**1. Introduction**

594 Mass Transfer - Advanced Aspects

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Compounds present in the microbial cells are of great commercial value. Their separation requires disintegration of cell walls and cytoplasmic membranes. The disintegration of microorganisms on an industrial scale is carried out, among the others, in bead mills (Chisti & Moo-Young, 1986; Geciowa et al., 2002; Hatti-Kaul & Mattiasson, 2003). A bead mill is a container filled with beads that are set in circular motion by a rotating stirrer. Microorganisms dispersed in the liquid are disrupted due to the impact of beads. The mechanism of cell destruction is a result of combined action of normal and tangential forces. There are three basic types of this mechanism: collisions of beads, grinding and rolling performed by the beads. Disintegration is a very complex process. In this process, cell walls of microorganisms are disrupted, intracellular compounds are released and dissolved in the continuous phase, cell walls are subjected to microgrinding, the released macromolecular compounds are cut and the released enzymes interact. Rheological properties of the suspension and its continuous phase are changed.

Currie et al. (1974) described disintegration kinetics of microorganisms in the bead mill. The authors developed a linear model by comparing experimental data with results obtained during the disintegration process carried out in a high-pressure homogenizer (Hetherington et al., 1971). A logical model based on the analogy to the theory of gas kinetics was proposed by Melendres et al. (1998). They presumed that microorganisms could be destroyed due to collisions of dispersed beads of the packing. A phenomenological model based on the flow of suspension between two volumes was developed by Heim & Solecki (1998). The authors assumed that cells were disrupted while moving from a safe volume to the one in which no living microorganisms could exist. Basing on the sequence of events: cell disruption – the release of intracellular compounds, Melendres et al. (1993) developed a nonlinear model of the release of selected intracellular enzymes. Heim et al. (2007) described nonlinearity of the kinetics caused by changes in the disintegration conditions which was a result of the process run. Nonlinearity of the disintegration process resulting from subsequent decline of the biggest fractions of yeast cells was observed by Solecki (2009). Earlier, Whitworth (1974) described nonlinearity of cell disintegration kinetics in a suspension containing microorganisms (*Candida lipolytica*) which belonged to the same species and occurred in two morphological forms. The recently developed theory of random transformations of dispersed matter makes it possible to include in the description the fundamental phenomena observed during the process (Solecki, 2011). The aim of the study was to

The Release of Compounds from Microbial Cells 597

On the other hand, volume Vγji can be limited by surface Fγβji when it is formed. This surface

*F V* γβ

*F V* γα

Space Vβji cannot include total unconverted material object belonging to set *N* for which an appropriate transformation volume is Vγji. Before being translocated to Vβji, every i-th object

Volume Vαi is safe for the unconverted i-th object from set *N*. We assume that intensive stirring takes place in it. Its aim is to make the concentration of unconverted objects from set *N* uniform within appropriate volumes which are safe for them. Volume Vαi is composed of two parts: Vαit which is part of volume Vαi whose subsets can be transformed to volumes Vγ<sup>i</sup> or Vβi; and Vαic which is part of volume Vαi whose subsets are never transformed to other volumes. In the case of the bead mill, volumes Vαic occur near all surfaces of the working chamber. These volumes are distant from the mill surface by the size of a cell, and the thickness of their layer is slightly smaller than the radius of the smallest bead in the packing. In volume Vαic the i-th cell is fully safe. The quotient of volumes Vαic to Vαi, the quotient of the sum of volumes Vγi and Vβi to Vαit and the rate of relative movement of the i-th cell to Vγ<sup>i</sup> are the factors that determine efficiency of the system of transformation of the i-th cell in a given technical device. An increase of the first factor causes a decrease of the transformation

For instance, studies were carried out to increase differences in the velocity of points on the surfaces of adjacent beads of the packing which circulates in the mill (Solecki, 2007). Such an effect was to be induced by the presence of immobile baffles between stirring disks of a classical mill. In many cases of geometric solution of the mill interior, because of introduction of the baffles the efficiency of microorganism disintegration was deteriorated (Solecki, 2007). The development of a narrow-clearance construction did not bring about elimination of classical mills equipped with multi-disk impellers from the market. Rate constants determined for the process of disintegration in the mill with a bell-shaped impeller were often lower than those determined for a classical mill in comparable process conditions (Solecki, 2007). Optimum operating conditions of narrow-clearance mills are obtained for smaller values of packing degree. In the case of a classical mill it is about 90%, while for the mill with a bell-shaped impeller 60%, and with a cylindrical stirrer only 40%. Part of space V is composed of volume Vδ which is safe for microorganisms from set *N* (Solecki, 2011). No stirring takes place in it and neither it nor its subsets are transformed to other volumes. Between volumes Vα and Vδ microbial cells can migrate freely. For a correctly constructed mill chamber, volumes Vδ (these can be slots in the place where two elements meet) are negligibly small and insignificant from the technology point of view, particularly when the device is sterilized between subsequent processes. In further studies it

efficiency. An opposite effect is caused by an increase of other factors.

The i-th object is transformed immediately after translocation of all its points to the transformation volume Vγji. If volume Vγji is bigger than the volume of the i-th object, then volume Vβji is formed in it. It is limited by surface F<sup>γ</sup>αji which does not belong to it according

 γ

 β

*ji* ∈ *ji* (6)

*ji* ∉ *ji* (7)

belongs to the transformation volume according to Eq. (6).

must be first transformed in volume Vγji.

was assumed that Vδ=0.

to Eq. (7)

develop, basing on this theory, a phenomenological model of microorganism disintegration in a bead mill and to present a mathematical description of the process which would include the effect of cell size on the rate of cell disintegration.
