How is packed-bed heat transfer calculated?

Find the heat transfer coefficient around one particle with the Wakao–Kaguei correlation Nu = 2 + 1.1Re^0.6Pr^(1/3), then multiply by particle surface area per unit volume a = 6(1−ε)/d to get the volumetric coefficient h·a. CFD treating the bed as a porous continuum uses h·a as the fluid–solid exchange coefficient.

What happens with finer particles?

Surface area and h both rise, so the volumetric coefficient grows a lot: in a rock bed with 0.3 m/s air, going from 40 mm to 5 mm raised h·a twentyfold, from 3,200 to 64,000 W/m³K. But pressure drop rose elevenfold, from 47 Pa to 536 Pa, increasing fan power.

How is particle size chosen for a thermal store?

In a 1 m bed even 40 mm particles give NTU 9, so outlet air nearly reaches bed temperature. With heat exchange ample, size is set mainly by pressure drop (fan power) and the width of the thermal front in the bed. At 40 mm the front width (length for NTU 3) is 0.33 m, a third of the bed, reducing usable storage.

Packed Bed Flow and Heat Transfer

Category: Fluid Analysis – Porous Media | Revised 2026-10-01

CAE visualization for packed bed theory - technical simulation diagram

Theory: heat exchange between particles and fluid

Overview

🙋

Some devices store heat by blowing air through beds of rocks or balls. How is particle size decided?

🎓

Smaller particles add surface and exchange heat better, but air passes less easily and pressure drop rises — size balances the two. Yet calculation shows that in a bed around 1 m long, even fairly large particles exchange heat amply, with outlet air nearly reaching bed temperature. So in practice fan power and how much the boundary between hot and cold zones (the thermal front) smears are the main drivers. Let's calculate a rock-bed store for several particle sizes.

Volumetric coefficient and NTU

$$ Nu_p = 2 + 1.1\,Re_p^{0.6}Pr^{1/3},\qquad a = \frac{6(1-\varepsilon)}{d_p},\qquad NTU = \frac{h\,a\,L}{\rho c_p u} $$

Re_p uses superficial velocity u (as if no particles) and diameter d_p. NTU measures how closely the fluid approaches solid temperature over length L; NTU = 3 closes 95% of the difference.

Pressure drop (Ergun)

$$ \frac{\Delta p}{L} = 150\,\frac{\mu u (1-\varepsilon)^2}{\varepsilon^3 d_p^2} + 1.75\,\frac{\rho u^2 (1-\varepsilon)}{\varepsilon^3 d_p} $$

Coffee Break Trivia

Schumann's packed-bed model

Schumann in the US first solved heat exchange between fluid and particles in a packed bed analytically in 1929, treating fluid and solid temperatures separately. This two-temperature model underpins packed-bed heat transfer analysis from blast-furnace stoves and chemical reactors to solar rock-bed storage and recent electricity-to-heat storage. Wakao and Kaguei compiled their particle heat transfer correlation in 1982.

Worked example: a rock-bed store

Air 25 °C, superficial velocity 0.3 m/s, bed length 1.0 m, porosity 0.40, rock conductivity 2.8 W/mK.

Particle size	Re	h	Specific area	Volumetric h·a	NTU	Pressure drop	Fan power
5 mm	96	89 W/m²K	720 m²/m³	64,200 W/m³K	180	536 Pa	161 W/m²
10 mm	191	65 W/m²K	360 m²/m³	23,400 W/m³K	66	221 Pa	66 W/m²
20 mm	383	48 W/m²K	180 m²/m³	8,600 W/m³K	24	99 Pa	30 W/m²
40 mm	765	36 W/m²K	90 m²/m³	3,200 W/m³K	9.0	47 Pa	14 W/m²

Thermal-front width

Particle size	Length for NTU = 3	Particle Biot number
5 mm	0.017 m	0.027
10 mm	0.046 m	0.039
20 mm	0.124 m	0.057
40 mm	0.334 m	0.085

🙋

If NTU of 9 already gives ample exchange, what's the drawback of bigger particles?

🎓

Outlet temperature alone looks fine, but in a store the temperature distribution in the bed matters. Hot air drives a hot–cold boundary (thermal front) through the bed; larger particles smear it, to 0.33 m — a third of the bed — at 40 mm. Once the front reaches the outlet, outlet temperature starts dropping, so a smeared front uses only part of the bed. Smaller particles sharpen the front but raise pressure drop and fan power; balancing the two often lands on about 10–20 mm. Particle Biot numbers are all below 0.1, so each rock can be represented by one temperature.

Analysis tips

Use a two-temperature model (no local thermal equilibrium).
If particle Biot exceeds 0.1, solve the temperature inside particles too.
Account for higher porosity and channelling near walls (wall effect).
Include axial conduction and dispersion that smear the front.
Find the fan operating point from pressure drop and evaluate at the real flow.

Coffee Break Trivia

“Only the walls heated early”

In an experimental thermal store, outlet temperature began rising earlier than expected. Arrays of sensors showed the thermal front advancing first along the container wall: particles pack less tightly there, so porosity is higher and air channels preferentially. The particles were too large relative to the container diameter; smaller particles or baffles along the wall fixed it. When the container-to-particle diameter ratio is small, this wall effect can't be ignored.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Assuming local equilibrium	Front too sharp	Two-temperature model
Mixing interstitial and superficial velocity	Wrong Re and Δp	Check definitions
Ignoring wall effect	Channelling missed	Near-wall porosity
Outlet temperature only	Capacity overrated	Check front width
Unchecked particle Biot	Errors for large particles	Check Bi < 0.1

🙋

I'd like to learn related topics.

🎓

Rate this article

Thank you for your feedback!

Helpful

More details

Report error

Helpful

More details

Report error

Written by the NovaSolver Editorial Team
Built with engineers and AI (editorial policy) — Sitemap

About the Authors

Packed Bed Flow and Heat Transfer

Theory: heat exchange between particles and fluid

Overview

Volumetric coefficient and NTU

Pressure drop (Ergun)

Schumann's packed-bed model

Worked example: a rock-bed store

Thermal-front width

Analysis tips

“Only the walls heated early”

Common mistakes

Mistakes and fixes

Related Topics

Related fields