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Flow Rates in Liquid Chromatography, Gas Chromatography and Open Research Software Supercritical Fluid Chromatography: a Tool for Optimization

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Flow Rates in Liquid Chromatography, Gas Chromatography and Open Research Software Supercritical Fluid Chromatography: a Tool for Optimization Journal of Meurs, J 2016 Flow Rates in Liquid Chromatography, Gas Chromatography and open research software Supercritical Fluid Chromatography: A Tool for Optimization. Journal of Open Research Software, 4: e32, DOI: http://dx.doi.org/10.5334/jors.126 SOFTWARE METAPAPER Flow Rates in Liquid Chromatography, Gas Chromatography and Supercritical Fluid Chromatography: A Tool for Optimization Joris Meurs Department of BioAnalytical Chemistry, VU University Amsterdam, NL [email protected] This paper aimed to develop a standalone application for optimizing flow rates in liquid chromatography (LC), gas chromatography (GC) and supercritical fluid chromatography (SFC). To do so, Van Deemter’s equation, Knox’ equation and Golay’s equation were implemented in a MATLAB script and subsequently a graphical user interface (GUI) was created. The application will show the optimal flow rate or linear veloc- ity and the corresponding plate height for the set input parameters. Furthermore, a plot will be shown in which the plate height is plotted against the linear flow velocity. Hence, this application will give opti- mized flow rates for any set conditions with minimal effort. Keywords: Gas Chromatography; Liquid Chromatography; Supercritical Fluid Chromatography; Optimiza- tion; Flow rate (1) Overview Theory Introduction Packed columns Liquid chromatography (LC), gas chromatography (GC) Van Deemter’s equation and supercritical fluid chromatography (SFC) cover a wide According to Van Deemter, the HETP (H) can be calculated area for quantitative analysis of organic compounds in as follows [8]: analytical chemistry [1]. For quantification of a compound the system needs a certain resolution and a suitable æö2 2 2gD f( kd) f( kd) p m ç f ÷ detector. In an ideal case, peaks are narrow and follow Hd=+2l p + + ÷ u (1) uDDç ÷ a Gaussian distribution. However, several factors are èøsm contributing to peak broadening like detector, column, mobile phase, injection, column temperature and flow Which is often reduced to: rate [2]. To get an optimal performance of the LC, GC or SFC system the theoretical plate height (HETP) should be BD Cud 2 minimized. Van Deemter [3], Knox [4] and Giddings [5] H=+ Ad m +p (2) p uD developed theories for calculating the contributions m to peak broadening in a packed column. Mostly, Van Deemters equation and Knoxs equation are used for Equation 1 describes band broadening by eddy diffusion calculating the HETP based on the flow velocity [6]. For (A), longitudinal diffusion (B), mass transfer (C) at capillary columns, Golay [7] developed an equation different flow velocities in a packed column. Eddy for HETP calculation. diffusion is dependent on the quality of the packing of Since these calculations are time-consuming and the column [8]. A well-packed column has an A-term value susceptible for errors, this paper aimed to develop an of about 1 [9]. application in which this calculation can be performed Longitudinal diffusion occurs when a concentration quickly. To do so, Van Deemter’s equation, Knox’ equation gradient exists. Analyte molecules will migrate from and Golay’s equation were chosen for calculating the a high concentration area to a low concentration area HETP for packed and capillary columns respectively. according to Ficks law. This process is dependent on the These equation were implemented in a MATLAB algo- diffusion coefficient of the mobile phase (DM) and a rithm and thereafter a Graphical User Interface (GUI) was constant, which is dependent on the quality of the developed. column [8]. A common value for the B-term is 2 [9]. Art. e32, p. 2 of 5 Meurs: Flow Rates in Liquid Chromatography, Gas Chromatography and Supercritical Fluid Chromatography Mass transfer is a complex process, which is described 13 B h= Av ++ Cv (10) by a function f, the derivative of the function f, the particle v size (dp), the diffusion coefficient of the mobile phase (DM), the diffusion coefficient of the stationary phase (Ds) and In contrast to the Van Deemter equation, the A-term the thickness of the liquid layer on the stationary phase is dependent on the reduced velocity (v) according to (df) [8]. The function of f is dependent on the retention Knox [11]. Common for the A-term, B-term and C-term are factor (k). In general, the values for C is 0.05 [9]. 1, 2 and 0.05 respectively according to Knox [9, 11]. To obtain the minimum value of the HETP, Equation 2 should be differentiated with respect to u and solved for Capillary columns dH/du = 0. Differentiation of Equation 2 resulted in: The separation efficiency of a capillary column can be calculated by using Golay’s equation [12]: Cd 2 ¶H p BDm =-2 (3) ¶uD u 22 m 2D du d H=+mc f( k) + gk( ) f (11) uDD The result of Equation 3 should then be used in Equation 2. mf To do so, has to be Equation 2 rewritten to a function in which the linear velocity is expressed as a function of the HETP: In which Dm is the diffusion coefficient of the mobile 22phase, u is the linear velocity, Df is the diffusion coefficient Z+- Dm H AD mp d Z -+ Dm H AD mp d of the film, dc is the diameter of the column and df is the u =-22 (4) 22Cd ppCd film thickness. Further, f and g are functions of the reten- tion factor (k). These functions are described below [12]: where, 1++ 6kk 11 2 (12) fk( )= 2 22 2 2 96( 1-k ) Z= Dmp A d -24 Ad p H -+ BCd p H (5) However, more often one uses reduced parameters for 2k gk( )= 2 (13) calculating the optimal flow rate since these parameters 31( -k ) can be easily transferred between system. The reduced Van Deemter equation is defined as follows [9]: To minimize the H, Equation 11 should differentiated with respect to u and the derived equation should be solved for B h=+ A + Cv (6) u = 0. The final result of differentiating Equation 11 with v respect to u is as follows: df22 k2D dgk The reduced plate height (h) is defined by respectively the ¶H cf( ) m ( ) (14) = -+2 A-term, B-term, C-term and the reduced velocity. As can ¶uDmfu D be derived from this equation, h is independent of the particle size and diffusion coefficient. The A-term, B-term In Table 1 the units per variable and common values for and C-term can be calculated according to the following the variables are shown respectively. equations [10]: Diffusion coefficients of the mobile phases used in LC, GC and SFC have a magnitude of 10–9 m2 . s–1, 10–6 m2 . s–1 Ad= 2l p (7) and 10–8 m2 . s–1 respectively [12]. BD= 2g G (8) Implementation and architecture 8 k d 2 First, a blank graphical user interface (GUI) was generated C = L (9) (MATLAB R2016a) in which a pop-up menu was added p2 1+kD ( ) L for accessing all calculation sections. After, the equa- tions described in Section 1 were implemented into cor- Where, λ is the correction factor for the irregularity of the responding section of the script. For example, Equation column packing, dp is the particle size, γ is the labyrinth 1 is implemented in the section ‘Liquid Chromatography’ factor (value between 0 and 1), DG is the diffusion (packed column). To ensure only relevant data can be coefficient of the analyte in the gas phase, k is the reten- given as input, limitations have been set per calcula- tion factor, dL is the thickness of the stationary phase layer tion section. The limitations for Equation 1 are shown and D is the diffusion coefficient of the analyte in the below. For calculation, first linear spaced data ( L lins- stationary phase [10]. pace) is generated representing the linear flow velocity. Subsequently, the theoretical plate height is calculated. Knox’ equation Thereafter, the minimum plate height is determined by Another well-used equation for calculating the contribu- using the min function. In order to find the correspond- tion of flow rates to band broadening is the Knox equa- ing linear flow velocity, the data of the linear flow velocity tion [4]. This equation only consists of reduced parameters and the corresponding theoretical plate heights are placed (Equation 2.7) [11]: into a matrix. Using find, the row of the minimal plate Meurs: Flow Rates in Liquid Chromatography, Gas Chromatography and Supercritical Fluid Chromatography Art. e32, p. 3 of 5 Type of column Variable Unit Internal Optimal flow Calculated flow rate Packed column diameter (mm) rate (mL/min) (mL/min) 2 0.15 – 0.50 0.28 A Dimensionless 3 0.60 0.63 B Dimensionless 4 0.80–2.0 1.11 C Dimensionless D m2 . s –1 Table 3: Results of test calculations compared to guide- m lines of [13]. – u m . s 1 Calculated values are rounded to two decimals. dp µm k Dimensionless from [13]. In two cases, the calculated results deviate, Capillary column but do not show unreasonable results. As is mentioned in the Implementation and Architecture dc mm section, limitations are set in order to enhance the quality 2 . –1 of the calculations. Furthermore, by clicking on the push Dm m s buttons, information can be obtained on common values. 2 . –1 Df m s A brief tutorial (How-To-Use.pdf) of the application u m . s–1 can be found in the code repository. d µm p (2) Availability µ df m Operating system k Dimensionless The GUI was originally designed in MATLAB R2016a Table 1: Overview of units per variable. (Windows) and later also tested in MATLAB R2010a (Windows) without any problems.
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