High-conductivity nanomaterials for enhancing thermal performance of latent heat thermal energy storage systems

Abstract

Dispersing high-conductivity nanomaterials into phase change materials (PCM) of latent heat thermal energy storage systems (LHTESS) is expected to solve the problem of poor thermal conductivity of PCMs. Accordingly, several metals, metal oxides and non-metals are employed as nanoadditives for PCMs by researchers. Besides thermal conductivity of PCMs, the other thermo-physical properties are also altered by nanoadditives. This paper provides comprehensive information on the effects of nanoadditives on the thermo-physical properties of PCMs through a critical review of related published works. The modified properties ultimately determine the charging and discharging rates of LHTESS. The extent of improvement in the thermal performance and the related issues are addressed. Further, the theoretical/empirical models developed so far for the evaluation of thermo-physical properties are deliberated.

Graphical abstract

Introduction

For the past few decades, scientific community has been trying to address the most serious issues of fossil fuels depletion and greenhouse gases production. The simultaneous solution of these problems remains in the form of renewable energy sources. Among various renewable energy sources, solar energy is given more attention mainly due to its abundant nature. However, even after many years of global research, large-scale utilization of solar energy has not been realized yet as the source availability is inconsistent. This drives the scientific community to look for effective devices to store solar thermal energy. Solar energy storage in the form of latent heat brings out some attractions which include high storage density, isothermal heat transfer, compact modules and availability of storage mediums for wide range of working temperatures.

Despite these favours, the engineering applications of latent heat thermal energy storage systems (LHTESS) are limited predominantly due to the poor thermal conductivity of the storage mediums available. Thus, the storage mediums which are the phase change materials (PCM) are not able to store or release the energy at a reasonable rate. The heat transfer rate can be accelerated by various techniques and the literature reveals few which are,

• Modules with fins

• Use of multiple PCMs

• Thermal conductivity enhancement

• Microencapsulation of PCM

The earlier works on the above techniques are extensively reviewed by Jegadheeswaran and Pohekar [2] and recently Tao and He [3]. Among the techniques, enhancing the thermal conductivity of PCMs using thermal conductivity promoters is simple and straightforward. As described by Fan and Khodadadi [4], there are two classes of promoters: stable structures and free-form enhancers. The high-conductivity stable structures are of different shapes/forms such as fins [5, 6], porous matrices [7], rings [8] and balls [9]. Representative images of those structures are shown in Fig. 1.

Fig. 1

Fixed thermal conductivity enhancers: a fins [5], b porous matrices [7], c rings [8], d balls [9]

As detailed by Jegadheeswaran and Pohekar [2], heat transfer during solidification is majorly by conduction whereas it is by natural convection in liquid PCM during melting except during the initial period. The melting rate may get suppressed as the presence of fixed structures would dampen out the natural convection. Hence, the increase in conduction heat transfer rate due to fixed structures must be much higher so that decrease in convection heat transfer rate would be negligible. This, of course needs careful design of structures and proper form. Further, the volume of structures affects the storage capacity of the system and the porosity of the metals employed plays vital role in altering the latent heat capacity of the PCM. According to Shiina and Inagahi [10], the porosity of metals should be as high as possible for high latent heat capacity. Fan and Khodadadi [4] have reviewed the application of fixed structures as thermal conductivity enhancer for PCMs.

The other option is use of free-moving high-conductivity materials. For long time, researchers have been investigating the augmentation of thermal conductivity of fluids used in heat transfer applications by suspending high-conductivity particles [11]. Conventionally, the dispersed solid additives are of micron sized or sometimes the dimension may be in few millimetres and research on fluids containing such solid particles was started almost 100 years back [12]. In case of PCMs, only few works have reported the employment of micron sized particles. For example, Mettawee and Assassa [13] and Chaichan et al. [14] have used aluminium particles and Jegadheeswaran et al. [15] have used copper particles for PCMs. All the authors have reported improved performance; however, the issues of particles settlement are not addressed. In addition to settlement related issues, large size particles may instigate abrasion problem [16]. In this perspective, it can be stated that employing particles of larger size may not be a good idea. Alternatively, materials of various shapes, i.e. particles, fibres, sheets and tubes with average dimension of less than 100 nm have been showing lot of promise in the field of enhanced heat transfer [17]. The nanomaterials are known for inherent attractive features such as high surface area-to-volume ratio, less particle momentum and high particle mobility. Thanks to these features, nanomaterials are expected to offer benefits like higher heat conduction (due to higher surface area), higher microconvection (due to mobility), better stability (due to reduced size) and reduced erosion (due to less momentum) [18]. The nanofluids can be prepared in two methods, namely single step method and two step method [19]. The two preparation methods are outlined in Fig. 2.

Fig. 2

Nanofluid preparation [19]: a single step method, b two step method

In case of heat transfer applications, use of nanomaterials started in the mid-1990s. However, it was limited to mainly fluids of single phase (liquids) and liquid–vapour two-phase fluids until early 2000 [20]. Since then, there has been growing interest among the researchers in nanosize high-conductivity materials added to PCMs. As reported in the earlier works, the nanoadditives can also be used for PCMs to augment the thermal conductivity. Despite good number of works, the full-scale employment of nanoadditives for PCM-based LHTESS needs further research. The relation between altered thermo-physical properties and thermal performance, appropriate material as nanoadditive, optimum concentration of nanoadditives and evaluation of thermo-physical properties are the key issues which need more clarity. In this perspective, this paper is aimed at reviewing the earlier works to present inclusive aspects of nano-PCM research and to identify what to be focussed in future works.

Nanomaterials and thermo-physical properties of composites

Present day nanoscience makes it possible for developing almost all the materials in nanosize. Among the available materials, it is important to choose appropriate materials for PCMs from the perspective that the resulting composites exhibit enhanced thermal performance. However, the thermal performance of PCMs depends on the thermo-physical properties and the addition of nanomaterials is expected to alter them. This necessitates the evaluation of thermo-physical properties of composites which would help in great deal in selecting appropriate materials as nanoadditives.

Thermal conductivity of composites

The nanoadditives are used mainly to overcome the poor thermal conductivity of available PCMs. Hence, it is obvious that the material should possess higher thermal conductivity. In this perspective, researchers have identified and employed quite a number of metals and metal oxides. Further, few non-metals are also in the picture. The list of various materials employed as nanoadditives with their thermal conductivity values is given in Table 1.

Table 1 List of nanomaterials with their thermal conductivities

Among metals, copper is widely employed as it is superior in terms of thermal conductivity. Lin and Al-kayiem [28] have shown that paraffin wax’s thermal conductivity could be increased by 30% with copper nanoparticles of 2% volume fraction. Zeng et al. [29] have used copper particles for tetradecanol and have reported 9 times higher thermal conductivity when 12% volume of copper nanowires was dispersed. Khodadadi and Hosseinzadeh [30], Hosseinzadeh et al. [21] and Sebti et al. [31] have not measured the thermal conductivity of nanocopper–PCM composites as these are basically numerical studies. However, these works reveal that the charging and discharging rates of composite PCMs are quite higher as compared to those of pure PCM. It is obvious that higher phase change rates are because of enhanced thermal conductivity. Hence, copper nanoadditives seem to be a potential candidate for enhancing the phase change rate. Apart from copper, aluminium and silver have also been utilized. For example, Constantinescu et al. [32] and Kalaiselvam et al. [33] have used aluminium and Wei [23], Parameshwaran et al. [34] and Zeng et al. [35] have used silver. The results are interesting as Parameshwaran et al. [34] could achieve 67% increase in thermal conductivity of organic ester with 5% mass fraction of silver particles.

The most tested metal oxides as nanoadditives for PCMs are copper oxide, aluminium oxide (alumina) and titanium oxide. All these metal oxides are also proven to be good thermal conductivity enhancers. Harikrishnan and Kalaiselvam [36] employed oleic acid with copper oxide. The increase in thermal conductivity was up to 100% when 2% (mass fraction) particles were dispersed. However, Jesumathy et al. [37] have given contradictory results as even 10% mass fraction of copper oxide could result in an increase of only 7.8%. In case of alumina, mass fraction of 10% could improve the thermal conductivity of paraffin by 17% [38]. Similarly, BaCl₂ solution ended with 16.74% improvement due to titanium oxide of 1.13% volume fraction [39]. The above outcomes indicate that the metal oxides can also work well for PCMs. However, TiO₂ is found to be superior to other metal oxides by Teng and Yu [40]. The superiority of TiO₂ over alumina is also demonstrated by Raja Jeyaseelan et al. [41]. Moreover, due to better suspension stability, TiO₂ could widen up the phase change range of paraffin. This allows melting process to occur even with relatively low temperature heat source.

The summarized studies indicate that all metal oxides are far inferior as compared to metals. This is because metals generally possess excellent thermal conductivity than their respective oxides, and one can come to the conclusion that metals are better choice than metal oxides. However, metals tend to oxidize which in turn reduces the thermal conductivity of PCMs over a period of time. Moreover, the metal additives would increase the mass and cost of the storage systems considerably. Hence, it is important to look for effective materials as nanoadditives other than metals and metal oxides.

Accordingly, nanoversions of carbon and its allotrope graphite are found to be potential candidates as carbon possesses low density compared to metals but comparable thermal conductivity. Kibria et al. [42] have indicated that carbon and graphite materials also possess some unique properties like resistance to corrosion/chemical attack and compatibility with all PCMs. In case of graphite, investigators have largely employed exfoliated graphite nanoplatelets (xGnP) although graphite nanofibres and graphene are also found to be suitable [43]. Shi et al. [44] have revealed the ineffectiveness of graphene due to more number of nanointerfaces which result in high interfacial thermal resistance, and hence, xGnP is suggested for effective thermal conductivity enhancement. At the same time, graphene would significantly enhance the shape stabilization performance due to its higher specific surface area than xGnP. Hence, the authors have suggested employing both the materials at right proportion but in smaller fraction. Kim and Drzal [45] have used xGnP up to 7% mass fraction for improving thermal conductivity of paraffin and have reported more than 200% increase. Jeon et al. [46] investigated the benefit of adding xGnP into three types of PCMs namely octadecane, heaxadecane and paraffin. All the composites showed remarkable results with a maximum of 100% when 5% mass fraction of xGnP was added into octadecane. However, Li [47] recently has claimed that nanographite powder has larger specific surface area and length–diameter ratio than xGnP and thus can form heat transfer network. Accordingly, the thermal conductivity of paraffin is increased by 250% with 7% mass fraction of nanographite powder as against 200% increase reported by Kim and Drzal [45]. The works discussed above highlight a near linear increase in PCMs’ thermal conductivity with graphite loading.

Carbon is another widely employed nanoadditives for PCMs in various forms. The use of carbon as nanoadditives for PCM was initiated by Elgafy and Lafdi [48] by employing carbon nanofibres (CNF) for paraffin wax. The thermal conductivity was improved by around 35% with 4% mass fraction of CNF. In another work by Sakalaukus et al. [49], the paraffin wax conductivity was observed to be increased by about 40% when 4% mass fraction of CNF was used.

However, the current research on nanomaterials takes special interest in carbon nanotubes (CNT). CNF with graphene layers wrapped into perfect cylinders are called CNTs for which extremely high thermal conductivity is predicted [50]. The high thermal conductivity is the result of large photon mean free path [51]. The CNTs can be either single walled (SWCNT) or multi-walled (MWCNT). The schematic representation of the structure of SWCNT and MWCNT is presented in Fig. 3. In general, CNTs possess very high thermal conductivity, i.e. fivefolds as compared to copper.

Fig. 3

Structure of carbon nanotubes (CNT) [50]: a graphite lattice, b single walled CNT, c multi-walled CNT

Using 0.6% volume fraction of MWCNT, Kumaresan et al. [52] could augment the thermal conductivity of paraffin by around 45%. Similarly, Wang et al. [53] have reported 30% higher thermal conductivity for palmite acid due to the addition of 1% mass fraction of MWCNT. The above-mentioned works have observed linear enhancement with CNT in both solid and liquid states. However, according to Lajvardi et al. [54], thermal conductivity of PCM has only nonlinear increase with MWCNT loading. Ji et al. [55] have recommended functionalized MWCNTs (oxidized MWCNTs adsorbed pyrogallol) for better thermal conductivity enhancement as this class has better dispersing capability than oxidized MWCNTs. On the other hand, Wu et al. [56] have suggested employing nanoencapsulated PCMs which are formed by impregnating PCMs in CNTs in place of nanocomposite PCMs. The authors have reported that the thermal conductivity of lauric acid could be increased by more than 20 times when the PCM was nanoencapsulated. Among carbon additives, graphite powder is preferred as it exhibits higher thermal conductivity enhancement than CNT and is also less expensive [57].

Besides non-metals, oxides of non-metals have also been studied as nanoadditives for PCMs. Selvaraj et al. [58] have shown thermal conductivity increase as 39% and 15% for deionized water and ethylene glycol, respectively, with 2% volume fraction of beryllium oxide (BeO). On the other hand, Manoj Kumar and Mylsamy [27] employed silicon oxide (SiO₂) for paraffin wax and have reported around 23% enhancement in thermal conductivity with 1% mass fraction although effect of volume fraction of nanoadditives was not investigated.

The concluded discussion clearly indicates that all the tested nanoadditives are capable of enhancing the thermal conductivity of various PCMs. Table 2 presents the role of different classes of nanomaterials in enhancing thermal conductivity of various PCMs. However, the increase is generally nonlinear with increase in concentration of nanoadditives which is shown in Fig. 4.

Table 2 Role of various nanomaterials in thermal conductivity enhancement of PCMs

Fig. 4

Nonlinear variation of thermal conductivity with particle loading: a CuO [36], b Ag [34], c CNT [52]

At this point, it is also imperative to address the consequence of particle size as far as thermal conductivity enhancement is concerned. To investigate the same, Zabalegui et al. [68] carried out experimental study on multi-walled carbon nanotube (MWNT) added to paraffin. Although the particle diameter is found to have negligible role, there are good number of works show the other way. According to Keblinski et al. [69], the mixture thermal conductivity is a function of not only particle volume fraction but also of particle size. Eastman et al. [70] have shown the inadequacy in calculation of thermal conductivity of nanofluids if only the concentration and shape of particles are taken into account. The need for comprehensive studies with samples of varying particle size with same composition is also stressed by the authors. Masuda et al. [71] have found 30% increase in thermal conductivity of water using Al₂O₃ nanoparticles of 4.3 vol% which has a mean diameter of 13 nm. On the other hand, Lee et al. [72] observed less than 10% enhancement only with same Al₂O₃ nanoparticle but of mean diameter of 38 nm. This indicates that decrease in particle size leads to increase in thermal conductivity. On the contrary, the increase in particle diameter of MWCNT resulted in increase in thermal conductivity of paraffin wax as reported by Temel et al. [73]. The authors have attributed this to the effective photon transfer due to larger particle size.

The relationship between particle size and thermal conductivity is due to Brownian-induced motion of the nanoparticles. Significant number of works focusing on the Brownian motion can be found in the literature [74,75,76,77,78,79]. Prasher et al. [80] have demonstrated that the Brownian motion is the prime contributor to the thermal conductivity enhancement. The Brownian motion is nothing but the irregular movement of the particles caused by the random collisions of the surrounding liquid molecules [81]. Figure 5 presents the schematic of Brownian motion of a particle.

Fig. 5

Brownian motion of a particle

Since nanoparticles possess surface energy due to higher surface area, the Brownian motion gets accelerated which in turn makes it nanoconvection dominant [53]. To quantify Brownian motion, the time averaged velocity of the particles (thermophoretic velocity) under a steady temperature gradient is calculated. Michaelides [22] has stated that the thermophoretic velocity depends on particle size. According to the order-of-magnitude analysis presented by Prasher et al. [80], the Brownian motion of smaller particles is stronger than that of relatively bigger particles. This shows that maximum benefit can be achieved if particles of least possible diameter are used. However, Shin and Banerjee [82] have stressed that the interfacial thermal resistance between particles and base fluid increases with decrease in particle size and thus improved thermal conductivity with particles of smaller size. Hence, the optimum size of the particles should be known to extract maximum benefit in terms of higher thermal conductivity. The optimum size can be estimated using the values of interfacial thermal resistance and thermal conductivities of base fluid and nanoparticles. The required expression can be found in the articles of Prasher et al. [80] and Shin and Banerjee [82]. The list of past works which have reported the relationship between particle size and thermal conductivity is presented in Table 3.

Table 3 Summary of works investigated the effect of particle size on effective thermal conductivity

Latent heat of composites

It is a matter of concern as the addition of nanomaterials may lead to reduction in latent heat value of PCM. According to Cai et al. [85] who prepared paraffin with high density polyethylene (HDPE), the additive in the composites makes the three-dimensional net structure more compact which reduces the molecular heat movement of PCM. Since the authors have not used nanoadditives, the validity of above phenomenon is not clear when it comes to nanoadditives-dispersed PCMs.

Latent heat value of PCMs is the direct measure of storage capacity of LHTESS modules. If the addition of nanomaterials results in significant reduction in latent heat of PCM, then the energy density of the storage unit would not be up to the mark. This makes no sense in employing PCM-based LHTESS units even if there is a drastic improvement in heat transfer. The reduction in latent heat due to metal nanoadditives is addressed by Parameshwaran et al. [34] and Zeng et al. [29]. Parameshwaran et al. [34] observed that addition of silver nanoparticles (5 mass%) into organic ester resulted in 11% decrease in the latent heat. On the other hand, 7.42% mass of copper nanoparticles led to reduction of latent heat of tetradecanol by 9.4% as reported by Zeng et al. [29].

The nanoversions of metal oxides are also found to be behaving like metals in reducing the latent heat. For example, 2% mass fraction of CuO led to a reduction in latent heat of oleic acid by 5.2% according to Harikrishnan and Kalaiselvam [36]. He et al. [39] have shown that 1.13% volume fraction of TiO₂ resulted in 9.5% reduction of latent heat of BaCl₂ solution, whereas 4% reduction for paraffin wax is reported by Nabhan [66] with 5% mass fraction of TiO₂. Similarly, addition of 5% and 10% mass fraction of Al₂O₃ resulted in 7% and 13% reduction, respectively, for the latent heat of paraffin according to Ho and Gao [38]. However, Saeed et al. [86] could observe the reduction only when mass fraction of Fe₃O₄ was 5% and above though there was slight improvement with 1 mass%.

Like metals and metal oxides, the non-metallic additives are found to be causing the reduction in latent heat. Li [47] has observed a maximum of 10% reduction in latent heat of paraffin with 10% mass fraction of graphite, whereas 3% mass fraction of CNT brings down the same by 3.69% [67, 87]. The MWCNT used by Parlak et al. [65] prompted less than 10% of reduction when the mass fraction was 5%. Similar effect is also observed by Kumaresan et al. [52] and Xiang and Drzal [88] with MWCNT and xGnP, respectively. Fang et al. [62] have used boron nitride nanosheets for paraffin and have found latent heat reduction of less than 12% even with 10% mass fraction of nanoadditives against the maximum thermal conductivity enhancement of 60%.

Contrary to all the above works, the results of Chieruzzi et al. [89] show an increase in latent heat due to nanoadditives. In spite of this, the above summary affirms that all types of nanoadditives cause reduction in latent heat in all cases (Fig. 6). However, considering the extent to which the thermal conductivity is enhanced, the reduction in latent heat is less significant. The same can be confirmed from the results of Risueno et al. [90] as the change in latent heat values of PCMs was found to be less significant when metal alloys were used as nanoadditives. Nevertheless, the quantity of nanoadditives should be kept as minimum as possible to minimize the adverse effect on latent heat.

Fig. 6

Variation of latent heat with particle loading: a CuO [36], b Ag [34], c graphite [47]

Besides the concentration of nanoadditives, the size also has a role in altering latent heat. Shaikh and Lafdi [91] have used SWCNT, MWCNT and CNF in paraffin wax and have shown that SWCNT which was the smallest among the three caused higher enhancement in latent heat. Accordingly, the authors have stated that the smaller size particles having higher molecular density would increase the latent heat of PCM as smaller particles exhibit strong intramolecular attraction with base fluid molecules. Since none of the nanomaterials employed in the work was tested with a range of diameters, it cannot be concluded that the latent heat gets increased as the size of the particles decreases. On the other hand, Zabalegui et al. [68] have proved that smaller particles affect significant reduction in latent heat.

The reduction in the latent heat is generally due the volume occupied by the nanoparticles which does not undergo phase change. However, Zabalegui et al. [84] have pointed that some more volume also does not contribute to the phase change as this additional volume possesses weak molecular bond structure. It is obvious that the energy required to break down the weak molecular bond would be less during melting. According to the authors, the strained volume could be a consequence of three interfacial mechanisms including Brownian motion and it is further stated that all the three mechanisms are highly pronounced when the particle size is small. Since the details of the three mechanisms are beyond the scope of this review, the essence of the same is presented in Table 4.

Table 4 Salient features of nanofluids interfacial mechanisms

Viscosity of composites

As stated already, heat transfer during melting of PCM is primarily governed by natural convection in the liquid PCM. This is because of the stronger buoyancy-driven flows existing in the molten PCM. The existence of buoyancy-driven flows is found in various types of configurations of LHTESS modules by earlier works [92,93,94]. If the presence of nanoadditives in the PCM by any means hampers the buoyancy-driven flows, then it would result in lesser convective coefficient. Hence, the net heat transfer rate may not be on higher side despite considerable enhancement in thermal conductivity. Since buoyancy-driven natural convection is determined by viscous force, viscosity of the PCM is critical parameter to be analysed. Moreover, it is important to understand how significantly nanoadditives affect the viscosity of PCM.

Kole and Dey [95] have emphasized that viscosity of nanofluids employed for heat transfer applications should be treated as significant as thermal conductivity. However, Mahbubul et al. [96] have expressed in their recent review that research on nanofluids primarily revolves around thermal conductivity only. Although the statement is derived from works on general heat transfer, it is also applicable in case of works concerning PCMs.

He et al. [39] have demonstrated that the viscosity of water increases with increase in volume fraction of TiO₂ nanoparticles. Though the increase is nonlinear, it is quite significant. As the concentration increases, the particles come closer, and hence, the frictional force between the particles and base fluid molecules increases. The results of Jesumathy et al. [37] are in agreement with those of He et al. [39] as viscosity of paraffin wax shows increasing tendency with mass fraction of CuO. Although Arasu et al. [60] have reported similar findings for Al₂O₃ and CuO in paraffin wax, it is found that the increase in dynamic viscosity is more pronounced with CuO than with Al₂O₃. Further, the viscosity of PCM with any concentration of both the nanoadditives is inversely proportional to temperature. At the same time, Mostafavinia et al. [97] have revealed that the effect of particle fraction on viscosity is more appreciable at relatively low temperature.

According to Ho and Gao [38], nanoadditives show much stronger influence on viscosity in comparison with the influence on thermal conductivity (four to ten times more). The increase in viscosity of nanofluids as additive loading increases is consistently reported by many researchers [96]. The same can be verified from Fig. 7. Nevertheless, roles of shape/size of the additives in altering the viscosity of PCMs are yet to be explored as none of the above-mentioned works has analysed the same. Even it is proved that more amounts of additives brings in benefit in terms of higher thermal conductivity, the amount should be limited from viscosity perspective. The clear understanding of relation between size/shape and viscosity would be more helpful in choosing optimum fraction of additives. This demands a lot more work in future.

Fig. 7

Variation of viscosity with particle loading: a Al₂O₃ [38], b CuO [37], c TiO₂ [39]

In addition to viscosity evaluation, it is crucial to get the insights of rheological behaviour to understand the convective heat transfer in nanofluids [98]. Sridhara and Satapathy [99] have indicated the possibility of non-Newtonian behaviour of nanofluids in their review article. In case of nano-PCM research, Kumaresan et al. [52] have studied the Newtonian behaviour of MWCNT-dispersed paraffin wax. It is found that only in low shear stress range (0–1 Pa), the viscosity of composites decreases with increase in shear stress. Thereafter, the increase in shear stress could not make any change in viscosity. It seems that the nano-PCM composites show shear thinning behaviour at low shear stress range and behave as Newtonian fluids thereafter. The shear thinning behaviour is found more severe when the fraction of nanoadditives is higher. Nevertheless, all tested composites are observed with considerable shear thinning behaviour at low shear stress range. Similarly, Motahar et al. [100] have stated that the non-Newtonian behaviour of nonofluids does not depend only on shear rate but also on particle concentration and viscosity of base fluid. It is proved by the same authors [64] in the other work in which the non-Newtonian behaviour of n-octadecane added with mesoporous silica particles is observed when the mass fraction of particles is greater than 3%. However, Pak and Cho [101] have found that the shear thinning behaviour of nanofluids begins only after some amount of nanoadditives loading and the volume fraction at which shear thinning behaviour begins varies depending upon the type of nanoadditives. Over and above, the non-Newtonian behaviour itself is questionable as some researchers claim that the nanofluids are generally Newtonian [83, 102].

In spite of having good number of publications on rheology of single-phase nanofluids, there is no firm conclusion yet on the same. As far as PCM-based nanofluids are concerned, only few studies could be found to understand the rheology. Hence, the future studies should have more focus on addressing the rheological issues.

Heat transfer enhancement

As far as PCMs are concerned, the heat transfer enhancement is generally gauged in terms of reduction in melting or solidification time after adding nanomaterials. As the nanoadditives are used as thermal conductivity enhancers, the melting or solidification rate of PCM is expected to be higher which means that the nanoadditives would lead to significant reduction in melting/solidification time. However, nanoparticles in the PCM not only alter the thermal conductivity but also other properties like viscosity and latent heat which is already elaborated. Moreover, the distribution of particles in the PCM may not be uniform and is indifferent in solid and liquid PCMs. One can suspect that these things affect the phase change process. Hence, it is also important to look into enhancement in melting/solidification rate rather making judgment based on only thermal conductivity enhancement.

Melting rate enhancement

The influence of Cu nanoparticles on the melting process of Erythritol stored in concentric cylinder configuration is studied by Abolghasemi et al. [24]. It is shown that the time required for complete melting could be substantially reduced by nano-Cu and the decrease in melting time seems to be monotonous with particle loading. However, a firm conclusion cannot be drawn from these findings as the results are contradictory to that of Arasu and Mujumdar [103] and Arasu et al. [59].

Arasu and Mujumdar [103] have observed that the melting rate of PCM in a square enclosure increases slightly only if small amount of AlO₂ is added (2 mass%). Beyond the said quantity, addition of particles tends to decrease the melting rate. Similarly, Arasu et al. [59] have shown that the enhancement in phase change rate of PCM in a concentric double pipe module is significant only during solidification, and thus, it is recommended to use lower fraction of nanoadditives for better cyclic performance.

Although all the above three works are based on numerical modelling, the discrepancy between the results of the former and those of other two is due to the inclusion of natural convection in the modelling. The numerical modelling by Abolghasemi et al. [24] has not taken into account the natural convection which means that the role of increased viscosity is ignored. For the applications wherein single-phase fluids are involved, the thermal conductivity improvement is given greater importance as it accelerates the conduction heat transfer. It is known that increase in viscosity due to the presence of nanoadditives suppresses the convection heat transfer. Since such applications mostly encounter forced convection, the diminished rate of convection heat transfer due to higher viscosity can be balanced by increased velocity of the flowing fluid. Hence, the overall heat transfer enhancement is expected to be in line with the enhancement in thermal conductivity. On the other hand, the case of PCMs looks different as the overall heat transfer rate may not be necessarily higher in the same order of thermal conductivity increase. It is clear that nanoadditives tend to suppress the natural convection during melting which can be validated by the findings of Ho and Gao [104]. Ho and Gao [104] conducted experiments on the melting of alumina nanoparticles-dispersed n-octadecane in a vertical enclosure. The heat transfer performance is reported in terms of surface-averaged Nusselt number at the hot wall. Although the surface-averaged Nusselt number is high in the beginning, the same becomes lesser as the natural convection dominates the heat transfer. This indicates the dampened natural convection which of course is due to particle addition.

Hence, the extent to which heat transfer rate can be augmented due to increase in thermal conductivity despite the handicapped thermal convection is to be quantified. Significant number of works is reported on the relation between particle loading and melting rate. Sciacovelli et al. [105] added copper nanoparticles to enhance the melting rate of paraffin wax in the LHTESS which is a shell and tube module. The computational modelling takes into account the convection in liquid PCM using the effective viscosity evaluated in terms of particle concentration. The enhancement in heat transfer rate is evaluated in terms of reduction in melting time and rise in liquid fraction. The maximum reduction in melting time with 4% volume fraction is reported as 15%. Similar numerical model was adopted by Hosseini et al. [106] to investigate the melting of copper particles added paraffin wax (RT50). Although both Sciacovelli et al. [105] and Hosseini et al. [106] have used shell and tube module, the former considered the vertically oriented system whereas the latter used horizontal one. Hosseini et al. [106] have reported heat transfer enhancement even up to 5% volume fraction of particles (maximum of 14.6%). It is also highlighted that the reduction in melting time and increase in liquid fraction are the result of increased penetration velocity of melt front. Although no attempt is made by both the works to determine the optimum volume fraction, it can be interpreted from the results that the enhancement in heat transfer with increase in particle addition is nonlinear.

Jourabian et al. [107] have investigated the melting of Cu particles added water (PCM) in an annulus. Although the melting rate enhancement is observed at all volume fraction of particles (0.01% and 0.02%), the enhancement effect is less pronounced at 0.02% as compared to 0.01%. Through series of numerical and experimental works, Dhaidan et al. [108,109,110] have observed that the enhancement in melting rate becomes less at higher particle fractions when the CuO added n-octadecane was melted whether in square container or in annular container or in cylindrical capsule. The nonlinear increase in melting rate of paraffin wax with Cu particles loading is also experimentally proved by Wu et al. [111]. The authors have revealed that the melting time could be reduced by 27% when 0.5% mass of Cu particles were added, but the reduction was only 32% when loading was doubled. Similarly, Patil and Dey [112] could achieve reduction of 6.52%, 10.86% and 13.0%, respectively, with CuO mass% of 0.33, 0.66 and 1 for paraffin wax. In case of alumina, 1 mass% resulted in 20% reduction, whereas it was only 29% even the mass% was increased to 5 [113].

Considering the preceding discussion, it can be stated that the melting rate of PCM cannot be monotonously increased by increasing the concentration of nanoadditives. However, the findings of Harikrishnan and Kalaiselvem [63] show that the monotonous heat transfer enhancement is possible only up to a small fraction of particle loading. The authors could achieve reduction in melting time of 6.43%, 14.62% and 21.05%, respectively, with 0.1, 0.2 and 0.3 mass% of TiO₂. Similar observations are made by Hajare and Gawali [114] who used TiO₂ and Al₂O₃ with maximum fraction of 0.075%.

Whether the melting rate enhancement is linear or nonlinear with nanomaterials loading, it would be appreciable as long as significant enhancement is achieved through nanomaterials. Figure 8 exhibits the enhancement in melting rate due to addition of nanomaterials. However, the enhanced viscosity due to nanomaterials might become dominant over enhanced thermal conductivity at some point of time which would result in adverse effect on melting rate. This necessitates the determination of the optimum fraction of nanoparticles beyond which reduction in convection heat transfer overcomes the enhancement in conduction heat transfer. This would be helpful in achieving enhanced overall melting rate.

Fig. 8

Melting rate enhancement due to nanoadditives: a melting status in a spherical container after 8 min [21], b melting status in a vertical shell and tube arrangement after 4000 s [105]

While investigating the impact of locations of two pairs of source–sink (positioned on sidewalls of a square container) on the melting performance of Al₂O₃ added paraffin wax, Ebrahimi and Dadvand [115] attempted to determine the optimum volume fraction of nanoadditives. Based on the locations of source–sink pairs, four cases were considered and among the two particle fractions considered (volume fractions 2% and 5%), 2% exhibited highest melting rate in all cases. Further, PCM with 5% volume fraction is proven to be inferior to pure PCM in terms of melting rate in two of the four cases considered. Mostafavinia et al. [97] have also carried out similar work with similar module, PCM and nanoparticles except the one with respect the location of the source–sink. These results also confirm the optimum fraction as 2%. On the other hand, Auriemma and Lazzetta [116] have shown that even volume fraction of 3% results in lower melting rate as compared to pure PCM. The numerical work focuses on melting of paraffin wax added with three different nanomaterials (Al₂O₃, CuO, ZnO) in a rectangular container. Among the three composites, CuO- and ZnO-added samples are observed with lower melting rate when the volume fraction is 3% than the pure PCM. Although 3% volume fraction Al₂O₃ has not reduced the melting rate, the difference in melting rate between pure PCM and PCM with Al₂O₃ of 3% volume fraction is insignificant. By looking at the similarity between the modules considered in the above works, 2% volume fraction may be considered as optimum volume fraction for highest melting rate. Murugan et al. [117] have predicted even lower value, i.e. 0.3% mass fraction as optimum value of MWCNT for paraffin wax.

However, higher fractions are also found as optimum values by few authors. Naeem et al. [26] have investigated the melting performance of Cu nanoparticle- and TiO₂ nanoparticle-dispersed paraffin wax. The melting process of PCM in spherical capsules was analysed with mass fractions of 1%, 2%, 3% and 5%, and highest enhancement is observed with 3 mass%. Lokesh et al. [118] have also observed 3 mass% as optimum fraction since MWCNT of 3 mass% could achieve highest reduction in melting time of paraffin wax in cylindrical capsule. Bashar [119] has reported even higher value as optimum fraction of CuO, i.e. 6% mass fraction. According to the findings, paraffin wax with this concentration has established highest melting rate throughout the charging process in a rectangular container. As a novel way of employing nanoadditives, Esfe et al. [120] investigated the thermal conductivity enhancement of ethylene glycol through hybrid nanoadditives (70% alumina and 30% SWCNT). In this case, the enhancement could be achieved up to a volume fraction of 2.5% whereas with single additive (SWCNT), the optimum fraction was only 1%.

The limited works reported on the optimum fraction of nanoparticles in PCM have recommended both lower and higher quantities as optimum quantity. The discrepancy between the findings of various works is due to the assorted influencing levels of natural convection in different cases during melting. It can be understood that the buoyancy force induced natural convection is dependent on thermo-physical properties of PCM and dimensions/configuration of the storage module [121]. Moreover, the orientation of the heat source has critical effect on the strength of natural convection. In fact, Sun et al. [122, 123] have observed that the natural convection is highly influential on melting rate when the PCM is heated from the side as compared to the case where the heat source is at the bottom. It is clear from the above that the melting process of pure PCM cannot be accelerated by natural convection in similar fashion in all cases. As a matter of fact, the natural convection is less pronounced in some cases. Hence, the quantity of nanoparticles added into the PCM should be kept lower if the natural convection is highly influential and vice versa. Overall, the optimum quantity should be decided based on the combination of PCM/nanomaterial, geometry/size of the container and orientation of the heat source. It should also be considered that the higher particles concentration may affect the stability of nano-PCM due to agglomeration. The summary of the reported works focusing on melting rate enhancement due to nanoadditives is given in Table 5.

Table 5 Summary of reports on enhancement of melting rate

Solidification rate enhancement

As it is known that the solidification of PCM is governed by conduction heat transfer, the high thermal conductivity of particles is expected to play more dominating role in affecting the solidification rate. According to Chandrasekaran et al. [124], even 0.1 mass% of CuO in deionized (DI) water could reduce the solidification time by 35%. The experimental work on solidification of this nano-PCM in spherical capsule is not extended to explore effect of varying concentration of nanoparticles on solidification rate.

Sebti et al. [125] have observed reduction in solidification time of 8% and 16%, respectively, with 2.5% and 5% volume fraction of Cu particles in water when the numerical test was carried out with horizontal concentric annulus. Kashani et al. [126] have also taken same combination of PCM and nanoparticles to numerically investigate the solidification in vertical enclosure. The authors could observe 15% reduction in solidification time with 5% volume fraction of Cu particles, whereas the reduction was about 32% when the volume fraction was 10%. Using the similar module, the solidification of paraffin wax under the influence of Cu particles was studied by Mahato et al. [127]. The time for complete solidification could be reduced by 9% with 2.5% volume fraction and by 18% when volume fraction was increased to 5%. Mehdi and Nsofor [128] have also found 8% reduction in solidification time by adding 3% of alumina particles in paraffin wax.

Hosseini et al. [129] too have used vertical enclosure for the numerical investigation of solidification of water dispersed with three different nanoparticles, i.e. Cu, TiO₂ and Al₂O₃. Although the results are not presented in the form of solidification rate/solidification time, the results of Nusselt number would prompt one to observe similar trends as that of Kashani et al. [126] even up to 20% volume fraction.

It is evident from the above findings that the solidification rate of PCM can be enhanced more or less linearly with increase in quantity of nanoparticles. This can be verified from the results of the earlier works which are compiled in Fig. 9. Besides these numerical studies, results of few experimental investigations which highlight the linear dependency of solidification rate on particle fraction seem to be noteworthy. Sathishkumar et al. [130] analysed the solidification rate of grapheme nanoplatelets-added DI water in spherical capsule. The authors have used nanoplatelets of 0.2%, 0.4% and 0.6% mass fraction and have confirmed the linear enhancement in solidification rate. The experimental work of Suresh Kumar and Kalaiselvam [131] has also shown linear enhancement in the solidification rate of palmitic acid which was added with CuO of 0.3 to 0.8% mass fraction in spherical/cylindrical capsules. Temirel [132] conducted experiments on the solidification of eicosane in spherical shell. Although the author has considered relatively higher concentrations of xGnP (1.5, 3 and 4.5%) as compared to Sathishkumar et al. [130], the enhancement rate exhibits similar trend.

Fig. 9

Solidification rate enhancement due to nanoadditives: a solidification status in a square cavity after 1500 S [30], b solidification status in annular space after 5 h [128]

On the other hand, the nonlinear enhancement in solidification rate can also be found in the results of few investigations. One such work is by Sharma and Ganesan [133] which is a part of other work by Sharma [134] in which water is the PCM and Cu nanoparticles are used for reducing the solidification time in trapezoidal cavity. As reported, the solidification time could be saved by 40% and 53%, respectively, with particle volume fraction of 10% and 20%. The nonlinear relation between solidification rate and particle fraction is also exposed experimentally by Fan and Khodadadi [61]. The authors prepared Cu particles added cyclohexane and have found non-monotonic enhancement of solidification rate once the mass fraction is more than 2%. This is because of nonlinear enhancement of solid phase thermal conductivity although the thermal conductivity of liquid phase exhibits monotonic enhancement with particle fraction.

Unlike melting, the dampening effect on natural convection due to nanoadditives does not pose any serious problem when it comes to solidification rate. This may prompt to use higher fraction of particles if solidification rate is prioritized. For example, in solar thermal applications, solar energy is stored during long day time whereas the retrieval of energy is to be done in the short late evening hours/night hours. This requires higher solidification rate of the PCM even if the melting rate is slower.

The linear enhancement of solidification rate is not always possible especially with higher fractions; for example, above mass fraction of 2% or volume fraction of 10%, the solidification rate may not be significant even though the same may keep increasing. Hence, it is better to use smaller quantity of nanoparticles from the perspective of achieving cost saving, higher energy storage density, stable nano-PCM and higher number of thermal cycles.

Apart from particle fraction, particle size has significant effect on solidification rate according to El Hasadi and Khodadadi [135]. The solidification of water dispersed with Cu particles of diameter 5 nm and 2 nm in a square cavity is analysed. The numerical results show that decreasing particle size leads to slower solidification as smaller particles tend to lower the solidus temperature as a result of constitutional supercooling. The earlier works concerning enhancement of solidification rate due to nanoadditives are listed in Table 6.

Table 6 Summary of reports on enhancement of solidification rate

Prediction of thermo-physical properties

As discussed in “Nanomaterials and thermo-physical properties of composites” section, the change in thermo-physical properties of PCMs due to nanoadditives determines the thermal performance improvement. Hence, characterization of nanomaterials added PCM remains as one of the critical activities of the concerned research. Although highly advanced instrumentation and well-established measurement techniques are available, testing nano-PCM for various quantities of nanoadditives is time-consuming and is highly expensive. On the other hand, employing theoretical/empirical relations results in reasonable prediction quickly but with zero cost. Moreover, the relations are more useful when it comes to numerical modelling-based research.

The thermo-physical properties of the nanofluid are a combination of the properties of the nanoadditive and the base fluid, and hence, the additive causes significant change in the properties of resultant fluid depending on the concentration. Hence, the relations for the properties of mixture should be expressed as a function of mass/volume fraction of additives in the mixture. However, the nanomaterial size, shape and more importantly the particle dynamics are also to be taken into account wherever applicable. In this perspective, researchers have developed various models for predicting the properties of nanofluids. This section reviews the models developed so far and their applicability for nano-PCMs.

Density and specific heat models

To be specific, density of the fluid plays a very important role wherever natural convection is involved like in the case of PCMs. Since buoyancy force is directly related to natural convection and is determined by density gradient, the magnitude of natural convection is influenced by density.

To determine density, a simple and straightforward model known as “Classical Mixture Model” is employed in most of the works concerning PCMs [20, 25, 136,137,138]. Here, the density of nanofluid (ρ_nf) is evaluated as follows.

$$\rho_{\text{nf }} = \varphi \rho_{{{\text{p}} }} + \left( {1 - \varphi } \right)\rho_{\text{f }}$$

(1)

where φ is the volume fraction of the nanomaterial, ρ_p and ρ_f are the density of the nanomaterial and the base fluid (PCM), respectively.

Equation (1) considers a linear relation between density of the mixture and the concentration of the nanoadditive. Further, it is expressed as a function of only the additive’s concentration.

However, Sharifpur et al. [139] claims that the mixture model over predicts the density and the deviation widens up as the volume fraction increases. The authors have further suggested taking into account the nanolayer in order to overcome the deviations. The nanolayer is nothing but the interfacial layer exists in the solid–liquid interface. The nanolayer is thus assumed as a void, and the void thickness is expressed in terms of particle size. The resulting equation would be an empirical relation. The inclusion of void thickness results in lesser volume fraction of particles than the usually calculated one. The effective volume fraction is given by

$$\varphi_{\text{eff }} = \frac{1}{{\frac{1}{\varphi } - 1 + \left( {r_{\text{p}} + t_{\text{v}} } \right)^{3} /r_{\text{p}}^{3} }}$$

(2)

Following Eq. (2), the modified equation for the density of nanofluid is written as

$$\rho_{{{\text{nf}}\left( {\text{modified}} \right) }} = \frac{{\rho_{\text{nf }} }}{{\left( {1 - \varphi } \right) + \varphi \left( {r_{\text{p}} + t_{\text{v}} } \right)^{3} /r_{\text{p}}^{3} }}$$

(3)

where r_p is the radius of the particles and t_v is the thickness of the nanolayer.

Equation (3) includes the effects of size and thickness of the particles besides the volume fraction. Although the model is proved to be more accurate and simple, the calculation is not as straightforward as with the case of mixture model. This is because the calculation of thickness needs experimental data. The model makes good prediction even up to a volume fraction of 1%.

The specific heat also plays a key influencing role when it comes to heat transfer performance. By adding nanoparticles in the heat transfer fluid, the ability to conduct the heat is enhanced. This would result in lower specific heat in comparison with the base fluid. Similar to density, specific heat of nanofluids (c_nf) is calculated using the law of mixture [140,141,142].

$$c_{\text{nf}} = \frac{{\left( {1 - \varphi } \right)\rho_{\text{f }} c_{\text{f }} + \varphi \rho_{\text{p }} c_{\text{p }} }}{{\left( {1 - \varphi } \right)\rho_{\text{f }} + \varphi \rho_{\text{p}} }}$$

(4)

where c_p and c_f are the specific heat of the nanoadditives and the base fluid, respectively.

The density and specific heat can simultaneously be determined by combining Eqs. (1) and (4) [143].

$$\rho_{\text{nf}} c_{\text{nf }} = \varphi \rho_{\text{p }} c_{\text{p }} + \left( {1 - \varphi } \right)\rho_{\text{f}} c_{\text{f}}$$

(5)

According to Wang et al. [144], the specific heat of nanofluid increases as the particle size decreases. The increased surface energy due to decreased size results in higher specific heat. Hence, one may demand expressing specific heat in terms of particle size along with volume fraction. However, further works [145, 146] have revealed that effect of particle size on specific heat of nanofluids is negligible.

Thermal conductivity models

Unlike density and specific heat, evaluation of effective thermal conductivity of nanofluids is bit challenging. Although plenty of models are developed for effective thermal conductivity, until now, reliable and complete solution is yet to be achieved.

In the simple approach, the effective thermal conductivity of nanofluids (k_nf) is predicted using the well-known Maxwell equation [147],

$$k_{\text{nf}} = k_{{{\text{f}} }} \left[ {\frac{{k_{\text{p }} + 2k_{\text{f }} + 2\varphi \left( {k_{\text{p }} - k_{\text{f }} } \right)}}{{k_{\text{p }} + 2k_{\text{f }} - \varphi \left( {k_{\text{p }} - k_{\text{f }} } \right)}}} \right]$$

(6)

where k_p and k_f are the thermal conductivity of the nanoadditives and the base fluid, respectively.

However, Eq. (6) holds good for the cases when larger (micro), spherical particles are added in small concentrations. Following Maxwell, Bruggeman [148] came out with a model which is again for the micron sized particles.

$$\varphi \left( {\frac{{k_{\text{p }} - k_{\text{nf }} }}{{k_{\text{p }} + 2k_{\text{nf }} }}} \right) + \left( {1 - \varphi } \right)\left( {\frac{{k_{\text{f }} - k_{\text{nf }} }}{{k_{\text{f }} + 2k_{\text{nf }} }}} \right) = 0$$

(7)

Unlike Maxwell’s one, Bruggeman’s model is found to be satisfactory even for higher concentrations [149]. In the later year, another model applicable to even non-spherical particles was proposed by Hamilton and Crosser [150]. The shape of the particles is accounted for by including a shape function (n).

$$k_{\text{nf}} = k_{\text{f }} \left[ {\frac{{k_{\text{p }} + \left( {n - 1} \right)k_{\text{f }} - \left( {n - 1} \right)\varphi \left( {k_{\text{f }} - k_{\text{p }} } \right)}}{{k_{\text{p }} + \left( {n - 1} \right)k_{\text{f }} + \varphi \left( {k_{\text{f }} - k_{\text{p }} } \right)}}} \right]$$

(8)

Here, the shape function is given as n = 3/ψ, where ψ is the sphericity.

$${\text{Sphericity = }}\frac{\text{Surface area of the sphere whose volume is equal that of the particles of actual size}}{\text{Surface area of the particles of actual size}}$$

Accordingly, researchers considered value for sphericity ranging from 0.3 to 1 [143].

Looking at the deliberated three models, it is understood that only the concentration and shape of the nanoadditives are taken into account. Hence, these models have measured the heat transfer mechanism at macroscopic level only (i.e. diffusive heat transfer in base fluid and particles).

As stated in “Latent heat of composites” section, the enhancement in heat transfer of nanofluids is governed by interfacial mechanisms and the models discussed above have not considered the effects of these mechanisms. In view of this, researchers have proposed quiet a number of models with inclusion of effects of heat transfer enhancement mechanisms.

Models with effects of interfacial layers

According to Yu and Choi [151], the thermal conductivity of nanoadditives is to be altered in order to take into account the effects of nanolayer. Accordingly, the thermal conductivity (k_p,m) is expressed as

$$k_{\text{p,m}} = k_{\text{p }} \left[ {\frac{{2\left( {1 - \gamma } \right) + \left( {1 + \beta } \right)^{3} \left( {1 + 2\gamma } \right)\gamma }}{{ - \left( {1 - \gamma } \right) + \left( {1 + \beta } \right)^{3} \left( {1 + 2\gamma } \right)}}} \right]$$

(9)

where γ is the ratio of the thermal conductivity of nanolayer to that of nanoadditives and β is the ratio of the thickness of the nanolayer to the radius of the particle.

Now, the expression for effective thermal conductivity is rewritten using Maxwell’s model

$$k_{\text{nf}} = k_{\text{f }} \left[ {\frac{{k_{\text{p,m }} + 2k_{\text{f }} + 2\varphi \left( {k_{\text{p,m }} - k_{\text{f }} } \right)\left( {1 + \beta } \right)^{3} }}{{k_{\text{p,m }} + 2k_{\text{f }} - \varphi \left( {k_{\text{p,m }} - k_{\text{f }} } \right)\left( {1 + \beta } \right)^{3} }}} \right]$$

(10)

As the above model is applicable to only spherical particles, the same authors [152] modified the Hamilton–Crosser model by including the effect of nanolayer in order to make it applicable to non-spherical particles.

$$k_{\text{nf}} = \left\{ {1 + \frac{{\left[ {\frac{1}{3}\mathop \sum \nolimits_{\text{j = a,b,c}} {\raise0.7ex\hbox{${\left( {k_{\text{pj}} - k_{\text{f}} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {k_{\text{pj}} - k_{\text{f}} } \right)} {k_{\text{pj}} + \left( {n - 1} \right)k_{\text{f}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${k_{\text{pj}} + \left( {n - 1} \right)k_{\text{f}} }$}}} \right]n\varphi_{\text{eff }} }}{{1 - \left[ {\frac{1}{3}\mathop \sum \nolimits_{\text{j = a,b,c}} {\raise0.7ex\hbox{${\left( {k_{\text{pj}} - k_{\text{f}} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {k_{\text{pj}} - k_{\text{f}} } \right)} {k_{\text{pj}} + \left( {n - 1} \right)k_{\text{f}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${k_{\text{pj}} + \left( {n - 1} \right)k_{\text{f}} }$}}} \right]\varphi_{\text{eff}} }}} \right\}k_{\text{f}}$$

(11)

Besides the above models, the literature presents few more thermal conductivity models which consider the effects of nanolayer [153,154,155]. The summary of these models along with their features can be found in Table 7.

Table 7 Thermal conductivity models with effects of interfacial layers

Models with effects of Brownian motion

As the heat transfer enhancement due to Brownian motion involves random motions of particles, the effective thermal conductivity should take into account the diameter of the particles/surface geometry (nanostructure) and the particles’ dynamics [156]. The representative models which include the Brownian motion effects [157,158,159,160] are presented in Table 8.

Table 8 Thermal conductivity models with effects of Brownian motion

In nano-PCM works, almost all have adopted either the Koo and Kleinstreuer [161] model or the one proposed by Amiri and Vafai [162]. In Koo and Kleinstreuer [161] model, the effective thermal conductivity is evaluated by adding the increase in thermal conductivity as a result of Brownian motion to the thermal conductivity estimated using Maxwell’s model. The first part is found to be a function of particles’ size/concentration, temperature and other properties of both particles and base fluid. Accordingly, the resultant equation is given as,

$$\begin{aligned} k_{\text{nf}} & = k_{\text{f }} \left[ {\frac{{k_{\text{p }} + 2k_{\text{f }} + 2\varphi \left( {k_{\text{p }} - k_{\text{f }} } \right)}}{{k_{\text{p }} + 2k_{\text{f }} - \varphi \left( {k_{\text{p }} - k_{\text{f }} } \right)}}} \right] \\ & \quad + 5 \times 10^{4 } \alpha \varphi \rho_{\text{f}} c_{\text{f}} f\left( {T,\varphi } \right)\sqrt {\frac{{K_{\text{B}} T}}{{\rho_{\text{f}} 2r_{\text{p}} }}} \\ \end{aligned}$$

(12)

where K_B = 1.381 × 10⁻²³ J K⁻¹ is the Boltzmann constant and α is the volume fraction of base fluid moving with a particle caused by Brownian motion. The function f(T, φ) in Eq. (12) includes the enhanced temperature (T) dependence. Evaluation of α and f needs extensive experimental data. For nano-PCM composites, the empirical relations given in Eqs. (13) and (14) are generally employed by researchers [25, 60, 163].

$$\alpha = X\left( {100\varphi } \right)^{Y}$$

(13)

$$f\left( {T,\varphi } \right) = \left( {A\varphi + B} \right)\frac{T}{{T_{\text{ref}} }} + \left( {C\varphi + D} \right)$$

(14)

where X, Y, A, B, C and D are the experimentally obtained constants and T_ref is the reference temperature = 273 K.

Besides the above model, the model proposed by Amiri and Vafai [162] also seems to be very popular among the nano-PCM works [21, 133, 140, 141]. The model measures the Brownian motion in terms of velocity and size of the particles. The Brownian motion enhanced thermal conductivity is then added into Maxwell’s equation. The final equation is

$$k_{\text{nf}} = k_{\text{f }} \left[ {\frac{{k_{\text{p }} + 2k_{\text{f }} + 2\varphi \left( {k_{\text{p }} {-}k_{\text{f }} } \right)}}{{k_{\text{p }} + 2k_{\text{f }} - \varphi \left( {k_{\text{p }} {-}k_{\text{f }} } \right)}}} \right] + C^{{\prime }} \rho_{\text{nf}} c_{\text{nf }} \left| V \right|\varphi 2r_{\text{p}}$$

(15)

where V is the velocity and the constant C′ can be empirically determined.

Particles clustering on effective thermal conductivity

Although various techniques are employed to produce stable nanofluids, problem of agglomeration cannot be completely prevented. Agglomeration is nothing but getting together by the particles which is a result of van der Waals forces [164]. The clusters are generally porous in nature and occupy larger space than the individual particles [165]. Hence, it is expected to increase the effective thermal conductivity. Wang et al. [166] predicted the effective thermal conductivity using a model based on fractal theory (improved effective medium theory) in which the effect of nanoparticles clustering is considered. Although the model works good up to a volume fraction of less than 0.5%, the calculation is complicated. However, Wu et al. [167] who have also used a fractal model could not find any improvement in thermal conductivity due to particles clustering as the enhancement is dampened by reduced convection. Moreover, the effect of clustering on effective thermal conductivity is more pronounced only for higher concentrations [168, 169]. In case of nano-PCM composites, the higher concentrations are generally not preferred in order to keep natural convection in the liquid PCM intact. Even with lower concentrations, particles clustering can be alleviated using techniques like surface treatment by surfactants [81]. It is also observed from the literature on nano-PCM research that none of the works has made an attempt to include the particles clustering in evaluating effective thermal conductivity. From this perspective, the models with particles clustering effects are not deliberated in this review.

Viscosity models

As far as viscosity is concerned, the development of models for particles suspended liquids started way back in 1906 by Einstein [170]. This first wok came out with a linear equation which is

$$\mu_{\text{nf}} = \mu_{\text{f }} \left( {1 + 2.5\varphi } \right)$$

(16)

where µ_nf and µ_f are the dynamic viscosity of the nanofluid and the base fluid, respectively.

This model was developed for rigid spherical particles and is applicable to non-interacting particles with a volume concentration of ≤ 2%.

Following Einstein [170], Brinkman [171] proposed a similar equation which is valid for volume fraction of up to 4%. Again this model is for only spherical, non-interacting particles. The model proposes the equation as

$$\mu_{\text{nf}} = \mu_{\text{f }} \left( {1 - \varphi } \right)^{ - 2.5}$$

(17)

For higher concentrations, a model which is very similar to Eq. (17) is given by Roscoe [172]. But this model is limited to the cases wherein particles of equal size are used.

In the later years, Krieger and Dougherty [173], Frankel and Acrivos [174] and Neilsen [175] developed models in terms of maximum particle volume fraction (φ_max) at which the flow is still possible. The Krieger and Dougherty [173] model is expressed as

$$\mu_{\text{nf}} = \mu_{\text{f }} \left( {1 - {\raise0.7ex\hbox{$\varphi $} \!\mathord{\left/ {\vphantom {\varphi {\varphi_{ \hbox{max} } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\varphi_{ \hbox{max} } }$}}} \right)^{{ - \eta \varphi_{ \hbox{max} } }}$$

(18)

where η is the intrinsic viscosity which is taken as 2.5 for mono dispersed rigid, spherical particles. This means that this model may not be applicable to practical cases as mono dispersion of particles is difficult to come across. The model given by Frankel and Acrivos [174] is expressed as

$$\mu_{\text{nf}} = \mu_{\text{f }} \frac{9}{8}\left( {\frac{{\left( {\varphi /\varphi_{ \hbox{max} } } \right)^{1/3} }}{{1 - \left( {\varphi /\varphi_{ \hbox{max} } } \right)^{1/3} }}} \right)^{ }$$

(19)

Similarly, Neilsen [175] recommended power law-based equation which is expressed as

$$\mu_{\text{nf}} = \mu_{\text{f }} \left( {1 + 1.5\varphi } \right){\text{e}}^{{{\raise0.7ex\hbox{$\varphi $} \!\mathord{\left/ {\vphantom {\varphi {\left( {1 - \varphi_{ \hbox{max} } } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {1 - \varphi_{ \hbox{max} } } \right)}$}}}}$$

(20)

All the models discussed above have not considered the Brownian motion. However, Batchelor [176] has suggested a Brownian motion-based model as

$$\mu_{\text{nf}} = \mu_{\text{f }} \left( {1 + 2.5\varphi + 6.2\varphi^{2} } \right)$$

(21)

It is reported that the above model is valid for a volume fraction up to 10%. Another model which includes the Brownian motion is the one given by Lundgren [177]. The equation of the model is expressed in the form of Taylor series.

$$\mu_{\text{nf}} = \mu_{\text{f }} \left( {1 + 2.5\varphi + \frac{25}{4}\varphi^{2} + f\left( {\varphi^{3} } \right)} \right)$$

(22)

Besides all these models, literature reveals some models expressed as empirical relations [178, 179]. The notable models are those proposed by Chen et al. [180], Hosseini et al. [1] and Corcione [181]. The viscosity models with the features are given in Table 9. For nano-PCM composites, the earlier works have majorly employed Brinkman model [133, 136, 137, 142, 182]. At the same time, Einstein model is also found a place in Faraji [182] and the one by Corcione [181] is used by Kashani et al. [140].

Table 9 Viscosity models

Latent heat models

It is generally believed that the there is a decrease in latent heat due to the volume occupied by the particles. Accordingly, all the numerical-based investigations have employed a straightforward expression to predict the latent heat of nano-PCM composites [20, 21, 30, 59, 61, 103, 136, 137, 140, 141]. The expression is written as

$$L_{\text{nf}} = \frac{{\left( {1 - \varphi } \right)\rho_{\text{f }} L_{\text{f }} }}{{\rho_{\text{nf }} }}$$

(23)

It is clear that Eq. (23) is written only in terms of volume fraction of particles. As discussed in “Latent heat of composites” section, the size of the particles may have a role in affecting the latent heat of PCM. Further, there may be additional volume of PCM which cannot undergo phase change due to weak molecular bond structure. To the best of Authors’ knowledge, none of the earlier works has considered the above while evaluating the latent heat of nano-PCM composites.

Conclusions

A review of literature concerning high-conductivity nanomaterials-added PCMs is presented. The effects of nanomaterials on thermo-physical properties of PCMs and subsequent effects on the thermal response are assessed. The models for evaluating the thermo-physical properties of nano-PCM composites are also discussed. The conclusions drawn out of this review are summarized below.

• Although the size of the nanoadditives should be as small as possible in order to avoid agglomeration, too smaller size may lead to reduction in thermal conductivity. This demands optimum size for thermal conductivity enhancement

• The size of the nanomaterial also has a role in latent heat reduction along with volume fraction

• The melting and solidification rates of PCMs cannot be increased linearly with increase in concentration of nanomaterials

• Rheology of nano-PCM composites still remains a least explored area and research on the same needs to be strengthened

• In PCM applications, the volume fraction of nanoadditives should be a trade-off between some important factors like, heat transfer enhancement, higher energy storage density, stability of nano-PCM, higher number of thermal cycles and cost saving.