Packed bed

A packed-bed heat battery is a column filled with capsules of phase-change material (PCM), with the heat transfer fluid flowing through the voids between them. The large surface of the capsules makes heat transfer easy, and the capsules are simple to make and replace. The family has one model, PackedBedModel, with options for the capsules.

Geometry

The packed bed. At each height, every capsule is the same, so one capsule stands for all those at that height.

Symbol	Parameter	Meaning
\(Z\)	"Bed length [m]"	length of the bed along the flow
\(A\)	"Bed cross-sectional area [m2]"	area of the bed cross-section
\(\varepsilon\)	"Bed porosity"	volume fraction of the bed taken by the fluid
\(R\)	"Capsule radius [m]"	outer radius of a capsule

The positions are \(z\), along the bed from the inlet, and \(r\), the distance from the centre of a capsule, from 0 to \(R\). The bed holds \((1 - \varepsilon) / (\tfrac{4}{3} \pi R^3)\) capsules per unit volume, with \(a = 3 (1 - \varepsilon) / R\) of capsule surface per unit volume.

The bed is a single flow path: the fluid flows through the voids, a flow area \(A_\mathrm{flow} = \varepsilon A\), carrying the whole mass flow rate \(\dot m\) (see the fluid). The model neglects the capsule shell (its thermal mass and resistance), conduction between capsules and along the bed in the PCM, dispersion in the fluid along the bed, and heat losses to the surroundings.

Equations

One capsule, melting from its surface inwards. Heat enters through the surface and is conducted radially.

The formulation follows Brosa Planella (2025), which gives the details. Resolved along the bed, it is also known as the concentric dispersion model (Ismail and Henríquez, 2002). At each height \(z\), heat is conducted radially in a capsule:

\[ \frac{\partial H}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 k \frac{\partial T}{\partial r} \right), \qquad 0 < r < R, \]

\[ \frac{\partial T}{\partial r} \bigg|_{r = 0} = 0, \qquad k \frac{\partial T}{\partial r} \bigg|_{r = R} = h_c \left( T_f - T_s \right), \]

with \(T_s = T|_{r = R}\), the temperature at the capsule surface. The fluid gains the heat that leaves the capsules at that height, \(q' = a A\, h_c (T_s - T_f)\) per unit length of bed.

Lumped capsules

With options={"storage element": "lumped"}, each capsule has a uniform temperature: one enthalpy per capsule, heated through its surface,

\[ \frac{\mathrm{d} H}{\mathrm{d} t} = \frac{3}{R}\, h_c \left( T_f - T \right). \]

This is much cheaper, and accurate when conduction inside a capsule is fast compared with heat transfer at its surface, that is when the capsule’s Biot number, \(\mathrm{Bi} = h_c R / k\), is small. "lumped with Jeffreson" puts part of the neglected internal resistance back, through an effective coefficient \(h_\mathrm{eff} = h_c / (1 + \mathrm{Bi} / 5)\) (Jeffreson), with \(k\) the conductivity at the capsule’s enthalpy. The correction is exact for slow changes in a sensible sphere, and an approximation while the PCM melts or freezes.

Options

Option	Values
"storage along flow path"	"resolved" (default), "averaged": see resolved or averaged
"storage element"	"resolved" (default), "lumped", "lumped with Jeffreson"
"heat transfer coefficient"	"constant" (default), "Wakao-Kaguei": \(\mathrm{Nu} = 2 + 1.1\, \mathrm{Re}^{0.6} \mathrm{Pr}^{1/3}\) (WakaoKaguei), with \(\ell = 2 R\) and the superficial velocity in \(\mathrm{Re}\)

The options combine: an averaged bed of lumped capsules is one well-mixed capsule for the whole bed. In this family, the cross-section averages along the bed (such as "Storage cross-section average temperature [K]") are the volume averages over a capsule. The model also gives the device totals per unit cross-sectional area of the bed, such as "Power per unit area [W.m-2]". The averaged bed is the reduced model of the encapsulated-LTES repository.

Parameter sets

The default parameter values are the experiments of Nallusamy et al. (2007): a 47 l tank of paraffin capsules, charged with water at 70 °C. A second set, "Raul2018", gives the experiments of Raul et al. (2018), with a heat transfer oil at 180 °C. See pytes.parameter_layers.

Examples

• Tutorial 3: packed-bed models

• Packed-bed model, against the measurements of Nallusamy et al. (2007)

• Lumped capsule

• Validation, against Schumann’s analytical solution

References

• F. Brosa Planella, A simple model for latent thermal energy storage systems with encapsulated phase-change material, International Journal of Heat and Mass Transfer 239, 126533 (2025), doi:10.1016/j.ijheatmasstransfer.2024.126533.

• K. A. R. Ismail and J. R. Henríquez, Numerical and experimental study of spherical capsules packed bed latent heat storage system, Applied Thermal Engineering 22, 1705–1716 (2002), doi:10.1016/S1359-4311(02)00080-7.