*2.6. Energy Balances*

## 2.6.1. Fluidized Bed

For each cell, a thermally fully developed mixture with a constant temperature is assumed. Therein, temperatures of solids and gases are equal; however, the temperature of the wake can differ, applying a vertical heat transfer between the cells.

Enthalpy fluxes for solids and gases are generally defined by mass flows: *H* = *M*·*h*. At the boundary of a cell, the enthalpy *h* of a mass flow of mixtures (e.g., gas inlet/outlet) is calculated as

$$h = \sum\_{\{\}} \mathbf{x}\_{\}^{\mathbf{y}} h\_{\}}(T). \tag{30}$$

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For solid mixtures, Equation (30) is written with index *k*. The temperature dependency of the enthalpy of a gas component *j* or a solid component *k* is calculated using polynomials from the software package FactSage®. In this approach, the enthalpies of the chemical reactions need not be considered additionally. The energy balance for a certain cell *n* is defined as

$$\frac{dH^n}{dt} = 0 = \dot{H}\_d^{n-1} - \dot{H}\_d^n + \dot{H}\_b^{n-1} - \dot{H}\_b^n + \dot{H}\_k^{n-1} - \dot{H}\_k^n + \dot{H}\_{dr}^{n-1} - \dot{H}\_{dr}^n + (1 - a)\dot{H}\_{n\lambda}^n - \dot{H}\_{n1\lambda}^n + \dot{H}\_{in}^n - k\_{\text{FaildB}}nd\_t dh(T^n - T\_\ell) \tag{31}$$

The last term of Equation (31) describes the heat loss through the reactor wall . *Q n L*. It is calculated by using the temperature of the cooling jacket *Tc* = 40 ◦C and, by adapting to experimental pilot plant data, a heat transfer coefficient of *kFluidB* = 12.9 Wm−2K−<sup>1</sup> was found.

2.6.2. Freeboard

> The enthalpy balance for the freeboard section of the reactor can be expressed by

$$\frac{dH\_f^m}{dt} = 0 = \dot{H}\_f^{m-1} - \dot{H}\_f^m - k\_f \pi d\_r dh(T^m - T\_c) + k\_p \frac{6}{d\_P p\_p} \dot{M}\_p^m \left(T\_p^m - T^m\right) \cdot v\_{p\epsilon} \tag{32}$$

where the term with the heat transfer coefficient, *kf*, describes the heat loss . *Q m L*, *f* through the reactor wall in the freeboard section. A value of 3.4 Wm−2K−<sup>1</sup> was selected by fitting the simulated temperature of the experimental temperature profile. According to the reactor design (see Figure 4a), hot solids from the regenerator ( . *MRegOut*) flow into the gasifier. However, the level of the inflow is located in the freeboard and above the surface of the fluidized bed (i.e., at position *HFluidB*). Hence, the particles pass the lower region of the freeboard before they dip in the fluidized bed. In this region, heat transfer

. *Q m p*, *f* between solid particles (*p*) and the gas phase (*f*) of the freeboard occurs. For this, a heat transfer coe fficient, *kp*, with a value of 160.7 Wm−2K−<sup>1</sup> was determined, which is well-aligned with values reported in the literature [32,33]. The temperature, *Tn p* , of the bed material in the region between the inlet and fluidized bed is calculated by solving the energy balance:

$$\frac{dH\_p^n}{dt} = 0 = \dot{M}\_{\text{Re}\,\text{Sout}} \varepsilon\_{p,p}^{n+1} T\_p^{n+1} - \dot{M}\_{\text{P}} \varepsilon\_{p,p}^n T\_p^n - k\_{pf} \frac{6}{d\_p p\_p} \dot{M}\_p \left(T\_p^n - T^n\right) \cdot \upsilon\_p \tag{33}$$

In this case, *T<sup>m</sup>*=*min p* = *TRegOut* and *vp* describes the velocity of a falling particle.

## *2.7. Chemical Reactions*

Table 1 lists all the chemical reactions considered in the model, including equations to calculate the reaction rates. The pyrolysis step (reaction 1) is modelled with a one-step reaction kinetic considering the products: Ash, char, H2O, gases (CO2, CO, CH4, and H2), non-condensable hydrocarbons (simplified as C2H4), and tars (simplified as Naphthalene: C10 H8). For the mass fractions <sup>ω</sup>*j*, experimental data from Fagbemi et al. [34] were used and interpolated to consider a temperature-dependent pyrolysis product composition. The values for the amounts of tar, however, originate from experiments with the 200 kW DFB system [19] and, hence, secondary pyrolysis reaction modelling was not required. With this assumption, catalytic e ffects of CaO on tar conversion are indirectly considered by measured concentrations. Residual char was handled as a mixture of C, H, and O and, according to the char analysis from Fagbemi et al. [34], the composition was also interpolated for di fferent gasification temperatures. It is worth noting that the used elemental analysis of wood pellets (C: 48.99 wt.-%waf, H: 6.97 wt.-%waf, O: 44.04 wt.-%waf) di ffered from the analysis of biomass in Fagbemi et al. [34]. Thus, yields from pyrolysis <sup>ω</sup>*j* needed to be adapted, in order to satisfy the elemental balance. Therefore, a linear equation system for the elements C, H, and O had to be solved. While coe fficients from C10 H8, CH4, C2H4, H2O, and char were fixed, the coe fficients of CO, CO2, and H2 were fitted to close the elemental balance of wood pellets.

Results for the mass fractions <sup>ω</sup>*j* are listed in Table A1. The reaction kinetics of ethene reformation (reaction 6) were adapted to fit the simulated H2 and CO concentrations to experimental values from the 200 kW DFB pilot plant. For the carbonation reaction, sorbent deactivation, which is dependent on the number of calcination–carbonation cycles, was taken into account through the parameter *Xave*. Details of the calculation of *Xave* are described in the subsequent section. For all gasification reactions, it is assumed that they occur only in the emulsion phase, due to the catalytic behavior of CaO and char, which enhance reaction rates compared to those in the gas phases [42].



## *2.8. Sorbent Deactivation*

If limestone is subjected to several calcination–carbonation cycles, its CO2 sorption capacity is reduced, due to sintering phenomena on the particle surface [43]. For limestone, Grasa and Abanades [44] described the decay of the CO2 carrying capacity *XN* as

$$X\_N = \frac{1}{\frac{1}{1 - X\_r} + kN} + X\_r. \tag{34}$$

Equation (34) depends on the number of calcination–carbonation cycles *N* and uses the empirical constants *k* = 0.52 and *Xr* = 0.075. However, in a fluidized bed system, particles have different residence times, which leads to a distribution of the average CO2 carrying capacity. According to the references [45–47], an average carrying capacity *Xave* is calculated by a population balance:

$$X\_{\rm wve} = \sum\_{N=1}^{\infty} \frac{F\_0}{F\_R} \left( 1 - \frac{F\_0}{F\_R} \right)^{N-1} \cdot X\_N \tag{35}$$

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In the current study, Equation (35) was integrated into the model to calculate the average CO2 carrying capacity of the particle system with regard to the carbonation reaction (Table 1, Reaction 4). Therein, *FR* is the molar-based flow of . *MRegOut*, describing the circulation flow of CaO from the regenerator into the gasifier. To compensate mass losses, mostly due to attrition, the reactor inventory was maintained by an input flow of raw limestone. In Equation (35), this input flow *F*0 was considered on a molar basis. As, in practice, the measured material flow of fresh limestone (by dosing units) contains particles which are small enough to be directly discharged, especially when feeding into the regenerator with fluidization velocities up to 5.5 m/s [11], it was assumed that *F*0 only represents the effective material flow that remains longer in the system. Fresh limestone and purge material are not considered in the mass and energy balance calculations, as their flow rates are very low compared to the circulating CaO mass flow. If this should be taken into account, detailed information is needed regarding the size of the particles and their degree of calcination when they are directly discharged from the regenerator due to their hydrodynamic properties.
