# Progressive Cellular Architecture in Microscale Gas Chromatography for Broad Chemical Analyses
## 2 Theoretical Rationale
For isothermal separation at the ambient temperature, the energy consumption resulted from the gas pump that provided the carrier gas flow, which was at a constant rate. Therefore, the energy consumption was proportional to the analysis time. Broadening the analysis, Cases 4-6 were used to compare heated separations in which all the columns were temperature programmed to have a ramp rate of 30 \({}^{\circ}\)C/min starting at 25 \({}^{\circ}\)C at the beginning of the separations. Similar to Cases 1-3, Case 4 incorporated a Column B, Case 5 incorporated a Column A, whereas Case 6 mimicked a 2-cell PCA that incorporated Column A for alkanes C\({}_{3}\)-C\({}_{7}\) and Column B for alkanes C\({}_{8}\)-C\({}_{11}\). The computed retention times of various alkanes and separation resolution between pairs are shown in Table 3 for these cases. In order to assess the implications of these calculations, it is necessary to first estimate the total energy consumption. The total energy required for heating a separation column (\(E_{total}\)) is the sum of the energy needed to heat the thermal mass (\(E_{mass}\)) and the heat dissipation (\(E_{diss}\)) to the ambient by conduction and convection. The radiative thermal dissipation is negligible. Hence \(E_{diss}\) can be expressed as:  
\[E_{diss}=\beta\cdot\Delta T_{avg}\cdot t_{a};\ \beta=K+h\cdot A \tag{4}\]  
where \(\beta\) is the overall power consumption coefficient; \(K\) is the effective thermal conductance between the separation column and the ambient; \(h\) is the convection heat transfer coefficient; \(A\) is the effective device area available for convection; \(\Delta T_{avg}\) is the average temperature \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Case 1** & & **Case 2** & **Case 3** \\ \hline Stationary Phase & & & & & \\ Thickness (\(\mu\)m) & & & & & \\ Total Analysis & & & & & \\ Time (s) & & & & & \\ \multirow{2}{*}{Propane} & \multirow{2}{*}{15.3} & \multirow{2}{*}{0.7 \({}^{\circ}\)} & \multirow{2}{*}{6.0} & \multirow{2}{*}{6.0} & \multirow{2}{*}{6.0} \\  & & & & & \\ \multirow{2}{*}{Butane} & \multirow{2}{*}{15.7} & \multirow{2}{*}{46.2} & \multirow{2}{*}{12.8} & \multirow{2}{*}{12.8} \\  & & & & & \\  & & & & & \\  & & 6.6 & & 24.0 & 24.0 \\ Hexane & 19.8 & & 242.4 & & 242.4 \\  & & 15.6 & & 33.8 & 33.8 \\ \multirow{2}{*}{Heptane} & \multirow{2}{*}{29.4} & \multirow{2}{*}{25.5} & \multirow{2}{*}{27.0} & \multirow{2}{*}{33.7} \\  & & & & & \\  & & 25.5 & & 27.0 & 33.7 \\ Octane & 53.4 & & 1787 & & 53.4 \\  & & 33.7 & & 38.2 & 33.7 \\ Nonane & 115.8 & & 4694 & & 115.8 \\  & & 35.2 & & 35.9 & 35.2 \\ Decane & 270.6 & & 11,868 & & 270.6 \\  & & 35.2 & & 35.9 & 35.2 \\ Undecane & 636.0 & & 28,704 & & 633.6 \\ \hline \end{tabular}  
* All cases computed the separation of alkanes C\({}_{3}\)-C\({}_{11}\) using a 3 m long, 250 \(\mu\)m ID separation column. \({}^{\mathrm{b}}\) In Case 3, Col. A separated propane-heptane, whereas Col. B separated octane–undecane. \({}^{\mathrm{c}}\) Red font indicates undesirable outcomes compared to other cases.  
\end{table} 
Table 2: Computed \({}^{\mathrm{a}}\) values of retention times of various alkanes and separation resolution between pairs of alkanes at 25 \({}^{\circ}\)C for Cases 1–3. difference between the separation column and the ambient during analysis; \(t_{a}\) is the analysis time [33]. The \(E_{mass}\) is proportional to maximum temperature difference between the separation column and the ambient (\(\Delta T_{max}\)):  
\[E_{mass}=C_{th}\cdot\Delta T_{\max} \tag{5}\]  
where \(C_{th}\) is the thermal capacitance. The heat dissipation in microcolumns has been previously analyzed by [PERSON] et al. [34]. Using graphs reported in that work, the values of \(\beta\) = 0.014 W/\({}^{\circ}\)C and \(C_{th}\) = 0.7 J/\({}^{\circ}\)C can be assumed for the 3 m long silicon-glass microcolumn in this case study. Table 4 shows the total analysis time, energy consumption, and column temperature for Cases 4-6.