Elenbaas Number
Computes the Elenbaas number of a channel formed by two heated vertical parallel plates, El = g·β·ΔT·b⁴ ÷ (ν·α·H), which is the Rayleigh number based on the plate spacing b multiplied by the ratio b/H. It governs natural convection in finned heat sinks, electronics enclosures and solar collectors: a very small El means a narrow, tall channel where the boundary layers merge and choke the flow, while a large El means the plates behave as isolated. The spacing that maximises a heat sink's total dissipation falls near El ≈ 46 for isothermal plates, a classic natural-convection fin design criterion. Gravity is taken as 9.80665 m/s², β is the fluid thermal expansion coefficient and H the plate height. Enter the expansion coefficient, the temperature difference, the spacing, the kinematic viscosity, the thermal diffusivity and the plate height.
Result
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Elenbaas Number: optimum spacing for vertical fins
A fanless heat sink runs into one trade-off on the first design pass: more fins give more area, yet they squeeze the channel until the boundary layers meet and the air stops rising. Anyone who has measured such a prototype knows the outcome — junction temperature climbs right after fins were added. The Elenbaas number tells you which side of that trade-off a chosen spacing lands on, and it reads the same way for a sealed enclosure, a finned transformer tank or a solar collector.
El = g·β·ΔT·b⁴ ÷ (ν·α·H) is the spacing-based Rayleigh number multiplied by b/H. Gravity is fixed at 9.80665 m/s², β is the fluid thermal expansion coefficient (1/T_film for an ideal gas), ΔT the plate-to-ambient difference, b the spacing, ν the kinematic viscosity, α the thermal diffusivity and H the plate height. That fourth power on the spacing rules everything: a 10 percent error in b moves El by 46 percent. The classic target for maximum total dissipation sits near El = 46 for isothermal plates, though different correlations place it anywhere from 28 to 55, so read it as a band rather than a magic value.
The correlation came from flat vertical isothermal plates with the channel open top and bottom, surrounded by still air, in laminar flow. It stops applying once a fan runs, once the top of the channel is blocked, or with tilted fins and pin-fin arrays. Radiation stays outside the model and is no footnote: on a black anodised sink it carries 20 to 40 percent of the heat, which shifts the real optimum spacing. Fluid properties belong at film temperature, and b and H belong in metres — entering b in millimetres inflates El by a factor of a trillion.
Frequently asked questions
Does the default El = 75.69 mean the spacing is right?
Where do the β, ν and α values in the form come from?
Can I use this with a plate colder than the surrounding air?
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