Analysis of the dimensions of physical quantities. Dimensional Analysis

Physical quantities, the numerical value of which does not depend on the chosen scale of units, are called dimensionless. Examples of dimensionless quantities are the angle (the ratio of the arc length to the radius), the refractive index of matter (the ratio of the speed of light in vacuum to the speed of light in matter).

Physical quantities that change their numerical value when the scale of units is changed are called dimensional. Examples of dimensional quantities are length, force, etc. The expression of a unit of a physical quantity in terms of basic units is called its dimension (or dimension formula). For example, the dimension of force in the CGS and SI systems is expressed by the formula

Considerations of dimension can be used to check the correctness of the answers obtained when solving physical problems: the right and left parts of the obtained expressions, as well as individual terms in each of the parts, must have the same dimension.

The method of dimensions can also be used to derive formulas and equations, when we know on what physical parameters the desired value may depend. The essence of the method is easiest to understand with concrete examples.

Applications of the method of dimensions. Consider a problem for which the answer is well known to us: with what speed will a body fall to the ground, freely falling without an initial velocity from a height, if air resistance can be neglected? Instead of a direct calculation based on the laws of motion, we will argue as follows.

Let's think about what the desired speed may depend on. It is obvious that it must depend on the initial height and on the acceleration of free fall It can be assumed, following Aristotle, that it also depends on the mass. Since only values of the same dimension can be added, the following formula can be proposed for the desired speed:

where C is some dimensionless constant (numerical coefficient), and x, y and z are unknown numbers to be determined.

The dimensions of the right and left parts of this equality must be the same, and it is this condition that can be used to determine the exponents x, y, z in (2). The dimension of speed is the dimension of height is the dimension of the acceleration of free fall is, finally, the dimension of the mass is equal to M. Since the constant C is dimensionless, formula (2) corresponds to the following equality of dimensions:

This equality must hold regardless of what the numeric values are. Therefore, it is necessary to equate the exponents at and M in the left and right parts of equality (3):

From this system of equations, we obtain Therefore, formula (2) takes the form

The true value of the speed, as is known, is equal to

So, the approach used made it possible to determine correctly the dependence on and and did not make it possible to find the value

dimensionless constant C. Although we have not been able to obtain an exhaustive answer, nevertheless, very significant information has been obtained. For example, we can state with complete certainty that if the initial height is quadrupled, the speed at the moment of falling will double and that, contrary to Aristotle's opinion, this speed does not depend on the mass of the falling body.

Choice of options. When using the method of dimensions, one should first of all identify the parameters that determine the phenomenon under consideration. This is easy to do if the physical laws describing it are known. In a number of cases, the parameters determining the phenomenon can be specified even when the physical laws are unknown. As a rule, you need to know less to use the dimensional analysis method than to write equations of motion.

If the number of parameters that determine the phenomenon under study is greater than the number of basic units on which the chosen system of units is built, then, of course, all the exponents in the proposed formula for the desired value cannot be determined. In this case, it is useful first of all to determine all independent dimensionless combinations of the chosen parameters. Then the desired physical quantity will be determined not by a formula like (2), but by the product of some (the simplest) combination of parameters that has the desired dimension (i.e., the dimension of the desired quantity) by some function of the found dimensionless parameters.

It is easy to see that in the above example of a body falling from a height, it is impossible to form a dimensionless combination from the quantities and the dimensionless combination. Therefore, formula (2) there exhausts all possible cases.

Dimensionless parameter. Let us now consider the following problem: we determine the range of the horizontal flight of a projectile fired in a horizontal direction with an initial velocity from a gun located on a mountain of height

In the absence of air resistance, the number of parameters on which the desired range may depend is equal to four: and m. Since the number of basic units is equal to three, a complete solution of the problem by the method of dimensions is impossible. Let us first find all the independent dimensionless parameters y that can be composed of and

This expression corresponds to the following equality of dimensions:

From here we get the system of equations

which gives and for the desired dimensionless parameter we obtain

It can be seen that the only independent dimensionless parameter in the problem under consideration is .

where is the yet unknown function of the dimensionless parameter. The method of dimensions (in the presented version) does not allow one to determine this function. But if we know from somewhere, for example, from experience, that the desired range is proportional to the horizontal velocity of the projectile, then the form of the function is immediately determined: the velocity must enter into it to the first power, i.e.

Now from (5) for the range of the projectile we get

which matches the correct answer

We emphasize that with this method of determining the type of function, it is enough for us to know the nature of the experimentally established dependence of the flight range not on all parameters, but only on one of them.

Vector units of length. But it is possible to determine the range (7) only from dimensional considerations, if we increase to four the number of basic units in terms of which the parameters are expressed, etc. Until now, when writing dimensional formulas, no distinction was made between units of length in the horizontal and vertical directions. However, such a distinction can be introduced based on the fact that gravity acts only vertically.

Let us denote the dimension of length in the horizontal direction through and in the vertical direction - through Then the dimension of the flight range in the horizontal direction will be the dimension of height will be the dimension of the horizontal speed will be and for acceleration

free fall we get Now, looking at formula (5), we see that the only way to get the right dimension on the right side is to consider it proportional. We again come to formula (7).

Of course, having four basic units and M, one can directly construct the value of the required dimension from four parameters and

The equality of the dimensions of the left and right parts has the form

The system of equations for x, y, z and and gives the values and we again come to formula (7).

The different units of length used here in mutually perpendicular directions are sometimes referred to as vector units of length. Their application significantly expands the possibilities of the dimensional analysis method.

When using the dimensional analysis method, it is useful to develop skills to such an extent that you do not make a system of equations for the exponents in the desired formula, but select them directly. Let's illustrate this in the next problem.

A task

Maximum range. At what angle to the horizontal should a stone be thrown to maximize the horizontal flight range?

Solution. Let's assume that we have "forgotten" all kinematics formulas and try to get an answer from dimensional considerations. At first glance, it may seem that the method of dimensions is not applicable here at all, since some trigonometric function of the throwing angle must enter into the answer. Therefore, instead of the angle a itself, we will try to look for an expression for the range. It is clear that we cannot do without vector units of length.

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. The general form of this regularity 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 BUT 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.

The use of dimensional analysis for solving specific practical problems is associated 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 final ratios obtained by him did not include the Reynolds similarity number, which plays an extremely important role in heat transfer.

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 rate of the working medium (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: the specific kinetic energy, a - 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.: Higher 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 .............................................................. ...... four

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 ............................................................... ............................ eight

2.1. Density................................................. ................................... eight

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............................................... .................................. eighteen

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. ................................................. ................................................. ..... twenty

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

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 properties of such systems, arguing that such 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. In terrestrial conditions constant and is 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 a certain system of 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 BUT 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), in 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 meaning and (if least squares method is to be used) concentration of uncertainty in one combination if possible. 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, changing the circumstances, until we extract from them general rules, because experience delivers the true rules.

Leonardo da Vinci

In physics... there is no place for confused thoughts...
Really understanding nature
This or that phenomenon should receive the main
Laws from considerations of dimension. E. Fermi

The description of this or that problem, the discussion of theoretical and experimental issues begins with a qualitative description and assessment of the effect that this work gives.

When describing a problem, it is necessary, first of all, to evaluate the order of magnitude of the expected effect, simple limiting cases, and the nature of the functional relationship of the quantities describing this phenomenon. These questions are called the qualitative description of the physical situation.

One of the most effective methods such an analysis is the method of dimensions.

Here are some advantages and applications of the dimensional method:

rapid assessment of the scale of the phenomena under study;
obtaining qualitative and functional dependencies;
restoration of forgotten formulas in exams;
fulfillment of some tasks of the exam;
verification of the correctness of the solution of problems.

Dimensional analysis has been used in physics since the time of Newton. It was Newton who formulated, closely related to the method of dimensions, the principle of similarity (analogy).

Students first encounter the dimensional method when studying thermal radiation in the 11th grade physics course:

The spectral characteristic of the thermal radiation of a body is spectral density of energy luminosity r v - energy of electromagnetic radiation emitted per unit of time per unit area of the body surface in a unit frequency interval.

The unit of spectral density of energy luminosity is the joule per square meter(1 J / m 2). The energy of the thermal radiation of a black body depends on temperature and wavelength. The only combination of these quantities with the dimension of J/m 2 is kT/ 2 ( = c/v). The exact calculation made by Rayleigh and Jeans in 1900, within the framework of the classical wave theory, gave the following result:

where k is the Boltzmann constant.

As experience has shown, this expression is consistent with experimental data only in the region of sufficiently low frequencies. For high frequencies, especially in the ultraviolet region of the spectrum, the Rayleigh-Jeans formula is incorrect: it differs sharply from experiment. The methods of classical physics turned out to be insufficient to explain the characteristics of black body radiation. Therefore, the discrepancy between the results of classical wave theory and experiment at the end of the 19th century called the "ultraviolet catastrophe".

Let us show the application of the dimension method on a simple and well-understood example.

Picture 1

Thermal radiation of a black body: ultraviolet catastrophe - discrepancy between the classical theory of thermal radiation and experience.

Imagine that a body of mass m moves in a straight line under the action of a constant force F. If the initial speed of the body is zero, and the speed at the end of the traveled section of the path of length s is equal to v, then we can write the kinetic energy theorem: Between the values F, m, v and s there is a functional connection.

Let us assume that the kinetic energy theorem is forgotten, but we understand that the functional dependence between v, F, m, and s exists and has a power law.

Here x, y, z are some numbers. Let's define them. The sign ~ means that the left side of the formula is proportional to the right side, that is, where k is a numerical coefficient, has no units of measurement and is not determined using the dimensional method.

The left and right parts of relation (1) have the same dimensions. The dimensions of v, F, m and s are: [v] = m/c = ms -1 , [F] = H = kgms -2 , [m] = kg, [s] = m. (Symbol [A] denotes the dimension of A.) Let us write the equality of dimensions in the left and right parts of relation (1):

m c -1 = kg x m x c -2x kg y m Z = kg x+y m x+z c -2x .

There are no kilograms at all on the left side of the equation, so there shouldn't be any on the right either.

It means that

On the right, meters are included in the powers of x + z, and on the left, in the powers of 1, so

Similarly, from a comparison of the exponents in seconds, it follows

From the obtained equations we find the numbers x, y, z:

x=1/2, y=-1/2, z=1/2.

The final formula looks like

By squaring the left and right sides of this relation, we get that

The last formula is a mathematical notation of the kinetic energy theorem, though without a numerical coefficient.

The principle of similarity, formulated by Newton, is that the ratio v 2 /s is directly proportional to the ratio F/m. For example, two bodies with different masses m 1 and m 2 ; we will act on them with different forces F 1 and F 2 , but in such a way that the ratios F 1 / m 1 and F 2 / m 2 will be the same. Under the influence of these forces, the bodies will begin to move. If the initial speeds are equal to zero, then the speeds acquired by the bodies on a segment of the path of length s will be equal. This is the law of similarity, which we arrived at with the help of the idea of the equality of the dimensions of the right and left parts of the formula, which describes the power-law relationship of the value of the final velocity with the values of force, mass, and path length.

The method of dimensions was introduced when building the foundations of classical mechanics, but its effective application for solving physical problems began at the end of the past - at the beginning of our century. A great merit in promoting this method and solving interesting and important problems with its help belongs to the outstanding physicist Lord Rayleigh. Rayleigh wrote in 1915: I am often surprised at the little attention given to the great principle of similarity, even by very great scientists. It often happens that the results of painstaking research are presented as newly discovered "laws", which, nevertheless, could be obtained a priori within a few minutes.

Nowadays, physicists can no longer be reproached with a neglectful attitude or insufficient attention to the principle of similarity and to the method of dimensions. Consider one of the classic Rayleigh problems.

Rayleigh's problem on vibrations of a ball on a string.

Let a string be stretched between points A and B. The tension force of the string F. In the middle of this string at point C there is a heavy ball. The length of segment AC (and, accordingly, CB) is equal to 1. The mass M of the ball is much greater than the mass of the string itself. The string is pulled and released. It is pretty clear that the ball will oscillate. If the amplitude of these x oscillations is much less than the length of the string, then the process will be harmonic.

Let us determine the frequency of vibrations of the ball on the string. Let the quantities , F, M and 1 be connected by a power law:

The exponents x, y, z are the numbers we need to determine.

Let us write out the dimensions of the quantities of interest to us in the SI system:

C -1, [F] = kgm s -2, [M] = kg, = m.

If formula (2) expresses a real physical regularity, then the dimensions of the right and left parts of this formula must match, that is, the equality

c -1 = kg x m x c -2x kg y m z = kg x + y m x + z c -2x

The left side of this equation does not include meters and kilograms at all, and seconds are included in the powers - 1. This means that for x, y and z the equations are satisfied:

x+y=0, x+z=0, -2x= -1

Solving this system, we find:

x=1/2, y= -1/2, z= -1/2

Consequently,

~F 1/2 M -1/2 1 -1/2

The exact formula for the frequency differs from the one found only by a factor of ( 2 = 2F/(M1)).

Thus, not only a qualitative, but also a quantitative estimate of the dependence of on the values of F, M, and 1 was obtained. In order of magnitude, the found power combination gives the correct value of the frequency. Evaluation is always of interest in order of magnitude. In simple problems, coefficients that are not determined by the method of dimensions can often be considered numbers of the order of unity. This is not a strict rule.

When studying waves, I consider qualitative prediction of the speed of sound by the method of dimensional analysis. We are looking for the speed of sound as the speed of propagation of a compression and rarefaction wave in a gas. Students have no doubts about the dependence of the speed of sound in a gas on the density of the gas and its pressure p.

We are looking for the answer in the form:

where С is a dimensionless factor, the numerical value of which cannot be found from the analysis of dimensions. Passing in (1) to the equality of dimensions.

m / s \u003d (kg / m 3) x Pa y,

m / s \u003d (kg / m 3) x (kg m / (s 2 m 2)) y,

m 1 s -1 \u003d kg x m -3x kg y m y c -2y m -2y,

m 1 s -1 \u003d kg x + y m -3x + y-2y c -2y,

m 1 s -1 \u003d kg x + y m -3x-y c -2y.

The equality of dimensions on the left and right sides of the equality gives:

x + y = 0, -3x-y = 1, -2y= -1,

x= -y, -3+x = 1, -2x = 1,

x = -1/2 , y = 1/2 .

So the speed of sound in a gas

Formula (2) at C=1 was first obtained by I. Newton. But the quantitative derivations of this formula were very difficult.

An experimental determination of the speed of sound in air was carried out in a collective work of members of the Paris Academy of Sciences in 1738, which measured the time it took the sound of a cannon shot to travel a distance of 30 km.

Repeating this material in the 11th grade, students' attention is drawn to the fact that the result (2) can be obtained for the model of the isothermal process of sound propagation using the Mendeleev-Clapeyron equation and the concept of density:

is the speed of sound propagation.

Having introduced the students to the method of dimensions, I give them this method to derive the basic MKT equation for an ideal gas.

Students understand that the pressure of an ideal gas depends on the mass of individual molecules of an ideal gas, the number of molecules per unit volume - n (the concentration of gas molecules) and the speed of movement of molecules -.

Knowing the dimensions of the quantities included in this equation, we have:

Comparing the dimensions of the left and right parts of this equality, we have:

Therefore, the basic equation of the MKT has the following form:

- this implies

It can be seen from the shaded triangle that

Answer: B).

We have used the dimension method.

The method of dimensions, in addition to carrying out the traditional verification of the correctness of solving problems, performing some tasks of the Unified State Examination, helps to find functional relationships between various physical quantities, but only for those situations where these dependencies are power-law. There are many such dependencies in nature, and the method of dimensions is a good helper in solving such problems.

In cases where the processes under study are not described differential equations, one of the ways to analyze them is an experiment, the results of which are best presented 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 value 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, is equal to 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 a specific example, 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 with the same bases, 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.