Prandtl Number: formula, example and interpretation

The two diffusivities Pr compares

The Prandtl number divides the rate at which momentum spreads through a fluid (kinematic viscosity ν) by the rate at which heat spreads (thermal diffusivity α). Both are diffusivities in m²/s, so the ratio is dimensionless. With no velocity or length in it, Pr is a property of the fluid at a given temperature and pressure, not of the flow.

You need Pr for two things: convection correlations give Nu as a function of Re and Pr, and Pr tells you whether the velocity or the thermal boundary layer is thicker, which guides the choice of correlation and near-wall CFD mesh.

Why μcₚ/k and ν/α give the same number

Pr = μcₚ / k = ν / α

Kinematic viscosity is ν = μ/ρ and thermal diffusivity is α = k/(ρcₚ). Dividing one by the other gives ν/α = (μ/ρ) × (ρcₚ/k) = μcₚ/k, and the density cancels. With dynamic viscosity μ (Pa·s), constant-pressure specific heat cₚ (J/(kg·K)) and thermal conductivity k (W/(m·K)) you can compute Pr without knowing ρ. If a property table lists only ν, you do need the density to convert it with μ = νρ.

Pa·s = kg/(m·s), J/(kg·K) = m²/(s²·K) and W/(m·K) = kg·m/(s³·K), so μcₚ has units of kg·m/(s³·K), and dividing by the units of k leaves 1. Entering specific heat in kJ or viscosity in cP breaks that cancellation and shifts the result by factors of 1000.

Names cause mistakes too. Tables call μ dynamic or absolute viscosity and ν kinematic viscosity, and the two are easy to swap. Go by the units: Pa·s, cP or kg/(m·s) means μ; m²/s or cSt means ν. Since 1 cSt = 10⁻⁶ m²/s, water at 20 °C has ν of about 1.0 cSt.

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Pr of water, air and sodium from a property table

Values from the IAEA property compilation. Water is saturated liquid, usable as an approximation for liquid water near atmospheric pressure; air is dry air at 98.1 kPa. The sodium table lists kinematic viscosity, 3×10⁻⁷ m²/s, so its μ is that value times the density of 844 kg/m³.

Representative properties and Pr from IAEA Tables 3.1, 3.3 and 3.10
Fluid and temperature	μ (Pa·s)	cₚ (J/(kg·K))	k (W/(m·K))	Tabulated Pr
Water 0 °C	1.7920 × 10⁻³	4220	0.5620	13.456
Water 20 °C	1.0000 × 10⁻³	4185	0.5995	6.981
Water 60 °C	4.677 × 10⁻⁴	4183	0.6510	3.005
Water 100 °C	2.807 × 10⁻⁴	4217	0.6777	1.747
Air 20 °C	1.819 × 10⁻⁵	1000	0.0258	0.71
Air 1000 °C	4.899 × 10⁻⁵	1180	0.0784	0.74
Liquid sodium 450 °C	2.532 × 10⁻⁴ (from νρ)	1272	71.2	0.0045

Between 0 °C and 100 °C the Pr of water falls from 13.456 to 1.747, a factor of about 7.70. Over the same range cₚ hardly moves and k rises by only 21%; most of the change comes from viscosity dropping by a factor of 6.4. Air stays almost constant, 0.71 at 20 °C and 0.74 at 1000 °C. The MIT 16.050 notes agree: gas Pr is close to unity, room-temperature air has Pr = 0.71, varying roughly 3% between 300 and 2000 K. So the wrong property temperature badly distorts Pr for water or oil but hardly for air. Liquid metals, with their high conductivity, fall below 0.01.

Checking the IAEA values by hand

Example 1. Water at 20 °C: undoing the column multipliers to reproduce Pr 6.981

Given (IAEA Table 3.3, saturated liquid at 20 °C): the viscosity column is headed 10⁴(Pa·s) and lists 10.000. Specific heat is 4.185 kJ/(kg·K). The conductivity column is headed 10³ W/(m·K) and lists 599.5. Density is 998.2 kg/m³ and kinematic viscosity is 1.0020×10⁻⁶ m²/s.

Unit conversion: μ = 10.000 ÷ 10⁴ = 1.0000×10⁻³ Pa·s, cₚ = 4.185 × 1000 = 4185 J/(kg·K), k = 599.5 ÷ 10³ = 0.5995 W/(m·K).

Calculation: Pr = 1.0000×10⁻³ × 4185 ÷ 0.5995 = 6.981. As a cross-check through ν/α: α = 0.5995 ÷ (998.2 × 4185) = 1.435 × 10⁻⁷ m²/s, which matches the table's thermal diffusivity column (multiplier 10⁷, value 1.435). Then ν/α = 1.0020×10⁻⁶ ÷ 1.435×10⁻⁷ = 6.983.

Interpretation: both routes agree with the tabulated 6.981. The last digit differs on the ν/α route only because α was rounded to four figures before dividing. Applying a multiplier the wrong way round (10.000 × 10⁴) puts the result off by 10⁸; the heading means "tabulated value = 10⁴ × μ".

Example 2. Water at 60 °C: how much a temperature change moves Pr and a correlation

Given (IAEA Table 3.3, saturated liquid at 60 °C): μ = 4.677×10⁻⁴ Pa·s, cₚ = 4183 J/(kg·K), k = 0.6510 W/(m·K).

Calculation: Pr = 4.677×10⁻⁴ × 4183 ÷ 0.6510 = 3.005, matching the table. Compared with 6.981 at 20 °C this is 2.32 times smaller. Broken down, viscosity falls by a factor of 2.14, conductivity rises by a factor of 1.086, and specific heat barely changes (4185 to 4183). The product 2.14 × 1.086 × 1.0005 = 2.32 recovers the ratio.

Interpretation: if you estimate h for a pipe carrying 60 °C water with 20 °C properties, the heating form of Dittus–Boelter (Nu ∝ Pr⁰·⁴) inflates Nu by 2.32⁰·⁴ = 1.40, about 40%, from the Pr term alone; viscosity also enters Re. Read μ, cₚ and k at the correlation's reference temperature, usually the bulk mean temperature for pipe flow.

Example 3. Air at 20 °C: should you use 0.705 or 0.71?

Given (IAEA Table 3.1, dry air at 98.1 kPa, 20 °C): the viscosity column, multiplier 10⁶(Pa·s), lists 18.19; specific heat is 1.00 kJ/(kg·K); the conductivity column, multiplier 10² W/(m·K), lists 2.58. The same row gives kinematic viscosity 15.61×10⁻⁶ m²/s, thermal diffusivity 22.0×10⁻⁶ m²/s and Pr 0.71.

Conversion and calculation: μ = 1.819×10⁻⁵ Pa·s, cₚ = 1000 J/(kg·K), k = 0.0258 W/(m·K), so Pr = 1.819×10⁻⁵ × 1000 ÷ 0.0258 = 0.705. From the same row, ν/α = 15.61 ÷ 22.0 = 0.710. The 98.1 kPa table pressure does not matter here: μcₚ/k has no density, and gas μ and k barely change over that pressure difference.

Interpretation: the 0.7% gap is table rounding: cₚ printed as 1.00 carries ±0.5% and k as 2.58 carries ±0.2%. Use 0.71 for estimating h, but 0.705, recomputed from the same μ, cₚ and k, when cross-checking an identity such as Pe = Re × Pr to the last digit.

What Pr says about boundary-layer thickness

In a laminar flat-plate boundary layer, Pr almost directly sets the ratio of the velocity boundary-layer thickness δu to the thermal thickness δT. The MIT 3.185 notes give δu/δT ≈ 0.975Pr^1/3 for fluids with Pr above about 0.6 and δu/δT ≈ 0.72Pr^1/2 for liquid metals with Pr below about 0.05. With the values from the table:

Water at 20 °C (Pr 6.981): δu/δT ≈ 0.975 × 6.981^1/3 = 1.86, so the thermal layer is a little more than half as thick as the velocity layer.
Water at 60 °C (Pr 3.005): δu/δT ≈ 1.41.
Air at 20 °C (Pr 0.71): δu/δT ≈ 0.87; the two layers are similar, with the thermal layer slightly thicker.
Liquid sodium at 450 °C (Pr 0.0045): δu/δT ≈ 0.72 × √0.0045 = 0.048, so the thermal layer is about 21 times thicker.

In a high-Pr fluid the temperature gradient sits in a thin layer at the wall, so a near-wall mesh sized from the velocity profile alone can miss the heat flux. In liquid metals the temperature field spreads well beyond the velocity layer and ordinary-fluid correlations fail; the INL RELAP-7 theory manual switches to a separate correlation, Nu = 5.0 + 0.025Pe^0.8, for liquid metals with Pr below 0.1. In turbulent flow eddies carry heat and momentum about equally, so MIT 3.185 puts the turbulent Prandtl number very close to one, with layers of similar thickness. The thickness ratios above apply to laminar plate flow only.

Pr also enters the size of the heat-transfer coefficient. By the laminar plate result in the same MIT notes, Nuₓ = 0.332Reₓ^1/2Pr^0.343, water at 20 °C has a local Nu (6.981/0.71)^0.343 = 2.19 times that of 20 °C air at the same Re and position. Converting to h = Nu·k/x adds the conductivity ratio of about 23 (0.5995 against 0.0258), so k dominates the gap in h and Pr adds a factor of about two.

Typical ways Pr comes out wrong

Specific heat entered in kJ: putting cₚ = 4.185 for 20 °C water gives Pr = 0.00698. Water below 0.01 means a unit error.
Viscosity entered as a cP number: 1 cP = 1 mPa·s = 0.001 Pa·s. Entering 20 °C water's 1.0 cP as 1.0 Pa·s gives Pr = 6981.
ν in the μ field: entering 1.0020×10⁻⁶ m²/s as μ gives 0.00699, smaller by the density factor of 998.2.
Properties from different temperatures: as Example 2 shows, the viscosity of water more than halves over 40 °C. Mixing rows describes no real state.
Engineering-unit tables mixed with SI: older handbooks list specific heat in kcal/(kg·°C), conductivity in kcal/(m·h·°C) and viscosity in kg/(m·h). One consistent unit system gives the same Pr: 20 °C water entered as μ = 3.6 kg/(m·h), cₚ = 0.9996 kcal/(kg·°C) and k = 0.5155 kcal/(m·h·°C) still gives Pr = 6.981 (using 1 kcal/(m·h·°C) = 1.163 W/(m·K)). Keep μ and cₚ in SI but enter k in kcal units, though, and you get 8.12. That lies between the Pr of water at 10 °C and 15 °C (9.432 and 7.937 in the IAEA table), so a range check will not flag it.
Molecular versus turbulent Pr: the turbulent Prandtl number Pr_t in a CFD turbulence model is a model constant linking eddy viscosity to eddy thermal diffusivity. It is not the property-table Pr.
Judging heat transfer from Pr alone: Pr gives neither h nor a heat rate; you also need Re and a correlation for the geometry.

Prandtl number questions

Why does the Pr of water change so much with temperature?

Water viscosity is very temperature-sensitive: in the IAEA table μ falls by a factor of 6.4 from 0 °C to 100 °C while cₚ and k change far less, so Pr drops from 13.456 to 1.747.

Can I just use Pr = 0.71 for air, or should I recompute it from the table?

For estimating h, 0.71 is enough; the 0.705 from the rounded IAEA inputs differs only within rounding. For last-digit cross-checks such as Re × Pr = Pe, use the Pr computed from the same inputs.

Can I find Pr without knowing the density?

Yes: Pr = μcₚ/k contains no density. You need ρ only for the ν/α form, or to convert a tabulated ν with μ = νρ, as for sodium.

Is this the same as the turbulent Prandtl number I enter in CFD?

No. The calculator returns the molecular Pr, a fluid property. The turbulent Prandtl number is a turbulence-model coefficient linking eddy viscosity to turbulent heat conduction, set by the model, not the fluid. Wall functions and the viscous sublayer still need the molecular Pr.

Pr is multiplied by flow numbers or compared with other diffusivity ratios.

Péclet number (Pe): Multiplying the Reynolds number by Pr gives Pe, which compares heat carried by the flow with heat conducted.
Nusselt number (Nu): In pipe correlations such as Dittus–Boelter, Pr is one of the two inputs that fix Nu.
Schmidt number (Sc): The mass-transfer counterpart of Pr, with the mass diffusivity in the denominator instead of the thermal diffusivity.

References

IAEA: Thermophysical Properties of Materials for Nuclear Engineering (2008). μ, cₚ, k and Pr for dry air (Table 3.1), saturated water (Table 3.3) and sodium (Table 3.10)
MIT OpenCourseWare (3.185): Transport Phenomena in Materials Engineering. Pr = ν/α as a fluid property, boundary-layer ratios 0.975Pr^1/3 and 0.72Pr^1/2, turbulent Prandtl number close to one
MIT OpenCourseWare (16.050): Introduction to Engineering Heat Transfer. Definition Pr = μcₚ/k, Pr = 0.71 for room-temperature air, about 3% variation from 300 to 2000 K
Michigan Technological University: CM3120 Module 2 Lecture IV. Pr and Sc as material-property ratios of the diffusivities ν, α and D; Pe = RePr
Idaho National Laboratory: RELAP-7 Theory Manual, Rev. 2. Pr exponent in Dittus–Boelter (0.4 heating, 0.3 cooling) and Nu = 5.0 + 0.025Pe^0.8 for liquid metals with Pr < 0.1