shear rate, for a Newtonian fluid, would produce a straight line, the slope of which would equal the viscosity. Such a plot is termed a rheogram (as shown in Fig. 9.10).
Thus, a Newtonian liquid has a constant viscosity regardless of shear, so that the viscosity of a Newtonian fermentation broth will not vary with agitation rate. However, a non-Newtonian liquid does not obey Newton's law of viscous flow and does not have a constant viscosity. The value for n (equation (9.9)) of such a fluid deviates from 1 and its behaviour is said to follow a power law model. Thus, the viscosity of a non-Newtonian fermentation broth will vary with agitation rate and is described as an apparent viscosity (/ia). A plot of shear stress against shear rate for a non-Newtonian liquid will deviate from the relationship depicted in Fig. 9.10, depending on the nature of the liquid. Several types of non-Newtonian liquids are recognized and typical rheograms of types important in the study of culture fluids are given in Fig. 9.11, and their characteristics are discussed below.
Bingham plastics are similar to Newtonian liquids apart from the fact that shear rate will not increase until a threshold shear stress is exceeded. The threshold shear stress is termed the yield stress or yield value, t0. A linear relationship of shear stress to shear rate is given once the yield stress is exceeded and the slope of this line is termed the coefficient of rigidity or the plastic viscosity. Thus, the flow of a Bingham plastic is described by the equation:
where n is the coefficient of rigidity and t0 is the yield stress.
There have been some claims of mycelial fermentation broths displaying Bingham plastic characteristics (Table 9.5). Everyday examples of these fluids include toothpaste and clay.
The apparent viscosity of a pseudoplastic liquid decreases with increasing shear rate. Most polymer solutions behave as pseudoplastics. The decrease in apparent viscosity is explained by the long chain molecules tending to align with each other at high shear rates resulting in easier flow. The flow of a pseudoplastic liquid may be described by the power law model, equation (9.9), i.e.:
K has the same units as viscosity and may be taken as the apparent viscosity. The flow-behaviour index is less than unity for a pseudoplastic liquid, the smaller the value of n, the greater the flow characteristics of the liquid deviate from those of a Newtonian fluid. Equation (9.9) may be converted to the logarithmic form as:
Thus, a plot of log shear stress against log shear rate will produce a straight line, the slope of which will
Table 9.5. Some examples of the rheologieal nature of fermentation broths
Organism
Rheologieal type
Reference
Pénicillium chrysogenum
Streptomyces kanamyceticus Pénicillium chrysogenum
Endomyces sp.
Pénicillium chrysogenum
Bingham plastic
Bingham plastic Pseudoplastic
Pseudoplastic
Casson body
Deindoerfer and Gaden (1955) Sato (1961) Deindoerfer and West (1960) Taguchi et al. (1968) Roels et al. (1974)
equal the flow-behaviour index and the intercept on the shear stress axis will be equal to the logarithm of the consistency coefficient.
Many workers have demonstrated that mycelial fermentation broths display pseudoplastic properties as shown in Table 9.5.
The apparent viscosity of a dilatant liquid increases with increasing shear rate. The flow of a dilatant liquid may also be described by equation (9.9) but in this case the value of the flow-behaviour index is greater than 1, the greater the value the greater the flow characteristics deviate from those of a Newtonian fluid. Thus, the values of K and n may be obtained from a plot of log shear stress against log shear rate. Fortunately this type of behaviour is not exhibited by fermentation broths — an everyday example is liquid cement slurry.
Casson (1959) described a type of non-Newtonian fluid, termed a Casson body, which behaved as a pseudoplastic in that the apparent viscosity decreased with increasing shear rate but displayed a yield stress and, therefore, also resembled a Bingham plastic. The flow characteristics of a Casson body may be described by the following equation:
where Kc is the Casson viscosity.
A plot of t/t against fy will give a straight line, the slope of which will equal the Casson viscosity and the intercept of the yV axis will equal -/r0.
Roels et al. (1974) claimed that the rheology of a penicillin broth could be best described in terms of a Casson body.
Therefore, to determine the rheologieal nature of a fluid it is necessary to construct a rheogram which requires the use of a viscometer which is accurate over a wide range of shear rates. Furthermore, the testing of mycelial suspensions may present special difficulties. These problems have been considered in detail by Van't Riet and Tramper (1991), whose book should be consulted for methods of assessing the rheologieal properties of mycelial fluids.
FACTORS AFFECTING K, a VALUES IN FERMENTATION VESSELS
A number of factors have been demonstrated to affect the KLa value achieved in a fermentation vessel. Such factors include the air-flow rate employed, the degree of agitation, the rheologieal properties of the culture broth and the presence of antifoam agents. If the scale of operation of a fermentation is increased (so-called 'scale-up') it is important that the optimum KLa found on the small scale is employed in the larger scale fermentation. The same KLa value may be achieved in different sized vessels by adjusting the operational conditions on the larger scale and measuring the K, a obtained. However, quantification of the relationship between operating variables and K, a should enable the prediction of conditions necessary to achieve a particular KLa value. Thus, such relationships should be of considerable value in scaling-up a fermentation and in fermenter design.
The effect of air-flow rate on K, a
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