Packed Bed Latent Heat Storage

Thermal energy storage (TES) units are used to accumulate thermal energy from solar, geothermal, or waste heat sources. The simplest TES units are built from water tanks, often found in households, where the solar energy is stored as sensible heat. These systems are called sensible heat storage (SHS) units. The thermal capacity of these tanks can be further increased by including latent heat, which gives rise to latent heat storage (LHS) units. Typically, LHS tanks contain spherical capsules filled with paraffin as phase change material. Paraffin is a suitable phase change material to include the effect of latent heat, as it is relatively inexpensive, reliable, and nontoxic, and it is commercially available for a wide range of melting temperatures.

This example is inspired by the experimental investigation found in Ref. 1. It models the flow through a packed-bed storage tank, and it includes the effects of heat transfer with phase change and local thermal nonequilibrium while charging the LHS unit.

Figure 1: Model setup.

Model Definition

The model geometry is shown in Figure 1. Geometry, material properties, and operating conditions are taken from Ref. 1. The thermo-physical properties of paraffin are listed in Table 1.

Table 1: Thermo-Physical Properties of Paraffin.
Material Property	Paraffin, Solid	Paraffin, Liquid
Melting temperature, Tm (°C)	60
Latent heat of fusion, L (J/kg)	213
Density, ρ (kg/m3)	861	778
Heat capacity, Cp (J/(kg·K))	1850	2384
Thermal conductivity, k (W/(m·K))	0.4	0.15

Paraffin-filled spherical capsules with a diameter of dp = 55 mm are stored in a tank of 36 cm in diameter and 47 cm in height. The porosity of this bed is εp = 0.49. The temperature is initially set to 32°C. Warm water flows through the tank with a flow rate of Vin = 2 l/min, and during thermal charging it is continuously heated up by a solar collector that delivers a power of Qu = 375 W. The temperature difference at the tank’s inlet and outlet is given by the relation

(1)

here, Tin and Tout are the inlet and outlet temperatures, and ρ and Cp are the density and heat capacity of water.

Ergun equation describes the flow through the packed bed, which estimates the pressure drop as a function of the velocity field u

Here, μ (Pa·s) and ρ (kg/m3) are the viscosity and density of water, dp (m) is the spheres’ diameter, and εp the porosity. The permeability κ (m2) of the packed bed is given by

The Reynolds number can be estimated as

(2)

The maximum velocity in the bed, v, is about 6 mm/s, which implies a Reynolds number of about 600. For this Reynolds number the flow field is assumed to be independent of the temperature distribution, such that a stationary field can be computed. This is a reasonable simplification that reduces the computational effort.

The relative large diameter of the capsules as compared to the tank dimensions suggests a significant temperature difference between the encapsulated paraffin and the surrounding water flow, thus a local thermal nonequilibrium (LTNE) approach is considered in this example.

The heat transferred from the paraffin-filled capsules to the water is modeled with a heat source

here, Ts and Tf are the paraffin and water temperatures, and qsf (W/(m3·K)) is the interstitial convective heat transfer coefficient, which for spherical capsules reads

The interstitial heat transfer coefficient hsf follows a Nusselt number correlation (see the section Local Thermal Nonequilibrium under Theory for Heat Transfer in Porous Media in the Heat Transfer Module User’s Guide for more information). Convection inside the capsules is neglected, thus paraffin is treated as a solid or immobile liquid.

Results and Discussion

The tank reaches a temperature of 70°C after approximately 11 hours. The resulting velocity and temperature distribution is shown in Figure 2.

Figure 2: Velocity field (streamlines) with the gray color indicating the pressure and temperature field (color contours) after 13 hours.

Figure 3 shows the evolution of the paraffin temperature, the water temperature, and the weighted average (porous-medium) temperature. During the phase change, the encapsulated paraffin is not in thermal equilibrium with the surrounding water. Measuring the water temperature at the inlet or the outlet does not give accurate information about neither the temperature inside the capsules nor the phase in which the paraffin wax is.

Figure 3: Evolution of water (dashed), paraffin (dotted) and average porous medium temperature (solid) during phase change for top (red), center (green) and low (blue) position.

Figure 4 shows the phase distribution after 7 hours. Near the walls, where the flow velocity is negligible, the phase transition has not yet begun while it is already completed in the center of the tank.

Figure 4: Distribution of solid phase (blue) and liquid phase (yellow) after 7 hours.

The evolution of the paraffin phase distribution is visualized in Figure 5. It starts at about 4 hours when water is heated up to the melting temperature of 60°C. Paraffin is completely molten after about 10 hours.

The latent heat storage tank is considered fully charged as soon as a temperature of 70°C is reached everywhere, which happens after approximately 11 hours.

Figure 5: Phase distribution after 4, 6, 8, and 10 hours.

Notes About the COMSOL Implementation

The time it takes to thermally charge the tank is not known a priori. To avoid calculating too many time steps, a stop condition is used which stops the simulation after a temperature of 70°C is reached everywhere inside the tank.

Reference

1. N. Nallusamy and others, “Study on performance of a packed bed latent heat thermal energy storage unit integrated with solar water heating system,” Journal of Zhejiang University-SCIENCE A, vol. 7, pp. 1422–1430, 2006.

Porous_Media_Flow_Module/Heat_Transfer/packed_bed_latent_heat_storage

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

Import the geometry from a file.

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file packed_bed_latent_heat_storage.mphbin.

Click

Global Definitions

Add parameters that will be used to set up the model.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
dp	55[mm]	0.055 m	Diameter of encapsulated PCM
por	0.49	0.49	Bed porosity
V_in	2[l/min]	3.3333E-5 m³/s	Flow rate
T0	32[degC]	305.15 K	Initial temperature
Qu	375[W]	375 W	Solar heating power
rho_av	(861[kg/m^3]+778[kg/m^3])/2	819.5 kg/m³	Average density of paraffin

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Paraffin, solid

In the window, under right-click and choose .

In the window for , type Paraffin, solid in the text field.

Paraffin, liquid

Right-click and choose .

In the window for , type Paraffin, liquid in the text field.

Glass Wool

Right-click and choose .

In the window for , type Glass Wool in the text field.

Select Domain 4 only.

Porous Material 1 (pmat1)

Right-click and choose .

Select Domain 2 only.

Continue with setting up the physics interfaces. After that you can fill the required material properties.

Free and Porous Media Flow (fp)

In the window, under click .

Select Domains 1–3 only.

Porous Medium 1

In the toolbar, click

and choose .

Select Domain 2 only.

According to Equation 2, Ergun’s equation gives the best description of the flow behavior in the bed.

In the window for , locate the section.

From the list, choose .

Porous Matrix 1

In the window, click .

In the window for , locate the section.

From the list, choose .

In the dp text field, type dp.

Heat Transfer in Solids and Fluids (ht)

Continue with setting up the heat transfer interface.

Fluid 1

In the window, under click .

Select Domains 1 and 3 only.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the T text field, type T0.

Porous Medium 1

In the toolbar, click

and choose .

Select Domain 2 only.

In the window for , locate the section.

From the list, choose .

In the dpe text field, type dp.

For a porous medium not in thermal equilibrium the initial temperatures for each component of the porous medium need to be specified.

Initial Values 1

In the window, expand the node, then click .

In the window for , locate the section.

In the Tf text field, type T0.

Continuity 1

The fluid phase temperature field is considered by default to be continuous with the surrounding domains.

Porous Matrix 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Initial Values 1

In the window, expand the node, then click .

In the window for , locate the section.

In the Ts text field, type T0.

The solid phase temperature field is considered by default to be insulated from the surrounding domains.

Porous Matrix 1

In the window, click .

Phase Change Material 1

In the toolbar, click

and choose .

In this solid-solid phase change process, the density is assumed to remain constant which is a reasonable simplification. The mean density value of liquid and solid paraffin is used, which is calculated in the parameter list.

In the window for , locate the section.

In the Tpc,1 → 2 text field, type 60[degC].

In the ΔT1 → 2 text field, type 2[K].

In the L1 → 2 text field, type 213[kJ/kg].

Locate the section. From the list, choose .

Multiphysics

Nonisothermal Flow 1 (nitf1)

In the toolbar, click

and choose , to couple the with the interface.

Materials

Now, fill out the remaining material properties. Because you have set up the physics, the software automatically detects which properties are required for the simulation.

Paraffin, solid (mat2)

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	0.4	W/(m·K)	Basic
Heat capacity at constant pressure	Cp	1850	J/(kg·K)	Basic

Paraffin, liquid (mat3)

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	0.15	W/(m·K)	Basic
Heat capacity at constant pressure	Cp	2384	J/(kg·K)	Basic

Glass Wool (mat4)

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	0.025	W/(m·K)	Basic
Density	rho	850	kg/m³	Basic
Heat capacity at constant pressure	Cp	1250	J/(kg·K)	Basic