Seasonal Solar Thermal Energy Storage

Getu Hailu

## Abstract

Solar intermittency is a major problem, and there is a need and great interest in developing a means of storing solar energy for later use when solar radiation is not available. Thermal energy storage (TES) is a technology that is used to balance the mismatch in demand and supply for heating and/or cooling. Solar thermal energy storage is used in many applications: buildings, concentrating solar power plants and industrial processes. Solar thermal water heaters capable of heating water during the day and storing the heated water for evening use are common. TES improves system performance by smoothing supply and demand and temperature fluctuations. Thermal energy storage has become a fast-growing business. According to a research report, the global thermal energy storage market is expected to reach USD 12.50 billion by 2025. The chapter describes different types of thermal energy storage systems. Brief history, current state of research and the future of thermal storage are presented. Types of thermal storages, classifications, advantages and disadvantages are discussed; important thermal and physical properties are tabulated. Advances in enhancement of thermal properties of materials are briefly discussed. Challenges, opportunities, market outlook, government incentives and polices that support deployment of energy storage systems are outlined.

Keywords: thermal energy storage, sensible heat storage, latent heat storage, phase change materials (PCM), energy storage

### 1. Introduction

Solar thermal energy storage is not a new concept. Early humans had realized the abundance of solar energy and devised many methods of capturing that energy. The Greek historian Xenophon wrote of the teachings of Socrates on how to orient a building as to keep it warm in the winter and cool in the summer. Romans would place many windows on a bath houses'south wall to provide heating for their baths and reduce the fuel needed for their hypocaust, or bath fire. Native Americans in the canyons of Arizona used the southern cliff exposure of a canyon to heat their adobe buildings cleverly placed in caves just so that the low winter Sun angle would soak them with sunlight while the summer angle would be higher and therefore missing the buildings [1].

Thermal energy storage dates to the times when humans lived in natural caves. Caves are warm in winter and cold in summer when compared to the outside temperature. Cave dwellers took advantage of deep underground caves (deep underground structures), which have almost negligible temperature variations with season. Historical records show that the oldest form of thermal energy storage is harvesting ice and snow for food preservation, cold drinks and space cooling [2]. Historical records also show that Romans, Greeks and Chinese explored the use of curved mirrors to concentrate the Sun's rays that could cause flames and explosion. According to a Greek legend, in 212 B.C., Archimedes used mirrors to focus sunlight on ships of an invading Roman fleet at Syracuse and destroyed the fleet [3].

In 1767, a French-Swiss scientist Horace Benedict de Saussure built the first solar heat trapper that could be used for cooking [4]. More than 100 years later, in 1876, Adams invented the first mass-produced solar thermal device, by adding solar energy concentrator to de Saussure's solar heat trapper [5, 6]. Adams' octagonal solar oven equipped with eight solar concentrators (mirrors), reportedly cooked rations for seven soldiers in 2 h. Adams reported, "The rations of seven soldiers, consisting of meat and vegetables, are thoroughly cooked by it in two hours, in January, the coldest month of the year in Bombay, and the men declare the food to be cooked much better than in the ordinary manner." His solar oven was mass produced in India and became quite popular. In the United States, the Adams'solar oven had become a popular product for camping and an educational device for teenage students [3, 6]. Cooking for soldiers using solar oven was also a concept a French mathematician Mouchot investigated. A year after Adams' invention, in 1877, Mouchot devised solar cookers for French soldiers in Algeria, including a shiny metal cone, made from a 105.5° section of a circle. He built a separate cooker to steam vegetables and wrote the first book on Solar Energy and its industrial applications [7]. Wang Xiping, a participant of the First World Conference in Solar Cooking, held in Stockton, California, in 1992, presented the use of solar cooking in China, with Peking duck roasted in Xiao's Duck Shop, Chengdu, China, in 1894 [8].

Clarence Kemp was credited with the first commercially available solar water heater patented in 1891 called "Climax" [9]. The invention of the Climax led to a brief explosion in solar thermal technologies. William Baily in 1909 developed a solar water heater called the "Day and Night," where the heater is separated from an insulated tank to extend the duration of available hot water [9]. The first commercial solar power unit produced steam and was built in Egypt in 1913, with its energy cost being compared to coal at the time [10]. However, these earlier technologies relied on the daily solar cycles and could only be used in climates where ambient temperatures remained above freezing.

Thermal energy storage has now become a fast-growing business. "I expect thermal will be bigger than batteries," said Ice Energy CEO Mike Hopkins, "because thermal loads are the large loads. They are the problematic loads; they are the loads that don't lend themselves to using electrical storage." [11]. Today, most of the thermal energy storage technologies have either been fully developed and commercialized or are in the demonstration and development stage. Figure 1 shows stages of different thermal storage technologies. In Figure 1, some key technologies are displayed with respect to their associated initial capital investment requirements and technology risk versus their current phase of development (i.e., R & D, demonstration and deployment or commercialization phases) [12].

Current research and development of thermal energy storage is mainly focused on reducing the costs of high-density storage, including thermochemical process and phase-change material (PCM) development [12]. Thermal storage systems have found applications worldwide. For example, cold-water storage tanks have been installed around the world to supply cooling capacity in commercial and industrial settings. In Canada, Germany, the Netherlands and Sweden, borehole and aquifer thermal energy storages provide both heating and cooling. In the United States, an estimated 1 GW of ice storage has been deployed to reduce peak energy consumption in areas with high numbers of cooling-degree days [13]. Borehole and aquifer

Figure 1.

Maturity of thermal energy storage technologies [12].

systems have been successfully deployed on a commercial scale to provide heating capacity in the Netherlands, Norway and Canada. The Drake Landing Solar Community in Okotoks, Canada is the first major implementation of borehole seasonal thermal energy storage in district heating in North America. It is also the first system of this type designed to supply more than 90% space heating with solar energy and the first operating in such a cold climate [14].

Thermochemical storage, in which reversible chemical reactions are used to store cooling capacity in the form of chemical compounds, is currently a focus in thermal storage R&D projects due to its ability to achieve energy storage densities of 5–20 times greater than sensible storage [12].

The focus of the following section will be on sensible and latent heat thermal storage technologies (i.e., technologies that have been fully developed and commercialized or are in the demonstration and deployment stage).

## 2. Types of thermal energy storage technology

Thermal energy storage is a technology that allows the transfer of heat and storage in a suitable medium. It is a technology that allows storage of thermal energy by heating or cooling a storage medium for a later use for either heating or cooling applications and power generation. Seasonal storage is defined as the ability to store energy for days, weeks or months to compensate for a longer term supply disruption or seasonal variability on the supply and demand sides of the energy system (e.g., storing heat in the summer for use in the winter via underground thermal energy storage systems) [12]. Advantages of using thermal energy storages include: increased overall efficiency, better reliability, better economics and less pollution of the environment (lower carbon dioxide - CO2) emissions [15]. The selection of thermal energy storage depends on the required storage duration, that is, diurnal or seasonal, economic viability, the type of energy source and operating conditions. Thermal energy storages can be classified based on the type of application, type of end user, type of technology and by the type of storage material used. Figure 2 shows the classification in detail.

The major characteristics of a thermal energy storage system are (a) its capacity per unit volume; (b) the temperature range over which it operates, that is, the temperature at which heat is added to and removed from the system; (c) the means of addition or removal of heat and the temperature differences associated therewith; (d) temperature stratification in the storage unit; (e) the power requirements

Figure 2. Classification of thermal energy storage systems.

for addition or removal of heat; (f) the containers, tanks or other structural elements associated with the storage system; (g) the means of controlling thermal losses from the storage system and (h) its cost [16].

#### 2.1 Sensible heat storage system

The most common type of thermal energy storage is sensible heat storage which utilizes both solid and liquid types of storage medium such as rock, sand, clay, earth, water and oil. In sensible heat storage change in temperature of the medium occurs, that is, the temperature is either increased or decreased. Heat is removed from the storage whenever required to satisfy a load, such as space heating or for domestic hot water. The removal of heat from the storage lowers its temperature. Although there are many possibilities of variations, sensible heat storage medium always consists of: an insulated container, heat storage medium and means for adding and removing heat.

In sensible hot heat storage systems, heat is added (i.e., the temperature is increased) to the storing medium, whereas in sensible cold storage systems, heat is removed thus lowering the temperature. In SHS system, the amount of energy stored is proportional to the difference between the storage medium's input and output temperatures, the mass of the storage medium and the medium's heat capacity [17]. The fundamental equation for calculating the amount of heat stored for sensible heat storage systems (SHS) is

$$Q = mC\_p \Delta T = \rho V C\_p \Delta T \tag{1}$$

where Q is the amount of heat stored [J], m is mass of the storage material [kg], Cp is specific heat of the storage material [J/kg K], ΔT is the temperature change [°C], ρ is the density of the storage material [kg/m<sup>3</sup> ] and V is the volume of the storage material [m3 ].

Heat loss from a sensible thermal storage is directly proportional to the temperature difference between the storage and the environment. An important consideration in sensible thermal storage systems is the rate at which heat can be released and extracted, which is a function of thermal diffusivity. Thermal conductivity,

which is material property of the thermal storage, affects charging and discharging rates of the storage. This relationship is expressed by the following equation [18].

$$
\lambda = \rho \mathbf{C}\_p a \tag{2}
$$

where λ is the thermal conductivity [W/m K], ρ is the density [kg/m3 ], Cp is the specific heat [J/kg K] and α is the thermal diffusivity [m<sup>2</sup> /s].

For a thermal energy storage system to be effective, certain requirements must be fulfilled. Requirements for the common sensible heat storage materials are high energy density (high density and specific heat) and good thermal conductivity (for residential applications usually above 0.3 W/m K). The ability of storing heat in a given container depends on the value of the quantity ρCp, the thermal capacity [19]. Thermal capacities of different storage materials are given in Tables 1 and 2. Most common sensible storage media include rocks, sand, pebbles packed in an insulated container. These materials have several advantages including non-toxicity, nonflammability and lower price. Thermal storage materials must be cheap and have good thermal capacity. Table 3 lists low cost thermal storage materials, the cost ranging from 0.05 to 5.00 \$/kg. The only drawback of these materials is their low heat capacities, ranging from 0.56 to 1.3 kJ/(kg °C), which can make the storage unit unrealistically large [20]. For example, one of the drawbacks of rock, sand and


#### Table 1.

Thermal capacities of selected solid storage materials [19, 20].


#### Table 2.

Thermal capacities of selected liquid storage materials [20].


#### Table 3.

Solid-state low-cost sensible thermal storage materials [20].

#### Figure 3.

Schematic of the evacuated tube solar collectors and the thermal storage.

pebbles is that larger amount is needed because of their lower heat storing capacity. The cost of the storage media per unit energy stored is, however, still acceptable for rocks [18].

## 2.1.1 Sensible solid heat storage system

Sensible solid heat storage media have advantages over liquids because of their higher temperature change. That is, solid thermal storages have the advantage of allowing higher temperature changes as compared to liquids. It should be noted that in sensible thermal storage, there is no phase change of the storing medium be it liquid or solid. Sensible solid storage media do not melt, therefore do not flow, hence no leakage is expected from the storage container. Figure 3 illustrates the use of solid sensible thermal storage. The schematic shows a sand-bed thermal storage underneath a garage floor. Underneath the garage slab is the solar thermal storage which contains fine sand and pit run gravel as a thermal storage medium. The sand bed was bordered underneath with <sup>20</sup> cm (8″) of polystyrene foam resulting in <sup>a</sup>

### Seasonal Solar Thermal Energy Storage DOI: http://dx.doi.org/10.5772/intechopen.79576

thermal resistivity of RSI-5.64 (US R-32) insulation barrier between the sand-bed and ground. The four sides of the sand-bed were insulated with 0.2 <sup>m</sup> (8″) of polystyrene foam board on both sides of <sup>a</sup> 0.2 <sup>m</sup> (8″) poured concrete foundation wall for <sup>a</sup> total of 0.4 <sup>m</sup> (16″) of insulating foam. Solar collectors heat <sup>a</sup> water-glycol solution that, during normal operation, passes through a heat exchanger to heat domestic hot water tank. When the domestic hot water tank is not calling for heat, the excess heat is sent to the sand bed (thermal storage) under the garage floor for heating [21]. The system has dual purpose: heating the garage by radiation and convection and heating domestic water.

An approximate rule of thumb for sizing is to use 300–500 kg of rock per square meter of collector area for space heating applications [18]. Rock or pebble-bed storages can also be used for much higher temperatures up to 1000°C [18].

## 2.1.2 Sensible liquid heat storage system

For liquid heat storage systems, the temperature range that can be reached is limited by their boiling points. The type of liquid used as a storage medium is determined by the desired storage temperature. Water, with its high specific heat, is the most common storage medium below 100°C temperature. Cost-effective, largescale thermal storage is possible by using naturally occurring confined underground water such as aquifers. Hot water is pumped to be stored into such aquifers, thereby displacing the existing cold ground water. This would lower the cost of the thermal storage as the only investment required is the cost of drilling openings for injecting and withdrawing water.

If water is used for higher temperature applications, (temperatures above 100°C), it must be pressurized, adding to the cost; for such a case the limitation of water is the critical point, that is, 374°C [17]. Organic liquids and high molecular weight oils are also effective at higher temperature. Although there are oils in the market, such as Terminol, which can be used without pressurization in the range of ˜10 to 320°C, they have the disadvantage of being of low specific heat (2.3 KJ/kg K vs. 4.19 kJ/kg K for water). In addition, oils are liable to high-temperature cracking, polymerization and formation of volatile products. The advantages and disadvantages of water as a storage medium are listed below [18].

#### Advantages:


Disadvantages:


Eq. (3) can be used to predict water storage temperature as a function of time [20].

$$\mathbf{t}\_s = \mathbf{t}\_i + \frac{\Delta \mathbf{z}}{m \mathbf{C}\_p} \left( \mathbf{Q}\_u - \mathbf{Q}\_L - \mathbf{U}\_s \mathbf{A}\_s \left( \mathbf{t}\_f - \mathbf{t}\_a \right) \right) \tag{3}$$

where Qu is rate of energy addition to the thermal collector, QL are rate of removal of energy from the collector, Us is the heat loss coefficient of storage tank, As is the storage tank surface area, tf is the final temperature, ta is the ambient temperature for the tank and τ is the time.

Figure 4 shows schematic of a typical water tank thermal storage system. In this system, a solar thermal collector supplies the input heat and a load is served by circulating hot water through the heat exchanger. In the schematic shown, the system can also be applied for domestic hot water systems since the heat exchanger prevents contamination of potable water in domestic hot water systems.

Hot water storage systems used as buffer storage for domestic hot water supply are usually in the range of 500 L to several cubic meters. This technology is also used in solar thermal installations for domestic hot water combined with building heating systems [20].

#### 2.1.3 Sensible cold storage system

In sensible cold storage systems, heat is removed from the storage medium. This has the effect of lowering the temperature of the storage medium. Cold storage has the potential to save operating costs. This is possible using cheaper electricity rate during off peak hours. A cold storage may consist of cold rocks or chilled water. An air-conditioning system can befit from heat sinks that can be used as cold storage to which heat is dumped. Coupling chillers to cold storages is a more efficient way of using them, although the initial investment cost is higher as compared to conventional air conditioning systems without cold storage. If water is used as cold storage medium, a large quantity is needed as its useful temperature is somewhat limited as compared to when it is used as sensible hot storage.

Figure 4. Schematic of a typical water thermal storage tank.

## 2.2 Latent heat storage system

Latent heat storage (LHS) is based on the heat absorption or release when a storage material undergoes a phase change from solid to liquid or liquid to gas or vice versa. Latent heat thermal storage system involves the storage of energy in phase change materials (PCMs). For example, when solid material melts and turns to liquid, it absorbs heat without changing its temperature. Thermal energy is stored and released with changes in the material's phase. Latent heat storage has the advantage of being compact, that is, for a given amount of heat storage, the volume of PCM is significantly less than the volume of sensible heat storage. This leads to use of less insulation material and applicability in places where space availability is a concern. Another advantage of phase change materials is that they can be applied where there is strict working temperature as the storage can work under isothermal conditions. Latent heat storage systems also have the advantage of having high storage density. Furthermore, small temperature changes in LHS result in storing large amount of heat.

A comparison of the LHS and SHS system shows that with LHS systems, about 5–10 times higher storage densities can be obtained [18]. The volume of PCM storage is two times lesser than that of water. LHS systems can also be used over a wide temperature range [18]. Phase change can be from solid to solid, from solid to liquid, from solid to gas, from liquid to gas and vice versa. When a phase change is from solid to solid, heat is stored as the material transitions from one crystalline arrangement to another. Solid-to-solid transitions have low latent heat. Solid-to-gas and liquid-to-gas transformations are associated with higher latent heat release and higher volume change; however, the large change in volume is an issue as a huge container is required making the system more complex and impractical. As a result, the most advantageous phase change is the transition from solid to liquid (small change in volume), although solid to liquid transitions have low latent heat as compared to liquid to gas. Solid-toliquid phase change materials are cost-effective as thermal energy storage media.

Since PCMs themselves cannot be used as heat transfer media, a separate heat transfer media must be used with heat exchanger in between to transfer energy from the source to the PCM and from PCM to the load. The heat exchanger to be used must be designed specially, in view of the low thermal diffusivity of PCMs in general. Generally, it is required that the PCM container must be compatible with the PCM and be able to handle volume changes.

The storage capacity of the LHS system with a PCM medium is given by [22]:

$$Q = m\left(\mathbb{C}\_{sp}(T\_m - T\_i) + a\_m \Delta h\_m + \mathbb{C}\_{lp}\left(T\_f - T\_m\right)\right) \tag{4}$$

where am is fraction melted, Clp is average specific heat between Tm and Tf [J/kg K], Csp is average specific heat between Ti and Tm [kJ/kg K], Δhm is heat of fusion per unit mass [J/kg], m is mass of heat storage medium [kg], Q is amount of heat stored [J],Tf is final temperature [°C],Ti is initial temperature [°C] and Tm is melting temperature [°C].

When a phase change material is heated, initially it behaves like sensible heat energy storage and there is a change in temperature. That is, at the beginning, the temperature of the PCM starts to rise (Figure 5). Once the phase change transition temperature is reached, the PCM continues to absorb heat without changing its temperature. The PCM starts to melt and transition from solid to liquid phase. The heat absorbed at constant temperature is called the latent heat of the phase transition. Figure 5 shows the transition process from solid liquid. It is seen from Figure 5 that the phase transformation process occurs at a constant temperature, and the amount of heat required to carry out the process is known as latent heat. Phase-transition enthalpy of PCMs is usually much higher (100–200 times) than sensible heat,

Figure 5. Latent energy storage for solid-liquid phase change.


#### Table 4.

Properties of some commercial PCM materials: organic, inorganic slats and eutectics [24, 25].

consequently latent heat storages have much higher storage density than sensible heat storages [23].

Table 4 gives thermophysical properties of available commercial PCMs: organic, inorganic salts and eutectics. No single material can have all the required properties for an ideal thermal storage medium. Therefore, in practice, available material is used, and designers try to make up for the poor physical property by an adequate system design.

#### 2.3 Classification of PCMs

Many phase change materials are available in any required temperature range. PCMs are generally divided into three main categories: organic PCMs, inorganic

## Seasonal Solar Thermal Energy Storage DOI: http://dx.doi.org/10.5772/intechopen.79576

PCMs and eutectics of organic and inorganic compounds. A detailed classification of PCMs is shown in Figure 6. Figure 7 shows the typical range of melting enthalpy and melting temperature of common material classes used as PCM.

## 2.3.1 Organic PCMs

Organic PCMs have several characteristics which make them useful for latent heat storage. They are chemically stable than inorganic PCMs. They have been found to be compatible with and suitable for absorption into various building materials. One of the drawbacks of organic compounds is their initial cost, which is higher than inorganic PCMs [27]. However, the installed cost of organic PCMs is

Figure 6. Classification of PCMs [22].

Figure 7.

Classes of materials that can be used as PCM and their typical range of melting temperature and melting enthalpy [26].


#### Table 5.

Comparison of organic and inorganic materials for heat storage [26, 28].

competitive to inorganic PCMs. Organic PCMs are flammable, and they may generate harmful fumes on combustion, which is undesirable. They have also been found to react with the products of hydration in concrete. Table 5 summarizes the advantages and disadvantages of organic and inorganic PCMs.

## 2.3.2 Inorganic PCMs

Inorganic PCMs are further classified as salts, salt hydrates and metallics. In general, inorganic PCMs have double the heat storage capacity per unit volume as compared to organic PCMs. They have a higher thermal conductivity, higher operating temperature and lower cost relative to organic phase change materials [27]. The advantages of these materials are: high latent heat values, inflammable, lowcost and availability. However, inorganic PCMs are corrosive, resulting in a short service life of the system and a higher cost [29]. Salts and salt hydrates can suffer from phase segregation and supercooling, which will reversibly affect the energy storage capacity [30]. The high storage density of salt hydrate materials is difficult to maintain and usually decreases with cycling. On the other hand, metals and metallic alloys do not suffer from phase segregation and supercooling, thus they have the potential for high temperature applications [31]. Supercooling (also called subcooling) is the process of chilling a liquid below its freezing point, without the liquid becoming a solid. This means that a temperature significantly below the melting temperature must be reached until the PCM begins to solidify and release heat. If that temperature is not reached, the PCM will not solidify at all and thus only store sensible heat [26]. Figure 8 illustrates the process of heating/melting and cooling/solidification including subcooling.

The best-known PCM is water and it has been used for cold storage for more than 2000 years [26]. Currently, cold storage with ice is state-of-the-art. For temperatures below 0°C, usually, water-salt solutions with a eutectic composition are used [26].

### 2.3.3 Eutectic PCMs

Eutectic compositions are mixtures of two or more components that solidify simultaneously out of the liquid at a minimum freezing point [26]. Therefore, none of the phases can sink down due to a different density. Eutectic compositions do not segregate during melting and freezing process because they freeze to a friendly blend of crystals [32]. Supercooling is observed with many eutectic solutions [33]. Also, some eutectic solutions may be susceptible to microbiological attack; therefore, they

### Figure 8.

Schematic of temperature change during melting and solidification of a PCM with subcooling (supercooling).

must be protected with biocides [34]. It has been reported that eutectic mixtures of fatty acid-alcohol have suitable phase change temperature, high latent heat, lower price and the potential as thermal storage materials for building energy storage [35].

## 2.4 Selection of appropriate PCM

The PCM to be used in the design of thermal storage systems should poses desirable thermophysical, kinetics and chemical properties [22]. As far as thermal properties are concerned, when selecting a PCM for a given application, the operating temperature of the heating or cooling should be matched to the transition temperature of the PCM. The latent heat should be as high as possible, especially on a volumetric basis, to minimize the physical size of the heat store. The following thermal properties need to be considered.


Phase stability during freezing/melting helps in setting heat storage. High density is desirable to achieve a smaller storage size. Small volume changes during phase transformation and small vapor pressure at operating temperatures help reduce containment issues. For these reasons, the following physical properties need to be considered:

i.Favorable phase equilibrium,

ii.High density,

iii.Small volume change and

iv.Low vapor pressure.

Supercooling has been an issue with PCM development, particularly for salt hydrates. Supercooling of more than a few degrees will interfere with proper heat extraction from the storage, and 5–10°C supercooling can prevent it entirely. Thus, the following kinetic properties need to be considered in selecting a PCM:

i.No supercooling and

ii.Sufficient crystallization rate.

PCMs can degrade, chemically decompose or they can be incompatible with materials of construction. PCMs should be non-toxic, nonflammable and nonexplosive for safety. Chemical properties need to be considered are as follows:

i.Long-term chemical stability,

ii.Compatibility with materials of construction,

iii.Non-toxicity and

iv.No fire hazards.

In terms of economics, for PCM to be cost-effective, in general, large-scale availability of the phase change materials is very important.

## 3. Improvement of PCM thermal properties

For PCMs to be cost-effective and efficient their thermal properties need to be enhanced. The enhancement of thermal properties such as thermal conductivity, latent heat and specific heat is important for the PCM under consideration to effectively transfer the heat and store more thermal energy during phase transition.

Thermal conductivity can be enhanced by: (1) inserting stationary metallic structure, (2) adding metallic nanoparticle, (3) by adding carbon additives such as graphite, graphene and carbon nanotubes (CNTs) and (4) by encapsulating the PCM [36]. Most of the researchers who studied insertion of stationary metallic structures focused on investigating configurations, shape, size and number of these insertions for optimization of thermal energy storage performance [37–39]. For example, in study conducted by Sheikholeslami et al. [40], fins and nano-enhanced phase change materials (NEPCM) were used as passive techniques to accelerate solidification process. They used finite element method to find the roles of radiation parameter, fin length and shape factor in minimizing solidification time. They used water as PCM and CuO as nanoparticles. They reported that using platelet nanoparticles leads to the greatest performance and that solid fraction of NEPCM radiation parameter has a direct relationship with solid fraction of NEPCM. As length of the fins increases, charging rate accelerates due to improvement in conduction mode [40]. In another study, Sheikholeslami investigated the effects of inner cylinder shape, diameter of nanoparticles and nanofluid volume fraction on solidification process of CuO nanoparticles and water mixture. They reported that adding nanoparticles could promote the PCM solidification with the optimum value of diameter for accelerating solidification being 40 nm [41].

Parameters such as varying volume fractions and percent weight of nanoparticles that enhance thermal performance (heat transfer) of thermal storage have been studied [42–44]. Carbon-based additives, namely graphite, graphene and Seasonal Solar Thermal Energy Storage DOI: http://dx.doi.org/10.5772/intechopen.79576

carbon nanotube (CNT) have attracted much attention and are one the most promising additives that can enhance the heat transfer of PCM [36]. It has been reported that the thermal conductivity of paraffin wax can be increased by saturating porous graphite matrices in paraffin [45, 46]. The low thermal conductivity of paraffins and fatty acids may also be enhanced by using thin encapsulation, maximizing the heat transfer area [47]. Encapsulation is a method of covering the PCM (that forms the core part of the encapsulated PCM) with an appropriate coating or shell material [36]. The purpose of encapsulation is to hold the liquid and/or solid phase of the PCM and keep it isolated from the surrounding, ensuring the correct composition of the PCM that would have otherwise changed due to the mixing of the PCM with the surrounding fluid [36]. Encapsulation has also the benefit of reducing the reaction of PCM with the surrounding, providing flexibility in frequent phase change processes, increasing mechanical stability of the PCM, improving the compatibility of hazardous PCMs that cannot be directly used or immersed in certain applications such as building cooling/heating systems [36]. Based on size, encapsulated PCMs can be classified as follows [48]:


Specific heat can be enhanced by improving crystallinity of the PCM [49, 50]. A number of studies have shown that the enhanced crystallinity of PCMs in certain composites can increase specific heat capacity of the PCM [51]. In the study conducted by [49], 26 nm SiO2 particles were added at 1% weight into a eutectic of alkali chloride salts (BaCl2, NaCl, CaCl2 and LiCl) with a melting point of 378°C. Addition of SiO2 resulted in an increase of the specific heat capacity of the PCM by an over 14%, as repeatedly shown by differential scanning calorimetry (DSC) measurements. Shin et al. [50] studied further the effect in detail using SiO2 particles having 2–20 nm diameter embedded in molten salt eutectic of Li2CO3-K2CO3. DSC measurements showed 38–54% and 118–124% increase in specific heat for the solid and the liquid phase of these composites, respectively.

Energy storage using PCM is directly dependent on the latent heat of the material. Therefore, it is always of great interest to develop materials with higher latent heat capacity. This allows either the storage of more energy within the same material mass or the use of reduced levels for a constant energy storage need [51]. Latent heat can be enhanced by increasing the crystallinity of the PCM. Warzoha and Fleischer studied the increase in latent heat resulting from the addition of multiwall carbon nanotubes (MWCNT), alumina or TiO2 to a base paraffin at levels of 20 vol. % concentration [52]. They found that the thermal energy that can be harnessed is 15–17% lower than the amount that can be extracted from the base paraffin during solidification; however, the thermal energy harnessed in the presence of graphene nanoparticles (15 nm thick, 15 μm diameter at 20 vol. %) is 11% higher than that for the base paraffin.

### 4. Challenges, opportunities and supporting polices

There are several reasons for increased deployment of energy storage technologies. The need to reduce greenhouse gas emission, the need to increase energy access and security, the need to replace aging energy infrastructure, and the need


Table 6.

Examples of government actions that have positively supported energy storage technology deployment [12].

#### Seasonal Solar Thermal Energy Storage DOI: http://dx.doi.org/10.5772/intechopen.79576

for decentralized energy production; all these are reasons for increased deployment of thermal energy storages. According to Grand View Research report, the global thermal energy storage market is expected to reach USD 12.50 billion by 2025 [53]. According to the report, increasing demand for access to efficient and cost competitive energy sources is expected to favor market growth. The expansion of thermal energy storage technologies is expected to be significant in Europe and Asia (particularly Japan) and somewhat lower (50%) in the United States. The global potential is estimated at approximately three times the European potential [54].

While there is great potential for thermal energy storages to be widely deployed, there are several obstacles that need to be overcome, of which the main two are cost and performance. Thermal storage market development and penetration varies considerably, depending on the application fields and regions. Penetration in the building sector is comparably slow in Europe where the construction of new buildings is around 1.3% per year and the renovation rate is around 1.5%; of course, the integration of thermal energy storage systems (TES) is easier during construction [54]. The estimate of the European potential is based on an implementation rate of TES systems in buildings of 5% [55]. Penetration could be higher in emerging countries with high rates of new buildings. TES potential for cogeneration and district heating is also associated with the building stock. The implementation rate of cogeneration is 10.2%, while the implementation of TES in these systems is assumed to be 15% [18]. As far as TES for power applications is concerned, a driving sector is concentrating solar power in which almost all new power plants in operation or under construction are equipped with TES systems, mostly based on molten salt [18].

Additional obstacles are related to material properties and stability, in particular for thermochemical systems. Each storage application needs a specific TES design to fit specific boundary conditions and requirements [18]. Most of such R&D efforts on TES technologies deal with materials (i.e., storage media for different temperature ranges), containers and thermal insulation development. For complex systems such as latent heat storage and chemical storages, more R&D is required in understanding system integration and process parameters as well as improving reacting materials.

A key to achieving widespread storage technology deployment is enabling compensation for the multiple services performed across the energy system [12]. Many governments have already acted in support of energy storage project development through efforts such as direct financial support of demonstration projects, comprehensive market transformations and mandates for energy storage projects (see Table 6) [12].

## 5. Conclusion

Thermal energy storage (TES) is a technology that works by soring thermal energy for later use. TES can be applied for heating, cooling, power generation and industrial processes. In the building area, TES are applied for use with single family houses, multi-user buildings, large commercial buildings and district heating. Most TES research has focused on materials, such as investigating storage media for different temperature ranges, containers and thermal insulation material development. PCM and thermochemical storage systems require further R&D work, for example, in the area of improving reacting materials, gaining better understanding of system integration and process parameters.

TES technologies face some barriers, cost being the key issue. Additional barriers are associated with material properties and stability, especially the thermochemical storage systems.

Penetration in the building sector is comparably slow in Europe where the construction of new buildings is around 1.3% per year and the renovation rate is around 1.5%; of course, the integration of TES is easier during construction [54]. Penetration could be higher in emerging countries with high rates of new buildings. TES potential for cogeneration and district heating is also associated with the building stock. The implementation rate of cogeneration is 10.2%, while the implementation of TES in these systems is assumed to be 15% [18].

## Author details

Getu Hailu Department of Mechanical Engineering, University of Alaska Anchorage, Anchorage, USA

Address all correspondence to: ghailu@alaska.edu

© 2018 The Author(s). Licensee IntechOpen. This chapteris distributed underthe terms oftheCreative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Seasonal Solar Thermal Energy Storage DOI: http://dx.doi.org/10.5772/intechopen.79576

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## Chapter 3

## Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling

Zhiguo Qu

## Abstract

Phase change materials (PCMs) have several advantages for thermal energy storage due to their high energy storage density and nearly constant working temperature. Unfortunately, the low thermal conductivity of PCM impedes its efficiency of charging and discharging processes. To solve this issue, different techniques are developed to enhance the heat transfer capability of PCMs. In this chapter, the common approaches, which include the use of extended internal fins, porous matrices or metal foams, high thermal conductivity nanoparticles, and heat pipes for enhancing the heat transfer rate of PCMs, are presented in details. In addition, mathematical modeling plays a significant role in clarifying the PCM melting and solidification mechanisms and directs the experiments. As a powerful mesoscopic numerical approach, the enthalpy-based lattice Boltzmann method (LBM), which is robust to investigate the solid-liquid phase change phenomenon without iteration of source terms, is also introduced in this chapter, and its applications in latent heat thermal energy storage (LHTES) unit using different heat transfer enhancement techniques are discussed.

Keywords: phase change materials, heat transfer enhancement, nanofluid, mathematical models, lattice Boltzmann method

### 1. Introduction to heat transfer enhancement techniques of PCMs

The development of renewable energy such as solar energy and wind energy has attracted lots of attention during the past decades due to the gap between the increased global energy demand and the decreased amount of fossil fuel in the world. However, one of the major drawbacks for renewable energy is its territorial, time-dependent, and intermittent characteristics. Under this circumstance, the energy storage techniques play an indispensable role for achieving a continuous and reliable supply of renewable energy [1, 2]. Thermal energy storage (TES), which stores the heat in the materials and generates the electricity with heat engine cycles later, is a promising energy storage technique. In general, TES could be classified into three different categories, namely latent, sensible, and thermochemical. With the advantages of high energy storage density and nearly constant charging/ discharging temperature, latent heat thermal energy storage (LHTES) using phase change materials (PCMs) is widely used in several renewable energy applications. However, a major issue of LHTES system is the low thermal conductivity of most

PCMs, which seriously impedes the energy storage efficiency. To handle this challenge, several heat transfer enhancement techniques are developed and discussed by researches during the past years. The existing effective approaches to ameliorate the thermal performance of PCMs include using extended internal fins, filling porous matrix or metal foams, adding high thermal conductivity nanoparticles, and using heat pipes [3].

With the characteristics of simple fabrication, low cost construction, and large heat transfer surfaces, fins are used in a majority of PCM-based LHTES systems. There are several different configurations of fins such as longitudinal, annular, circular, plates, pins, tree shape, and other novel geometries as shown in Figure 1 [4]. By applying extended internal fins, the average thermal conductivity and heat transfer depth of LHTES system are improved, so that the melting and solidification rate of PCMs are accelerated. However, there exists a tradeoff between the increased PCM charging/discharging rate in LHTES unit and its corresponding reduced energy storage capacity with the existence of internal fins. An optimum design of fin configuration and arrangement becomes significant for achieving high energy storage efficiency of LHTES unit with PCMs. Under this circumstance, lots of researches are numerically and experimentally carried out to investigate the conjugate heat transfer between fins and PCMs, and the enhancement of PCM thermal performance with different type of fins is deeply understood and optimized during the recent years [5–21].

Due to the high surface area to volume ratio generated by the tortuosity of metal foams as shown in Figure 2, the PCM charging/discharging rates could be highly improved by inserting metal foams into the LHTES unit [22]. As air inevitably exists

Figure 1. Different configuration of fins used in LHTES system with PCMs [4].

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

Figure 2. A piece of FeCrAlY foam sample manufactured with the sintering route [22].

in the porous structure of metal foams, the infiltration of PCMs into metal foams is hindered, which correspondingly reduces the impregnation ratio of PCMs. As a consequence, the energy storage capacity of LHTES system using combination of PCMs and metal foams is affected. To handle this difficulty, a vacuum impregnation method is generally used to prepare the PCM/metal foam composite materials. Figure 3 shows the apparatus and procedures of impregnation treatment for PCMs with vacuum assistance, and the detailed steps could be found in Ref. [23]. As the PCM melting and solidification are actually accelerated due to the interconnected heat transfer channel inside the metal foams, the porosity and pore size of metal foams are the most significant factors, which affect the energy storage efficiency of LHTES. The copper foam with different porosities and pore sizes is displayed in Figure 4 as an example [24]. The conduction heat transfer in the LHTES system could be consolidated with the decrement of porosity and pore sizes because of the increased density of high speed heat transfer channels inside the metal foams. However, natural convective heat transfer of liquid PCMs through the metal foams is hampered due to the reduced void space caused by the decreased metal foam porosity and pore size. In addition, when the porosity of metal foams decreases, the amount of pure PCMs in the LHTES unit is reduced, which decreases the energy storage capacity. Due to the above reasons, metal foams with appropriate porosity, pore size, and filling ratio, which balance the conduction and natural convection, are essential for achieving the optimum heat transfer rate of PCMs and the most efficient energy storage of LHTES unit. Hence, the mechanisms of PCM melting and solidification processes inserted with various metal foams are studied by several researches at both macroscopic and pore scales [25–44]. Besides, the heat transfer rate of PCMs could also be ameliorated by applying other additives such as graphite [45, 46], carbon nanotubes [47], and graphene [48].

As the nanotechnology has achieved rapid development during the past decades, adding high thermal conductivity nanoparticles becomes a new technique to improve the low heat transfer rate of PCMs [49]. Khodadadi and Hosseinizadeh first investigated the enhancement of PCM heat transfer capability using nanoparticles [50], and their results demonstrated that nanoparticle-enhanced PCMs (NEPCMs) have a great potential in TES applications. The SEM micrographs of nanoparticle-enhanced PCM (NaNO3-KNO3) with different nanoparticles and mass fractions are shown in Figure 5 [51]. With the existence of nanoparticles, the thermophysical properties of PCMs such as thermal conductivity and latent heat capacity are varied. The mechanism of the effects of surface, chemical, and physical

#### Figure 3.

Schematic of the apparatus and procedure for the preparation of composite PCMs using a vacuum impregnation method [23].

Figure 4. Copper foam samples with different pore sizes and porosities [24].

properties of nanoparticles on the thermal properties of PCMs is reviewed in Ref. [52]. In the recent years, many researches related to NEPCMs are carried out, which mainly focus on enhancing the charging/discharging speed of PCMs with nanoparticles [53–73]. However, although the effective thermal conductivity of PCMs is ameliorated by adding nanoparticles, the energy storage capacity of LHTES unit is decreased. Furthermore, the use of nanoparticles increases the viscosity of PCMs, which impedes the development of natural convective heat transfer. Under this circumstance, the total heat transfer rate of PCMs may decrease especially for the cases under high temperature with dominant convective heat transfer. Compared with melting/solidification rate of PCMs, the energy storage rate of LHTES system is the essential goal of storing heat using PCMs. Hence, more investigations, which concentrate on the energy storage rate of NEPCM, should be completed in

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

Figure 5.

SEM photos of nanoparticle-enhanced NaNO3-KNO3 with 0.5 wt.% (a–d), 1.0 wt% (e–h), and 1.5 wt% (i–l) of nanoparticles. Silica (a, e, i), alumina (b, f, j), titania (c, g, k), and alumina-silica nanoparticles (d, h, l) [51].

the future research, and the technique of adding nanoparticles for enhancing the thermal performance of LHTES needs to be further compared with other heat transfer improvement technologies in order to realize the optimum energy storage efficiency.

As the most commonly used heat exchanger devices, heat pipes are widely used to amplify the charging/discharging processes of PCMs and to transfer heat from a source to the storage or from the storage to a sink with heat transfer fluid (HTF) [74]. Although increasing the heat transfer area on the PCM side using extended fins or metal foams is the most efficient and simple method to ameliorate the energy storage rate of LHTES system as previously discussed, when there exists high temperature HTF passing through the LHTES tank such as the waste heat recovery, heat pipes are indispensable for achieving the high efficiency energy storage. The transient charging/discharging processes of PCMs in a LHTES unit with heat pipes are shown in Figure 6 [75]. The configuration and arrangement of heat pipes play a significant role in the energy storage rate of PCMs. To optimize the thermal performance of heat pipe–assisted LHTES systems, lots of experimental and numerical works are carried out during the past few years [76–96].

The significant research progress of PCM heat transfer enhancement using a single technique is discussed and summarized in the above paragraphs. Recently, to further improve the heat transfer capability of PCM and compare the effectiveness of different approaches (use of fins, metal foams, nanoparticles, or heat pipes), the charging/discharging processes of PCMs with hybrid heat transfer enhancement techniques are investigated. Although adding nanoparticles could ameliorate the effective thermal conductivity of PCMs, the heat transfer area on the PCM side is not improved. Based on this, the extended fins are considered to be used for

Figure 6.

The transient charging/discharging processes of PCM in a LHTES tank with HTF passing through heat pipes [75]. (a) Charging process and (b) discharging process.

enhancing the heat transfer depth of NEPCM in a LHTES unit [97–100]. Darzi et al. studied the melting and solidification of PCM enhanced with radial fins and nanoparticles in cylindrical annulus, and they found that adding fins on the hot or cold tubes is the best approach to expedite the heat transfer rate [98]. Lohrasbi et al. optimized the copper nanoparticle-PCM solidification process in a fin-assisted LHTES system [99]. The results indicate that immersing fin in LHTES unit increases the solidification rate more significantly than dispersing nanoparticles. Parsazadeh and Duan investigated the effects of fins and nanoparticle in a shell and tube LHTES unit [100]. They found that adding Al2O3 nanoparticles even decreases the overall heat transfer rate because the thermal conductivity enhancement with nanoparticles could not compensate for the natural convection reduction. Similarly, porous matrices are also inserted into the LHTES unit to improve the thermal performance of NEPCM [101, 102]. Hossain et al. studied the melting process of NEPCM inside the porous medium [101], and it is observed that the movement of PCM melting front is more significant under the influence of porous medium than that of nanoparticles. Tasnim et al. investigated the convection effect on the melting process of NEPCM filled in porous enclosure [102]. The results showed that both the conduction and convection heat transfer are degraded by the presence of nanoparticles. From these researches, it could be found that using extended fins or porous matrices is more effective than adding nanoparticles for enhancing the charging/discharging rate of LHTES system. Besides, other hybrid heat transfer enhancement techniques for enhancing the energy storage rate of LHTES with PCMs such as combination of fins and metal foams [103] or using combined three techniques [104–107] are also recently studied and analyzed.

In this chapter, the mathematical models for PCM charging and discharging processes with different heat transfer enhancement techniques are shown. In addition, the lattice Boltzmann method (LBM) for solid-liquid phase change

phenomenon is reviewed with some classical analytical and numerical validation cases, and the implementation of graphic processor unit (GPU) computing is also presented. Furthermore, the applications of LBM modeling for LHTES system with various heat transfer improvement approaches are discussed.

## 2. Mathematical models

## 2.1 Governing equations of fluid flow and solid-liquid phase change

To simulate the charging and discharging processes of PCMs, the following assumptions are usually made to simplify the mathematical models: (1) the fluid flow of liquid PCMs is Newtonian, laminar, and incompressible with negligible viscous dissipation and (2) the thermophysical properties of PCMs, nanoparticles, fins, and metal foams are constant, except the PCM density ρ, which is a linear function of temperature T using the Boussinesq approximation. Based on the above assumptions, the flow of liquid PCMs is governed by the continuity equation and the momentum equation expressed in the Cartesian coordinate as:

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial \mathbf{z}} = \mathbf{0} \tag{1}$$

$$
\rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial \mathbf{y}} + w \frac{\partial u}{\partial \mathbf{z}} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \mu \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial \mathbf{y}^2} + \frac{\partial^2 u}{\partial \mathbf{z}^2} \right) \tag{2}
$$

$$\rho \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = -\frac{\partial p}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) + (\rho \theta) \mathbf{g} (T - T\_r) \tag{3}$$

$$
\rho \left( \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} \right) = -\frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) \tag{4}
$$

where ρ is the density; p is the pressure; μ is the dynamic viscosity; t is the time; β is the thermal expansion coefficient of PCMs; g is the magnitude of gravitational acceleration; T is the temperature; Tr is the reference temperature; x is the horizontal coordinate; y is the vertical coordinate; z is the coordinate, which is orthogonal with x and y coordinates; and u, v, and w are the fluid velocities in the x, y, and z directions, respectively.

The solid-liquid phase change process of PCMs is governed by the enthalpy equation as:

$$\frac{\partial(\rho H)}{\partial t} + \rho c\_p \left( u \frac{\partial T}{\partial \mathbf{x}} + \nu \frac{\partial T}{\partial \mathbf{y}} + w \frac{\partial T}{\partial \mathbf{z}} \right) = k \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial \mathbf{y}^2} + \frac{\partial^2 T}{\partial \mathbf{z}^2} \right) \tag{5}$$

where cp and k are the specific heat and thermal conductivity of PCMs. The enthalpy H of PCMs is defined as:

$$H = c\_p(T - T\_r) + f\_l L \tag{6}$$

where fl is the PCM liquid fraction, and L is the latent heat of PCMs. By calculating the enthalpy H of PCMs, the liquid fraction fl and temperature T could be updated by the following equations:

Thermal Energy Battery with Nano-enhanced PCM

$$f\_l = \begin{cases} 0, & H \triangleleft H\_s \\ \frac{H - H\_s}{H\_l - H\_s}, & H\_s \triangleleft H\_l \triangleleft \\ 1 & H \triangleleft H\_l \end{cases} \tag{7}$$

$$T = \begin{cases} T\_m - \frac{H\_s - H}{c\_p}, & H \triangleleft H\_s \\\\ T\_m, & H\_s \triangleleft H\_l \triangleleft \\\\ T\_m + \frac{H - H\_l}{c\_p} & H \triangleleft H\_l \end{cases} \tag{8}$$

where Hs is the enthalpy of solid state PCMs, Hl is the enthalpy of liquid state PCMs, and Tm is the melting/solidification temperature of PCMs.

#### 2.2 Nanofluid models

For simulating the fluid flow and heat transfer of nanoparticle-enhanced PCMs (NEPCMs), the nanoparticle is assumed to be spherical in shape, so that the Brinkman model and the Maxwell model for nanofluid are valid. In addition, the NEPCMs are considered as continuous media with the thermal dispersion being neglected. Based on this, the effective viscosity of NEPCMs is computed using the Brinkman model as [108]:

$$
\mu\_{\rm nf} = \frac{\mu\_{\rm PCM}}{\left(\mathbf{1} - \boldsymbol{\Phi}\right)^{2.5}} \tag{9}
$$

where Φ is the volume fraction of nanoparticles, μPCM is the dynamic viscosity of pure PCM, and μnf is the dynamic viscosity of NEPCMs. The thermal conductivity of NEPCMs is calculated according to the Maxwell model as [109]:

$$\frac{k\_{\rm nf}}{k\_{\rm PCM}} = \frac{k\_p + 2k\_{\rm PCM} - 2\left(k\_{\rm PCM} - k\_p\right)\Phi}{k\_p + 2k\_{\rm PCM} + \left(k\_{\rm PCM} - k\_p\right)\Phi} \tag{10}$$

where kPCM, kp, and knf are the thermal conductivities of pure PCMs, nanoparticles, and NEPCMs, respectively. Furthermore, the density of nanofluid ρnf is computed using interpolation as:

$$
\rho\_{\rm nf} = (\mathbf{1} - \Phi)\rho\_{\rm PCM} + \Phi\rho\_p \tag{11}
$$

� � where ρ are the densities of pure PCM and nanoparticles. The heat PCM and ρ<sup>p</sup> capacitance of NEPCMs ρcp is defined as: nf

$$\left(\rho c\_p\right)\_{\mathfrak{n}f} = \left(1 - \Phi\right) \left(\rho c\_p\right)\_{\text{PCM}} + \Phi \left(\rho c\_p\right)\_p \tag{12}$$

� � � � where ρcp is the heat capacitance of pure PCM, and ρcp is the heat PCM <sup>p</sup> capacitance of nanoparticles. The thermal expansion volume of NEPCMs ρβ nf ð Þ is given as:

$$(\rho \beta)\_{\eta f} = (\mathbf{1} - \Phi)(\rho \beta)\_{\text{PCM}} + \Phi(\rho \beta)\_p \tag{13}$$

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

where ρβ PCM and ð Þ<sup>p</sup> ð Þ ρβ are the thermal expansion volume of pure PCM and nanoparticles, respectively. The latent heat of NEPCMs ρL nf ð Þ is computed as:

$$(\rho L)\_{\mathfrak{n}f} = (\mathbf{1} - \Phi)(\rho L)\_{\text{PCM}} \tag{14}$$

where ð Þ ρL PCM is the latent heat of pure PCM. Then, the corresponding enthalpy of NEPCMs Hnf is given as:

$$H\_{\mathfrak{nf}} = c\_{p\_{\mathfrak{nf}}}(T - T\_r) + f\_l L\_{\mathfrak{nf}} \tag{15}$$

#### 2.3 Conjugate heat transfer between PCMs and fins or metal foams

When the extended fins or metal foams are used as the heat transfer enhancement techniques for LHTES unit, the conjugate heat transfer occurs between the PCMs and the fins or metal foams. The heat transfer inside the fins or metal foams is governed by the conduction equation as:

$$
\rho\_s c\_{p\_s} \frac{\partial T}{\partial t} = k\_s \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial \mathbf{y}^2} + \frac{\partial^2 T}{\partial \mathbf{z}^2} \right) \tag{16}
$$

ρs, cps , and ks are the density, specific heat, and thermal conductivity of fins or metal foams, respectively. On the interface of PCMs and fins or metal foams, the coupled Dirichlet-Neumann boundary conditions for conjugate heat transfer should be satisfied:

$$-k\_{\rm PCM} \frac{\partial T\_{\rm PCM}}{\partial n} = -k\_s \frac{\partial T\_s}{\partial n} \tag{17}$$

$$T\_{\text{PCM}} = T\_s \tag{18}$$

TPCM and Ts are the temperature of PCMs and fins or metal foams, kPCM is the thermal conductivity of PCMs, and n is the normal unit of the interface. The above boundary conditions could be satisfied in the lattice Boltzmann method automatically by using the harmonic mean value of thermophysical properties of PCMs and fins or metal foams as discussed in the following section. To investigate the melting and solidification processes of PCMs filled with metal foams at pore scale, the morphology of the real metal foam structures could be reconstructed using the

Figure 7. Reconstructed PCM filled with metal foams using QSGS [111].

quartet structure generation set (QSGS) as shown in Figure 7 [110, 111], where εave and dp are the porosity and the pore size of metal foams. Besides, the metal foams could also be reconstructed with scanning electron microscopy (SEM).

## 3. Lattice Boltzmann method for solid-liquid phase change

## 3.1 The development history of enthalpy-based LBM for solid-liquid phase change

As a mesoscopic numerical approach developed during the past more than two decades, the lattice Boltzmann method (LBM) has become a powerful tool for simulating complex heat transfer and fluid flow problems such as single-phase flows, multiphase flows, turbulence flows, flows in porous media, heat transfer, phase change, and transport in microfluidics [112–115]. The mesoscopic nature of LBM makes it appropriate for tackling the evolvement of solid-liquid interface during the phase change process. The existing LBM for solid-liquid phase change problems could be generally categorized as follows: (1) the phase-field based method, (2) the enthalpy-based method, and (3) the immersed boundary method. For the phase-field method, an auxiliary parameter, which varies smoothly across the diffusive phase interface, is used to track the solid-liquid interface implicitly [116, 117]. Unfortunately, the extremely finer grids are indispensable in the interfacial region, which increases the computational effort. In addition, Huang and Wu developed the immersed boundary-thermal lattice Boltzmann method for modeling the solid-liquid phase change phenomenon [118]. The melting interface is explicitly tracked by the Lagrangian grids, and the temperature and fluid flow are solved on the Eulerian nodes. However, an interpolation between the Lagrangian nodes and the Eulerian nodes should be carried out using the Dirac delta function, which reduces the computation accuracy. Compared with other methods, the enthalpy-based method becomes attractive for solid-liquid phase change due to its high efficiency and robustness. Jiang et al. first used LBM with enthalpy formulation to investigate the phase change problems [119]. However, the convective heat transfer is not considered in this work. Huber et al. developed a LBM model using double distribution functions to simulate the solid-liquid phase change process with convective heat transfer [120], and the numerical results are analyzed and compared with scaling laws and previous numerical work. Eshraghi and Felicelli developed an implicit LBM scheme to investigate the conduction with phase change [121]. Different from the previous enthalpy-based LBM, the iteration of nonlinear latent heat source term in the energy equation is avoided by solving a linear system of equations. Feng et al. further extended this implicit model with a consideration of natural convection to investigate the melting process of nanoparticle-enhanced PCMs [122]. To improve the computational efficiency, Huang et al. modified the equilibrium distribution function of temperature for enthalpy, so that the iteration of heat source term or solving a linear system of equations is not indispensable [123]. Unfortunately, the stability range of relaxation time for this model is narrow, which limits its applications for real scientific and engineering problems. To handle this drawback, Huang and Wu improved their work by using a multiple-relaxation-time (MRT) collision scheme instead of the single-relaxation-time (SRT) one, so that the numerical stability is highly ameliorated [124]. In addition, the thermal conductivity and the specific heat capacity are decoupled from the relaxation time and equilibrium distribution Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

function, which make this model appropriate for satisfying the Dirichlet-Neumann boundary conditions for conjugate heat transfer. As the model developed by Huang et al. is limited in two-dimensional cases, Li et al. recently developed the SRT and MRT models for axisymmetric and three-dimensional solidliquid phase change problems [125, 126]. Besides, to speed up the computation, Su and Davidson worked out a mesoscopic scale timestep adjustable non-dimensional LBM [127], and the time steps can be adjusted independent of mesh size by changing the transient mesoscopic Mach number. Furthermore, although the key advantage of LBM is to carry out the pore-scale numerical modeling of heat transfer in porous media, a few lattice Boltzmann models for solid-liquid phase change in porous media at the representative elementary volume (REV) scale are also developed [128–130]. The classical enthalpy-based MRT lattice Boltzmann model developed by Huang and Wu is reviewed in this section because of its simplicity and wide application by researchers for latent heat thermal energy storage problems.

#### 3.2 Multiple-relaxation-time (MRT) method

#### 3.2.1 MRT LBM for fluid flow

The general two dimensional nine-velocity (D2Q9) MRT LBM model is presented in this part for simulating the fluid flow. In the D2Q9 model, lattice velocities e<sup>i</sup> are given by:

$$\mathbf{e}\_{i} = \begin{cases} \mathbf{0} = (0, 0) \\ \mathbf{f}\_{i} = c(\cos[(i-1)\pi/2], \sin\left[(i-1)\pi/2\right]), \quad i = 1, 2, 3, 4 \\ \mathbf{f}\_{i} = \sqrt{2} \langle (\cos\left[(2i-9)\pi/4\right], \sin\left[(2i-9)\pi/4\right]), \quad i = 5, 6, 7, 8 \end{cases} \tag{19}$$

where c is the lattice speed, and the collision step carried out in the momentum space is given as:

$$\mathfrak{m}\_f(\mathbf{x}, t + \delta\_t) = \mathfrak{m}\_f(\mathbf{x}, t) - \mathfrak{S}\left[\mathfrak{m}\_f(\mathbf{x}, t) - \mathfrak{m}\_f^{eq}(\mathbf{x}, t)\right] + \delta\_t \left(1 - \frac{\mathfrak{S}}{2}\right) \mathfrak{F}\_{\mathfrak{m}}(\mathbf{x}, t) \tag{20}$$

where x is the location vector, t is the time, δ<sup>t</sup> is the time step, I is the unit matrix, and mf is the distribution function in momentum space defined as:

$$m\_f(\mathbf{x}, t) = \begin{bmatrix} m\_{f0}(\mathbf{x}, t), m\_{f1}(\mathbf{x}, t), \dots, m\_{f8}(\mathbf{x}, t) \end{bmatrix}^T \tag{21}$$

The equilibrium distribution function in momentum space meq is [131–133]: <sup>f</sup>

$$\mathbf{m}\_f^{eq} = \rho,\\ -2\rho + 3\rho \frac{\left|\mathbf{u}\right|^2}{c^2},\\ \rho - 3\rho \frac{\left|\mathbf{u}\right|^2}{c^2},\\ \rho \frac{u\_\mathbf{x}}{c}, -\rho \frac{u\_\mathbf{x}}{c},\\ \rho \frac{u\_\mathbf{y}}{c}, -\rho \frac{u\_\mathbf{y}}{c},\\ \rho \frac{u\_\mathbf{x}^2 - u\_\mathbf{y}^2}{c^2},\\ \rho \frac{u\_\mathbf{x}u\_\mathbf{y}}{c^2} \tag{22}$$

where ρ is the density, u is the fluid velocity vector, ux is the horizontal velocity, and uy is the vertical velocity. The diagonal relaxation matrix S is defined as:

$$\mathbf{S} = \text{diag}\left(\mathbf{s}'\_{\varnothing}, \mathbf{s}\_{\epsilon}, \mathbf{s}\_{\iota}, \mathbf{s}\_{\jmath}, \mathbf{s}\_{\varrho}, \mathbf{s}\_{\jmath}, \mathbf{s}\_{\varrho}, \mathbf{s}\_{\varrho}, \mathbf{s}\_{\varrho}\right) \tag{23}$$

where s0, se, sε, sj, sq, and sp are the parameters related to relaxation time, which could be chosen as described in Ref. [134]. The discrete force term in the momentum space Fmðx; tÞ is given by [135, 136]:

$$F\_{\mathbf{m}}(\mathbf{x},t) = \left(0, 6\frac{\mathbf{F}\bullet\mathbf{u}}{c^2}, -6\frac{\mathbf{F}\bullet\mathbf{u}}{c^2}, \frac{\mathbf{F}\_{\mathbf{x}}}{c}, -\frac{\mathbf{F}\_{\mathbf{x}}}{c}, \frac{\mathbf{F}\_{\mathbf{y}}}{c}, -\frac{\mathbf{F}\_{\mathbf{y}}}{c}, 2\frac{\mathbf{F}\_{\mathbf{x}}u\_{\mathbf{x}} - \mathbf{F}\_{\mathbf{y}}u\_{\mathbf{y}}}{c^2}, \frac{\mathbf{F}\_{\mathbf{x}}u\_{\mathbf{y}} + \mathbf{F}\_{\mathbf{y}}u\_{\mathbf{x}}}{c^2}\right)^T \tag{24}$$

where F is the body force of fluid flow, Fx is the body force in the horizontal direction, and Fy is the body force in the vertical direction. After the collision process, the post collision distribution function in the velocity space fi ðx; t þ δtÞ is calculated through inverse transformation:

$$f\_i(\mathbf{x}, t + \delta\_t) = \mathbf{M}^{-1} \mathfrak{m}\_f(\mathbf{x}, t + \delta\_t) \tag{25}$$

where the dimensionless orthogonal transformation matrix M is chosen as [137]:

$$\mathbf{M} = \begin{Bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ -4 & -1 & -1 & -1 & -1 & 2 & 2 & 2 & 2 \\ 4 & -2 & -2 & -2 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & -1 & 0 & 1 & -1 & -1 & 1 \\ 0 & -2 & 0 & 2 & 0 & 1 & -1 & -1 & 1 \\ 0 & 0 & 1 & 0 & -1 & 1 & 1 & -1 & -1 \\ 0 & 0 & -2 & 0 & 2 & 1 & 1 & -1 & -1 \\ 0 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 \\ \end{Bmatrix} \tag{26}$$

Then, the streaming step is carried out as:

$$f\_{\,i}(\mathfrak{x} + \mathfrak{e}\_{\,i}\delta\_t, t + \delta\_t) = f\_{\,i}(\mathfrak{x}, t + \delta\_t) \tag{27}$$

The nonslip velocity condition on the diffusive interface and in the solid phase is tackled by recalculating the density distribution function fi through a linear interpolation as [138]:

$$f\_i = f\_l f\_i + (\mathbf{1} - f\_l) f\_i^{eq}(\rho, \mathfrak{u}\_\mathfrak{s}) \tag{28}$$

fl is the liquid fraction of PCM, f eq is the equilibrium distribution function, <sup>i</sup> <sup>¼</sup> <sup>M</sup>�<sup>1</sup> eq which could be calculated by inverse transformation as <sup>f</sup> eq <sup>m</sup> . For the solid <sup>i</sup> <sup>f</sup> phase, there is us¼0. Hence, the macroscopic variables, density ρ and velocity u, are defined as:

$$\rho = \sum\_{i=0}^{8} f\_i, \rho \mathbf{u} = \sum\_{i=0}^{8} \mathbf{e}\_i \mathbf{f}\_i + \frac{\delta\_t}{2} \mathbf{F} \tag{29}$$

As mentioned in Ref. [138], the density ρ in the term f eqðρ; us<sup>Þ</sup> in Eq. (28) should <sup>i</sup> be first calculated by Eq. (29) in order to ensure the mass conservation. Then, for the liquid phase of fl ¼ 1, the above lattice Boltzmann model recovers to the standard scheme for incompressible flow. On the other hand, for the solid phase of fl ¼ 0, the above model could satisfy that fi ¼ f eqðρ; us<sup>Þ</sup> indicating that the nonslip <sup>i</sup> velocity u¼us is ensured.

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

#### 3.2.2 MRT LBM for solid-liquid phase change

The MRT lattice Boltzmann equation (LBE) for the total enthalpy H distribution function mgðx; tÞ is expressed as [124]:

$$\mathfrak{m}\_{\mathfrak{g}}(\mathfrak{x}, t + \delta\_t) = \mathfrak{m}\_{\mathfrak{g}}(\mathfrak{x}, t) - \mathfrak{S}\left[\mathfrak{m}\_{\mathfrak{g}}(\mathfrak{x}, t) - \mathfrak{m}\_{\mathfrak{g}}^{eq}(\mathfrak{x}, t)\right] \tag{30}$$

where mg is the distribution in momentum space given as:

$$m\_{\mathbf{g}}(\mathbf{x},t) = \begin{bmatrix} m\_{\mathbf{g}^0}(\mathbf{x},t), m\_{\mathbf{g}^1}(\mathbf{x},t), \dots, m\_{\mathbf{g}^8}(\mathbf{x},t) \end{bmatrix}^T \tag{31}$$

eq The equilibrium moment mg is given by:

$$\begin{split} m^{eq} &= \left( H, -4H + 2c\_{p, \text{ref}}T + 3c\_{p}T \frac{\mathfrak{u}^{2}}{c^{2}}, 4H - 3c\_{p, \text{ref}}T - 3c\_{p}T \frac{\mathfrak{u}^{2}}{c^{2}}, \\ c\_{p}T \frac{\mathfrak{u}\_{\text{x}}}{c}, -c\_{p}T \frac{\mathfrak{u}\_{\text{x}}}{c}, c\_{p}T \frac{\mathfrak{u}\_{\text{y}}}{c}, -c\_{p}T \frac{\mathfrak{u}\_{\text{y}}}{c}, c\_{p}T \frac{\mathfrak{u}\_{\text{x}}^{2} - \mathfrak{u}\_{\text{y}}^{2}}{c^{2}}, c\_{p}T \frac{\mathfrak{u}\_{\text{x}}\mathfrak{u}\_{\text{y}}}{c^{2}} \right)^{T} \end{split} \tag{32}$$

where T isthe temperature, and cp isthe specific heat calculated by interpolation as:

$$\mathbf{c}\_{p} = (\mathbf{1} - f\_{l})\mathbf{c}\_{p,s} + f\_{l}\mathbf{c}\_{p,l} \tag{33}$$

cp,s is the specific heat of PCM at solid state, and cp,l is the specific heat of PCM at liquid state. To achieve good numerical stability, the reference specific heat cp,ref is defined by the harmonic mean of cp,s and cp,l as:

$$\mathcal{L}\_{p,ref} = \frac{2\mathcal{L}\_{p,s}\mathcal{L}\_{p,l}}{\mathcal{L}\_{p,s} + \mathcal{c}\_{p,l}} \tag{34}$$

The parameters in the diagonal relaxation matrix <sup>S</sup> satisfy <sup>s</sup><sup>0</sup> <sup>¼</sup> 1, se <sup>¼</sup> sp, sj <sup>¼</sup> <sup>1</sup> τ , and 0<se, <sup>ε</sup>, <sup>q</sup><2, where the relaxation time τ is given as:

$$\frac{k}{\rho c\_{p,ref}} = c\_s^2 (\pi - 0.5) \delta\_t \tag{35}$$

where <sup>k</sup> is the thermal conductivity, and cs <sup>¼</sup> <sup>p</sup><sup>c</sup>ffiffi is the sound speed. To reduce <sup>3</sup> the numerical diffusion, a "magic" relationship is found by Huang and Wu [124] as:

$$
\left(\frac{1}{s\_\epsilon} - \frac{1}{2}\right)\left(\frac{1}{s\_\circ} - \frac{1}{2}\right) = \frac{1}{4} \tag{36}
$$

Similar to the computation of fluid flow, the post-collision distribution function in the velocity space gi could be calculated by inverse transformation as:

$$\mathbf{g}\_i(\mathfrak{x}, t + \delta\_t) = \mathbf{M}^{-1} \mathfrak{m}\_{\mathbf{g}}(\mathfrak{x}, t + \delta\_t) \tag{37}$$

The streaming process is completed as:

$$\mathbf{g}\_i(\mathbf{x} + \mathbf{e}\_i \delta\_t, t + \delta\_l) = \mathbf{g}\_i(\mathbf{x}, t + \delta\_l) \tag{38}$$

Then, the enthalpy H is calculated as:

$$H = \sum\_{i=0}^{8} \mathbf{g}\_i \tag{39}$$

Furthermore, it should be pointed out that the nonequilibrium extrapolation method developed by Guo et al. could be applied for the boundary conditions of fluid flow and enthalpy on the surfaces of LHTES unit [139, 140].

#### 3.3 Classical examples for code validation

#### 3.3.1 One-dimensional transient conjugate heat transfer

To validate the capability of MRT LBM for tackling the differences in thermophysical properties, the one-dimensional transient conjugate heat transfer in two regions without phase change is used to compare the numerical results with analytical solutions. Initially,T is equal to 1 in the region A at x>0, and T is equal to 0 in the region B at x<0. The analytical solution for this problem is given as [141]:

$$T^{A}(\mathbf{x},t) = \frac{1}{1 + \sqrt{\left(\rho \mathbf{C}\_{p}\right)^{B} k^{B} / \left(\rho \mathbf{C}\_{p}\right)^{A} k^{A}}} \left[1 + \sqrt{\left(\rho \mathbf{C}\_{p}\right)^{B} k^{B} / \left(\rho \mathbf{C}\_{p}\right)^{A} k^{A}} \text{erf}\left(\frac{\mathbf{x}}{2\sqrt{k^{A}t / \left(\rho \mathbf{C}\_{p}\right)^{A}}}\right)\right] \tag{40}$$

$$T^{\mathcal{B}}(\mathbf{x},t) = \frac{1}{1 + \sqrt{\left(\rho \mathbf{C}\_p\right)^B k^B / \left(\rho \mathbf{C}\_p\right)^A k^A}} \text{erfc}\left(-\frac{\mathbf{x}}{2\sqrt{k^B t / \left(\rho \mathbf{C}\_p\right)^B}}\right) \tag{41}$$

When the aluminum is used for region A while the liquid water is chosen as the material in region B, the comparison between LBM results and analytical solutions is presented in Figure 8. A good agreement is observed, and the maximum L2 error with <sup>200</sup> <sup>x</sup> <sup>200</sup> lattice grids is only <sup>6</sup>:<sup>7983</sup> <sup>x</sup> <sup>10</sup>-<sup>5</sup> [142].

#### 3.3.2 One-dimensional melting by conduction

In order to verify the MRT LBM for solid-liquid phase change phenomenon, the one-dimensional melting by conduction at a constant phase change temperature Tm is simulated. Initially, the substance is uniformly solid at a temperature T<sup>0</sup> (T0<Tm). The melting process begins at time t ¼ 0 when the temperature of the left wall is at a high temperature of Th (Th > Tm). Then, the analytical solution for the temperature in this problem is given as [143]:

$$T(\mathbf{x},t) = T\_h - \frac{T\_h - T\_m}{\text{erf}(\lambda)} \text{erf}\left(\frac{\mathbf{x}}{2\sqrt{k\_l t/(\rho c\_{p,l})}}\right), 0 \le t \le X\_i(t), \text{liquid} \tag{42}$$

$$T(\mathbf{x},t) = T\_0 + \frac{T\_m - T\_0}{\text{erfc}\left(\lambda/\sqrt{R\_{\text{ac}}}\right)} \text{erfc}\left(\frac{\mathbf{x}}{2\sqrt{k\_\text{t}t/(\rho c\_{p,t})}}\right), \mathbf{x} \text{>} X\_i(t), \text{solid} \tag{43}$$

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

#### Figure 8.

One-dimensional transient conjugate heat transfer [142].

where kl and ks are the thermal conductivities of liquid PCM and solid PCM, <sup>½</sup>kst<sup>=</sup>ð<sup>ρ</sup>cp,sÞ� respectively. The parameter Rac <sup>¼</sup> is the ratio of thermal diffusivity <sup>½</sup>klt<sup>=</sup>ð<sup>ρ</sup>cp,lÞ� between the solid and liquid phases of PCM, and the location of phase interface Xi is calculated as:

$$X\_i = 2\lambda \sqrt{k\_l t / (\rho c\_{p,l})} \tag{44}$$

The parameter λ is the root of the transcendental equation:

$$\frac{\text{Ste}\_l}{\exp\left(\lambda^2\right)\text{erf}\left(\lambda\right)} - \frac{\text{Ste}\_s\sqrt{R\_{ac}}}{\exp\left(\lambda^2/R\_{ac}\right)\text{erfc}\left(\lambda/\sqrt{R\_{ac}}\right)} = \lambda\sqrt{\pi} \tag{45}$$

where the Stefan numbers, Stel and Stes, are defined as:

$$\text{Ste}\_l = \frac{c\_{p,l}(T\_h - T\_m)}{L}, \text{Ste}\_s = \frac{c\_{p,s}(T\_m - T\_0)}{L} \tag{46}$$

The comparison of temperature T between the analytical solutions and the SRT or MRT LBM with different relaxation times is presented in Figure 9. It could be observed that the numerical diffusion exists for the small ðτ<sup>l</sup> close to 0:5Þ and large ðτl>2Þ relaxation times when the SRT model is applied. However, the numerical diffusion is highly reduced once the MRT model with a "magic" parameter relation shown in Eq. (36) is used.

#### 3.3.3 Two-dimensional melting by convection

The natural convection with melting in a square cavity heated from the side wall is usually used to validate the code for solid-liquid phase change. First, the

Figure 9. (a) SRT LBM (b) MRT LBM. Comparison of temperature T between analytical solutions and SRT or MRT LBM results with different relaxation times [124].

solid-liquid phase change with convection in a cavity at the Rayleigh number Ra ¼ 25000, the Prandtl number Pr ¼ 0:02, and the Stefan number Ste ¼ 0:01 is compared with the results by Mencinger [144]. The average Nusselt number Nuave at the hot wall in terms of the Fourier numbers Fo is plotted in Figure 10(a), and the melting interface positions at Fo ¼ 10 and Fo ¼ 20 are shown in Figure 10(b). In addition, the average melting fraction fl is also presented in Figure 10(c). It is

Figure 10. Natural convection with melting in a cavity [142].

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

obvious that the current LBM results are consistent with the work of Mencinger [142]. Besides, for the cases of Pr>1, the MRT LBM for natural convection with melting in cavity could be calibrated by the scaling laws and correlations derived by Jany and Bejan [145]. The average Nusselt number Nuave at hot wall is given as:

$$Nu\_{\text{ave}} = \left(2Fo \text{Ste}\right)^{-1/2} + \left[0.33Ra^{1/4} - \left(2Fo \text{Ste}\right)^{-1/2}\right] \left\{1 + \left[0.0175Ra^{3/4} \text{(FoSte)}^{3/2}\right]^{-2}\right\}^{-1/2} \tag{47}$$

As shown in Figure 10(d), the average Nusselt number Nuave at Pr ¼ 6:1989 and Ste ¼ 0:1 agrees well with the results of scaling law correlations at different Rayleigh number Ra.

## 3.4 GPU acceleration

The characteristic of highly parallel nature is a significant advantage of LBM over other traditional macroscopic numerical methods. In the CUDA programming platform, the CPU and GPU work as the host and the devices, respectively, as shown in Figure 11 [146]. It means that the parallel tasks are executed on GPU, while the CPU is responsible for the initial conditions and all the sequential

Figure 11. Schematic diagram of CUDA platform [146].

commands. First, the initial conditions are set up in the host memory, and then the data are moved to the memory of GPU. The threads are grouped into blocks, which are the component of grids as displayed in Figure 11. For instance, in the real simulation, if the lattice grid number is (Mx, My, Mz) and the block size is (Nx, Ny, ˜ ° Nz), the corresponding grid size is Mx=Nx; My=Ny; Mz=Nz . Furthermore, <sup>a</sup> kernel is a function, which is executed on the concurrent threads on GPU. The collision step, streaming step, tackling of boundary conditions, and computation of macroscopic variables are completed in different kernels. It is important to note that all the kernels should be synchronized between CPU and GPU. Finally, the data on the GPU should be copied to the host memory of CPU for printing at any specific time when the results are needed. During the recent years, LBM is demonstrated to be appropriate for GPU computing [147–151], and it has been applied to solve several different physical problems [152–155].

## 4. Applications of LBM modeling in latent heat thermal energy storage

### 4.1 Enhancement of PCM performance with fins

As presented in the previous sections, the extended fins could be used to increase the heat transfer depth in LHTES system and accelerate the energy storage efficiency. In the recent years, LBM is applied by several researches to study the solid-liquid phase change of PCMs with extended fins [142, 156, 157]. Jourabian et al. studied the melting process of PCM in a cavity with a horizontal fin heated from the sidewall by enthalpy-based D2Q5 LBM [156]. The results indicated that adding a fin enhances the melting rate for all positions and different lengths compared with the LHTES cavity without fin. They also found that although varying the position of the fin from the bottom to the middle has a negligible effect on melting rate, the melting time is increased once the fin is mounted on the top of LHTES cavity. Talati and Taghilou used an implicit LBM to study the PCM solidification in a rectangular finned container [157]. It was found that the optimum aspect ratio of container for solidification equals to 0:5, and changing the fin material from aluminum to copper has no significant influence on the solidification rate. Ren and Chan applied enthalpy-based MRT LBM to investigate the PCM melting performance in an enclosure with internal fins and finned thick walls with GPU acceleration [142]. The transient PCM melting process with different number of fins at the Fourier number Fo ¼ 0:15 is shown in Figure 12 in terms of temperature contours, liquid fraction, and streamlines. It could be found that the PCM melting rate is obviously enhanced by adding more internal fins in the cavity. However, the energy storage capacity of LHTES system is reduced when the number of internal fins increases, so that the appropriate fin configuration and number should be designed for engineering applications. They found that using a less number of longer fins is more effective than applying shorter fins for enhancing the thermal performance of PCMs. Besides, compared with the LHTES cavity with horizontal fins heated from side walls, the LHTES enclosure using vertical fins heated from the bottom surface has a better charging rate. From the above researches, it should be concluded that the enthalpy-based LBM is successful for simulating the conjugate heat transfer with solid-liquid phase change for melting and solidification of PCMs accelerated with fins.

#### 4.2 Nanoparticle-enhanced PCM

The nanoparticles are commonly used to ameliorate the low thermal conductivity of most PCMs. With some appropriate assumptions as presented in Section 2,

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

Figure 12. Transient PCM melting process with different number of fins [142].

the thermal performance of PCMs with nanoparticles could be modeled using enthalpy-based LBM. Jourabian et al. studied the convective melting process of Cunanoparticle enhanced water-ice in a cylindrical-horizontal annulus [158, 159]. It was demonstrated that the melting rate of NEPCM is accelerated due to the improvement of thermal conductivity and the reduced latent heat. When the heated cylinder located in the bottom section of the annulus, the increment effect on NEPCM melting rate by adding more nanoparticles decreases because of the augmentation of NEPCM viscosity, which weakens the convective heat transfer. Feng et al. investigated the melting of water (ice)-copper nanoparticle NEPCM in a bottom-heated rectangular cavity by treating the latent heat source term with an implicit scheme [122]. They also found that the heat transfer rate of NEPCM increases with respect to the increment of nanoparticle volume fractions. However, the energy storage rate is the most significant parameter for a LHTES unit. Although adding nanoparticles into PCMs could increase their thermal conductivity, it increases the viscosity of PCM, which weakens the convective heat transfer. Due to this reason, the energy storage rate of LHTES system may even decrease with the increasing nanoparticle volume fractions, especially for the case with large temperature gradient. On the other hand, the increment of nanoparticle volume fraction decreases the energy storage capacity of LHTES unit, so that the energy storage rate could also be affected. Under this circumstance, more future research attentions should be paid on the influences of nanoparticles on the energy storage rate of NEPCM. In addition, the solid-liquid phase change model of NEPCM using LBM could be extended to study the charging and discharging of NEPCM under hybrid heat transfer enhancement techniques such as nanoparticle-fin or nanoparticle-metal foam combinations.

### 4.3 PCM filled with metal foams

Due to its intersected and connected heat transfer channels, metal foam is inserted into the LHTES system for enhancing the melting and solidification rates of PCMs. The numerical modeling of PCM solid-liquid phase change phenomenon filled in metal foams using enthalpy-based LBM could be categorized as the representative elementary volume (REV) scale modeling [128–130, 160] and pore-scale modeling [111, 161]. Tao et al. investigated the performance of metal foams/paraffin composite PCM in a LHTES cavity using LBM at REV scale [160]. They found that increasing the metal foam PPI (number of pores per inch) could enhance the conduction heat transfer, while the convective heat transfer is weakened. In addition, although decreasing the metal foam porosity could accelerate the PCM melting rate, the energy storage density of LHTES unit is dramatically reduced. Due to the above two tradeoffs, the optimum metal foam structure with the porosity of 0.94 and PPI of 45 is highly recommended. However, the REV scale modeling requires the use of some empirical relations, and the influences of metal foam morphology on energy storage rate are difficult to be analyzed. As a comparison, with the advantage of

#### Figure 13.

Transient PCM melting process filled in metal foams with different pore sizes [111]. (a) εave ¼ 0:95 and dp ¼ 1 mm and (b) εave ¼ 0:95 and dp ¼ 0:75 mm.

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

LBM for tackling complex boundary conditions, pore-scale modeling without using any empirical equations achieves the detailed fluid flow and heat transfer information inside the metal foams, so that the energy storage efficiency of LHTES unit could be further optimized. Ren et al. investigated the effects of metal foam characteristics on the PCM melting rate through pore-scale LBM modeling [111]. The transient PCM melting process inside the metal foams of different pore sizes at the Fourier number Fo ¼ 0:04 is plotted with respect to temperature field, melting interface, and fluid velocity vectors in Figure 13. It could be observed that the charging rate of PCM is accelerated when the pore size decreases from dp ¼ 1 mm to dp ¼ 0:75 mm at a porosity of εave ¼ 0:95. Furthermore, the temperature field in the melted region of LHTES with smaller pore size dp ¼ 0:75 mm is found to be more uniform than that of LHTES with pore size dp ¼ 1:0 mm, which means that the thermal conductivity and its corresponding conduction heat transfer in LHTES unit are improved with a decreasing pore size (increasing PPI). Besides, they also concluded that an appropriate metal foam porosity should be chosen in the real engineering applications in order to balance the PCM melting speed and the energy storage density of LHTES unit. By reconstructing the microstructure of metal foam with X-ray computed tomography, Li et al. investigated the solid-liquid phase change phenomenon of PCM inserted with metal foams under different gravitational acceleration conditions [161]. The results indicated that the transition of the dominant heat transfer mechanism from convection to conduction occurs when the gravity gradually decreases. Due to the attenuated convection effect with decreasing gravity, the PCM melting development is dramatically hindered. Besides, they also concluded that the decreasing metal foam porosity could enhance the effective thermal conductivity of LHTES unit because of the extended heat transfer area. However, the above pore-scale modeling of PCM charging and discharging processes enhanced by metal foams is limited in two-dimensional cases. The three-dimensional pore-scale modeling, which is indispensable for optimizing the complicated metal foam structures, should be carried out in the future work.

#### 4.4 PCM with heat pipes

The heat pipes are used to transfer heat between PCMs and heat transfer fluid (HTF), so that the charging and discharging of LHTES system could be accelerated. The configuration, arrangement, and number of heat pipes in the LHTES unit are essential for the PCM melting and solidification speed. Luo et al. applied LBM to study the convection melting in complex LHTES system with heat tubes [162]. The effects of inner heat pipe arrangement on PCM melting process are illustrated in Figure 14 with respect to temperature, flow, and phase fields at the Fourier number Fo ¼ 3, the Prandtl number Pr ¼ 0:2, the Stefan number Ste ¼ 0:02, and the Rayleigh number Ra <sup>¼</sup> <sup>5</sup> � <sup>10</sup>4. For the case using centrosymmetric inner heat pipes, the conduction heat transfer is dominant because the inner heat pipe is surrounded by solid PCM at the melting temperature, so that its melting rate is faster than the LHTES system with inline or staggered heat tubes as displayed in Figure 15. In addition, there is no obvious difference between the melting rates of LHTES systems using inline or staggered heat pipes. Although using heat pipe individually could enhance the thermal performance of LHTES to some extent, other heat transfer enhancement techniques are usually coupled with heat pipes such as fins or nanoparticles to further improve the PCM charging and discharging rate.

#### Figure 14.

Temperature, flow, and phase fields at Fo ¼ 3 with different arrangements of inner tubes: (a) centrosymmetric, (b) inline, and (c) staggered for the Prandtl number Pr ¼ 0:2, the Stefan number Ste ¼ 0:02, and the Rayleigh number Ra <sup>¼</sup> <sup>5</sup> � 104 [162].

#### Figure 15.

The total liquid fraction fl versus the Fourier number Fo for different arrangements of inner tubes at the Prandtl number Pr <sup>¼</sup> <sup>0</sup>:2, the Stefan number Ste <sup>¼</sup> <sup>0</sup>:02, and the Rayleigh number Ra <sup>¼</sup> <sup>5</sup> � <sup>10</sup><sup>4</sup> [162].

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

## 4.5 PCM with hybrid heat transfer enhancement techniques

In this section, the melting and solidification of PCMs enhanced using combined heat transfer enhancement techniques modeled by LBM are discussed, and the effectiveness of different approaches for improving the heat transfer capability of LHTES unit is compared. Huo and Rao carried out an investigation of NEPCM solid-liquid phase change process in a cavity with a heat pipe and separate plates using LBM [163]. It was found that the NEPCM of the case with separate located in the middle of cavity melts fastest because of the weakened heat accumulation. The results also showed that when the location of separate is less than 0.3, the melting rate of NEPCM is even slowed down due to the heat accumulation around the separate plate. Gao et al. developed an enthalpy-based MRT LBM model with a free parameter in the equilibrium distribution function for solidliquid phase change in porous media and conjugate heat transfer with highcomputational efficiency and stability [164]. Then, they investigated the PCM melting process in porous media with a conducting fin, and the results indicated that the heat transfer rate could be further improved by adding a fin into the porous medium. It was also observed that the vertical position of the fin has no remarkable impact on the PCM melting speed when there exists porous media. Jourabian et al. investigated the constrained ice melting around a cylinder using three heat transfer enhancement techniques by LBM [165]. They pointed out that adding nanoparticles may increase the dynamic viscosity of the base PCM, which

#### Figure 16.

Transient nanoparticle-enhanced PCM melting process filled in metal foams of pore size dp ¼ 1 mm and heat pipe radius R ¼ 2 mm [166]. (a) Φ ¼ 0:09, εave ¼ 0:99 and (b) Φ ¼ 0:01, εave ¼ 0:91.

has a negative effect on natural convective heat transfer. Besides, the tradeoff between consolidating the conduction heat transfer and weakening the convective heat transfer through decreasing the metal foam porosity needs further attention and investigation. Ren et al. completed a pore-scale comparative study of nanoparticle-enhanced PCM melting in a heat pipe–assisted LHTES unit with metal foams [166]. The melting process of NEPCM in a LHTES cavity filled with metal foams using a heat pipe at the Fourier number Fo ¼ 0:06, pore size dp ¼ 1 mm, and heat pipe radius R ¼ 2 mm is shown in Figure 16. Although different combinations of nanoparticle volume fraction Φ and metal foam porosity εave are used for the LHTES unit, it should be noted that the volume fraction of pure PCM is kept at 90%, so that the energy storage capacity is unchanged for a fair comparison. It could be observed that the NEPCM with less nanoparticle volume fraction and lower metal foam porosity melts faster than that with more nanoparticles in the higher porosity metal foam. This finding indicated that using metal foams is more effective than adding nanoparticles for enhancing the charging rate of NEPCMs. They also found that there exists an optimum heat pipe radius for achieving the best energy storage rate in LHTES unit. As discussed in the above sections, LBM is demonstrated to be appropriate for simulating the charging and discharging processes of PCMs in LHTES system with different kinds of heat transfer enhancement technologies. Under this circumstance, the enthalpy-based LBM will definitely play a more significant role in the future research of thermal energy storage using PCMs.

## 5. Summary

In this chapter, different heat transfer enhancement techniques of PCMs for LHTES unit are discussed and compared. As the numerical modeling plays a significant role in clarifying the mechanism of complicated physical processes, the mathematical models for fluid flow of liquid PCMs and the PCM solid-liquid phase change phenomenon are presented. In order to investigate the PCM charging and discharging processes enhanced by nanoparticles, fins, or metal foams, the empirical relations for nanofluids and the mathematical formula for conjugate heat transfer are shown. The development history of the lattice Boltzmann method for solidliquid phase change problems is carefully reviewed, and the enthalpy-based multiple-relaxation-time LBM is discussed in detail due to its simplicity and robustness. Besides, the implementation of GPU computing is briefly discussed to accelerate the computational efficiency of LBM modeling. Then, the applications of LBM modeling in LHTES system with different heat transfer enhancement approaches are presented, which demonstrate that the mesoscopic and highly parallel LBM is powerful for understanding the melting and solidification processes of PCMs.

## Acknowledgements

This work was financially sponsored by the National Key Research and Development Program of China (No. 2017YFB0102703), the National Natural Science Foundation of China (No. 51536003), 111 Project (grant number B16038), and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51721004).

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling DOI: http://dx.doi.org/10.5772/intechopen.80574

## Author details

Zhiguo Qu

Key Laboratory of Thermo-Fluid Science Engineering of MOE, School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi, China

\*Address all correspondence to: zgqu@mail.xjtu.edu.cn

© 2018 The Author(s). Licensee IntechOpen. This chapteris distributed underthe terms oftheCreative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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## Chapter 4
