*2.2. Equilibrium Model*

The equilibrium model based on the assumption of each plate in the absorption column is considered as a theoretical plate (equilibrium plate) which means that the vapor and the liquid leave any plate at thermodynamic equilibrium [19]. According to Seader et al. [28], the main assumptions of equilibrium model are as follows:


In practice, this equilibrium takes place only at the interfaces between the vapor and liquid phases, so the e fficiencies such as point and Murphree e fficiencies are used in equilibrium model to account deviations from real equilibrium state [26]. For modelling the whole packed column, the packed bed is divided into stags (equilibrium stage). Figure 3 illustrates the typical entry and exit parameters of equilibrium plate stage. The equilibrium model of absorption process consists of well-known and accepted correlations called MESH equations that include the equations of component material balance, the equations of phase equilibrium, summation equations, and energy balance for each stage as following [21,28,29]:

**Figure 3.** Equilibrium stage model [21].

Overall mass balance for stage *j*:

$$\mathbf{L}\_{j-1} - \mathbf{L}\_j - V\_j + V\_{j+1} = \mathbf{0} \tag{1}$$

Component mass balance for stage *j*:

$$\mathbf{L}\_{j-1}\mathbf{x}\_{i,j-1} - \mathbf{L}\_j\mathbf{x}\_{i,j} - V\_j y\_{i,j} + V\_{j+1} y\_{i,j+1} = 0 \tag{2}$$

Energy balance equations for stage *j*:

$$\left(\mathbf{L}\_{j-1}\mathbf{H}\_{j-1}^{L} - \mathbf{L}\_{j}\mathbf{H}\_{j}^{L} - V\_{j}\mathbf{H}\_{j}^{V} + V\_{j+1}\mathbf{H}\_{j+1}^{V} = 0\tag{3}$$

Phase equilibrium

$$y\_i - \mathbb{K}\_i \mathbf{x}\_i = \mathbf{0} \tag{4}$$

Summation equations

$$\sum y\_i = 1\tag{5}$$

$$\sum x\_i = 1\tag{6}$$

where *L*, *V* are the molar flow rate of liquid and vapor respectively, *K* is equilibrium ratio, *H* is enthalpy, *xi*, *yi* are the mole fractions of component *i* in liquid and vapor phases respectively.

#### *2.3. Rate-Based Model Description*

The rate-based model consists of a set of well-accepted equations, which are modelled to calculate the mass and energy transfer across the interface using rate equation and mass transfer coefficients [26]. For the calculation of the mass transfer coefficient and interfacial area, the correlation by Billet and Schultes [30] can be applied.

#### *2.4. Assumptions and Mathematical Model for Rate-Based Model*

Assumptions for the rate-based approach according to Afkhamipour et al. [26] are summarized below:


#### *2.5. Material and Energy Balances*

For easy and accurate calculations, the packed-bed column with a height of *Z* is divided into some stages. Figure 4 shows a stage *j* of the column, which represents a differential height of the column (*j* refers to the stage number, where *i* refers to the compounds). The material and energy balances around stage *j* are performed by using the MERSHQ equations (Equations of Material, Energy balances, Rate of mass and heat transfer, Summation of composition, the hydraulic equation of pressure drop, and equilibrium relation) presented by Taylor et al. [31] as the following:

**Figure 4.** Rate-based stage model (Reproduced from ASPEN PLUS software manual).

Material balance for bulk liquid:

$$F\_j^L \mathbf{x}\_{i,j}^F + L\_{j-1} \mathbf{x}\_{i,j-1} + N\_{i,j}^L + r\_{i,j}^L - L\_j \mathbf{x}\_{i\bar{j}} = 0 \tag{7}$$

Material balance for bulk vapor:

$$\boldsymbol{F}\_{j}^{V}\boldsymbol{y}\_{i,j}^{F} + \boldsymbol{V}\_{j+1}\boldsymbol{y}\_{i,j+1} + \boldsymbol{N}\_{i,j}^{V} + \boldsymbol{r}\_{i,j}^{V} - \boldsymbol{V}\_{j}\boldsymbol{y}\_{i,j} = \boldsymbol{0} \tag{8}$$

Material balance for liquid film:

$$N\_{i,j}^{I} + r\_{i,j}^{fL} - N\_{i,j}^{L} = 0 \tag{9}$$

Material balance for vapor film:

$$N\_{i,j}^V + r\_{i,j}^{fV} - N\_{i,j}^I = 0 \tag{10}$$

Energy balance for bulk liquid:

$$\left(F\_j^L H\_j^{\text{FL}} + L\_{j-1} H\_{j-1}^L + Q\_j^L + q\_j^L - L\_j H\_j^L = 0\right) \tag{11}$$

Energy balance for bulk vapor:

$$\left\| F^V\_j H^{FV}\_j + V\_{j+1} H^V\_{j+1} + Q^V\_j - q^V\_j - V\_j H^V\_j = 0 \tag{12}$$

Energy balance for the liquid film

$$q\_j^l - q\_j^L = 0 \tag{13}$$

Energy balance for vapor film:

$$q\_j^V - q\_j^I = 0\tag{14}$$

Phase equilibrium at the interface:

$$\mathbf{y}\_{i,j}^{I} - \mathbb{K}\_{i,j}\mathbf{x}\_{i,l}^{I} = \mathbf{0} \tag{15}$$

Summations:

$$\sum\_{i=1}^{n} x\_{i,j} - 1 = 0 \tag{16}$$

$$\sum\_{i=1}^{n} y\_{i,j} - 1 = 0 \tag{17}$$

$$\sum\_{i=1}^{n} x\_{i,j}^{l} - 1 = 0\tag{18}$$

$$\sum\_{i=1}^{n} y\_{i,j}^{l} - 1 = 0 \tag{19}$$

where *F* is the molar flow rate of feed. *L*, *V* are the molar flow rates of liquid and vapor respectively, *N* molar transfer rate, *K* is equilibrium ratio, *r* is reaction rate, *H* is enthalpy, *Q* is heat input to a stage, *q* is heat transfer rate, *xi* , *yi* are the mole fraction of component *i* in liquid and vapor phases respectively.

#### *2.6. Mass Transfer through the Interfacial Area*

The rate-based model is based on the two-film theory that describes the mass transfer between gas and liquid [26]. The film theory is based on the assumption that when two fluid phases are coming in contact with each other, a thin layer of stagnant fluid exists on each side of the interface [32]. From Figure 5, the partial pressure of component *i* drops from *Pi* at gas bulk to *PIi* at the interface [32,33]. This pressure difference creates a driving force *Pi* − *PIi* for component *i* to transfer it from the gas bulk to gas film and then from the gas film to liquid film [33]. The accumulation of the component *i* at the liquid film creates a concentration difference between the liquid film and the liquid bulk. Similarly, this concentration difference creates a driving force *CIi* − *Ci* for component *i* to transfer from the liquid film to the liquid bulk [32,33]. The molar flux *N* of component A from the bulk of one phase to the interface is written as below [33]:

$$N = k \Delta c \tag{20}$$

where *N* is the molar flux of the component (moles per unit area per unit time) and Δ*c* is the driving force for mass transfer between the bulk and the interface. Consequently, Equation (11) can be written for phase and liquid phase [33] as:

$$N\_{A,G} = K\_G \left( P\_i - P\_i^l \right) \tag{21}$$

$$N\_{A,L} = K\_L \left(\mathbf{C}\_i^l - \mathbf{C}\_i\right) \tag{22}$$

**Figure 5.** Concentration profiles based on the two-film model [34], \* refers to the conditions at the equilibrium state.

At steady state, the flux of *i* from bulk gas to the interface must be equal to the flux of *i* from the interface to the bulk liquid [33]:

$$N = N\_{L,i} = N\_{G,i} \tag{23}$$

$$N = K\_{\mathbb{G}} \left( P\_i - P\_i^l \right) = K\_{\mathbb{L}} \left( \mathbb{C}\_i^l - \mathbb{C}\_i \right) \tag{24}$$

*KG* and *KL* are two different overall mass transfer coefficients (with different units). If equilibrium conditions exist at the interface, the overall mass transfer coefficient can be calculated as follows [35]:

$$\frac{1}{K\_G} = \frac{1}{k\_G} + \frac{H}{Ek\_L} \tag{25}$$

where *H* is Henry's law constant, *E* is the enhancement factor. *kG* and *kL* are the mass transfer coefficients without reaction in the gas and liquid phase. The mass transfer coefficients and the wetted interfacial area for mass and heat transfer were calculated according to the correlations proposed by several studies [30,35,36] as following:

$$k\_{L,i}a\_{eff} = C\_L \left(\frac{\mathcal{g}}{v\_L}\right)^{1/6} \left(\frac{D\_{L,i}}{d\_{l}}\right)^{1/2} a^{2/3} \mathcal{U}\_L^{1/3} \left(\frac{a\_{eff}}{a}\right) \tag{26}$$

$$k\_{\rm G,i} a\_{\varepsilon f} = \mathbb{C}\_{\rm G} \left(\varepsilon - h\_{\rm L}\right)^{-1/2} \left(\frac{a^3}{d\_h}\right)^{1/2} D\_{\rm G,i} \left(\frac{\mathrm{l}L\_{\rm G}}{a\nu\_{\rm G}}\right)^{3/4} \left(\frac{\upsilon\_{\rm G}}{D\_{\rm G,i}}\right)^{1/3} \left(\frac{a\_{\varepsilon f}}{a}\right) \tag{27}$$

$$\frac{a\_{eff}}{a} = 1.5 \left( a d\_h \right)^{-0.5} \left( \frac{\mathcal{U}\_L d\_h}{\upsilon\_L} \right)^{-0.2} \left( \frac{\mathcal{U}\_L^2 \rho\_L d\_h}{\sigma} \right)^{0.25} \left( \frac{\mathcal{U}\_L^2}{\mathcal{g} d\_h} \right)^{-0.45} \tag{28}$$

$$d\_h = 4\frac{\varepsilon}{a} \tag{29}$$

where *kL*.*<sup>i</sup>* and *kG*,*<sup>i</sup>* and are the mass transfer coefficients of component *i* in the liquid and gas phase respectively, *aeff* is the effective interfacial area per unit packed volume, *a* is the total surface area per unit packed volume, *CG* and *CL* are the gas and liquid and transfer coefficient parameter respectively; characteristic of the shape and structure of the packing, *g* is gravitational constant, *vL* and ν*G* are

kinematic viscosity of the liquid and gas phase respectively, *DL*,*<sup>i</sup>* and *DG*,*<sup>i</sup>* are diffusivity of component *i* in the liquid and gas phase respectively, *dh* is hydraulic diameter of the dumped packing, ρ*L* is density of liquid, *UL* and *UG* are velocity of liquid and gas phase respectively with reference to free column cross section, ε is void fraction of the packing, *hLa*is column liquid holdup, σ is liquid surface tension.

Theoretical liquid holdup correlation of Billet and Schultes [30] is given below:

$$h\_L = \left(\frac{12}{\mathcal{g}} \upsilon\_L \mathcal{U}\_L a^2\right)^{1/3} \tag{30}$$

#### *2.7. Heat Transfer through the Interfacial Area*

In order to determine the heat transfer through the interfacial area, the Chilton-Colburn-Analogy is used [35]. The expressions for the analogy are taken from many studies [35,37]. These expressions lead to a heat transfer coefficient *h* as [35]:

$$h = k\_G \left(\frac{\rho\_G \left(c\_p / M\_{\overline{w}, L}\right) \lambda^2}{D^2}\right)^{1/3} \tag{31}$$

where *kG* is mass transfer coefficient, ρ*G* is the density of the gas, *cp* is specific molar heat capacity, *Mw*,*<sup>L</sup>* is molecular weight of the liquid phase, λ is Thermal conductivity, *D* is the diffusion coefficient.

#### *2.8. Thermodynamics Approaches for Prediction of Phase Behavior*

The accuracy of equilibrium and rate-based models depend on the accurate prediction of the phase behavior properties of chemical species and their mixtures [19]. There are two conventional approaches for the prediction of phase behavior: approach (ϕ,ϕ) and approach (ϕ/γ) [38]. By approach (ϕ,ϕ), the fugacity coefficient ϕ. is applied for predicting the non-ideal behavior of both vapor and liquid phases [19]. The fugacity coefficient is calculated at the equilibrium condition according to Gebreyohannes et al. [38] as below:

$$f\_i^V = f\_i^L \tag{32}$$

$$f\_i^V = \varphi\_i^V y\_i \mathbf{P} \tag{33}$$

$$f\_i^L = \varphi\_i^L \mathbf{x}\_i \mathbf{P} \tag{34}$$

$$
\ln \eta\_i^a = -\frac{1}{RT} \int\_{\alpha \alpha}^{Va} \left[ \left( \frac{\partial p}{\partial n\_i} \right)\_{T, V, n\_{i,j}} - \frac{RT}{V} \right] dV - \ln Z\_m^a \tag{35}
$$

where *f Li* and *f Vi* are Fugacity of component *i* in the liquid and gas phase respectively, ϕ*Li* and ϕ*Vi* are Fugacity coefficient of the component *i* in liquid and gas phase respectively, *P* is pressure, *xi* and *yi* mole fraction of the *f* component *i* in liquid and gas phase respectively, ϕ*ai* is Fugacity coefficient of the component *i* where *a* refer to liquid or gas phase, *V* is total volume, *R* is Gas constant it has the value 0.08314 L bar <sup>K</sup>−<sup>1</sup>, *T* is temperature, *ni* is mole number of component *i, Zam* Compression factor.

The approach (ϕ/γ) uses the fugacity coefficients (ϕ) and the activity coefficients (γ) to account the non-ideal behavior of vapor and liquid phase. The equations related to (ϕ/γ) approach are listed as below [38]:

$$f\_i^V = f\_i^L \tag{36}$$

$$f\_i^V = \varphi\_i^V y\_i \mathbf{P} \tag{37}$$

$$f\_i^L = \mathbf{x}\_i \mathbf{y}\_i f\_i^{\ast, l} \tag{38}$$

where *f*∗, *I i* is liquid fugacity of pure component *i* at mixture temperature, γ*i* is the liquid activity coefficient of component *i*. While ϕ*Vi* is calculated according to Equation (26), the activity coefficients can be calculated from non-random two liquid model (NRTL) as below [39]:

$$\ln \gamma\_i = \frac{\sum\_{j=1}^{n} \mathbf{x}\_j \tau\_{ji} \mathbf{G}\_{ji}}{\sum\_{k=1}^{n} \mathbf{x}\_k \mathbf{G}\_{ki}} + \sum\_{j=1}^{n} \frac{\mathbf{x}\_j \mathbf{G}\_{ij}}{\sum\_{k=1}^{n} \mathbf{x}\_k \mathbf{G}\_{kj}} \left( \tau\_{ij} - \frac{\sum\_{m=1}^{n} \mathbf{x}\_m \tau\_{mj} \mathbf{G}\_{mj}}{\sum\_{k=1}^{n} \mathbf{x}\_k \mathbf{G}\_{kj}} \right) \tag{39}$$

$$G\_{i\bar{j}} = \exp(-a\_{i\bar{j}}\tau\_{i\bar{j}}) \tag{40}$$

$$
\pi\_{ij} = a\_{ij} + \frac{b\_{ij}}{T} + e\_{ij}ln T + f\_{ij}T \tag{41}
$$

$$a\_{i\rangle} = c\_{i\rangle} + d\_{i\rangle}(T - 273.315K) \tag{42}$$

Here, *aij* is NRTL non-randomness constant for binary interaction, *aij*, *bij*, *cij*, *dij*, *eij*, and *fij* are binary parameters. The ASPEN PLUS physical property system has extensive property databanks for binary parameters for the model.
