Constant-volume vapor-liquid equilibrium for thermal energy storage: Generalized analysis of pure fluids

Thermal energy storage is of great interest both for the industrial world and for the district heating and cooling sector. Available technologies present drawbacks that reduce the margin of application, such as low energy density, limited temperature range of work, and investment costs. Phase transition is one of the main phenomena that can be exploited for thermal energy storage because of its naturally high energy density. Constant-volume vapor-liquid transition shows higher flexibility and increased heat transfer properties with respect to available technologies. This work presents a description of the behavior of these types of systems. The analysis is carried out through a generalized approach using the Corresponding State Principle. Variation of internal energy as a function of temperature over a fixed range is calculated at constant volume at different values of specific volume. It is shown that, for lower specific volumes, larger temperature ranges of work can be achieved without occurring in the steep pressure increase typically given by the expansion of liquid. Maximum operating temperature range is increased by up to 20% of the critical temperature with minimal energy loss. In optimal subsets of these ranges of temperature, the energy storage capacity of vapor-liquid systems increases at lower volumes, with energy storage capacity increasing to up to 40% with a 50% increase of the reduced volume. This is especially valid for more complex fluids, which are more interesting for these applications because of their higher heat capacity.


INTRODUCTION
The development of energy saving technologies is a key factor for the deployment of sustainable processes. Thermal energy storage systems have been widely investigated because of their role in primary energy saving [1]. The interest for these technologies arises from a number of reasons, such as the need to cover the mismatch between the energy supply and the energy demand or the possibility to exploit a cyclic thermal load that could be present in a process [2].
The main thermal energy storage mechanisms are sensible heat storage, latent heat storage and thermochemical heat storage [3]. Latent thermal energy storage systems are those in which the latent heat released or absorbed by the phase change of a storage medium is exploited to discharge or charge the stored energy in the medium itself. Solid-liquid phase change materials have been developed already, with a wide range of materials being available [4]. However, the applications of these technologies are limited by the low conductivity of these solid-liquid phase change materials, which results in poor heat exchange performances [5,6]. On the other hand, vapor-liquid phase change is expected to show much better heat exchange performances with respect to solid-liquid phase change.
Scientific literature shows scarce research over these types of systems. Vapor-liquid phase transition has not been considered for two main reasons. The analysis of Sharma et al. [4] shows that although on mass basis the vapor-liquid latent heat is higher than the solid-liquid latent heat, on volume basis this property is greatly reduced by the much lower density of the vapor phase. In his work, Abhat [7] states that the necessity of pressure-controlled vessels that store a two-phase system rules out the exploitation of this transition because of the increased practical complexity.
This work quantifies the performances of pure fluid vapor-liquid systems in closed and constant volumes from an energy storage point of view. Closed and constant volumes are chosen because they represent best the envisioned application for these energy storage systems. Internal energy of the two-phase volumes as well as their energy storage capacities are evaluated at different reduced densities. Moreover, the variation of internal energy is considered in order to evaluate the change in storage capacity with temperature at different reduced densities.
The conceptual approach of this work is to consider the behavior of pure fluids at different specific densities in vaporliquid equilibrium at constant volume. This volume can be schematized as a cylinder filled with a two-phase pure fluid. The free surface level identifies the vapor fraction χ of the two-phase systems, and thus the overall density ρ. A schematic of the system is given in Figure 1.
In this work, thermodynamic properties are usually on volume basis and not mass basis because they represent better the heat storage capacity of closed and constant volumes. This means that whenever internal energy specific to volume is mentioned, it is referred to as u � (measured for instance in J/m 3 ).
In order to perform a generalized fluid-independent analysis, the three-parameter Corresponding State Principle is used [8]. The principle states that properties of fluids only depend on reduced thermodynamic coordinates and complexity of the fluids themselves. This allows evaluation of the energy stored by different types of fluids in two-phase vapor-liquid systems over a fixed reduced temperature range at the same overall reduced specific volumes. This leads to a good generalization of the results obtained in this work for most fluids.

FLUID MODEL
This section gives an overview of the formulation of the Corresponding State Principle that is used in this work to calculate the fluid properties. The formulation of the principle is the Lee-Kesler one [9,10], in which the acentric factor is used as a measure of the molecular complexity of the fluid. This approach consists in the calculation of the properties of a simple and a reference complex fluid at the requested reduced temperature and pressure. The difference between reference complex fluid property and the simple fluid property is divided by the acentric factor of the reference fluid ω r , which in the Lee-Kesler formulation is n-octane, and then multiplied by the acentric factor of the actual fluid of interest ω. The scaled difference is added to the calculated simple reference fluid property. Mixing rules for critical properties are not needed because focus is on pure fluids. Mathematically, the compressibility factor Z is where Z 0 and Z 1 are two different functions in terms of reduced temperature and pressure. These two functions represent the compressibility factor of a simple fluid, Z 0 , and the difference between compressibility factor of the reference complex fluid and the simple fluid, Z 1 . The Lee-Kesler Equation of State needs to be expressed in temperature and volume to be able to describe the stated problem. The Z 0 and Z 1 functions are not per se functions of the actual specific volume v, which is only included in the mathematical expression through the compressibility factor definition. This means that, even though the function has the same mathematical form of a Benedict-Webb-Rubin-Starling Equation of State and could be solved with similar algorithms [11,12], it cannot be solved directly for volume, because it is not explicit in it. It rather needs a double numerical method, one for the solution of each function Z 0 and Z 1 in temperature and pressure and one for the solution of the overall system in temperature and volume. The factors of the solution Z, i.e. Z 0 and Z 1 , are used to calculate the residual properties of the fluids, i.e. the difference between the value of the property π and the value of the same property in a fictitious ideal gas state π ig . In the Lee-Kesler approach, any thermodynamic residual π res can be written with the same formulation as for Z: In this case, internal energy residual on mass basis is calculated from the enthalpy residual, which is a relatively simple function of Z 0 and Z 1 . From the definitions of residual property and enthalpy it can be found that (4) Δh = Δh ig + Δh res = Δu ig + Δu res + pv = Δh ig -R g T + Δu res + pv (5) Δu res = Δh res -pv + R g T (6) Vapor-liquid equilibrium needs to be accounted for in the calculation of phase properties for two-phase systems. For pure fluids saturation pressure is easily calculated. Since each function Z 0 and Z 1 represent the volumetric behavior of reference fluids in reduced temperature and pressure, they show a sharp variation around specific values of the reduced pressure. This information combined leads to a simple formulation of the saturation pressure, which is provided by Lee and Kesler themselves [9]. Once the saturation pressure is calculated, saturated liquid and saturated vapor compressibility factors can be obtained with a simpler temperature-pressure formulation of the Equation of State. For pure fluids, this simplifies the calculation of the vapor fraction χ greatly. This is because the dew and bubble pressures are equal, and the two phases have same unity composition in the saturated phases.
To calculate the properties of the overall vapor-liquid system, calculation of residual properties needs to be done twice, once for the Z 0v and Z 1v pair to calculate the saturated vapor compressibility factor Z v , and once for the Z 0l and Z 1l pair to calculate the saturated liquid compressibility factor Z l . Once the vapor fraction and the phase residual properties are calculated, overall properties can be obtained through the leverage rule. The internal energy residual function can be thus evaluated at various temperatures and fixed specific volume.
The Lee-Kesler Equation of State does not present a way to calculate the ideal gas properties, but only provides the residual properties. The calculation of the ideal gas internal energy is a widely studied topic [13], and is usually treated with polynomial approximations that are fluid-specific. In order to maintain the general approach of the Lee-Kesler Equation of State, the assumption of constant specific heat is done. This leads to an important simplification while keeping a generalized approach, accounting for different behavior of different fluids in the ideal gas state.

GENERALIZED STUDY
This section shows the adopted procedure for generalizing the proposed analysis and the considered thermodynamic conditions. According to the generalized Lee-Kesler Equation of State, fluid properties are only dependent on the reduced coordinates of its state and its molecular complexity. Classes of fluids can be defined based on their molecular complexity to study the change of behavior of these fluids. Different values of the acentric factor ω can be assigned to different hypothetical fluids, representing the various complexity classes they could have. Given that n-octane is used as a reference complex fluid, with ω r equal to 0.398, an acentric factor of 0.4 is chosen to refer to complex fluids, 0 to simple fluids, and 0.2 to fluids with intermediate behavior. Some reference fluids are used to test the validity of this approach, as listed in Table 1. These fluids are chosen to cover as uniformly as possible the reasonable ω spectrum.
Fluid complexity has a major impact on the ideal gas behavior of the fluids. The same classes of fluids are used to define the ideal gas specific heat c v ig for continuity of results. The classes of fluids defined in Table 1 overlap the classical monoatomic-linear-polyatomic molecule classification that is used to account for the effect of molecular degrees of freedom Pure fluids are tested in the range of v r between 0.3 and 500 as well as of T r between 0.4 and 0.95. This temperature range allows coverage of a significant portion between the triple point temperature and the critical temperature for all fluids. Going below the triple point temperature or above the critical temperature would be meaningless for this work because vapor-liquid equilibrium does not occur outside this window. The volume range is chosen in order to allow studying the effect of vapor liquid equilibrium in the whole spectrum. The value of 0.3 for reduced volume is considered to have a comparison with a one-phase liquid fluid, which will not undergo phase change.

RESULTS
This section provides the overall results obtained by this analysis. An analysis of the countercondensation phenomenon is given. An energy storage analysis is then given, first in the extended volume range of interest and then in the more specific area of overall reduced volume lower than the critical volume.

Phenomenon description and countercondensation effect
Constant-volume processes are represented well in the volume-temperature diagram, as in Figure 2. On this diagram there are two distinct areas: one at an overall specific volume lower than the critical one (v<v c ), on the left, and one at higher (v>v c ), on the right. These two regions show different phase change behavior. Figure 2 depicts the processes at reduced volumes of 0.45 and 45. The first crosses the saturated liquid while the second the saturated vapor line. Figure 3 shows the behavior of vapor fraction as a function of temperature for the two cases. In volume-constant processes higher than critical, increasing the temperature leads to an increase in the vapor fraction and, eventually, to a complete vapor phase. At lower than critical, increasing the temperature leads to an increase of the vapor fraction at first, but close to saturation to a decrease in vapor fraction. The phenomenon for which an increase in temperature eventually leads to a decrease of the vapor fraction will be called countercondensation in this work. It is the reason for which when saturation temperature is eventually reached in this process, the system behaves as pressurized liquid tanks at temperatures above their maximum operating temperature. While on the right side of the diagram pressure in the two phase grows exponentially with temperature and then grows more or less linearly after saturation, on the left side of the diagram it keeps growing steeply. After the saturation curve is crossed, systems that go through countercondensation experience an increase in pressure that is up to two orders of magnitude than the ones that do not. This is because for liquids, the higher the temperature, the stronger the pressure energy needs to be in order to contain the thermal expansion forces and keep specific volume constant [14].

Extended volume range analysis
This section focuses on the whole reduced volume range, from 0.3 to 500, for an intermediate complexity fluid by three diagrams. The v r -T r diagram shows the different density levels of the analysis, defining the target range of temperature and reduced volume. The T r -Δu � r diagram shows the thermal energy storage capacity on a volume basis, indicating how much thermal energy can be stored over the temperature range. Lastly, the T r -∂u � r /∂T r shows the local heat capacity at a specific temperature, indicating which system stores better the heat at a given temperature. Figure 5 shows the various processes on the first type of diagrams, with six reduced density levels. Figure 6 shows how much energy is stored for each of those processes at each of those different levels of reduced volumes. Increasing the specific volume does not improve heat storage capacity. A sharp decrease can be seen between the first two curves at v r equal to 0.3 and 1 because, although the mass specific storage capacity does not vary dramatically (it even increases, as can be seen in Figure 6), density variation has a predominant effect on the heat storage capacity.

Analysis at overall volume below critical
The results from the extended volume range indicate that systems at the right of the critical volume v c , are not of interest. This result is not limited to the case of intermediate complexity fluids, but can be extended to any fluid. Hence, this section focuses on the values of reduced volume only between 0.30 and 0.90. Approaching closely the critical volume v c would be useless because the conditions of the fluid would get too close to the critical point and the margin for exploitation of the phase change would be limited, since the two phases get more and more similar. Figure 7 shows the v r -T r diagram for all tests run in this range. The diagrams are fairly similar to each other, since the reduced coordinates are the same. A change in the slope of the saturated liquid curve can be seen, where for more complex fluids the saturation temperature goes higher. This leads to a change in the point of countercondensation, in particular moving it at higher reduced temperatures for more complex fluids.  Figure 8 shows the internal energy storage capacity on volume basis as a function of temperature for the three hypothetical fluids. A discontinuity corresponding to countercondensation can be seen for all curves that cross the saturation curve in Figure 7, e.g. the curve with reduced volume equal to 0.45 for simple fluids has a discontinuity at a temperature that is at 20% of the critical temperature lower than the one at reduced volume equal to 0.60. This discontinuity leads to a sharp change in the slope of the curve, which means less energy stored per unit temperature increment. Figure 8 also shows how the curves at higher reduced volume start showing higher energy storage capacity with respect to the purely liquid condition, which is the one at reduced volume equal to 0.30. This is despite having up to 50% higher specific volume, as in the case of complex fluids, where the curve at reduced volume equal to 0.45 shows up to 70% more energy stored as the curve at reduced volume equal to 0.30, which is the pure liquid.  Figure 9 shows the local variation of internal energy as a function of reduced temperature, with additional information on the pressure of the system itself through a logarithmic colormap. The pressure change after countercondensation moves to higher temperatures for more complex fluids. Internal energy shows the same pattern as in Figure 8, with low reduced volume systems having up to twice the capacity than the pure liquid systems at high complexity fluids for specific reduced temperatures. Lower volume systems show lower pressures at same temperature, which means that some energy storage capacity could be sacrificed to have wider operating ranges of temperature regardless of the complexity of the fluid itself. Figure 9. Variation of reduced internal energy on volume basis ∂u � r /∂T r as a function of reduced temperature T r for hypothetical fluids, with acentric factor ω equal to 0 (top), equal to 0.2 (middle) and equal to 0.4 (bottom) with reduced volume v r from 0.3 to 0.9 and with colorbar for reduced pressure P r .