A comprehensive assessment of temperature difference and natural convection on phase transition in different latent heat storage systems accompanied by porous media

doi:10.21203/rs.3.rs-2248047/v1

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A comprehensive assessment of temperature difference and natural convection on phase transition in different latent heat storage systems accompanied by porous media

https://doi.org/10.21203/rs.3.rs-2248047/v1

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With the constant increasing demand for energy and the challenges arising from the growing population, securing a reliable energy supply seems critical. Latent energy storage is introduced as one of the most efficient solutions to harness the renewable and waste energy and secure a constant supply. The phase change material (PCM) in this type of energy storage suffers from low responsiveness, slowing down the industrialization of latent storage. This study proposed different conventional PCM-based storage units to numerically analyze the most influential determinant in each system during the melting evolution. The working fluid temperature, natural convection and porous media at different porosities are the factors selected for comparison between four configurations. The temperature of working fluid is of similar significance to three types of units as an enhancement of 22% has been obtained by 5 degrees increase in inlet temperature, while the pipe model has benefited almost 26% improvement under the same condition. Triplex tube heat exchanger (TTHX) in all scenarios outperforms other units since the heat transfer area for this type is greater that that of its counterparts. Natural convection phenomenon is most effective in the pipe model as its absence decelerate the melting rate by 236%. Porous media integration produced, by average, 93% faster systems thanks to its extra heat transfer surface, although it suppressed the gravitational movement.

Energy storage is one of the basic fields for research and development in this century. The main reason for this high concentration in this field can be traced to the increasing need for energy and fossil fuels, leading scientists to move from conventional resources to renewable resources. Nature, in various forms such as solar energy, can provide the earth with clean, low-cost, widely harvestable and endless energy. This energy can be obtained in three main ways, which include: latent heat thermal energy storage (LHTES), sensible heat thermal energy storage (SHTES) and thermochemical energy storage (TCES). In latent energy storage systems, the presence of phase change materials (PCMs) makes it possible to keep the temperature constant during transition course, which is due to the fact that the at the melting or freezing temperature, the PCM temperature remains constant. PCM-based systems are able to solve many common energy storage problems well. This feature is possible due to the possibility of storing large amounts of energy at low temperatures. The existence of this type of function has led to the emergence of many applications using this type of mechanism in solar collectors [1], heating and cooling of electronic products [2], heat storage for building systems and HVAC [3], hot water equipment in the building [4] and recovery of thermal waste [5].

During the melting evolution of an energy storage system, natural convection phenomenon is considered as the dominant method of heat transfer. The dominant buoyancy force in this type of phase transition creates a vortex in the system, which in turn improves heat transfer. The geometry of the system can also affect the impactfulness of natural convective heat transfer originating from the gravity. The most ubiquitous problem in this type of energy storage relates to inadequate thermal conductivity of PCM, which can slow down the process and the lengthen the operations. Therefore, the vast majority of research in this field have been fixated on improving this aspect of latent energy storage systems. One possible solution to speed up phase transition in systems using shell and tube configuration is to use multi-tube heat exchangers [6]. Agyenim et al. [7] used Erythritol as a PCM in a shell-and-tube system and experimentally evaluated the efficiency of this latent horizontal energy storage system. Instead of using a single tube, they divided it into four smaller tubes and compared the results with the original configuration by temperature gradients in the radial, axial, and angular directions. The evaluated results showed an improvement in radial and axial heat transfer of up to 2.5% and 3.5%, respectively. Similarly, the effect of multiplexing shell and tube systema in a numerical investigation by Esapour et al. [8] was studied. While paraffin RT35 was chosen as the PCM, the results showed an increase in heat transfer rate of up to about 30% in case of using 4 tubes compared to shell and tube design. In the same study [8], the authors also examined the effect of heat transfer fluid (HTF) temperature on heat transfer and efficiency. They concluded that by increasing the HTF temperature from 50 to 60 ° C, in all the cases studied, the rate of reduction of the complete melting time would be reduced by up to 33%.

The melting process for RT27 paraffin as a PCM was also performed numerically by Liu et al. [9] to evaluate the positive or negative effects of different compositions and designs of large and small tubes in the shell. In addition, the effect of placement of multi-tube units and the effect of radial ratio were also investigated. The fastest solid-to-liquid phase change operation was achieved by a state consisting of a large tube and two small tubes with a radial ratio of 2. In another study, Liu et al. [10] used paraffin as a PCM and air as a HTF, and showed that a heterogeneous layout of pipes in a multi-pipe system could yield a significant 57% improvement in performance. In an experimental study by Jesumathy et al. [11], the authors made a detailed assessment of the effect of the inlet temperature of the working fluid as well as the inlet flow rate in a two-pipe storage system. They found that a 2 ° C change in HTF temperature could be up to 25% effective on the performance of the respective system. Kousha et al. [12] simultaneously investigated melting and freezing in an experimental study. The authors revealed that by applying 4 tubes, compared to the conventional shell and tube system, the time required for melting and freezing is reduced by 43% and 50%, respectively. Joybari et al. [13] experimentally compared the melting process of RT60 paraffin in a shell and simple finless tube unit and a 5-tube system. The final results of this study showed the positive effect of multi-pipe energy storage system on convective heat transfer, which resulted in an increase in efficiency of nearly 74%.

Another important aspect that can be effective in the thermal performance of a LHTES is associated with the discussion of the design and geometry effect of heat exchangers. Pourakabar et al. [14] numerically studied the effects of conventional methods of improving thermal efficiency in a system containing Icosane as a PCM mixed with copper porous material. They deployed finless two-tube and four-tube configurations, with two circular and elliptical geometries as the shape of the shell, for their evaluation. For the melting operation in the circular shell, the authors introduced the two-tube placement in the vertical direction as the best mode for this type of shell geometry, while for the elliptical shell geometry, the four-tube mode, located in the horizontal direction was approved by the authors as the best scenario. Faghani et al. [15] performed numerical simulations to evaluate the effects of the orientation of circular and elliptical shells inside a heat exchanger under melting operation. They were able to achieve a maximum improvement of 62% in reducing the melting duration for the case where the elliptical tube containing HTF was placed vertically and the elliptical shell was placed horizontally. Li et al. [16] numerically analyzed the melting operation of the PCM in a spherical shell by studying the effects of HTF temperature, sphere radius, conduction heat transfer coefficient of the PCM and the material used for the shell. 40% better efficiency was achieved when the shell radius was 40 mm and the fluid temperature was 60 °.

Another determinant of great importance that can improve the efficiency and performance of heat transfer in LHTES is the porous media, which can significantly scale up the thermal conductivity of PCM [17, 18]. Zhao et al. [19] took advantage of the porous media inside the shell of the energy storage unit and were able to improve thermal performance by 3 to 10 times. The reason for this effective improvement can be attributed to the presence of pores or voids between the porous medium, which can dramatically increase the level of heat transfer between the PCM and the energy source while occupying very little space in the annulus. Thus, the need to change dimensions to maintain energy storage capacity is rendered futile. Zheng et al. [20] investigated and analyzed a rectangular cavity containing PCM and porous media numerically and two-dimensionally. The results revealed that 20% improvement and acceleration of phase-change is obtainable in the presence of porous media. In a numerical and experimental study, Atal et al. [21] attempted to accurately measure the effect of the porosity of a porous media on a horizontal LHTES containing paraffin. They found that the porosity coefficients of 95% and 77% gave completely different results, while the lower porosity coefficients led to better results and better thermal conductivity. Mancin et al. [22] also experimentally measured the effectiveness of different porous materials for paraffins with melting temperatures of 53˚, 57˚ and 59˚. In this study, sample foams with a porosity density of 5,10,40 PPI with a constant porosity coefficient of 95% were used. The porous material network was capable of playing a good unifying role as a heat dissipator and minimize melting duration. In a numerical study, Deng et al. [68] compared the performance and thermal efficiency of an energy storage system based on PCMs with the presence and absence of porous materials. The results proved that the length of melting and freezing operations is reduced by 95% and 93%, respectively, if porous materials are applied.

Reviewing the studies conducted, despite extensive reviews and evaluations of LHTES and variables affecting its performance, a study that comprehensively compares the performance of conventional systems is nonexistent. Most studies evaluated one or more variables in a selected system, so in this paper, a comprehensive comparison of the effects of HTF inlet temperature, natural convection, and the implementation of porous media in four types of storage units are evaluated according to the variable under study. The 4 types of storage systems being modelled in the current study include cylinder model, pipe model, triplex-tube and multi-tube designs. The amount of PCM in each given model is identical so that a fair comparison can be made with regard to various considered parameters. The implementation of porous media is done at three different porosities for each given units and copper has been adopted as porous material. Plus, numerical simulations of the respective configurations, which have been validated against experimental results, are done using ANSYS Fluent 18.2 software package.

The different configurations considered in this study can be divided into four different categories, which are: shell and tube system where the PCM is trapped in the shell (Cylinder design), shell and tube system where the PCM is trapped in the tube (Pipe model), the triplex tube heat exchanger (TTHX) in which the PCM is housed in the middle tube, and the multi-tube unit in which the PCM material is immured in the shell. To better illustrate and understand the types of schemas studied, Fig. 1 demonstrates 2-D representations of these four types of storage units. For having a fair comparison, the amount of PCM or in another word the storage capacity of eighter configurations are similar. Thus, the dimensions and diameters of them are different. The cylinder unit is 36 mm and 60 mm in diameter from the inner tube to the outer tube, respectively while the diameter of the inner tube housing PCM in the pipe unit equals 48 mm. As for the multi-tube, the four inner tubes diameter is set to 16 mm as the outer tube has a diameter of 60 mm across. Lastly, The TTHX dimensions are similar to that of the cylinder model, but it has an extra tube in the vicinity of PCM whose dimension equals 80 mm in diameter.

The material deployed in this study, RT82 paraffin, was selected because it has many suitable properties, including physical and chemical stability, non-toxicity and non-corrosiveness during multiple phase-change cycles. Copper has also been chosen as porous material due to its outstanding thermal conductivity and its capability to improve on the overall thermal conductivity of the respective system. The thermophysical properties of the materials used are tabulated in Table 1.

Table 1

Thermophysical Properties of materials
a	b
c	d
Figure 1 2-D outlines of a) Cylinder (Shell) model b) Pipe model c) Multi-Tube model d) TTHX

Property	RT82	Cu
ρm (kg/m3)	770	8920
Cp (kJ/kg K)	2	380
k (W/m K)	0.2	400
µ (N s/m2)	0.03499	-
L (kJ/kg)	176	-
Ts (K)	350	-
Tl (K)	358	-
β (1/K)	0.001	-

According to studies, so far, none of the studies related to the comparison of different types and designs of energy storage arrangements have not examined as to how the temperature difference between PCM and HTF can play out juxtaposed to different heat transfer surface between PCM and HTF in various configurations. Therefore, two HTF inlet temperatures of 368 and 373 K are used, which leads to a more accurate identification of the effect of this component with the thermal performance in the four proposed arrangements. In addition, considering the multi-pipe unit in the problem design, it is possible to simultaneously make a more accurate assessment of the importance of tube placement against the temperature difference. Natural convection plays a major role, especially in the melting evolution where Buoyancy forces are more dominant. Hence, to examine the impactfulness of this phenomenon during melting and compare its importance in different configurations, natural convection is not applied in some cases. The study of the implementation of porous materials and how it affects the performance of energy storage, due to the evaluation of other effective variables, has been less studied in detail. Therefore, the systems introduced are mixed with copper as porous material to improve the poor thermal conductivity of the PCM in this study, which is paraffin RT82. All cases considered for evaluation are tabulated in Table 2 and Table 3.

Table 3

Proposed case studies – Porous-Media-included Scenarios
Table 2 Proposed case studies – Plain Scenarios
#Case	Configuration	Material	HTF inlet temperature	Natural Convection	Initial temperature
A1	Cylinder	RT82	368 K	Applied	300 K
A2	Pipe
A3	TTHX
A4	Multi-Tube
A5	Cylinder		373 K	Applied
A6	Pipe
A7	TTHX
A8	Multi-Tube
A9	Cylinder		368 K	Not Applied
A10	Pipe
A11	TTHX
A12	Multi-Tube

#Case	Configuration	Materials	HTF inlet temperature	Natural Convection	Porosity	Initial temperature
P1	Cylinder	RT82 & Copper	368 K	Applied	99%	300 K
P2	Cylinder				97%
P3	Cylinder				95%
P4	Pipe				99%
P5	Pipe				97%
P6	Pipe				95%
P7	TTHX				99%
P8	TTHX				97%
P9	TTHX				95%
P10	Multi-Tube				99%
P11	Multi-Tube				97%
P12	Multi-Tube				95%


Figure 2 2-D Computational Field


Figure 3 Grid network quality assessment


Figure 4 Validation Analysis

	Cylinder (Shell)	Pipe	TTHX	Multi-Tube
300 s
600 s
900 s
1200 s

Figure 5 Liquid fraction contours for 4 configurations at HTF temperature of 368 K and with natural convection


Figure 6 Liquid fraction plot against time for A1-A4 cases – HTF temperature of 368 k


Figure 7 Liquid fraction plot against time for A5-A8 cases – HTF temperature of 373 k

	Cylinder (Shell)	Pipe	TTHX	Multi-Tube
300 s
600 s
900 s
1200 s

Figure 8 Liquid fraction contours for 4 configurations at HTF temperature of 368 K and without natural convection


Figure 9 Liquid fraction plot against time for A9-A12 cases – HTF temperature of 368 k – without natural convection

	Cylinder (Shell)	Pipe	TTHX	Multi-Tube
30 s
60 s
90 s
200 s

Figure 10 Liquid fraction contours for 4 configurations at HTF temperature of 368 K and with Porous Media at 99% porosity


Figure 11 Liquid fraction plot against time for P1, P4, P7 and P10 cases – HTF temperature of 368 K – with Porous Media at 99% porosity


Figure 12 Liquid fraction plot against time for P2, P5, P8 and P11 cases – HTF temperature of 368 K – with Porous Media at 97% porosity


Figure 13 Liquid fraction plot against time for P3, P6, P9 and P12 cases – HTF temperature of 368 K – with Porous Media at 95% porosity
Nomenclature
Amush mushy zone constant (kg m-3 s)
Cp specific heat (J kg-1 K-1)
Ci inertia coefficient
D pipe diameter (m)
df fiber diameter (m)
dp pore diameter (m)
f friction factor
g acceleration of gravity (m s-2)
h heat transfer coefficient (W m-2 K-1)
K permeability (m2)
L latent heat of fusion (kJ kg-1)
m mass (kg)
P pressure (Pa)
T temperature (K)
Tm melting point (K)
t time (s)
u, v fluid velocity in Cartesian coordinates (m s-1)
x, y Cartesian coordinates (m)
Greek symbols
$\beta$ melting fraction
$\varepsilon$ porosity
$\gamma$ thermal expansion coefficient (K^− 1)
$\lambda$ thermal conductivity
$\mu$ dynamic viscosity (kg m^− 1 s^− 1)
$\rho$ density (kg m^− 3)
Subscripts
e effective value
i inner tube
m middle tube
PCM phase change material
por porous media
ref reference value

To avoid additional computational costs in this study while keeping the error under satisfactory thresholds, the following hypotheses and simplifications are introduced for this purpose:

Melting evolution calculations are performed by enthalpy method implemented on ANSYS Fluent software package.
The integration of porous materials inside the PCM and its modeling is done by Brinkman–Forchheimer–extended Darcy correlation.
To avoid solving the system of equations due to the presence of porous media and combining equations related to the PCM and porous media, local thermal equilibrium modelling by defining the effective thermal conductivity can integrate the system of equations.
A uniform and isotropic porous media is assumed.
The thermophysical properties of the metal foam is unchanged during the melting operation.
The motion of the PCM in the fluid phase is assumed to be transient, laminar and incompressible.
Viscosity changes and temperature fluctuations of the HTF are ignored.
There is a no-slip condition on the system’s wall.
Apart from density, the other thermophysical properties of paraffin are assumed to be constant.
Due to temperature changes, the density of the PCM fluctuates. To account for this change, the Boussinesq approximation is used to accurately predict the density.
Volumetric changes due to solid-to-liquid phase-change in RT82 paraffin are negligible.
The thermal energy storage system is completely insulated from the surrounding.

2.1 Governing Equations

The thermal performance and behavior of the melting evolution for a PCM, taking into account the conduction heat transfer and natural convection, can be accurately determined by the equations of continuity, the momentum equation and the energy equation as the following:

$$\frac{\partial \rho }{\partial t}+\nabla .\left(\rho v\right)=0$$

Eq. 1 represents the continuity equation in which ρ and v represent density and velocity vectors, respectively. The following equation can be used for the momentum equation:

$$\rho (\frac{{\partial u}}{{\partial t}}+u\frac{{\partial u}}{{\partial x}}+v\frac{{\partial u}}{{\partial y}})=\mu (\frac{{{\partial ^2}u}}{{\partial {x^2}}}+\frac{{{\partial ^2}u}}{{\partial {y^2}}}) - \frac{{\partial P}}{{\partial x}}+{S_u}$$

$$\rho (\frac{{\partial v}}{{\partial t}}+u\frac{{\partial v}}{{\partial x}}+v\frac{{\partial v}}{{\partial y}})=\mu (\frac{{{\partial ^2}v}}{{\partial {x^2}}}+\frac{{{\partial ^2}v}}{{\partial {y^2}}}) - \frac{{\partial P}}{{\partial y}}+{S_v}$$

The momentum equation introduced above is defined by the terms P, ρ and µ, which are the sign of pressure, density and dynamic viscosity, respectively. Source terms can also be calculated in Eqs. 2 and 3 from the following equations.

$${S_u}={A_{mush}}\frac{{{{(1 - \beta )}^2}}}{{{\beta ^3}+\delta }}u - \frac{\mu }{K}u - \frac{{\rho {C_i}}}{{\sqrt K }}\left| u \right|u$$

$${S_v}={A_{mush}}\frac{{{{(1 - \beta )}^2}}}{{{\beta ^3}+\delta }}v - \frac{\mu }{K}v - \frac{{\rho {C_i}}}{{\sqrt K }}\left| v \right|v+{\rho _{ref}}g\gamma (T - {T_{ref}})$$

Where β represents the liquid fraction of the PCM, which is defined as follows:

$$\left\{ \begin{gathered} \beta =1{\text{ T > }}{{\text{T}}_m} \hfill \\ 0<\beta <1{\text{ T = }}{{\text{T}}_m} \hfill \\ \beta =0{\text{ T < }}{{\text{T}}_m} \hfill \\ \end{gathered} \right.{\text{ }}$$

Relationships related to porous material can also be defined and calculated from the ensuing equations.

(7)
(8)
(9)

The first parameter shown to the right of Eq. 4 denotes the momentum-limiting factor for the fluid stream during phase-change and predicts its behavior when passing through the porous media, which is marked as A_mush and is called mushy zone constant. The common range considered for this factor is in the range of 10⁴ to 10⁷ kg / m3 / s, which plays a key role in determining the behavior of melting durations. For this research, 10⁵ kg / m3 / s, according to several studies in this field, has been considered for it [23–25].

The parameter β, which is the fraction of liquid in the annulus, is zero for temperatures below the melting temperature, which renders the expression ${A_{mush}}\frac{{{{(1 - \beta )}^2}}}{{{\beta ^3}+\delta }}u$ mathematically undefinable. To circumvent this mathematical error, a very small numeric value, equal to 0.001 and denoted by δ, is added to the denominator of the expression. The second and third expressions to the right of Eq. 4 are terms related to the resistance factor due to the presence of porous media. In addition, due to the melting of the PCM, temperature fluctuations and temperature expansion, the buoyancy forces have a direct impact on the behavior of the system, the relationship of which is shown in fourth term of Eq. 5. From Eqs. 7 and 8, two terms K and C_i can be calculated as the permeability coefficient and the inertia coefficient, respectively. Eq. 9 is associated with 3 effective variables related to the porous media, which include: ε as the porosity coefficient of the porous media, df and dp, which are the fiber radius and the pore radius, respectively.

The energy equation can be defined as follows:

$$\left[(1-\epsilon )(\rho {C}_{p}{)}_{por}+\epsilon (\rho {C}_{p}{)}_{PCM}\right]\frac{\partial T}{\partial t}+(\rho {C}_{p}{)}_{PCM}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)={\lambda }_{e}\left(\frac{{\partial }^{2}T}{\partial {x}^{2}}+\frac{{\partial }^{2}T}{\partial {y}^{2}}\right)-\epsilon \rho L\frac{\partial \beta }{\partial t}$$

Where λ_e represents the effective heat transfer coefficient for the combination of the PCM and the porous media, whose value, from the experimental correlations [26] can be calculated as follows:

$${\lambda }_{e}=\frac{\sqrt{3}}{2}{\left[\frac{0.09\psi }{{\lambda }_{PCM}+\frac{1}{3}(1+\psi )({\lambda }_{por}-{\lambda }_{PCM})}+\frac{0.91\psi }{{\lambda }_{PCM}+\frac{2}{3}\psi ({\lambda }_{por}-{\lambda }_{PCM})}+\frac{\frac{\sqrt{3}}{2}-\psi }{{\lambda }_{PCM}+\frac{0.12}{\sqrt{3}}\psi ({\lambda }_{por}-{\lambda }_{PCM})}\right]}^{-1}$$

$$\psi =\frac{{ - 0.09+\sqrt {0.0081+\frac{{2\sqrt 3 }}{3}(1 - \varepsilon )\left[ {2 - 0.09(1+\frac{4}{{\sqrt 3 }})} \right]} }}{{\frac{2}{3}\left( {1.91 - \frac{{0.36}}{{\sqrt 3 }}} \right)}}$$

2.2 Initial & Boundary Conditions

As the symmetry line across vertical direction make the system exactly identical due to the presence of natural convection, the 2-D computational domain of this study is half of the whole unit to minimize the computational cost. Figure 2 presents the respective domain in which the vertical line, outer surface of the outer tube and middle tube represent symmetry, insulation and constant temperature boundary conditions. As stated in the simplifications considered for this numerical assessment, the fluctuations of HTF across the storage unit is somewhat negligible, leading to constant temperature condition at the interface between PCM and HTF in all 4 given units. Plus, as for the initial condition upon melting procedure, all the system is considered to be uniformly at 300 K.

The enthalpy-porosity methodology has been adopted for solving the numerical analysis by ANSYS Fluent 18.2 in which a rigid grid network is applied where both phases can be simulated in high precision using above-mentioned governing equations. A pseudo-porous-media is assumed to exist in the annulus in case of two phase being concurrently present within the system and the porosity of such porous media is determined by liquid fraction. To solve the equation, SIMPLE approach which is a built-in finite volume CFD methodology has been adopted while PRESTO is applied for discretization of pressure and momentum and energy equations are taken into account by QUICK scheme. When it comes to residuals threshold, energy equation’s value is set to 10^− 6 and 10^− 4 is applied for remaining variables. Plus, as this is a transient solution, an adequate time step of 0.1s is adopted for the unsteady discretization of governing equations.

2.3 Grid Independency Assessment

One of the significant aspects of solving a numerical simulation is the grid network quality as it poses a conspicuous effect on the outputs. Thus, three different meshing networks are applied and compared accordingly to come up with the best quality and less burdensome computational cost. Figure 3 demonstrates a plot of liquid fraction against time for 6100, 10700 and 15425 grid elements. Based on the result, the distinction in output arising from various grid network is somewhat negligible, however, the difference between 10700 and 157425 is smaller than difference between 6100 and 10700 networks. Therefore, average of 10700 elements is applied across all storage units for this study as it offers a high accuracy while keeping cost at acceptable threshold.

2.4 Validation of the Numerical Analysis

Validation is of prime importance in a computational study of any kind to substantiate the reliability of respective results. To this end, the experimentally conducted investigation of Mat et al. [27] has been picked out for validation purposes. Figure 4 presents a temporal plot of average temperature for comparing the results of experimental and numerical studies. As depicted, the discrepancy between two outputs is somewhat negligible, despite the fact that a 2-D computational domain has been adopted for numerical analysis against the 3-D real unit. Thus, the veracity and dependability of current study has been proved as the 5% error threshold holds true for the entire period of validation plot.

This section expounds on the importance of HTF temperature, natural convection and incorporation of porous media in the 4 different variations of storage systems. To analyze the resultant data, this part is subcategorized into multiple sections eighter of which delves into each parameter by presenting various figures and plots of liquid fraction which is the main criterion of storage systems performance and responsiveness.

3.1 HTF Temperature effectiveness across various units

HTF temperature has been proved significant when it comes to the thermal assessment of a storage unit. Thus, this needs to be more deeply analyzed as to how much effective this parameter is across different units and how exactly this effectiveness might change with different configurations. Two temperatures of 368 K and 373 K with 5 degrees temperature shift have been selected to shed a light on this determinant. Figure 5 illustrates liquid fraction of 4 proposed configurations at 4 temporal stages during the solid-to-liquid phase-change. At the first depicted stage, at 300 s, the difference between various units is somewhat evident, especially for the TTHX in which two ring-shaped melted area has been formed near the vicinity of inner and outer tubes. The cylinder is witnessing just a small fraction melted at 300 s while with the time passing by, the upper half of annulus is taking some pace in getting liquidated compared to the lower half. This holds true for other variations as well where natural convection effectiveness becomes more significant with time elapsing. Among various units, the TTHX is introduced as the best one to get fully melted before 1200 s while other designs are trailing far behind. This stems from the fact that the heat transfer area between PCM and HTF is greater than any other configurations in TTHX as both the inner and outer tubes contain hot HTF, leading to the fastest melting rate among the 4 units. The next fastest responsive unit is the pipe model which performs better compared to the cylinder and multi-tube configurations as there is not any tube in the middle to suppress the movement of liquid across annulus due the gravitational forces and this uniform annulus leads to a superior responsiveness. This observation may be surprising after looking at the results of Fig. 5, but in the next paragraph the reason is explained. The multi-tube design in which 4 smaller tubes is responsible for housing the HTF in place of a big tube seen in cylinder or TTHX model stands as the third fastest performer. This extra heat transfer surface thanks to four small tubes gives rise to the best performing unit after TTHX and pipe. After these three models, and make the cylinder the least desirable unit for thermal storage purposes.

To delve into more detail as to how each unit melted during the phase-change procedure at HTF temperature of 368 K, Fig. 6 depicts the temporal changes of liquid fraction for different units. The first impression upon looking into the respective plot is associated with the close performance of multi-tube and TTHX at the early stages of melting evolution, but after the passage of time, these performers drifted apart to the point where multi-tube stands at the third place with marginal latency compared to the pipe configuration. It takes almost 5800 s for the cylinder unit to get fully melted which is the worst case by far compared to other units under the given conditions. Next, multi-tube unit melting duration exceeds 2700 s, preceded by pipe model at roughly 2400 s despite multi-tube’s superior performance in almost entire period of phase-change procedure. This underperformance of multi-tube is attributed to the natural convection effect, especially at the late stages of melting phenomenon. Due to the presence of 4 tubes in the annulus, the suppression of upward trend stirred by gravity is more potent in the multi-tube design than a uniform annulus seen in the pipe model and this even worsens as the time goes by since in the melting procedure the natural convection effect takes hold at the late stages where the majority of PCM is liquidated. Hence, although multi-tube design offers faster melting rate in the major periods of its phase-change, the fully melting time is slightly lower for the pipe model. Besides, TTHX takes almost 1000 s to be liquified completely, outperforming other units by a sizeable difference.

The next plot presented as Fig. 7 shows the liquid fraction figures at HTF temperature of 373 K. What stands out in this plot compared to Fig. 6 is the similarity in overall trends for all respective units. The next significant point in this plot concerns how faster each unit get melted completely compared to HTF temperature of 368 K. From looking into the data, except for pipe configurations, other three remaining models registered a roughly 22% improvement with 5 degrees shift in HTF in temperature while this figure for pipe model reached approximately 26%. This extra enhancement for the pipe model is associated with natural convection being more significant at higher HTF temperature and since pipe model is the one configuration taking the most advantage from this phenomenon, that explains as to why by scaling up the HTF temperature from 368 K to 373 K, the most increase occurred in the pipe model.

3.2 Natural Convection effectiveness across various units

As can be seen from the previous section results, natural convection is of critical importance in each configuration and acts as a determinant with regard to thermal performance of each unit. This section examines the precise effectiveness that gravitational force may enforce on the storage unit. To this end, in the numerical simulations of this part, the natural convection effect is omitted to measure how its absence change the performance of 4 given configurations. Figure 8 displays the liquid fraction contours for all units at 4 stages of melting process. Most importantly, the absence of natural convection led to a completely symmetrical melting front as there is not any upside movement to distinguish the upper half from the lower half. Under this set of conditions similar to prior studied conditions, TTHX proved superior, but unlike the results in presence of natural convection, multi-tube claims the second-best unit in absence of gravity. This is manifested more clearly in Fig. 9 where the order of best to worst cases has undergone a noteworthy change. In the prior section, under HTF temperatures of 368 and 373 K and applied natural convection, the pipe model slightly outperformed multi-tube in terms of full melting time. Interestingly, without natural convection condition, the multi-tube configuration stands as the second-best performer. Although the A12 which corresponds to multi-tube has the second position, compared to applied gravity, it takes almost 3100 s to get fully melted, a 14% lengthening of phase-change period when Buoyancy forces were applied. This figure for TTHX unit registered a 6% increase in melting duration while surprisingly, the absence of natural convection for the shell or cylinder unit decreases the duration by 7%, implying the detrimental impact of natural convection for this configuration. The worst-case scenario in absence of gravity is clearly the pipe model whose performance takes a huge toll as it takes almost 8300 s to finish the procedure, a spike of 236% in melting duration. This illustrates the outstanding importance of natural convection on the pipe unit and how much difference Buoyancy forces can impose contingent upon the design and configurations of a storage unit. On the other hand, in the shell design, natural convection proved undesirable completely.

3.3 Porous Media incorporation effectiveness across various units

The integration of metallic foams into the storage unit has gained attraction in recent years. However, a comprehensive evaluation of such technique within LHTES seemed lacking. To this end, copper as porous material at three different porosities of 99%, 97% and 95% is selected to cast light on its effectiveness in different types of LHTES. Figure 10 showcases the liquid fraction contours of cases with 99% porosity and temperature of 368 K. A major observation deducible from this figure concerns the order of magnitude of temporal stages presented here which tops at 200 s. This is attributed to the substantial effect that porous media impose on the overall performance of all units. The next point of importance is associated with uniformity of respective contours for all configurations. Despite the applied gravitational force, unlike contours presented in the first section, the upper half of the annulus does not differ from the lower half by sizeable margins. This suppression of natural convection, while only 1% of the annulus is designated to porous media, indicates the extended network of connected cells which in turn curbs the upward movement stirred by gravity. This may seem detrimental to the thermal performance of this units, but due this extended connected network, the conductive heat transfer is bolstered by a great extent and compensates for the weakened convective heat transfer.

Figure 11 to 13 represent the temporal alteration of liquid fraction for P1-P12 cases at porosities of 99%, 97% and 95%. Compared to plain cases of A1-A4, the pipe design won the smallest enhancement share of 70%, 86% and 90% at porosities of 99%, 97% and 95%, respectively which are still outstanding improvements. But, the corresponding figures for other configurations are quite similar, registering approximately 90%, 95% and 97% enhancements at the given porosities. This discrepancy between the pipe and other models stems from the fact that natural convection effectiveness is most advantageous in the pipe unit, as proved in prior section, and since the presence of porous media takes an immense toll on natural convection, thus the enhancement provided by metallic foam is less significant.

This current paper examined 2-D computational simulations of various storage units to cast light on their differing responsiveness during the melting operation to multiple heat promoters namely, HTF temperature, natural convection effectiveness and porous media of different porosities. The resultant data on 24 proposed scenarios in 4 configurations of pipe, cylinder (shell), TTHX and multi-tube have been in the form of liquid fraction contours and plots have been presented. From the results and assessment of respective figures and plots, some major observations standing out are given as the following:

TTHX configuration registered the best performance irrespective of condition whereas there was not a single unit registering the worst performance in all conditions.
Shifting the HTF temperature from 368 K to 373 K, the pipe design took the most advantage by decreasing the melting time by almost 26% as opposed to 22% improvement in the remaining three units.
The natural convection effectiveness proved greatly superior in the pipe model as not considering gravity in the simulation led to a staggering slow-down of melting evolution of 236%. On the other hand, omitting the gravity helped the cylinder configuration to take pace in phase-change procedure and resulted in roughly 7% faster melting rate.
Porous media incorporation into the annulus of the 4 units greatly enhanced the functionality of phase-change process by an average of 93% at different porosities. Although the extended network of connected cells curbed the natural convection vortices and upward trends, the added heat transfer surface area in the form of conduction facilitated the melting rate by a sizeable extent.
Installation of metal foam in the pipe model resulted the least improvements at given porosities compared to other units as the suppression of natural convection was mostly affected this configuration.

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No competing interests reported.

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A comprehensive assessment of temperature difference and natural convection on phase transition in different latent heat storage systems accompanied by porous media

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Abstract

1. Introduction

2. Problem Description

2.1 Governing Equations

2.2 Initial & Boundary Conditions

2.3 Grid Independency Assessment

2.4 Validation of the Numerical Analysis

3. Results and Discussions

3.1 HTF Temperature effectiveness across various units

3.2 Natural Convection effectiveness across various units

3.3 Porous Media incorporation effectiveness across various units

4. Conclusion

References

Additional Declarations

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