1 Introduction

With progression in technology, electronic components are shrinking in size while their heat dissipation quest is increasing day by day. Additionally, such devices need to be operated at safe temperature levels (≤80 °C) to ensure their proper functionality. Conventional cooling techniques like air and water based cooling, thermoelectric and heat pipes have almost been pushed to their applicable limits. They are now becoming outpaced to cater the ever increasing demands of cooling [1]. These developments led the interest of researchers to explore more efficient innovative cooling technologies.

Tuckerman and Pease [2] proposed microchannel heat sink for the first time and predicted their cooling capacity to be 1000 W/m2. Later on, microchannel heat exchangers have gained much attention due to their compact size and more substantial heat transfer to volume ratio. To further enhance the efficiency of microchannels, mPCM slurries as heat transfer fluid and thermal storage devices are potential candidates due to their high heat absorption and almost constant temperature during phase change process. In recent years, due to the advancement in manufacturing technologies, new frontiers have been opened for mPCM slurries. They have proved to be promising candidates for improving the thermal performance of air conditioning and refrigeration systems, building applications, textile industry, photovoltaic cells, mine cooling, and microelectronics. Despite its higher thermal performance, mPCM slurry suffers from various drawbacks which include phase segregation, agglomeration, degree of subcooling, low thermal conductivity, rupturing of mPCM particles, and poor stability. Several ways of improving the thermal performance and combating the demerits of mPCM particles involve additions of nucleating agents, nanomaterials, carbon nanotubes, metal coatings, surfactants, and thickeners into the mPCM slurry [3,4,5,6,7,8,9,10,11,12].

Numerous studies have been reported in this area to review the manufacturing technologies, thermophysical properties, heat transfer characteristics, and applications of mPCM slurry in various fields [6, 13,14,15,16,17,18,19,20,21]. A large number of investigations regarding heat transfer and pressure drop characteristics, for mPCM slurry metlting, in micro/mini channels have been reported. Overall, these studies indicated that mPCM slurry as heat transfer fluid has a positive impact on thermal performance of heat sinks.

In 1994, Choi et al. [22] developed a three region model to determine the local bulk temperature for mPCM slurry in the direction of fluid flow. The model has been widely employed to estimate the local bulk fluid temperature of mPCM slurry. The effects of mass concentration and the diameter of mPCM particles, their melting range and homogeneity on heat transfer characteristics of mPCM slurry was experimentally determined by Goel et al. [23]. A decrease in Stefan number (Ste) and melting range resulted higher Nusselt number (Nu) and lowered dimensionless wall temperature. Similar results were also reported in other studies as well [24, 25]. Lateron, based on the experimental setup of Goel et al. [26], Roy and Avanic [27] analyzed the effect of mPCM shell wall on the heat transfer performance. They reported that the heat transfer characterisics of mPCM slurry were not affected significantly by the presence of shell wall. In an experimental study, Rao et al. [28] investigated heat transfer characteristics of mPCM slurry in minichannels. Their findings revealed that a 5% mass concentration of mPCM slurry showed a higher Nusselt number than pure water.

In another experimental study, Inaba et al. [29] reported heat transfer enhancement by utilizing slurry consisting of mPCM particles of two different sizes for laminar and turbulent regimes. Nusselt number increased by 2.8% when compared with pure water. This was attributed to enhanced particle–particle interaction as well as the highler latent heat. Huang et al. [30] experimentally further analyzed thermal performance of mPCM slurry by the addition of a magnetic field in the test section. They reported that heat transfer coefficient increased whereas the surface temperature decreased in the region of external magnetic field. This was attributed to the delay of the development of the boundary layer. Song et al. [1] proposed a new method for heat transfer enhancement using mpCM slurry. A mixture of liquid metal and mPCM particles were introduced to improve thermal conductivity and effective specific heat capacity. This mixture resulted in 1.58 to 2.58 times higher heat transfer coefficient than that of single phase fluid. In an experimental study Babapor [31] investigated different combinations of paraffin and nanocomposites for the enhancement of thermal conductivity during the solidification process.They observed that thermal conductivity enhanced upto 150% but on the other hand specific heat capacity was degraded to 39% maximum. Howard et al. [32] observed high resolution local measurements during the melting of mPCM slurry flow in standard tubes by infrared thermography. Their findings were enhanced heat transfer with mPCM slurry as compared to the pure water.

Compared to experimentation numerical simulation is less expensive, though computationally intensive method to investigate a range of operating conditions. The flow of mPCM slurry in microchannels of diferent aspect ratios was studied by Kondle et al. [33] under constant surface temperature and heat flux boundary conditions. They found that the Nusselt number initially increased and then decreased. The increase in Nusselt number was attributed to the melting of mPCM slurry whereas the drop in Nusselt number was due to the lower Cp of the complelety melted mPCM slurry. Similar findings were reported by Zeng et al. [25] and Ravi [24]. In a numerical study, Sabbah et al. [34] employed a microchannel heat sink to investigate the flow of mPCM slurry using a 3D conjugated model. They reported that the heat transfer performance enhanced by 30–50% when compared to pure water. Sabbah et al. [35] reported that the develepoment of the boundary layer was delayed when mPCM particles were added into the base fluid. Similar findings were reported by Seyf et al. [36]. Far et al. [37] investigated the thermal performance of microchannel heat sink with oblique fins and operating on mPCM slurry. They reported that Nusselt number was improved when the ratio of tip clearance (Tp) to width of channel (W) was increased from 0.20 to 0.37. In another study, Far et al. [38] investigated the thermal performance of different working fluids (water, mPCM slurry and nanofluids) in a double layer microchannel coupled with heat sink. Higher cooling performance of the heat sink was reported. Song et al. [39] numerically investigated the effect of twisted tape inserts in the microchannel. The heat transfer coefficient was found to be higher for twisted tape inserts because of delay in thermal boundary layer development caused by the swirl flow. Different shapes of microchannels (Planar spiral coil, coiled square tubes, helical coil heatexchanger and wavy channels) were investigated [40,41,42]. They had the similar findings of higher heat tranfer as compared to base fluid but on the other hand the pressure drop worsened.

The potential application of this freezing study is in the areas of energy storage, minimizing cooling/heating demands of built environment, heating ventilation and air conditioning system and other such applications where there is demand of thermal energy management in the system under consideration. Majority of experimental as well as numerical studies reported in the literature with focus on the heat transfer and pressure drop performance during the melting of mPCM slurry in mini/micro channels has been reported for the constant heat flux boundary condition. Cyclic operation of mPCM slurry does require melting as well as solidification process. Absorbed heat is rejected and mPCM slurry is made ready for the next operating cycle in this freezing process. There hardly exists any study with a focus on freezing heat transfer characteristics of mPCM slurry employed in microchannels. Even the previously reported literature on the melting heat transfer characteristics of the mPCM slurry is restricted to constant heat flux boundary conditions. In practical heat transfer applications, the heat flux does not remain constant in the direction of fluid flow. In this work, the heat transfer and pressure drop characteristics of mPCM slurry in microchannel heat sink has been investigated during the freezing cycle under the convective boundary condition. The convective boundary condition has been incorporated by employing pure water at the top and bottom surface of the test section as a secondary heat transfer medium. The effect of mPCM slurry mass concentration on local Nusselt number, pressure drop, bulk and surface temperature under the laminar regime has been investigated.

2 Physical Model and Boundary Conditions

Figure 1 shows the schematic of the test section employed in this study (details on materials and operating conditions can be traced from [43]). Working fluid (water/mPCM slurry) flows inside six microchannels of length 450 mm. For solidification, mPCM slurry rejects heat to the cold water flowing in counter current direction over the top and bottom surface of the microchannels. The secondary heat transfer medium (cold water) flows in two channels of dimensions (430 mm × 15 mm × 3 mm) at the upper and lower surfaces of an aluminum block.

Fig. 1
figure 1

Physical model and coordinates

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Some of the assumptions have been taken to simplify the numerical model. The flow inside the channel is taken as laminar and incompressible. The boundary conditions applied to the test section are summarized as follows (Table 1).

Table 1 Boundary conditions and their locations
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The phase change process was implemented by applying the governing equations for mass (Eq. 1), momentum (Eqs. 2–4) and energy equation for fluid and channel wall (Eqs. 5–6). The subscript f stands for both the working fluids (water and mPCM). A detailed comparison of pure water and mPCM slurry properties are shown in Table 2.

$$\frac{\partial u}{{\partial x}} + \frac{\partial v}{{\partial y}} + \frac{\partial w}{{\partial z}} = 0$$
(1)
$$\rho_{f} \left( {u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}} + w\frac{\partial u}{{\partial z}}} \right) = - \frac{\partial P}{{\partial x}} + \mu_{f} \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }} + \frac{{\partial^{2} u}}{{\partial z^{2} }}} \right)$$
(2)
$$\rho_{f} \left( {u\frac{\partial v}{{\partial x}} + v\frac{\partial v}{{\partial y}} + w\frac{\partial v}{{\partial z}}} \right) = - \frac{\partial P}{{\partial y}} + \mu_{f} \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }} + \frac{{\partial^{2} v}}{{\partial z^{2} }}} \right)$$
(3)
$$\rho_{f} \left( {u\frac{\partial w}{{\partial x}} + v\frac{\partial w}{{\partial y}} + w\frac{\partial w}{{\partial z}}} \right) = - \frac{\partial P}{{\partial z}} + \mu_{f} \left( {\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} w}}{{\partial y^{2} }} + \frac{{\partial^{2} w}}{{\partial z^{2} }}} \right)$$
(4)
$$\rho_{f} C_{f} \left( {u\frac{{\partial T_{f} }}{\partial x} + v\frac{{\partial T_{f} }}{\partial y} + w\frac{{\partial T_{f} }}{\partial z}} \right) = k_{f} \left( {\frac{{\partial^{2} T_{f} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{f} }}{{\partial y^{2} }} + \frac{{\partial^{2} T_{f} }}{{\partial z^{2} }}} \right)$$
(5)
$$k_{s} \left( {\frac{{\partial^{2} T_{s} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{s} }}{{\partial y^{2} }} + \frac{{\partial^{2} T_{s} }}{{\partial z^{2} }}} \right) = 0$$
(6)
Table 2 Correlations for thermophysical properties of mPCM slurry
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3 Numerical Method

The numerical model was developed in ANSYS 15. The partial differential equations were converted into algebraic equations using Finite Volume Method. Pressure velocity coupling was solved by SIMPLE algorithm. The turbulent flow of cold water inside the upper as well as the lower channels was incorporated using k − ε turbulent model with enhanced wall treatment funtions. The Laminar flow model had been incorporated for mPCM slurry flow inside the microchannels. The convergence criteria for governing equations of mass, momentum and energy were set at 10–6. The fluid properties of mpCM slurry were implemented by user defined funtions. The specific heat capacity of the mPCM slurry for solid and liquid phase was modelled with piecewise polynomial profile.The computational domain consists of half of the test section because of the geometrical symmetry. A grid independence test was performed to ensure the grid independence of numerical results for three different mesh sizes (Mesh 1 (77 × 6), Mesh 2 (88 × 8) and Mesh 3 (109 × 12) divisions in y and x directions) as shown in Fig. 2.

Fig. 2
figure 2

Mesh independence for three different mesh sizes

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4 Fluid Properties and Flow Parameters

For this study, the MPCM-37 with mass concentration in 5–15% range was used as mPCM wehereas water was used as a base fluid. The differential scanning calorimeter [43] curve for MPCM-37 was obtained from Microtek Lab [44] and is shown in Fig. 3. The DSC curve shows that during the melting cycle, the melting phase change transition range was from 32.87 to 39.9 °C. The Fig. 2 shows that mPCM particles completely freezed in the temperature range of 25 °C–33.14 °C with a subcooling of around 7.8 °C. While considering the DSC curve the freezing phase change range of 31 °C to 25 °C was selected for numerical simulation. To ensure the complete melting of mPCM slurry before entering the microchannel the inlet temperature of slurry was fixed at 33 °C. The inlet temperature of 15 °C and velocity of 0.37 m/s were set for cold water in the upper and lower channels.

Fig. 3
figure 3

DSC curve of mPCM-37 Slurry [44]

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Figure 3 shows the thermophysical properties of mPCM-37 slurry; properties were estimated by using standard methodology, as suggested in the literature [33] is shown in Table 2.

The thermophysical properties for mPCM slurry calculated with the standard correlations given in Table 2 are illustrated in Table 3.

Table 3 Themophysical properties of mPCM-37 slurry
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5 Results and Discussion

To study freezing charactersitics of mPCM slurry, problem was numerically simulated for pure water and mPCM slurry in 5–15% mass concentrations range, both combinations were tested under similar operating conditions. For this study, slurry was tested with 5–15% concentration and an increase in thermal performance was noticed with increase in concentration. However it is clarified that higher concentration (>25%) are normally not considered due to to their non Newtonian behaviour in a thermofluid system as well as increase in pressure drop. The data reduction procedure for circumferentially averaged local temperature, Tw,z, and local heat flux, qw,z, at the inner surface along the channel, local bulk temperature, local Nusselt number, and pressure drop has been detailed in [45] and is not duplicated here.

The reliability of any numerical study highly relies on its validation with experimental evidences, data presented in this study was validated with the experimental data of Chen et al. [46]. The results for validation of local Nusselt number are shown in Fig. 4 where maximum deviation between experimental and numerical results was noticed to be 14.6%. The observed error is within experimental uncertainty and this confirms the validation of the numerical approach followed in this study.

Fig. 4
figure 4

Comparison of local Nusselt number with dimensionless axial distance for validation of numerical results

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Figure 5 shows obtained temperature contours at z = 0.5 (channel outlet) for water and mPCM slurry for tested mass concentration (5–15%). It is clear from the presented figures that in all cases particles in close contact with wall freezes quickly whereas central core remained in melted phase. A clear depiction of aforestated evidence can be observed from the results shown (Fig. 5) for 15% mass concentration. Figure 6 shows variation of bulk temperature along the channel, in comparison with pure water, the addition of mPCM particles slows down development of thermal boundary layer. This retardation further increased with increase in mass concentration of mPCM particles.

Fig. 5
figure 5

Temperature variations at the outlet of microchannel. \(w_{{f,{\text{in}}}} = 1.2\;{\text{m/s}},\;w_{{c,{\text{in}}}} = 0.37\;{\text{m/s}},\;T_{{c,{\text{in}}}} = 15\;^{ \circ } {\text{C}},\;T_{{f,{\text{in}}}} = 33\;^{ \circ } {\text{C}}\)

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Fig. 6
figure 6

Variation of bulk temperature along the channel. \(w_{{f,{\text{in}}}} = 1.2\;{\text{m/s}},\;w_{{c,{\text{in}}}} = 0.37\;{\text{m/s}},\;T_{{f,{\text{in}}}} = 15\;^{ \circ }{\text{C}},\;T_{{c,{\text{in}}}} = 33\;^{ \circ } {\text{C}}\)

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Variation of local bulk temperature for pure water and tested concentrations of mPCM slurries for different velocities (0.55–1.2 m/s) are shown in Fig. 7. For all tested cases higher bulk temperatures were noticed with mPCM as compared with pure water. For fluid velocity of 1.2 m/s, a difference of 0.7 K, 1.24 K and 1.64 K was observed with 5, 10 and 15% concentration of mPCM respectively at the outlet of the channel. Comparatively high effective specific heat capacity of mPCM slurry could be the possible reason for their higher wall temperatures, mPCM particles absorbs more heat than water during their freezing process. Additionally it was noticed that this difference in temperature increases with decrease in fluid flow velocity. Figure 7e shows 3.12 K higher temperature with 15% mPCM than water at the outlet of the channel. As fluid flow velocity decreases its corresponding residence time in channel increases and this facilitates in high heat absorption.

Fig. 7
figure 7

Variation of local bulk temperature along the fluid flow direction. a w = 1.2 m/s, b w = 1.0 m/s, c w = 0.83 m/s, d 0.65 m/s, e w = 0.55 m/s

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The local surface temperature along channel length are shown in Fig. 8, results are shown for pure water and mPCM slurry with 5–15% concentration and for different flow velocities (0.55–1.2 m/s). Qualitatively all cases showed similar trend for decrease in surface temperature along channel length. It is clarified that the channel has two adiabatic sections (at the inlet and outlet) while heat exchange process is happening in between these adiabatic sections. The bumps/kinks in temperature plot (at z = 0.03 m and z = 0.47 m) in all plots of Fig. 7 is possibly due to presence of these adiabatic sections, a gradual decrease was observed in between aforementioned adiabatic sections. Detailed scan of Fig. 7a revealed that at z = 0.1 m higher surface temperatures were noticed with pure water than mPCM slurries. This could be possibly due to incomplete freezing of mPCM at this stage, the specific heat capacity of mPCM slurry in its solid and in liquid phase is relatively less than that of pure water. The enlarged image (Fig. 8a) showed maximum difference with 15% mass concentration of mPCM slurry. With an increase in mass concentration more mPCM particles are there in the slurry, this eventually results in poor thermal performance than water, and maximum deviation is observed with the highest tested concentration of mPCM. Beyond z = 0.1 m, slurry particles start freezing process. This is also noticeable from the higher surface temperature because of the higher latent heat absorption of mPCM slurry. Variation in the surface temperature was more evident for low fluid velocities. Figure 8e showed that higher surface temperature was noticed at 0.55 m/s flow velocity with 15% mass concentration of mPCM slurry.

Fig. 8
figure 8

Variation of local surface temperature along the fluid flow direction. At fluid velocity a 1.2 m/s, b 1.0 m/s, c 0.83 m/s, d 0.65 m/s, e 0.55 m/s

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For analysis of thermal response local Nusselt numbers of both fluids (pure water and mPCM slurry) were compared under similar operating parameters. Figure 9 graphically shows results from this analysis where data for all test cases is presented. As it can be seen from Fig. 8, Nusselt number was higher at the inlet of the channel and thereafter decreased. The hydrodynamic and thermal boundary layer are in developing region near inlet of the test section and this results in higher heat transfer in this region whereas development of the same in later segments deteriorated heat transfer. Further downstream of the aforementioned segments, the local Nusselt number was noticed to have fixed values and this could be explained with fully developed flow conditions in this segment of the test section.

Fig. 9
figure 9

Variation of local Nusselt number along the fluid flow direction. At fluid velocity a 1.2 m/s, b 1.0 m/s, c 0.83 m/s, d 0.65 m/s, e 0.55 m/s

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For all tested flow rates relatively better heat transfer was observed with mPCM slurry and an increase in its mass concentration further helped in heat transfer intensification. For low flow velocities (0.55 to 0.65 m/s) and for all tested concentrations Nusselt number collapsed near the end of channel. This is possibly due to increased residence time inside the channel. This results in complete freezing of mPCM particles near the end of the channel.

The variation of average Nusselt number with respect to the fluid velocity is shown in Fig. 10. The average Nusselt number increased with the increase in mass concentration and fluid velocity. At v = 1.2 m/s the Nusselt number for 15% mass concentration was found to be 21% higher than pure water.

Fig. 10
figure 10

Variation of average Nusselt number with respect to fluid velocity

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Higher viscosity of the mPCM slurry causes higher pressure drop which is a potential drawback. Figure 11 illustrates the variation in pressure drop for water and different tested concentrations of mPCM slurry. An increase in pressure drop was observed as the flow rates were increased. This trend was observed for all tested concentrations of the mPCM slurry. For w = 1.2 m/s, when compared to the pure water, the pressure drop with mPCM slurry was 5.1%, 18.6%, and 30.6% higher when the mass concentrations were 5%, 10% and 15% respectively.

Fig. 11
figure 11

Pressure variation with respect to fluid velocity Tc,in = 15 °C, Tf,in = 33 °C

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It was noticed that the at v = 0.55 m/s, the Nusselt number with mPCM slurry was 7.7%, 14.4%, and 21.38% higher than pure water, whereas an increase in pressure drop of the order of 4.9%, 18.6% and 30.62% for 5, 10 and 15% mass concentrations respectively was noticed. The results showed that mPCM slurry with 5% mass concentration behaves more efficiently as compared to the higher concentration slurries.

6 Conclusions

Numerical investigation on heat transfer characteristics during freezing of mPCM slurry was reported in this paper. The heat transfer characteristics were explored under convective boundary conditions and findings were compared with thermo-hydraulic performance observed with pure water. The key results are summarized below.

It has been observed that an increase in the mass concentration of mPCM has a positive effect on the local Nusselt number. Nusselt number and heat transfer performance increases with an increase in mPCM concentration.

During the freezing process, the bulk temperature and surface temperature were higher with mPCM slurry as compared to pure water.

The pressure drop of mPCM slurry was found to be higher as compared to water due to its higher viscosity.

In order to fully utilize the latent heat of mPCM slurry, the inlet temperature, flow rate and its concentration should be optimized. This shall ensure the complete freezing of mPCM particles during the flow inside the channels.

7 Future Work

The future work will consider the heat transfer and pressure drop analysis of mPCM slurry with the addition of nanoparticles to further enhance the heat transfer performance. It will also focus on the cost effectiveness.