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Nanomaterials 2020, 10, 2202 6 of 16 80 ≤ Re < 1000–2000: Wavy-laminar flow Re ≥ 1000–2000: Turbulent flow regime However, this particular definition of Reynolds number does not encompass the rotational aspect of the SDR. For this reason, another dimensionless number will also be incorporated into the model to characterise rotation of the disc. Two dimensionless groups will be considered: The rotational Reynolds number, Reω, and the Rossby number, Ro. The merits and applicability of each are discussed below. The rotational Reynolds number is a dimensionless number which may be used to describe the rotational aspect of flow on the film. Similar to the conventional Reynolds number, it is the ratio of inertial (centrifugal) to viscous forces and is expressed in the following form: ωr2 Reω = ν . (6) The rotational Reynolds number provides an alternative to the conventional Reynolds number for characterising flow regime in the reactor as Re does not take angular velocity into consideration [49,61]. Similarly, Reω does not include flow rate in its expression. Ozar et al. (2003) states that both flow rate and disc speed play a major role in flow transition from laminar to turbulent [62]. Rotational Reynolds number criteria for categorising flow regimes in a spinning disc reactor are as follows [61]: Reω< 104: Laminar regime 104≤ Reω<105: Flow instabilities increase and flow is in transition to turbulent regime Reω ≥ 105: Turbulent regime Often the rotational Reynolds number is identified as the Taylor number, Ta. Saw et al. (1985) used the Taylor number along with the conventional Reynolds number to develop a predictive model for liquid film thickness on a rotating disc [63]. This was further applied by Khan (1986) and by Mohammadi (2014), the latter proposing that TiO2 particle size is directly proportional to the dimensionless form of liquid film thickness [64,65]. Their models assumed negligible Coriolis forces on the film, provided Re2/Ta is less than unity. For the present work, Re2/Ta is in the range of 10−3 to 10−5 at the edge of the disc, though tends to get greater towards the centre of the disc. As 90% of the disc has a value less than 1, it may be assumed that Coriolis forces are negligible in our experimental work. Alternatively, the Rossby number has been used in previous work to characterise liquid flow on a spinning disc [66–68]. It is defined as the ratio of inertial (centrifugal) to Coriolis forces. The Rossby number is presented in Equation (7), where ui is the inlet velocity calculated from total flow rate of the antisolvent and solvent/solute streams. The Rossby number is estimated to lie in the range of 0.045 to 0.405 for the operating conditions used in the present work, indicating dominance of Coriolis forces over inertial when Ro < 1, whereas centrifugal forces dominate when Ro << 1 [67]. These values have been estimated at the edge of the disc in order to correlate with particle size measurements for particles collected at the edge of the disc. The Rossby number is often applied in combination with the Ekman number, Ek, as presented in Equation (8) [69]. It implies that for a small Ro value, Ek will be large in order to maintain an order of magnitude of 1. In circumstances where Ro << 1, the Ekman number is significantly greater to satisfy Equation (8), implying negligible Coriolis forces [70]. Based on this, the Ro values in the present work are considered to be within the region where Coriolis forces are negligible, as Ek >> 1 at the outer region of the disc [70]. As with the Rossby number, towards the centre of the disc, Ekman number too tends towards values where Coriolis forces come into play. Ek of less than 2 is where flow deviates from the centrifugal model [49]. For the current system, Ek is greater than 2 for the majority of the reactor. Ro = ui (7) ωr Ek = ν . (8) ωδ2PDF Image | Hydro Starch Nanoparticles Precip Spinning Disc Reactor
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