*2.2. The Theoretical Basis*

Research on materials and calculations, design and manufacture of laboratory equipment model, and the studied mathematical can be seen in [1–14], which are based on the balance of materials, energy, and momentum equations to find equations to change pressure over time and the height of the bed by analytic transformations. However, to study the rule of changing the technological parameters of a single bed in the equipment, it is necessary to implement the experimental plan as follows:


Now we have some governing equations as follows


$$\frac{\partial \mathbf{C}\_i}{\partial t} - D\_L \frac{\partial^2 \mathbf{C}\_i}{\partial z^2} + \frac{\partial (\mathbf{C}\_i u)}{\partial z} + \rho\_p \left(\frac{1-\varepsilon}{\varepsilon}\right) \overline{\frac{\partial q\_i}{\partial t}} = 0 \tag{1}$$

Equation (1) can be completely solved by Matlab to simulate the change of the concentration according to the height of the column when determining the porosity parameters (ε), axial diffusion coefficient (*DL*) and speed apparent (*u*) by analysis, calculation, and experiment. But testing by experiment is difficult because it is hard to install a device that measures the exact concentration according to the height of the column.

For ideal gas (*Ci* = *yiP*/*RT*), Equation (1) is transformed into following forms:

$$\begin{cases} -D\_L \frac{\partial^2 C\_i}{\partial z^2} + y\_i \frac{\partial u}{\partial z} + u \left( \frac{\partial y\_i}{\partial z} + y\_i \left( \frac{1}{p} \frac{\partial P}{\partial z} - \frac{\partial T}{\partial z} \right) \right) \frac{\partial y\_i}{\partial z} + \frac{\partial y\_i}{\partial t} \\ + y\_i \left( \frac{1}{p} \frac{\partial P}{\partial z} - \frac{1}{T} \frac{\partial T}{\partial z} \right) + \left( \frac{\rho\_P RT}{P} \right) \frac{1 - \varepsilon}{\varepsilon} \right) \frac{\overline{\partial q\_i}}{\partial t} = 0 \end{cases} \tag{2}$$

$$\frac{\partial P}{\partial t} + P \frac{\partial u}{\partial z} + u \frac{\partial P}{\partial t} + PT \left( \frac{\partial}{\partial t} \left( \frac{1}{T} \right) - u \frac{\partial}{\partial z} \left( \frac{1}{T} \right) \right) - 2D\_L R \frac{\partial P}{\partial z} \frac{\partial}{\partial z} \left( \frac{1}{T} \right) + \rho\_p RT \left( \frac{1-\varepsilon}{\varepsilon} \right) \sum\_{i=1}^{n} \frac{\overline{\partial q\_i}}{\partial t} = 0 \tag{3}$$

Equations (2) and (3) describe the change of concentration, pressure, and temperature over time and the height of the column. Solving these equations is very complex; it is often considered that the process is isothermal (for the PSA cycle) and isometric (for the TSA cycle).

In this work, the studied column working under the PSA cycle is considered as an isothermal adsorption process. In order to simulate and verify this model, we must build an equation that describes the change of pressure over time and the height of the column. Respond to the empirical model developed to measure the pressure according to the height of the column.

Results of the analytic changes from changes in concentration to pressure changes as follows For adsorption process we have:

$$\frac{\partial p\_i}{\partial t} \left( 1 + \frac{1 - \varepsilon}{\varepsilon} K \right) = -\varepsilon.\mu. \frac{\partial p\_i}{\partial z} + \varepsilon.D\_\mathsf{L} \cdot \frac{\partial^2 p\_i}{\partial z^2} \tag{4}$$

For desorption process we have:

$$\frac{\partial p\_{\rm des}}{\partial t} \left( 1 + \frac{1 - \varepsilon}{\varepsilon} K \right) = -\mu . \frac{\partial p\_{\rm des}}{\partial z} + D\_{\rm des} \cdot \frac{\partial^2 p\_{\rm des}}{\partial z^2} \tag{5}$$

Equations (4) and (5) describe the change of pressure over time and according to the height of the column, which depend on the adsorption constant, porosity, axial diffusion coefficient, and speed. These parameters are determined by analysis, calculation, and experiment. This equation will be solved by Matlab. Equation (5) is one case of Equation (4) when porosity ε = 1 in the case of desorption.


$$-K\_i \frac{\partial^2 T}{\partial z^2} + \varepsilon \rho\_i \mathbb{C}\_{p, \mathfrak{J}} \Big( \mu \frac{\partial T}{\partial z} + T \cdot \frac{\partial u}{\partial z} \Big) + \left( \varepsilon\_l \rho\_\delta \mathbb{C}\_{p, \mathfrak{J}} + \rho\_b \mathbb{C}\_{p, \mathfrak{s}} \right) \frac{\partial T}{\partial t} - \rho\_b \sum\_{i=1}^n \left( \overline{\frac{\partial q\_i}{\partial t}} (\Delta H\_i) \right) + \frac{2h\_i}{R\_{Ri}} (T - T\_w) = 0 \tag{6}$$

Loss of heat through the column wall is defined as

$$
\rho\_W \mathcal{C}\_{p,W} A\_{\bar{w}} \frac{\partial T\_{i,w}}{\partial t} = 2\pi R\_{\text{Bi}} h\_i (T - T\_{\bar{w}}) - 2\pi R\_{\text{Bo}} h\_o (T\_{\bar{w}} - T\_{atm}) \tag{7}
$$

where *Aw* = <sup>π</sup>(*R*2*B*,*<sup>o</sup>* − *<sup>R</sup>*2*B*,*<sup>i</sup>*).

The amount of heat expected during the adsorption process is very small, the heat generated during the adsorption process and the steaming process completely passes through the column wall.Herein, the temperature of the column will change insignificantly over time and along the height of the column, so the process is considered as an isothermal state. Then, Equations (6) and (7) are only theoretical and not experimental.


$$-\frac{dP}{dZ} = a.\mu.\mu + b.r.\mu.|\mu|; a = \frac{150(1-\varepsilon)^2}{4R\_p^2\varepsilon^2}; b = \frac{1.75(1-\varepsilon)}{2R\_p\varepsilon} \tag{8}$$

Equation (8) describes the change of pressure according to the height of the column, in addition, it depends not only on parameters such as particle size and porosity, but also it depends on the velocity of the gas flow through the column. In the theoretical calculation, it is usually taken at a constant speed however, in practice, this experimental speed can change up to 20% caused by O2 being adsorbed. So, the error between theory data and experiment data can reach 20%.


$$q\_i^\* = \frac{B\_i q\_{mi} P\_i^{ni}}{1 + \sum\_{j=1}^n B\_j P\_j^{ni}} \tag{9}$$

in which

$$q\_{\rm mi} = K\_1 + K\_2 \cdot T; B\_i = K\_3 \exp(\frac{K\_4}{T}); \newline n\_i = \frac{k\mathfrak{z} + k\mathfrak{z}}{T} \frac{\partial q\_i}{\partial z} = \omega\_i \{q\_i^\* - \overline{q\_i}\}; \newline \omega\_i = \frac{15D\_{\rm ci}}{r\_c^2} = \bigcirc\_{r=1}^5 (1 + B\_i P\_i)^2 \sum\_{i=1}^5 \omega\_i = \omega\_i \{q\_i^\* - \overline{q}\_i\}$$

Equation (9) describes the adsorption load depending on the equilibrium constant, the equilibrium adsorption load and the pressure, temperature. These parameters can be determined experimentally and calculated in the case of isothermal.

In this work, the simulation of the change of pressure according to the height of the column and with the Equations (4) and (5) are established. We must determine the parameters of the model such as porosity, axial diffusion coefficient, and speed to find the solution of the above equations with the initial conditions and given boundary conditions.

For the adsorption process, Equation (4) has some fundamental conditions as follows:

+ Initial condition:

> The partial pressure at (*z* = 0, *t* = 0) is *p*(*<sup>z</sup>*, *t* = 0) = 0.

+ Boundary conditions: According to Danckwerts standards, we have:

The partial pressure at all times in input position and output of adsorption column are: at *z* = *z*;

$$p(z=0,t) = \begin{cases} \left.p^d + \frac{D\_l}{\mu\_c} \frac{\partial p}{\partial z}\right|\_{z=0,t}; & 0 \le t \le t^c\\ \left. + \frac{D\_l}{\mu\_c} \frac{\partial p}{\partial z}\right|\_{z=0,t}; & t \ge t^c \end{cases}$$

where *pd* is the partial pressure of the adsorbed material in the adsorption column at *z* = *L*;

$$\frac{\partial p}{\partial t}|\_{\mathbf{x}=L} = 0$$

where *tc* is the duration of the adsorption cycle.

For the desorption process, Equation (5) has some fundamental conditions as follows:

+ Initial condition:

> The partial pressure at (*z* = 0, *t* = 0) is:

$$p(z, t=0) = p\_{\text{des}}^{\rho}$$

+ Boundary conditions: According to Danckwerts standards, we have:

The partial pressure at all times in input position and output of adsorption column are: at *z* = *z*;

$$p(z=0,t) = \begin{cases} \left.p\_{\rm des}^{d} + \frac{D\_{\rm L}}{\mu\_{\rm des}} \frac{\partial p\_{\rm dem}}{\partial z}\right|\_{z=0,t}; & 0 \le t \le t^c\\ \left.p\_{\rm des}^{c}\right| & t \ge t^c \end{cases}$$

where *pd* is the partial pressure of the adsorbed material in the adsorption column at *x* = *L*;

$$\left. \frac{\partial p\_{\text{des}}}{\partial t} \right|\_{z=L} = 0$$

where *t c* is the duration of the adsorption cycle.

In this study, we will not simulate the adsorption process. Because when the simulation time is set, this process is very similar to the adsorption process.

#### **3. Calculation and Simulation Results**

Calculation to determines the porosity of the column [16–19] includes some steps as follows

+ Determining bulk density ρ*b* = 0.676 g/cm3, particle density ρ*p* = 0.78 g/cm<sup>3</sup> and solid density ρ*s* = 2.174 g/cm<sup>3</sup> by means of volume, mass weighing, and pycnometer.

The porosity of the column is determined in the following Table 1.


**Table 1.** Results of calculation porosity of a single fixed bed.

+ Determining the air velocity through the column [16–19] can be expressed as:

$$
\mu\_c = \frac{V\_{kk}}{\varepsilon\_l \frac{\pi D^2}{4}} \tag{10}
$$

When the total pressure reaches the critical pressure, the free step of the gas molecule is approximately the capillary size. Di ffusion follows both the Knusen di ffusion and molecular di ffusion mechanisms:

+ Di ffuse in transition zone [16–19] is:

$$\frac{1}{D\_{\rm L}} = \frac{1}{D\_{\rm AB}} + \frac{1}{D\_{\rm k}} \tag{11}$$

+ Molecular di ffusion [16–19] is:

$$D\_{AB} = 0.00158 \frac{T^{3/2} \left(\frac{1}{\mathcal{M}\_A} + \frac{1}{\mathcal{M}\_B}\right)^{1/2}}{P \cdot \sigma\_{AB}^2 \Omega\_{AB}} \tag{12}$$

+ Flow di ffusion (Knusen) [16–19] is:

$$D\_K = \theta 700 \cdot R\_p \cdot \left(\frac{T}{M}\right)^{1/2} \tag{13}$$

From (11)–(13) we determine the basic parameters of the models (4) and (5) following pressure and temperature:

From the results from Tables 1 and 2 we can simulate adsorption column according to the change of partial pressure of adsorbed material over time and in column height in case of unstable working column V1, V2 are open, the installation time is 460 s to observe the maximum capacity of the column at a pressure from 1 bar to 8 bar. In this paper we only give results at pressures of 5 bar, 5.5 bar and 8 bar for discussion.


**Table 2.** Results of calculation parameters of a single fixed bed model at different pressure.
