1. Introduction
The share of renewable energy in total primary energy supply is on the rise with the need to move away from the dependency on fossil fuels and to reduce CO
emissions. To achieve the target of the Paris Agreement, CO
emissions need to be reduced significantly [
1]. The total energy usage of the residential and service sector accounts for 35.5% of the total global energy consumption, from which 75% is used for space and domestic heating [
2]. Solar energy can provide heating without the need for fossil fuels by using solar boilers. However, the major problem of solar energy is the gap between peak power production and the peak power demand.
To bridge this gap, there is an urgent need for introduction of heat storage systems. Two common forms of heat storage are Sensible Heat Storage (SHS) and Latent Heat Storage (LHS). SHS uses sensible heat to store thermal energy, meaning that the energy stored in the Thermal Energy Storage (TES) is simply used to heat up a material from which this heat can later be extracted. This version of TES is cheaper and simpler than LHS, which uses both sensible heat and latent heat [
3]. When LHS is used, the storage medium undergoes phase change in the temperature range in which the storage system operates. The advantage of this is that, for similar storage capacity, LHS offers a higher thermal energy storage density compared to SHS. Another advantage is that during phase change the temperature is constant, thereby allowing a constant temperature difference between heat transfer fluid (HTF) and energy storage material.
Various LHS systems exist which employ phase change material (PCM) in a variety of ways [
4,
5,
6]. Of these, a packed bed of PCM particles provides the highest heat transfer area and thus allows for both quick charging and discharging. These LHS systems are of interest for use in both large-scale concentrated solar power (CSP) plants and small-scale domestic applications. In the case of CSP, the TES is a way to store high temperature heat used to drive a generator to produce electricity. Mohammadnejad et al. [
7] studied the use of different flow velocities and bed porosity during discharging and the effect of different configurations of particles. Senthil et al. [
8] studied the positioning of the rectangular solar rectangular solar receiver in a CSP setup.
In domestic applications, on the other hand, TES is used to store heat purely for hot water consumption in the form of domestic heating and sanitary applications. For example, Pakrouh et al. [
9] studied a system that uses water as HTF and paraffin wax as PCM. This form of domestic energy storage is often done by simple SHS systems. However, the use of PCM can increase the efficiency of the TES by storing the energy not just in sensible heat but also in the latent heat of the PCM.
To help in the design of the LHS, multiple experimental studies were conducted, for instance, by Barrientos et al. and Nullasamy et al. [
10,
11]. Conducting such experiments is often costly, while the use of numerical models can offer a cheaper method to provide design criteria for LHS system, limiting the number of experiments needed. Another advantage is the amount of detailed information that can be extracted from the model. Furthermore, modeling allows direct control over all variables within a system without a concern for deviations and measurement accuracy. Besides, modeling an unstructured packed bed gives the possibility to measure the temperature of every single particles, which is practically unfeasible in experimental measurements.
Various methods can be used to model a beds of phase changing particles (e.g., single-phase modeling, two-phase modeling, and continuous or discreet solid phase), as shown in the overview by Garcia et al. [
12]. Mao and Zhang [
13] combined three PCMs for use as a heat storage medium. They studied numerically the impact of of particle diameter, porosity, and height-to-diameter ratio of the storage tank on the total storage energy, storage capacity ratio, axial temperature curve, and utilization ratio of the PCM. Khor et al. [
14] investigated both numerically and experimentally the effect of the fillers in the void space between the macro-encapsulated PCM of a packed bed to increase the storage capacity of the tank. The PCM particles are placed in the tank in a structured way. They found void space reduction with the granular materials allows maintaining the storage efficiency while achieving an increase in phase change material encapsulation size to reduce its overall cost. Duan et al. [
15] used a numerical model to find the suitable structured packing for a packed bed of PCM capsule. They found that FCC arrangement (face center cubic packing) has superior characteristics, such as high charging rate, low investment cost, large total heat storage energy, and stable output temperature, compared with BCC arrangement (body center cubic packing) or simple cubic packing.
The response time in a heat storage system is an essential parameters that should be as high as possible. Therefore, the system must be designed in such a way to enhance the heat transfer between heat transfer fluid and PCM particles. The bed structure has a significant impact on the heat transfer through a packed bed. Therefore, the effect of this parameter must be well studied to find the optimum structure for a faster response time and a higher energy density. However, there is a gap in the literature to address these challenges. The objective of this work is to investigate the effect of the bed structure on the charging and discharging of a packed bed of PCM. For this purpose CFDEM
® coupling, from the open-source CFDEM
® project [
16], is used to calculate the heat transfer between particles (i.e., PCM) and the surrounding fluid phase. Moreover, the model has been further developed to calculate the phase transition within the particle. The outcome of this study provides detailed information that can be used by the designer to further optimize the PCM storage tank.
2. The Mathematical Model
To model an unstructured bed of PCM particles, the model first proposed by Schumann is used [
17]. This model is a two-phase model meaning that the particle and fluid phases are treated by separate equations which are coupled. Within the present model, a packed bed is considered as an ensemble of a finite number of particles (i.e., discreet phase). Transient conservation equations for energy is solved for PCM particles individually. Applying this model to all particles of a packed bed forms the entire packed bed process as a sum of the individual particle processes. The flow through the void space in the bed is modeled as a flow passing through a porous media (i.e., CFD domain), while the interaction between the solid and the fluid phases by heat and momentum transfer is taken to account. To avoid porosity equal to zero in the CFD domain, particles must be smaller that the CFD cells. Water is used as the HTF in all cases and is considered as an incompressible fluid and continuous phase; the governing equations therefore can be written as follows.
Continuity equation for the fluid: Momentum equation for the fluid:
where
is the void fraction,
t is the time,
is the fluid velocity vector,
is the pressure,
is the fluid density, and
is the stress tensor. This particular form of the momentum equation is based on the so-called Model III as given by Zhou et al. [
18]. The last term on the right hand side is the coupling term of momentum transfer between the particles and the fluid with:
where
is the particle velocity vector and
is the sum of all the drag forces in the computational cell with volume
.
Energy equation for the fluid:
where
is the fluid temperature,
is the fluid heat specific heat capacity, and
is the thermal conduction coefficient of the fluid. The last term on the right-hand side is the heat transfer between the particles and the fluid due to convection. Radiation from the particles are neglected, thus it is assumed that convection is the only mode of heat transfer between particles and fluid.
The particles have their separate equations to govern conservation of momentum and energy.
Momentum equations for the particles:
where
is the mass of the
ith particle;
is the acceleration vector of the particle;
and
are, respectively, the force in normal direction and the force in tangential direction due to particle-particle interaction;
is the drag force on the particle;
is the gravitational acceleration vector;
is the particle density; and
is the particle volume.
where
is the moment of inertia of the particle,
is the angular momentum vector, and
is the moment arm vector.
Energy equation particles:
where
is the particle heat capacity,
is the particle temperature, and
is the heat source/sink therm.
It is assumed that the particles act as a lumped system and therefore have a single temperature. Furthermore, the particles are frozen in place during fluid–particle simulations and therefore the bed is a fixed bed. Thus, all forces in Equations (
5) and (
6) are set to zero during particle–fluid simulations. However, the drag force is still calculated and taken into account in the fluid momentum equation. Freezing the particles in place is a valid assumption since in reality their movement would be limited.
It is also assumed that the only mode of heat transfer which the particles experience is convection with the fluid. Particle-to-particle conduction and radiation are neglected, as these are small compared to the convection term. Lastly, it is assumed that the walls of the container are well insulated so losses to the environment are neglected as well.
The heat transfer coefficient between particles and fluid can be determined using Nusselt correlation from Li and Mason [
19] with coefficient variable
n being 3.5:
where the Reynolds number is
and the Prandtl number is
Through Equation (
8), the heat transfer coefficient
can be determined which can then be used to calculate the specific heat flux between a particle and the fluid due to convection.
To take into account phase change, the enthalpy method is used by taking the heat capacity of the particles as a function of temperature [
20,
21]. To ease the modeling of this transition, the heat capacity curve has to be approximated; for this, multiple models exist [
22], as shown in
Figure 1.
In the validation with measurements [
11], the exact profile of the Cp as a function of the temperature for the used PCM is not known. However, it was found that the right triangle method was the most agreeable with experimental results. Since the same PCM is used for validation as is used in the models presented here, the right triangle method is used, leading to the heat capacity as follows:
This change in has been incorporated into the CFDEM® coupling program, thus allowing for the modeling of phase transition in a packed bed of particles.
Validation
The model was validated using the data available from experiments carried out by Nallusamy et al., which concerns a structured simple cubed bed filled with Paraffin wax particles [
11]. The results of the model compared to the experiment are given in
Figure 2 and
Figure 3.
In these figures, it can be seen that their general behavior is the same, and they are in a relatively good agreement. The particle heating curve in
Figure 2 shows that initially the particle temperature is overestimated by the model. This is due to the experiment measuring with temperature sensors inside the particle. Since the model assumes a lumped system, the temperatures of modeled particles are overestimated compared to the interior of the actual particle. The particle centers experience a delay in the heating up due to the thermal resistance.
After a while, the numerical curve flattens and the phase change start in the model. Since the heating curve in the experiment is also not straight at this point, we can conclude that in actuality there is also some phase change taking place. However, the model seems to overestimate the amount of latent heat storage occurring between about 320 and 330 K, which is due to the curve being only an approximation. This means that effect of phase change at lower temperatures is exaggerated in the model. This can also be seen at 334 K, where the experimental temperature line is almost flat while the numerical line is climbing; this suggests that a greater part of the phase change occurs at higher temperatures in the experimental case.
This effect of exaggerated phase change at lower temperatures can also be seen in the temperature of the HTF in
Figure 3. The increase of latent heat storage at lower temperatures causes a greater amount of heat transfer between fluid and particle. This is because a greater difference in temperature between particles and HTF is maintained for a longer time. Therefore, the numerical model temperature is higher. The flattening effect on the temperature curve of the HTF due to phase change is also spread out more due to the overestimation of latent heat storage at lower temperatures.
From this, one can conclude that the model is still quite accurate; however, the effect of the approximation of the curve and the lumped system assumption can be seen. Using a more accurate curve, which requires dedicated measurements, could alleviate this difference. Nevertheless, if the effects of the approximation of specific heat capacity and the lumped system assumption are kept in the back of ones mind, the general behavior of the system can still be modeled accurately and conclusions about the systems behavior can be drawn.
3. Model Setup
The particles are made of paraffin wax with the thermodynamic properties as given in
Table 1. All particles are assumed to be spherical with a diameter of 55 (for large particles) or 27.5 mm (for small particles). Water is used as the HTF with the inlet temperature of 343.15 K during the charging phase and 293.15 K during discharging. The initial temperatures of the particles are 293.15 and 343.15 K for charging and discharging cases, respectively.
In all cases, the flow rate is constant at 2 L per minute at the inlet and move from bottom to the top. The energy storage required is based on heat demand of a single household, which is approximated to be 120 L of water at 60 C or K. By considering an overcapacity of 20% and the stored energy in the PCM as well as the sensible heat stored in the water around the particles, 670 large (55 mm) particles are required to meet storage demand.
An overview of the different configurations is given in
Figure 4. The unstructured beds are generated using DEM (Discrete Element Method) by free falling of particles. In all cases, the total volume and consequently the total mass of the PCM are the same. During all simulations, it is assumed that the wall is perfectly insulated.
For the CFD domain, a tank of 0.5-m diameter and 11-m height is defined. To store all the particles, a smaller tank would have been sufficient but adding extra CFD cells before and after the bed increases the stability of the model. The combined CFD and DEM domain is shown in
Figure 5. Here, one can see half of the CFD domain with the DEM particles. The flow goes from the inlet at the bottom to the outlet at the top for charging (heating up) cases and the reverse for discharging (cooling) cases.