Effect of fluid velocity and particle size on the hydrodynamic diffusion layer thickness

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Introduction
The dissolution kinetics of pharmaceutical solids are important in the evaluation of the performance of a drug product [1].Moreover, understanding the mass transfer mechanisms at the solid-liquid interfaces is essential to be able to fully grasp the complicated dissolution process [2].The mass transport occurring at this interface can be explained by the relative movement of the fluid surrounding the particle.For a solid moving in a liquid or a stationary solid in a flowing liquid, there will be a gradient of the relative velocity between the solid surface and the liquid, ranging from zero at the surface of the solid material [3,4] and increasing to a maximum value far from the interfacial region.The transport of dissolved substance from the surface of the solid will thus be carried out by diffusion, as well as by convective flow.One could consider two domains of fluid surrounding the solid, i.e. an inner domain in the form of a fluid layer coating the particle, where the fluid velocity is "low" and the mass transfer is dominated by diffusion [3,5] and an outer domain, where the fluid velocity is "high" and the mass transfer is dominated by convection.
This process was addressed a century ago with the well-known work of Noyes and Whitney [6], followed by Nernst [7] and Brunner [8] by a simplification, considering the inner low convection domain as a stagnant layer and the outer high convection domain as a perfectly stirred bulk.The "stagnant layer" is also called the effective diffusion boundary layer or hydrodynamic diffusion layer (HDL) [2,9].The simplicity and applicability of the diffusion layer model have made it widely used.
Here, it is worth emphasising that the HDL should not be interpreted as a de facto stagnant layer around the solid.It should rather be understood as a stagnant layer surrounded by a well stirred domain in an equivalent system, where the dissolution rate is the same as that in the actual system.Further, the HDL thickness may not exactly be the same as the (average) thickness of the domain where diffusion is the dominating transport process, although a functional relationship between the two is expected to exist, at least in principle.
When dissolution from particles is modelled, a consideration of their diminishing surface area is required.In addition to the HDL, the curvature of the particle will also affect the release rate.In most of the models mentioned below, the effects of curvature of the particle and the HDL are intentionally or unintentionally merged and described by a lumped parameter denoted as an apparent or effective diffusion layer.In other works [10,14,16], the respective effect on dissolution rate of the HDL and the curvature of the particle are clearly separated.Different denotations have been used in the literature cited above, such as h app and h eff , both for the thickness of the HDL as for the lumped efficient thickness due to the HDL and curvature effects.In this paper, we use h HDL and h lump , respectively, to clearly distinguish them and to reduce confounding when comparing with earlier literature.
For example, Hixson and Crowell [10] derived the cube-root law, stating that the cube root of the solid weight decreases linearly with time.The derivation was based on the assumption that the dissolution rate is proportional to the surface area of spherical particles, resulting in a linear decrease in particle size with time, which is sound when the particle size is much larger than the thickness of the HDL and the curvature effect is negligible [11].
Considering the effect of the particle curvature on the dissolution rate, Higuchi and Hiestand [12] derived a two-thirds-root expression, stating that the solid mass to the power of 2/3 decreases linearly with time.This applies when the particle size is much smaller than the thickness of the HDL.
Wang and Flanagan derived a dissolution model [10], considering both the curvature of the particle surface and the thickness of the HDL (h HDL ), which, in turn, was assumed to be independent of the particle radius.In that way, they obtained a model that was applicable to a wider range of particle sizes.The model could incorporate both the cube-root, and two-thirds-root particle dissolution models mentioned above as well as an intermediate square-root law, as suggested by Niebergall et al. [13].
In a later work [14], Wang and Flanagan studied single particle dissolution of benzocaine in water at a constant liquid flow rate.The specific dissolution rate (rate normalized by surface area) was found to exhibit a particle radius dependency, and these results were welldescribed by their particle dissolution model with an h HDL of 110 µm and a benzocaine diffusion coefficient of 1.4 × 10 − 5 cm 2 /s.
More recent studies focused on different drug particle radii, suggesting that the relationship between particle size and h HDL or h lump depends on particle size and can be divided into different particle size intervals [14].Hintz and Johnson [15] modelled the dissolution of polydisperse powders and proposed that a critical particle radius of 30 µm exists above which the h lump is a constant and thus independent of particle size, while h lump is equal to the particle radius for particles with a radius less than 30 µm.The critical value of h lump of 30 µm was derived based on a rotating disc, where the actual particle size is not taken into consideration.However, the derived value correlated well with powder dissolution profiles, and their work is supported by other studies [16,17].Bisrat et al. [18] and deAlmeida et al. [19] both used an electrical sensing zone method to study dissolution of polydisperse solids.Bisrat et al. [18] proposed that for griseofulvin and oxazepam powders, the h lump decreased significantly, with the particle size dropping below a critical particle diameter of 15 µm.For larger particles, the effect of particle size on the h lump became less significant.In a similar study, deAlmeida et al. [19] determined a critical particle diameter of 22 µm for ibuprofen with the following conclusions; 1) h lump was linearly proportional to the diameter when the diameter was less than 22 µm and 2) h lump was a constant when the diameter was above 22 µm.In the studies mentioned above, the dependence of the h lump on particle size was studied, but the dependence of the hydrodynamics received little attention.
In a more recent study performed by Sheng et al. [14], the h HDL was evaluated as a function of both particle size and hydrodynamics using a USP dissolution apparatus II.In this study, fenofibrate was used as a model compound.Suspensions of five size fractions were used (less than 20, 20-32, 32-45, 63-75, and 90-106 µm), and their dissolution was investigated at two different paddle speeds (50 and 100 rpm).They found that the critical particle radius was dependent on the fluid velocity, being 37.7 µm at 50 rpm and 23.7 µm at 100 rpm.Furthermore, a linear relationship was observed between h HDL and particle radius for particles smaller than the critical sizes.For particles larger than the critical size of 23.7 µm at 100 rpm, a constant h HDL of approximately 43.5 µm was observed.At 50 rpm, the h HDL increased with particle size but at a slower rate.Because of the difficulties in preparing spherical and crystalline fenofibrate particles, irregular shaped particles were used where the shape factor was considered, i.e. the ratio of particle surface radius to volume radius, assuming that the particles would dissolve in an isotropic manner with a constant shape factor.One possible limitation that is pointed out by Sheng et al. [14] is that the h HDL determined from the powder dissolution measurement in the USP dissolution apparatus is an average value from all particles.Here, the hydrodynamic conditions and the fluid dynamics around each individual particle are considered to be the same, even though this is typically not the case for all particles in a multi-particulate dissolution experiment.Moreover, even though the particles are fractionated into narrow size fractions, monodisperse particles are difficult to obtain.In a recent study, Salehi et al. [20] studied dissolution in an USP II apparatus, applying a hierarchical mass transport model and accounting for the impact of pH and buffer effects for S.B.E.Andersson et al. ionizable drugs as well as the impact of convection and shear rate.They propose e.g. that convection is an important mechanism for dissolution of particles and validate the model outcome with experiments.This study refer to another recent study by Wang and Brasseur [21], who apply CFD to model flow and dissolution in the intestine.Wang and Brasseur propose, contrary to the conclusion by Salehi et al., that in their system, convection is less important compared to shear rate as transport and dissolution mechanism.
The examples described above accentuate the importance of using sound approximations of the HDL thickness as well as of understanding the relationship between the velocity of the surrounding liquid and the HDL thickness when dissolution of particulate solids is modelled.Indirectly, sound models also give the possibility of determining the diffusion coefficients and values of solubility.In order to derive improved knowledge of the thickness of the HDL during dissolution, it is thus important to conduct fundamental studies on particle dissolution under well-controlled hydrodynamic conditions.
A well-controlled hydrodynamic condition may be obtained by a rotating disc method; however, the particle-size effects cannot be studied using this approach.A single particle dissolution technique has previously been presented [22] that is considered to satisfy the requirement of using a well-controlled, qualified experimental situation.In this paper, the same technique is used to study the dependence of h HDL on fluid flow velocity.The values of h HDL extracted from experiments are compared to the corresponding parameter obtained by theoretical computational fluid dynamics (CFD) simulations.In addition, simulations are made to theoretically explore relations between h HDL and particle radius in the particle size (radius) range 5-40 µm at flow rates between 10 and 100 mm/s.
The experimental h HDL values were determined under either of three distinct assumptions: (1) that h HDL is independent of particle radius; (2) that h HDL is proportional to particle radius to the power of 0.5, according to the Frössling/Ranz-Marshall equation [23][24][25]; and (3) that h HDL is proportional to particle radius to the power of 0.6, based on the outcome of the simulations reported in this study.To our knowledge, this is the first study using single particles to determine the h HDL values in different fluid velocities, evaluating the data using CFD simulations.

Materials
Ibuprofen was provided by Orion (Espoo, Finland).Carbamazepine and indomethacin were purchased from Sigma-Aldrich (Steinheim, Germany).The medium used for dissolution measurements was milli-Q water (Millipore, Purelab Flex 2, Elga LabWater, Lane End, UK).The diffusion coefficients and the solubility in water are presented in Table 1.

Preparation of particles
An excess amount of particles was stirred in milli-Q water at room temperature overnight to create a saturated solution.Thereafter, the excess solid material was removed from the suspension by filtration using a filter with a pore size of 0.2 µm.A slurry was subsequently prepared by adding approximately 2 mg of particles to 5 mL of the saturated solution, a small amount of which was introduced into a petri dish (1-2 drops from a transfer pipette).Using a micropipette, a single particle was drawn from the added particles which was used in the single particle dissolution experiment (see below).The procedure was repeated for all compounds used.

Single particle dissolution measurement
A micropipette-assisted microscopy technique [29] was used to determine the intrinsic dissolution rate of the compounds, in order to calculate the h HDL in different fluid flow velocities.In the experiments, single particles of diameters of approximately 100 µm were used and a series of particle properties, such as the mass, surface area and projected area diameter of the particle, were determined and used to calculate the intrinsic dissolution rate.Further experimental details are given in an earlier work [22].The single particle dissolution experiment is illustrated in Fig. 1.

CFD simulation models for single particle dissolution
To compare with dissolution experiments, CFD simulations were conducted using the experiment's typical particle dimensions and with the same diffusion coefficients and solubilities as for the substances used in the experimental study.Momentum balance was expressed by the incompressible Navier-Stokes equation, as appropriate for an incompressible linear viscous (i.e.Newtonian) fluid when gravity is disregarded.Here, v is the material time derivative of the fluid velocity v, ∇p is the spatial pressure gradient and ∇ 2 v is the spatial Laplacian of the fluid velocity.The fluid density ρ f and dynamic viscosity μ f are constants (Table 2).Mass balance of dissolved drug was expressed by the (convection) diffusion equation,

Table 1
The solubility in water and the diffusion coefficients of carbamazepine, ibuprofen and indomethacin.a The diffusion coefficients of carbamazepine [26], ibuprofen [27] and indomethacin [27] were extracted from the literature.b The solubility in water for carbamazepine was extracted from the literature [28].c The solubility of ibuprofen and indomethacin was experimentally determined in a previous study [22].as appropriate for dilute solutions for which Fick's law is applicable, and the diffusion coefficient D is constant (Table 2).Here, Ċ is the material time derivative of the drug concentration C and ∇ 2 C is the spatial Laplacian of the drug concentration.
The geometry was the same as in our previous study [22], with a small cylindrical domain (diameter 1.6 mm and length 3 mm) that mimicked the flow-pipette and a larger cylindrical domain (diameter 4 mm and length 2 mm) that represented the ambient liquid.A fully developed (i.e.parabolic) inlet velocity profile was prescribed at one end of the flow pipette (the smaller cylindrical domain; average velocity v and maximum velocity v max = 2v).The other end of the flow pipette emptied into the larger cylindrical domain.A pressure boundary condition on the opposing face of the larger cylindrical domain enabled liquid outflow.An oblique conical holder kept a particle on the centre axis of the flow-pipette, at a distance of 1 mm from its outflow end.The mirror symmetry of the system was used to reduce the size of the computational domain.
Three different particle shapes were studied: First, rectangular particles (110 × 50 × 30 μm 3 ), with rounded edges and corners (radius 5 μm).Second, equivalent spherical particles of the same surface area (diameter d eq = 76.5 µm).Third, a free spherical particle of diameter d eq = 76.5 µm, i.e. the same geometrical setup as the second simulation but without the conical holder present.Simulations were performed for a range of inlet flow velocities v (including the ones investigated experimentally) using COMSOL Multiphysics 5.6 (COMSOL AB, Stockholm, Sweden).Drug dissolution was studied by prescribing the dissolved drug concentration at the solubility C s at the particle boundary (i.e.C = C s at the boundary; Table 2).
To investigate the relation between h HDL and particle radius (r) and flow velocity (v), a second series of CFD simulations were made.In these simulations, a second geometry was used, as illustrated in Fig. 2, comprising a cylindrical domain (diameter 20 × r and length 2 × 20 × r) with the particle located on the cylinder axis in the middle of the cylinder (Fig. 2a).A slip boundary condition was enforced at the cylinder wall, and a constant normal inflow velocity was postulated at one end of the cylinder, and a static pressure outlet was used at the other end.
Utilizing the rotational symmetry of the problem, an axisymmetric setup was used, with a graded mesh consisting of about 580 000 triangular elements (Fig. 2a and b).Simulations were performed for a range of flow rates between 10 and 100 mm/s and four particle radii (5, 10, 20 and 40 μm), utilising parameters appropriate for ibuprofen.

Equations for diffusion layer thickness determinations 2.4.1. Equations for diffusion layer thickness determinations from experiments
In the following, different equations for particle decrease over time are reported, assuming diffusion-controlled release through a diffusion layer and different relations between h HDL and particle radius of the form h HDL = k α •r α .
Wang et al. [11] derived an expression (Eq.( 3)) for the change of radius over time by dissolution of a spherical particle, assuming diffusion-controlled release through a diffusion layer of thickness h HDL independent of particle radius, i.e.
where t is the time, ρ s the density of the solid drug, r 0 the initial radius of the particle and r(t) the radius of a particle at time point t.
Assuming that the Frössling/Ranz-Marshall correlation [23][24][25] is valid, it can be shown that h HDL is expected to be proportional to the square root of particle radius, i.e. h HDL = k 1/2 ⋅r 1/2 (see Appendix A).The following expression, relating particle radius with time, is then obtained (see Appendix B): Simulations made in this study indicate a h HDL proportional to r 3/5 (see results section below), i.e. h HDL = k 3/5 ⋅r 3/5 for the radii and flow velocities investigated.The following expression, relating the particle radius with time, is then obtained (see Appendix B): From the particle dissolution experiments, where r(t) was monitored over time, the factors k 0 , k 1/2 and k 3/5 were determined from Eqs. (3)-( 5) respectively by the Levenberg-Marquardt curve fit algorithm, minimising the sum of squared differences of the left and right side of the equations.The fittings were implemented by an in house Visual Basic macro in the software Excel.Although the particles were not spherical, Eqs.(3)-( 5) were assumed to provide satisfactory approximations for a diffusion layer determination and an equivalent radius, r, was determined as the radius of the sphere with the same surface area as the particle.

Equations for hydrodynamic diffusion layer thickness determinations from CFD simulations
From the CFD simulations, the dissolution rate could be extracted  and averaged over the particle surface.Under the same assumptions as for the derivation of Eq. ( 3), it can be shown [11] that the dissolution rate per unit area (i.e. the mass flux) ϕ, of a spherical particle with radius r is given by: which leads to the following closed expression for the determination of.hHDL Thus, inserting the averaged ϕ into Eq.( 7), an estimate of HDL thickness could be obtained from the simulations.

Results and discussion
Understanding the mass transport process is one of the key aspects when evaluating and determining drug dissolution.Consequently, numerous studies have focused on the relationship between drug particle size and h HDL .However, most of the studies use multi-particulate dissolution measurements, i.e. polydisperse powders or suspension experiments.The disadvantage of using these methods is that the hydrodynamic conditions around each individual particle are assumed to be identical.In reality, the multi-particulate system is more complex, as different particle sizes exist, including also different fluid dynamics around each individual particle.Another method used to determine h HDL is the rotating disc; here, the hydrodynamic conditions are controlled, but different particle sizes and dissolution kinetics cannot be assessed.

Result of dissolution experiments of the three substances and comparison to corresponding simulations
In this study, fixed single particles in various steady fluid velocities were used to create a controlled approach, where the exact particle size as well as the fluid dynamics around each single particle are known.The radius r of each individual single particle in different fluid velocities (46 -103 mm/s) is presented in Table 3.The radius is monitored throughout the dissolution experiment and used as an input in Eqs. ( 3)-( 5) in order to calculate h HDL [11].In our calculations, we have used diffusion coefficients extracted from the literature [26,27].The h HDL values obtained from the dissolution measurements, together with the h HDL values calculated from the simulations, are presented in Table 4.To illustrate the quality of the fit, a typical example with experimentally measured radii and radii determined from Eq. ( 3) with the obtained h HDL is shown in Fig. 3.
As can be observed in Table 4, the h HDL values decrease with an increase in fluid velocity, where the h HDL decreases from 16.4 to 3.98 µm for carbamazepine, from 7.65 to 4.34 µm for ibuprofen and from 3.01 to 2.09 µm for indomethacin, which is qualitatively expected.Those values can be compared to the h HDL values obtained from CFD simulations (Table 4), which are close to the calculated h HDL values from single particle dissolution experiments.We thus found a good concordance between experimental and simulated determined h HDL , where extraction of the experimental h HDL utilised an equivalent spherical particle.
The calculated h HDL values obtained from both dissolution experiments and dissolution simulations were plotted against the fluid flow velocities, see Fig. 4. As mentioned earlier, there is a good agreement between the h HDL values from simulations and experiments, and a similar trend can be noticed when it comes to a decrease in h HDL with an increase in fluid velocity.
For indomethacin, a good concordance is shown between the simulated values and the values obtained from an analysis of experimental data; the data analysed with the assumption that h HDL is proportional to the square root of particle radius, in particular, show an almost perfect concordance with the simulated values.For ibuprofen, the experimentally determined h HDL is higher than the h HDL obtained from the simulations, but within a factor of 2.5.The determined h HDL from carbamazepine experiments exhibited the highest discrepancy from the h HDL obtained from the simulations.The carbamazepine experiments were also the most cumbersome to conduct, in the sense of a high

Table 3
The initial and final radius (r) for each individual dissolution experiment, with four different fluid velocities: 46, 66, 88 and 103 mm/s.The measurements were run for 5 min (carbamazepine), 15 min (ibuprofen) and 60 min (indomethacin) if nothing else is stated.

Carbamazepine Final r(µm)
Ibuprofen Initial r (µm) variation acknowledged by the generally higher standard deviation of h HDL as well as the high average value of h HDL at the lowest flow rate.The latter is due to the fact that the analysis of one of the triplicates gave a considerably larger thickness (33.7 µm for constant h HDL and 48.0 µm for a square-root dependence on r) and might be regarded as an outlier.However, the h HDL at higher flow rate still shows a reasonable concordance with the simulated values.
The results show that the different models used to extract h HDL from the measurements provide values that are comparable in size to those obtained from simulations.However, at this stage, it is difficult to judge whether the h HDL is best: a) regarded as independent of the particle radius, as, for instance, suggested by Wang and Flanagan [30]; b) is proportional to the square root of the particle radius, as suggested in this work based on the Frössling/Ranz-Marshall correlation; or c) is proportional to the particle radius to the power of 3/5, as suggested by the simulations in this work.All three models fit the measured data equally well.This could be explained by the fact that the measured radii are in the range where the radius versus time characteristics, in practice, are linear and therefore can be fitted equally well to the linear parts of Eqs.
(3)-( 5) as shown in Fig. 5.To distinguish between the models, nonlinear behaviour needs to be observed, i.e. the experiments have to be designed so that the reduction in particle size could be followed down to radii of about 10 -5 µm according to Eqs. (3)-( 5) and also illustrated in Fig. 5.For the current setup, unfortunately, it was difficult to measure the dissolution of particles with an r below approximately 15 µm, since below this r, the particle either detached or rearranged on the micropipette.As mentioned previously [22], one possible explanation for this is that a relatively large micropipette tip diameter (approximately 50 µm) was used.A reduction of the tip diameter could thus potentially enable attachment of particles for a longer time-period.

Table 4
The hydrodynamic diffusion layer thicknesses ± S.D. of single particles determined from simulations of box shaped particles attached to the holder, equivalent sphere attached to the holder, free equivalent sphere and from dissolution experiments in different fluid velocities (46 -103 mm/s) for carbamazepine, ibuprofen and indomethacin.Experiments were analysed assuming constant h HDL (α = 0) and h HDL proportional to square root of radius (α = 1/2) and proportional to radius to the power of 3/5 (α = 3/5).In the two latter cases, the values were recalculated for the radius 38.6 µm.Discrepancies between h HDL from simulations and measurements could probably also, to some extent, be attributed to the used values of diffusion coefficients.The literature tends to give different values of diffusion coefficients for the same substance, and other values of h HDL would have been obtained if different diffusion coefficients had been used in the models used to extract h HDL expressed by Eqs. ( 3)-( 5) as well as in the simulations.Further, in this study imaging of the particle in just one direction was used for estimating its volume and surface area and their changes.This rather simple way to determine the particle size entails most probably a portion of error in the determination, which in turn may be a part of the observed discrepancy between the values of h HDL obtained from the simulations and the measurements.A more sophisticated size estimation, e.g. by imaging from different directions, will probably give more accurate volume and surface area determinations and may be a subject for future studies.
The dissolution of indomethacin and ibuprofen was simulated with almost identical diffusion coefficients (see 1) but a relatively large difference in solubility (see Table 1).Still, in practice, indomethacin and ibuprofen show identical h HDL versus flow characteristics (Fig. 4).The dissolution of carbamazepine was simulated with a diffusion coefficient approximately 25 per cent higher, resulting in h HDL values approximately 10 per cent higher for the different flows.Hence, it seems like the h HDL is independent of the solubility, while it has a diffusion-coefficient dependence.It can also be seen from the simulations that the different particle/holder geometries have a minute impact on the h HDL , i.e. the area-equivalent spherical particle actually approximates the actual particle well.
This behaviour is consistent with Eq. (A.9) in Appendix A, where h HDL has no dependence on solubility but a power-law dependence on the diffusion coefficient.It also coincides with the qualitative picture of the meaning of h HDL , being or being linked to the thickness of the domain, where diffusional transport is dominating over the convective transport.One could, for example, consider two substances with the same diffusion coefficient, but different solubility, as it is in practice in this study with indomethacin and ibuprofen.Both the concentration profile and its derivative are proportional to the solubility.The convective flow is, amongst other things, proportional to the concentration, and the diffusional flow is proportional to the derivative (gradient) of the concentration.Thus, the convective and diffusional flows are just scaled up or down in the same quantitative way when solubility changes (Fig. 6a).Hence, the ratio between these flows will be independent of solubility, and solubility will not affect where the crossover from a diffusion-dominated to a convection-dominated transport occurs.On the contrary, the diffusion coefficient will affect the extent of the diffusion-dominated domain.For example, a larger diffusion coefficient, as with the case of carbamazepine, will increase the diffusional transport; thus, a higher convective flow is required to equal the convective transport with diffusional transport.A higher convective flow is found at a larger distance from the particle surface and therefore, h HDL is expected to be larger (Fig. 6b).
We find it valuable to investigate dissolution under these relatively controlled conditions, i.e. well controlled relative velocity between the particle and medium, as well as directly observe the dissolution of the particle.For in vivo systems, the relative velocities between the medium and particle are, in most cases, considerably lower [31].A similar study at lower flow velocities would therefore be desirable, and the experience from this study could be of value for such a study.From a wider perspective, such studies can provide a valuable contribution of knowledge when studying more complicated systems like dissolution of suspensions, where relative velocity has to be modelled and the dissolution rate for a multi-particle system is acknowledged in a collective way e.g. by concentration changes in the dissolution medium.
In this study, the bulk is de-ionized water kept under sink conditions and the derivations of the used equations were done with these assumptions.In cases where the bulk could not be maintained under sink condition e.g.simulating more in vivo like conditions, considerations must be made for that and will most probable decrease the diffusion layer thickness, as shown by Wang et al [32] in a similar situation.In cases when the bulk solution is buffered the ion composition will by diffusion and acid-base reactions most probable, as e.g.described by Mooney et al [33] and Salehi et al, [20], affect the pH in the solution in immediate contact with the particle surface.A changed pH will affect the saturation solubility in the solution in immediate contact with the particle surface, which in turn will affect the release rate.

Simulation of dissolution from free spheres of different radii under different flow velocities
For a spherical particle shape and the idealised flow conditions assumed, the dependence of h HDL on particle size could be summarised by an expression of the form.
where k α is a proportionality constant and α is a non-dimensional exponent.To obtain k α and α, linear fits on logarithmised r and h HDL values were determined, as shown in Fig. 7, and gave accurate fits (R 2 > 0.9999).In summary, the obtained proportionality constant k α decreased with increasing flow rate, as expected.However, the exponent α exhibited a modest flow-rate dependence, and ranged from a value of 0.61 for a flow rate of 10 mm/s to a value of 0.59 for a rate of 100 mm/s.In the same way as for the particle radius, for a spherical particle shape and the idealised flow conditions assumed, the dependence of h HDL on fluid velocities could be summarised by an expression of the form.
where k β is a proportionality constant and β is a non-dimensional exponent.To obtain k β and β, linear fits on logarithmised v and h HDL values were determined and gave accurate fits (R 2 > 0.9999).In summary, the obtained proportionality constant k β decreased with decreasing particle size.However, the exponent β exhibited a modest particle-size dependence, ranging from a value of − 0.41 for a particle size of 40 µm to a value of − 0.39 for a particle size of 5 µm.
Considering both the radial (r) and fluid velocity (v) dependence of h HDL , the following equation was assumed.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.An illustration of the single particle dissolution experiment.A single particle is attached to a micropipette and inserted into a flow-pipette with constant fluid flow velocities, varying from 46 to 103 mm/s.The decrease of the single particle is observed using a microscope.

Fig. 2 .
Fig. 2. (a) Geometry and full axisymmetric mesh and (b) magnification of the mesh in the vicinity of the particle.

Fig. 6 .
Fig. 6.Schematic illustration of the effect of (a) solubility, C s and (b) diffusion coefficient, D on the thickness of the hydrodynamic diffusion layer, h HDL .As seen in (a), both the diffusional and convective transport rates increase in proportion to the solubility, leaving the h HDL independent of solubility.As seen in (b), only the diffusional transport rate increases when the diffusion coefficient increases, resulting in a larger h HDL .

h
HDL = k αβ r α v β (10) A multi-linear fit on logarithmised h HDL values vs logarithmised r and v values was determined (R 2 > 0.9998) and gave k αβ = 2.05, α = 0.602 and β = − 0.395.These values are close to 3/5 and − 2/5 respectively, which were used in the derivation of Eq. (5) above and in Appendix A. There is thus a relation between h HDL and r and v that is of the same type as the relation derived from the Frössling/Ranz-Marshall relation (See Visualization, Writingoriginal draft.

Table 2
Parameter values used in the numerical simulation.