Cross-check Re, Pr and Pe for pipe flow

Start with the question

Consider a circular pipe with a 10 mm internal diameter and a mean fluid speed of 0.2 m/s. We want to compare inertia with viscosity and then compare thermal advection with diffusion. These are different questions: a Reynolds number alone cannot answer the heat-transfer question. This exercise connects three calculators and checks whether their outputs are consistent.

The constant properties below are illustrative inputs, not a certified water-property table. For an experiment, replace them with properties evaluated at the same temperature and pressure. Use the internal diameter as L throughout this exercise.

1. Prepare the inputs in SI units

Entering 10 for diameter tells this calculator to use 10 metres. Re and Pe would both be 1000 times too large, but their relationship would still hold. An algebraic cross-check therefore does not replace a unit check.

2. Calculate Reynolds number

Check it with kinematic viscosity: ν = μ/ρ = 10⁻⁶ m²/s, giving Re = vL/ν = 0.002/10⁻⁶ = 2000. Entering ν into a field labelled μ changes the physical meaning because their units differ.

For fully developed flow in a smooth circular pipe, 2000 is in the usual laminar range. Entrance effects, disturbances and roughness still need separate consideration; a displayed regime label does not measure those conditions.

3. Calculate Pr and thermal diffusivity

Momentum diffusivity is seven times thermal diffusivity for these inputs. With properties held constant, doubling the flow speed leaves Pr unchanged. If Pr changes when only speed changes, check whether another property or unit changed as well.

4. Check Pe by two routes

The second route gives Re × Pr = 2000 × 7 = 14000. Expanding the product gives (ρvL/μ)(μcₚ/k) = ρcₚvL/k = vL/α. This identity requires the same length, speed and property conditions. If you round α to 1.43×10⁻⁷ first, the direct result is approximately 13986. Check input precision before treating this small difference as a calculator error.

The timescale interpretation offers another check: t_advection = L/v = 0.05 s and t_diffusion = L²/α = 700 s. Their ratio is 14000. These are scales based on the chosen diameter; they do not predict outlet temperature or the time required to heat the whole pipe.

Troubleshoot a mismatch

1. Check whether one calculation uses diameter while another uses total pipe length.
2. Distinguish μ from ν and convert specific heat from kJ to J if needed.
3. Derive α from the same ρ, cₚ and k.
4. Separate thermal Pe from species Pe. With a species diffusivity D, the corresponding identity is Re × Sc.

Changing only v to 0.4 m/s gives Re = 4000, Pr = 7 and Pe = 28000. This demonstrates sensitivity of the formulas with fixed properties; it is not a solution for a heated flow whose properties vary.

What these outputs cannot determine

These ratios do not provide pressure loss, a heat-transfer coefficient h, wall temperature or heat rate. To evaluate Nu = hL/k, first obtain h from a measurement or a correlation appropriate to the geometry and boundary conditions. Record property sources, evaluation temperature and the definition of L with your results so another calculation can be compared on the same basis.