Biochemical Engineering Fundamentals Part 1 - Bailey J., Ollis D. 1989

Transport Processes in Biotechnological Systems
Mass Transfer Between Gas and Liquid Phases in Cellular Systems
Basic Principles of Mass Transfer Theory

The solubility of oxygen in aqueous media at 1 atm air pressure and room Temperature is approximately 10 parts per million (ppm^-1) (Table 8.1). During active Respiration, a Yeast Cell population can consume oxygen at a rate of about 0.3 g O₂ per hour (per 1 g of dry cell mass). The maximum oxygen utilization rate at a population density of 10⁹ Cells/ml can be estimated by assuming that The Cell volume is 10^-10 ml, 80% of which is Water. Then the absolute oxygen demand will be

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FIG. 8.2. Various modes of contact between the gas and liquid phases: a — freely rising gas bubbles, freely falling solid particles, and liquid droplets; b — mechanical agitation.

Consequently, during active respiration, the cell population consumes oxygen at a rate equivalent to approximately 750 times the oxygen saturation per hour. Since dissolved oxygen reserves are quite meager, a viable cell population can only survive if oxygen is continuously introduced into the liquid medium. This task is far from simple, as the low solubility of oxygen results in a small concentration gradient, which acts as the main driving force for mass transfer from one phase to the other.

For sparingly soluble substances, such as oxygen and Hydrocarbons in aqueous media, the relationship between the two equilibrium concentrations at the phase boundary, c_gi (on the gas phase side) and c_li (on the liquid phase side), can generally be expressed by an equation reflecting a linear distribution law, such as Henry's law:

Mc_li = с_gi      (8.1)

This equation holds true if the solute exchange rate across the phase boundary is significantly higher than the overall mass transfer rate; this condition is usually satisfied because at an air pressure of 1 atm and 25 °C, the collision frequency of O₂ molecules with the surface is approximately 10²⁴ per square centimeter per second, which far exceeds the total oxygen demand in the typical microbiological process example given above.

At steady state, The rate of oxygen transfer to the gas-liquid interface equals the rate of its transport through the film on the liquid phase side (Fig. 8.1). If c_g and c_i are the oxygen concentrations in the gas and liquid phases, respectively, we can write the equality of the two transfer rates:

Here, k_l and k_g are the Mass transfer coefficients in the liquid and gas phases, respectively.

Since interfacial concentrations cannot be measured directly during mass transfer studies, mass transfer is usually quantified using the overall mass transfer coefficient K_l and the overall driving force (concentration difference) c_l* - c_l, where c_l* is the concentration of the dissolved

substance in the liquid phase in equilibrium with the gas phase:

Using these parameters, the solute flux density can be expressed as:

Flux density = K_l(c_l*- с_l)      (8.4)

From equations (8.1) through (8.4), it follows the well-known expression relating the overall mass transfer coefficient K_l to the physical parameters of the two-film model k_g, k_l, and M:

For sparingly soluble substances, M is much greater than unity. Furthermore, k_g is typically much greater than k_l. It follows that K_l is approximately equal to k_l, and thus practically all the mass transfer resistance is concentrated on the liquid-side film.

The rate of oxygen transfer per unit volume of the Reactor, Q_O2, is defined as

Here, a' = A/V is The ratio of the gas-liquid interfacial area to the volume of the liquid phase; equation (8.6) adopts the approximation K_l ≈ k_l just discussed. Since this chapter focuses primarily on aeration processes, we will subsequently consider mainly oxygen transfer and therefore use k_l instead of K_l as the mass transfer coefficient. The symbol a appearing in some equations denotes the gas-liquid interfacial area per unit volume of the biological reactor (i.e., per unit volume of the gas-liquid dispersed system). The gas located above the liquid phase is not taken into account.

Keep in mind that Q_O2 represents a parameter reflecting the local volumetric O₂ absorption rate; the average volumetric oxygen utilization rate (expressed in moles per unit time per unit volume) over the entire liquid volume V is defined as

In the general case, is equal to Q_O2 if and only if the hydrodynamic conditions, the specific interfacial area, and the oxygen concentration are identical at every point within the reactor volume.

The observed average mass transfer rate is determined by a variety of phenomena and factors, including the power consumed by the reactor (per unit volume), the rheology of the liquid phase and dispersed system, the CHARACTERISTICS OF THE sparging device, and the overall flow Structure within the reactor. Fig. 8.3 illustrates the numerous relationships between the observed average transfer rate and the phenomena governing this transfer. Since we generally lack critical fundamental data on coalescence and re-dispersion rates, bubble size distribution, and residence times within the reactor, we are typically forced to develop the necessary equations using appropriate average values: average bubble size, gas holdup (the volume fraction of gas dispersed in the liquid), average residence times of the gas and liquid phases in the reactor, etc.

The data presented in Table 8.1 indicate that c_i* depends on temperature and medium composition. The dependence on medium composition becomes more complex if the dissolved gas can react with the liquid phase. This situation is typical, for example, of carbon dioxide, which can exist in the liquid phase in any of four forms: СО₂, Н₂СО₃, НСО₃^-, and СO₃^2-. The corresponding equilibrium constants at 25°С are:

which demonstrate that the total concentration c_T of carbon dissolved as СO₂ strongly depends on pH:

FIG. 8.3. Relationship between agitation intensity and gas transfer rate. [Reproduced by permission from: Resnick W., Gal-Or В., Adv. Chem. Eng., 7, 295 (1968).]

This relationship is shown graphically in Fig. 8.4; it is easy to see that at pH < 5 almost all carbon is present as dissolved molecular СO₂, bicarbonate dominates in the range 7 < pH < 9, and carbonate prevails at pH > 11. Only dissolved СO₂ crosses the gas–liquid interface; its transfer rate can be described by equation (8.2).

FIG. 8.4. Equilibrium concentrations of СО₂, НСО₃-, СО₃^2-, and Н₂СО₃ in solution (c_T is the total concentration of СO₂ in all four forms; p_CO2 = 10^-3.5 atm (corresponding to the partial pressure of СO₂ in air); the pH was adjusted to the desired value using a strong acid or strong base). (From: Stumm W., Morgan J. J., Aquatic Chemistry, p. 127, John Wiley and Sons, N.Y., 1970.)

In neutral or alkaline media, both mass transfer effects and chemical processes play a significant role. For instance, the rate of the reversible reaction (8.11) is very high;

Conversely, reaction (8.13) proceeds much more slowly (k₁ = 20 s^-1; k_-1 = 0.03 s^-1 at 25°С):

Consequently, depending on the conditions, the rate-determining step for The transfer of СO₂ into the gas phase can be either the chemical reaction [equation (8.13)] or the physical process [СO₂(aq)→СO₂(g)].