**2. Methods**

#### *2.1. Method Description*

Figure 2 illustrates the industrial reactors for ethylene carbonate hydrogenation considered and the modeling scheme used in this study. Two types of industrial reactors are modeled, namely, adiabatic and multi-tubular heat-exchange reactors; for the heatexchange reactors, two different coolants are considered: boiling water and heat conduction oil (Figure 2a). The adiabatic reactor is 2 m in diameter and 8 m in length, whereas the heatexchange reactor comprises 1682 tubes of 0.05 m diameter and 8 m length, corresponding to a total capacity of 3 × <sup>10</sup><sup>4</sup> t/a methanol and ethylene glycol. Key geometric parameters and operating conditions are listed in Table 1. The 2D asymmetrical steady-state multi-scale reactor models are built under the following assumptions:

**Figure 2.** Schematic diagram of (**a**) industrial adiabatic and heat-exchange reactors for ethylene carbonate hydrogenation and (**b**) the multi-scale reactor model.

#### **Table 1.** Model parameters.


Details of the multi-scale reactor models, including governing equations, heat and mass transfer correlations, chemical kinetics and model implementation are introduced in the following text.


#### *2.2. Governing Equations*

The fluid computational mass conservation equation in the bed is

$$\frac{\partial(\varepsilon\_{\rm b}\rho\_{\rm f})}{\partial t} + \nabla \cdot (\varepsilon\_{\rm b}\rho\_{\rm f}\sigma) = 0 \tag{1}$$

The momentum Equation (2), the energy Equation (7) and the species transport equation Equation (10) for flow in the catalyst bed are listed below.

Momentum equation:

$$\frac{\partial(\varepsilon\_{\mathsf{b}}\rho\_{\mathsf{l}}\boldsymbol{\sigma})}{\partial t} + \nabla \cdot (\varepsilon\_{\mathsf{b}}\rho\_{\mathsf{l}}\boldsymbol{\sigma}\boldsymbol{\sigma}) = -\varepsilon\_{\mathsf{b}}\nabla p + \nabla \cdot (\varepsilon\_{\mathsf{b}}\boldsymbol{\pi}) + S\_{\mathsf{b}} \tag{2}$$

where *p* is the static pressure and *τ* is the stress tensor. *S*<sup>φ</sup> is the momentum source term for fluid flow in porous media,

$$\mathcal{S}\_{\Phi} = \mathcal{B}\_{\mathbf{f}} - \left(\frac{\varepsilon\_{\mathbf{b}}^{2}\mu}{K}\mathbf{v} + \frac{\varepsilon\_{\mathbf{b}}^{3}\mathcal{C}\_{2}}{2}\rho\_{\mathbf{f}}|\mathbf{v}|\mathbf{v}\right) \tag{3}$$

representing the viscous and inertial drag forces imposed on the fluid flow by the pore walls within the porous media, in which *C*<sup>2</sup> is the inertial loss coefficient.

$$C\_2 = \frac{3.5}{d\_\text{P}^\circ} \frac{(1 - \varepsilon\_\text{b})}{\varepsilon\_\text{b}^{3}} \tag{4}$$

The *k*-*ω* SST turbulence model is used with the turbulence kinetic energy *k* and the specific dissipation rate *ω* obtained from the following equations:

$$\frac{\partial(\rho\_l k)}{\partial t} + \frac{\partial(\rho\_l k u\_i)}{\partial x\_i} = \frac{\partial}{\partial x\_j} \left(\Gamma\_k \frac{\partial k}{\partial x\_j}\right) + G\_k - \Upsilon\_k + S\_k + G\_{k\mathbf{b}} \tag{5}$$

$$\frac{\partial(\rho\_l \omega)}{\partial t} + \frac{\partial(\rho\_l \omega \omega\_i)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_j} \left(\Gamma\_\omega \frac{\partial \omega}{\partial \mathbf{x}\_j}\right) + \mathbf{G}\_\omega - \mathbf{Y}\_\omega + \mathbf{S}\_\omega + \mathbf{G}\_{\omega \mathbf{b}} \tag{6}$$

Energy equation:

$$\begin{split} \frac{\partial}{\partial t} \left( \varepsilon\_{\rm b} \rho\_{\rm f} E\_{\rm f} + (1 - \varepsilon\_{\rm b}) \rho\_{\rm s} E\_{\rm s} \right) + \nabla \cdot \left( \boldsymbol{\sigma} \left( \rho\_{\rm f} E\_{\rm f} + p \right) \right) \\ = S\_{\rm f}^{\rm h} + \nabla \cdot \left[ k\_{\rm c} \nabla T - \left( \sum\_{i} h\_{i} I\_{i} \right) + \left( \boldsymbol{\pi} \cdot \boldsymbol{\nu} \right) \right] \end{split} \tag{7}$$

where *E*<sup>f</sup> is total fluid energy and *E*<sup>s</sup> is total solid medium energy. The energy source *S*<sup>h</sup> f represents the chemical reaction heat.

$$S\_{\rm f}^{\rm h} = -\sum\_{j} \frac{h\_j^0}{M\_{\rm w,j}} R\_j \tag{8}$$

where *h*<sup>0</sup> *<sup>j</sup>* is the enthalpy of formation of species and *Rj* is the volumetric rate of creation of species *j*. The effective heat conduction of bed *k*e is computed in Equation (9).

$$k\_{\mathbf{c}} = \begin{cases} \begin{array}{c} k\_{\mathbf{c}, \text{ax}} = \frac{u\_0 p\_{\mathbf{f}} c\_{p, \mathbf{f}} d\_{\mathbf{f}}^{\mathbf{V}}}{P c\_{\mathbf{f}, \text{ax}}}\\\ k\_{\mathbf{c}, \mathbf{r}} = \frac{k\_{\mathbf{r}}^0}{k\_{\mathbf{f}}} + k\_{\mathbf{f}} \cdot \frac{P c\_{\mathbf{f}}^0}{P c\_{\mathbf{f}, \mathbf{r}}^{\mathbf{V}}} \end{array} \end{cases} \tag{9}$$

Species equation:

$$\frac{\partial}{\partial t}(\rho\_\mathbf{f} \mathbf{y}\_i) + \nabla \cdot (\rho\_\mathbf{f} \mathbf{v} \mathbf{y}\_i) = -\nabla \cdot \mathbf{J}\_i + \mathbf{S}\_i \tag{10}$$

where *Yi* is mass fraction of each species, and *J<sup>i</sup>* is the diffusive flux of species *i* arising from gradients of species concentration and temperature.

$$J\_i = -\rho\_\text{f} D\_\text{e} \sum\_{j=1}^{N-1} \nabla Y\_j - D\_{T,i} \frac{\nabla T}{T} \tag{11}$$

*Si*, the net source of species *i* due to chemical reactions, *Si* is computed as the sum of the reaction rates:

$$S\_{\bar{i}} = M\_{\mathbf{w},i} \sum\_{r=1}^{N\_{\mathbb{R}}} \mathbb{R}\_{i,r} \tag{12}$$

where *M*w,*<sup>i</sup>* is the molecular weight of species *i* and *R*ˆ*i*,*<sup>r</sup>* is the molar rate of creation/destruction of species *i* in reaction *r*. In Equation (11), the calculation of thermal diffusion coefficients *DT*,*<sup>i</sup>* adopts the following empirically based expression [46].

$$D\_{T,i} = -2.59 \times 10^{-7} T^{0.659} \left[ \frac{M\_{\rm w,i}^{0.511} X\_i}{\sum\_{i=1}^{N} M\_{\rm w,i}^{0.511} X\_i} - Y\_i \right] \cdot \left[ \frac{\sum\_{i=1}^{N} M\_{\rm w,i}^{0.511} X\_i}{\sum\_{i=1}^{N} M\_{\rm w,i}^{0.489} X\_i} \right] \tag{13}$$

where *Xi* is mole fraction of species *i*; the effective mass transfer coefficient of bed *D*<sup>e</sup> is computed as:

$$D\_{\mathbf{c}} = \begin{cases} \ D\_{\mathbf{c}, \mathbf{x}\mathbf{x}} = \frac{d\_{\mathbf{p}}^{\mathbf{r}} u\_{0}}{P \epsilon\_{\mathbf{m}, \mathbf{x}}} \\ \qquad D\_{\mathbf{c}, \mathbf{r}} = \frac{d\_{\mathbf{p}}^{\mathbf{r}} u\_{0}}{P \epsilon\_{\mathbf{m}, \mathbf{r}}} \end{cases} \tag{14}$$

#### *2.3. Bed Voidage and Pressure Drop*

When the ratio of tube diameter to the catalyst's volume-equivalent diameter (denoted by *R* in the following paper) is less than 10, the wall effect cannot be ignored [47]. In this work, *R* values for the adiabatic fixed bed (2 m in diameter) and the heat-exchange reactor (single tube 0.05 m in diameter) filled with 5 mm spherical catalyst are 400 and 10, respectively. Therefore, without the wall effect, the bed voidage of the adiabatic reactor is set as constant in the radial direction (Supplementary Materials Table S5). Meanwhile, the fluctuations of bed voidage in the radial direction is considered for the heat-exchange reactors (Table S5) [48–50].

The bed pressure drop is computed by the Ergun equation in Equation (15) [51].

$$\frac{|\Delta P|}{L} = \frac{150}{d\_{\rm P}^{\rm s}} \frac{(1 - \varepsilon\_{\rm b})^2 \mu}{\varepsilon\_{\rm b}^{\rm 3}} u\_0 + \frac{1.75 \rho\_{\rm f}}{d\_{\rm P}^{\rm s}} \frac{(1 - \varepsilon\_{\rm b})}{\varepsilon\_{\rm b}^{\rm 3}} u\_0 \tag{15}$$

#### *2.4. Heat and Mass Transfer in the Catalyst Bed*

The axial effective heat conduction is computed as [52]:

$$k\_{\mathbf{e},\mathbf{ax}} = \frac{u\_0 \rho\_{\mathbf{f}} c\_{p,\mathbf{f}} d\_{\mathbf{f}}^{\mathbf{v}}}{P c\_{\mathbf{h},\mathbf{ax}}} \tag{16}$$

in which the Peclet number for axial heat conduction *Pe*h,ax equals 2. The radial effective heat conduction is computed by:

$$k\_{\rm e,r} = \frac{k\_{\rm r}^0}{k\_{\rm f}} + k\_{\rm f} \cdot \frac{Pe\_{\rm h}^0}{Pe\_{\rm h,r}^{\infty}} \tag{17}$$

in which the molecular Peclet number *Pe*<sup>0</sup> <sup>h</sup> is [53]:

$$Pe\_{\rm h}^{0} = RePr = \frac{\mu\_{0} \rho\_{\rm f} c\_{p,\rm f} d\_{\rm P}^{\rm v}}{k\_{\rm f}} \tag{18}$$

and *Pe*<sup>∞</sup> h,r is [54]:

$$P e\_{\rm h,r}^{\infty} = 8[2 - (1 - \frac{2}{R})^2] \tag{19}$$

The following expression for the effective thermal stagnant conductivity *<sup>k</sup>*<sup>0</sup> r *<sup>k</sup>*<sup>f</sup> is obtained by Equation (20) [55]:

$$\begin{split} \frac{k\_{\mathbf{r}}^{0}}{k\_{\mathbf{f}}} &= \left(1 - \sqrt{1 - \varepsilon\_{\mathbf{b}}}\right) + \frac{2\sqrt{1 - \varepsilon\_{\mathbf{b}}}}{1 - B\kappa^{-1}} \\ &\times \left[\frac{B\left(1 - \kappa^{-1}\right)}{\left(1 - B\kappa^{-1}\right)^{2}} \ln\left(\frac{\kappa}{B}\right) - \frac{B - 1}{1 - B\kappa^{-1}} - \frac{B + 1}{2}\right] \end{split} \tag{20}$$

where *B* = *C*<sup>f</sup> 1−*ε*<sup>b</sup> *ε*b 1.11 ; *C*<sup>f</sup> = 1.25 (sphere), *C*<sup>f</sup> = 2.5 (cylinder).

Under turbulent conditions (*Re* > 100), axial mixing of mass can be approximated as mixing in a cascade of *L*/*d*p ideal mixers [56]. The Peclet number for axial mass dispersion approximately equals 2.

$$P e\_{\text{m,ax}} = \frac{d\_{\text{p}}^{\text{v}} u\_0}{D\_{\text{e,ax}}} = 2 \tag{21}$$

The effective radial mass transfer coefficient of bed *D*e,r is computed as [57]:

$$Pe\_{\mathbf{m,r}} = \frac{d\_{\mathbf{p}}^{\mathbf{v}} \mu\_0}{D\_{\mathbf{e,r}}} = C \left( 1 + \frac{19.4}{R^2} \right) \tag{22}$$

where *<sup>C</sup>* <sup>=</sup> 8.65 *<sup>d</sup>*<sup>v</sup> p *d*a <sup>p</sup> [58].
