Heat Transfer and Residence Time Distribution in Plug Flow Continuous Oscillatory Baffled Crystallizers

Heat transfer coefficients in a continuous oscillatory baffled crystallizer (COBC) with a nominal internal diameter of 15 mm have been determined as a function of flow and oscillatory conditions typically used under processing conditions. Residence time distribution measurements show a near-plug flow with high Peclet numbers on the order of 100–1000 s, although there was significant oscillation damping in longer COBC setups. Very rapid heat transfer was found under typical conditions, with overall heat transfer coefficients on the order of 100 s W m–2 K–1. Furthermore, poor mixing in the COBC cooling jacket was observed when lower jacket flow rates were implemented in an attempt to decrease the rate of heat transfer in order to achieve more gradual temperature profile along the crystallizer length. Utilizing the experimentally determined overall heat transfer coefficients, a theoretical case study is presented to investigate the effects of the heat transfer rate on temperature and supersaturation profiles and to highlight potential fouling issues during a continuous plug flow cooling crystallization.


Continuous Oscillatory Baffled Crystalliser Theory and Operation
The basic principle of the continuous oscillatory baffled crystalliser (COBC) stems from a series of periodically spaced orifice baffles on to which an oscillation is applied. As the flow interacts with the baffles, eddies are created. This repeating oscillation cycle ensures strong radial mixing. Figure S1 illustrates the flow interactions inside the COBC. mixing intensity in a non-baffled and non-oscillated tubular system of the same dimensions, the net flow rate would need to be increased from 50 g min -1 to 1200 g min -1 , which would result in an approximate residence time of only 4 minutes i.e., 5% of oscillatory baffle flow.
When utilizing oscillatory flow technology there are namely three variables that may be selected for operation for a fixed reactor length, oscillatory frequency, oscillatory amplitude and net flow rate. 1, 2 Selection of optimal conditions will be dictated by the process, for example oscillatory conditions will be selected which are capable of suspending material at completed desupersaturation, also for a fixed length of a system a particular residence time is required, to achieve sufficient growth and size of the resulting particle. To achieve this various oscillatory conditions, residence times and net flow rates will be employed depending on the compound of interest investigated and associated crystallisation kinetics. Thus, a range of operating conditions within the system needs to be studied to provide an operation design space for crystallisation applications.

Residence Time Distribution Measurements in Literature
There have been many studies measuring residence time distributions (RTDs) in oscillatory flow systems with a wide range of geometries and operating conditions. From these studies it has been found that ideal operating conditions include working within velocity ratios between 2 -10 and Re o ≥100. Table S1 shows a summary of these previous reports on RTD behaviour within oscillatory flow systems. The range of geometries makes it difficult to make appropriate comparisons.
where is the experimental time with a given concentration, . The variance, is calculated using: Given experimentally a perfect pulse of tracer cannot be achieved, the imperfect pulse method is applied to provide a more realistic description of the experimental tracer profile.
The imperfect pulse method for determining axial dispersion accounts for the finite dispersion of tracer concentration over the duration of the injection. The basic differential equation representing the dispersion model, used previously to describe dispersion in oscillatory flow mixing, is represented by: The imperfect pulse model was fitted to experimental data, and completed using procedures described in detail elsewhere. 5 Concentrations were normalised using the following equation: The mass of tracer flowing through the system was calculated from the area under the RTD curve (summation of ), where is the mass of tracer (kg) and is fluid velocity m 3 s -1 . ∆ This material balance allows confirmation of tracer recovery once the experiment is complete.

S6
A transfer function, , was applied to convert the upstream response to the downstream ( ) response, described by: Using the number, the axial dispersion coefficient was calculated. The model response 2 at position 2 was calculated by convolving the measured input response and the where is the time response downstream and is the time response from the upstream response. The model calculated response was then compared to the experimental 2 ( ) ′ 2 ( ) using the target function defined as: The number was varied to find the value resulting in minimal difference between the calculated and predicted responses, this value was taken as the number of the system. A representative result obtained using this fitting procedure is shown in Figure S2. Using the Pe number obtained from the minimal difference between predicted and observed values, an overlay of the raw data with the predicted response can be plotted ( Figure S2, right panel).

Studies
The full set of experimental conditions for the heat transfer studies are given in Table S3. The full experimental temperature data obtained from single and double oscillations setups can be found in Figure S3 and Figure S4 respectively.

Full Description of Temperature Profile Model
A method for determining the overall heat transfer coefficient ( ) for oscillatory flow in baffled tubes from experimental temperature data has been outlined previously. 16 Firstly, it considers that U can be determined from the following equation for the heat transfer rate ( ): Secondly, it considers that the log mean temperature difference ( ) over a COBC section can be given by: By substituting the expression for the log mean temperature difference into the expression for , the overall heat transfer coefficient for a COBC section can be determined by the following equation: According to Newton's law of cooling/heating, the temperature of the solution ( ) will vary 1 along the COBC length ( ) according to the following differential equation: When the solution flows through a straight represents the temperature of the jacket fluid The values in these expressions are the heat exchange area per unit volume ( ), the cross-  Using the overall heat transfer coefficients, and making certain assumptions, the temperature profile along the COBC can be modelled. Depending on which assumptions are made, the temperature profile model can have three levels. Level 1 assumes that the jacket fluid S11 temperature is constant and that there are no heat losses from the jacket. Level 2 assumes that the jacket fluid varies and that there are no heat losses from the jacket. Level 3 assumes that the jacket fluid varies and that there are heat losses from the jacket. For each level, the relevant differential equations must be solved either numerically or analytically in order to model the temperature profile along the COBC. In order to implement the model, the solubility of paracetamol in this solvent system must be known over the temperature range which is explored. Solubility data for this system have been obtained previously 17 and are plotted in Figure S5. S14 Figure S5. Experimental solubility of paracetamol in water/IPA (60/40 wt%).
The equation used to fit this solubility curve is as follows: The experimental procedure involves cooling a saturated paracetamol solution from 70 °C to a certain extent over a fixed COBC length (5.5m). Utilising a particular overall heat transfer coefficient (cooling method) will result in the solution being cooled to a particular final temperature over the COBC length. Using a basis of 1 kg solvent, the values of the constants involved in modelling the temperature and supersaturation profiles throughout the crystallisation process are listed in Table S4.

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The variables which will change over the course of the crystallisation process as they are being determined by the model are given in Table S5.