7.4. MASS TRANSFER TO SUSPENSIONS OF SMALL PARTICLES

7.4A. Introduction

Mass transfer from or to small suspended particles in an agitated solution occurs in a number of process applications. In liquid-phase hydrogenation, hydrogen diffuses from gas bubbles, through an organic liquid, and then to small suspended catalyst particles. In fermentation, oxygen diffuses from small gas bubbles, through the aqueous medium, and then to small suspended microorganisms.

For a liquid–solid dispersion, increased agitation over and above that necessary to freely suspend very small particles has very little effect on the mass-transfer coefficient kL to the particle (B2). When the particles in a mixing vessel are just completely suspended, turbulence forces balance those due to gravity, and the mass-transfer rates are the same as for particles freely moving under gravity. With very small particles of, say, a few μm or so, which is the size of many microorganisms in fermentations and some catalyst particles, their size is smaller than eddies, which are about 100 μm or so in size. Hence, increased agitation will have little effect on mass transfer except at very high agitation.

For a gas–liquid–solid dispersion, such as in fermentation, the same principles hold. However, increased agitation increases the number of gas bubbles and hence the interfacial area. The mass-transfer coefficients from the gas bubble to the liquid and from the liquid to the solid are relatively unaffected.

7.4B. Equations for Mass Transfer to Small Particles

1. Mass transfer to small particles <0.6 mm

Equations for predicting mass transfer to small particles in suspension have been developed which cover three size ranges of particles. The equation for particles <0.6 mm (600 μm) is discussed first.

The following equation has been shown to hold for predicting mass-transfer coefficients from small gas bubbles such as oxygen or air to the liquid phase or from the liquid phase to the surface of small catalyst particles, microorganisms, other solids, or liquid drops (B2, C3):

Equation 7.4-1


where DAB is the diffusivity of the solute A in solution in m2/s, Dp is the diameter of the gas bubble or the solid particle in m, μc is the viscosity of the solution in kg/m · s, g = 9.80665 m/s2, Δρ = (ρcρp) or (ρpρc), ρc is the density of the continuous phase in kg/m3, and ρp is the density of the gas or solid particle. The value of Δρ is always positive.

The first term on the right in Eq. (7.4-1) is the molecular diffusion term, and the second term is that due to free fall or rise of the sphere due to gravitational forces. This equation has been experimentally checked for dispersions of low-density solids in agitated dispersions and for small gas bubbles in agitated systems.

EXAMPLE 7.4-1. Mass Transfer from Air Bubbles in Fermentation

Calculate the maximum rate of absorption of O2 in a fermenter from air bubbles at 1 atm abs pressure having diameters of 100 μm at 37°C into water having a zero concentration of dissolved O2. The solubility of O2 from air in water at 37°C is 2.26 × 107 g mol O2/cm3 liquid or 2.26 × 104 kg mol O2/m3. The diffusivity of O2 in water at 37°C is 3.25 × 109 m2/s. Agitation is used to produce the air bubbles.

Solution: The mass-transfer resistance inside the gas bubble to the outside interface of the bubble can be neglected since it is negligible (B2). Hence, the mass-transfer coefficient outside the bubble is needed. The given data are


At 37°C,


Substituting into Eq. (7.4-1),


The flux is as follows, assuming for dilute solutions:


Knowing the total number of bubbles and their area, the maximum possible rate of transfer of O2 to the fermentation liquid can be calculated.


In Example 7.4-1, kL was small. For mass transfer of O2 in a solution to a microorganism with Dp ≅ 1 μm, the term 2DAB/Dp would be 100 times larger. Note that at large diameters the second term in Eq. (7.4-1) becomes small and the mass-transfer coefficient kL becomes essentially independent of size Dp. In agitated vessels with gas introduced below the agitator in aqueous solutions, or when liquids are aerated with sintered plates, the gas bubbles are often in the size range covered by Eq. (7.4-1) (B2, C3, T1).

In aerated mixing vessels, the mass-transfer coefficients are essentially independent of the power input. However, as the power is increased, the bubble size decreases and the mass transfer coefficient continues to follow Eq. (7.4-1). The dispersions include those in which the solid particles are just completely suspended in mixing vessels. Increase in agitation intensity above the level needed for complete suspension of these small particles results in only a small increase in kL (C3).

Equation (7.4-1) has also been shown to apply to heat transfer and can be written as follows (B2, C3):

Equation 7.4-2


2. Mass transfer to large gas bubbles >2.5 mm

For large gas bubbles or liquid drops >2.5 mm, the mass-transfer coefficient can be predicted by

Equation 7.4-3


Large gas bubbles are produced when pure liquids are aerated in mixing vessels and sieve-plate columns (C1). In this case the mass-transfer coefficient or kL is independent of the bubble size and is constant for a given set of physical properties. For the same physical properties the large-bubble Eq. (7.4-3) gives values of kL about three to four times larger than Eq. (7.4-1) for small particles. Again, Eq. (7.4-3) shows that the kL is essentially independent of agitation intensity in an agitated vessel and gas velocity in a sieve-tray tower.

3. Mass transfer to particles in transition region

In mass transfer in the transition region between small and large bubbles in the size range 0.6–2.5 mm, the mass-transfer coefficient can be approximated by assuming that it increases linearly with bubble diameter (B2, C3).

4. Mass transfer to particles in highly turbulent mixers

In the preceding three regions, the density difference between phases is sufficiently large to cause the force of gravity to primarily determine the mass-transfer coefficient. This also includes solids just completely suspended in mixing vessels. When agitation power is increased beyond that needed for suspension of solid or liquid particles and the turbulence forces become larger than the gravitational forces, Eq. (7.4-1) is not followed, and Eq. (7.4-4) should be used where small increases in are observed (B2, C3):

Equation 7.4-4


where P/V is power input per unit volume as defined in Section 3.4. The data deviate substantially by up to 60% from this correlation. In the case of gas–liquid dispersions it is quite impractical for agitation systems to exceed gravitational forces.

The experimental data are complicated by the fact that very small particles are easily suspended, and if their size is on the order of the smallest eddies, the mass-transfer coefficient will remain constant until a large increase in power input is added above that required for suspension.

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