1

where pb has the units of g L-1. Once again, the advantage of using this system of units is that it will give the correct numerical value of the packing density in SI (since g L-1 is equivalent to kg m-3). Note that the packing density can also be the calculated if both the porosity of the bed (s, m3-voids m-3-bed) and the density of the substrate particles themselves (ps, kg m-3) are known:

where pa is the density of air. Typically the second term on the right hand side will make only a minor contribution, since the air density is typically two to three orders of magnitude smaller than the density of the substrate particles. However, it is not a simple matter to determine the porosity and, as shown in the next section, it would be more common to use this equation to give an estimate of the porosity.

The value of pb will change during the fermentation, for a variety of reasons:

• agitation will affect bed structure;

• the growth of hyphae between particles will affect bed structure;

• the growth process will typically affect particle size and density;

• changes in the moisture content of the bed will affect particle size and density.

Little work has been done to characterize such changes during SSF processes. It is often assumed that the total bed volume does not change. This might occur if the major nutrient is not a structural polymer of the substrate particle, such that the particle size changes little during the fermentation, even as nutrients are converted into CO2. However, if both particle size and density change, the packing density will change in a complex way; these changes must be measured experimentally.

Weber et al. (1999) studied the effect of moisture content on a parameter related to the bed density, namely the specific packed volume on a dry basis (VP, m3 kg-dry-matter-1). They prepared a known mass of moist solid substrate (mb, g), packed it in a manner identical to the packing used for the fermentation, measured the volume occupied by this moist bed (Vm, L), dried it, and then measured the volume occupied by the dried bed (Vd, L). The shrinkage factor (S, m3 -dry-bed m-3-moist-bed) can be calculated directly as the ratio of the two volumes (i.e., Vd/Vm) since the amount of dry matter in the moist and dry samples is identical. Shrinkage was negligible or minor for solid supports designed to be impregnated with a nutrient solution, such as hemp (S = 1.0), bagasse (S = 1.0), and perlite (S = 0.9). In the case of oats the shrinkage was quite significant (S = 0.55). Of course, in an actual fermentation the bed would not be allowed to dry out completely, but Oostra et al. (2000) did show that the value of VP of oat particles can fall by as much as 30% (from about 0.0020 to about 0.0015 m3 kg-dry-matter-1) as the moisture content falls from 1.1 kg-water kg-dry-matter-1 to 0.57 kg-water kg-dry-matter-1, a fall of this magnitude being expected during a packed-bed fermentation with this substrate for the production of spores of the biocontrol fungus Coniothy-rium minitans.

19.2.4 Porosity (Void Fraction)

The way that the substrate bed as a whole packs is important in determining the effectiveness of aeration. The packing will affect the size and continuity of the inter-particle spaces, and it is through these inter-particle spaces that O2 is made accessible to the organism at the particle surface. These effects are characterized by the porosity (s), that is, the fraction of the total bed volume that is comprised by the void spaces (Fig. 19.2):

V V Vb where Vb is the total volume occupied by the bed (m3), Va is the volume within the bed occupied by air (m3), and Vs is the volume within the bed occupied by the substrate particles (m3).

The smaller that particles are, the smaller will be the size of the inter-particle spaces. However, note that the overall porosity does not change significantly, especially for spherical particles, for which the porosity is independent of particle size. Of course, the smaller size of the particles causes larger pressure drops when air is being forced through the substrate mass. This is due to the larger overall surface area of solids that is present, which causes the static gas film that occurs on solid surfaces to occupy a greater proportion of the void volume.

Uniformly sized spheres cannot pack in such a way as to exclude air, even when they are tightly packed (Fig. 19.3(a)). For solid spheres, the porosity can be predicted reasonably easily depending on the way the substrate was packed. However, substrate particles might deform within the bed due to the overlying weight of the bed or due to agitation, even if they were originally spherical. With irregularly sized particles, smaller particles can tend to fill the inter-particle spaces that would otherwise be vacant (Fig. 19.3(b)). This can happen when the substrate releases fines during movement and handling of the dry substrate. For irregularly sized and shaped solids it is not possible to predict the porosity with any accuracy with simple equations and it must be measured experimentally. Particles with large flat surfaces will tend to lie with the flat surfaces touching, excluding O2 and therefore greatly limiting the amount of growth (Fig. 19.3(c)).

The bed porosity typically appears as a key parameter within bioreactor models. However, it is not necessarily an easy thing to measure, especially in SSF processes. If the density of the solid particle is known, it may be possible to estimate the porosity from the bed packing density. A container of known volume is filled with a substrate bed in the same manner in which a bioreactor would be packed, and this is reweighed to give the bed weight (mb, kg). Writing the volume terms in Eq. (19.4) as a mass divided by a density and assuming that the mass of air in the bed (ma, kg) makes a negligible contribution, it is possible to arrive at an equation for the porosity in terms of the overall bed density (pb, kg m-3) and the substrate particle density (ps, kg m-3):

Fig. 19.3. Effect of particle size and shape on the porosity of the bed. (a) Regular spheres cannot pack in such a manner as to exclude void spaces even when they are packed in the most tightly packed conformation possible. (b) However, if spheres are not uniform in size, then smaller spheres can pack within the void spaces between larger spheres. (c) Particles with flat surfaces can tend to lie side-by-side, excluding air

Fig. 19.3. Effect of particle size and shape on the porosity of the bed. (a) Regular spheres cannot pack in such a manner as to exclude void spaces even when they are packed in the most tightly packed conformation possible. (b) However, if spheres are not uniform in size, then smaller spheres can pack within the void spaces between larger spheres. (c) Particles with flat surfaces can tend to lie side-by-side, excluding air

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