Since the diameter of the CoDIR increases by a factor of approximately 9 (from 0.5–4.6 mm) along the reactor, it should be discussed to what extent the model assumptions are appropriate and which limitations exist.
A critical parameter in our model is the Nusselt number. As the Reynolds number for the initial diameter of 0.5 mm is 1.4, the laminar flow edge case for a Nusselt number of 3.656 can be assumed. This results in a heat transmittance of 998 W/(m2 K) for the initial diameter. In the case of heat transfer in laminar flow regions, heat transport only occurs by thermal conduction. Therefore, the Nusselt number for a laminar system is the lowest possible value. Throughout the reactor, the Reynolds number further decreases as the volumetric flow rate is held constant and the diameter increases. Thus, safe operation as a result of the ninefold diameter increase is not expected. Furthermore, for a real reactor system, the Nusselt number can be easily enhanced by inducing secondary flow with internals such as static mixers, chicanes, or zigzag channels.
Assuming fully developed hydrodynamic and thermal profiles for the entire reactor is applicable if the hydrodynamic and thermal entry lengths are significantly smaller than the total length of the reactor. Using Eq. (5), a hydrodynamic entry length of 0.035 mm is calculated, and using Eq. (6), a thermal entry length of 1.6 mm is calculated. Both values are several orders of magnitude smaller than the reactor length of approximately 89 mm [11]. With regard to heat transfer, compared to the fully developed region, the thermal entrance region shows an intensified heat transfer rate.
Notably, for a reactive system, the Nusselt number would be higher than that predicted for the laminar edge case [3]. As a higher Nusselt number leads to a faster drop in temperature, the risk of thermal runaway further decreases.
Plug flow can only be assumed for tubular channels if radial compared to axial diffusion is sufficiently high [16]. In general, plug flow provides a good representation of the prevailing flow regime in microreactors due to small tube dimensions and therefore short radial diffusion paths compared to axial ones. The dispersion model describes the degree of local back mixing for laminar flow in circular microchannels [16]. Axial dispersion can be calculated according to Taylor & Aris as a function of the tube radius, the mean velocity, and the coefficient of molecular diffusion [17,18,19]. As a result of the ninefold diameter increase throughout the CoDIR, the radial diffusion time increases by approximately a factor of 81 toward the end of the reactor [20]. Especially in the later stages of the CoDIR, plug flow behavior can therefore probably only be achieved by enhancing radial dispersion, e.g., by installing additional mixing elements such as static mixers [21, 22], chicanes [23], coiled flow inverters [24, 25] or zigzag channels [26, 27].
Regarding mass transfer, the model assumes one ideally mixed reactant phase. Therefore, the reaction rate is not limited by mass transport and is only dependent on the intrinsic reaction kinetics. Since liquid phase nitration reactions often involve multiphase systems, mass transfer limitations affect the reaction kinetics and would have to be taken into account [13, 28].
Furthermore, a simplified fictitious second-order reaction is used without side reactions taking place. The principle of the CoDIR can thus be illustrated in a manner that is decoupled from the chemistry. However, the improvements in productivity obtained in this way do not accurately represent the results obtained experimentally. A tendency toward increased productivity, however, is also given in practice.
An independent parameter variation was performed to evaluate the degree of parametric sensitivity. The results show no thermal runaway among any of the tested variables. A sensitivity analysis, according to Semenov, Gray and Renken, is a widely accepted approach to quantify this particular problem [29, 30]. Because the traditional approach cannot represent the sensitivity of the CoDIR, an adapted approach can be found in the supporting material.
Finally, the concept of a continuous diameter expansion can be used as an additional modular element in the design of microchannels. Compared to a sudden increase in diameter, a continuous increase offers significant advantages, as the productivity increases while the occurrence of temperature peaks is reduced. Additionally, the proposed approach even offers the option to optimize the selectivity of complex reaction systems. Different temperature zones could be created to inhibit specific side reactions based on deviating activation energies.
The drawback is higher simulation efforts since the optimal reactor profile must be tailored to the dedicated reactions/process. As a result, one becomes somewhat inflexible since small changes in the process require a new optimal wall profile to be simulated. Therefore, we propose a possible application of the CoDIR in ongoing production processes to boost productivity while reducing temperature fluctuations. Therefore, with technologies such as metal 3D printing, a new flexible module can be derived.