MATHEMATICAL MODELING OF SPREAD OF CONCENTRATION IN STRATOSPHERE FROM EXPLOITATION OF CARRIER ROCKETS

 

ISAKHOV A., ZHAKEBAEV D., ZHUBAT K.

 

In this paper the modeling of distribution of rocket fuel components in the atmosphere is studied. On the basis of the Navier-Stokes equations a mathematical model of impurity migration process in the stratified medium was developed. The numerical algorithm was worked out using a scheme of splitting on physical parameters. Modeling of rocket fuel components distribution in surface layer of the atmosphere and in stratosphere was carried out. Results of modeling are presented in the form of three dimensional graphs.

 

Components of modern rocket-and-space equipment, especially carrier rockets, constitute a serious danger for environment due to considerable reserves of high-energy chemical fuel. For example, in the process of launch of “Proton-M” carrier rocket the projected scope of emission to the atmosphere of remnants of crude heptyl from stage 1 and 2 amounts to 1,7 tons, and in case of the carrier rocket breakdown the emissions of such highly toxic fuel to the atmosphere total tens of tons. In the surface atmosphere layer up to 1 km from the ground the emissions resulting from launching of spaceships can lead to acid rains and weather changes within the area of launching covering up to 200km2. In the stratosphere at the elevation of 40-60km the processes of mixing are less intensive what causes contaminations generating at such levels to last longer. So, aerosol particles emitted by stage 1 of carrier rockets can retain in the stratosphere for the period of up to one year or longer possibly affecting thermal balance of the atmosphere.

The subject of the research work is modeling of distribution of rocket fuel components in the surface layer of the atmosphere and modeling of the dynamics of movement and transformation of aerosol cloud in the stratosphere formed in the process of draining of the first stage of the carrier rocket.

For description of processes of migration, diffusion and transformation of impurities they should be considered on the basis of a physically rich model accounting for daily course of changes of dispersion depending on meteorological fields, orographic, thermal heterogeneities of underlying surface, turbulent features of the atmosphere [1] and so on. In mathematical modeling impurities dispersion processes a very important stage is development and choice of the corresponding computational algorithm and approximation of equation of migration.

Mathematical model. Large-scale movements in the surface layer of the atmosphere are approximately described by a system of equations including motion equations, equations of continuity and equations of concentration. This model ensures computation of fields of velocities and concentration. An advanced turbulent spatial flow is considered [2,3]. The equations are the following:

                                                     (1)

                                                                                             (2)

where -                                                                                                 (3)

For modeling of distribution of rocket fuel components in the surface layer of the atmosphere the following equation was used:

                                                                     (4) where ui are velocity components; D diffusions factor; αT = vt/Pr

Smagorinsky dynamic model was used as a model of turbulence [4]. For application of the dynamic model double averaging was conducted with the filter length ∆ = 2∆ then

 

                                       (5)

Equation averaged for two times with two filters having length [∆ and] has correspondingly the following design:

 

                                                         (6)

where. From (4) and (5) it follows that  then  has the following:  and tensions

From formula (6) using least-squares method when value C in the form of  where

 

Boundary conditions. For the task of distribution of rocket fuel components in the surface layer of the atmosphere the following boundary conditions were set:

at the upper boundary of the air mass:

 

on the ground surface:

 at lateral boundaries:

 

For the task of dynamics and migration of aerosol cloud resulting from draining of the stage 1 of the carrier rocket in the stratosphere the boundary conditions are the following:

at the upper boundary of the air mass:

 

at the lower boundary:

at lateral boundaries:

 

Numerical algorithm. For solution of the task with the account of the proposed model a scheme of splitting on physical parameters is used [5]. At the first stage it is supposed that carrying over of movement amount takes place only due to convection and diffusion. Intermediate field of velocity is determined by rhythmic steps using double-sweep method. At the second stage the determined intermediate field of velocity is used for determination of the field of pressure. Poisson equation for the field of pressure is solved using Fourier method in combination with matrix double-sweep method used for determination of Fourier factors. At the third stage it is supposed that migration takes place only due to pressure gradient. Task algorithm is paralleled on a high-performance system [6,7]:

I)   

II)  

 III) 

Results of modeling. In Fig.1 contour lines are depicted denoting concentrations of toxic rocket fuel components (RFC) in the western wind after regular falling of stage 1 of “Proton-M” carrier rocket at different time moments.

 

Fig.1 contour lines are depicted denoting concentrations of toxic rocket fuel components (RFC) in the western wind after regular falling of stage 1 of “Proton-M” carrier rocket at different time moments

 

As is seen in the figures, the excitation caused by diffusion-convection flows, extends to the boundary of the computed area and reaches it in 12-15 hours. Results of modeling of migration of RFC in the surface layer of the atmosphere show that migration of heptyl and its spreading rate depend on the direction and force of the wind. The main contamination site is the area of falling of the carrier rocket’s stage and the adjacent 100-120 m wide and 150-180 m long ellipse-shaped territory with integral concentration of 0,25mg/m2. Carrying of heptyl by winds out of the territory does not exceed maximum permitted concentrations. Highly concentrated heptyl is found at the site of falling of the carrier rocket having integral concentration of 1,2 – 1,5mg/m2.

In the second task a launch of carrier rocket is modeled in the presence of tail wind with the velocity of 2 m/s. Calculations were executed within rectangular area with dimensions along both horizontal directions being 20km, and the altitude totaling 40-60 km. The results of the calculations are shown in fig.2 and 3. On fig.2 contour lines are depicted showing concentrations of RFC (view from above) after draining of stage 1 of “Proton-M” carrier rocket.

 

Fig.2 Contour lines showing concentrations of RFC (view from above) 1 hour (a) and 6 hours (á) after draining of stage 1 of “Proton-M” carrier rocket. Altitude - 50km, tail wind, wind velocity amounting to 2m/s.

 

On Fig.3 equiscalar surfaces are depicted showing concentrations of RFC after draining of stage 1 of “Proton-M” carrier rocket at the altitude of 50 km at different time periods. The results obtained in the course of solving task two make it noticeable that the concentration of combustion products is spread over a territory bigger than the dynamic excitation field. As the time passes, the dynamic field fades away, and concentration field changes its state to the state of passive impurity and keeps migrating in the stratosphere for a long period of time. It is also problematic to trace further migration of concentrations in the stratosphere after their carrying over from the computed area since it necessitates modeling in larger scales.

 

Fig.3 Equiscalar surfaces showing concentrations of RFC (view from above) 1 hour (a) and 6 hours (á) after draining of stage 1 of “Proton-M” carrier rocket. Altitude - 50km, tail wind, wind velocity amounting to 2m/s

 

Thus, basing on Navier-Stokes equations a mathematical model of RFC distribution was designed allowing for modeling the processes of migration either within the surface layer of the atmosphere or within stratosphere. The obtained results can be used for monitoring of ecological situations within the areas where carrier rockets’ falling occur, as well as for prediction of scopes of atmosphere contamination and assessment of ecological damage caused in relation to the environment.

 

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