COMPUTING THE PARTICLE SETTLING VELOCITY TAKING INTO ACCOUNT THE COEFFICIENT OF DYNAMIC VISCOSITY Текст научной статьи по специальности «Физика»

COMPUTING THE PARTICLE SETTLING VELOCITY TAKING INTO ACCOUNT THE COEFFICIENT OF DYNAMIC

VISCOSITY

Muradov F.A.

Doctoral student, 1Digital technologies and artificial intelligence research institute https://doi.org/10.5281/zenodo.14024015

Abstract. Using a spherical coordinate system, the article creates a mathematical model of the transfer and diffusion of dangerous gaseous chemicals andfine particles in the atmosphere. The mathematical model that has been built considers the modes of operation of sources that emit dangerous compounds and gas pollutants, together with their impact on the process being studied. In contrast to the writings of other authors, each layer in time and each spatial variable's rate of particle deposition was taken into account independently. Using an implicit finite-difference technique with a high level of approximation in the time and spatial variables, a numerical solution has been constructed to solve the problem. A software tool has been constructed to conduct computer experiments based on the developed mathematical apparatus. Numerical models have demonstrated that the distribution of fine aerosols and gaseous chemical particles in the atmosphere is primarily determined by climatic conditions. The study method is impacted by various factors, including the type of vegetation present, the topography, and the characteristics of the pollutant. According to the analysis, there is a reduced absorption of pollutant particles in the summer and a higher absorption of pollutant particles during seasons with high humidity.

Keywords: mathematical model; numerical algorithm; approximation; implicit finite difference scheme.

INTRODUCTION

A crucial physical characteristic that makes it possible to describe the friction of a fluid experiences due to molecular attractions that prevent it from flowing is called viscosity [1].

This characteristic is typically employed in the quality control of goods like paints, oils, and cosmetics [2].

A variety of techniques have been put forth to quantify viscosity. Rao has measured the viscosity of several fluids using the falling ball viscometer [3].

Specifically, food goods, silicate glass, pharmaceutical beverages, and petroleum products. In order to determine the fluid's viscosity, the experiment examines the ball's fall velocity as it rolls and slides inside a cylindrical tube filled with the liquid.

A double capillary tube scanning viscometer (SDCV) was created by Kim et al. and is used to test the viscosity of both Newtonian and non-Newtonian fluids, including blood [4]. This kind of viscometer measures the amount of time it takes for a specific amount of liquid to pass through a capillary, or the emptying time.

An axial flow has been postulated by Heuser G et al. Blood was subjected as a studied fluid to fixed shear loads of brief duration using a Couette viscometer [5], and the results allowed for the regulation of flow parameters to enhance fluid quality.

The suggested system is based on the investigation of the deformation and flow of the material under external stress by imparting shear to the material, as done by Münstedt et al. [6]

using a universal extensional rheometer. The system's operational modes are evaluated using a polystyrene material, and during the course of three decades of tensile stress, the viscosity and recoverable strain in the stable elongation state were measured.

A viscometer known as a "Sine Wave Vibro Viscometer" was created by Izumo et al. [7] based on their analysis of the amortized oscillatory motion of a mass attached to a spring. This particular type of sinusoidal wave Vibro-viscosimeter has a wide measurement range without the need to replace the sensor and can measure a small amount of material.

This Vibro-viscometer has been used to evaluate the viscosity of a variety of materials, including the coagulation process, the concentrations of ethanol solutions, the cloud point of surfactants, and the polymerization of silicone adhesives. In [8-20], more experimental techniques are suggested.

STATEMENT OF THE PROBLEM

As discussed in [21, 22, 23, 24, 26], to determine the initial sedimentation velocity of particles, it is necessary to take into account three main forces that act on particles when moving in the atmosphere: gravity Mt, buoyancy force Nv, resistance force Rs, which are determined using:

r2

PWg

M. = mg; Nv =mpg; Rs =kcS- s

Here mp - mass of air in volume; kc - air resistance coefficient.

Since, in general, aerosol particles emitted from production facilities have a spherical shape with a diameter equal to d, these three forces are calculated using the following formulas:

Tid3 „ Tid3 „ , Tid2 Wg

Mt =^—pg; Nv pg; Rs p g

6 ' " v ^ ° " 4 ' 2

Using the equilibrium equation, we can find the initial sedimentation velocity of particles

Wg (0)=

4dg(pH -p )

3kcp

where pH - particle density.

A mathematical model based on the laws of hydromechanics has been developed for the problem of transfer and diffusion of aerosol particles in the atmosphere, taking into account the settling velocity of fine particles wg and weather and climate factors:

dwg mg - 6mirwg - 0,5cpsw2g

dt m

where is the initial value:

(1)

t=o=(2)

Here c=0,5 - dimensionless quantity; p - particle density; r - particle radius; s - cross-sectional area of a particle; g - free fall speed; m - particle mass; t] - coefficient of dynamic viscosity of a particle.

When determining the settling velocity of particles, three main forces acting on a particle moving in the atmosphere are taken into account: gravity, compression force, and drag force.

It should be noted here that for the numerical solution of equation (1), it is necessary to set

the initial condition and the initial iteration value for wg (0), wg . As was considered in the work

[22, 23, 25], the initial values and the initial iteration for these variables are specified by the following relations:

a) with stable stratification

(0) = d 2 g (P-PZ ) . „I = d lg(P-Pz Z

g y 1 Q g

18k g 18k

b) with indifferent stratification

„ \0.714 _ „ \0.714

w (0) = cd (P-Pz)_. W = c d (p-pz)_

rvg^> ^ 0.286t 0.43 ' g 1 0.286, 0.43

pz k pz k

c) with unstable stratification

Wg (0) = . wg, = .

Here pz - particle density; Cj = 0,78.

In the differential equation (1), the dynamic viscosity coefficient of a particle t] is determined in the following ways:

1. The coefficient of dynamic viscosity of a particle at temperature T can be determined using the Sutherland formula in the following form:

n = V0 0

3

T + C ( Tv

T

v T0 y

(3)

T + C

where rj0 - the value of the particle at K Tc=273,15i, S- Sutherland constant value for a particle.

2. In practice, the dynamic viscosity coefficient of a particle is also expressed in the following exponential form:

1 = %

4

3

i

T

vT0 y

(4)

3. Also, the coefficient of dynamic viscosity of a particle can be expressed as follows [1]:

77 = A + BT + CT2 + DT3. (5)

Here, the coefficients A, B, C and D are considered as empirical coefficients and their values are taken from the experimental values.

NUMERICAL METHOD FOR SOLVING THE PROBLEM

In the differential equation (1), we apply the substitution w2 = 2w w - w2 and linearize:

dwg mg - 67it]rw:< - 0,5cps (2wgwg - w2g )

(6)

dt m

Solving the differential equation (6) analytically is difficult. Therefore, we use an implicit time scheme for this differential equation. Below, we present the sequence of applying the finite difference for each layer, dividing the particle deposition velocity into three time layers: a) in the direction 0x

w 3 - w„

mgx - ßmjrw 3 - 0,5cps

i

n+— 3

1

«H--

2ww 3 - w~

V

At m

Let's open the brackets, simplify the like terms and get the following:

(m + 6

i

n +—

w 3 =

(\ "— 2 m + 67Ti]rAt + cpsAtw,; )wg 3 = mw'g + mgxAt + 0,5cpsAtn>g ;

mgxAt + 0,5cpsAtwi

m

-w +-

m + ßTrrjrAt + cpsAtwg m + ßTrrjrAt + cpsAtwg b) in the direction 0y

(7)

2

n+—

f

n+2 n+\ mgv - 6mirws 3 - 0,5cps

wg 3 - wg 3

2

«H—

2™gWg 3

V

At m

Let's open the brackets, simplify the like terms and get the following:

2 1

m + 67rrjrAt + cpsAtw) w 3 = mwp 3 + mgvAt + 0,5cpsAt\\\,;

6 / 6 6 / o

2

n+—

w 3 = ■

m

i

n+—

m + 67iî]rAt + cpsAtwg v) in the direction 0z

■ w„ J + ■

mg Ai + 0,5cpsAl\r m + ô/rtjrAl + cpsAtw

(8)

wn+1 - .. g g

z

3 mgy - 67Tr/rw'g+1 - 0,5cps (2wgw'g+1 - w2g )

At m

Let's open the brackets, simplify the like terms and get the following:

2

(m + GjrqrAt + cpsAtw ) w"+l = mw 3 + mg At + 0,5cpsAtvi>2;

\ 6/6 6 / o

wn+1 =■

m

- if

2

n+— 3

mg_At + 0, 5c/;>vA/ir

(9)

m + 6 7ii] r At. + cp.vA/iT'^ m + 6 7ii] r At. + cp.vA/iT'^

The obtained finite-difference nonlinear equations (7)-(9) are solved by a simple iteration

method. The approximation of the iteration process is verified using the condition

wf+1 - w

<s,

where s is the accuracy of the iteration process and G is the number of iterations.

ANALYSIS OF RESULTS

Below is the change in the coefficient of dynamic viscosity of various aerosols, such as ammonia, nitrogen and carbon dioxide, depending on the temperature. The dynamics of the change in the density of these aerosols at different temperatures and their sedimentation rate are also presented.

CONCLUSION

The following points can be noted as conclusions. In contrast to numerous studies by other authors, in this work the rate of particle deposition is considered as a variable quantity that has a significant impact on the process under study. Also, computer experiments have established that the most important factor influencing the process of distribution of harmful gaseous compounds and fine aerosols in the atmosphere is the vegetation growing in the area under consideration.

Dynamic viscosity coefficient of different aerosols

- Ammonia - Nitrogen - Carbon dioxide L

260 280 300 320 340 360

Temperature in Kelvin

Fig. 1. Dynamics of change in the coefficient of dynamic viscosity of some aerosols at

different temperatures

Dynamic viscosity coefficient of carbon dioxide

260 280 300 320 340 360

Temperature in Kelvin

Fig. 2. Dynamics of change in the coefficient of dynamic viscosity of carbon dioxide in different formulas at different temperatures

ie_5_Dynamic viscosity coefficient of ammonia

1.31 - I I

260 280 300 320 340 360

Temperature in Kelvin

Fig. 3. Dynamics of change in the coefficient of dynamic viscosity of ammonia in different

formulas at different temperatures

Fig. 4. Dynamics of change in the coefficient of dynamic viscosity of nitrogen in different

formulas at different temperatures

Fig. 5. Dynamics of changes in the densities of some aerosols at different temperatures

Graph of deposition rates of different aerosols

- Ammonia

- larbon dioxide

300 320

Temperature in Kelvin

Fig. 6. Dynamics of changes in the sedimentation rates of some aerosol particles at different

temperatures

This paper proposes a mathematical model based on three-dimensional differential equations based on the laws of fluid mechanics. The model was developed using a spherical coordinate system. The motivation for the transition to spherical coordinates was the possibility of reducing the computer load, the direct use of the original geospatial data and the relatively simple cartographic visualization of the calculation results. To solve the problem, a numerical algorithm has been developed using an implicit finite-difference scheme with a high order of approximation in time and spatial variables.

Using a software-implemented mathematical apparatus, it is possible to assess and predict the spatiotemporal evolution of industrial emissions of harmful substances in the atmospheric boundary layer, which is of practical value for engineers, ecologists and researchers.

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