Boundary-Layer Thickness and Hydrodynamic Effects

Why the Diffusion Layer Matters

A solid dissolves at its surface, but molecules must then move from the solid–liquid interface into the surrounding fluid. Agitation changes this transport: it continually moves fluid near a particle and can reduce the effective distance over which dissolved molecules must diffuse. That change can increase dissolution rate even when the solid’s exposed surface area and its solubility have not changed.

To interpret an observed rate increase, it is important to separate this hydrodynamic effect from a change in particle surface area. For example, stronger mixing may alter both the effective diffusion layer and the degree to which particles are dispersed or wetted. A rate change alone does not identify which mechanism changed.

The Film Model and Its Assumptions

Let \(C_s\) be the dissolved concentration at the solid–liquid interface and \(C_b\) the concentration in the well-mixed bulk liquid. In the film model, the concentration falls across a near-surface layer from \(C_s\) to \(C_b\). If the concentration profile across that layer is approximated as linear, Fick’s law gives the outward mass flux per unit surface area:

Here \(J\) has units of mass per area per time, \(D\) has units of area per time, and \(h\) has units of length. The total dissolution rate from a wetted surface area \(A\) is therefore:

In this equation, \(M_s\) is the mass remaining in the solid. The negative sign indicates that solid mass decreases during dissolution. If concentrations are expressed in \(\mathrm{kg/m^3}\), \(A\) in \(\mathrm{m^2}\), and \(k_L\) in \(\mathrm{m/s}\), the rate is in \(\mathrm{kg/s}\).

This expression is useful when the interface is close to local equilibrium, so that \(C_s\) can be specified from the relevant solubility, and transport through the liquid is the rate-limiting process being represented. It also assumes a meaningful bulk concentration, a sufficiently uniform and wetted area, and conditions that do not change substantially during the calculation. The film is an effective representation, not necessarily a sharply bounded physical slab. In a real vessel, flow may be turbulent and the local layer may vary across a particle.

How Agitation and Fluid Properties Enter

In the simple film model, increasing agitation is represented by a smaller effective \(h\), and therefore a larger \(k_L\). This is a useful directional expectation, not a universal formula connecting stir rate to layer thickness. The relationship depends on the vessel, impeller, particle motion, and flow regime. A given increase in rotational speed does not imply a fixed proportional decrease in \(h\).

A more general way to organize hydrodynamic effects is with the Sherwood number, a dimensionless mass-transfer coefficient. For a characteristic length \(L_c\), fluid density \(\rho\), characteristic fluid speed \(u\), and dynamic viscosity \(\mu\), define:

Correlations between \(Sh\), \(Re\), and \(Sc\) are specific to a geometry and flow regime; there is no single correlation that applies to every dissolution apparatus. These dimensionless quantities help explain why agitation and fluid properties cannot always be treated as independent adjustments. Agitation changes fluid speed and thus \(Re\); viscosity and density affect \(Re\) and \(Sc\); and \(D\) affects both \(Sc\) and the conversion from \(Sh\) to \(k_L\).

At fixed temperature and for a dilute solute in a Newtonian liquid, the Stokes–Einstein relation gives a useful approximation: \(D\) decreases as viscosity increases, when solute size is unchanged. This does not mean that viscosity alone determines the rate. It may also change the flow field and the effective layer thickness. If \(D\) and \(h\) both change, the relevant comparison is the resulting \(k_L=D/h\).

Separating Hydrodynamic and Surface-Area Effects

For two conditions, the film model gives a rate ratio that makes the separate contributions explicit:

This factorization is valid for positive rates under the film model. If the concentration difference is unchanged and \(D\) is unchanged, then \(k_{L,2}/k_{L,1}=h_1/h_2\). In that restricted comparison, a rate increase can be decomposed into an area ratio and a layer-thickness ratio. The decomposition is a model-based calculation: it is not evidence that the factors were independently measured.

Fluid-Property Changes and Interpretation

Fluid properties can affect the rate through more than one term. The following calculation illustrates why a change in \(D\) should not automatically be described as an agitation effect. Suppose a hypothetical fluid change reduces \(D\) from \(4.00\times10^{-10}\) to \(3.00\times10^{-10}\ \mathrm{m^2/s}\), and the effective layer changes from \(80.0\) to \(100\ \mathrm{\mu m}\). Let \(A=0.0100\ \mathrm{m^2}\) and \(C_s-C_b=0.250\ \mathrm{kg/m^3}\) in both conditions.

In fluid A, \(k_{L,A}=4.00\times10^{-10}/8.00\times10^{-5}=5.00\times10^{-6}\ \mathrm{m/s}\), and \(R_A=(0.0100)(5.00\times10^{-6})(0.250)=1.25\times10^{-8}\ \mathrm{kg/s}=0.0450\ \mathrm{g/h}\). In fluid B, \(k_{L,B}=3.00\times10^{-10}/1.00\times10^{-4}=3.00\times10^{-6}\ \mathrm{m/s}\), and \(R_B=(0.0100)(3.00\times10^{-6})(0.250)=7.50\times10^{-9}\ \mathrm{kg/s}=0.0270\ \mathrm{g/h}\). The rate ratio is \(0.0270/0.0450=0.600\), equal to the mass-transfer coefficient ratio. Both the lower diffusivity and the thicker effective layer contribute to the predicted decrease.

If only viscosity changes, Stokes–Einstein can help estimate the direction and approximate change in \(D\), subject to its assumptions. But a prediction that holds \(h\) fixed while changing viscosity isolates only one model component. In an actual apparatus, viscosity may also alter the flow and therefore \(h\). Measuring or otherwise justifying the mass-transfer coefficient is preferable to attributing the full observed rate difference to diffusivity alone.

Derivation, Diagnostics, and Alternatives

The film-model derivation is short but its assumptions matter. Across a layer of thickness \(h\), take the concentration gradient to be approximately \((C_b-C_s)/h\). Fick’s law gives \(J=-D(dC/dx)\); choosing the coordinate \(x\) to increase away from the surface, this gradient is negative when \(C_s>C_b\). Therefore the outward flux is positive and equals \(D(C_s-C_b)/h\). Multiplying flux by wetted area gives total rate. This argument also shows why the model uses the product \(A k_L\): rate data alone may not distinguish a larger area from a larger mass-transfer coefficient.

Useful diagnostics compare predictions with measurements made while changing one factor at a time. If \(A\), \(D\), and \(C_s-C_b\) are controlled, thinning \(h\) by a factor of two predicts a twofold rate increase. If the observed rate does not respond as predicted, possible explanations include an incorrect effective layer estimate, incomplete or changing wetting, an altered concentration difference, or a different rate-limiting process. A rate that stops increasing with agitation can indicate that liquid-side transport is no longer the only important resistance; it does not prove that the model is wrong in every condition.

Other approaches answer related but different questions. An empirical \(k_L\) fitted to a particular apparatus can summarize measured transfer without resolving the local flow, but its value may not transfer to another geometry or agitation setting. A hydrodynamic correlation for \(Sh\) can predict how transfer varies with flow when its geometry and regime are appropriate, but using an unrelated correlation can introduce systematic error. Computational fluid dynamics can represent spatial flow and transfer in greater detail, at the cost of additional inputs and validation. The film model is simpler and transparent, but its single \(h\) averages over local variation.

The effective-layer approach is most useful for organizing mechanisms and comparing controlled conditions. It is limited when area, surface concentration, diffusivity, or local flow changes substantially during dissolution, or when several transport and reaction processes contribute comparably. In those cases, the model may still be a useful approximation, but fitted values of \(h\) or \(k_L\) should be interpreted as effective parameters for the specified conditions rather than universal material constants.