# Efficient Simulation of Chromatographic Processes Using the Conservation Element/Solution Element Method

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Models of Chromatographic Processes

^{−1}), D

_{ax}is the axial dispersion coefficient (m

^{2}·s

^{−1}), k

_{m}is the mass transfer coefficient between the two phases (s

^{−1}), C is the liquid-phase concentration (g·L

^{−1}), q is the solid-phase concentration (g·L

^{−1}), q* is the solid-phase concentration at the interphase boundary in equilibrium with the liquid phase (g·L

^{−1}), t and z are time and spatial coordinates, respectively, (s) and (m), and i and k are the component and column indices, respectively. This model formulation assumes isothermal operation, a constant void fraction, and a constant mobile-phase velocity inside each of the columns. The apparent axial dispersion coefficient D

_{ax}lumps together all effects leading to band broadening in addition to the finite mass transfer resistance, which is known to have a similar effect [2]. Below, only the limiting case when D

_{ax}$\to $ 0 is considered which is valid for efficient columns with a high number of theoretical stages N. The algebraic equation describes the thermodynamic equilibrium between the solid and the liquid phase and represents the adsorption isotherm. In the present work, two types of adsorption isotherms are used:

^{−1}), and r is the component index. In the limiting case where the mass transfer is instantaneous (i.e., k

_{m,i}$\to \infty $ and ${q}_{i,\text{}k}\to {q}_{i,\text{}k}^{*}$) and there is negligible dispersion, the ideal equilibrium model [1] is obtained.

## 3. Conservation Element/Solution Element (CE/SE) Method

_{ax}$\to $ 0 can be presented in the following vector form (the indices are omitted for simplicity):

**u**is vector of state variables,

**f**is vector of fluxes, and

**p**is vector of source terms. Equation (1) can then be written in the following form:

**u**at the j-th spatial point and n-th time instant is

**u**

_{z}and

**f**

_{t}are the discrete analogues of the derivatives $\partial \mathit{u}/\partial z$ and $\partial \mathit{f}/\partial t$. The current value of ${\mathit{u}}_{j}^{n}$ is calculated from the already available values of ${\mathit{u}}_{j\pm 1/2}^{n-1/2},{\mathit{p}}_{j\pm 1/2}^{n-1/2},{\mathit{u}}_{z,j\pm 1/2}^{n-1/2},{\mathit{f}}_{j\pm 1/2}^{n-1/2},$ and ${\mathit{f}}_{t,j\pm 1/2}^{n-1/2}$ at the previous time instant. The different versions of the CE/SE method only differ in the way the spatial derivative

**u**

_{z}of the state variables is calculated. In the current work, the proposition of [21] is used.

_{0}is

_{t}is the numerical analogue of $\partial \mathit{u}/\partial \mathit{t}$ and is

## 4. Results

#### 4.1. Single Column with the Ideal Equilibrium Model

- (i)
- Specify the simulation time t
_{sim}; - (ii)
- Specify the number of time steps N
_{t}, i.e., the number of conservation elements; - (iii)
- Calculate the time step size $\u2206t$;
- (iv)
- Specify $\overline{\mathrm{CFL}}$ ($0<\overline{\mathrm{CFL}}\le 1)$;
- (v)
- Calculate the spatial step size $\u2206z$ from Equation (20);
- (vi)
- Continue with the reversed CE/SE method.

**ode45**which uses the Runge–Kutta method [25]. Doubled computational time was found using

**ode23**, which is also a Runge–Kutta method but of lower order. As an alternative for stiff systems, the

**ode15s**solver was also used but it was threefold slower than ode45. This indicates that the present system is nonstiff.

#### 4.2. Single Column with the LDF Model

_{m}and the CE/SE method parameters are given in Table 2. The values of the mass transfer coefficients were chosen to be high enough such that physical mass transfer was negligible.

_{m},

_{i}concentration fronts were less steep compared to those in Figure 3a. In consequence, a lower number of grid points was required to resolve the fronts in both cases (CE/SE and MOL), leading to lower computational times.

#### 4.3. Binary SMB Process with the LDF Model

- − Desorbent node

- − Extract node

- − Feed node

- − Raffinate node

_{sw}is the switching time (s), V

_{col}is the column volume (m

^{3}), and p is the zone index. From these values, the internal flowrates Q

_{int}(m

^{3}·s

^{−1}) in each zone of the SMB plant could be calculated.

_{i}and b

_{i}, the component A (blue) had lower affinity to the solid phase and flowed out of the raffinate port, while the component B (red) had higher affinity to the solid phase and flowed out from the extract port.

#### 4.4. Ternary Center-Cut Eight-Zone SMB Process with Linear Isotherms and the Ideal Equilibrium Model

- − First desorbent node

- − First extract node

- − First feed node

- − First raffinate node

- − Second desorbent node

- − Second extract node

- − Second feed node

- − Second raffinate node

#### 4.5. Ternary Center-Cut Eight-Zone SMB Process with Langmuir Isotherms and the LDF Model

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

b_{i,k} | retention factor in the Langmuir isotherm expression (L·g^{−1}) |

C_{i,k} | liquid phase concentration (g·L^{−1}) |

$\mathrm{CFL}$ | Courant–Friedrichs–Lewy number |

$\overline{\mathrm{CFL}}$ | Courant–Friedrichs–Lewy number in the reversed CE/SE method |

D_{ax} | axial dispersion coefficient (m^{2}·s^{−1}) |

D_{col} | column diameter (m) |

D_{pipe} | pipe diameter (m) |

f | vector of fluxes |

H_{i,k} | adsorption Henry coefficient |

${k}_{m}{}_{i,k}$ | mass transfer coefficient (s^{−1}) |

L_{col} | column length (m) |

m | dimensionless flowrate ratio of liquid and solid phases |

N | number of theoretical stages of the cell model |

N_{col} | number of columns in the SMB plant |

N_{comp} | number of components |

N_{t} | number of time steps |

N_{z} | number of spatial steps |

p | vector of source terms |

q_{i,k} | solid phase concentration (g·L^{−1}) |

${q}_{i,k}^{*}$ | solid phase concentration at the interphase in equilibrium with the liquid phase (g·L^{−1}) |

Q | volumetric flowrate (m^{3}·s^{−1}) |

t | time coordinate (s) |

t_{sim} | simulation time (s) |

t_{sw} | switching time (s) |

u | vector of state variables |

V_{col} | column volume (m^{3}) |

v_{k} | liquid phase velocity (m·s^{−1}) |

z | spatial coordinate (m) |

Δt | time step size (s) |

Δz | spatial step size (m) |

ε | column void fraction |

τ | adjusted time |

A, B, C | different components |

De | desorbent stream |

Ex | extract stream |

Fe | feed stream |

i, r | component indices $\left(i,r=1,2,\dots ,{N}_{\mathrm{comp}}\right)$ |

in | column inlet |

int | internal flowrate |

j | spatial coordinate index $\left(j=1,2,\dots ,{N}_{z}\right)$ |

k | column index $\left(k=1,2,\dots ,{N}_{\mathrm{col}}\right)$ |

n | time coordinate index $\left(n=1,2,\dots ,{N}_{t}\right)$ |

out | column outlet |

p | zone index $\left(p=\mathrm{I},\text{}\mathrm{II},\text{}\mathrm{III},\text{}\mathrm{IV}\right)$ |

Ra | raffinate stream |

## Appendix A. Direct Conversion of the Equilibrium Model to the Form Given by Equation (10)

**f**instead of

**u**according to

**q**. For the Langmuir isotherm, this is always possible for any number of components.

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**Figure 1.**Computational schemes of the conservation element/solution element (CE/SE) method: (

**a**) standard CE/SE method; (

**b**) reversed CE/SE method.

**Figure 2.**Single-column binary chromatographic process with Langmuir isotherms—ideal equilibrium model. (

**a**) Concentration profiles along the column calculated using the two numerical methods and the analaytical solution. (

**b**) Comparison of the computational times for each of the numerical methods.

**Figure 3.**Single-column binary chromatographic process with Langmuir isotherms—LDF model. (

**a**) Concentration profiles along the column calculated by two numerical methods and the analaytical solution. (

**b**) Comparison of the computational times for each of the numerical methods.

**Figure 4.**Binary SMB chromatographic process with Langmuir isotherms—LDF model. (

**a**) Process configuration. (

**b**) Comparison of the computational times for each of the methods. (

**c**) Concentration profiles along the SMB plant calculated using the CE/SE method. (

**d**) Concentration profiles along the SMB plant calculated using MOL. Dashed curves in (

**c**,

**d**) are at the beginning of each cycle, while solid curves are at the end.

**Figure 5.**Ternary center-cut eight-zone SMB chromatographic process with raffinate recycle with linear isotherms—ideal equilibrium model. (

**a**) Process configuration. (

**b**) Comparison of the computational times for each of the methods. (

**c**) Concentration profiles along the SMB plant calculated using CE/SE method. (

**d**) Concentration profiles along the SMB plant calculated using MOL. Dashed curves in (

**c**,

**d**) are at the beginning of each cycle, while solid curves are at the end.

**Figure 6.**Ternary center-cut eight-zone SMB chromatographic process with raffinate recycle with Langmuir isotherms—LDF model. (

**a**) Concentration profiles along the SMB plant calculated using CE/SE method. (

**b**) Concentration profiles along the SMB plant calculated using MOL. (

**c**) Comparison of the computational times for each of the methods. Dashed curves in (

**a**,

**b**) are at the beginning of each cycle, while solid curves are at the end.

**Table 1.**Simulation parameters and reversed CE/SE method parameters for Example 1 (single-column binary process with Langmuir isotherms described by the ideal equilibrium model).

Quantity | Value | Quantity | Value | Quantity | Value |
---|---|---|---|---|---|

L_{col} (m) | 1.0 | H_{A} | 2 | H_{B} | 4 |

v (m·s^{−1}) | 0.1 | b_{A} (L·g^{−1}) | 2 | b_{B} (L·g^{−1}) | 4 |

ε | 0.8 | C_{Fe,A} (g·L^{−1}) | 0.9 | C_{Fe,B} (g·L^{−1}) | 0.8 |

t_{sim} (s) | 10 | N_{t} | 501 | $\overline{\mathrm{CFL}}$ | 0.4 |

**Table 2.**Simulation parameters and CE/SE method parameters for Example 2 (single-column binary process with Langmuir isotherms described by the linear driving force (LDF) model).

Quantity | Value | Quantity | Value |
---|---|---|---|

k_{m},_{A} (s^{−1}) | 10 | N_{z} | 101 |

k_{m},_{B} (s^{−1}) | 10 | CFL | 0.4 |

**Table 3.**Simulation parameters for Example 3 (binary SMB process with Langmuir isotherms described by the LDF model).

Quantity | Value | Quantity | Value | Quantity | Value | Quantity | Value |
---|---|---|---|---|---|---|---|

L_{col} (m) | 0.5 | m_{I} | 5 | H_{A} | 2 | H_{B} | 4 |

D_{col} (m) | 0.02 | m_{II} | 1.8 | b_{A} (L·g^{−1}) | 0.2 | b_{B} (L·g^{−1}) | 0.4 |

ε | 0.8 | m_{III} | 2.8 | k_{m},_{A} (s^{−1}) | 10 | k_{m},_{B} (s^{−1}) | 10 |

D_{pipe} (m) | 0.002 | m_{IV} | 1.3 | C_{Fe,A} (g·L^{−1}) | 0.9 | C_{Fe,B} (g·L^{−1}) | 0.7 |

t_{sw} (s) | 40 |

**Table 4.**Simulation parameters for Example 4 (ternary center-cut eight-zone SMB process with linear isotherms described by the ideal equilibrium model).

Quantity | Value | Quantity | Value | Quantity | Value | Quantity | Value | Quantity | Value |
---|---|---|---|---|---|---|---|---|---|

L_{col} (m) | 0.5 | m_{I,1} | 2.55 | m_{I,2} | 1.82 | H_{A} | 1.1 | C_{Fe,A} (g·L^{−1}) | 0.9 |

D_{col} (m) | 0.02 | m_{II,1} | 1.57 | m_{II,2} | 1.22 | H_{B} | 1.7 | C_{Fe,B} (g·L^{−1}) | 0.8 |

ε | 0.75 | m_{III,1} | 2.19 | m_{III,2} | 2.55 | H_{C} | 2.5 | C_{Fe,C} (g·L^{−1}) | 0.7 |

D_{pipe} (m) | 0.002 | m_{IV,1} | 0.86 | m_{IV,2} | 1.01 | t_{sw} (s) | 60 |

**Table 5.**Simulation parameters for Example 5 (ternary center-cut eight-zone SMB process with Langmuir isotherms described by the LDF model).

Quantity | Value | Quantity | Value | Quantity | Value | Quantity | Value | Quantity | Value |
---|---|---|---|---|---|---|---|---|---|

m_{I,1} | 2.55 | m_{I,2} | 2.10 | H_{A} | 1 | H_{B} | 2 | H_{C} | 2.5 |

m_{II,1} | 1.893 | m_{II,2} | 0.928 | b_{A} (L·g^{−1}) | 1 | b_{B} (L·g^{−1}) | 2 | b_{C} (L·g^{−1}) | 2.5 |

m_{III,1} | 1.90 | m_{III,2} | 1.99 | k_{m},_{A} (s^{−1}) | 10 | k_{m},_{B} (s^{−1}) | 10 | k_{m},_{C} (s^{−1}) | 10 |

m_{IV,1} | 0.915 | m_{IV,2} | 0.85 | C_{Fe,A} (g·L^{−1}) | 0.5 | C_{Fe,B} (g·L^{−1}) | 5.0 | C_{Fe,C} (g·L^{−1}) | 1.5 |

t_{sw} (s) | 40 |

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**MDPI and ACS Style**

Chernev, V.P.; Vande Wouwer, A.; Kienle, A.
Efficient Simulation of Chromatographic Processes Using the Conservation Element/Solution Element Method. *Processes* **2020**, *8*, 1316.
https://doi.org/10.3390/pr8101316

**AMA Style**

Chernev VP, Vande Wouwer A, Kienle A.
Efficient Simulation of Chromatographic Processes Using the Conservation Element/Solution Element Method. *Processes*. 2020; 8(10):1316.
https://doi.org/10.3390/pr8101316

**Chicago/Turabian Style**

Chernev, Valentin Plamenov, Alain Vande Wouwer, and Achim Kienle.
2020. "Efficient Simulation of Chromatographic Processes Using the Conservation Element/Solution Element Method" *Processes* 8, no. 10: 1316.
https://doi.org/10.3390/pr8101316