.2 W/(m·K); the tabulated Pr is 0.0045.
Unit conversion: μ = νρ = 3×10⁻⁷ × 844 = 2.532 × 10⁻⁴ Pa·s and cₚ = 1.272 × 1000 = 1272 J/(kg·K). Enter 0.0002532 in the μ field, 1272 for cₚ and 71.2 for k.
Calculation: Pr = 2.532×10⁻⁴ × 1272 ÷ 71.2 = 0.00452, consistent with the tabulated 0.0045. The table gives ν to one significant figure, so only the first two digits of the result mean anything.
Interpretation: sodium's viscosity is about 4 times lower than that of water, but its conductivity is more than 100 times higher. The result is a Pr roughly 1/1500 of that of 20 °C water. In this range the pipe-flow correlations written for water and air are not used as they stand; INL RELAP-7 applies a separate Nu correlation based on Pe to liquid metals with Pr < 0.1.
Example 3. Reproducing unit mistakes with 60 °C water
Correct inputs (IAEA Table 3.3, 60 °C): μ = 0.0004677 Pa·s, cₚ = 4183 J/(kg·K) and k = 0.6510 W/(m·K) give 3.005.
Specific heat entered in kJ: typing 4.183 in the cₚ field gives 0.003005. The digits 3005 survive; only the decimal point moves three places.
Viscosity entered in cP: typing the same water's viscosity, 0.4677 cP, straight into the μ field gives 3005.
Interpretation: both mistakes keep the significant digits of the correct answer, so the result screen looks plausible. Comparing every result against the size ranges above is the fastest way to catch them.
The pipe-flow note builds Pr = 7 from cₚ = 4200, recomputes the same state as ν/α to confirm 7, and then checks that multiplying by Re = 2000 gives the 14000 returned by the Péclet calculator. Keeping one property set as you pass this result to other calculators makes that kind of check straightforward.
Using the Prandtl number calculator
My result differs from the tabulated Pr in the last digit. What went wrong?
Usually nothing; it is rounding. In a table such as the IAEA air table, where specific heat is printed to three figures as 1.00, μcₚ/k gives 0.705 against a tabulated 0.71, a gap of about 0.7%. If the gap is a factor of 10 or 1000, then recheck the unit multipliers.
How do I handle a mixture such as a glycol solution?
Enter μ, cₚ and k of the mixture itself, all at the same temperature. Do not average the component Prandtl numbers by concentration. Viscosity in particular varies non-linearly with concentration and temperature, so use a property table or measurement for that specific concentration.
What do I do with Pr once I have it?
Usually you put it into a convection correlation together with the Reynolds number to get the Nusselt number, then obtain the heat-transfer coefficient from h = Nu × k ÷ D. Check that the reference temperature the correlation asks for is the one at which you computed Pr.
References
- IAEA: Thermophysical Properties of Materials for Nuclear Engineering (2008). Properties and Pr of saturated water at 20 °C and 60 °C (Table 3.3), dry air (Table 3.1) and sodium at 450 °C (Table 3.10)
- MIT OpenCourseWare (3.185): Transport Phenomena in Materials Engineering. Pr = ν/α as a fluid property and the liquid-metal regime below Pr of about 0.05
- Idaho National Laboratory: RELAP-7 Theory Manual, Rev. 2. Separate Nu correlation for liquid metals with Pr < 0.1
Related guides and calculators
Guide
Calculation notes
- Cross-check Re, Pr and Pe for pipe flow: Shows the Pr from this calculator multiplied by the Reynolds number and matched against the Péclet calculator.
- Thermal and mass diffusion: the distinct roles of Pr, Sc, Pe and Nu: How this result feeds into judging thermal boundary-layer thickness, and how it parallels the Schmidt number.
Related calculators
- Péclet number (Pe): Multiply this result by Re and compare with the Pe calculator to check your inputs.
- Nusselt number (Nu): Uses the same k when you turn a correlation Nu or a measured h into the other.
- Schmidt number (Sc): The mass-diffusion counterpart; in liquids it comes out far larger than Pr.