Rapid multistep kinetic model generation from transient flow data† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c6re00109b Click here for additional data file.

SNAr reaction profiles were generated using an automated reactor, collected in less than 3 hours, and allowed accurate estimation of kinetic parameters.


S1 Experimental details
A schematic of the reactor configuration can be seen in Figure 1 and a labelled photograph is shown in Figure S1. Three Jasco PU-980 dual piston pumps were connected separately to solutions of (A) 1 M 2,4-difluoronitrobenzene 1 and internal standard N,N-

HPLC analysis
HPLC quantitative analysis was performed on an Agilent 1100 series LC using an Ascentis Express C18 reverse phase column (5 cm length, internal diameter 4.6 mm and 2.7 μm particle size), a water/acetonitrile mobile phase, and a 254 nm wavelength detector. A VICI Valco internal sample injector (0.06 μL volume) extracted aliquots of neat reaction for HPLC analysis without prior quench or dilution. All reported values were nonisolated and attained by HPLC based on normalisation of response factors using N-N-dimethylbenzamide as an internal standard. The HPLC was calibrated for the different reaction components. A typical HPLC is shown in Figure S2. For the pyrrolidine 2 reaction system the compounds were separated using an isocratic (51% water/49% acetonitrile), 1.5 mL min −1 total flow rate and a 2 min run time. For the morpholine 6 reaction system the compounds were separated using an isocratic (60% water/40% acetonitrile), 1.5 mL min −1 total flow rate and a 1.66 min run time.

Model determination
The experimental data was used to compare five different proposed kinetic models, with different concentration dependencies with respect to the aromatic component and pyrrolidine for each step in Scheme 1.
Reaction step rate = k x [aromatic] y [pyrrolidine] z Based on different statistical analysis techniques, the second order model M 1,1 gave a significantly better fit (Table S1)

S4 Joint confidence intervals
The covariance matrix measures how much two random variables change together, and was used to estimate the joint confidence intervals. The parameter joint confidence regions were examined to observe the multi-parameter dependence of one parameter upon another, a change in one parameter value will cause a change in the other parameter value ( Figure S3).
An elongated elliptical confidence region shows that the rate constant and activation energy for each pathway are correlated. A reduction in size of the joint confidence regions is achieved through exploration of sensitive regions of the experimental space. There will always be some correlation between the activation energy and rate constant for a particular step. The joint uncertainty was minimised for each parameter through conducting sufficient experiments where change would be observed; the uncertainties for k 3 and E a3 and for k 4 and E a4 were minimised by using a high molar excess of pyrrolidine 2 and operating the reactor at high temperature. Figure S3. 68, 95 and 99% joint confidence regions for the estimated parameters after all experiments for pyrrolidine 2 system.

S5 Parameter correlation
The correlation matrix is a normalised form of the covariance matrix, and describes the strength and direction of the relationship between two parameters. The correlation matrix is generated as an output as part of the DynoChem fitting report ( Figure S4). A correlation value near zero means one parameter can be changed without expecting a change in another parameter; −1 negatively correlated, +1 positively correlated. The highest correlation observed was between each rate constant and its activation energy as shown by the joint confidence intervals and the correlation matrix. The correlation matrix clearly shows that a sufficient experimental design space had been explored to minimise the correlation between different parameters. Figure S4. Correlation matrix for the S N Ar pyrrolidine reaction kinetic parameters.

S6 Exploring a wide design space
The dimensionless time plot, see Figure S5, represents the relative reactivity of all the input conditions used in the experimental study, a higher number indicates higher reactivity. The plot shows that experimental data were collected from very mild to aggressive conditions, within the constraints of the equipment, were studied thus allowing the kinetics for the whole reaction scheme to be fitted. A parity plot, a plot of the fitted responses vs. the observed responses to assess the quality of the fit is shown in Figure S6.

S7
Kinetic parameters for morpholine S N Ar system Scheme S1. S N Ar reaction of 2,4-difluoronitrobenzene 1 with morpholine 6.

S8
Experimental data and model fit to morpholine S N Ar system Figure S7. Concentration-time profiles for morpholine reaction system from simultaneous parameter fitting using

Linear gradient flow ramps for 2,4-fluoronitrobenzene
To establish the validity of the first order with respect to 2,4-difluoronitrobenzene 1, linear flow ramp gradients were developed. The relative ratio of the pump flow rates for P1 to P2 were varied to obtain different 2,4-difluoronitrobenzene 1 concentration levels whilst keeping the concentration of morpholine 6 constant ( Figure S9). A concentration-time profile was generated for each concentration level of 2,4-difluoronitrobenzene 1 ( Figure S10). These experiments confirmed the first order dependency with respect to 2,4-difluoronitrobenzene 1.   Table S1.

S10 Comparison to measurements under steady-state conditions
The steady-state experimental data was compared with the flow ramp gradient data. Each   Table S1.

S11 Influence of dispersion on kinetic parameter estimation
In this work we use the residence time in a coiled tube reactor to access kinetics of the S N AR system studied. If the coil reactor may be described by piston flow (also plug flow) the conversion for a first order reaction is: However, it is well established that dispersion can have a significant effect on the progress of a reaction (see for instance Rosas 1 ). In the 1950s Taylor  mm ID tubes. They found that the dispersion ratio reduces significantly below 1.
Theoretical work by Janssen 9 and Johnson and Kamm 10 showed dispersion ratio in coils may be correlated by DeSc 0.5 . Shetty and Vasudeva 6 fit their data to ln(DeSc 0.5 ) and Iyer and Vasudeva 7 extend the correlation with a second order term to describe data for which Re>100. These, and more recent data from others in small bore tube have been plotted in Figure S12, and it appears that the data is not consistent. We found that the data sets for Re<100 and could be correlated with Eq (3) This is similar to the correlation described by Van den Berg and Deelder 8 where ( is a critical number below which the coil behaves as a straight tube; it appears this value is unique for each literature data set. For the data measured in wide bore tube by Trivedi and Vasudeva 4 and Iyer and Vasudeva 7 the required value of α is lower than that given by eq(3).
In the 1960s the effect of dispersion was coupled to reaction system. Wehner and Wilhem 11 solved the general equation for the effect of dispersion on the conversion X of a solute due to a first order reaction with rate constant k: Eq (4) In which dispersion is characterised with the dimensionless Péclet number Pe . Eq(3) can = be simplified for low values of 1 : 1 This is done by applying a second order Taylor series expansion of ≈ 1 + 2 -2( Eq (5) As I very small, the second term in the denominator is much smaller and may be neglected so that: Eq (6) Here we assumed so (7) Combining equation (2) and (7) results and observing (i) in liquid systems the term D m becomes negligible compared to D s and (ii) it follows that: Eq (8) This shows that in a coil the observed rate constant k obs may be given as: Where the Damköhler number for radial diffusion may be defined as . The relative . For a second order reaction the reaction at a particular excess of pyrrolidine the rates are a factor 10 lower, and this will result in significantly lower errors, Table S4, thus the error in the rate constants with dispersion is very low for the overreaction pathways.
In conclusion, the estimated rate parameters may be up to 10% higher than those in Table 1, for the fastest rate up to 20%. This is based on a worst case scenario of a dispersion ratio of 1.   Two pump reservoir solutions were prepared with biphenyl (Sigma Aldrich 99.9%) solution in ethanol (9.75 mmol L -1 ) (VWR 99.96%) and acetone (VWR 99.9%). The pump feeds were mixed using a Swagelok tee-piece and fed through a PTFE coil of 6.1 mL (0.79 mm internal diameter, length 1232 cm, volume 6.1 mL from the tee-piece to the sample loop) before online HPLC analysis using a 4-port microvolume sampling loop (0.06 µL volume). The pumps were calibrated before running the experiment. The volume was measured by recording the time it taken for a dye tracer segmented by an air bubble to flow from the teepiece to the sample loop at 1 mL min −1 .The solvent pump was set to pump at 1.017 mL min -1 for 2.5 min followed by alternate periods of the biphenyl and solvent pump at 1.017 mL min -1 for 6 min intervals. The sample loop was set to acquire at 11.5 minute intervals, thus obtaining samples at different points through the pulse giving a representation of multiple samples at 30 s intervals of a single pulse ( Figure S13). Figure S14. Predicted F curve and RTD E(t) curve using the experimental data.