Mass transfer in agitated vessels usually involves dispersed gases absorbing into and often reacting with a continuous liquid phase, e.g., in oxidation or chlorination. Interfacial area and contact time depend on the operating conditions. Even when the relevant mass transfer is to or from suspended solids, the rate may be affected by segregation. The rate of mass transfer NB of a component B between phases 1 and 2 can be expressed simply as an overall mass transfer coefficient K multiplied by the total interfacial contact area A and the overall equivalent concentration driving force.
In the latter case a is the specific interfacial area, i.e., the total contact area per unit volume of the dispersion and V is the volume of the reactor. The notation cB2 refers to the actual concentration of component B in phase 2 while c*B1 refers to the concentration of B that would be present in phase 2 if it were in equilibrium with the bulk concentration in phase 1.
The K term embodies the resistances to mass transfer on each side of the interface which are controlled by the local turbulence and the local rates of diffusion of the transferred material, 1/K = 1/k1 + 1/k2. In processes involving contaminated interfaces there may be a third term to allow for a diffusional resistance in a surface film.
Depending on the nature of the system, particularly the ease of saturating one of the phases, or the presence of relatively fast reactions, it is usual to simplify the model and consider the controlling resistance as being on one side or the other. For most physical absorption systems (like oxygen dissolving in clean water) liquid phase conditions are most important so kL is the "film coefficient" of interest. With reacting systems (e.g., atmospheric oxygen into a waste water with a significant concentration of a strong reducing agent like sulfite) kG may be more relevant. In general, liquid phase agitation has little influence on the gas side coefficient, though it does affect the contact area term of course.
The area A results from a balance between the dispersing actions of the fluid forces in the system and complex coalescence processes that depend on the frequency and duration of bubble or droplet collisions. The physical and chemical nature of the system is very relevant, as are the agitation conditions, usually characterized in terms of the local turbulence or energy dissipation rate. Unfortunately, the critical influence of contamination on dispersion size distribution and the cleanliness of interfaces have not been quantified and most of the reliable data available in the literature refer to clean air-water systems.
Since it is difficult to separate the effects of agitation on K and A, it has become usual to group the terms and express values of kLa or kGa as functions of the process variables. The concentration driving force is also difficult to define accurately, especially in large vessels which may be very far from homogeneous and for which it is usually necessary to select some appropriate model that allows for possible segregation in either or both phases.
Mass transfer processes in agitated vessels are almost invariably turbulent, with mixing Reynolds numbers above 104 in the uniform ungassed Power number regime (Re = ND2ρ/η, where D and N are the impeller diameter and speed, and ρ and η are the liquid density and viscosity, respectively, (see Agitation Devices). Energy dissipation controls both surface area and turbulent mass transfer coefficients, so it is necessary to know the power demand of gassed agitated systems, a parameter that is considerably influenced by operation in a two-phase regime.
Radial flow turbines (see Agitation Devices) are usually used for gas dispersion. Efficient dispersion is not sensitive to the details of sparger design providing that the supplied gas reaches and is distributed by the impeller. Down pumping axial flow impellers are less suitable because rising gas tends to disrupt the liquid outflow unless the sparger ring is larger than the agitator so that only gas that has circulated around the tank reaches the impeller.
Gas-liquid dispersion can be characterized in terms of the gas flow number and the Froude Number Fr (= N2D/g) in which is the gas supply rate, g, gravitational acceleration. It is probable that the Froude Number ought also to include a density ratio term for the two phases though since Δρ/ρ ~ 1 in gas/liquid systems this is usually ignored. At low values of FlG, common in small scale laboratory investigations, the trailing vortices behind the blades of radial flow impellers accumulate and redisperse gas, Figure 1a. Beyond a well defined value of FlG gas accumulates in large cavities behind the blades, Figure 1b. This regime is the usual at industrial scale.
Moderately high impeller speeds are needed to recirculate rising gas to the impeller or to carry bubbles down to the bottom of the vessel, Figure 2a and Figure 2b.
Very high gas rates, or insufficient impeller speeds, can lead to the pumping action of the agitator being overwhelmed by buoyancy forces, a condition known as flooding, Figure 2c. The flow transitions are characterized by the following equations; cavities are of the vortex form if FlG < 0.18 (D/T)2, flooding will occur if FlG > 30Fr (D/T)35. Recirculation requires that FlG < 13Fr2(D/T)−5. If conditions are too gentle, Fr < ~ 0.045, buoyancy forces allow the gas to escape and no stable cavities are retained. These equations are used as the basis for Figure 3, which shows a flow regime map for Rushton turbines with D/T = 0.4.
Flow maps can ensure that hydrodynamic conditions met in a full-scale plant match those used in laboratory investigations.
For almost all designs of impeller the power required to maintain a given rate of rotation is less when loaded with a dispersion of gas or vapor in liquid than when operating in liquid alone. It has been shown that this is not merely a density effect. The relative power demand, (RPD), generally falls smoothly until the impeller is finally swamped by gas. Figure 4 shows typical results for a Rushton turbine loaded with different gas rates while maintaining constant speeds.
The RPD levels off at about 0.25 × Fr−0.3 as the flooding transition is approached. The moment of large cavity development corresponds to a point of inflection in the curve, about halfway towards the final level of RPD. On this basis accurate estimates can be made of the power input of a Rushton impeller for any operating conditions.
Gas supplied beneath a downward pumping axial impeller opposes the agitator action as it rises against the downflow. The impellers have less intense tip vortices than a Rushton so that vortex cavities are less effective in their dispersing action. The development of large cavities, which severely affect the efficiency of pitched blade turbines and narrow blade hydrofoils, is hastened by direct loading if gas is supplied from a small sparger mounted close to the impeller.
Figure 5 shows a set of RPD curves for a 30° pitched Blade Turbine. Operating near the catastrophic fall in power level when large cavities develop would be highly undesirable from the mechanical viewpoint. Beyond this point the competing actions of pumping and buoyancy make the impeller function as a rather inefficient radial impeller.
Many reactors use more than one impeller on a common shaft, usually in an attempt to improve the quality of mixing throughout the vessel (see Figure 6). The favored geometry is that of one or more downward pumping axial impellers above a single radial flow impeller, which is mounted just above a sparger. In modern practice the upper impellers are usually wide blade hydrofoils. RPD in multiple impeller equipment can be estimated by treating the lowest impeller as if it were alone in a "standard" tank. The upper impellers are only loaded with about 40% of the sparged gas throughput because of by-passing [Warmoeskerken and Smith (1988)].
Power dissipation is the key to mass transfer and gas holdup. In terms of overall performance a widely used correlation for clean ("coalescing") systems [Harnby et al. (1992)] is :
In some solutions with reduced coalescence, e.g., aqueous ionic solutions above 2 mole% concentration, the smaller bubble sizes lead to mass transfer rates perhaps twice that represented by this equation. Other solutions that generate stable small bubble dispersions, notably those of proteins and surfactants, do not produce the opposite effect if transfer through the interface is hindered.
Mass transfer rates in multiple impeller installations are much more difficult to characterize because of the uncertainties in the local concentration driving force. One basis for design is to consider the gas either to be well mixed (implicitly then everywhere at its outlet concentration) or in plug flow, with the liquid compartmentalized into regions centered on each impeller with appropriate estimates of the local energy dissipation rate.
Many stirred tank reactors involve boiling systems. In the absence of sparged gas the RPD can be estimated on the basis of an Agitation Cavitation number defined as CAgN = (2Sg/ ) in which S is the submergence to the impeller mid plane and vt is the tip velocity. Providing this Agitation Cavitation number is less than some critical value, characteristic for each design of impeller, the RPD can be expressed to a sufficient accuracy by RPD = A × CAgN0.4 in which the constant A has the value of 1.2 for a Rushton Turbine and 0.8 for a 6 blade PBT [Smith and Katsanevakis (1993)]. The power draw in sparged conditions can be estimated by assuming that the total loading provided by vapor and gas in a sparged hot system is that corresponding to complete saturation of the sparged gas at the bulk liquid temperature. This will always be lower than the boiling point at the reactor pressure. It seems likely that in boiling systems the mass transfer will be gas film controlled so the conventional equations for predicting mass transfer rates, which are based on kLa rather than kGa, need to be used with caution.
Harnby, N., Nienow, A. W., and Edwards, M. F. (1992) Mixing in the Process Industries, Massachusetts: Butterworth Heinemann.
Smith, J. M. and Katsanevakis, A. (1992) Trans. I. Chem. E.
Warmoeskerken, M. M. C. G. and Smith, J. M. (1988) Proc. 2nd. Conf. Bioreactor Fluid Mechanics, Elsevier, 79-197.