*3.1. Reactions of CO2 in Sodium Hydroxide Solution*

When CO2 is absorbed into the NaOH solution, the reactions between CO2 with NaOH take place. These reactions can be expressed as follows [20]:

$$\rm{NaCO\_2 + NaOH} \leftrightarrow \rm{NaHCO\_3} \tag{1}$$

$$\text{NaHCO}\_3 + \text{NaOH} \leftrightarrow \text{Na}\_2\text{CO}\_3 + \text{H}\_2\text{O} \tag{2}$$

The rate of the above reactions can be calculated by the following second-order reaction rate equation [21]:

$$r\_{\rm CO\_2} = k\_2 C\_{\rm NaOH} C\_{\rm CO\_2} \tag{3}$$

The reaction can be regarded as a pseudo-first-order one when in Equation (4) is satisfied [20,22].

$$\sqrt{1 + \frac{D\_{\rm CO\_2} k\_2 \rm C\_{NaOH}}{k\_{\rm L}^2}} - 1 \ll \frac{\rm C\_{NaOH}}{2\rm C\_0} \tag{4}$$

where *D*CO2 is the diffusion coefficient of CO2 in NaOH solution.

In this study, in Equation (4) held because the left-side of in Equation (4) was over 100 times smaller than the right-side. Therefore, the rate constant of the pseudo-first-order reaction can be written as:

$$k\_{\rm app} = k\_{\rm 2} \text{C\_{NaOH}} \tag{5}$$

#### *3.2. Gas and Liquid Flow in the RZB Rotor*

Gas and liquid flows in the RZB rotor are presented in Figure 3. Gas flows inwards from the periphery to the center of the RZB rotor, and the gas velocity can be divided into tangential component (*v*θ), radial component (*v*r), and axial component (*v*z) in this process. The ratio of the tangential velocity to the total velocity is 81.6% to 99.3% [23], suggesting that gas mainly moves in a spiral between the rotating and static baffles.

**Figure 3.** Side view of gas and liquid flow in RZB rotor: (**1**) static disk; (**2**) liquid inlet; (**3**) static baffle; (**4**) stagnant filmy liquid; (**5**) perforations; (**6**) gas stream; (**7**) shaft; (**8**) rotating disk; (**9**) flying filmy liquid; (**10**) turbulent filmy liquid; (**11**) flying fine droplets; (**12**) rotating baffle; (**13**) filmy liquid climbing up on the internal surface of rotating baffle; (**14**) filmy liquid on the rotating disk. (Red and blue symbols represent gas and liquid flows, respectively.).

The liquid flow moves outwards through the zigzag passage in the rotor. At the center of the rotating disk, liquid flows tangentially outwards under the action of centrifugal force until it reaches the internal surface of the rotating baffle and then climbs up to the perforated area. The liquid is dispersed into fine droplets when passing through the perforations in the upper section of the rotating baffle and flies along a tangential direction to the static baffle. The fine droplets are captured by the static baffle to form a filmy liquid, which flows spirally downwards on the internal surface of the static baffle due to gravity and tangential movement of the liquid. The filmy liquid departs tangentially from the static baffle as a sheet and reaches the next rotating baffle without falling onto the rotating disk because the tangential liquid velocity is much larger than the axial liquid velocity caused by gravity. The liquid is then captured by the next rotating baffle and repeats the above process in the zigzag passage until leaving the RZB rotor [10].

In terms of the above gas and liquid flow characteristics in the RZB rotor, the gas– liquid contact area can be divided into three different zones, as shown in Figure 3, as zones I, II, and III, and all of them make contributions to the mass transfer [3]. In zone I, fine droplets contact with gas in a spiral movement in the space formed by the rotating and static baffles, contributing to the gas–liquid mass transfer. In zone II, mass transfer is scarcely attributed to contact between gas and the stagnant filmy liquid formed by droplets on the upper section of the static baffle, but mainly results from the surface renewal caused by droplets continually impinging on the turbulent filmy liquid on the lower section of the static baffle [2]. In zone III, the liquid exists as an intact sheet when there is no gas flow, but some fine droplets may result from rupture of the sheet when gas flows through the passage. The contact of the flying liquid and gas in zone III also contributes to mass transfer. Hence, the rotor of the RZB can be regarded as a series of wetted-wall columns with the centrifugal field [3].

It has been reported that mass transfer in zone II contributes most to the overall mass transfer because the continued horizontal impingement by liquid jets on the turbulent filmy liquid on the static baffles brings about a great surface renewal rate. Mass transfer in zones I and III is at least one order of magnitude less than that in zone II because the renewal time of the liquid surface in zones I and III is longer than that in zone II according to the penetration theory [2].

#### *3.3. Model Development*

According to the flow characteristics of gas and liquid in the RZB rotor, model assumptions for CO2 absorption into the NaOH solution in the RZB were made as follows:


For the chemical absorption of CO2 into NaOH solution, CO2 is absorbed from the gas phase to the liquid phase via a gas–liquid interface. During this process, the gas–liquid mass transfer depends both on the liquid-phase and gas-phase mass transfer. Therefore, the gas–liquid effective interfacial area (*a*), liquid-side mass-transfer coefficient (*k*L), and gas-side mass-transfer coefficient (*k*G) should be considered for calculating *K*G*a*.

The space in the rotor of the RZB was divided into certain annular regions for model development (Figure 4). Nine rotating baffles (1, 2, ... , *i*a, *i*<sup>a</sup> + 1, ... , 9) and ten static baffles (0 , 1 , ... , *i*b, *i*<sup>b</sup> + 1, ... , 9 ) were radially alternately arranged. Region *i* was the annular region between the rotating baffle *i*<sup>a</sup> of radius *r*a,*<sup>i</sup>* and static baffle *i*<sup>b</sup> of radius *r*b,*i*, and region *i*' was the annular region between the static baffle *i*<sup>b</sup> of radius *r*b,*<sup>i</sup>* and rotating baffle *i*<sup>a</sup> + 1 of radius *r*a,*i*+1. Mass transfer zones I and II were located in region *i*, while zone III was in region *i*'.

**Figure 4.** Top view of liquid and gas flows in the rotor of RZB.

In zone I of region *i*, droplets left the rotating baffle *i*<sup>a</sup> of radius *r*a,*<sup>i</sup>* with a tangential velocity *u*a,*<sup>i</sup>* along the tangential direction to the static baffle *i*b. The lifetime of droplets in zone I *t*I,*<sup>i</sup>* is written as:

$$t\_{\mathbf{I},i} = \frac{\sqrt{r\_{\mathbf{b},i}^2 - r\_{\mathbf{a},i}^2}}{\mu\_{\mathbf{a},i}} \tag{6}$$

$$
\mu\_{\mathfrak{a},i} = \omega r\_{\mathfrak{a},i} \tag{7}
$$

where *ω* represents the angular velocity of the RZB rotor.

The liquid holdup of droplets *ε*I,*<sup>i</sup>* in this area can be calculated by the following equation:

$$\varepsilon\_{\rm I,i} = \frac{Q\_{\rm L} t\_{\rm I,i}}{\pi \left(r\_{\rm b,i}^2 - r\_{\rm a,i}^2\right) z} = \frac{Q\_{\rm L}}{\pi \omega \varepsilon r\_{\rm a,i} \sqrt{r\_{\rm b,i}^2 - r\_{\rm a,i}^2}}\tag{8}$$

where *Q*<sup>L</sup> is the liquid volumetric flow rate in the RZB rotor and *z* is the axial depth of the rotor.

The correlation of the effective mass transfer area of droplets *A*I,*<sup>i</sup>* in zone I of region *i* is as follows [16]:

$$A\_{\rm I,i} = a\_{\rm I,i} \pi \left(r\_{\rm b,i}^2 - r\_{\rm a,i}^2\right) z = \frac{6\varepsilon\_{\rm I,i}}{d\_i} \pi \left(r\_{\rm b,i}^2 - r\_{\rm a,i}^2\right) z \tag{9}$$

where *di* is the average diameter of droplets in zone I of region *i*.

In zone II of region *i*, the surface area of the turbulent filmy liquid is equal to the effective mass transfer area in this zone, which can be obtained by the following equation:

$$A\_{\rm II,i} = 2\pi r\_{\rm b,i} A\_2 h\_{\rm II} \tag{10}$$

where *A*<sup>2</sup> is a turbulent coefficient used to correct the variation of surface area of the filmy liquid caused by the impingement of droplets (with a value of 2.19) [25] and *h*II represents the axial length of the turbulent filmy liquid.

In zone III of region *i*', the liquid remained in the form of an intact sheet between the static and rotating baffles according to assumption (4). The filmy liquid tangentially left the static baffle of radius *r*b,*<sup>i</sup>* with a tangential velocity of *u*b,*i*, and thus the lifetime of the flying filmy liquid in zone III of region *i*' *t*III,*i*' is expressed as follows:

$$t\_{\rm III,i'} = \frac{\sqrt{r\_{\rm a,i+1}^2 - r\_{\rm b,i}^2}}{u\_{\rm b,i}} = \frac{\sqrt{r\_{\rm a,i+1}^2 - r\_{\rm b,i}^2}}{u\_{\rm a,i} \frac{r\_{\rm a,i}}{r\_{\rm b,i}}} \tag{11}$$

Thus, the surface area of the flying filmy liquid equals the gas–liquid effective mass transfer area in this zone:

$$A\_{\rm III,i'} = \left(2\pi r\_{\rm b,i} + 2\pi r\_{\rm a,i+1}\right)\sqrt{\left(r\_{\rm a,i+1} - r\_{\rm b,i}\right)^2 + \left(u\_{\rm g,i}t\_{\rm III,i'} + \frac{1}{2}gt\_{\rm III,i'}^2\right)^2} \tag{12}$$

where *u*g,*<sup>i</sup>* is the axial component of the turbulent filmy liquid velocity in zone II.

Therefore, the gas–liquid effective interfacial area in the RZB rotor can be obtained as follows:

$$a = a\_{\rm I} + a\_{\rm II} + a\_{\rm III} = \frac{\sum\_{i=1}^{9} A\_{\rm I,i} + \sum\_{i=1}^{9} A\_{\rm II,i} + \sum\_{i'=1}^{8} A\_{\rm III,i'}}{\pi (r\_{\rm o}^2 - r\_{\rm i}^2)z} \tag{13}$$

The mass transfer in the liquid phase consists of that in the droplets and filmy liquid. Because the droplets were treated as rigid balls without inside circulation during the flight due to a small distance in zone I of region *i* [26], the following mass partial differential equation can be used to present the CO2 diffusion process from gas bulk to droplets based on the pseudo-first-order irreversible chemical reaction:

$$\frac{\partial \mathcal{C}\_{\text{CO}\_2}}{\partial t\_{\text{I},i}} = D\_{\text{CO}\_2} \frac{\partial^2 \mathcal{C}\_{\text{CO}\_2}}{\partial R^2} + \frac{2D\_{\text{CO}\_2}}{R} \frac{\partial \mathcal{C}\_{\text{CO}\_2}}{\partial R} - k\_{\text{PP}} \left( \mathcal{C}\_{\text{CO}\_2} - \mathcal{C}\_{\text{CO}\_2}^\* \right) \tag{14}$$

$$\text{I.C. } t\_{\text{I},i} = 0, \ R \ge 0: \ \mathcal{C}\_{\text{CO}\_2} = \mathcal{C}\_{\text{CO}\_2}^\*;$$

$$\text{B.C. } R = 0, \ t\_{\text{I},i} \ge 0: \ \mathcal{C}\_{\text{CO}\_2} = \mathcal{C}\_{\text{CO}\_2}^\*;$$

$$R = \frac{d\_i}{2}, \ t\_{\text{I},i} \ge 0: \ \mathcal{C}\_{\text{CO}\_2} = \mathcal{C}\_{\text{O}}; \tag{15}$$

where *C*<sup>0</sup> (=*P*CO2,0/*H*) and *C*CO2*\** are the molar concentration of CO2 at the gas–liquid interface and the equilibrium molar concentration of CO2 in the liquid bulk, respectively. Letting *C*<sup>A</sup> = *C*CO2 − *C*CO2*\**, the following expressions can be obtained:

$$\frac{\partial \mathcal{C}\_{\mathcal{A}}}{\partial t\_{\mathcal{I},i}} = D\_{\text{CO}\_2} \frac{\partial^2 \mathcal{C}\_{\mathcal{A}}}{\partial R^2} + \frac{2D\_{\text{CO}\_2}}{R} \frac{\partial \mathcal{C}\_{\mathcal{A}}}{\partial R} - k\_{\text{SPP}} \mathcal{C}\_{\mathcal{A}} \tag{16}$$

$$I.\text{C. } t\_{\mathcal{I},i} = 0, \ R \ge 0: \ \mathcal{C}\_{\mathcal{A}} = 0;$$

$$B.\text{C. } R = 0, \ t\_{\mathcal{I},i} \ge 0: \ \mathcal{C}\_{\mathcal{A}} = 0;$$

$$R = \frac{d\_i}{2}, \ t\_{\mathcal{I},i} \ge 0: \ \mathcal{C}\_{\mathcal{A}} = \mathcal{C}\_0 - \mathcal{C}\_{\mathcal{CO}\_2}^\*. \tag{17}$$

After Laplace transform, the following ordinary differential equation is derived:

$$\frac{d^2u}{dR^2} + \frac{2}{R}\frac{du}{dR} - \frac{k\_{\rm app} + s}{D\_{\rm CO\_2}}u = 0\tag{18}$$

Letting *β* = *u* × *R*, Equation (22) can be obtained by Equations (19)–(21).

$$\frac{d\mathcal{B}}{dR} = \frac{du}{dR}R + u \tag{19}$$

$$\frac{d^2\beta}{dR^2} = \frac{d^2u}{dR^2}R + 2\frac{du}{dR} \tag{20}$$

$$\frac{1}{R}\frac{d^2\beta}{dR^2} = \frac{d^2u}{dR^2} + \frac{2}{R}\frac{du}{dR} \tag{21}$$

$$\frac{d^2\beta}{dR^2} = \frac{k\_{app} + s}{D\_{CO\_2}}\beta \tag{22}$$

The general solution of *β* can be calculated by Equation (23):

$$\mathcal{B} = \boldsymbol{\mu} \times \boldsymbol{R} = \boldsymbol{C}\_1 \boldsymbol{e}^{-\boldsymbol{R}\sqrt{\frac{k\_{\rm app} + s}{D\_{\rm CO\_2}}}} + \boldsymbol{C}\_2 \boldsymbol{e}^{\boldsymbol{R}\sqrt{\frac{k\_{\rm app} + s}{D\_{\rm CO\_2}}}} \tag{23}$$

Then, the following equation is obtained by Laplace inverse transform:

$$\begin{array}{l} \mathbf{C}\_{\text{A}} = \frac{\mathbf{C}\_{\text{I}}}{2\mathbb{R}} \exp \left( R \sqrt{\frac{k\_{\text{app}}}{D\_{\text{CO}\_{2}}}} \right) \operatorname{erfc} \left( -\frac{R}{2\sqrt{D\_{\text{CO}\_{2}}} t\_{\text{I},i}} - \sqrt{k\_{\text{app}}} t\_{\text{I},i} \right) \\ \quad + \frac{\mathbf{C}\_{\text{I}}}{2\mathbb{R}} \exp \left( -R \sqrt{\frac{k\_{\text{app}}}{D\_{\text{CO}\_{2}}}} \right) \operatorname{erfc} \left( -\frac{R}{2\sqrt{D\_{\text{CO}\_{2}}} t\_{\text{I},i}} + \sqrt{k\_{\text{app}}} t\_{\text{I},i} \right) \\ \quad + \frac{\mathbf{C}\_{\text{I}}}{2\mathbb{R}} \exp \left( -R \sqrt{\frac{k\_{\text{app}}}{D\_{\text{CO}\_{2}}}} \right) \operatorname{erfc} \left( \frac{R}{2\sqrt{D\_{\text{CO}\_{2}}} t\_{\text{I},i}} - \sqrt{k\_{\text{app}}} t\_{\text{I},i} \right) \\ \quad + \frac{\mathbf{C}\_{\text{I}}}{2\mathbb{R}} \exp \left( R \sqrt{\frac{k\_{\text{app}}}{D\_{\text{CO}\_{2}}}} \right) \operatorname{erfc} \left( \frac{R}{2\sqrt{D\_{\text{CO}\_{2}}} t\_{\text{I},i}} + \sqrt{k\_{\text{app}}} t\_{\text{I},i} \right) \end{array} \tag{24}$$

where *erfc*(*x*) is the excess error function.

To simplify Equation (24), *B* is defined as follows:

$$B = \exp\left(\frac{d\_i}{2}\sqrt{\frac{k\_{\rm app}}{D\_{\rm CO\_2}}}\right) \text{erf}\left(\frac{\frac{d\_i}{2}}{2\sqrt{D\_{\rm CO\_2}t\_{\rm I,i}}}+\sqrt{k\_{\rm app}t\_{\rm I,i}}\right) + \exp\left(-\frac{d\_i}{2}\sqrt{\frac{k\_{\rm app}}{D\_{\rm CO\_2}}}\right) \text{erf}\left(\frac{\frac{d\_i}{2}}{2\sqrt{D\_{\rm CO\_2}t\_{\rm I,i}}}-\sqrt{k\_{\rm app}t\_{\rm I,i}}\right) \tag{25}$$

where *erf*(*x*) is the error function.

Letting *C*<sup>1</sup> = −*C*<sup>2</sup> and substituting Equation (17) into Equation (24), *C*<sup>1</sup> and *C*<sup>2</sup> can be expressed as follows:

$$C\_1 = \frac{C\_0 - C\_{\text{CO}\_2}^\*}{2B} d\_i \tag{26}$$

$$\mathcal{C}\_2 = -\mathcal{C}\_1 = -\frac{\mathcal{C}\_0 - \mathcal{C}\_{\mathcal{CO}\_2}^\*}{2B} d\_i \tag{27}$$

Therefore, Equation (28) can be obtained by combining Equations (24), (26) and (27):

$$\begin{array}{ll} \mathbf{C\_{A}} = \frac{\left(\mathbf{C\_{0}} - \mathbf{C\_{CO\_{2}}^{\*}}\right)d\_{i}}{2RB} & \exp\left(R\sqrt{\frac{k\_{\text{app}}}{D\_{\text{CO}\_{2}}}}\right)erf\left(\frac{R}{2\sqrt{D\_{\text{CO}\_{2}}}t\_{\text{I},i}} + \sqrt{k\_{\text{app}}}t\_{\text{I},i}\right) \\ & + \frac{\left(\mathbf{C\_{0}} - \mathbf{C\_{CO\_{2}}^{\*}}\right)d\_{i}}{2RB}\exp\left(-R\sqrt{\frac{k\_{\text{app}}}{D\_{\text{CO}\_{2}}}}\right)erf\left(\frac{R}{2\sqrt{D\_{\text{CO}\_{2}}t\_{\text{I},i}}}\right) \\ & - \sqrt{k\_{\text{app}}t\_{\text{I},i}} \end{array} \tag{28}$$

In terms of Fick's first law, the rate equation describing mass transfer at the gas–liquid interface can be written as:

$$k\_{\rm L-I,i} \left( \mathcal{C}\_{\rm CO\_2} - \mathcal{C}\_{\rm CO\_2}^\* \right) = D\_{\rm CO\_2} \frac{\partial \mathcal{C}\_{\rm A}}{\partial R} \Big|\_{R=\frac{d\_i}{2}} \tag{29}$$

It was assumed that CO2 was completely consumed by chemical reaction of CO2 and NaOH in the liquid bulk. Thus, *<sup>C</sup>*CO2*\** can be ignored, and *<sup>k</sup>*L−I,*<sup>i</sup>* in zone I is given by Equation (30):

$$k\_{\rm L-I,i} = \sqrt{k\_{\rm app} D\_{\rm CO\_2}} - \frac{2D\_{\rm CO\_2}}{d\_i} \tag{30}$$

In regard to liquid film in the turbulent filmy liquid in zone II of region *i*, CO2 diffusion into the liquid film with a pseudo-first-order reaction can be expressed as:

$$\frac{\partial \mathbb{C}\_{\rm CO\_2}}{\partial t\_{\rm II,i}} = D\_{\rm CO\_2} \frac{\partial^2 \mathbb{C}\_{\rm CO\_2}}{\partial \mathbf{x}^2} - k\_{\rm app} \left( \mathbb{C}\_{\rm CO\_2} - \mathbb{C}\_{\rm CO\_2}^\* \right) \tag{31}$$

$$\text{I.C.} \ t\_{\text{II},i} = 0, \ x \ge 0: \ \mathsf{C}\_{\text{CO}\_2} = \mathsf{C}\_{\text{CO}\_2}^\*;$$

$$\text{B.C.} \ x = 0, t\_{\text{II},i} \ge 0: \ \mathsf{C}\_{\text{CO}\_2} = \mathsf{C}\_0;$$

$$\mathsf{x} = \delta(\to \infty), t\_{\text{II},i} \ge 0: \ \mathsf{C}\_{\text{CO}\_2} = \mathsf{C}\_{\text{CO}\_2}^\*;\tag{32}$$

where *δ* is the average thickness of the liquid film in zone II and *t*II,*<sup>i</sup>* is the lifetime of the liquid film, which is the average time consumed for renewing the liquid film once.

The following equation can be obtained from Equations (31) and (32):

$$\frac{\partial \mathcal{C}\_{\mathcal{A}}}{\partial t\_{\Pi,i}} = D\_{\mathcal{CO}\_2} \frac{\partial^2 \mathcal{C}\_{\mathcal{A}}}{\partial x^2} - k\_{\text{app}} \mathcal{C}\_{\mathcal{A}} \tag{33}$$

$$I.\text{C. } t\_{\Pi,i} = 0, \ x \ge 0: \ \mathcal{C}\_{\mathcal{A}} = 0;$$

$$B.\text{C. } x = 0, t\_{\Pi,i} \ge 0: \ \mathcal{C}\_{\mathcal{A}} = \mathcal{C}\_0 - \mathcal{C}\_{\mathcal{CO}\_2}^\*;$$

$$x = \delta(\to \infty), t\_{\Pi,i} \ge 0: \ \mathcal{C}\_{\mathcal{A}} = 0; \tag{34}$$

By Laplace transform, the following equation is derived:

$$\frac{d^2u}{dx^2} - \frac{k\_{\rm app} + s}{D\_{\rm CO\_2}}u = 0\tag{35}$$

$$B.\text{C. } \mathfrak{x} = 0, \text{ s} \ge 0: \ \mathfrak{u} = \frac{\mathrm{C} \mathrm{o} - \mathrm{C}\_{\mathrm{CO}\_2}^\*}{\mathrm{s}};$$

$$\mathfrak{x} = \delta(\to \infty), \text{ s} \ge 0: \ \mathfrak{u} = 0; \tag{36}$$

Substituting the boundary conditions in Equation (36) into Equation (35), the general solution of *u* can be obtained by Equation (37):

$$u = \frac{C\_0 - C\_{\rm CO\_2}^\*}{s} \exp\left(-\mathbf{x}\sqrt{\frac{k\_{\rm app} + s}{D\_{\rm CO\_2}}}\right) \tag{37}$$

By the Laplace inverse transform, the relationship of the CO2 concentration distribution, lifetime, and liquid film thickness can be described by the following equation:

$$\begin{array}{l} \text{C}\_{\text{A}} = \frac{\left(\text{C}\_{0} - \text{C}\_{\text{CO}\_{2}}^{\*}\right)}{2} \; \exp\left(\text{x}\sqrt{\frac{k\_{\text{app}}}{D\_{\text{CO}\_{2}}}}\right) \text{erfc}\left(\frac{\text{x}}{2\sqrt{D\_{\text{CO}\_{2}}t\_{\text{II},i}}} + \sqrt{k\_{\text{app}}t\_{\text{II},i}}\right) \\ \qquad + \frac{\left(\text{C}\_{0} - \text{C}\_{\text{CO}\_{2}}^{\*}\right)}{2} \text{exp}\left(-\text{x}\sqrt{\frac{k\_{\text{app}}}{D\_{\text{CO}\_{2}}}}\right) \text{erfc}\left(\frac{\text{x}}{2\sqrt{D\_{\text{CO}\_{2}}t\_{\text{II},i}}}\right) \\ \qquad - \sqrt{k\_{\text{app}}t\_{\text{II},i}} \end{array} \tag{38}$$

Hence, the mass transfer rate at the gas–liquid interface in zone II can be obtained based on Fick's first law.

$$R\_{\rm CO\_2\cdots H,i} = D\_{\rm CO\_2} \frac{\partial \mathcal{C}\_{\rm A}}{\partial x}|\_{x=0} = \left(\mathcal{C}\_0 - \mathcal{C}\_{\rm CO\_2}^\*\right) \left(\sqrt{k\_{\rm app} D\_{\rm CO\_2}} \,\,\,\mathrm{erf}\sqrt{k\_{\rm app} t\_{\rm II,i}} + \sqrt{\frac{D\_{\rm CO\_2}}{\pi t\_{\rm II,i}}} \exp\left(-k\_{\rm app} t\_{\rm II,i}\right)\right) \tag{39}$$

According to assumptions (5) and (6), the renewal of the liquid film in zone II was caused by the impingement of droplets, which came from the lowest circle of the perforated area in the rotating baffle. It was assumed that the droplets formed by every perforation had the same amount, lifetime, velocity, and diameter. Therefore, the renewal frequency, *Si*, defined as the number of droplets leaving the rotating baffles per unit time, in zone II of region *i* is expressed as:

$$S\_i = \frac{36Q\_{\rm L}}{\pi \mathfrak{m}\_{\rm a,i} d\_i^{\rm 3}} \tag{40}$$

where *n*a,*<sup>i</sup>* is the number of perforations in the rotating baffle *i*a.

Then the lifetime of the liquid film in the turbulent filmy liquid can be calculated by Equation (41) [18]:

$$t\_{\Pi,i} = \frac{1}{S\_i} \tag{41}$$

It was assumed that *t*II,*<sup>i</sup>* caused by every droplet was the same. Thus, Equation (42) can be used to express the Higbie distribution function of the lifetime of the liquid film [16]:

$$
\psi(t\_{\mathrm{II},i}) = \frac{1}{t\_{\mathrm{II},i}} \tag{42}
$$

Hence, the relationship between the liquid-side mass-transfer coefficient in zone II (*k*L−II,*i*) and the mass transfer rate of liquid film at the gas–liquid interface can be written as:

$$\int\_{0}^{t\_{\rm II,i}} R\_{\rm CO\_{2}-\rm II,i} \psi(t\_{\rm II,i}) dt = k\_{\rm L-\rm II,i} \left(\mathbb{C}\_{0} - \mathbb{C}\_{\rm CO\_{2}}^{\*}\right) \tag{43}$$

Based on the same assumption for droplets, *<sup>C</sup>*CO2*\** can be ignored. Therefore, *<sup>k</sup>*L−II,*<sup>i</sup>* related to *k*app, *D*CO2, and *t*II,*<sup>i</sup>* can be deduced as follows:

$$k\_{\rm L-II} = \frac{\sqrt{k\_{\rm app} D\_{\rm CO\_2}}}{t\_{\rm II,i}} \left[ t\_{\rm II,i} \text{erf} \left( \sqrt{k\_{\rm app} t\_{\rm II,i}} \right) + \sqrt{\frac{t\_{\rm II,i}}{\pi k\_{\rm app}}} \exp \left( -k\_{\rm app} t\_{\rm II,i} \right) + \frac{1}{2k\_{\rm app}} \text{erf} \left( \sqrt{k\_{\rm app} t\_{\rm II,i}} \right) \right] \tag{44}$$

As for the liquid film in the flying filmy liquid in zone III of region *i*', the model development process of *k*L−III,*i*' was the same as that of *k*L−II,*I*, and thus the expression of *k*L−III,*i*' was similar to that of *k*L−II,*i*, except for the lifetime of the liquid film in the flying filmy liquid, which was calculated by Equation (11).

Hence, *k*L*a* in the RZB rotor is written as follows:

$$k\_{\rm L}a = k\_{\rm L-1}a\_{\rm I} + k\_{\rm L-\Pi}a\_{\rm II} + k\_{\rm L-\Pi}a\_{\rm III} = \frac{\sum\_{i=1}^{9} k\_{\rm L-1,i} A\_{\rm I,i} + \sum\_{i=1}^{9} k\_{\rm L-\Pi,i} A\_{\rm II,i} + \sum\_{i'=1}^{8} k\_{\rm L-\Pi,i'} A\_{\rm III,i'}}{\pi (r\_0^2 - r\_{\rm i}^2)z} \tag{45}$$

To date, there is no available expression to calculate *k*<sup>G</sup> in the RZB rotor. Thus, a surface renewal model reported by Guo et al. [27] was employed to calculate *k*<sup>G</sup> in the RZB rotor as follows:

$$k\_{\rm G} = \sqrt{D\_{\rm G} k\_{\rm s} v\_{\rm G}^2} \tag{46}$$

where *D*<sup>G</sup> is the diffusion coefficient of CO2 in gas phase, *k*<sup>s</sup> is the proportionality coefficient, and *v*<sup>θ</sup> is the average tangential gas velocity in the RZB rotor.

The *K*G*a* of the RZB can be calculated by [15]:

$$\frac{1}{\mathcal{K}\_{\rm G}a} = \frac{1}{k\_{\rm G}a} + \frac{H}{k\_{\rm L}a} \tag{47}$$

The experimental *K*G*a* can be determined by Equation (48), which was obtained in the previous study on CO2 absorption [12]:

$$\begin{array}{l} \text{K}\_{\text{G}}a = \frac{G'}{\pi Pz\left(r\_{\text{o}}^{2} - r\_{\text{i}}^{2}\right)} \quad \left[ \ln \frac{y\_{\text{CO}\_{2}-\text{in}}\left(1 - y\_{\text{CO}\_{2}-\text{out}}\right)}{y\_{\text{CO}\_{2}-\text{out}}\left(1 - y\_{\text{CO}\_{2}-\text{in}}\right)} \right. \\ \quad + \left(\frac{y\_{\text{CO}\_{2}-\text{in}}}{1 - y\_{\text{CO}\_{2}-\text{in}}} - \frac{y\_{\text{CO}\_{2}-\text{out}}}{1 - y\_{\text{CO}\_{2}-\text{out}}}\right) \right] \end{array} \tag{48}$$

where *G'* is the inlet gas flow rate of inert gas (without reaction or dissolution), *P* is the total pressure, and *y*CO2−in and *y*CO2−out denote the molar fraction of CO2 in the inlet and outlet gas streams, respectively.

By substituting the experimental *K*G*a* from Equation (48) and the calculated *k*L*a* and *a* from Equations (45) and (13), respectively, into Equation (47), *k*<sup>G</sup> can be obtained. Consequently, *k*<sup>s</sup> can be obtained, which is necessary for the establishment of the mass transfer model. Thus, the calculated *k*G, *k*L, *a*, and *K*G*a* for the RZB were obtained from this mass transfer model.

The density and viscosity values were obtained from reference [28], while the surface tension value was from reference [29]. The equilibrium, kinetics, and transport parameters used for modeling are tabulated in Table 2.


**Table 2.** Parameters for model development.

#### **4. Results and Discussion**

#### *4.1. Model Validation*

The RZB had an average value for *<sup>k</sup>*<sup>s</sup> of 6.17 × <sup>10</sup>−<sup>8</sup> kmol<sup>2</sup> s/kPa2 <sup>m</sup><sup>8</sup> under the experimental conditions. By using *k*L, *k*G, and *a* obtained from the above model and the average value of *k*s, the calculated *K*G*a* of the RZB was obtained according to Equation (47). The comparison of *k*L, *k*G, and *a* between this work and reference [10] can be found in Table S1 in the Supplementary Materials file.

Figure 5 is the diagonal diagram of the predicted and experimental values of *K*G*a*. It is shown that this model provided good predictions on *K*G*a* of CO2 absorption in the RZB, with deviations generally less than 10% in comparison with the experimental results. Additionally, in the Figures shown in Sections 4.2–4.6, the curves for the predicted values were consistent with those for the experimental results. In addition, the values of individual and overall mass transfer parameters have been presented in Table S2 in the Supplementary Materials file.

**Figure 5.** Diagonal diagram of experimental and predicted *K*G*a*.
