Graetz Number

Fluid entering a heated tube does not arrive with a fully developed temperature profile; it arrives flat, and the heat has to work inward from the wall as the fluid travels. For the first stretch of the tube the centreline has not heard about the wall at all, the temperature gradient at the surface is steep, and the local heat transfer coefficient is far above what it will eventually settle to. The Graetz number measures how far into that story you are: $\mathrm{Gz} = (D/L)\,\mathrm{Re}\,\mathrm{Pr}$, which compares the time the fluid spends in the tube with the time heat needs to diffuse from the wall to the centre.

The consequences are practical and often large. In fully developed laminar flow the Nusselt number is a constant — 3.66 at a constant wall temperature, 4.36 at a constant wall heat flux — and it depends on nothing at all, not the velocity, not the fluid. In the entry region it can easily be two or three times that, and it is still falling. Using 3.66 on a short tube therefore UNDERSTATES the transfer, which is the flattering direction for a design and the dangerous one for a rating. The entry-length correlations that fix this, Hausen and Sieder–Tate, are both written directly in Gz for exactly this reason. As a rule of thumb the thermal entry length in laminar flow is about $0.05\,D\,\mathrm{Re}\,\mathrm{Pr}$, which is Gz ≈ 20 — and for water at Re = 1500 in a 20 mm tube that is 21 m of pipe. Small-bore laminar exchangers are essentially all entry region.

Graetz is a composite. The product $\mathrm{Re}\cdot\mathrm{Pr}$ is the thermal Péclet number, advection over thermal diffusion, so $\mathrm{Gz} = (D/L)\,\mathrm{Pe}$ — Péclet scaled by the duct's aspect ratio. Reading it that way makes the meaning transparent: Péclet says how strongly the flow outruns diffusion, and $D/L$ turns that into a statement about this particular tube. The catalog carries the mass-transfer Péclet number already, and the thermal one is the same group with the thermal diffusivity underneath.

Three cautions, and the first is the one that catches people. Conventions differ, and by a lot. Many texts define the INVERSE, $x^{*} = L/(D\,\mathrm{Re}\,\mathrm{Pr})$, and call THAT the Graetz number or the dimensionless axial distance — so a large Gz in one book is a small Gz in another. A third convention inserts $\pi/4$ and writes the group as $\dot{m}c_p/(kL)$. All three describe the same physics, and a threshold carried across without checking is off by whatever separates them. Second, the classical Graetz problem assumes the VELOCITY profile is already developed while the thermal one is not, which is a good approximation for large Prandtl numbers and a poor one for liquid metals, where heat outruns momentum and the combined entry problem has to be solved. Third, this is a laminar tool. In turbulent flow the entry length is short — ten to twenty diameters, not a thousand — which is precisely why the Dittus–Boelter correlation can get away with a single blanket restriction of $L/D > 10$ and never mention Graetz at all.