Air pollution modelling in the case of accidents with toxic chemical substances Текст научной статьи по специальности «Математика»

y^K 514.18

M. Biliaiev, P.Horsev, V. Nochvai, E. Gunko

AIR POLLUTION MODELLING IN THE CASE OF ACCIDENTS WITH

TOXIC CHEMICAL SUBSTANCES

Dnepropetrovsk National University of Railway Engineering, Ukraine

A new code to simulate the 3-D process of pollutant dispersion among buildings was developed. This code is based on the numerical integration of the 3-D transport equation and equation of the potential flow. The implicit difference schemes are used for the numerical integration.

Introduction. To predict the damage after toxic gas emissions during the different accidents the special model is used in Ukraine. This model is approved by Ukrainian Government. But the model has a lot of lacks and is widely criticized in scientific circles. The model gives only the information about the square of contaminated area. This model doesn't take into account the possibility of wind velocity or wind direction change. The main lack of this model is that it cannot predict the change of the concentration of the toxic gas in the atmosphere after emission. Now days the implication of the numerical models is of great interest [1,2]. The main purpose was to give the engineers the tool which is more effective than the standard model.

Mathematical model. To simulate the process of pollutant transfer in the atmosphere (toxic chemical substances) the transport equation is used.

CC CuC CvC CwC „ C ( CC} C

H-1-h

Ct Cx Cy Cz Cx y ^ Cx ) Cy I ' y dy

( dC >

CC

faC=— I mx—1+— My— I Mz-C- \+L Qi (t)(x-x y>(y~y1 )s(z-z )

where u, v, w are the velocity components in x, y and z direction respectively; C is the concentration of toxic substance; a is the parameter taking into account the process of toxic gas decay; ^x, ^y, ^z are the coefficients of turbulent

diffusion in x, y and z direction respectively; xi, yi, zi are the coordinates of point source of emission; Q/t) is the intensity of pollutant emission; 8(x -xi)(y - yt)ô(z - zt) is Dirac's delta-function.

In the developed numerical model the following profile of velocity component u and coefficient of diffusion ^z is used:

f \ n / \

z , M z = k1 z

u = u1 — -

1 z1 ) 1 z1 J

where u1 is the velocity at height z1; k1=0,2; «=0,16; m « 1.

Also the following formulae were used in the numerical model = 0,11u;

P- x My.

The transport equation is used with the following boundary conditions:

m

• inlet boundary: C\Met = CE, where CE is the known concentration (very often CE = 0);

• outlet boundary: in numerical model the condition C (i + l,j,k ) = C (i,j,k) is used (this boundary condition means that we neglect the process of diffusion on this plane);

• top boundary and ground surface — = 0.

On

Hydrodynamic model. To simulate the 3-D wind flow over buildings the model of potential flow is used. In this case the governing equation is

O2 P O2 P O2 P n

-Y +-T +-T = 0 (1)

Ox Oy Oz

where P is the potential of velocity.

The boundary conditions for Eq. (2) are as following

dP

- at the "solid" boundaries we have : -r— = 0 ,

on

where n is a normal to the boundary ;

Op

- at the inlet boundary we have: = VM,

on n

where Vn is the known meaning of the speed;

- at the outlet boundary we have : P = P0 + const .

The components of velocity are calculated as follows

O P O P O P u = -, v = -, w =

O x Oy O z

Numerical integration of the equations. To develop the numerical

model the following splitting of Eq.1 is carried out:

O c O uc O vc O wc

— +-+-+-+ a c = 0,

O t O x O x O z

O c O ( O c O ( O c O ( O c ^

O t O x

v O xy

+

O y V^y O y J

O +—

O z

. O z J

Oc = Z Q (i )8(r " r).

ot

The time dependent derivative is approximated as follows:

OC rn+1 - rn-OC rlj rlj

Oi M

The convective derivatives are written in the following form:

Our Ou+r Ou-r Ovr Ov+r Ov-r Owr dw+r Ow-r -=-+-; -=-+-; -=-+-

Ox Ox Ox Oy Oy Oy Oz Oz Oz

where + u + \u\ . - u -\u\ + v + |v|. - + w + \w\. - w-|w| where u =-L_I ; u =-LL, -L-L; v =-^, w =-L-1; w =-i—1■.

2 2 2 2 2 2 The approximation of the convective derivatives is as following:

^ + nn+1 + nn+1

du C - ui+1, j,k Cijk - uijk Ci-1, j,k = L+ cn+1 dx Ax x

- - rn+1 - ^n+1

du C - ui+1, j,k Ci+1, j,k ui jk Ci jk = L- cn+1 d x Ax x

V+ Cn+1 - yi.JX Vc^ijk y+ cn+1 ijk i j-1 k

A+

y c^ 1 y i.j+1.k^i.j+1.k - V Cn+1

A-

wi, J,кx 1Cjk - wLcJ-1

Az

w- Cn 1 wi. j. k+i. j k+1 - w^cn;1

= L+vCr

dy A, +

V ^ i,j +1,k i.J + 1.k iJk ilk n

'LyC

dy

- w+.,rn+ - w+.,Cn+\ ,

= L+Cr

dz

- w-, k Cn+k- w-cn:1 ,

L-zCn+1.

dC ^ x C n+1 c n+1 Ci+1, j,k ~ Cijk

dx | Ax 2

dC ^ - Cn+1 cn+1 ci,j+1,k - cijk

d+ | A- 2

dC ^ - P z Cn 1 - Cn 1 ci, J, k+1 cijk

dz j Az 2

dz Az

The second order derivatives are approximated in such way:

C n+1 _ C n+1

P x ^«P x 2 - P x 1,],k Ax 2 '-1,7,k = ^¿Cn+1 + M+cCn+1,

d i dC 1 C'"\ , - C"r C"+\ - Cn+1 ,

d y dy 1« ~ y Ci,J+1,k 2 Cjk - ~ y Cj C2 i,J-1,k = M~-yC"+1 + M++yCn+\

Ay

Cn+1 - Cn+1

- ~z i,j,k - 2 ij,k= M~-zCn+1 + M+zCn+1. Az 2

After the approximation of the derivatives the solution of the governing equation is split in four steps:

- at the first step k = 1 the difference equation is

Cn +k - Cn W N.

+2 (L+xCk+L+-Ck+L+Ck)) qk = o;

- at the second step k = n + 2; c = n + 4 the difference equation is

Ck s^iC 7 Hr - C

jk ~Cjk + 2LCk + L+Ck + L-Ck))Ck = 0;

At 2

3 1

- at the third step k = n + — ■ c = n + — the difference equation is

4' 2

Ck - Cc i i

iJk iJk = -M- Cc + M++xCk + M- Cc + M + Ck + M- Cc + M + Ck

2 xx yy yy ZZ zz

- at the fourth step k = n + r c = n +3 the difference equation is

4

ck — Cc i i \

-J*-= !M— Ck +M+xCc +M— Ck +M+ Cc +M— Ck +M+ Cc I.

2 xx yy yy zz zz !

At the fifth step (at this step the influence of the source of pollutant ejection is taken into account) the following approximation is used:

+

5 n+1 5 n

cj,u-cj,k = (t5n) 5.

At l=1 AxAjAz 1'

The function Sl is equal to zero in all the cells except the ones where the source of emission is situated.

The considered difference scheme is implicit and absolutely steady but the unknown concentration C is calculated using the explicit formulae at each step of splitting ( so called "method of running calculation") .

To solve Eq. (2) A.A. Samarskii's change-triangle difference scheme is used. In this case instead of equation (1) the 'time-dependent' equation for the potential of velocity is used in the model

dP d2P d2P d2P

2 + —t + —T , (2)

d^ dx2 dy2 dz2

where ^ is the 'fictitious' time.

For ^ ^ œ the solution of this equation tends to the solution of Laplas

equation (1) .

According to A.A. Samarskii's change-triangle difference scheme the solution of equation (2) is split in two steps:

-at the first step the difference equation is

nn+V2 _ pn pn _ pn _pn+l2 + pn+12 pn _ pn

ri,j,k ri,j,k = r,+\, j,k ri,j ,k + ', j k ' 1, j,k \ ri,J,k +

0,5A^ Ax2 Ax2 Ay2

_ pn+12 , pn+1/2 pn _ pn _pn+1]2 , pn+12

+ ri, j k J_ 1,k {ri,jM1 i, j ,k ri ,j k ,k_1

Ay2 Az2 Az

- at the second step the difference equation is

pn+1 pn+1j2 pn + 1 pn + 1 pn+1/2 ^ pn+1j2 pn + 1 p

i, j ,k i, j ,k _ i+1, j, k i, j, k +__i, j ,k_i_ 1, j ,k + i, j+1,k_i, j, k +

n + 1

0,5A^ Ax2 Ax2 Ay2

pn + ^2 + pn + 1/2 pn+1 pn+1 pn + 1/2 + pn + 1/2

+__i, j, k_i, j _1,k + i, j ,k+1_i, j ,k + i, j, k i, j, k _1

+ 2 "

Aj2 Az2 Az2

From these expressions the unknown value P is calculated using the explicit formulae at each step (the "method of running calculation"). The

calculation is completed if the condition P"+l - Ptnj u ^ s is fulfilled (where

s is a small number, n is the number of iteration ). The components of velocity vector are calculated on the sides of computational cell as follows

u = P', J,k - P'-^ J,k v = P', J,k - P', J-',k P', J ,u - Pi, j ,k-1

U<, J, k a , ', J , k ~ A ,, , W .k = -1-1- .

A x Aj ', J ,k a z

Results. On the base of the numerical model the code 'AirPollut' was developed. The developed code was used to solve some practical problems. In Fig.1, 2 the results of the numerical simulation of the air pollution near two buildings are shown.

Fig.1 Pattern of toxic gas concentration near two buildings for time

t = 15 sec after accident

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Fig.2 Pattern of the toxic gas concentration near two buildings for time t = 23

sec after accident

As it is clear the developed model allows to calculate the toxic gas propagation with account of buildings influence.

Concluding Remarks. Nowadays the work is carried out to develop the numerical model for the application in the case of arbitrary terrain and using Navier - Stokes equations.

References

1. Belayev N.N. , Kazakevitch M. I., Khrutch V. K . Computer simulation of the pollutant dispersion among buildings. // Wind Engineering into 21st Century. Proceedings of the Tenth International Conference on Wind

Engineering . Copenhagen / Denmark / A. A. BALKEMA / ROTTERDAM / BROOKFIELD .-1999. P. 1217-1220.

2. Belayev N.N., Khrutch V.K. An engineering approach to simulate the 3-d wind flows over buildings // Proceedings of the Fourth International Colloquium on Bluff Body Aerodynamics & Applications . Sept. 11-14 , 2000 , Ruhr-Universitat, Bochum , Germany. Volume of Abstracts. P. 471-475.

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