Phenomenological method

The complexity of food production processes and the variety of operating factors are the objective basis for the widespread use of the so-called phenomenological dependencies. Historically, a large number of phenomena of the transfer of energy and matter are approximated by dependencies of the form

I = aX, (1)

where I - the speed of the process; a - constant; X - the driving force behind the process.

The class of such phenomena includes: deformation of a rigid body (Hooke's law); movement of electric current along a conductor (Ohm's law); molecular heat transfer (Fourier's law); molecular mass transfer (Fick's law); generalized (not only molecular) laws of heat and mass transfer; energy losses when a fluid moves through a pipeline (Darcy and Weisbach laws); motion of a body in a continuous medium (Newton's law of friction), etc. In the laws describing these phenomena, the constants have a physical meaning and are named accordingly: elastic modulus, electrical resistance, molecular thermal conductivity, molecular diffusion coefficient, convective thermal conductivity or turbulent diffusion coefficient, Darcy coefficient of friction, viscosity, etc.

Having drawn attention to this, the Belgian physicist of Russian origin I. Prigogine, the Dutch physicists L. Onsager, S. de Groot and others generalized these phenomena in the form of relation (1), which was called phenomenological, or the relation of the logic of phenomena. It formed the basis of the phenomenological research method, the essence of which is briefly formulated as follows: for small deviations from the equilibrium state, the rate of flow I any complex process is proportional to the driving force of this process X.

The main laboriousness of research using this method is to identify the factors or parameters that are the stimuli of this process, and the factors that characterize its result. Having identified them, the relationship between them is represented in the form of dependence (1), and the numerical value of the coefficient connecting them a determined experimentally. For example, if the driving force of the extraction process is the difference between the concentrations ΔС of the extractable substance in the raw material and in the extractant, and the rate of the process is characterized by the time derivative of the concentration of this substance C in the raw material, then we can write:

BΔC,

where B— extraction rate coefficient.

You can always name a number of parameters that characterize both the driving force and the effectiveness of the process. As a rule, they are unambiguously related. Therefore, the phenomenological equation can be written in many versions, that is, for any combination of parameters that characterize the driving force and effectiveness of the process.

The phenomenological method, being formal, does not reveal the physical essence of the ongoing processes. However, it is widely used due to the simplicity of describing the phenomena and the simplicity of using experimental data.

Experimental method

On the basis of a preliminary analysis of the problem under study, factors are selected that have a decisive or significant effect on the desired result. The factors that have little influence on the result are discarded. Discarding factors is associated with the search for compromises between the simplicity of analysis and the accuracy of describing the phenomenon under study.

Experimental studies are carried out, as a rule, on a model, but an industrial installation can also be used for this. As a result of experimental studies carried out according to a specific plan and with the required repetition, the dependences between factors are revealed in graphical form or in the form of calculated equations.

The experimental method has the following advantages:

• the ability to achieve high accuracy of the derived dependencies
• a high probability of obtaining dependencies or physical characteristics of the research object, which cannot be found by any other method (for example, thermophysical characteristics of products, degree of emissivity of materials, etc.).

At the same time, the experimental research method has two significant drawbacks:

• high labor intensity, due, as a rule, to a significant number of factors influencing the phenomenon under study
• the found dependencies are private, related only to the investigated phenomenon, which means that they cannot be extended to conditions other than those for which they were obtained.

Analytical method

This method consists in the fact that on the basis of the general laws of physics, chemistry and other sciences, differential equations are compiled that describe a whole class of similar phenomena.

For example, the differential Fourier equation determines the temperature distribution at any point of the body through which heat is transferred by thermal conductivity:

A 2 t, (2)

where a is the coefficient of thermal diffusivity, m 2 / s; t - Laplace operator;

2 t = + +.

Equation (2) is valid for any stationary medium.

The advantage of the analytical method is that the obtained differential equations are valid for the entire class of phenomena (heat conductivity, heat transfer, mass transfer, etc.).

However, this method has significant drawbacks:

• the complexity of the analytical description of most technological processes, especially processes accompanied by heat and mass transfer; this explains the fact that such calculation formulas are known today little
• impossibility in many cases to obtain the solution of differential equations analytically using formulas known in mathematics.

9. Cutting.

Cutting is one ofmain technological processes of the food industry.

A variety of materials are subjected to cutting, such as: candy mass in the confectionery industry, dough mass in the bakery industry, vegetables and fruits in the canning industry, sugar cakes in the beet sugar industry, meat in the meat industry.

These materials have a variety of physical and mechanical properties, which is determined by the variety of cutting methods, the type of cutting tools, cutting speed, cutting devices.

An increase in the capacity of food industry enterprises requires an increase in the productivity of cutting machines, their efficiency, the development of rational cutting modes.

The general requirements for cutting machines can be formulated as follows: they must provide high productivity, ensure high quality products, high wear resistance, ease of operation, minimum energy costs, good sanitary condition, small dimensions.

Classification of cutting devices

Food cutting devices can be divided intogroups on the following grounds:

by purpose: for cutting brittle, hard, elastic-viscous-plastic and inhomogeneous materials;

according to the principle of action: periodic, continuous and combined;

by type of cutting tool: lamellar, disk, string, guillotine, rotary, string (fluid and pneumatic), ultrasonic, laser;

Rice. 1. Types of cutting tools:
a - rotor; b— guillotine knife; c - circular knife; g - string

by the nature of the movement of the cutting tool: with rotary, reciprocating, plane-parallel, rotary, vibration;

by the nature of the movement of the material during cutting and by the type of its attachment.

In fig. 1 shows some types of cutting tools: rotary, guillotine, disk, jet.

Cutting theory

Cutting has the task of processing the material by separating it in order to give it a given shape, size and surface quality.

In fig. 2 shows a diagram of material cutting.

Fig2. Cxe m a pe knowledge of the material:
1- pa cut material; 2 - cutting tool, 3 - plastic deformation zone, 4 - elastic deformation zone, 5 - boundary zone, 6 - fracture line

For pe 3 a materials are separated into parts as a result of the destruction of the boundary layer. Failure is preceded by elastic and plastic deformation, as shown in the figure. These types of deformations are created by applying force to the cutting tool. Failure of a material occurs when the stress becomes equal to the ultimate tensile strength of the material.

Cutting work is spent on creating elastic and plastic deformation, as well as overcoming the friction of the tool against the material being cut.

The cutting work can be determined theoretically as follows.

Let us denote the force that must be applied to the edge of a knife 1 m long to destroy the material through R (hN / m). Work A (in J) is spent on cutting the material with an area l - l (in m 2) we will

A - (Pl) l - Pl 2

By referring the work to 1 m 2 , we get the specific work of cutting (in J / m 2 ).

Some types of cutting

Beet cutters and vegetable cutters... In sugar factories, beet chips of a grooved or lamellar farm are obtained by cutting. In the canning industry, carrots, beets, potatoes, etc., are subjected to cutting.

The action of the cutters is based on the relative movement of the cutters - knives and material. This relative motion can be accomplished in a variety of ways.

The main types of cutting are disc and centrifugal. Disc cutting for beets is shown in fig. 3. It consists of a horizontal slotted rotating disc and a stationary drum located above it. Frames with knives are installed in the slots of the disc (Fig. 4). The disc rotates on a vertical shaft with a rotational speed of 70 rpm. The average linear speed of the knives is about 8 m / s.

The drum is filled with beets to be chopped. When the disc rotates, the beets, pressing against the knives under the influence of gravity, are cut into shavings, the shape of which depends on the shape of the knives.

In addition to disc cutting, centrifugal cutting is also used. In these x cutting knives are fixed in the slots of the walls of the stationary vertical cylinder. The material to be cut is set in motion by the blades of a snail rotating inside the cylinder. The centrifugal force pushes the product against the knives, which cut it open.

P is. 5. Diagram of a rotary cutting device

In fig. 5 shows rotary cutting for confectionery products. Candy mass, decorated in bundles 3from the die 1 of the forming machine falls on the receiving tray 2 and is fed through it to the cutting device. Cutting e the device consists of a set of rotors freely rotating on the axis 4 with knives attached to them. Each harness has its own rotor. It is driven by a moving tourniquet. 5 cut candies fall onto 6 conveyor belt.

In fig. 6 shows two types of machines for cutting frozen and unfrozen meat, bread, potatoes, beets, etc., called tops.

The design of tops used inindustry copied from meat grinders, xopo sho famous and common in everyday life. Three types of cutting tools are used in tops: fixed scoring knives, knife grates and movable flat knives.

Cutting is done with a pair of cutting tools - flat m rotating knife and knife grid. The material is fed by the auger, pressed against the knife grate, material particles are pressed into the holes of the grate, and continuously rotating flat kniveswith blades pressed against the gratings, cut off material particles.

Rice. 6. Two types of tops:
a - without forced supply of material; b — forced feed

The frequency of rotation of the screw for low-speed tops is 100-200, for high-speed ones over 300 rpm.

29. Homogenization.

The essence of homogenization. Homogenization (from the Greek homogenes - homogeneous) - creation of a homogeneous homogeneous structure that does not contain parts that differ in composition and properties and are separated from each other by interfaces. Homogenization is widely used in the canning industry, when the product is brought to a finely dispersed mass with particles of 20 ... 30 μm in diameter at a pressure of 10 ... 15 MPa. In confectionery industries, due to homogenization, which consists in processing the chocolate mass in conch machines, emulsifiers or melangers, an even distribution of solid particles in the cocoa butter is ensured and the viscosity of the mass is reduced.

Particles of emulsions, suspensions, suspensions are significantly smaller in size than the working bodies of any mechanical mixing devices. The particle sizes are smaller than the vortexes generated by the mixing devices, and smaller than the sizes of other inhomogeneities in the flow of a continuous medium. Due to the movement of the medium initiated by mechanical mixers, the associations of particles move in it as a whole without a relative displacement of the components of the dispersed phase and the dispersion medium. Such a movement cannot provide mixing of the components of the medium on the required scale.

The extent to which it is advisable to mix food particles is determined by the conditions of food assimilation. At present, the boundaries of the scale to which it is advisable to homogenize food mixtures have not been identified. There are, however, a number of studies showing the feasibility of homogenizing foods down to the molecular level.

For homogenization of products, the following physical phenomena are used: crushing of liquid particles in a colloid mill; throttling of the liquid medium in the valve clearances; cavitation phenomena in liquid; motion of ultrasonic waves in a liquid medium.

Crushing liquid particles in a colloid mill.Between the carefully processed hard conical surfaces of the rotor and stator of a colloid mill (Fig. 7), the emulsion particles can be crushed to a size of 2 ... 5 microns, which is often sufficient for homogenization.

Rice. 7. Diagram of a colloid mill:
1- rotor; 2 — stator; h - clearance

Throttling of the liquid medium invalve clearances.If a liquid medium, compressed to 10 ... 15 MPa, is throttled, passing through a small-diameter nozzle or through a throttle (throttle washer), then the spherical formations in it, when accelerated in the nozzle, are drawn into long threads. These threads break into pieces, which is the reason for their crushing (Fig. 8).

The stretching of spherical formations into filamentary ones is determined by the fact that the flow acceleration is distributed along the direction of motion. The frontal elements of the formations are subject to acceleration earlier than their rear parts and are under the influence of increased speeds for a longer time. As a result, the spherical liquid particles are elongated.

Cavitation phenomena in liquid.They are realized by passing the flow of a continuous medium through a smoothly tapering channel (nozzle) - Figure 8. In it, it accelerates, and the pressure decreases in accordance with the Bernoulli equation

where p - pressure, Pa; ρ is the density of the liquid, kg / m 3; v - its speed, m / s; g - free fall acceleration, m / s 2; H— liquid level, m

When the pressure drops below the saturated vapor pressure, the liquid boils. With a subsequent increase in pressure, the vapor bubbles "collapse". The high-intensity, but small-scale pulsations of pressure and velocity of the medium generated in this case homogenize it.

Similar phenomena arise when bluff bodies move (rotate) in a fluid. In the aerodynamic shadow behind the bluff bodies, the pressure decreases and cavitation cavities appear, moving together with the bodies. They are called attached cavities.

The movement of ultrasonic waves in a liquid medium. V In ultrasonic homogenizers, the product flows through a special chamber, in which it is irradiated by an ultrasonic emitter (Fig. 10).

With the propagation of traveling waves in the medium, relative displacements of the components occur, repeating with the frequency of the generated oscillations (above 16 thousand times per second). As a result, the boundaries of the components of the medium are blurred, the particles of the dispersion phase are crushed, and the medium is homogenized.

Rice. 8. Scheme of crushing a fat particle when passing through the valve clearance

Rice. 9. Scheme of operation of the valve homogenizer:
1 - working chamber; 2 - seal; 3 - valve; 4 - body

During homogenization of milk by ultrasonic waves and other disturbances, the limiting sizes of milk particles are established, below which homogenization is impossible.

Milk fat particles are round, almost spherical particles 1 ... 3 microns in size (primary balls or nuclei), combined by 2 ... 50 pieces or more into conglomerates (aggregates, bunches). In the composition of conglomerates, individual particles retain their individuality, that is, they remain clearly distinguishable. Conglomerates are in the form of chains of individual particles. The integrity of the conglomerate is determined by the adhesive forces of the rounded particles.

Rice. 10. Scheme of an ultrasonic homogenizer with generation of pulsations directly in its volume:
1 - homogenization cavity, 2— vibrating plastic; 3 - liquid jet nozzle

All practical methods of homogenization provide crushing of conglomerates, at best, to the size of primary balls. In this case, the surfaces of the adhesion cohesion of the primary drops are broken under the action of the difference in the dynamic pressures of the dispersion medium acting on individual parts of the conglomerate. The fragmentation of primary drops by ultrasonic waves can take place only by the mechanism of the formation of surface waves on them and the breakdown of their crests by the flow of the dispersion medium. Fragmentation occurs at the moment when the forces that cause it exceed the forces that hold the original shape of the particles. At this moment, the ratio of these forces will exceed the critical value.

The forces leading to the fragmentation of both primary particles and their conglomerates are the forces (N) created by the dynamic pressure of the dispersion medium:

where Δp d - dynamic pressure of the dispersion medium, Pa; ρ is the density of the medium, kg / m 3; u, v - respectively, the velocity of the medium and the particle, m / s; F = π r 2 - mid-section area, m 2; r - radius of the primary particle, m

Particle speed v (t ) is calculated by the formula reflecting Newton's second law (equality of the product of the mass of a particle and the acceleration to the force of the frontal resistance of the medium flowing around it):

where C x —The coefficient of frontal resistance to the droplet movement; t is its mass, kg;

where ρ to - particle density, kg / m 3 .

Now the particle speed v (t ) is found by integrating the equation

With sinusoidal oscillations with a frequency f (Hz) and amplitude p a (Pa) at the speed of sound in a dispersive medium c (m / s) the speed of the medium u (t) (m / s) is determined by the expression

The original shape of the particles is held by the forces:

for a spherical particle it is the surface tension force

where σ is the coefficient of surface tension, N / m;

for a conglomerate of particles, this is the adhesive strength of the primary particles

where a is the specific force, N / m 3; r e - equivalent radius of the conglomerate, m

The ratio of the forces R and R p, called the crushing criterion, or the Weber criterion ( We ), is written in the form:

for a spherical particle

for particle conglomerate

If the current (time-dependent) value of the Weber criterion exceeds the critical value, i.e., at We (t)> We (t) cr , the radius of the primary particle r (t) and the equivalent radius of the conglomerate r e (t ) decrease to a value at which We (t) = We (t) Kp. As a result, a mass of substance is detached from the primary particle or from their conglomerate, corresponding to a decrease in the radius within the specified limits. In this case, the following relations are valid

In the presented calculation expressions for crushing particles, the only factor causing crushing is the difference between the velocities of the particles and the environment [ u (t) - v (t )]. This difference increases with decreasing density ratio ρ / ρ To ... When fat particles in milk are crushed, this ratio is greatest and their crushing is most difficult. The situation is aggravated by the fact that milk fat particles are covered with a more viscous shell of swollen proteins, lipids and other substances. For each cycle of ultrasonic vibrations, a small amount of small droplets break off from the crushing droplets, and for the crushing to proceed as a whole, repeated application of external loads is necessary. Therefore, the duration of crushing is many hundreds and even thousands of oscillation cycles. This is observed in practice with high-speed video filming of oil droplets crushing by ultrasonic vibrations.

Interaction of particles with shock waves.Under the influence of ultrasonic vibrations of normal intensity, it is possible to crush only conglomerates of drops. For crushing the primary droplets, pressure perturbations with an intensity of about 2 MPa are required. This is unattainable with modern technology. Therefore, it can be argued that milk homogenization to a particle size of less than 1 ... 1.5 microns is not implemented on any operating equipment.

Further crushing of droplets is possible under the influence of a series of shock impulses created in a homogenized medium by a special stimulator, for example, a piston connected to a hydraulic or pneumatic impulse drive. High-speed filming of droplets, which are affected by such impulses, shows that in this case fragmentation is realized by the mechanism of "blowing off the smallest droplets from their surface." In this case, the disturbance of the speed of the environment leads to the formation of waves on the surface of the drops and the disruption of their combs. Repeated repetition of this phenomenon leads to significant crushing of droplets or fat particles.

73. Requirements for the grain drying process.

Heat drying of grain and seeds in grain dryers is the main and most efficient method. On farms, at state grain-receiving enterprises, tens of millions of tons of grain and seeds are dried annually. Huge funds are spent on the creation of grain drying equipment and its operation. Therefore, drying must be properly organized and carried out with the greatest technological effect.

Practice shows that drying grain and seeds in many farms is often much more expensive than in the state system of grain products. This happens not only because less efficient dryers are used there, but also due to an insufficiently clear organization of grain drying, improper operation of grain dryers, non-compliance with the recommended drying regimes, and the absence of production lines. The current recommendations for drying agricultural seeds provide for responsibility for the preparation of grain dryers and their operation on collective farms of chairmen and chief engineers, and on state farms - directors and chief engineers. Agronomists and grain dryers are responsible for the drying process. State seed inspectorates exercise control over the sowing quality of seeds.

In order to most efficiently organize the drying of grain and seeds, it is necessary to know and take into account the following basic provisions.

1. The maximum permissible heating temperature, that is, to what temperature a given batch of grain or seeds should be heated. Overheating always leads to deterioration or even complete loss of technological and sowing qualities. Insufficient heating reduces the drying effect and increases its cost, since at a lower heating temperature less moisture will be removed.
2. Optimum temperature of the drying agent (heat carrier) introduced into the grain dryer chamber. When the heat carrier temperature is lower than the recommended temperature, the grain is not heated to the required temperature, or to achieve this, it will be necessary to increase the residence time of the grain in the drying chamber, which reduces the productivity of the grain dryers. The drying agent temperature higher than the recommended one is unacceptable, as it will cause overheating of the grain.
3. Features of drying grain and seeds in grain dryers of various designs, since these features often entail a change in other parameters and, above all, the temperature of the drying agent.

The maximum permissible temperature for heating grain and seeds depends on:
1) culture; 2) the nature of the use of grain and seeds in the future (i.e., intended use); 3) the initial moisture content of grain and seeds, i.e., their moisture before drying.

The grains and seeds of different plants have different heat resistance. Some of them, other things being equal, can withstand higher heating temperatures and even for a longer time. Others, even at lower temperatures, change their physical state, technological and physiological properties. For example, seeds of fodder beans and beans at a higher heating temperature lose the elasticity of their shells, crack, and their field germination decreases. Wheat grain intended for the production of bakery flour can only be heated up to 48-50 ° C, and rye grain - up to 60 ° C. When wheat is heated above the specified limits, the amount of gluten is sharply reduced and its quality deteriorates. Very fast heating (at a higher temperature of the coolant) also negatively affects rice, corn and many legumes: (the seeds crack, which makes it difficult to further process them, for example, into cereals.

When drying, the intended purpose of the batches must be taken into account. So, the maximum heating temperature of wheat seed grain is 45 ° С, and food grain is 50 ° C ... The difference in heating temperature for rye is even greater: 45 ° C for seed and 60 ° for food (for flour). (In general, all grain and seed lots that need to be viable are heated to a lower temperature. Therefore, brewing barley, malt rye, etc. are dried using seed modes.

The maximum permissible temperature for heating grain and seeds depends on their initial moisture content. It is known that the more free water in these objects, the less heat-resistant they are. Therefore, when the moisture content in them is more than 20% and especially 25%, the temperature of the coolant and the heating of the seeds should be reduced. So, with the initial moisture content of peas and rice 18% (Table 36), the permissible heating temperature is 45 ° C, and the temperature of the heat carrier is 60 O C. If the initial moisture content of these seeds is 25%, then the permissible temperature will be 40 and 50 ° C, respectively. At the same time, a decrease in temperature also leads to a decrease in evaporation (or, as they say, removal) of moisture.

It is even more difficult to dry large-seeded legumes and soybeans, when, at high humidity (30% and higher), drying in grain dryers has to be carried out at a low temperature of the heat carrier (30 ° C) and heating the seeds (28-30 ° C) with insignificant moisture removal during the first and second pass.

The design features of grain dryers of different types and brands determine the possibilities of their use for drying seeds of various crops. For example, drum dryers do not dry legumes, corn and rice. The movement of grain in them and the temperature of the drying agent (110-130 ° C) are such that the grains and seeds of these crops are cracked and severely injured.

Considering the issues of thermal drying in grain dryers, one must remember about the unequal moisture-yielding ability of grain and seeds of various crops. If the moisture yield of wheat, oats, barley and sunflower seeds is taken as a unit, then, taking into account the applied heat carrier temperature and moisture removal in one pass through the grain dryer, the coefficient (K)will be equal: for rye 1.1; buckwheat 1.25; millet 0.8; corn 0.6; peas, vetch, lentils and rice 0.3-0.4; forage beans, beans and lupine 0.1-0.2.

Table 1. Temperature conditions (in ° C) for drying seeds of various crops on grain dryers

Culture

Mine

Drum

Culture

Seed moisture content before drying within,%

Number of passes through the grain dryer

Mine

Drum

drying agent temperature, in o C

o C

limiting temperature of seed heating, in o C

drying agent temperature, in o C

limiting temperature of seed heating, in o C

limiting temperature of seed heating, in o C

Wheat, rye, barley, oats

Peas, vetch, lentils, chickpeas, rice

over 26

Buckwheat, millet

Corn

over 26

It should also be borne in mind that due to a certain moisture-yielding capacity of grain and seeds, almost all dryers used in agriculture provide moisture removal in one pass of the grain mass only up to 6% under modes for food grain and up to 4-5% for seed ... Therefore, grain masses with high humidity have to be passed through dryers 2-3 or even 4 times (see Table 1).

Problem number 1.

Determine the suitability of a drum sieve with specified parameters for sieving 3.0 t / h of flour. Initial data:

The penultimate digit of the cipher		The last digit of the cipher
ρ, kg / m 3		n, rpm
α, º		R, m
		h, m	0,05

Solution

Given:

ρ - bulk material weight, 800 kg / m 3 ;

α - angle of inclination of the drum to the horizon, 6;

μ is the coefficient of loosening of the material, 0.7;

n - the number of revolutions of the drum, 11 rpm;

R - drum radius, 0.3 m;

h - the height of the material layer on the sieve, 0.05 m.

Rice. 11. Scheme of a drum sieve:
1 - drive shaft; 2 - drum-box; 3 - sieve

where μ is the coefficient of material loosening μ = (0.6-0.8); ρ - bulk material weight, kg / m 3 ; α — angle of inclination of the drum to the horizon, deg; R - drum radius, m; h - the height of the material layer on the sieve, m; n - the number of revolutions of the drum, rpm.

Q = 0.72 0.7 800 11 tg (2 6) =
= 4435.2 0.2126 = 942.92352 0.002 = 1.88 t / h

Let us compare the obtained value of the productivity of the drum sieve with 3.0 t / h given in the condition: 1.88< 3,0 т/ч, значит барабанное сито с заданными параметрами непригодно для просеивания 3,0 т/ч муки.

Answer: unsuitable.

Problem number 2.

Determine the dimensions (length) of a flat gyratory screen for sorting 8000 kg / h of material. Initial data:

The penultimate digit of the cipher		The last digit of the cipher
r, mm		ρ, t / m 3
α, º		h, mm
	0 , 4

Solution

r - eccentricity, 12 mm = 0.012 m;

α - angle of inclination of the spring screen to the vertical, 18º;

f - coefficient of friction of the material on the sieve, 0.4;

ρ - bulk material weight, 1.3 t / m 3 = 1300 kg / m 3;

h - the height of the material layer on the sieve, 30 mm = 0.03 m;

φ is the filling factor, taking into account the incomplete loading of the bearing surface with material, 0.5.

Rice. 12. Diagram of a gyratory screen:
1 - spring; 2 - sieve; 3 - shaft vibrator; 4 - eccentricity

Rotational speed of the gyratory screen:

rpm

The speed of material advance over the sieve:

M / s,

where n - frequency of rotation of the shaft of the screen, rpm; r - eccentricity, m; α - angle of inclination of the spring screen to the vertical, deg .; f Is the coefficient of friction of the material against the sieve.

M / s.

Cross-sectional area of ​​the material on the screen S:

Kg / h,

where S - cross-sectional area of ​​the material on the screen, m 2; v - speed of material advance along the screen, m / s; ρ - bulk material weight, kg / m 3 ; φ is the filling factor, taking into account the incomplete loading of the bearing surface with material.

M 2.

Screen length b:

h - the height of the material layer on the sieve.

Answer: screen length b = 0.66 m.

Problem number 3.

Determine the power on the shaft of the suspended vertical centrifuge for separating sugar massecuite, if the inner diameter of the drum D = 1200 mm, drum height H = 500 mm, outer drum radius r 2 = 600 mm. The rest of the initial data:

The penultimate digit of the cipher		The last digit of the cipher
n, rpm		τ p, s
m b, kg		ρ, kg / m 3	1460
d, mm		m s, kg

D - inner diameter of the drum, 1200 mm = 1.2 m;

H - drum height, 500 mm = 0.5 m;

r n = r 2 - outer radius of the drum, 600 mm = 0.6 m

n - drum rotation frequency, 980 rpm;

m b - drum weight, 260 kg;

d - shaft neck diameter, 120 mm = 0.12 m;

τ p - drum acceleration time, 30 s;

ρ - massecuite density, 1460 kg / m 3 ;

m with - suspension weight, 550 kg.

Rice. 13. Scheme for determining the magnitude of pressure on the walls of the drum

Converting the drum rotation frequency to angular velocity:

glad / s.

Capacities N 1, N 2, N 3 and N 4:

KW

where m b - mass of the centrifuge drum, kg; r n - outer radius of the drum, m;τ p - drum acceleration time, s.

The thickness of the annular layer of massecuite:

where m c - the mass of the suspension loaded into the drum, kg; H - the height of the inner part of the drum, m.

The inner radius of the massecuite ring (according to Figure 13):

r n = r 2 Is the outer radius of the drum.

Power for transferring kinetic energy to massecuite:

KW

where η - coefficient of efficiency (for calculations, takeη = 0.8).

Separation factor in the centrifuge drum:

where m - the mass of the drum with suspension ( m = m b + m s), kg; F - separation factor:

Power to overcome friction in bearings:

KW

where p ω - angular speed of rotation of the drum, rad / s; d - shaft neck diameter, m; f - coefficient of friction in bearings (for calculations, take 0.01).

KW.

Power to overcome the friction of the drum on the air:

KW

where D and H - drum diameter and height, m; n - drum rotation frequency, rpm.

Substitute the obtained power values ​​into the formula:

KW.

Answer: centrifuge shaft power N = 36.438 kW.

Problem number 4.

The penultimate digit of the cipher		The last digit of the cipher
t, ºС	32,55	φ , %

R - total air pressure, 1 bar = 1 10 5 Pa;

t - air temperature, 32.55 ºС;

φ - relative humidity, 75% = 0.75.

According to Appendix B, we determine the saturated steam pressure ( r us ) for a given air temperature and translate into the SI system:

for t = 32.55 ºС p sat = 0.05 at · 9.81 · 10 4 = 4905 Pa.

Air moisture content:

where p - total air pressure, Pa.

Enthalpy of humid air:

where 1.01 is the heat capacity of air at ρ = const kJ / (kg K); 1.97 - heat capacity of water vapor, kJ / (kg · K); 2493 - specific heat of vaporization at 0 С, kJ / kg; t - air temperature by dry bulb, S.

Wet air volume:

Wet air volume (in m 3 for 1 kg of dry air):

where is the gas constant for air, equal to 288 J / (kg · K); T - absolute air temperature ( T = 273 + t), K.

M 3 / kg.

Answer: moisture content χ = ​​0.024 kg / kg, enthalpy I = 94.25 kJ / kg and the volume of humid air v = 0.91 m 3 / kg dry air.

Bibliography

1. Plaksin Yu. M., Malakhov NN, Larin VA Processes and apparatuses for food production. - M .: KolosS, 2007 .-- 760 p.

2. Stabnikov V.N., Lysyansky V.M., Popov V.D. Processes and apparatus for food production. - M .: Agropromizdat, 1985 .-- 503 p.

3. Trisvyatsky L.A. Storage and technology of agricultural products. - M .: Kolos, 1975 .-- 448 p.

Physical processes can be investigated by analytical or experimental methods.

Analytical dependencies allow studying processes in general form based on functional analysis of equations and are a mathematical model of a class of processes.

A mathematical model can be represented as a function, an equation, as a system of equations, differential or integral equations. Such models usually contain a lot of information. A characteristic feature of mathematical models is that they can be transformed using a mathematical apparatus.

So, for example, functions can be examined for extremum; differential or integral equations can be solved. At the same time, the researcher receives new information about the functional relationships and properties of the models.

The use of mathematical models is one of the main methods of modern scientific research. However, it has significant drawbacks. In order to find a particular solution from the whole class that is inherent only to this process, it is necessary to set unambiguity conditions. Establishing the boundary conditions requires a reliable experiment and careful analysis of experimental data. Incorrect acceptance of the boundary conditions leads to the fact that not the planned process, but a modified one, is subjected to theoretical analysis.

In addition to the indicated lack of analytical methods, in many cases it is either impossible or extremely difficult to find analytical expressions, taking into account the unambiguous conditions, which most realistically reflect the physical essence of the process under study.

Sometimes, investigating a complex physical process under well-grounded boundary conditions, the original differential equations are simplified due to the impossibility or excessive cumbersomeness of their equation, which distorts its physical essence. Thus, it is very often difficult to implement analytical dependencies.

Experimental methods allow you to deeply study the processes within the accuracy of the experimental technique and focus on those process parameters that are of greatest interest. However, the results of a particular experiment cannot be extended to another process, even one that is close in physical essence, because the results of any experiment reflect the individual characteristics of only the investigated person.

process. From experience it is still impossible to conclusively establish which of the parameters have a decisive influence on the course of the process and how the process will proceed if various parameters are changed simultaneously. With the experimental method, each specific process must be investigated independently.

Ultimately, experimental methods make it possible to establish particular dependencies between individual variables in strictly defined intervals of their change.

Analysis of variable characteristics outside these intervals can lead to distortion of dependence, gross errors.

Thus, both analytical and experimental methods have their own advantages and disadvantages, which often complicate the effective solution of practical problems. Therefore, the combination of the positive aspects of analytical and experimental research methods is extremely fruitful.

Phenomena, processes are studied not in isolation from each other, but in a complex manner. Various objects with their specific variable values ​​are combined into complexes characterized by uniform laws. This makes it possible to extend the analysis of one phenomenon to others and to a whole class of similar phenomena. With this principle of research, the number of variables decreases, they are replaced by generalized criteria. As a result, the desired mathematical expression is simplified. Methods of combining analytical methods of research with experimental methods of analogy, similarity, dimensions, which are a variety of modeling methods, are based on this principle.

Let us consider the essence of the analogy method using an example. Heat flux depends on temperature difference (Fourier's law)

Here is the thermal conductivity coefficient.

Mass transfer or transfer of a substance (gas, vapor, moisture) is determined by the difference in the concentration of a substance WITH(Fick's law):

where is the mass transfer coefficient.

The transfer of electricity along a conductor with linear resistance is determined by the voltage period (Ohm's law):

where is the coefficient of electrical conductivity.

All these phenomena under consideration are characterized by different physical processes, but have identical mathematical expressions, i.e. they can be investigated by analogy.

Depending on what is taken as the original and the model, there can be different types of analogy modeling. So, if the heat flow is studied on a model with fluid movement, then the modeling is called hydraulic; if the heat flow is investigated on an electrical model, the simulation is called electrical. Simulation can be mechanical, acoustic, etc.

The identity of the mathematical expressions of the processes of the original and the model does not mean that these processes are absolutely analogous. In order to model the studied process of the original on the model as much as possible, it is necessary to observe the criterion of analogies. So, compare and, the coefficients of thermal conductivity and electrical conductivity, temperature t and tension u it makes no sense. To eliminate this incompatibility, both equations must be represented in dimensionless quantities: each variable NS represent as a product of constant dimension NS N to a variable without

dimensional NS b:

Bearing in mind (26), we write the expressions for and in the form:

After simple transformations, we have

Both expressions are written in dimensionless form and can be compared.

The equations will be identical if

This equality is called the analogy criterion. With its help, the parameters of the model are set according to the original equation of the object.

The number of analogy criteria is one less than the number of terms of the studied initial expression. Since the number of unknowns is greater than the number of equations, then some parameters of the model are set. Usually this is the time of observation or the course of the process on the model. It should be easy to observe for the operator.

Electrical simulation is now widespread. Let's look at an example of it.

It is necessary to study the patterns of mass fluctuations m suspended parallel by an elastic spring and a damper to the plane. For this system, the differential equation has the form

where is the damping coefficient;

- mechanical movement;

- coefficient characterizing the elasticity of the spring (deformation of the spring under the action of a unit of force);

Is the force applied to the system.

To determine the parameters, equation (27) can be investigated by the method of electrical analogies. For the electrical model of the circuit, the equation has the form

where is the capacitance of the capacitor;

- magnetic flux;

- process time in the power grid;

- resistor, inductance;

- the current of the mains.

After appropriate transformations (see the example above), we write the dimensionless equations as follows

The choice of criteria (29) presents certain difficulties. To simplify the construction of the model, a system of scale equations is used.

Since the mechanical (original) and electrical (model) processes are similar, the variable values ​​of these systems change over time in a regular manner in a certain ratio - the scale.

Scale factor one or another variable is the ratio of the variables of the model and the original

where are the scales of variables.

Taking into account the scale variables, the equations for the model and the original are as follows:

These equations are identical if

Scale systems (30) are identical to the criteria of analogues (29), but in a simpler form.

Using the system of scale equations (30), the parameters of the model are calculated, and based on the maximum deviations of the variables of the original and the model, scale factors are calculated.

Given the average values ​​of the original parameters, according to (30), the average values ​​of the model parameters are calculated and the electric circuit is designed. Next, the original is examined on the model. Varying, the parameters of the original are studied on the model.

With the help of electrical modeling, you can study, analyze various physical processes that are described by mathematical dependencies. This simulation is versatile, easy to operate, and does not require bulky equipment.

Analogue machines (AVMs) are used in electrical modeling. AVM is understood as a certain combination of various electrical elements in which processes occur that are described by mathematical dependencies similar to those for the studied object (original). In this case, the scale factors of the independent and variables must be observed

analog and original values.

AVM is used to study a certain class of problems. The solution of problems is carried out in such a way that it is possible to simultaneously obtain the value of the required quantities in different zones (points) of the system. With the help of AVM, it is possible to solve problems in different scale times, including accelerated ones, which in some cases is of great scientific interest. Ease of solving problems, fast processing of information, the ability to solve complex problems determine the widespread use of AVM. There are general and special purpose AVMs. General purpose AVMs solve high-order differential equations (more than 50) and are intended for various purposes: calculations of network diagrams, base voltages, etc.

When solving problems with equations up to the 10th order, low-power machines MN-7 are used; MN-10; EMU-6 and others; up to the 20th order - the average power of the MN-14; EMU-10, etc.

For simple tasks, the continuous media method is usually used with the use of electrically conductive paper (plane problem) or electrolytic baths (volume problem). The model is made of conductive paper of the same electrical conductivity. The geometry of the object is modeled at a certain scale. Electrodes simulating the boundary conditions are attached to the ends of the figure. When simulating processes with conductive liquids (electrolytes), the baths are filled with weak solutions of salts, acids, alkalis, etc. An inhomogeneous field is simulated using an electrolyte of different concentrations. The method of continuous media is intended for solving problems of heat conduction, stress distribution, etc. It is simple, but limited to solving the Laplace boundary value problems.

In the method of electric grids, differential equations are transformed into a system of linear ones, solved by the method of finite differences. With the help of grid models on electric integrators, stationary and non-stationary problems can be investigated.

An electrohydrodynamic analogy is a widely used modeling method. It is based on the electrical simulation of the movement of liquid, vapor or gas and is widely used to study the water regime of the foundations of buildings, structures, dams, etc.

The method of hydraulic modeling on hydrointegrators is also often used. Hydrointegrators are devices in which water moves through a system of pipes and nodes connected to each other. The studied constant and variable values ​​are modeled by the head, levels and flow rates of water in the vessels.

The integrator consists of many nodes T(fig. 7).

At each such node, the water balance is

where is the cross-sectional area of ​​the vessel;

- water levels in vessels;

- hydraulic resistance (pressure difference for passing a single flow rate);

- water consumption.

At a constant water level in a vessel or a constant area of ​​this vessel,

If given at the initial moment of time T= 0, the definition of the function takes place the integration of equation (31), i.e., the registration of the heads and water levels on the hydrointegrator. For a particular case (32), integration is reduced to solving algebraic expressions on a hydrointegrator.

If there are multiple nodes N, then the solution of the system with N equations for the transfer of heat, moisture, matter on the integrator is reduced to observing the water levels in the vessels.

The parameters of the equations can be relatively easily changed by changing the number of nodes, cross-sections of vessels, hydraulic resistances, and water flow rates on the integrator. It is very easy to set different initial and boundary conditions,

changing the initial water levels in the vessels.

The method of hydraulic modeling allows solving various problems: stationary and non-stationary; one-, two and three-dimensional; with constant and variable coefficients; for a homogeneous and non-uniform field; those. is versatile. It is widely used in solving various problems in the field of construction: calculating temperatures and stresses in various structures of buildings and structures; analysis of the process of humidification and moisture accumulation in the foundations of buildings, roads, etc .; analysis of the processes of deformation and destruction of structures; assessment of the temperature field when steaming reinforced concrete products; determination of physical and thermal characteristics of materials and structures; calculation of the thermal regime of buildings, roads and other structures under climatic influences to study water filtration in hydraulic structures; calculation of freezing of soils of canvas and foundations of structures and in other cases.

This method is characterized by the availability of programming, ease of solving complex problems, good visibility of the ongoing processes, sufficiently high accuracy of calculations, the ability to stop and repeat the process on the model. However, the equipment for this method is cumbersome and is produced in limited quantities.

Similarity theory- this is the doctrine of the similarity of phenomena. It is most effective when it is impossible to find dependencies between variables on the basis of solving differential equations. Then it is necessary to conduct a preliminary experiment and, using its data, draw up an equation (or a system of equations) using the similarity method, the solution of which can be extended beyond the boundaries of the experiment. This method of theoretical study of phenomena and processes is possible only on the basis of a combination with experimental data.

Let us consider the essence of the similarity theory using a simple example. Let there be a row of rectangles. This is a class of flat figures, since they are united by common properties - they have four sides and four right angles. From this class, only one figure can be distinguished that has a specific meaning of the sides. l 1 and l 2. Numerical values l 1 and l 2 define the conditions for unambiguity. If the parties l 1 and l 2 times the value TO e, which can be given any meaning, then we get a series of similar flat figures, combined into a certain group:

The quantities TO e called similarity criteria.

This method of bringing the similarity is applicable not only for flat, united figures, but also for various physical quantities: time, pressures, viscosities, thermal diffusivity, etc.

Similarity criteria are created within a given class of group phenomena by transforming unambiguity conditions into similar systems. All phenomena included in one group are similar and differ only in scale. Thus, any differential equation is characteristic of a class of dissimilar phenomena. The same equation with boundary conditions and similarity criteria is characteristic only for a group of similar phenomena. If the boundary conditions are presented without a similarity criterion, then the differential equation can be used to analyze only a particular case.

Similarity theory is based on three theorems.

Theorem 1(M.V. Kirpichev and A.A. Gukhman.). Two physical phenomena are similar if they are described by the same system of differential equations and have similar (boundary) conditions of uniqueness, and their defining similarity criteria are numerically equal.

Theorem 2. If physical processes are similar, then the criteria for the similarity of these processes are equal.

Theorem 3. The equations describing physical processes can be expressed by a differential relationship between similarity criteria.

In a group of similar phenomena, differing only in scale, it is possible to disseminate the results of a single experiment.

When using the theory of similarity, it is convenient to operate with similarity criteria, which are denoted by two Latin letters of the names of scientists.

Let's consider some similarity criteria.

When studying fluid flows, the Reynolds criterion is applied

where is the dynamic viscosity;

- movement speed;

l- distance, thickness, diameter of the pipeline.

Criterion Re is an indicator of the ratio of inertial forces to friction forces.

Euler's criterion

Here is the period of pressure when the fluid moves in the pipeline due to friction;

- density.

Various criteria are used in heat and mass transfer.

Fourier criterion

where a- criterion of temperature or moisture conductivity;

- time;

l- the characteristic size of the body (length, radius).

This criterion characterizes the rate of heat equalization in a given body.

Lykov's criterion

Here a, a 1 - coefficients of heat and mass transfer.

This criterion characterizes the rate of change in mass transfer (moisture, steam) relative to heat transfer. It varies widely (from 0 to 1000).

Kirpichev criterion

- heat flow.

This criterion characterizes the ratio of the heat flux supplied to the body surface to the heat flux removed into the body.

All of the above, as well as other criteria, are dimensionless. They are independent from each other, so their combination gives new criteria.

When studying phenomena and processes, it is convenient to use similarity criteria. The experimental data are processed in the form of generalized dimensionless variables and equations are compiled in criterial form, i.e. into differential equations instead of variables, etc. set similarity criteria. Next, they begin to solve the theoretical equation in a criterial form. The obtained analytical solution makes it possible to extend the results of a single experiment to a group of similar phenomena and to analyze variables outside the experiment.

Similarity criteria are used to solve differential equations in many variables. In this case, it is expedient to represent the equations and boundary conditions in a criterial dimensionless form, although this is sometimes not easy. Solving equations in dimensionless form is less laborious, since the number of variables decreases, the analytical expression is simplified, and the volume of calculations is significantly reduced. All this simplifies the drawing up of graphs and nomograms. Therefore, the ability to compose differential equations in a criterial form, solve them and analyze them is of great interest for a scientist.

In a number of cases, there are processes that cannot be directly described by differential equations. The relationship between variables in such processes, ultimately, can only be established experimentally. To limit the experiment and find the connection between the main characteristics of the process, it is effective to use the method of dimensional analysis, which combines theoretical research with experiments and allows you to compose functional dependencies in a criterion form.

Let the function F for any complex process

Values ​​have a specific dimension of units of measurement. The method of dimensions provides for a choice from the number To three main independent units of measurement. Rest To - The three quantities included in the functional dependence (34) must have dimensions expressed in terms of three basic ones. In this case, the main values ​​are chosen so that the rest To- 3 were introduced in function F as dimensionless, in the similarity criteria.

In this case, function (34) takes the form

Three ones means that the first three numbers are a ratio to correspondingly equal values.

Expression (40) is analyzed in terms of the dimensions of the quantities. As a result, the numerical values ​​of the exponents are established and the similarity criteria are determined. For example, when water flows around a bridge support at a speed V. Moreover, n 5 - Froude criterion Fr.

As a result, the function under study takes the form

This formula will make it possible to study the process of flow around the bridge support in different variants of velocity sizes, provided that the similarity criteria are equal. It can also be used to analyze a process using similarity theory on models.