Elsevier

Chemical Engineering Science

Volume 190, 23 November 2018, Pages 190-219
Chemical Engineering Science

Streamline-averaged mass transfer in a circulating drop

https://doi.org/10.1016/j.ces.2018.02.042Get rights and content

Highlights

  • Liquid-liquid extraction for a high Peclet number circulating drop is considered.

  • A boundary layer model for solute mass transfer is applied at early times.

  • A streamline-averaged model for solute mass transfer is applied at later times.

  • Boundary layer is switched to streamline-averaged model after one streamline orbit.

  • Solutions are obtained without a need to solve stiff advection-diffusion equations.

Abstract

Solute mass transfer is considered from the outside to the inside of a circulating drop in the context of liquid-liquid extraction. Specifically an internal problem is treated with resistance to mass transfer dominated by the liquid inside the drop. The Peclet number of the circulation is large, on the order of tens of thousands. A model is proposed by which the mass transfer into the drop begins in a boundary layer regime, but subsequently switches into a so called streamline-averaged regime. Solutions are developed for each regime, and also for the switch between them. These solutions are far easier to obtain than those of the full advection-diffusion equations governing this high Peclet number system, which are very stiff. During the boundary layer regime, the rate at which solute mass within the drop grows with time depends on Peclet number, with increases in Peclet number implying faster growth. However larger Peclet numbers also imply that the switch to the streamline-averaged regime happens sooner in time, and with less solute mass having been transferred to date. In the streamline-averaged regime, solute concentration varies across streamlines but not along them. In spite of the very large Peclet number, the rate of mass transfer is controlled diffusively, specifically by the rate of diffusion from streamline-to-streamline: sensitivity to the Peclet number is thereby lost. The model predictions capture, at least qualitatively, findings reported in literature for the evolution of the solute concentration in the drop obtained via full numerical simulation.

Introduction

Liquid-liquid extraction is a versatile chemical engineering separation technique applicable in diverse fields, including metals processing (Jyothi et al., 2009, Nishihama et al., 2001), oil processing (Yahaya et al., 2013), biomolecule processing (Mazzola et al., 2008, Silva and Franco, 2000) and food processing (Moreno-Gonzalez and Garcia-Campana, 2017). The separation is realised (Richardson et al., 2002) via diffusive transfer of a solute dissolved in one solvent to another immiscible solvent down a gradient of chemical potential. Usually extraction proceeds by dispersing drops of the first solvent (dispersed phase) in the second one (continuous phase) and mass transfer either occurs into the drop (in the event that the solute is initially in the continuous phase) or out of the drop (if the solute is initially in the dispersed phase).

Although the extraction process actually involves a multitude of droplets contained within an extraction column (Mohanty, 2000), so as to understand what is happening at a fundamental level, a starting point is to consider a single drop (Wegener et al., 2014). Mass transfer to and/or from individual drops has been a widely studied topic in chemical engineering (Brodkorb et al., 2003, Handlos and Baron, 1957, Johns and Beckmann, 1966, Korchinsky et al., 2009, Kumar and Hartland, 1999, Negri and Korchinsky, 1986, Negri et al., 1986, Piarah et al., 2001, Ubal et al., 2011, Waheed et al., 2002), and the field becomes wider still if one accounts for analogous systems including heat transfer to/from drops (see e.g. Prakash and Sirignano, 1978, Sadhal et al., 1997, Sirignano, 2010) as well as mass transfer to/from bubbles (see e.g. Juncu, 2005, Juncu, 2011).

In the case of liquid-liquid extraction, regardless of the direction of mass transfer (whether to or from the drop), it is useful to be able to predict how long the mass transfer process takes. This determines the residence time for which drops need to be present in an extraction column. Since drops will migrate through such a column at a speed determined (Wegener et al., 2014) by a balance between buoyancy and viscous drag forces, the residence time needed for mass transfer determines the required column height. Estimating the mass transfer time scale accurately is thereby important. If the estimate of the time scale required to achieve mass transfer is too low, then one is at risk of designing an extraction column that is too short, and hence that does not attain the target amount of mass to be transferred. By contrast, if the estimate of the time scale required to achieve mass transfer is too high, a column is likely to be over-designed to be taller than it needs to be, with higher cost implications.

As mentioned above, the driving force for liquid-liquid extraction is diffusive, i.e. transport down a gradient of chemical potential. However estimating the mass transfer time scale can be complicated by convective effects (Uribe-Ramirez and Korchinsky, 2000a, Uribe-Ramirez and Korchinsky, 2000b, Ruckenstein, 1967). A drop of one solvent experiences shear stresses as it migrates through another immiscible solvent. These shear stresses set up fluid flow past the drop and a circulation pattern within the drop itself. Liquid-liquid extraction is thereby a convective-diffusive process rather than just a purely diffusive one.

Convection is beneficial to the extraction process (Ubal et al., 2010, Uribe-Ramirez and Korchinsky, 2000b) and the more complex the flow pattern is, the more beneficial convection tends to be (Edelmann et al., 2017). In the case, for instance, of transfer of a solute from the outside of a drop to the inside, the fluid flow past the drop (see Fig. 1(a)) ensures that the drop surface is continually exposed to a new source of solute (rather than the solute concentration immediately outside the drop starting to become depleted, as would happen in the absence of any fluid flow). Moreover the circulation pattern inside the drop (again see Fig. 1(a)) ensures that material which was originally near the drop surface (and which has therefore acquired solute from the outside by diffusion) is removed from the drop surface and replaced by fresh material (of low solute concentration) from the drop interior: this can keep solute concentration gradients confined to a sharp boundary layer (Uribe-Ramirez and Korchinsky, 2000b) near the drop surface (see Fig. 1(b)), hence speeding up the rate of mass transfer quite substantially, although that situation does not necessarily last indefinitely (for reasons to be explained shortly).

Despite the evident benefits to the liquid-liquid extraction process, the presence of circulation complicates the computations that must be performed to determine exactly how liquid-liquid extraction proceeds. Although convection-diffusion equations are conceptually simple to set up (including being amenable to solution via commercial software packages) a significant issue in this particular system is that the circulation is usually rapid compared to diffusion (i.e. the relevant Peclet number is much larger than unity, often on the order of tens of thousands Uribe-Ramirez and Korchinsky, 2000a, Uribe-Ramirez and Korchinsky, 2000b). This means that fluid must circulate around the drop many times before the extraction process is complete. Numerical simulations of the process, whilst possible (Edelmann et al., 2017, Ubal et al., 2011), are computationally very expensive owing to the need to resolve each individual circulation: convective-diffusive problems at high Peclet number are, in numerical terms, exceedingly stiff (Press et al., 1992).

A way to understand liquid-liquid extraction in circulating drops has been considered by Ubal et al. (2010) based on ideas originally proposed by Kronig and Brink (1950) and by Abramzon and Borde, 1980, Brignell, 1975, Oliver et al., 1985, Prakash and Sirignano, 1978. Convection is, by definition, along streamlines, so that (at least in a system with steady and laminar fluid flow) the only way to transport material across streamlines is via diffusion, no matter how fast the flow. Since in high Peclet number flows, convection along streamlines is much faster than diffusion across them, it is expected (Ubal et al., 2010) that the solute concentration field should very rapidly become uniform along streamlines, but with diffusion-driven concentration variations from streamline to streamline (see Fig. 1(c)). This then constitutes a streamline-averaged formulation of liquid-liquid extraction. Even though the mass transport rate into the drop is ultimately diffusively controlled, a benefit is still derived from convection. In order to fill the entire drop, the distance over which solute must diffuse is no longer the entire drop radius (as it would be in a system with no convection whatsoever Korchinsky et al., 2009, Negri and Korchinsky, 1986, Negri et al., 1986), but rather the distance from the drop surface to an internal stagnation point about which all the other streamlines circulate.

The exact location of that stagnation point depends on the exact flow field in the drop, but an earlier study using a plausible circulation pattern (Uribe-Ramirez and Korchinsky, 2000b) found it to be roughly one third of the drop radius beneath the surface. Since the time scale for diffusive processes is sensitive to distance (as can be established on dimensional grounds), convective-diffusive liquid–liquid extraction should proceed over a substantially shorter time scale than a comparable process without any convection. This has been verified by Ubal et al. (2010) using full numerical simulations of the convective-diffusive mass transfer process.

For the reasons pointed out above however (i.e. the Peclet number is very large Uribe-Ramirez and Korchinsky, 2000a, Uribe-Ramirez and Korchinsky, 2000b and so the governing equations are stiff), these numerical simulations proved extremely expensive. It is expected that a streamline-averaged theory will be much more amenable computationally, as it does not need to resolve the (very short) timescale associated with individual streamline circulations, but instead can focus exclusively on the longer diffusive time-scale over which mass transfer actually occurs. The necessary equations were in fact formulated by Ubal et al. (2010) but the solution was not implemented in that work, albeit there are previous implementations in literature for very a specific streamline layout (Brignell, 1975, Kronig and Brink, 1950, Prakash and Sirignano, 1978). Our purpose here is to revisit the implementation of the streamline-averaged theory, and to analyse the predictions it makes.

Solving the streamline-averaged theory is not however without difficulties. The theory assumes that solute concentration is uniform or near uniform along streamlines. This assumption is however invalid early on in the evolution (Ruckenstein, 1967, Uribe-Ramirez and Korchinsky, 2000b) for reasons we now explain. Consider for instance the situation as described earlier whereby solute diffuses into the drop from the outside. Circulating streamlines inside the drop that pass very near to the drop surface (and which are therefore exposed to the solute external to the drop) can and do acquire mass diffusively (Uribe-Ramirez and Korchinsky, 2000b), but those same streamlines also penetrate very close to the drop axis, deep inside the drop where solute may not yet have reached. Until these streamlines undergo one complete circulation then (with material elements on the streamline having had the opportunity to pass along both the drop surface and the drop axis), it is a poor approximation to say the solute concentration is uniform along them. Instead what is required is an early-time theory valid up until the time of one complete circulation. The early-time theory must keep proper account of the solute mass entering the drop during this stage, so that the streamline-averaged theory which follows on from it is taken to start off with the correct amount of solute mass.

Suitable early-time theories have been proposed by Uribe-Ramirez and Korchinsky, 2000b, Ruckenstein, 1967, Levich et al., 1965, Vorotilin et al., 1965, and have also been discussed by Ubal et al. (2010). They fall into the general class of boundary layer theories (Leal, 2007) since at early times only streamlines passing near the drop surface acquire solute from the outside (see Fig. 1(b)), those same streamlines subsequently being the ones that transport this solute along the drop axis (again see Fig. 1(b)). Our secondary aim in this paper is to match the early-time boundary layer theory with the later time streamline-averaged theory: exactly how and when to switch between these theories has not been established previously.

Interestingly the work of Uribe-Ramirez and Korchinsky (2000b) treated an extension of the boundary layer theory which assumed that the solute acquired by streamlines as they passed close to the drop surface could subsequently be well mixed in the drop interior when carried along the parts of those same streamlines that passed close to the drop axis. This implied that when those same streamlines arrived back at the drop surface, they did so (by assumption) with a solute concentration far lower than the one with which they had formerly departed from it, having left behind a substantial amount of mass in the drop interior. Solute concentration gradients for fluid elements arriving back at the surface were thereby kept artificially large (Uribe-Ramirez and Korchinsky, 2000b), and the consequent predicted rate of mass transfer into the drop was likewise far too large.

Full numerical simulation of the (exceedingly stiff, and hence numerically expensive) convection–diffusion equations revealed that this assumption of solute mixing between streamlines within the drop interior was not valid (Ubal et al., 2010). Instead, for a streamline passing both very near the drop surface and very near the drop axis, the only material elements on that streamline which could arrive at the surface with low solute concentrations were those that had not yet been in close contact with the surface. Such material elements can however only survive up to one entire orbit of a streamline, implying that significant non-uniformities in solute concentrations along streamlines can likewise only survive that long. A further implication was that the large cross-stream concentration gradients near the drop surface which are predicted by the boundary layer theory (and the rapid mass transfer into the drop that these large gradients imply) cannot last indefinitely, but rather only up to the time at which one entire streamline orbit is complete (as indeed the work of Brignell (1975) recognised). The conclusion of Ubal et al. (2010) then was not that the boundary layer theory itself was inherently incorrect, merely that it was inappropriate to apply it for times far beyond a single orbit time.

To reiterate, once an entire streamline orbit is complete, all elements on the streamline in question must have spent part of their life near the surface (acquiring solute from the exterior of the drop) and part of their life near the drop axis (but barely mixing solute with other streamlines). This means that, not only do gradients in solute concentration normal to the drop surface (and hence mass transfer rates into the drop) start to decrease from one orbit time onwards, the solute concentration on the entire near surface streamline should be comparatively uniform, being set by the concentration immediately outside the drop. This however is the criterion for the previously mentioned streamline-averaged theory (Ubal et al., 2010) to begin to apply. In other words the upper time limit for applicability of the early-time boundary layer theory should coincide with the lower time limit for applicability of the streamline-averaged theory.

It is plausible then that the entire evolution of the mass transfer process in a circulating drop can be described by selecting a suitable time to switch between the two aforementioned theories. This then is the hypothesis that we investigate here. Moreover, as we will discuss shortly, one of the features of the streamline-averaged theory is that it can be set up retaining just very incomplete information about the streamline pattern, rather than requiring complete knowledge of all details of the geometry and kinematics of the flow field in the drop. Thus one of the key questions we shall consider is whether it is feasible to predict evolution of concentration fields in the drop using just incomplete information about the circulation.

This work is laid out as follows. Section 2 discusses the equations governing convective-diffusive mass transfer within a drop during liquid-liquid extraction, and how these reduce to either boundary layer theories or streamline-averaged theories in relevant limits. Section 3 details the methods used to solve the equations governing each of the above mentioned theories, as well as the parameter values selected in each case. After that, Section 4 presents results obtained for each theory and the process of switching between them. Section 5 presents conclusions.

Section snippets

Theory and governing equations

This section is laid out as follows. The general equations for mass transfer into a circulating liquid drop are given in Section 2.1. A boundary layer theory of the mass transfer process is described in Section 2.2, building upon material which is already presented in Ubal et al. (2010). Streamline orbit times are computed in Section 2.3: these are needed in order to determine the time to switch from a boundary layer theory to a streamline-averaged one. As we will see, defining this time scale

Solution methods

Section 2 concerned itself with formulating the model used to describe mass transfer to a circulating drop during liquid-liquid extraction. The present section highlights the solution methodology that we use to implement the model. Since the theory of Section 2 involves a number of diverse elements combined together into a single overall model, the solution methods we use likewise require us to bring together a number of diverse techniques. That said, each solution technique that we employ is

Results

This section presents and discusses the predictions of the model for solute mass transfer in a circulating drop. The structure of this section mirrors the structure already used in Section 2 to describe the model itself. In other words, first a boundary layer theory is considered and then this is subsequently switched to a streamline-averaged theory. Specifically in what follows Section 4.1 examines data for θ0(θ,T) which is the initial location of a material point which currently finds itself

Conclusions

We have presented a technique for predicting the time evolution of solute mass transfer into a circulating drop in the context of liquid-liquid extraction. This is a challenging system upon which to perform full numerical simulations: despite the fact that the full numerical simulations are conceptually easy to set up, owing to the Peclet number being exceedingly large (on the order of tens of thousands), the governing equations are stiff. We therefore sought to find a technique to solve for

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