Engineering task and calculation objective
This module calculates the heat transfer in forced convection through packed beds (particle beds and fixed beds) according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section G9). Packed beds of spheres, cylinders, cubes, or other packing elements with fluid flowing through them are ubiquitous in process engineering: fixed-bed reactors with catalyst packings, adsorbers, regenerators and thermal storage units, drying and cooling processes for bulk solids, or packed columns with heat exchange between gas and packing.
The calculation follows the well-established approach of Gnielinski: the heat transfer between fluid and particle surface is traced back to the correlation for a single body in cross flow. For this purpose, the Reynolds number is formed with the velocity in the void space — superficial velocity divided by the void fraction — and the characteristic particle dimension. An arrangement factor, which depends on the void fraction and the particle shape (sphere, cylinder, cube), raises the single-body value to the packed-bed value.
If you want to calculate the heat transfer coefficient in a fixed bed — for example for the heat-up time of a thermal storage unit or the temperature control of a catalyst bed — this module delivers the Nusselt number and the fluid-to-particle heat transfer coefficient according to the VDI Heat Atlas.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define particle shape and bed data: The inputs are the particle shape (sphere, cylinder, cube, or other packing elements), the characteristic particle dimension (for non-spherical particles, the diameter of a sphere with the same surface area), and the void fraction of the bed.
- Form the interstitial velocity and Reynolds number: The approach velocity referred to the empty cross section (superficial velocity) is divided by the void fraction; together with the particle dimension and the kinematic viscosity, this yields the Reynolds number of the packed bed.
- Calculate the Nusselt number of the single body: For a single sphere in cross flow, the Nusselt number is formed as the sum of the pure-conduction limit (Nu = 2) and the quadratic superposition of the laminar and turbulent boundary layer contributions according to Gnielinski.
- Apply the arrangement factor of the bed: The shape factor f_a of the bed — a function of the void fraction for beds of spheres, with shape-specific values for cylinders and cubes — converts the single-body Nusselt number to the packed-bed value. It accounts for the mutual interaction of the particles in the bed.
- Evaluate the heat transfer coefficient and exchange area: The Nusselt number, the thermal conductivity of the fluid, and the particle dimension give the heat transfer coefficient between fluid and particle surface. Combined with the volume-specific surface area of the bed, this yields the volumetric heat transfer of the bed.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Prandtl number | Pr | - |
| Thermal conductivity | λ | W/(m·K) |
| Kinematic viscosity | ν | m²/s |
| Inlet temperature | ϑe | °C |
| Outlet temperature | ϑa | °C |
| Particle temperature | ϑw | °C |
| Form factor | fa | - |
| Equivalent diameter of a sphere | dk | m |
| Velocity in free cross-section | wfree | m/s |
| Void fraction | ψ | - |
| Area | Ap | m² |
| Volume of the vessel containing the bed | V | m³ |
| Particle volume | VF | m³ |
| Free cross-section of the unfilled vessel | A | m² |
| Mass flow | m | kg/s |
| Density | ρ | kg/m³ |
| Form of the particles | Formfaktor | - |
| Specific heat capacity | cp | J/(kg·K) |
| Dynamic viscosity | η | mPa·s |
| Volume flow | V | m³/s |
| Mean temperature | ϑa | °C |
| Pressure | p | Pa |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Wärmestromdichte | Nu Re | W/m² |
| Wärmeübergangskoeffizient | α | W/(m²·K) |
| Nusselt-Zahl | Nu Re | - |
| Reynolds-Zahl | Nu Re | - |
Calculation options
Form of the particles
Bed consisting of spheres of the same size · Cylindrical particles with 0.24 < l/d < 1.2 · Cubes · Raschig rings · Berl saddles
Worked example
A fixed bed of spheres with d = 10 mm and void fraction ψ = 0.40 is traversed by air at 20 °C with a superficial velocity of w = 1 m/s. The heat transfer coefficient between air and particle surface is required (wall correction neglected) — a worked example of packed-bed heat transfer.
Given values
| Particle diameter d | 10 mm |
| Void fraction ψ | 0.40 |
| Superficial velocity w | 1 m/s |
| Kinematic viscosity of air (20 °C) ν | 15.32 · 10⁻⁶ m²/s |
| Thermal conductivity λ | 0.0259 W/(m·K) |
| Prandtl number Pr | 0.71 |
Solution
Reynolds number with the interstitial velocity
Reψ = w · d / (ψ · ν) = 1 · 0.01 / (0.40 · 15.32 · 10⁻⁶) ≈ 1,632
Nusselt number of the single sphere
Nulam = 0.664 · Reψ1/2 · Pr1/3 ≈ 23.9
Nuturb = 0.037 · Reψ0.8 · Pr / [1 + 2.443 · Reψ−0.1 · (Pr2/3 − 1)] ≈ 12.8
Nusphere = 2 + (23.9² + 12.8²)1/2 ≈ 29.1
Arrangement factor of the sphere bed
fa = 1 + 1.5 · (1 − ψ) = 1 + 1.5 · 0.60 = 1.90
Nu = fa · Nusphere = 1.90 · 29.1 ≈ 55.4
Heat transfer coefficient
α = Nu · λ / d = 55.4 · 0.0259 / 0.01 ≈ 143 W/(m²·K)
Result
| Reynolds number Re_ψ | 1,632 |
| Nusselt number, single sphere | 29.1 |
| Arrangement factor f_a | 1.90 |
| Nusselt number, packed bed | 55.4 |
| Heat transfer coefficient α | ≈ 143 W/(m²·K) |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
What void fraction is realistic for beds of spheres?
Random packings of equal-sized spheres reach void fractions of about 0.36 to 0.42; loose packings lie above this range, vibrated packings below it. Near the wall, the void fraction increases systematically (wall channeling), which at small ratios of vessel diameter to particle diameter (below about 10) leads to bypass flow along the wall and poorer mean heat transfer — strictly speaking, the correlation applies only to extended beds.
How are non-spherical particles treated?
The characteristic dimension used is the diameter of a sphere with the same surface area as the particle. The influence of shape on the flow around the particles is captured by the shape-specific arrangement factor, which the VDI Heat Atlas provides for spheres, cylinders, cubes, and other packing elements. For strongly anisometric particles (rings, saddles), the values should be regarded as approximations.
Does the calculated coefficient also describe the heat transport to the vessel wall?
No. The module delivers the heat transfer between fluid and particle surface within the bed. The wall heat transfer of a wall-cooled or wall-heated fixed bed and the effective radial thermal conductivity of the packing are separate quantities with their own correlations in the VDI Heat Atlas; for the design of wall-cooled reactors, both transport resistances must be combined.
Does the calculation also apply to fluidized beds?
No, the correlation assumes a stationary fixed bed operated below the minimum fluidization point. Above the minimum fluidization velocity the bed fluidizes, and particle motion and bubble formation fundamentally change the heat transport; the dedicated fluidized-bed sections of the VDI Heat Atlas apply there.