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52
Sectuin 3. Technical sciences
Azimov R. K., Department of Metrology, Standardization and Certification, Abdurakhmonov A. A., Department of Metrology, Standardization and Certification, Talipov A. R.
Department of Radio Devices and Systems
Makhmudov M. M.
Technology of Manufacturing of Electronic Equipment
Faculty of Engineering Systems of Tashkent State Technical University E-mail: [email protected].
INVESTIGATION OF THERMOPHYSICAL PROPERTIES AND CHARACTERISTICS OF DISPERSED MATERIALS BASED ON EXPERIMENT PLANNING METHODS
Abstract. Thermal physical characteristics of dispersed materials are studied on the basis of experiment planning methods, their thermal conductivity, thermal diffusivity, heat capacity, thermal absorption, which are thermal characteristics and mass transfer characteristics: moisture conductivity, thermal conductivity, mass capacity, etc., have been determined. Thermal processes of materials as well as the structure of the material as a quasi-homogeneous body have been studied.
Keywords: heatphysical characteristic, disperse materials, thermal process, experiment, planning of an experiment.
Thermophysical characteristics are usually of great interest. They represent a set of parameters that characterize simultaneously the reaction of the material to the processes of heat and mass transfer. The latter includes, first of all, thermal characteristics:
- coefficient of thermal conductivity X,
- coefficient of thermal diffusivity k,
- volumetric heat capacity C,
- heat absorption b = JIc as well
As mass transfer characteristics:
- coefficient of moisture permeability K,
- coefficient of thermal and moisture permeability ST,
- coefficient of mass intensity SM, etc.
In this work, the main emphasis is placed on a comprehensive study of only the complex of thermal characteristics of A, k, C and b.
The research was motivated by the following assumptions. On the one hand, the authors had
studied, in addition to typical dispersed materials, also complex solid non-metallic structures. In such objects, as a rule, there is no mass transfer. On the other hand, dispersed systems are characterized by the simultaneous flow of organically interrelated processes of heat and moisture exchange.
The study of thermal processes in such materials is carried out in two key directions.
1. Based on data gathering, solving and analyzing a unified system of equations of heat and mass transfer. It is necessary to know the thermal and mass transfer characteristics of the material. This alternative finds more and more applications; however, its practical implementation at present is associated with considerable difficulties: the need to take into account a large number of characteristics that vary widely depending on the structure of the material and the conditions of experience.
Currently, rather complex nature of these dependencies is far from being fully understood. Otherwise, when precise quantitative relationships are established between the complex of thermal characteristics and the properties and structure of the material, complications arise in solving non-linear heat conduction problems. As a rule, to bring the solution to a productive end, it is necessary to take the thermal characteristics of the substance within certain limits constant.
2. Based on the representation of the structure of the material as a quasi-homogeneous body. In this case, the task of finding the temperature field is reduced to solving one heat conduction equation, complicated by the action of internal sources and, first of all, by the presence of variable thermal characteristics. With this approach, it is necessary to take into account only the thermal characteristics of the material such as X, k, b and C .
In this case, the characteristics are not taken as constant ones: they reflect mutual impact of all possible processes in the material, in addition to thermal conductivity, such as convection, radiation and, above all, mass transfer. Therefore, it is rather
unjustified to call them thermal characteristics and the term thermophysical characteristics is proper to describe them. We emphasize that such a narrowing of the problems solved on the basis of the second principle as compared with the first one is undoubtedly a disadvantage, since it is not possible to simultaneously find the humidity and temperature fields.
However, other obvious advantages of the second principle: its relative simplicity, a significantly smaller number of necessary parameters, the ability to bring solutions up to the operational view - all of the above allow recommending it in a significant number of engineering-type tasks.
This technique is especially effective when it is necessary to jointly evaluate heat and mass transfer, if boundariers are set within which it ends and the boundaries of the temperature and humidity under study are precisely defined, as well asthe structural features of the material under study are noted.
Thus, we believe that in order to assess the thermal properties of dispersed materials and to understand the basic thermal processes occurring in them under the most different modes and conditions, it is necessary and sufficient knowledge of four parameters X, k, b and C, and reflected as effective characteristics of the entire set of heat exchange and mass transfer processes occuring in the material. The choice of the first or second option is not an alternative. On the contrary, they should complement each other.
To facilitate the selection, we put forward a factor that distinguishes the extremum-type tasks. The task is extreme if its goal is to search for the extremum of a certain function. To establish which of the two problems is extreme, one must turn to their formulations and find out where the requirements of extremality are satisfied. In task 1, it is required to establish a relationship between the moisture content of the material and three factors. It is not determined here what thermal conductivity is optimal, and it is not required to optimize it.
In task 2, it is necessary to increase the reliability conditions under which its values will increase. Tasks
of the device. The very formulation of the problem of type 1 will be called interpolational, and of type
indicates that the existing reliability does not satisfy 2 - extremal. the experimenter and requires the search for such
Thermalphysical characteristics of measurement results of humidity of materials y -►
X2 b
-► X4 b
x, - humidity; x2 - gaseous phase; x3- solid phase; x4 - temperature Figere 1. "Black box" schematic
To describe the object of study, it is convenient to use the concept of a cybernetic system, which is schematically shown in Fig. 1. Sometimes such a system is called a "black box". The arrows on the right depict the numerical characteristics of the research objectives. We denote them by the letter "y" and call them optimization parameters.
For experiments, one must be able to influence the behavior of the "black box". All methods of such exposure, we denote by letter "x" and call them factors. In solving the problem, we will use mathematical models of the object, i.e. the equation relating the interrelation between optimization parameter and the factors. This equation in general form can be written as follows:
y = 4>(x1, x2, ...,xk), (1)
where (^ ), means: "as a function of".
Such a function is called a response function. Later we will look at how this function can be selected and built. Now we can comprehend how the conditions for conducting experiments are obtained in the experiment that we are going to conduct.
Each factor in the experience can take one of several values. Such values will be called levels. It may turn out that the factor is capable of taking infinitely many values (continuous series). However, in practice, the accuracy with which a certain value is established is not infinite.
Therefore, we can assume that every factor has a certain number of discrete levels. This assumption
greatly facilitates the construction and analysis of the "black box" and experiment, and also simplifies the assessment of their complexity.
A fixed set of levels of factors determines one of the possible states of the "black box". At the same time, this is the essence of the conditions for conducting one of the possible experiments. If we enumerate all possible sets of states, then we get a complete set of various experiments.
To find out the number of different states, the number of levels of factors (if it is the same for all factors) is enough to raise to the power of the number of factors k: pk,: where p is the number of levels. In addition, it is obvious that the real objects that we encounter every day, have significant complexity. So, at first glance, a simple system with four factors on four levels is very complex.
In these conditions, it is necessary to abandon such experiments, which include all possible experiments: the search is too large. The question arises: how many and what kind of experiments should be included in the experiment to solve the problem? This is where experiment planning comes to the rescue.
However, it should be borne in mind that when planning an experiment, it does not matter what properties the object of a study has. We indicate two basic requirements that have to be considered.
First of all, it is important whether the experimental results are reproduced on the object. We will
select some levels for all factors and in these conditions we will conduct an experiment. Then we repeat it several times at unequal intervals and compare the values of the optimization parameter.
The scatter of these values characterizes the reproducibility of the results. If it does not exceed a certain predetermined value (our requirements for the accuracy of the experiment), then the object satisfies the requirement of reproducible results, and if it exceeds, it does not satisfy this requirement. We will consider only those objects for which the requirement of reproducibility is satisfied.
Planning an experiment implies active intervention in the process and the possibility of choosing in each experience the levels of factors that are of interest. Therefore, this experiment is called active. The object on which an active experiment is possible is called a managed one. This is the second requirement for the object of study.
In practice, there are no absolutely controllable objects. A real object is usually affected by both controlled and unmanaged factors. Uncontrollable factors affect the reproducibility of the experiment and cause its violation. If the requirement of reproducibility is not met, one must turn to the active - passive experiment.
Perhaps poor reproducibility is explained by the action of a factor that is systematically changing (drifting) in time. Then you need to refer to a special method of planning. Experiment, finally, it is possible that all factors are uncontrollable. In this case, the problem arises of establishing a connection between the optimization parameter and the factors from the results of observations of the object's behavior, or, as they say, from the results of a passive experiment (7). We will not consider these cases. Our goal is to present the methods of planning an extreme experiment for reproducible controlled static objects.
Planning an extreme experiment is a method of choosing the quantity and conditions for conducting experiments that are minimally necessary for find-
ing the optimal conditions, that is, for solving the problem posed.
Starting to get acquainted with the planning of an extreme experiment, one must keep in mind that when optimizing, the so-called deterministic approach is widespread, especially widely used in chemistry. In this case, it is proposed to build a physical model of the process based on a thorough study of the mechanism of the phenomena (for example, kinetics, hydrodynamics), which makes it possible to obtain a mathematical model of the object in the form of a system of differential equations.
Undoubtedly, the deterministic and static (associated with the planning of the experiment) approaches should reasonably complement each other, and not be contrasted.
The use of all possible experiments to obtain a model leads to absurdly large experiments. The task of choosing the experiments necessary for the experiment, methods of mathematical processing of their results and decision-making - this is the task of planning an experiment. A special case of this task is the planning of an extreme experiment, that is, an experiment designed to find the optimal conditions for the functioning of an object.
Using the principles of regression and correlation analysis in the processing of experimental data, it is possible to find a relationship between the variables and optimum conditions. In both cases, the mathematical model is the response function, which relates the optimization parameter characterizing the results of the experiment, with the variable parameters with which the experimenter varies during the experiments: y = Q(xv x2,..., xk). (2)
It is customary to call independent variables as xi, x2 , ..., X'k
factors, the coordinate space with the coordinates X1, X2, ..., Xk
- the factor space, and the geometric image of the response function in the factor space - the response surface.
This surface can be represented as a contour diagram (Fig. 1), reflecting, for example, the dependence of the reaction yield (in%) on temperature
and concentration. In this case, the optimal output is concentrated in a small area of the surface. If the experiments and their processing were carried out by the traditional method (when only one variable changes, and all the others are kept constant) there is a high probability of falling into a false optimum.
When using statistical methods, the mathematical model is represented as a polynomial - a segment of the Taylor series into which the unknown relation (2) decomposes:
y=p0pjxj + £ pujxuxj+£px+...,(3)
j =1 u,j=1u^1 j=1
where 5 =-
1 dX:
PUk =
d2$
dx„ ÔX:
h =
d2$
dx 2
Due to the fact that in a real process there are always uncontrollable and uncontrollable variables, the change in magnitude is random. Therefore, when processing the experimental data, so-called sample regression coefficients b0,bj,b . ,b. are obtained, which are estimates of theoretical coefficients
A), Pj, Puj, Pjj.
The regression equation derived from the experience will be written as follows:
k k k
(4)
y = b0 + V b1x1 + V b-xx: +V b
J 0 1 1 Z—i uj u j _
X 2 + .
11 1
j =1 u =1 j=1
The coefficient b0 is called the free term of the regression equation; the coefficients b. are linear effects; the coefficients b.. are quadratic effects; coefficients b - interaction effects.
u>
The coefficients of equation (4) are determined by the least squares method from the condition:
N
p = - y)2 = min- (5)
i=1
Here: N - the sample size from the entire set of values of the parameters studied. The difference between the sample size N and the number of links imposed on this sample l is called the number of degrees of freedom of the sample f :
f = N-l (6)
When finding a regression equation, the number of links is equal to the number of coefficients to be determined.
When studying the dependence on one variable parameter, it is useful to determine the type of regression equation by constructing an empirical regression line. For this, the entire range of variation on the correlation field (Fig. 2) is divided into different intervals v. All points in this interval Ax, belong to its middle xj. To do this, calculate the private average for each interval y.:
j nj
S xi
y, = (7)
n
Here: n■ - number of points in interval Ax. .
i^n, = N (8)
j=1
where k - number of intervals of division; N - selection volume.
Then the points ( x,, y,) are connected in series with straight line segments. The resulting broken line is called an empirical regression line ofy by x. By the form of the empirical regression line, you can choose a regression equation y = f (x).
The task of determining the parameters of the regression equation is reduced practically to the determination of the minimum of the function of several variables. If a
y = f (x ,bo,bpb2,...) (9)
there is a function that is differentiable and requires bg, b, b,, ... to choose so that
N
V = - f(x,,bo,bi,b2,...)]2 = min. (10)
i=1
A necessary condition for the minimum <p(b0,bpb2,...) is the fulfillment of equations:
(11)
= o, dç= o, dç= 0 _
dbn - -
db1 db1
or
X2[y - f X Ab^,..)]2 ^ = 0
i=1 dbo
N 2[yi - f x AAA,...)] ^ = 0 X db1
(12)
Thereafter
we receive:
x=0
x =0
x =0
i=1
enters the regression equation and is called in mathematical statistics the system of normal equations.
The value & > 0 for any bg) b1, by ..therefore, it must necessarily have at least one minimum. Therefore, if the system of normal equations has a unique
solution, then it is the minimum for the quantity &. It is impossible to solve the system (12) in general. To do this, you need to specify a specific form of the function f.
References:
1. Chudnovsky A. F. Thermophysical characteristics of dispersed materials. State publishing house of physical and mathematical literature.- M.: 1962.- 456 p.
2. Berliner M. A. Moisture measurement. Ed. 2nd, Pererab. And add.- M., "Energy", 1973.- 400 p.
3. Osipenko N. B. Experiment planning and experimental data processing: texts of lectures for students of mathematical specialties: texts of lectures for students of specialty 1-31 03 01-02 - "Mathematics (scientific and pedagogical activity" / NB Osipenko; Education of the Republic of Bashkortostan, F. Skaryna Gomel State University - Gomel: F. Skaryna State University, 2010.- 439 p.
4. Scott E. Maxwell, Harold D. Delane, Ken Kelley. Designing Experiments and Analyzing Data: A Model Comparison Perspective, Second Edition (Avec CD) 2nd Edition. Kluwer Academic Publishers, - Boston. 2010.
5. Bonamente Massimiliano. Statistics and Analysis of Scientific Data. Springer. 2013.
* df(x ) * df(x )
i=i dbo i=i dbo
A df(x ) , , , ,df(x )
i=1 dbi tl dbi
. (13)
The system of equations (13) contains as many equations as the unknown coefficients b0, b , by ...
