In cases where the processes under study are not described by differential equations, one of the ways to analyze them is an experiment, the results of which are best represented in a generalized form (in the form of dimensionless complexes). The method of compiling such complexes is dimensional analysis method.

The dimension of any physical quantity is determined by the ratio between it and those physical quantities that are taken as the main (primary). Each system of units has its own basic units. For example, in the International System of Units SI, the units of length, mass and time are respectively taken to be the meter (m), kilogram (kg), second (s). Units of measurement of other physical quantities, the so-called derived quantities (secondary), are adopted on the basis of laws that establish a relationship between these units. This relationship can be represented in the form of the so-called dimension formula.

Dimension theory is based on two assumptions.

• 1. The ratio of two numerical values ​​of any quantity does not depend on the choice of scales for the main units of measurement (for example, the ratio of two linear dimensions does not depend on the units in which they will be measured).
• 2. Any relationship between dimensional quantities can be formulated as a relationship between dimensionless quantities. This statement represents the so-called P-theorem in dimension theory.

From the first position it follows that the formulas for the dimension of physical quantities should have the form of power dependences

where are the dimensions of the basic units.

The mathematical expression of the P-theorem can be obtained based on the following considerations. Let some dimensional quantity A 1 is a function of several independent dimensional quantities , i.e.

Hence it follows that

Let us assume that the number of basic dimensional units through which all can be expressed P variables, equals T. The P-theorem states that if all P variables expressed in terms of basic units, then they can be grouped into dimensionless P-terms, i.e.

In this case, each P-term will contain a variable.

In problems of hydromechanics, the number of variables included in the P-terms must be four. Three of them will be decisive (usually these are the characteristic length, the fluid flow velocity and its density) - they are included in each of the P-terms. One of these variables (the fourth) is different when passing from one P-term to another. Degree indicators of defining criteria (let us denote them by x, y , z ) are unknown. For convenience, we take the exponent of the fourth variable equal to -1.

The relations for P-terms will look like

The variables included in the P-terms can be expressed in terms of the basic dimensions. Since these terms are dimensionless, the exponents of each of the basic dimensions must be equal to zero. As a result, for each of the P-terms, it is possible to compose three independent equations (one for each dimension) that relate the exponents of the variables included in them. The solution of the resulting system of equations makes it possible to find the numerical values ​​of unknown exponents X , at , z. As a result, each of the P-terms is determined in the form of a formula composed of specific quantities (environment parameters) in the appropriate degree.

As case study we will find a solution to the problem of determining the pressure loss due to friction in a turbulent fluid flow.

From general considerations, we can conclude that the pressure loss in the pipeline depends on the following main factors: diameter d , length l , wall roughness k, density ρ and viscosity µ of the medium, average flow velocity v , initial shear stress, i.e.

(5.8)

Equation (5.8) contains n=7 members, and the number of basic dimensional units. According to the P-theorem, we obtain an equation consisting of dimensionless P-terms:

(5.9)

Each such P-term contains 4 variables. Taking as the main variables the diameter d , speed v , density, and combining them with the rest of the variables in Eq. (5.8), we obtain

Composing the dimension equation for the first П-term, we will have

Adding the exponents at same grounds, we find

In order for the dimension P 1 was equal to 1 ( P 1 is a dimensionless quantity), it is necessary to require that all exponents be equal to zero, i.e.

(5.10)

System algebraic equations(5.10) contains three unknown quantities x 1, y 1,z 1. From the solution of this system of equations, we find x 1 = 1; at 1=1; z 1= 1.

Substituting these values ​​of the exponents into the first P-term, we obtain

Similarly, for the remaining P-terms we have

Substituting the resulting P-terms into equation (5.9), we find

Let's solve this equation for P4:

Let's express it from here:

Taking into account that the loss of head due to friction is equal to the difference between the piezometric heads, we will have

Denoting the complex in square brackets by, we finally get

The last expression represents the well-known Darcy-Weibach formula, where

Formulas for calculating the coefficient of friction To discussed in paragraphs 6.13, 6.14.

It should be emphasized that the ultimate goal in the case under consideration remains the same: finding similarity numbers for which modeling should be carried out, but it is solved with a significantly smaller amount of information about the nature of the process.

To clarify what follows, we will briefly review some of the fundamental concepts. A detailed presentation can be found in the book by A.N. Lebedev "Modeling in scientific and technical research." - M.: Radio and communications. 1989. -224 p.

Any material object has a number of properties that allow quantitative expression. Moreover, each of the properties is characterized by the size of a certain physical quantity. The units of some physical quantities can be chosen arbitrarily, and with their help represent the units of all the others. Physical units chosen arbitrarily are called main. In the international system (as applied to mechanics), this is the kilogram, meter and second. The rest of the quantities expressed in terms of these three are called derivatives.

The base unit can be denoted either by the symbol of the corresponding quantity or by a special symbol. For example, the units of length are L, units of mass - M, unit of time - T. Or, the unit of length is the meter (m), the unit of mass is the kilogram (kg), the unit of time is the second (s).

Dimension is understood as a symbolic expression (sometimes called a formula) in the form of a power monomial, connecting the derived value with the main ones. General form this pattern has the form

Where x, y, z- Dimension indicators.

For example, the dimension of speed

For a dimensionless quantity, all indicators , and hence .

The next two statements are quite clear and do not need any special proofs.

The ratio of the sizes of two objects is a constant value, regardless of the units in which they are expressed. So, for example, if the ratio of the area occupied by windows to the area of ​​walls is 0.2, then this result will remain unchanged if the areas themselves are expressed in mm2, m2 or km2.

The second position can be formulated as follows. Any correct physical relationship must be dimensionally uniform. This means that all terms included in both the right and left parts of it must have the same dimension. This simple rule is clearly implemented in everyday life. Everyone realizes that meters can only be added to meters and not to kilograms or seconds. It must be clearly understood that the rule remains valid when considering even the most complex equations.

The method of dimensional analysis is based on the so-called -theorem (read: pi-theorem). -theorem establishes a connection between a function expressed in terms of dimensional parameters and a function in a dimensionless form. The theorem can be more fully formulated as follows:

Any functional relationship between dimensional quantities can be represented as a relationship between N dimensionless complexes (numbers) composed of these quantities. The number of these complexes , Where n- number of basic units. As noted above, in hydromechanics (kg, m, s).

Let, for example, the value A is a function of five dimensional quantities (), i.e.

(13.12)

It follows from the -theorem that this dependence can be transformed into a dependence containing two numbers ( )

(13.13)

where and are dimensionless complexes composed of dimensional quantities.

This theorem is sometimes attributed to Buckingham and is called - Buckingham's theorem. In fact, many prominent scientists contributed to its development, including Fourier, Ryabushinsky, and Rayleigh.

The proof of the theorem is beyond the scope of the course. If necessary, it can be found in the book of L.I. Sedov "Methods of similarity and dimensions in mechanics" - M .: Nauka, 1972. - 440 p. A detailed justification of the method is also given in the book by V.A. Venikov and G.V. Venikov "Theory of similarity and modeling" - M.: Higher school, 1984. -439 p. A feature of this book is that, in addition to issues related to similarity, it includes information about the methodology for setting up an experiment and processing its results.

Using dimensional analysis to solve specific practical tasks is connected with the need to compile a functional dependence of the form (13.12), which at the next stage is processed by special techniques that ultimately lead to obtaining numbers (similarity numbers).

The main creative stage is the first stage, since the results obtained depend on how correct and complete the researcher's understanding of the physical nature of the process is. In other words, how functional dependence (13.12) correctly and fully takes into account all the parameters that affect the process under study. Any mistake here inevitably leads to erroneous conclusions. The so-called "Rayleigh's error" is known in the history of science. Its essence is that when studying the problem of heat transfer in turbulent flow, Rayleigh did not take into account the influence of flow viscosity, i.e. did not include it in the dependency (13.12). As a result, the Reynolds similarity number, which plays exclusively important role in heat exchange.

To understand the essence of the method, consider an example, illustrating both the general approach to the problem and the method of obtaining similarity numbers.

It is necessary to establish the type of dependence that makes it possible to determine the pressure loss or head loss in turbulent flow in round pipes.

Recall that this problem has already been considered in Section 12.6. Therefore, it is of undoubted interest to establish how it can be solved using dimensional analysis and whether this solution gives any new information.

It is clear that the pressure drop along the pipe, due to the energy spent to overcome the forces of viscous friction, is inversely proportional to its length, therefore, in order to reduce the number of variables, it is advisable to consider not , but , i.e. pressure loss per unit length of the pipe. Recall that the ratio , where is the pressure loss, is called the hydraulic slope.

From the concept of the physical nature of the process, it can be assumed that the resulting losses should depend on: the average flow velocity working environment(v); on the size of the pipeline, determined by its diameter ( d); from physical properties transported medium, characterized by its density () and viscosity (); and, finally, it is reasonable to assume that the losses must be somehow related to the state of the inner surface of the pipe, i.e. with roughness ( k) of its walls. Thus, dependence (13.12) in the case under consideration has the form

(13.14)

This is the end of the first and, it must be emphasized, the most important step in the analysis of dimensions.

In accordance with the -theorem, the number of influencing parameters included in the dependence is . Consequently, the number of dimensionless complexes , i.e. after appropriate processing (13.14) should take the form

(13.15)

There are several ways to find numbers. We will use the method proposed by Rayleigh.

Its main advantage is that it is a kind of algorithm that leads to the solution of the problem.

From the parameters included in (13.15) it is necessary to choose any three, but so that they include the basic units, i.e. meter, kilogram and second. Let them be v, d, . It is easy to verify that they satisfy the stated requirement.

Numbers are formed in the form of power monomials from the selected parameters multiplied by one of the remaining ones in (13.14)

; (13.16)

; (13.17)

; (13.18)

Now the problem is reduced to finding all exponents. At the same time, they must be selected so that the numbers are dimensionless.

To solve this problem, we first determine the dimensions of all parameters:

; ;

Viscosity , i.e. .

Parameter , And .

And finally, .

Thus, the dimensions of the numbers will be

Similarly, the other two

At the beginning of Section 13.3, it was already noted that for any dimensionless quantity, the dimensional exponents . Therefore, for example, for a number we can write

Equating the exponents, we obtain three equations with three unknowns

Where do we find; ; .

Substituting these values ​​into (13.6), we obtain

(13.19)

Proceeding similarly, it is easy to show that

And .

Thus, dependence (13.15) takes the form

(13.20)

Since there is a non-defining similarity number (Euler number), then (13.20) can be written as a functional dependence

(13.21)

It should be borne in mind that the analysis of dimensions does not and in principle cannot give any numerical values ​​in the ratios obtained with its help. Therefore, it should end with an analysis of the results and, if necessary, their correction based on general physical concepts. Let us consider expression (13.21) from these positions. Its right side includes the square of the speed, but this entry does not express anything other than the fact that the speed is squared. However, if we divide this value by two, i.e. , then, as is known from hydromechanics, it acquires an important physical meaning: specific kinetic energy, and - dynamic pressure due to the average speed. Taking this into account, it is expedient to write (13.21) in the form

(13.22)

If now, as in (12.26), we denote by the letter , then we arrive at the Darcy formula

(13.23)

(13.24)

where is the hydraulic coefficient of friction, which, as follows from (13.22), is a function of the Reynolds number and relative roughness ( k/d). The form of this dependence can be found only experimentally.

LITERATURE

1. Kalnitsky L.A., Dobrotin D.A., Zheverzheev V.F. Special course of higher mathematics for higher educational institutions. M.: graduate School, 1976. - 389s.

2. Astarita J., Marruchi J. Fundamentals of hydromechanics of non-Newtonian fluids. - M.: Mir, 1978.-307p.

3. Fedyaevsky K.K., Faddeev Yu.I. Hydromechanics. - M.: Shipbuilding, 1968. - 567 p.

4. Fabrikant N.Ya. Aerodynamics. - M.: Nauka, 1964. - 814 p.

5. Arzhanikov N.S. and Maltsev V.N. Aerodynamics. - M.: Oborongiz, 1956 - 483 p.

6. Filchakov P.F. Approximate methods of conformal mappings. - K .: Naukova Dumka, 1964. - 530 p.

7. Lavrentiev M.A., Shabat B.V. Methods of the theory of functions of a complex variable. - M.: Nauka, 1987. - 688 p.

8. Daly J., Harleman D. Fluid Mechanics. -M.: Energy, 1971. - 480 p.

9. A.S. Monin, A.M. Yaglom "Statistical hydromechanics" (part 1. -M .: Nauka, 1968. -639 p.)

10. Schlichting G. Theory of the boundary layer. - M.: Nauka, 1974. - 711 p.

11. Pavlenko V.G. Fundamentals of fluid mechanics. - L.: Shipbuilding, 1988. - 240 p.

12. Altshul A.D. hydraulic resistance. - M.: Nedra, 1970. - 215 p.

13. A.A. Gukhman "Introduction to the theory of similarity." - M.: Higher school, 1963. - 253 p.

14. S. Kline "Similarities and Approximate Methods". - M.: Mir, 1968. - 302 p.

15. A.A. Gukhman “Application of the theory of similarity to the study of heat and mass transfer processes. Transfer processes in a moving medium. - M.: Higher scale, 1967. - 302 p.

16. A.N. Lebedev "Modeling in scientific and technical research". - M.: Radio and communications. 1989. -224 p.

17. L.I. Sedov "Methods of similarity and dimensions in mechanics" - M .: Nauka, 1972. - 440 p.

18. V.A.Venikov and G.V.Venikov "Theory of similarity and modeling" - M.: Higher school, 1984. -439 p.

1. MATHEMATICAL APPARATUS USED IN FLUID MECHANICS .............................................................. ................................................. ..... 3

1.1. Vectors and operations on them .............................................................. ...... 4

1.2. Operations of the first order (differential characteristics of the field). ................................................. ................................................. ..... 5

1.3. Operations of the second order............................................................... ......... 6

1.4. Integral Relations of Field Theory............................................... 7

1.4.1. Vector field flow .............................................................. ... 7

1.4.2. Circulation of the field vector ............................................... 7

1.4.3. Stokes formula .................................................. ............. 7

1.4.4. Gauss-Ostrogradsky formula............................. 7

2. BASIC PHYSICAL PROPERTIES AND PARAMETERS OF THE LIQUID. FORCES AND STRESSES ............................................................... ............................ 8

2.1. Density................................................. ................................... 8

2.2. Viscosity................................................. ...................................... 9

2.3. Classification of forces .................................................. .................... 12

2.3.1. Mass forces .................................................................. ............. 12

2.3.2. Surface forces .................................................................. .... 12

2.3.3. Stress tensor .............................................................. ...... 13

2.3.4. Equation of Motion in Stresses .................................. 16

3. HYDROSTATICS............................................... .................................. 18

3.1. Fluid Equilibrium Equation............................................... 18

3.2. Basic equation of hydrostatics in differential form. ................................................. ................................................. ..... 19

3.3. Equipotential surfaces and surfaces of equal pressure. ................................................. ................................................. ..... 20

3.4. Equilibrium of a homogeneous incompressible fluid in the field of gravity. Pascal's law. Hydrostatic law of pressure distribution... 20

3.5. Determination of the force of liquid pressure on the surface of bodies .... 22

3.5.1. Flat surface................................................ .... 24

4. KINEMATICS............................................... ...................................... 26

4.1. Steady and unsteady motion of a fluid ...... 26

4.2. Continuity (continuity) equation............................................... 27

4.3. Streamlines and trajectories ............................................................... ............ 29

4.4. Stream tube (stream surface).................................................. ... 29

4.5. Jet flow model ............................................................... ............ 29

4.6. Continuity equation for a trickle............................................... 30

4.7. Acceleration of a liquid particle ............................................................... ...... 31

4.8. Analysis of the movement of a liquid particle .............................................. 32

4.8.1. Angular deformations .................................................................. ... 32

4.8.2. Linear deformations .................................................................. .36

5. VORTEX MOTION OF A LIQUID .............................................................. .38

5.1. Kinematics of vortex motion............................................... 38

5.2. Vortex intensity .............................................................. ................ 39

5.3. Circulation speed .................................................................. ............... 41

5.4. Stokes' theorem................................................... ......................... 42

6. POTENTIAL LIQUID MOVEMENT .............................................. 44

6.1. Speed ​​Potential .............................................................. ................. 44

6.2. Laplace equation .................................................. ................... 46

6.3. Velocity circulation in a potential field.................................... 47

6.4. Plane flow current function .................................................................. .47

6.5. Hydromechanical meaning of the current function .............................. 49

6.6. Relationship between the speed potential and the current function .............................. 49

6.7. Methods for Calculating Potential Flows .............................................. 50

6.8. Superposition of Potential Flows.................................................... 54

6.9. Non-circulating flow past a circular cylinder .................. 58

6.10. Application of the theory of functions of a complex variable to the study of plane flows of an ideal fluid ..... 60

6.11. Conformal mappings .................................................................. ..... 62

7. HYDRODYNAMICS OF AN IDEAL LIQUID .............................. 65

7.1. Equations of motion for an ideal fluid.................................... 65

7.2. Gromeka-Lamb transformation............................................... 66

7.3. Equation of motion in the form of Gromeka-Lamb .............................. 67

7.4. Integration of the equation of motion for a steady flow.................................................................. ................................................. ........... 68

7.5. Simplified derivation of the Bernoulli equation............................... 69

7.6. Energy meaning of the Bernoulli equation .............................. 70

7.7. Bernoulli's equation in the form of heads............................................... 71

8. HYDRODYNAMICS OF A VISCOUS LIQUID .............................................. 72

8.1. Model of a viscous fluid ............................................................... ........... 72

8.1.1. Linearity hypothesis .................................................................. ... 72

8.1.2. Homogeneity hypothesis .................................................................. 74

8.1.3. Hypothesis of isotropy .............................................................. .74

8.2 Equation of motion of a viscous fluid. (Navier-Stokes equation) ............................................... ................................................. ........... 74

9. ONE-DIMENSIONAL FLOWS OF INCOMPRESSIBLE LIQUID (fundamentals of hydraulics) ........................................................ ................................................. ................. 77

9.1. flow rate and average speed........................................... 77

9.2. Weakly deformed flows and their properties....................... 78

9.3. Bernoulli equation for the flow of a viscous fluid .............................. 79

9.4. The physical meaning of the Coriolis coefficient .............................. 82

10. CLASSIFICATION OF LIQUID FLOWS. STABILITY OF MOVEMENT............................................................... ............................................. 84

11. REGULARITIES OF THE LAMINAR FLOW IN ROUND PIPES ............................................................... ................................................. .......... 86

12. MAIN REGULARITIES OF TURBULENT MOTION. ................................................. ................................................. .............. 90

12.1. General Information................................................... ....................... 90

12.2. Reynolds equations................................................... ............ 92

12.3. Semi-empirical theories of turbulence............................................... 93

12.4. Turbulent flow in pipes .............................................. 95

12.5. Power Laws of Velocity Distribution....................... 100

12.6. Loss of pressure (pressure) during turbulent flow in pipes. ................................................. ................................................. ..... 100

13. FUNDAMENTALS OF THE THEORY OF SIMILARITY AND MODELING .......... 102

13.1. Inspection Analysis of Differential Equations..... 106

13.2. The concept of self-similarity ............................................................... .110

13.3. Dimensional Analysis .................................................................. ............ 111

Literature …………………………………………………………………..118

In cases where there are no equations describing the process, and it is not possible to create them, it is possible to use the analysis of dimensions to determine the type of criteria from which the similarity equation should be compiled. Beforehand, however, it is necessary to determine all the parameters essential for the description of the process. This can be done on the basis of experience or theoretical considerations.

The method of dimensions subdivides physical quantities into basic (primary), which characterize the measure directly (without connection with other quantities), and derivatives, which are expressed through the basic quantities in accordance with physical laws.

In the SI system, the basic units are assigned designations: length L, weight M, time T, temperature Θ , current strength I, the power of light J, amount of substance N.

Derived value expression φ through the main is called the dimension. The formula for the dimension of a derived quantity, for example, with four basic units of measurement L, M, T, Θ, looks like:

Where a, b, c, d are real numbers.

In accordance with the equation, dimensionless numbers have zero dimension, and basic quantities have dimension equal to one.

In addition to the above principle, the method is based on the axiom that only quantities and complexes of quantities that have the same dimension can be added and subtracted. From these provisions it follows that if any physical quantity, for example ∆ p, is defined as a function of other physical quantities in the form ∆ p= f(V, ρ, η, l, d) , then this dependence can be represented as:

,

Where C- constant.

If we then express the dimension of each derived quantity in terms of the main dimensions, then we can find the values ​​of the exponents x, y, z etc. Thus:

In accordance with the equation, after substituting the dimensions, we obtain:

Grouping then homogeneous members, we find:

If in both parts of the equation we equate the exponents with the same basic units, we get the following system of equations:

There are five unknowns in this system of three equations. Therefore, any three of these unknowns can be expressed in terms of the other two, namely x, y And r through z And v:

After substituting the exponents
And into a power function it turns out:

.

The criterion equation describes the fluid flow in the pipe. This equation includes, as shown above, two criteria-complexes and one criterion-simplex. Now, using the analysis of dimensions, the types of these criteria are established: this is the Euler criterion Eu=∆ p/(ρ V 2 ) , Reynolds criterion Re= Vdρ/η and parametric criterion of geometric similarity G=l/ d. In order to finally establish the form of the criterion equation, it is necessary to experimentally determine the values ​​of the constants C, z And v in the equation.

1. Experimental determination of the constants of the criterion equation

When conducting experiments, the dimensional quantities contained in all similarity criteria are measured and determined. According to the results of the experiments, the values ​​of the criteria are calculated. Then they make up tables, in which, according to the values ​​of the criterion K 1 enter the values ​​of the defining criteria K 2 , K 3 etc. This operation completes the preparatory stage of processing experiments.

To generalize tabular data as a power law:

logarithmic coordinate system is used. The selection of exponents m, n etc. achieve such an arrangement of experimental points on the graph so that a straight line can be drawn through them. The straight line equation gives the desired relationship between the criteria.

Let us show how to determine in practice the constants of the criterion equation:

.

In logarithmic coordinates lgK 2 – lgK 1 this is the straight line equation:

.

Putting the experimental points on the graph (Fig. 4), draw a straight line through them, the slope of which determines the value of the constant m= tgβ.

Rice. 4. Processing of experimental data

It remains to find a constant . For any point on a straight line on the graph
. Therefore, the value C find by any pair of corresponding values K 1 And K 2 counted on the straight line of the graph. For the reliability of the value determined by several points of a straight line and the average value is substituted into the final formula:

At more criteria, the determination of the equation constants is somewhat more complicated and is carried out according to the method described in the book.

In logarithmic coordinates, it is not always possible to arrange experimental points along a straight line. This happens when the observed dependency is not described power equation and we need to look for a function of a different kind.

WITH BELIEVABLE "FROM END TO BEGINNING" REASONS IN ASSESSING TECHNOLOGICAL PROCESS FACTORS

General information about the dimensional analysis method

When studying mechanical phenomena a number of concepts are introduced, for example, energy, speed, voltage, etc., which characterize the phenomenon under consideration and can be given and determined using a number. All questions about motion and equilibrium are formulated as problems of determining certain functions and numerical values ​​for the quantities characterizing the phenomenon, and when solving such problems in purely theoretical studies, the laws of nature and various geometric (spatial) relationships are presented in the form of functional equations - usually differential.

Very often, we do not have the opportunity to formulate the problem in a mathematical form, since the studied mechanical phenomenon is so complex that there is no acceptable scheme for it yet and there are no equations of motion yet. We encounter such a situation when solving problems in the field of aircraft mechanics, hydromechanics, in problems of studying strength and deformations, and so on. In these cases, the main role is played by experimental research methods, which make it possible to establish the simplest experimental data, which subsequently form the basis of coherent theories with a strict mathematical apparatus. However, the experiments themselves can be carried out only on the basis of a preliminary theoretical analysis. The contradiction is resolved during the iterative process of research, putting forward assumptions and hypotheses and testing them experimentally. At the same time, they are based on the presence of similarity of natural phenomena, as a general law. The theory of similarity and dimensions is to a certain extent the "grammar" of the experiment.

Dimension of quantities

Units of measurement of various physical quantities, combined on the basis of their consistency, form a system of units. Currently, the International System of Units (SI) is used. In the SI, independently of one another, the units of measurement of the so-called primary quantities are chosen - mass (kilogram, kg), length (meter, m), time (second, sec, s), current strength (ampere, a), temperature (degree Kelvin, K) and the strength of light (candle, sv). They are called basic units. The units of measurement of the remaining, secondary, quantities are expressed in terms of the main ones. The formula that indicates the dependence of the unit of measurement of a secondary quantity on the main units of measurement is called the dimension of this quantity.

The dimension of a secondary quantity is found using the defining equation, which serves as the definition of this quantity in mathematical form. For example, the defining equation for speed is

.

We will indicate the dimension of a quantity using the symbol of this quantity taken in square brackets, then

, or
,

where [L], [T] are the dimensions of length and time, respectively.

The defining equation for force can be considered Newton's second law

Then the dimension of the force will have the following form

[F]=[M][L][T] .

The defining equation and the formula for the dimension of work, respectively, will have the form

A=Fs and [A]=[M][L] [T] .

In the general case, we will have the relationship

[Q] =[M] [L] [T] (1).

Let's pay attention to the record of the relationship of dimensions, it will still be useful to us.

Similarity theorems

The formation of the theory of similarity in the historical aspect is characterized by its three main theorems.

First similarity theorem formulates the necessary conditions and the properties of similar systems, arguing that similar phenomena have the same similarity criteria in the form of dimensionless expressions, which are a measure of the ratio of the intensity of two physical effects that are essential for the process under study.

Second similarity theorem(P-theorem) proves the possibility of reducing the equation to a criterion form without determining the sufficiency of conditions for the existence of similarity.

Third similarity theorem points to the limits of the regular distribution of a single experience, because similar phenomena will be those that have similar conditions for uniqueness and the same defining criteria.

Thus, the methodological essence of the theory of dimensions lies in the fact that any system of equations that contains a mathematical record of the laws governing the phenomenon can be formulated as a relationship between dimensionless quantities. The determining criteria are composed of mutually independent quantities that are included in the uniqueness conditions: geometric relationships, physical parameters, boundary (initial and boundary) conditions. The system of defining parameters must have the properties of completeness. Some of the defining parameters can be physical dimensional constants, we will call them fundamental variables, in contrast to others - controlled variables. An example is the acceleration of gravity. She is a fundamental variable. Under terrestrial conditions - a constant value and - a variable in space conditions.

For the correct application of dimensional analysis, the researcher must know the nature and number of fundamental and controlled variables in his experiment.

In this case, there is a practical conclusion from the theory of dimensional analysis and it lies in the fact that if the experimenter really knows all the variables of the process under study, and there is still no mathematical record of the law in the form of an equation, then he has the right to transform them by applying the first part Buckingham's theorems: "If any equation is unambiguous with respect to dimensions, then it can be converted to a relation containing a set of dimensionless combinations of quantities."

Homogeneous with respect to dimensions is an equation whose form does not depend on the choice of basic units.

PS. Empirical patterns are usually approximate. These are descriptions in the form of inhomogeneous equations. In their design, they have dimensional coefficients that "work" only in certain system units of measurement. Subsequently, with the accumulation of data, we come to a description in the form of homogeneous equations, i.e., independent of the system of units of measurement.

Dimensionless combinations, in question, are products or ratios of quantities, drawn up in such a way that in each combination of dimensions are reduced. In this case, the products of several dimensional quantities of different physical nature form complexes, the ratio of two dimensional quantities of the same physical nature - simplices.

Instead of varying each of the variables in turn,and changing some of them can causedifficulties, the researcher can only varycombinations. This circumstance greatly simplifies the experiment and makes it possible to present in graphical form and analyze the obtained data much faster and with greater accuracy.

Using the method of dimensional analysis, organizing plausible reasoning "from the end to the beginning".

Having become familiar with the general information, particular attention should be paid to the following points.

The most efficient use of dimensional analysis is in the presence of one dimensionless combination. In this case, it is sufficient to experimentally determine only the matching coefficient (it is enough to set up one experiment to compile and solve one equation). The task becomes more complicated with an increase in the number of dimensionless combinations. Compliance with the requirement of a complete description of the physical system, as a rule, is possible (or perhaps they think so) with an increase in the number of variables taken into account. But at the same time, the probability of complication of the form of the function increases and, most importantly, the amount of experimental work increases sharply. The introduction of additional basic units somehow relieves the problem, but not always and not completely. The fact that the theory of dimensional analysis develops over time is very encouraging and orients to the search for new possibilities.

Well, what if, when searching for and forming a set of factors to be taken into account, i.e., in fact, recreating the structure of the physical system under study, we use the organization of plausible reasoning "from end to beginning" according to Pappus?

In order to comprehend the above proposal and consolidate the foundations of the dimensional analysis method, we propose to analyze an example of establishing the relationship of factors that determine the efficiency of explosive breaking during underground mining of ore deposits.

Taking into account the principles of the systems approach, we can rightfully judge that two systemic interacting objects form a new dynamic system. In production activities, these objects are the object of transformation and the subject instrument of transformation.

When breaking ore on the basis of explosive destruction, we can consider the ore massif and the system of explosive charges (wells) as such.

When using the principles of dimensional analysis with the organization of plausible reasoning "from end to beginning", we obtain the following line of reasoning and a system of interrelations between the parameters of the explosive complex and the characteristics of the array.

d m = f 1 (W ,I 0 ,t deputy , s)	d m = k 1 W(st deputy ¤ I 0 W) n (1)
I 0 = f 2 (I c ,V Boer ,K And )	I 0 = k 2 I c V Boer K And (2)
I c = f 3 (t deputy ,Q ,A)	I With = k 3 t air 2/3 Q 2/3 A 1/3 (3)
t air = f 4 (r zab ,P Max l well )	t air = k 4 r zab 1/2 P Max –1/2 l well (4)
P Max = f 5 (r zar D)	P Max = k 5 r zar D 2 (5)

The designations and formulas for the dimensions of the variables used are given in the Table.

VARIABLES	Designation	dimensions
Maximum crushing diameter	d m	[ L]
Line of least resistance		[ L]
Compressive strength of rocks
Period (interval) of deceleration of blasting	t deputy	[ T]
Explosion impulse per 1 m 3 of the array	I 0
Specific consumption of drilling, m / m 3	V Boer	[ L -2 ]
The utilization rate of wells under charge	TO is
Explosion impulse per 1 m of well	I c
Explosion energy per 1 m of charge
Acoustic hardness of the medium (A=gC)
The impact time of the explosion in the well	t air	[ T]
stemming density	r zab	[ L -3 M]
Well length	l well	[ L]
Maximum initial well pressure		[ L -1 M T -2 ]
Charge density in the well	r zar	[ L -3 M]
Explosive detonation speed		[ L T -1 ]

Passing from formula (5) to formula (1), revealing the established relationships, and also keeping in mind the previously established relationship between the diameter of the average and the diameter of the maximum piece in terms of collapse

d Wed = k 6 d m 2/3 , (6)

we obtain the general equation for the relationship of factors that determine the quality of crushing:

d Wed = kW 2/3 [ s t deputy / r zab 1/3 D -2/3 l well 2/3 M zar 2|3 U centuries 2/3 A 1/3 V Boer TO is W] n (7)

Let us transform the last expression in order to create dimensionless complexes, while keeping in mind:

Q= M zar U centuries ; q centuries =M zar V Boer TO is ; M zab =0.25 p r zab d well 2 ;

Where M zar is the mass of the explosive charge in 1 m of the well length, kg/m;

M zab – mass of stemming in 1 m of stemming, kg/m;

U centuries – calorific value of explosives, kcal/kg.

In the numerator and denominator we use [M zar 1/3 U centuries 1/3 (0.25 pd well 2 ) 1/3 ] . We will finally get

All complexes and simplices have a physical meaning. According to experimental data and practice data, the power exponent n=1/3, and coefficient k is determined depending on the scale of simplification of expression (8).

Although the success of dimensional analysis depends on a correct understanding of the physical meaning of a particular problem, after the choice of variables and basic dimensions, this method can be applied completely automatically. Therefore, this method can be easily stated in prescription form, bearing in mind, however, that such a "recipe" requires the researcher to correctly select the constituent components. The only thing we can do here is to give some general advice.

Stage 1. Select independent variables that affect the system. Dimensional coefficients and physical constants should also be considered if they play an important role. This is the most responsibleny stage of the whole work.

Stage 2. Choose a system of basic dimensions through which you can express the units of all selected variables. The following systems are commonly used: in mechanics and fluid dynamics MLq(Sometimes FLq), V thermodynamics MLqT or MLqTH; in electrical engineering and nuclear physics MLqTO or MLqm., in this case, the temperature can either be considered as a basic quantity, or expressed in terms of molecular kinetic energy.

Stage 3. Write down the dimensions of the selected independent variables and make dimensionless combinations. The solution will be correct if: 1) each combination is dimensionless; 2) the number of combinations is not less than that predicted by the p-theorem; 3) each variable occurs in combinations at least once.

Stage 4. Examine the resulting combinations in terms of their acceptability, physical sense and (if the least squares method is to be used) concentrations of uncertainty, if possible, in one combination. If the combinations do not meet these criteria, then one can: 1) get another solution to the equations for the exponents in order to find the best set of combinations; 2) choose another system of basic dimensions and do all the work from the very beginning; 3) check the correctness of the choice of independent variables.

Stage 5. When a satisfactory set of dimensionless combinations is obtained, the researcher can plan to change the combinations by varying the values ​​of the selected variables in his equipment. The design of experiments should be given special consideration.

When using the method of dimensional analysis with the organization of plausible reasoning "from the end to the beginning", it is necessary to introduce serious corrections, and especially at the first stage.

Brief conclusions

Today it is possible to form the conceptual provisions of research work according to the already established normative algorithm. Step-by-step following allows you to streamline the search for a topic and determine its stages of implementation with access to scientific provisions and recommendations. Knowledge of the content of individual procedures contributes to their expert evaluation and selection of the most appropriate and effective.

Progress of scientific research can be presented in the form of a logical scheme, determined in the process of performing research, highlighting three stages that are characteristic of any activity:

Preparatory stage: It can also be called the stage of methodological preparation of research and the formation of methodological support for research. The scope of work is as follows. Definition of the problem, development of a conceptual description of the subject of research and definition (formulation) of the research topic. Drawing up a research program with the formulation of tasks and the development of a plan for their solution. Reasonable choice of research methods. Development of a methodology for experimental work.

Main stage: - executive (technological), implementation of the program and research plan.

final stage: - processing of research results, formulation of the main provisions, recommendations, expertise.

Scientific provisions are a new scientific truth - this is what needs and can be defended. The formulation of scientific provisions can be mathematical or logical. Scientific provisions help the cause, the solution of the problem. Scientific provisions should be targeted, i.e. reflect (contain) the topic for which they were solved. In order to carry out a general linkage of the content of R&D with the strategy for its implementation, it is recommended to work on the structure of the R&D report before and (or) after the development of these provisions. In the first case, work on the structure of the report even has heuristic potential, contributes to the understanding of R&D ideas, in the second case, it acts as a kind of strategy test and feedback for R&D management.

Let's remember that there is a logic of searching, doing work and lo geek presentation. The first is dialectical - dynamic, with cycles, returns, difficult to formalize, the second is the logic of a static state, formal, i.e. having a strictly defined form.

As a conclusion it is desirable not to stop working on the structure of the report during the entire time of the research and thus episodically "check the clocks of TWO LOGICS".

The systematization of modern problems of mining at the administrative level contributes to the increase in the efficiency of work on the concept.

In the methodological support of research work, we often encounter situations where the theoretical provisions on a specific problem have not yet been fully developed. It is appropriate to use methodological "leasing". As an example of such an approach and its possible use, the method of dimensional analysis with the organization of plausible reasoning "from end to beginning" is of interest.

Basic terms and concepts

Object and subject of activity	Relevance	mining technology
Concept		Mining technology facility
Purpose and goal setting		Mining Technology Tools
problem problem situation	Structure	Physical and technical effect
Stages and stages of research		Scientific position
Similarity theorems	Dimension	Basic units

Experience is the explorer of nature. He never deceives... We must make experiments by changing the circumstances until we extract general rules from them, because experience provides true rules.

Leonardo da Vinci

Basic concepts of modeling theory

Modeling is a method of experimental study of a model of a phenomenon instead of a natural phenomenon. The model is chosen so that the results of the experiment can be extended to the natural phenomenon.

Let the quantity field be modeled w. Then, in the case of exact modeling at similar points of the model and the full-scale object, the condition

where is the scale of the simulation.

In the case of approximate modeling, we obtain

The ratio is called the degree of distortion.

If the degree of distortion does not exceed the measurement accuracy, then the approximate simulation does not differ from the exact one. It is impossible in advance to make sure that the value does not exceed some predetermined value, since in most cases it cannot even be determined in advance.

analogy method

If two physical phenomena of different physical nature are described by identical equations and uniqueness conditions (boundary or, in the stationary case, boundary conditions) presented in a dimensionless form, then the phenomena are called analogous. Under the same conditions, phenomena of the same physical nature are called similar.

Despite the fact that similar phenomena have different physical nature, they refer to one individual generalized case. This circumstance made it possible to create a very convenient method of analogies for studying physical phenomena. Its essence is as follows: it is not the studied phenomenon, for which it is difficult or impossible to measure the desired values, that is subjected to examination, but a specially selected one similar to the one being studied. As an example, consider the electrothermal analogy. In this case, the phenomenon under study is a stationary temperature field, and its analogy is a stationary electric potential field

Heat equation

(9.3)

Where absolute temperature,

and the electric potential equation

(9.4)

where the electric potential are similar. In dimensionless form, these equations will be identical.

If the boundary conditions for the potential are created, similar to those for the temperature, then in the dimensionless form they will also be identical.

The electrothermal analogy is widely used in the study of heat conduction processes. For example, the temperature fields of gas turbine blades have been measured by this method.

Dimensional Analysis

Sometimes it is necessary to study processes that are not yet described by differential equations. The only way to study is experiment. It is advisable to present the results of the experiment in a generalized form, but for this it is necessary to be able to find dimensionless complexes characteristic of such a process.

Dimensional analysis is a method of compiling dimensionless complexes under conditions when the process under study is not yet described by differential equations.

All physical quantities can be divided into primary and secondary. For heat exchange processes, the following are usually chosen as primary: length L mass m, time t, quantity of heat Q excess temperature . Then the secondary values ​​will be such quantities as the heat transfer coefficient thermal diffusivity a and so on.

Dimension formulas for secondary quantities have the form of power monomials. For example, the dimension formula for the heat transfer coefficient is

(9.5)

Where Q-quantity of heat.

Let all physical quantities essential for the process under study be known. It is required to find dimensionless complexes.

Let us compose a product from the formulas of dimensions of all physical quantities essential for the process in some degrees that are still undefined; obviously, it will be a power monomial (for the process). Suppose that its dimension (of a power monomial) is equal to zero, i.e., the exponents of the primary quantities included in the formula of dimensions have decreased, then the power monomial (for the process) can be represented in the form of a product of dimensionless complexes of dimensional quantities. Hence, if we compose a product from formulas of dimensions that are essential for processes of physical quantities in indefinite degrees, then from the condition that the sum of the exponents of the powers of the primary quantities of this power monomial is equal to zero, we can determine the desired dimensionless complexes.

Let us demonstrate this operation using the example of a periodic heat conduction process in a solid body washed by a liquid heat carrier. We will assume that differential equations for the process under consideration are unknown. It is required to find dimensionless complexes.

The essential physical quantities for the process under study are the following: characteristic size l(m), thermal conductivity solid body, (J/(m K)), specific heat of a solid With(J / (kg K)), density of a solid body (kg / m 3), heat transfer coefficient (heat transfer) (J / m 2 K)), period time , (c), characteristic excess temperature (K). From these quantities, we compose a power monomial of the form

The exponent of the primary quantity is called the dimension of the secondary quantity with respect to the given primary.

Let us replace into physical quantities (except Q) their dimension formulas, as a result we obtain

In this case, the exponents have values ​​at which Q falls out of the equation.

We equate the exponents of the monomial to zero:

for length

a - b - 3i - 2k = 0; (9.8)

for the amount of heat Q

0; (9.9)

for time

for temperature

for mass m

There are seven significant quantities in total, there are five equations for determining indicators, which means that only two indicators, for example, b and km can be chosen arbitrarily.

Let us express all the exponents in terms of b And k. As a result, we get:

from (8.8), (8.9), (8.12)

f = -b - k; (9.14)

r=b + k; (9.15)

from (8.11) and (8.9)

n=b+f+k=b+(-b-k) + k = 0; (9.16)

from (8.12) and (8.9)

i = f = -b -k. (9.17)

Now the monomial can be represented in the form

Since the indicators b And k can be chosen arbitrarily, let's say:

1. at the same time we write