*3.2. Plant Modeling*

A few simplifications had to be made in the thermodynamic modeling of the SA due to lack of better data. While there are thermodynamic models and simulations available for the prediction of the hydrate–liquid–vapor curve (see the compilation by Khan et al. [44]), accurate predictions of enthalpy changes due to combinations of both phase transition and change of state are still missing. Therefore, the specific enthalpy change Δ*h*tr of phase transition (formation and dissociation) is modeled using the available data of the hydrate–liquid–vapor equilibrium (cf. Figure 1) and the Clapeyron equation

$$
\Delta h\_{\rm tr} = T \Delta v (\mathbf{d} \, p/\mathbf{d} \, T). \tag{1}
$$

The exact procedure is described in great detail by Anderson [45,46]. The change of enthalpy caused by a change of state without phase transition can be described by

$$\mathbf{d}h = \left(\frac{\partial h}{\partial T}\right)\_p \mathbf{d}T + \left(\frac{\partial h}{\partial p}\right)\_T \mathbf{d}p = c\_p \, \mathbf{d}T + \left[v - T\left(\frac{\partial v}{\partial T}\right)\_p\right] \mathbf{d}p.\tag{2}$$

Both the variation of *cp* [29] as well as the thermal expansivity [47] are small over the range of temperatures considered. Further, SA is nearly incompressible [48]. Consequently, Equation (2) simplifies to

$$\mathbf{dd} \approx \mathbf{c}\_p \, \mathbf{d}T + \boldsymbol{\upsilon} \, \mathbf{d}p,\tag{3}$$

where both *cp* = 2.1 kJ/kg/K [29] and *v* = 1.07 cm3/g [49] are assumed to be constant for all conditions considered.

The modeling of air and water does not require such simplifications and is done using the Python wrapper to the software CoolProp [50], an open-source database for fluid and humid air properties. The discrepancy between the simplifications necessary in the modeling of SA and the accurate modeling of water and air naturally results in uncertainties with respect to both the thermodynamic states and the energy balances. In view of other unknowns of much larger magnitude (e.g., the degree of SP, losses due to technical challenges in the handling of SA, etc.) these can be neglected, though.

The modeling of the plant components is done using enthalpy balances according to the parameters listed in Table 1.

Except for the storage units, heat losses are generally neglected. Compressors, pumps, and turbines are described via the isentropic efficiencies *η*comp, *η*pump, and *η*turb, respectively. For ease of comparison, the isentropic efficiencies are chosen in agreement with those used for the modeling of an LAES plant [13]. The expansion valve is assumed to be isenthalpic. The refrigeration machines and heat pumps are calculated with fixed coefficients of performance *β*rm and *β*hp, respectively. Heat exchangers are modeled isobarically with fixed minimum temperature differences Δ*T*sensible and Δ*T*latent for the sensible and the

latent heat transfer, respectively. The heat losses of the sensible and the latent heat storage are modeled using the storage efficiencies *η*sensible and *η*latent. *h*latent and *c*sensible denote the specific phase change enthalpy for the latent heat storage and the specific heat capacity for the sensible heat storage, respectively. The storage efficiencies and volumetric capacities are loosely calibrated to gravel beds for the sensible heat storage and to paraffin for the latent heat storage (see ref. [2] for typical values). In general, a pronounced degree of SP for the storage of SA is assumed, but some "boil-off" losses due to an intake of heat cannot be avoided. These losses are modeled by the hydrate storage efficiency *η*hs. The hydration number efficiency *η*hn allows deviations from the stoichiometric hydration number to be accounted for and reduces the air storage capacity of the SA. The void fraction *η*vs is used to model the fraction of air-filled void space in the SA pellet with respect to its volume.

**Table 1.** Plant parameters of reference case.


In the simulations of SAES plant operation, all parameters listed in Table 1 can be selected within physically acceptable ranges, while all other states are derived from the enthalpy balances as described above. Typically, efficiencies are fixed before the start of the simulation and the input parameters environment temperature *T*env, formation temperature *T*form, dissociation temperature *T*diss, and SA storage temperature *T*storage are varied to find optimal conditions of operation.
