Conservation Equations for the Gas Phase

For the three bioreactor operation modes (batch, fed-batch and continuous), the continuously flowing gas phase is ubiquitous. The conservation equation for a specie i in the gas phase (e.g., i — O2, CO2) can then be expressed as d{ClGVG)/dt = QgfQgf - QcCic - N.aV, VG = VT-V (2.3)

with cig denoting concentration of specie 1 in the gas phase, Qg the volumetric gas phase flow rate, Vq the gas phase holdup in the bioreactor (volume of gas phase in bubbles and head space), the subscript F the gas feed, Ni the flux of specie i from the gas phase to liquid phase, a the gas-liquid interfacial area per unit culture volume, V the culture volume, and Vt the volume of empty reactor. Equation 2.3 provides a volume-averaged description of the gas phase. In a bioreactor, the gas phase is introduced at the bottom of the bioreactor using spargers. As is evident from Equation 2.3, the rate of transport of a specie i depends on the gasliquid interfacial area, which is higher, the smaller the size of gas bubbles (say bubble diameter). Although the bubble size near the sparger is more or less uniform, there are variations in characteristics of gas bubbles (such as bubble shape and size, gas-liquid interfacial area and concentrations of various gaseous species) during their ascent through the culture. Further, these characteristics of the gas phase in the head space above the culture may be substantially different from those of the gas phase in ascending gas bubbles. A detailed accounting of (bubble-to-bubble and gas bubbles to headspace) heterogeneity in the gas phase, although possible, can certainly divert one's attention from description of culture behavior. For this reason, Equation 2.3 is commonly used for representation of events in the gas phase. It must be realized that while the volume of the head space in batch and continuous cultures is essentially time-invariant [and therefore so is Vg as per Eq. 2.3 ] since the culture volume is essentially unchanged, an increase in culture volume (V) during a fed-batch operation implies a reduction in the head space and Vg [Eq. 2.3].

The transport of a specie % from the gas phase to the liquid phase and vice versa occurs through boundary layers on each side of the gas-liquid interface in the two phases. The dominant mechanism for transport of specie i in each boundary layer in a direction orthogonal to the gas-liquid interface is molecular diffusion. Assuming that specie i does not participate in any chemical reactions in the gas-side boundary layer, the flux Nt in Eq. 2.3 can be expressed as

with C*G denoting the concentration of specie i in the gas phase at the gasliquid interface, kiG the gas-side mass transfer coefficient for specie i, DiG the molecular diffusivity of i in the gas phase, and §G the thickness of the gas-side boundary layer. On the other side of the gas-liquid interface, one must consider in the liquid-side boundary layer the transport of specie i by molecular diffusion. Such transport occurs in parallel with consumption or generation, as appropriate, of specie i as a result of cellular metabolism by cells present in the liquid-side boundary layer and is therefore influenced by the latter. Precise description of events occurring in the liquid-side boundary layer then requires solution of conservation equations which account for diffusion of specie i and its participation in one or more reactions within cells leading to its consumption or generation. Similar conservation equations must also be considered for all species that are non-volatile and participate in cellular reactions. These conservation equations are typically nonlinear second-order (spatially) ordinary (partial) differential equations and simultaneous solution of these can be a computationally challenging task. For this reason, it is assumed that cellular reactions occur to negligible extents in the liquid-side boundary layer. Since the gas-liquid interface has infinitesimal capacity to retain specie i, flux of specie i must be continuous at the gas-liquid interface and the gas and liquid phases must be at equilibrium with respect to species i at the gas-liquid interface. When the two phases are dilute with respect to i, the equilibrium is described by Henry's law. The following relations then apply at the gas-liquid interface.

In Eq. 2.5, Ci and C* denote the concentrations of specie i in the bulk liquid and the liquid phase at the gas-liquid interface, Hi the Henry's law constant for specie i, kn the liquid-side mass transfer coefficient for specie i (kn — DiL/&L), Da, the molecular diffusivity of i in the liquid phase, and Sl the thickness of the liquid-side boundary layer. The thicknesses of the boundary layers in the gas and liquid phases are dependent, among other things, on flow patterns in the two phases. Since the interfacial concentrations are not easily tractable, in view of Eq. 2.5, the flux of specie i across the gas-liquid interface can be expressed in terms of easily tractable variables, CiG and ci j 8ls

kiL + HikiG

The volumetric flow rate of gas feed is usually many-fold greater than that of liquid feed. A pseudo-steady state hypothesis is therefore invoked as concerns Eq. 2.3, with the term on the left side (accumulation term) being much less than any of the three terms on the right side (as concerns absolute magnitudes). Eq. 2.3 then reduces to an algebraic relation. The conservation equations for the culture and its constituents are discussed next.

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