**2. Theory**

## *2.1. Kinetic Models*

Generally, external mass transport from the gas phase to the outer particle surface, the intra-particle diffusion and/or the chemical reaction at the char surface determine the rate of char–gas reactions, depending on temperature and particle properties. For temperatures below 1000 ◦C and particles in the order of magnitude of 0.1 mm, the reaction rate is controlled by the chemical reaction [9].

Equation (1) is a general expression for the chemical reaction rate, given by Lu et al. [10].

$$\frac{dX}{dt} = k(T, \overline{p}\_{\mathbb{S}}) \, f(X) \tag{1}$$

Here, *k* is the apparent reaction rate depending on temperature *T* and the partial pressures, of the gasifying agents and gas phase products, described by the vector *pg*, according to a reaction model. *f*(*X*) describes the change in physical or chemical properties of the char with ongoing char conversion, *X*, according to a conversion model.

A simple representation of the apparent reaction rate during gasification is the Arrhenius reaction model, which only considers the partial pressure *pg* of the gasifying agen<sup>t</sup> and the temperature.

$$k\_{Arr}(T, p\_\mathcal{g}) = p\_\mathcal{g} \ k\_0 \ e^{-\frac{E\_\mathcal{g}}{kT}} \tag{2}$$

The kinetic parameters for this model are the pre-exponential factor *k*0 and the activation energy *Ea*. The inhibitive influence of the product, which has been observed in several studies [11,12], is considered when applying the Langmuir–Hinshelwood reaction model (L–H model) to the gasification mechanism [13]. Here, the rate-determining step is the formation of occupied sites on the carbon surface. Equations (3) and (4) apply to the CO2 and H2O gasification respectively.

$$k\_{\rm LH,CO\_2}(T,\overline{p}\_{\mathcal{J}}) = \frac{p\_{CO\_2}k\_1\ e^{-\frac{E\_{g,1}}{RT}}}{1 + p\_{CO}k\_2\ e^{-\frac{E\_{g,2}}{RT}} + p\_{CO\_2}k\_3\ e^{-\frac{E\_{g,3}}{RT}}}\tag{3}$$

$$k\_{LH,H\_2O}(T,\overline{p}\_\mathcal{J}) = \frac{p\_{H\_2O} \, k\_1 \, e^{-\frac{E\_{g,1}}{RT}}}{1 + p\_{H\_2} \, k\_2 \, e^{-\frac{E\_{g,2}}{RT}} + p\_{H\_2O} \, k\_3 \, e^{-\frac{E\_{g,3}}{RT}}}\tag{4}$$

Here, the kinetic parameters are the three pre-exponential factors *k*1, *k*2 and *k*3 and the three activation energies *Ea*,1, *Ea*,<sup>2</sup> and *Ea*,3. They have to be determined separately for steam and CO2 gasification.

In this work, four conversion models for the change in char properties with progressing char conversion are investigated with respect to their applicability for the fuel sample: the volumetric model (VM), the grain model (GM), the random pore model (RPM), which are the most common models used in gasification kinetics [14], and the Johnson model (JM). According to Equation (5), the VM assumes a decreasing reaction surface proportional to the remaining volume or mass of the particle.

$$\frac{dX}{dt} = k(T, \overline{p}\_{\mathcal{S}}) \begin{pmatrix} 1 - X \end{pmatrix} \tag{5}$$

In Equation (6), the GM or shrinking core model considers the particles as an assembly of nonporous spheres with constant density and decreasing diameter [15]. The reaction only takes place at the surface.

$$\frac{dX}{dt} = k \{ T, \overline{p}\_{\,\,\,\beta} \, (1 - X)^{\frac{2}{3}} \} \tag{6}$$

The RPM was proposed as an semi-empirical model by Bhatia and Perlmutter [16]. It considers arbitrary pore size distributions in the reacting solid and is able to predict a first increasing and then decreasing reaction rate due to the growth and later the coalescence of pores. In the according equation, Equation (7), ψ is a parameter related to the pore structure of the unreacted sample.

$$\frac{dX}{dt} = k \{ T, \overline{p}\_{\overline{\mathbb{S}}} \} \left( 1 - X \right) \left( 1 - \psi \ln(1 - X) \right)^{0.5} \tag{7}$$

The JM is another semi-empirical approach by Johnson [17].

$$\frac{dX}{dt} = k \{ T, \overline{p}\_{\mathcal{S}} \} \left( 1 - X \right)^{\frac{2}{3}} e^{\alpha X^2} \tag{8}$$

In Equation (8), the term (1 − *X*) 2 3 is proportional to the e ffective surface area, as in the shrinking core model, and the term *e*α*X*<sup>2</sup> represents the relative reactivity of the e ffective surface area, which decreases with increasing conversion levels.
