Simulation Study of the Liquid–Solid Multistage Adsorption Process

Abstract

In the present study, a computational model to simulate the separation of the multi-stage device is developed and used to verify an arbitrary-shaped adsorption isotherm and a limited mass transfer rate. The model’s governing equations are solved numerically by the MATLAB computing platform. For a specific separation, a suitable design must take into account the concentration record of the effluent solutions in the separation device. Further, since the experimental investigation has many limitations, an accurate mathematical description of a system could be viewed as an alternative approach to understanding it comprehensively. The usefulness of the simulation code depends heavily on how well it matches the experimental results and predicts them with minor adjustments and improvements. Here, the model is validated and used to investigate how changing the system’s parameters can affect its performance. The study found that increasing the size of the system (unit number and pore volume of the adsorbent) resulted in more solutions. Adsorption effectiveness was also investigated and it was found to be relatively unaffected by dividing the total amount of solution adsorption over many units, as long as slurrying was maintained at an adequate level. The model not only provides the prediction of the discharge concentration record but also the evaluation of the separation effectiveness attained by the multistage device.

1. Introduction

Estimating the concentration history of chemicals in the effluent is crucial when designing equipment to carry out a specific separation [1,2]. There seems to be a ceiling on the use of experimental findings. In order to better understand and ultimately optimize the process’ many mechanisms, a mathematical understanding of the process is necessary. Adsorption treatments using a network of linked tanks are seldom discussed in the literature. However, the overall performance of a series of sequential vessels and column adsorption has been extensively investigated in certain publications [3,4,5,6].

Since the 1990s, several ideas have been published on multiphase CFD-based simulations of liquid–solid two-phase flows [7,8]. The Eulerian–Eulerian (EE) approach, often known as the two-fluid model (TFM), and the Eulerian–Lagrangian (EL) method are the two main types of methods used to solve liquid–solid flow. Because it requires less computing power to apply, TFM is now the method of choice for studying two-phase flow when liquids and particles are present. The kinetic theory of granular flow, first proposed by Lun et al. [9] and refined since then, is often used to model solids’ viscosity and stresses by incorporating the conceptual equations of solid pressure.

Many configurations of numerous flow regions, including plug flow regions, fully mixed regions, bypassing regions, and dead water regions, have been examined for imperfect mixing in tanks. The coverage of column adsorption has been exhaustive. The differential equations characterizing the process are inserted and solved in kinetic models under a variety of conditions (finite or infinite mass transfer rate, nonlinear or linear isotherms, and finite diffusions), taking into account various control methods [10,11,12,13]. Additionally, adsorption columns were modeled using tank-in-series models. Backflow superposition and a two-dimensional array of stirring tanks to promote axial and radial diffusion are two examples of how this concept has been modified [14,15,16].

Samei and Raisi [17] researched multi-stage CO₂/CH₄ gas separation using rubbery and glassy membranes. Gas diffusion and sorption in rubbery and glassy membranes were explored using transfer, thermodynamic, and non-equilibrium lattice fluid models. They examined single- and multi-stage CH₄ recovery, product purity, capital, and yearly costs. High-permeability membranes outperformed high-CO₂/CH₄-selectivity membranes economically. The single-stage process failed to extract 99% of CH₄ and purify 98% of the product. Two- or three-stage membrane separation may work.

Rao and Kapur [18] used equipment for bivariate partition functions, statistical assumptions, and circuit structure to create general equations for the multi-stage separator’s bivariate partition function, component partition function, and yield stochastic partition function. The simulation results and analyses show that all two-stage separators provide a sharper edge than the constituent equipment operating alone. The overall bypass increases, decreases, or remains unchanged depending on the configuration. The circuit configuration can manipulate the cut density over a wide range.

Colussi et al. [19] developed a powerful supercritical multi-stage multicomponent separation column simulation program. Newton’s simultaneous convergence and the Fibonacci search method were used to discover the ideal damping factor. The band structure and Jacobian of the model equations boost performance in the program. The structure of thermodynamic functions and derivatives lowered computing times.

Sadek et al. [20] used adsorption variables to model and quantify the effectiveness of Cu(II) elimination from aqueous solutions using nano-zero-valent aluminum. The extraction performance of Cu(II) exceeded 50%. Langmuir isotherms and pseudo-second-order kinetic models characterized the adsorption data effectively. The best modeling approach for predicting Cu(II) extraction performance was an artificial neural network, followed by support vector regression and linear regression. The artificial neural network’s exceptional accuracy suggested that it could optimize the nano zero-valent aluminum performance for removing Cu(II) from polluted water at large scales and under a variety of operating conditions.

The adsorption rate has been extensively explored in theory and experiments for gas–solid adsorption. This is not yet the case with liquid–solid adsorption. This is due partly to the fact that studying a molecule as it approaches a surface from the liquid phase is challenging, both experimentally and theoretically, due to the ever-changing arrangements of solvent molecules.

A novel MatLab simulation program is implemented using the substance conservation equations as a starting point to simulate oil separation on the previously studied multi-stage adsorption equipment. The MatLab program designed for the simulated solution is outlined and verified by comparing it with experimental results. Furthermore, the impact of simulation parameters was studied. The MatLab simulation provides a sustainable

2. Experiment

The distillation of Qurna crude oil produced the oil fraction used in this study, which has a 365 to 375 °C boiling point range (F_365–375). Methanol was employed to remove the alumina from compounds that iso-octane could not dissolve. It was discovered that iso-octane was the better solvent for separating mineral oils. Alumina was employed with the following properties: a 100–200 mesh number, 0.261 mL/g pour volume, 70 Å pore diameter, 145 m²/g surface area, and 1.041 g/mL bulk density. A rotary film evaporator linked to a water bath maintained at 60–70 °C evaporated the solvent from the samples. A flask was filled with the condensed solvent. The oil samples were tested by measuring their refractive index with an Abbe-60 ED refractometer.

Five glass units (stages) measuring 6″ in height and 4″ in diameter make up the multi-stage apparatus. The bottom part goes to a two-way glass tap. Each unit had a stirrer attached to the end plate, and a speed of 250 rpm was deemed adequate. Two cylindrical solvent feed tanks and a spherical solvent reservoir made up the solvent feed system. The apparatus received nitrogen from a cylinder. The F_365–375 oil sample was introduced to the top unit. It then allowed overflow with the solution to the other units after being agitated for 160 min. For every experimental run, a constant flow rate of the solvent was continually delivered to the top unit. The samples were taken from the bottom unit, cleaned of the solvent, and weighed, and their densities and refractive indices were determined. The properties of the oil fraction and the multi-stage apparatus’s schematic diagram are provided in detail in previous literature [21].

3. Modeling Parameters and Equations

This simulation was performed in MATLAB by placing alumina and a total bulk liquid phase in a series of vessels (not counting the liquid in pores) [22,23]. The oil solution in the tank is hypothesized to consist of equal parts of the desired adsorbed component (nonparaffinic components) and paraffinic components [24,25].

An isothermal operation was assumed to streamline the simulation code for this process. It was also assumed that the tanks were perfectly mixed [24]. Therefore, the concentration of each tank’s output was the same as in the bulk liquid. Further, a uniform adsorbent concentration was assumed throughout the system. The isotherm for the system is used to derive a uniform distribution assumption for the adsorbed phase concentration relative to the fluid concentration. Parameters going into and coming out of the nth intermediate tank are listed in

Table 1. Simulation derivatives and parameters for an intermediate nth tank.

Simulation Parameters	Symbol at Time t	Symbol at Time t + δt
Volume of bulk liquid phase	$V_t$	$V_t$
Liquid flow rate	$V_{t-1}$	$V_{t-1}$
Input concentration	$C_{i-1}$	$C_{i-1}+\frac{d{C}_{i-1}}{dt} \delta t$
Output concentration	$C_i$	$C_i+\frac{d{C}_i}{dt} \delta t$
Pore concentration	$C_{p,i}$	$C_{p,i}+\frac{d{C}_{p,i}}{dt} \delta t$
Available pore volume	$m V_p$	$m V_p+m\frac{d{V}_p}{dt} \delta t$

.

The nonparaffinic components are subjected to a material balance for the time interval of δt.

Input − Output = Accumulation

where

Input = $V\left(C_{i-1}+\frac{d{C}_{i-1}}{dt} \delta t\right)\delta t$

Output = $V\left(C_i+\frac{d{C}_i}{dt} \delta t\right)\delta t$

Accumulation in the bulk liquid phase = $V_t\left(C_i+\frac{d{C}_i}{dt} \delta t\right)-V_t C_i$

Accumulation in the pores = $\left(m V_p+m\frac{d{V}_p}{dt} \delta t\right)\left(C_{p,m}+\frac{d{C}_{p,i}}{dt} \delta t\right)-m V_p C_{p,i}$

therefore,

$$\begin{array}{c}V C_{i-1} \delta t+V\frac{d{C}_{i-1}}{dt} \delta t \delta t-V C_i+V\frac{d{C}_i}{dt} \delta t \delta t\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}=V_t C_i+V_t\frac{d{C}_i}{dt} \delta t-V_t C_i+m V_p C_{p,i}+m V_p\frac{d{C}_{p,i}}{dt} \delta t+m{C}_{p,i}\frac{d{V}_p}{dt} \delta t\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+m\frac{d{V}_p}{dt} \frac{d{C}_{p,i}}{dt} \delta t \delta t-m V_p C_{p,i}\hfill \end{array}$$

We divide by δt, take the limits as δt → 0, and although V_p is a time-dependent variable, it will be treated as a constant here, i.e., $\frac{d{V}_p}{dt}=0$. The resulting equation is:

$$V \left(C_{i-1}-C_i\right)=V_t\frac{d{C}_i}{dt}+m V_p\frac{d{C}_{p,i}}{dt}$$

(1)

The instantaneous equilibrium assumption permits $\frac{d{C}_p}{dt}$ to be written as $\frac{d{C}_p}{dC} \frac{dC}{dt}$ where $\frac{d{C}_p}{dC}$ is the equilibrium isotherm slope. The above formula gives:

$$\frac{d{C}_i}{dt}=\frac{V \left(C_{i-1}-C_i\right)}{V_t\left(1+\frac{m V_p}{V_t} \frac{d{C}_{p,i}}{d{C}_i}\right)}$$

(2)

The system represents the nth series of completely mixed tanks in Equation (2). For the general situation of an arbitrarily shaped isotherm, the analytical solution of this set on the nth nonlinear ordinary differential equation is not achievable, and the numerical approach will be utilized for integration.

4. Description of MatLab Simulation

The nth set of equations is solved using a MatLab simulation program for the adsorption process. Figure 1 depicts the software’s high-level scheme. Applying a pulse of similar amplitude to the starting concentration of the non-paraffinic elements in the top unit is a good analogy for the operational mode of the simulation trials. However, these are the result of applying the equilibrium associated with the mass balance equation over the device, i.e.,

$$C_p^{*}=f\left(C_m\right)-available as an equilibrium isotherm$$

(3)

The immediate equilibrium indicates that $C_p^{*}=C_p$. The starting concentration of the second and later adsorption devices and the steady iso-octane stream concentration are both zero. The input to SepProg, which employs the ODE45 MatLab function, is recipes, initial conditions, and a database. SepProg then calls the derivative equations in the SepDeriv function and resolves the initial condition problem of the ordinary differential equations (ODEs) with regard to time.

The derivative $\frac{d{C}_i}{dt}$ is solved at every time interval for each unit. This solution requires the evaluation of $\frac{d{C}_p}{dC}$ The equilibrium diagram’s slope was found from the experimental results (C_p vs. C). The three regions of the C_p vs. C slope may be represented in the linear equation, as illustrated below.

$$\begin{array}{cc}{C}_p=4.75 C& for C<0.0041\\ C_p=11.86 C-0.0297& for 0.0041<C<0.0101\\ C_p=3.13 C+0.05& C>0.0101\end{array}\}$$

(4)

From the above equations, the values of $\frac{d{C}_p}{dC}$ are 4.758, 11.87, and 3.148 for the three regions, respectively.

Measurement of the sum of the areas under the curves depicting the effluent concentration vs. time at every position (indicating the quantity of the nonparaffinic chemicals) for each unit discovered in the PAC subfunction is calculated by calculating the integral from the following equation used as the effluent concentration–time curve computed in SepProg:

$$ZV\left(t\right)=V {{\displaystyle \int }}_0^{t}C dt$$

(5)

in which ZV(t) is the nonparaffinic compound’s weight at a specific time.

5. Results

5.1. Verification of the Computer Program

Before implementing the MatLab simulation software, the numerical consistency of the system is evaluated. This has nothing to do with the simulation’s applicability, which is solely considered in light of the findings of the experiments. In the event of a sudden variation in the process input, the following statement (assuming no adsorption occurs) yields the output concentration, C_n, from a sequence of n similarly sized, fully mixed tanks at any given time:

$$\frac{C_n}{C_0}=E_{\theta i}=\frac{{\theta }_i^{N-1}}{\left(N-1\right)!} e^{-{\theta }_i}$$

(6)

where ${\theta }_i=\frac{t}{t_i}=t \frac{V_{t_i}}{V}$.

A step increase is represented by the following affirmation:

$$\frac{C_n}{C_0}=f\left(t\right)=1-e^{\frac{-V_{t_i}}{V_t}}\left[1+\frac{V.t}{V_t}+\frac{1}{2!}\left(\frac{V.t}{V_t}\right)+\dots +\frac{1}{\left(n-1\right)!}{\left(\frac{V.t}{V_t}\right)}^{n-1}\right]$$

(7)

The MatLab Program simulated the same conditions, and experimental results and the simulation output are presented in Figure 2 and Figure 3.

It is feasible to use an analytical approach if the equilibrium data could be expressed as a computable integral equation, such as a Langmuir or polynomial-type isotherm. For the polynomial,

$$\begin{gathered} C_p=a_0+a_{1}C+a_2{C}^2+a_3{C}^3+\dots \\ \frac{d{C}_p}{dC}=a_1+2a_{2}C+3a_3{C}^2+\dots \end{gathered}$$

(8)

The integration of Equation (2) over one tank gives:

$${{\displaystyle \int }}_0^{t}dt={{\displaystyle \int }}_{C_m}^{C_1}\frac{V_t\left(1+\frac{m V_p}{V_t} \frac{d{C}_{p,i}}{dt}\right)}{V\left(C_0-C_1\right)} dC$$

(9)

where C₁ = C_m at t = 0 and C₁ = C₁ at t = t.

The outputs may conveniently be integrated into the concentration–time curve of the effluent by substituting the polynomial equation into the previous equation.

The F_365–375 oil, iso-octane, and aluminum isothermal data have been shown to be relatively well suited for three straight lines, i.e., dC_p/dC is, of course, continuously changing in value for each of the three regions over the time frame of the above equation’s integration. This results in additional simplification. Therefore, integrating Equation (9) is,

$$t=\frac{V_t\left[1+\frac{m V_p}{V_t} {\left(\frac{d{C}_{p,i}}{dt}\right)}_{constant}\right]}{-V}\ln \left(\frac{C_0-C_m}{C_0-C_1}\right)$$

(10)

The simulated outcomes predicted by MatLab software were contrasted with the equations above under a comparable set of conditions, as illustrated in

Table 2. Comparison of simulation results and the solution of Equation (10) for one unit (V = 2.5 mL/min., m = 300 g, Vt = 722 mL).

Time, min	Concentration, g/mL
	Simulation Results	Solution of Equation (10)	% Difference
0	4.1999 × 10−3	4.2024 × 10−3	0.06%
500	1.3387 × 10−3	1.3403 × 10−3	−0.12%
1000	4.2695 × 10−4	4.2752 × 10−4	−0.13%
1500	1.3579 × 10−4	1.3592 × 10−4	−0.09%
2000	4.5652 × 10−5	4.5722 × 10−5	−0.15%
2500	1.3384 × 10−5	1.3393 × 10−5	−0.07%
3000	2.4566 × 10−6	2.4604 × 10−6	−0.16%
3500	1.7757 × 10−5	1.7789 × 10−5	−0.18%
4000	2.4485 × 10−5	2.4486 × 10−5	−0.01%
4500	−3.1723 × 10−5	−3.1772 × 10−5	−0.15%
5000	2.5146 × 10−4	2.5189 × 10−4	−0.17%

. A superb agreement was again reached, verifying the MatLab program’s accuracy.

5.2. Simulation of Adsorption Parameter

The impact of changing the key variables of the effluent concentration profile’s simulation has been analyzed. Figure 4 depicts the effects of three runs varying the adsorbent’s pore volume. Comparatively low pore volume values suggest less oil was absorbed into the pores, leaving more significant oil discharging in the output solution.

Effluent percentage against total discharge quantity is displayed in Figure 5 for a three-unit setup with 300 g of alumina and the equivalent weight of “desirable aromatics.” The solution amount was mimicked in a variety of ways among units, but the overall liquid volume was kept constant.

Multiple distributions of the same amount of alumina were modeled across five units for similar systems. Figure 6 displays the concentration curves that were produced for the different distributions.

Figure 7 presents the simulation’s findings on the flow rate and discharge concentration relationship. It presents the results of 2, 3, and 5 mL discharge volumes. If the assumption of quick equilibrium is adopted, identical concentration–time discharge curves will ultimately arise from flow changes. This is something that is going to take place independently of whether or not high flow rates or low flow rates were implemented.

The most critical parameter in the adsorption process is the adsorbent/oil ratio. Higher adsorbent/oil ratio values are intended to result in more successful separations as opposed to situations with low adsorbent/oil ratio values when overloading the adsorbent is a significant problem. There are numerous techniques to model the adsorbent/oil ratio. The modeled adsorbent/oil ratios of 20:1, 100:3, and 100:1 were achieved by varying the quantity of oil introduced in the upper device. Their corresponding simulated discharge concentration curves are displayed in Figure 8 for a five-stage process employing 300 g of alumina.

6. Discussion

The sorption-desorption process causes influences that are analogous to those of a nonlinear isotherm when there are significant heat exchanges. For the system in place, this was deemed to be of little significance. The adsorbable material is quantitatively shifted into the succeeding units when there is substantial overloading in the top unit. Due to this, the concentration of these components in the discharge is greater than what the model would predict if the top unit were not overloaded. In order to provide a significant comparison between the theoretical and experimental findings, this data deviation is minimized. In order to accomplish this goal, the theoretical simulation of the system used the experimental discharge concentrations collected from the top unit as the input for the simulation of the subsequent units.

Figure 2 and Figure 3 show the simulation output and experimental of the pulsed and stepped inputs nonadoptive system for units 1 to Unit 5, respectively. It presents the values of MatLab used to model these similar circumstances. There is much consistency across the datasets. If the two numbers line up, then the simulation is mathematically accurate. If the equilibrium isotherm is written as a polynomial or as a Langmuir isotherm, then the adsorption problem in a single tank may be solved analytically.

The model is not significantly constrained by assuming isothermal operations since the apparatus (and other genuine adsorption systems) only have a small heat of adsorption. Prewetting the iso-octane adsorbent often reduces any further heat emission. Total mixing is seldom achieved in actuality. Therefore, the model’s stipulation will cause an overestimation of the separation efficiency. It is well known that the adsorbent’s pore volume changes with the mixture’s composition in contact with the solid. It might be not very easy to predict such variation when dealing with complex mixtures such as the oil solutions detailed in the present study. However, the heterogeneity of the oil might lead to less variation in the pores of an alumina structure that results from an average impact. Although the assumption of immediate equilibrium has often been considered a restriction of the model, it is implied by the low concentrations of solutions and the slow flow rates utilized here in this large model. Equation (1) is the same as simulating stirred tank reactors for the bulk liquid phase and solid phase.

The results of three separate runs, each with the pore volume of the adsorbent changed, are shown in Figure 4. The relatively low pore volume values show that a lower amount of oil was absorbed into the pores, resulting in more oil discharging into the output solution. When the pore volume of the adsorbent is high, it has the opposite effect and generates low concentration profiles for all units.

Lower concentration profiles are the consequence of the opposite effect, which is indicated by the higher value of the adsorbent pore volume. It has been demonstrated that variations in particle size throughout regeneration and reactivation might account for the anticipated value of the porous aluminum structure. These variations have only a minor impact on the projected concentration patterns. However, when the pore volume changes significantly, the effluents’ concentration profiles change significantly.

The observed changes in the particle size or humidity levels that might have occurred throughout the reactivation process are thought to have only a minor impact on the expected concentration curves. Nevertheless, when there is a significant change in the pre-volume, there is a noticeable change in the discharge concentration patterns.

Figure 5 depicts the fraction of discharge released relative to the overall amount of discharge. The assumed homogeneity of these shapes is shown as straight lines (all units of 800 mL). The effluent concentration gradient from unit 4 is typically shifted toward the origin as the volume of solution in each unit is progressively increased (400/700/900/1200 mL). Reducing the fluid volume from 1200 mL to 900 mL, 700 mL, and 400 mL produces reverse motion (dash-point lines in Figure 5). According to the effluent concentration on the cascade’s first, second, and third units, earlier and higher concentration peaks are caused by a smaller portion of the total solution. This is expected because a system with less overall solution will have higher concentrations, even if the amount of oil and adsorbent stays the same.

Figure 6 illustrates the concentration graphs that were generated for the various distributions. It has been shown that, in contrast to a uniform distribution of alumina, increasing the amount of alumina produces smaller concentration trends in the discharge of the following unit. It lessens alumina, leading to an increase in the concentrations. Because more nonparaffinic chemicals are maintained in the adsorbed phase with higher alumina utilization, concentration profiles would be lower. This is evident when looking at the concentration shapes of the effluent coming from, for example, the third unit. In this situation, the ratio of adsorbent to oil is relatively high.

Eventually, whether high or low flow rates were used, equal concentration–time effluent curves emerge from flow changes if the hypothesis of immediate equilibrium is assumed (Figure 7). When a constant mass transfer rate is considered, the flow rate fluctuation will affect these diffusivity process features, affecting the mass transfer coefficient value.

Figure 8 presents the findings of a simulation that examined the influence of the initial concentration on the concentration of the discharge. The beginning concentration levels of the upper devices were 0.0451, 0.0251, and 0.0721 g/mL. Higher adsorbent/oil ratios are related to a reduction in the concentration profiles, as seen in Figure 8. Adjusting the total volume of adsorbents in otherwise equivalent procedures is another approach to simulating the effects of the adsorbent/oil ratio.

As an example, Figure 5 depicts the effluent concentration from three-unit systems, which include, respectively, 620 g (160 + 200 + 260), 900 g (300 + 300 + 300), and 1140 g (540 + 340 + 260) of the total alumina amount employed in the device. All methods provide the same amount of oil. As adsorbent/oil ratios increase, decreasing concentration trends were produced (i.e., higher alumina) (Figure 6). Another method to illustrate the effects of the shift in the ratio of adsorbent/oil is to use more units. Figure 5’s solid lines show that as the number of units in a sequence increases, the discharge concentration curves exhibit characteristics indicative of more efficient separations, including progressively lower peaks and later peak timing.

In engineering, sustainability is about ensuring people can live within their means. Consequently, fewer resources must be utilized. Engineers can apply their knowledge and cutting-edge techniques to lessen the effects of consumption in order to achieve this goal. Computer simulation is one of these tools. Computer simulation helps create more efficient processes to decrease material use and provide the optimal material for a specific procedure. In addition to anticipating the record of discharge concentration, the model permits an investigation of the separation performance obtained by the multistage device.

7. Conclusions

Computational modeling simulating the separation process of the multi-stage adsorption device was developed as part of this project. The validity of a finite mass transfer frequency and an arbitrarily formed adsorption isotherm were confirmed. To apply the model to a specific system, the equilibrium isotherm, overall mass transfer coefficient, and pore volume of the adsorbent must all be known in the multi-stage apparatus. Thus, the accuracy of these requirements would determine how accurate the predicted outcomes would be. The model’s performance was examined by comparing its predictions to experimental data, which was then used to study how the system’s efficiency responds to different parameters. The research revealed that higher efficiencies were attained with more units, a bigger adsorbent pore volume, and more adsorbent and solution in the system. The efficiency was not significantly affected by the distribution of equal solution amount adsorption as long as enough slurrying was maintained in each unit.

This research helped shed light on the separation process of multi-stage adsorption. However, more work has to be performed to establish an empirical connection between the total mass transfer coefficient and the actual concentration. The isotherm data and the discharge concentration record will enable the kinetic parameters developed in this work to forecast the operation of the multi-stage device under a wide variety of situations.

It is possible to investigate several different adsorbent, oil, and solvent configurations. This might be performed for the purpose of determining whether or not the model is accurate in identifying how the multi-stage device would operate. Experiments would need to be run on each process in order to obtain the necessary data for the equilibrium isotherms and mass transfer coefficient. The performance of the adsorption equipment was evaluated and determined to be adequate. Any kind of advancement in the regulation of the flow rates between the stages would be helpful in increasing the precision of the achievement predictions.