# Theoretical Analysis of Forced Segmented Temperature Gradients in Liquid Chromatography

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## Abstract

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## 1. Introduction

## 2. Non-Isothermal Equilibrium Model

## 3. Analytical Solution

#### Performance of the Chromatographic Process

## 4. Illustration and Discussion of Results

#### 4.1. Analysis of Single-Component Injection

#### 4.2. Analysis of Consecutive Injections of Multicomponent Mixtures

## 5. Materials and Methods

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Case (

**a**) A single-solute pulse is injected one time into the column. The pulse is allowed to face a low temperature in segment II. As a result, the pulse becomes narrower and more concentrated. Case (

**b**) Unlike case (

**b**), the injected pulse faces a high temperature in segment II. Then, the pulse becomes broader and less concentrated.

**Figure 2.**(

**a**) Illustration of a mixture of three components having three different Henry constants, periodically injected at $z={z}_{0}=0$. The cycle time is denoted by ${t}_{\mathrm{c}}$. (

**b**) Effluent concentrations at ${z}_{\mathrm{max}}$ for isothermal conditions and corresponding cycle time ${t}_{\mathrm{c},\mathrm{iso}}$ (case “Late Eluter”). In this case the retention times of the first two components are significantly smaller than the retention time of the third component. (

**c**) The same mixtures are injected using suitable segmented temperature gradients (non-isothermal operation). The pulses then elute closer to each other, leading to a reduction in the cycle time, that is, ${t}_{\mathrm{c},\mathrm{niso}}<{t}_{\mathrm{c},\mathrm{iso}}$.

**Figure 3.**Illustration of the solution behavior in $\alpha $ and $\gamma $ states for cooling of Type I: Switching is done from reference temperature, ${T}_{\mathrm{R}}$, to low temperature, ${T}_{\mathrm{L}}$. Concentrations, associated with left axes, are plotted along the axial coordinate z at particular times inside the column and plotted against the time coordinate t at different locations in the column. The total mass, ${\mathrm{N}}_{\mathrm{total}}$, associated with the right axes of z-plot and t-plot, are also plotted to guarantee the mass balance. The grey area represents segment II that is kept at a low temperature. Every point in the t-plot corresponds to two points in the z-plot and every point in the z-plot corresponds to two points in the t-plot. The blue profile in t-plot is showing the reaching times of the adsorption and desorption fronts to ${z}_{0}$. Corresponding to it is a blue profile in z-plot at ${\mathrm{t}}_{\mathrm{I}}={t}_{\mathrm{inj}}=20s$. The black profile in t-plot is showing the profile when the fronts reach ${z}_{\mathrm{m}}$. This crossing is plotted in the z-plot at several positions, that is, black one when the adsorption front reaches ${z}_{\mathrm{m}}$ and orange one when the desorption front reaches ${z}_{\mathrm{m}}$. The pink profile in the t-plot is plotted at ${z}_{\mathrm{max}}$. Whereas, the pink profile in the z-plot illustrates the situation when the adsorption front reaches ${z}_{\mathrm{max}}$. As $\beta $ is the crucial state, we have marked the area covering this sate and shown it in Figure 4 to delve the transition more clearly.

**Figure 4.**Illustration of the solution behavior in the intermediate state $\beta $ for cooling of Type I: The state $\beta $, highlighted by blue colored dashed boundary in Figure 3, is rotated clockwise along with the corresponding portion of the z-plot. The pulse is shown at two places by cyan color, using dashed line when most of its part is in segment I and, by dotted line when most of its part is in segment II. As the sapce-bandwith gradually changes in state $\beta $, the corresponding concentration also changes accordingly. All the space-bandwidths (${\mathrm{Z}}^{\alpha}$, ${\mathrm{Z}}^{\beta ,\mathrm{I}}$, ${\mathrm{Z}}^{\beta ,\mathrm{II}}$, ${\mathrm{Z}}^{\gamma}$) can be calculated using Equation (13).

**Figure 5.**Illustration of the solution behavior for heating of Type I: Concentrations are plotted along the left axes and the total mass of the component, ${\mathrm{N}}_{\mathrm{total}}$, showing the achievement of mass conservation, is plotted along the right axes of z-plot and t-plot. Segment II is kept at a high temperature. (

**a**) The pulse is accelerated but in segment II, space-bandwidths become broader as compared to segment I. The time-bandwidth stays the same as in the previous case. (

**b**) The blue color is showing the position of the pulse at ${\mathrm{t}}_{\mathrm{I}}={t}_{inj}=20\phantom{\rule{3.33333pt}{0ex}}s$, in black color, when its adsorption front reaches ${z}_{\mathrm{m}}$, in orange color, when its desorption front reaches ${z}_{\mathrm{m}}$ and it is plotted in pink color when its adsorption front reaches ${z}_{\mathrm{max}}$. (

**c**) The blue color profile is showing the arrival times of both the fronts at ${z}_{0}$, black color, when they reach ${z}_{\mathrm{m}}$ and the pink color profile is showing when they reach ${z}_{\mathrm{max}}$.

**Figure 6.**Illustration of the solution behavior for both cooling and heating of Type II: Concentrations and mass are again plotted along the left and right axes, respectively. (

**a**) The grey color is representing segment II with low temperature. The temperature is switched from reference to low at time ${t}_{1}$ when the desorption front of the pulse also enters segment II. The space-bandwidth is marked by black color and stays unchanged. The time-bandwidth of the pulse is marked by orange color which becomes longer. (

**b**) Here, the grey color is representing segment II with high temperature. Unlike case (a), here the time-bandwidth decreases. (

**c**,

**d**) are z-plots for cooling and heating respectively. Whereas. (

**e**,

**f**) are t-plots for cooling and heating respectively.

**Figure 7.**The space-time trajectories for three components under isothermal (3 injections) and the non-isothermal (4 injections) conditions. In the non-isothermal case, the first switching of the temperature is done at ${t}_{1,\mathrm{max}}$ and second switching at ${t}_{3,\mathrm{max}}$ and then they shift by ${t}_{\mathrm{c},\mathrm{niso}}$ alternatively.

**Table 1.**Reference parameters used in Section 4.1 (single-component injection).

Symbol | Quantity | Value Used in Simulation |
---|---|---|

${z}_{\mathrm{max}}$ | Length of the column | 0.1 m |

A | Cross-sectional area of the column | 0.0000196 ${\mathrm{m}}^{2}$ (diameter d = 0.5 cm) |

$\u03f5$ | Porosity of the column | 0.4 |

u | Interstitial velocity | 0.00167 m/s |

${t}_{\mathrm{inj}}$ | Injection time | 20 s |

${c}_{1,\mathrm{inj}}$ | Feed concentrations | 1 mmol/L |

${a}_{1}({T}_{\mathrm{R}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}298\phantom{\rule{3.33333pt}{0ex}}\mathrm{K})$ | Henry constant at reference temperature, ${T}_{\mathrm{R}}$ | 0.75 |

${a}_{1}({T}_{\mathrm{L}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}270\phantom{\rule{3.33333pt}{0ex}}\mathrm{K})$ | Henry constant at lower temperature, ${T}_{\mathrm{L}}$ | 1.73 |

${a}_{1}({T}_{\mathrm{H}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}360\phantom{\rule{3.33333pt}{0ex}}\mathrm{K})$ | Henry constant at higher temperature, ${T}_{\mathrm{H}}$ | 0.18 |

**Table 2.**Reference parameters used in Section 4.2 (Consecutive injections of multicomponent mixtures).

Symbol | Quantity | Value Used in Simulation |
---|---|---|

${z}_{\mathrm{max}}$ | Length of the column | 0.1 m |

A | Cross-sectional area of the column | 0.0000196 ${\mathrm{m}}^{2}$ (diameter d = 0.5 cm) |

$\u03f5$ | Porosity of the column | 0.4 |

u | Interstitial velocity | 0.00167 m/s |

${t}_{\mathrm{inj}}$ | Injection time | 10 s |

${c}_{i,\mathrm{inj}}$ | Feed concentrations | [1, 1, 1] mmol/L |

${a}_{1}({T}_{\mathrm{R}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}298\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 1 at ${T}_{\mathrm{R}}$ | 0.25 |

${a}_{1}({T}_{\mathrm{L}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}258\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 1 at ${T}_{\mathrm{L}}$ | 0.9 |

${a}_{1}({T}_{\mathrm{H}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}360\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 1 at ${T}_{\mathrm{H}}$ | 0.06 |

${a}_{2}({T}_{\mathrm{R}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}298\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 2 at ${T}_{\mathrm{R}}$ | 1.0 |

${a}_{2}({T}_{\mathrm{L}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}258\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 2 at ${T}_{\mathrm{L}}$ | 3.6 |

${a}_{2}({T}_{\mathrm{H}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}360\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 2 at ${T}_{\mathrm{H}}$ | 0.25 |

${a}_{3}({T}_{\mathrm{R}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}298\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 3 at ${T}_{\mathrm{R}}$ | 3.0 |

${a}_{3}({T}_{\mathrm{L}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}258\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 3 at ${T}_{\mathrm{L}}$ | 10.8 |

${a}_{3}({T}_{\mathrm{H}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}360\phantom{\rule{4pt}{0ex}}\mathrm{K})$ | Henry constant for comp. 3 at ${T}_{\mathrm{H}}$ | 0.74 |

**Table 3.**Results for Section 4.2 (Consecutive injections of multicomponent mixtures).

Symbol | Quantity | Value Obtained |
---|---|---|

${t}_{\mathrm{c},\mathrm{iso}}$ | Cycle time in isothermal case | 257 s |

${t}_{\mathrm{c},\mathrm{niso}}$ | Cycle time in non-isothermal case | 197 s |

${t}_{\mathrm{c},\mathrm{niso},\mathrm{hyp}}$ | Hypothetical cycle time in non-isothermal case | 129 s |

${t}_{1,\mathrm{m}}$ | Time taken by the adsorption front component 1 to reach ${z}_{\mathrm{m}}$ | 41 s |

${t}_{1,\mathrm{max}}$ | Time taken by the desorption front component 1 to reach ${z}_{\mathrm{max}}$ (non-isothermal case) | 120 s |

P${}_{i,\mathrm{iso}}$ | Production rate of each component under isothermal conditions | $5.1\times {10}^{-7}$ mmol/s |

P${}_{i,\mathrm{niso}}$ | Production rate of each component under non-isothermal conditions | $6.6\times {10}^{-7}$ mmol/s |

${\mathrm{P}}_{\mathrm{inc}}$ | Increase in the overall production rate | 30% (approx.) |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hayat, A.; An, X.; Qamar, S.; Warnecke, G.; Seidel-Morgenstern, A.
Theoretical Analysis of Forced Segmented Temperature Gradients in Liquid Chromatography. *Processes* **2019**, *7*, 846.
https://doi.org/10.3390/pr7110846

**AMA Style**

Hayat A, An X, Qamar S, Warnecke G, Seidel-Morgenstern A.
Theoretical Analysis of Forced Segmented Temperature Gradients in Liquid Chromatography. *Processes*. 2019; 7(11):846.
https://doi.org/10.3390/pr7110846

**Chicago/Turabian Style**

Hayat, Adnan, Xinghai An, Shamsul Qamar, Gerald Warnecke, and Andreas Seidel-Morgenstern.
2019. "Theoretical Analysis of Forced Segmented Temperature Gradients in Liquid Chromatography" *Processes* 7, no. 11: 846.
https://doi.org/10.3390/pr7110846