13. Consider the mathematical notions and tell what you know on the topic.

Maxima and Minima

A mininnim value of a function is not necessarily least value which it assumes. In cases for which a least value actually exists, it has already been called the “lower bound” of the function. We say, for example, that, the function

ƒ(x) = x² cos x

has a minimum at x = 0, since ƒ(0)= 0 and ƒ(x)>0 for the values round

about 0; but this function oscillates infinitely and has no lower bound.

A precise definition can be formulated in the following way:

Definition: The set of numbers x such that 0<│x ─ a │<η (i.e. the interval (a ─ η, a + η) with the number a excluded) is called the neighbourhood of a.

Text 2

1. Read and discuss the text. Hydromechanic Processes and Apparatuses. Industrial Centrifuge

Centrifuges are used in­dustrially to cast metal pipe, to recover crystalline precipitates such as table sugar, to separate cream from milk or small pieces of lean meat from particles of tat, to concen­trate slurries, and to dry laundry.

Working principle. According to Newton's first law, a moving ob­ject continues in a straight line unless a force is exerted to deflect it. When an object is moved in a circular path, the reaction against the deflecting force (the pull of the rope on the bucket) is the centrifugal force (which also holds

the water in the bucket). The centrifugal force exerted on a particle in a centrifuge is given by:

F = 0.00001117 (rpm) - x R x M where the force (F) is expressed in grains, the speed (rpm) is in revolutions per minute, the radius R (distance between the axis of rotation and the par­ticle) is in centimeters, and the particle mass (M) is given in grams. Note that the force is directly proportional to the radius (which means that at a fixed speed if the radius is doubled or halved, the force is correspondingly doubled or halved) and is proportional to the square of the speed (doubling the speed increases the centrifugal force at a given radius by a factor of four). For highspeed centri­fuges the diameter is limited by the strength of the alloys used. As the speed is increased, the diameter must be decreased, and to a close approxi­mation the maximum speed is inversely proportional to the diameter. The fact that centrifugal force is proportional to the square of the speed allows very high gravitational fields to be reached in small diameter rotors.

Preparative Laboratory Centrifuges are used in research and in clinical studies, to separate particles ranging in size from blood cells and flocculant precipitates to protein mole­cules or low molecular weight substances on the basis of cither sedimentation rate or buoyant density. Zonal Centrifugation. This technique, also termed isogenic centrifugation, is widely used to separate subceilular particles, viruses, and DNA mole­cules. Swinging-bucket centrifuges are frequentlv used for separating large particles, widely used for blood fractionation and for the sedimentation of coarse precipitates. In the filter-drum centrifuge, liquid passes through the perfo­rations, while solid particles remain in the drum.

Analytical Uttracentrifuge are optical centrifuges cell fitted with transparent quartz or sapphire windows and scanning photo-multiplier cell for precise measure­ments and recording the rate of sedimenta­tion, sedimentation

equilibrium, and buoyant density.

Fig. 32. Centrifugue.