Fig. 9.15. A typical power curve for a baffled vessel agitated by a flat-blade turbine.
Reynolds number so that the value of x is zero ,nd the value of the Reynolds number is in excess of l»4-
,f the values of the exponent, * are substituted into ^ M1 (9 21) for the zones of viscous and turbulent (hm.'l'licn the following terms are given:
For turbulent flow P = cpN3D5. (9.23)
I „mi dic-e equations it may be seen that power con-sU„,piion is dependent only on the viscosity of the liquid in the region of viscous flow and that increased ,,vcd "i agitation, or an increase in the impeller di-uiieiei. u-sults in a proportionally greater increase in power transmission to a liquid in turbulent flow than to !.ik- in mucous How. Conditions of viscous flow are rare in'k-niK iuation processes, the majority of fermenta-iinn- exhibiting flow characteristics in either the turbulent or transition zones. If turbulent flow is demonstrated to occur in a fermentation then equation (9.23) may be used to predict its power requirements and to predict the operating conditions of different sized vessels to achieve the same agitation conditions, as outlined b\ Hanks (1979). Power consumption on the small scale may be represented as:
sm and on the large scale as:
PL = cpNlDl wheie i lie subscripts sm and L refer to the small and large scales respectively. Maintaining the same power input per unit volume:
where V is the volume.
Assuming the vessels to be geometrically similar then c will be the same regardless of scale and as the same broth would be employed p would remain the same for both systems
K,n/VL = (N?mDfm)/(N*Dl) (9.26) For geometrically similar vessels
Dsm/DL = (Vsm/VL)1/3 Therefore, substituting for Djm/DL in (9.26)
If transient flow conditions occur in a fermentation then it is necessary to construct a complete power curve for such predictions and this is discussed later in the chapter.
The work of Rushton et al. (1950) was carried out using ungassed liquids whereas the vast majority of fermentations are aerated. It is widely accepted that aeration of a liquid decreases the power consumption during agitation because an aerated liquid, containing suspended air bubbles, is less dense than an unaerated one and large gas-filled cavities generated behind the agitator blades decrease the hydrodynamic resistance of the blades. A number of workers have produced correlations of gassed power consumption, ungassed power consumption and operating variables, that of Michel and Miller (1962) being widely used:
where Q is the volumetric air flow rate.
However, more recent correlations have been elucidated which are applicable over a wider range of operating conditions than that of Michel and Miller. Hugh-mark (1980) produced the following correlation from 248 sets of published data:
where Q is the volumetric air flow rate, g is the acceleration due to gravity, and W is the impeller blade width. Using dimensional analysis:
Pg/P = 0.0312 • Fr^1-6 - Re0 064Na~ 0 38 • (T/D)0 H
where Na is the aeration number and equals Q/ND and T is the vessel diameter.
Provided it is remembered that these expressions are not particularly accurate they may be used to predict power consumption in gassed systems where turbulent flow is known to be operating. However, it should be remembered that in non-mycelial fermentations the greatest power demands often occur during agitation when the system is not gassed, that is during the sterilization of the medium in situ or if the air supply were to fail. Thus, in designing the system care must be taken to ensure that the agitator motor is sufficiently powerful to agitate the ungassed system and for fixed speed motors the operating speed should be specified with respect to the ungassed power draw (Gbewonyo et al, 1986).
From the foregoing account it may be seen that reasonable techniques exist to relate operating vari ables to power consumption and, hence, to the degree of agitation which may be shown to have a proportional effect on KLa. However, these techniques apply to Newtonian fluids and are not directly applicable to the study of non-Newtonian systems. Non-Newtonian fluids do not have constant viscosities, which creates difficulties in utilizing relationships which rely on being able to determine the fluid viscosity. These difficulties may be avoided if the agitation system is capable of maintaining turbulent-flow conditions during the fermentation, because under such conditions power consumption is independent of the Reynolds number and, hence, of viscosity. However the high viscosities of the majority of mycelial fermentation broths make fully turbulent flow conditions impossible, or extremely difficult, to achieve. Such fermentations tend to exhibit transient zone flow conditions which necessitate the construction of complete power curves to correlate power consumption with operating variables. The fact that the viscosity of a non-Newtonian liquid is affected by shear rate means that the viscosity of a non-Newtonian fermentation broth will not be uniform throughout the fermenter because the shear rate will be higher near the agitator than elsewhere in the vessel. Thus, the determination of the impeller Reynolds number is made difficult by not knowing the viscosity of the fermentation broth. Metzner and Otto (1957) proposed a solution to this paradox by introducing the concept of average shear rate (y) related to the agitator shaft speed in the vessel, by the equation:
where k is a proportionality constant.
Metzner and Otto determined the value of the proportionality constant to be 13 for pseudoplastic fluids in conventional, baffled reactors agitated by single, flat-blade turbines. Several groups of workers have determined values of k under a wide range of operating variables; the values range from approximately 10 to 13. Metzner et al. (1961) suggested that a compromise value of 11 could be used for calculation purposes, with relatively little loss of accuracy, which would obviate the necessity to determine k for each circumstance. Therefore, provided that the rheological properties of a fermentation broth are known, an apparent average viscosity of the fluid may be calculated using the average shear rate which would enable the calculation of the impeller Reynolds number for each value of the impeller rotational speed, thus enabling a power curve to be constructed. Such a power curve may be used to predict the power requirements of a fermentation and to scale up a fermentation on the basis of power consumption per unit volume. Metzner and Otto's approach has not been widely applied but there are some recent examples of its adoption. Nienow and Elson (1988) suggested that a repetition of their work would be very valuable using the more sophisticated instrumentation now available. An example of the use of the technique is considered in a later section considering the operation of viscous polysaccharide fermentations.
The effect of medium and culture rheology on K, a
As can be seen from the previous section, the rheology of a fermentation broth has a marked influence on the relationship between KLa and the degree of agitation. The objective of this section is to discuss the effects of medium and culture rheology on oxygen transfer during a fermentation. A fermentation broth consists of the liquid medium in which the organism grows, the microbial biomass and any product which is secreted by the organism. Thus, the rheology of the broth is affected by the composition of the original medium and its modification by the growing culture, the concentration and morphology of the biomass and the concentration and rheological properties of the microbial products. Therefore, it should be apparent that fermentation broths vary widely in their rheological properties and significant changes in broth rheology may occur during a fermentation.
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