Introduction

Possibility of saving large amount of energy and great energy density are main characteristics of PCMs. Solar air heating units are one of the nice applications of them. In recent years, NEPCM was offered to use instead of pure PCM. In this way, is storage energy enhanced. Bondareva et al. [1] utilized alumina to enhance thermal properties of PCM in a cooling unit. They concluded that performance is augmented with adding alumina. Innovative shape of heat storage system has been suggested by Sheikholeslami et al. [2]. They also utilized nano-sized particles to expedite discharging. Raizah et al. [3] scrutinized non-Newtonian nanofluid free convection within an open porous tank. They employed power law model for working fluid. Inorganic nanomaterial has been dispersed into water by Sheikholeslami and Mahian [4] during solidification. Influence of magnetic field has been employed by them. Aly et al. [5] investigated thermo-diffusion impacts on nanomaterial convective flow inside a porous annulus. Innovative uses of nanomaterial for renewable energy have been simulated by Sajid and Ali [6]. Simulation of discharge phenomena has been carried out by Sheikholeslami et al. [7]. They assumed that conduction is the main mode for solidification. Alsabery et al. [8] scrutinized two-phase nanofluid convection within a tank with two-lid wall. Ali et al. [9] simulated the transient problem of blood flow in an artery. They assumed the nano-pharmacodynamic migration of working fluid. Soomro et al. [10] utilized the Prandtl fluid model for nanomaterial flow over a plate. Yadav et al. [11] investigated the pulsating nanomaterial flow inside a porous domain in appearance of electric field. Reaching the effective working fluid is a challenge in recent years [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29].

The focus of the present work is on discharging process of PCM in a wavy duct. Using nanoparticles and wavy walls makes the solidification rate to intensify. Such system can be used for heating of buildings. FVM is utilized to simulate various cases to clarify the impact of concentration of nanomaterial and amplitude of wavy wall on solidification.

Mathematical model and problem demonstration

In the current paper, 2D air heat exchanger with wavy walls has been considered. As shown in Fig. 1, the annulus space is filled with paraffin with CuO nanoparticles. The properties of both PCM and nanoparticles are reported in Table 1. The laminar air flow was considered in inner channel. The cold air becomes warmer due to solidification. Unsteady forced convection has been considered. Free convection can be neglected in discharging process. The geometry was symmetric, and so there is no need to simulate the whole domain. The unsteady governing flow and temperature equations are expressed as:

$$ \frac{\partial v}{\partial t} + \mathop V\limits^{ \to } \cdot \nabla v = \frac{1}{{\rho_{\text{nf}} }}\left( {\mu_{\text{nf}} \nabla^{2} v - \nabla P} \right) + Cv\frac{{\left( {1 - \lambda } \right)^{2} }}{{\lambda^{3} + \varepsilon }} $$
(1)
$$ \frac{\partial u}{\partial t} + \mathop V\limits^{ \to } \cdot \nabla u = \frac{1}{{\rho_{\text{nf}} }}\left( { - \nabla P + \mu_{\text{nf}} \nabla^{2} u} \right) + Cu\frac{{\left( {1 - \lambda } \right)^{2} }}{{\lambda^{3} + \varepsilon }} $$
(2)
$$ \nabla \cdot \mathop V\limits^{ \to } = 0 $$
(3)
$$ \frac{\partial T}{\partial t} + \mathop V\limits^{ \to } \cdot \nabla T + \frac{{\partial \left( {\rho L\lambda } \right)_{\text{nf}} }}{\partial t} = \frac{{k_{\text{nf}} }}{{\left( {\rho C_{\text{p}} } \right)_{\text{nf}} }}\nabla^{2} T $$
(4)

\( C = 10^{5} ,\varepsilon = 10^{ - 3} \) are mushy zone and small number constant.

Fig. 1
figure 1

Present channel and mesh sample

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Table 1 Properties of CuO and paraffin
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\( k_{\text{nf}} ,\mu_{\text{nf}} ,\left( {\rho C_{\text{p}} } \right)_{\text{nf}} ,L_{\text{nf}} , \) and \( \rho_{\text{nf}} , \) can be calculated as:

$$ k_{\text{nf}} = k_{\text{f}} \frac{{ - 2\phi \left( {k_{\text{f}} - k_{\text{s}} } \right) + 2k_{\text{f}} + k_{\text{p}} }}{{\phi \left( {k_{\text{f}} - k_{\text{s}} } \right) + k_{\text{p}} + 2k_{\text{f}} }} $$
(5)
$$ \mu_{\text{nf}} \left( {1 - \phi } \right)^{2.5} = \mu_{\text{f}} $$
(6)
$$ \left( {C_{\text{p}} } \right)_{\text{nf}} = \left[ {(1 - \phi )\left( {\rho C_{\text{p}} } \right)_{\text{f}} + \phi \left( {\rho C_{\text{p}} } \right)_{\text{s}} } \right]\rho_{\text{nf}}^{ - 1} $$
(7)
$$ \left( {\rho L} \right)_{\text{nf}} = (1 - \phi )\left( {\rho L} \right)_{\text{f}} $$
(8)
$$ \rho_{\text{nf}} = \rho_{\text{f}} (1 - \phi ) + \phi \rho_{\text{s}} $$
(9)

Liquid fraction and total enthalpies are defined as:

$$ \lambda = \left\{ {\begin{array}{*{20}l} 0 \hfill & {T < T_{\text{s}} } \hfill \\ {\frac{{T - T_{\text{s}} }}{{T_{\text{l}} - T_{\text{s}} }}} \hfill & {T_{\text{s}} < T < T_{\text{l}} } \hfill \\ 1 \hfill & {T < T_{\text{l}} } \hfill \\ \end{array} } \right. $$
(10)
$$ H_{\text{e}} = h + \lambda L $$
(11)
$$ h = h_{\text{ref}} + \int_{{T_{\text{ret}} }}^{T} {\left( {C_{\text{p}} } \right)_{\text{nf}} } {\text{d}}T $$
(12)

Results and discussion

The focus of the current article is on solidification of PCM inside thermal storage for building application. CuO nanoparticles are added into pure PCM (n-octadecane). PCM exists on outer channel and air flows inside the sinusoidal duct. During solidification, air becomes warmer and exits the channel with higher temperature. Finite volume method based on ANSYS Fluent 18.2 is employed to model mathematical equations which are obtained with assumption of neglecting free convection and homogenous model for nanomaterial. The first assumption is valid for solidification because conduction mode is dominant in this kind of phase change.

The enthalpy porosity method [30] is employed for zones which are full of PCM. The value of 0.2 has been selected as velocity under-relaxation factor. To be sure about the correctness of the present model and code, we reproduced the outputs of previous experimental paper [31]. Solidification front was compared and reached nice agreement. Besides, validation figure indicates that free convection cannot affect discharging process (Fig. 2).

Fig. 2
figure 2

Validation for solidification [31]

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Figure 3 illustrates the impact of \( a,\,\phi \) on liquid fraction of NEPCM. During this transient phenomena, liquid is converted to solid. In liquid fraction contours, red and blue colors signify the liquid and solid phases, respectively. Thus, liquid fraction profile is reduced with time. This graphs demonstrated that dispersing CuO into PCM leads to greater solidification rate. Supportive effect of \( \phi \) becomes lower with the use of wavy surface. Increasing amplitude of sinusoidal wall results in augmentation of discharging rate. Minimum full solidification time is 36.2 h which belongs to case with the greatest \( a,\,\phi \). Distributions of liquid fraction and temperature during solidification are depicted in Figs. 4–7. Liquid starts to freeze in zones near the inlet air flow. As time passes, solidification front propagates along the duct. The top right side of channel becomes solid in greater time. As amplitude of wavy channel rises, discharging rate is enhanced and solidification completed in lower time. The left side has lower temperature than the right side. As time reaches full time discharging, temperature of airflow and left-side PCM becomes equal.

Fig. 3
figure 3

Impact of \( a,\,\phi \) on liquid fraction

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Fig. 4
figure 4

Contours of liquid fraction for various times when \( a = 5\;{\text{mm}},\,\phi = 0.05 \)

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Fig. 5
figure 5

Contours of temperature for various times when \( a = 5\;{\text{mm}},\,\phi = 0.05 \)

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Fig. 6
figure 6

Contours of liquid fraction for various times when \( a = 10\;{\text{mm}},\,\phi = 0.05 \)

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Fig. 7
figure 7

Contours of temperature for various times when \( a = 10\;{\text{mm}},\,\phi = 0.05 \)

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Conclusions

In the current simulation, discharging of NEPCM has been modeled via FVM. Wavy duct has been considered, and CuO nanoparticles have been dispersed into paraffin. Cold airflow enters the inner duct and becomes warmer in outlet section. Zones near the outlet of airflow require longer time to complete discharging. Employing wavy channel with greater amplitude needs lower time in comparison with smaller amplitude. Liquid fraction reduces with rise of time. Reaching zero value of liquid fraction indicates the full solidification. Volume fraction of CuO has reverse relationship with liquid fraction.