2.1.2. Single Reactor Model

The thermodynamic model for a single reactor is based on [20], which is a simplified 0-D model with energy balances on absorber and window surfaces. The main heat fluxes in and from the receiver are shown in Figure 3.

**Figure 3.** Thermodynamic solar reactor model, based on [20] with own adaptations.

The reactor is composed of the absorber, i.e., the metal oxide active material, and of a quartz window. In reality, the absorber is bell-shaped, however, in the model this form is approximated as a half-sphere in order to simplify the calculations. In addition, the absorber is assumed to behave as a black-body.

The available solar power (straight yellow arrow) hits the quartz window and is either reflected, absorbed or transmitted toward the absorber. The absorber reflects towards the window (straight red arrow), and also in this case the radiation is either absorbed, transmitted or reflected by the window. The absorber also loses a share of power through the insulation (curly red arrow). Also the quartz window itself loses heat by radiation and convection to the environment (straight and curly dark blue arrows). During the thermal reduction phase, nitrogen is pre-heated and injected while the absorber temperature is kept constant. The nitrogen flows through the absorber pores with an initial temperature lower than the one of the absorber, and reaches thermal equilibrium with it. The heat absorbed from the absorber to heat up the nitrogen is simply calculated knowing the gas specific heat. As nitrogen flows through the active material, this latter releases oxygen. During the water splitting phase steam is injected instead of nitrogen, while hydrogen is released, but the modelling approach is the same. The details of the calculation of each single term in the heat fluxes can be found in [20].

Nitrogen is obtained by air separation in a nitrogen generator consuming electricity, while the electricity consumption for its pumping in the system is calculated based on a fixed pressure ratio.

A heat exchanger between nitrogen and steam inlet and outlet flows fulfils the overall heat requirements (boiling and overheating). When hot nitrogen exits from the reactor, it is stored in a container with a certain thermal efficiency. This operation is not performed with water and hydrogen exiting the reactor. Whenever heat is required to heat up nitrogen or to vaporize and over heat steam, the required amount is subtracted from the value available in the storage, also in this case considering a certain efficiency to simulate the losses.

The heat balance on the absorber and on the quartz window are performed at each time step by applying a forward Euler method. The energy balance on the absorber is:

$$\frac{dT\_a}{dt} = \frac{\dot{Q}\_{sun,\tau} + 0.5 \cdot \dot{Q}\_{w,rad} - \dot{Q}\_{a-w,rad} - \dot{Q}\_{a-ins,cond} - \dot{Q}\_{a-fl,conv} - \dot{Q}\_{h,rec}}{m\_{abs} \cdot c\_{p,abs}}$$

The energy balance on the window is:

$$\frac{dT\_w}{dt} = \frac{\dot{Q}\_{\text{sun},a} - \dot{Q}\_{w,rad} + \dot{Q}\_{a-w,a} - \dot{Q}\_{w-f l,conv} - \dot{Q}\_{w-env,conv}}{m\_w \cdot c\_{p,w}}$$

2.1.3. Kinetic Models for Nickel-Ferrite and Ceria

The duration of each reaction phase (TR and WS) is determined by the kinetics. For nickel-ferrite a unimolecular decay law has been used as proposed by [19], based on [30]. The non-stoichiometric factor *δ* can be calculated linearly interpolating the results of [19] for the reaction temperature (1400 ◦C for the TR phase) and the oxygen partial pressure (10−<sup>5</sup> bar).

From *δ*, the maximum value of the oxygen moles that can be extracted at equilibrium *ψO*2,*max* can be assessed, considering the mass and the molar mass of the absorber:

$$
\psi\_{O\_2,\text{max}} = \frac{\delta \cdot m\_a}{2 \cdot MM\_a}.
$$

After that, the oxygen release at each time step can be assessed by integration of the kinetic differential equation:

$$\frac{d\psi\_{O\_2}}{dt} = -k\_{r\chi} \cdot \psi\_{O\_2}$$

So it possible to compute the moles of oxygen released up to a certain instant *nO*<sup>2</sup> (*t*) by integration, and to obtain the molar production rate . *nO*<sup>2</sup> (*t*) as the difference of the value of *nO*<sup>2</sup> (*t*) in two consecutive time steps:

$$\dot{m}\_{O\_2}(t) = \frac{n\_{O\_2}(t) - n\_{O\_2}(t-1)}{\delta t} = \frac{\psi\_{O\_2,\max}}{\delta t} \cdot \left(e^{-k\_{\text{reg}} \cdot (t-1)} - e^{-k\_{\text{reg}} \cdot (t)}\right)$$

For the WS phase, a similar procedure applies, but also the steam concentration is accounted for in the differential equation:

$$\frac{d\psi\_{O\_2}}{dt} = -k\_{\text{ws}} \cdot c\_{H\_2O} \cdot (\psi\_{O\_2,\text{max}} - \psi\_{O\_2})$$

The integration follows the same procedure as for the TR phase. The enthalpy of reaction associated to the oxygen release and to the oxygen absorption are considered fixed values and have been taken from [31], while the specific heat capacity of the ferrite have been taken from [32].

For ceria the kinetic calculation is similar, with the only difference that in this case oxygen can be released also during the transition phase between WS and TR. The value of *δ* is given in [33] as a function of the temperature and of the oxygen partial pressure, while the constants *kreg* and *kws* has been interpolated from [34]. The enthalpy of reaction for the regeneration phase is not constant, as suggested by [33], while the enthalpy of oxygen absorption has been considered equal to the nickel-ferrite case, since no data was available in the literature.

#### 2.1.4. Plant Model

Due to the fact that the temperature requirements for the two phases of the cycle are different—i.e., for nickel-ferrite 1400 ◦C and 900 ◦C for TR and WS, respectively—the power needed by each single reactor strongly varies over time, as shown in Figure 4 (left). The power requirement is the highest during the transition from WS to TR phase, when a heat-up takes place. During the TR phase the temperature of the absorber remains constant, while the power requirement falls to an intermediate value. The power requirement of the TR phase is not constant due to the impact of the reaction kinetics on the thermal balance. After that a cool down takes place. During this phase the power requirement falls to zero. The last phase is the WS, when the power requirement slightly increases.

**Figure 4.** Heat requirement of a single nickel-ferrite reactor (**left**), and of a module (**right**) during a complete cycle.

If only one reactor is used, a large amount of the available solar energy (i.e., the difference between the peak power and the current value of the required power) would be wasted, since the solar field would need to be sized on the peak of the reactor power curve. A constant power requirement makes a more efficient use of the available solar power, (red line in Figure 4). This ideal behavior can be at least approached by running several reactors together in a module, each reactor starting is cycle with a given time displacement (Figure 4, right plot). The solar power consumption of a module is then the sum of the power consumption of its single reactors, considering the time displacement of each reactor power curve. The optimal number of reactors per module allows approaching a constant value for the module consumption curve. It is a function of the duration of the heat-up phase, which in turn is a function of the chosen absorber material. Finally, a complete plant consists of a series of modules, and in each hour of the year, the number of activated modules in the plant depends on the currently available sun power. The main results of the technical model are—for each time step—temperature value of absorber and quartz window, heat flows and solar heat requirements for a single reactor as well as for an entire module, oxygen and hydrogen production rates. In the end, the size of the main plant components is evaluated.

#### *2.2. Economic Model*

The results of the technical model are fed into the economic model in order to calculate the levelized cost of hydrogen (LCOH). The LCOH is defined as the ratio between the total annual cost and the amount of produced hydrogen, considering the energy cost for hydrogen pumping in the long-term storage:

*LCOH* <sup>=</sup> *total annual cost mH*2, *annual production*

The total annual cost is calculated as a sum of a series of items. The most important one is the annual instalment [Mio. €/y], which is assumed to be completely covered by the loan, according to:

$$inst\_{ann} = inv \cdot \frac{i \cdot (1 + i)^{(t\_{dcbt} - t\_{construation})}}{(1 + i)^{(t\_{dcbt} - t\_{construation}) - 1}}$$

Another cost item is the metal oxide substitution, as it is assumed that the active material is prone to deterioration and only can withstand a certain number of cycles before being substituted. Annual cost of water, technical nitrogen, nitrogen compression and a general factor for operation and maintenance and insurance are taken into account. The main assumptions used for the calculation of such cost are listed in Table 1.


**Table 1.** Assumption of main assumptions for the economic model (base case).

The total investment *inv* is the sum of the investment for all plant components, i.e., heliostat field, thermal storage for nitrogen, hydrogen underground storage, reactor and solar tower. Some important data and the main cost assumptions for the base case are reported in Table 2. Also the hydrogen efficiency storage is reported, which considers a loss of 10% of the energy content in the gas for its pumping and extraction in the storage [35,36].


**Table 2.** Investment assumptions (base case).

The tower specifications (e.g., tower height as a function of receiver capacity) required for the cost calculation is based on [27]. The total reactor mass is based on internal consideration at DLR, and is supposed to be fixed for simplicity reasons. The volume available for the absorber is also fixed, since it depends on the reactor configuration based on [20]. The absorber mass depends on the density and porosity of the material, however, due to the small fraction of reactor mass in the absorber material, the total mass of the reactor is not a function of it, but it is fixed. The absorber specific cost refers to an eventual industrial scale production, and is the same for the two materials.

### **3. Results and Discussion**
