Читать книгу Agitator Design for Gas-Liquid Fermenters and Bioreactors - Gregory T. Benz - Страница 38

Agitation systems, just as any other system producing or modifying fluid flow, must obey the laws of physics. In terms of mathematical models, they obey the equations of continuity and the Navier–Stokes equations. Unfortunately, those equations can usually only solve problems analytically in relatively simple geometries, such as flow in a pipe, and, often, only in laminar flow. Such equations can be supplemented by various turbulence models.

An agitated tank, however, is a very complex geometry. Most would agree that it is all but impossible to solve the equations of motion for an agitated tank by analytical methods. In modern times, there have been many successful attempts to model agitated tanks by using numerical methods, which in essence convert differential equations into a series of algebraic approximations. Those approximations can be very good, depending on the skill of the modeler and the computational power used. These methods are often called CFD (Computational Fluid Dynamics) and sometimes called CFM (Computational Fluid Mixing.) Chapter 14 describes some of the uses of CFD as applied to fermenter design.

The traditional way of solving agitation problems is quite different. The approach that has been used in most studies, and which is still the staple of agitator design, is to use the equations of motion to derive dimensionless number groups and then correlate experimental data in terms of those dimensionless numbers. That is the approach we will take for the majority of this book.

We will not show the derivation of the dimensionless numbers, but will describe the ones important for our use in designing agitators, and how they are used, especially for fermenter design.

Some readers may be unfamiliar with the concept of dimensionless numbers, so we will give a brief description here, prior to getting into the commonly used dimensionless numbers.

A dimensionless number is a ratio of quantities such that the dimensions and units in the numerator exactly match the dimensions and units in the denominator, thereby canceling all dimensions and units. The resulting dimensionless number has no units or dimensions; it is just a scalar number. It also does not depend on what units are used, though converting dissimilar units to a consistent set of units will assist with the math.

A rather trivial example is the concept of aspect ratio of a cylinder, which equals its height or length divided by its diameter. A 5‐ft. tall cylinder with a 12 in. diameter has an aspect ratio of 5. That is because 5 ft. is 5 times as much, in terms of its dimension (length), as 12 in. But the math would be more obvious and less prone to error if we first converted the diameter to feet by dividing by 12, or, alternatively, converting the height of the cylinder to inches by multiplying by 12. But the important point is that it is the ratio of the actual physical dimensions and is not unit dependent. We could have stated the dimensions as meters, microns, or cubits; the dimensionless number we are calling aspect ratio would still be 5.

In the next several sections, we will cover the major dimensionless numbers used in fermenter agitator design, and then we will show how some of them are used.