It is most logical to build equipment as large as possible because of the economy of scale. There is an empirical relationship between cost and size of an item of equipment. According to this relationship, as facility size increases, its cost increases thus:
cost j /sizej\"
where n is an exponent or scale factor. Scale factors have been estimated to be 0.6 for brewing (Pratten, 1071), 0.7 to 0.8 for a single cell protein plant (Humphrey, 1975; MacLennan, 1976), 0.6 for antibody production (Birch et al., 1987), 0.75 for fermentation processes (Bartholomew and Reisman, 1979) and 0.6 for waste water treatment (Eckenfelder, 1989).
In brewing, when cylindrico-conical vessels are used, there would appear to be no economic advantage in scaling up above 108,000-dm3 capacity (Hoggan, 1977), although vessels of 360,000 dm3 have been installed. In some breweries a number of smaller vessels have been installed to allow for fluctuating demand of different beers.
The operational vessel volume is a critical factor when considering high volume-low cost products. In processes carried out at volumes greater than 100 m3, the use of an air-lift fermenter is more economical as the relative investment costs per unit of output decreased more rapidly than for a stirred type of fermenter (Table 12.4).
However, a number of restraints have to be considered before deciding on the scale of operation. Such restrictions include cooling and aeration requirements and the method of fermentation vessel construction. The need for cooling provisions in many fermentations has already been described (Chapter 7). At this stage it is important to remember that the volume of a fermenter is proportional to r3 (where r is the fermenter radius), whereas the increase in surface area is proportional to r2. Therefore the scaling up of a vessel will lead to a decrease in the surface area to volume ratio and therefore a decrease in the effectiveness of a cooling jacket. There may be a fermenter volume above
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