*2.5. Heat and Mass Transfer of Catalyst Particles*

Due to mild exothermicity of the ethylene carbonate hydrogenation reaction, the temperature difference between the catalyst surface and the gas phase is estimated to be less than 1 K in Equation (23). Consequently, the temperature of catalyst surface and the gas phase is taken as identical. The gas–solid heat transfer coefficient *α*<sup>f</sup> is calculated as [59]:

$$
\Delta T\_{\rm ex} = \frac{Q\_{\rm reaction}}{\alpha\_{\rm f} \cdot \text{S}\_{\rm cat}} \tag{23}
$$

$$Nu = 2 + 1.1 Re^{0.6} \cdot Pr^{\frac{1}{5}} \tag{24}$$

$$N\mu = \frac{\mathfrak{a}\_{\rm f} d\_{\rm P}^{\rm V}}{k\_{\rm f}} \tag{25}$$

For heat transfer inside the catalyst particle, the temperature gradient within the catalyst particle is related to the heat of reaction *Q*reaction and the effective thermal conductivity of the catalyst particle *λ*eff,cat (Equation (26)) and can be calculated by Equation (27) [60,61].

$$
\Delta T\_{\rm in} = \frac{D\_{\rm eff,EC} \cdot (C\_{\rm EC,s} - C\_{\rm EC,center}) \cdot (-\Delta\_{\rm r}H)}{\lambda\_{eff,cat}} \tag{26}
$$

$$
\lambda\_{\rm eff,cat} = k\_{\rm f} \left( \frac{k\_{\rm s}}{k\_{\rm f}} \right)^{1 - \varepsilon\_{\rm cat}} \tag{27}
$$

Under the reaction conditions considered in this article, Δ*T*in is estimated to be less than 1 K. Therefore, the catalyst particle is considered isothermal.

The mass transfer between the catalyst surface and the gas phase is described by a mass transport coefficient *k*g:

$$k\_{\text{g}} \cdot a \cdot \left(\text{C}\_{\text{EC,gas}} - \text{C}\_{\text{EC,s}}\right) = R\_{\text{EC,s}} \tag{28}$$

where *R*EC,s is the effective consumption rate of EC at the particle surface. kg is related to the catalyst shape, the Reynolds number and the Schmidt number, as shown in Equations (29)–(31) [62].

$$Sh = 2 + 0.6 Re^{0.5} Sc^{\frac{1}{3}} \text{ (sphere)} \tag{29}$$

$$Sh = 0.61 Re^{0.5} Sc^{\frac{1}{3}} \text{ (cylinder)}\tag{30}$$

$$Sh = \frac{k\_{\rm g} d\_{\rm P}^{\rm v}}{D\_{\rm EC,m}} \tag{31}$$

The effect of mass transfer inside the catalyst particles, namely, internal mass transfer, is accounted for with the generalized Thiele modulus approach, which is applicable to a broad range of rate equations [63]. The generalized Thiele modulus with respect to EC consumption rate (*φ*gen,EC) is expressed as

$$\phi\_{\text{gen,EC}} = \frac{V\_{\text{cat}}}{S\_{\text{cat}}} \sqrt{\frac{k\_{\text{v}}}{D\_{\text{eff,EC}}} \cdot \frac{n+1}{2} \cdot C\_{\text{EC,s}}{2}^{n-1}} \tag{32}$$

The effectiveness factor for internal mass transfer is then computed as:

$$\eta\_{\rm EC} = \frac{\tanh\left(\phi\_{\rm gen, EC}\right)}{\phi\_{\rm gen, EC}}\tag{33}$$

Note that Equations (28)–(31) must be solved simultaneously in an iterative manner because the two sides in Equation (28) are related by the EC concentration at catalyst particle surface, which is unknown in advance.
