Diffusion of Radioactive Materials in the Atmosphere

American Journal of Environmental Sciences (): 3-7, 009 ISSN 3-34X 009 Science Publications Diffusion of Radioactive Materials in the Atmosphere Abdul-Wali Ajlouni, Manal Abdelsalam, Hussam Al-Rabai ah and Abdullah M.S. Ajlouni Department of Applied Phsics, Tafila Technical Universit, Tafila, Jordan Department of Science, Husun Technical College, Amman, Jordan Abstract: Problem statement: The dispersion of radioactive materials in the environment related to escaping of noble gases, halogens and aerosols of non-volatile radioactive materials, from the reactor containment during normal operations, or in the event of a sever reactor accident. Approach: radionuclide dispersion in the environment is demonstrated b mathematical tools which are the partial differential equations, mainl the diffusion equation. A mathematical model to calculate the concentration of nuclear pollutants (radioactivit) with certain boundar conditions is constructed. Results: Solving the mathematical model and using some approimations lead to a distribution represents a model for plume of radioactive pollutants dispersed in two dimensions normal to the wind direction in which the plume moves as an entire non-dispersible unit. Conclusion: The obtained result theoreticall are ver close to those achieved eperimentall. Ke words: Radioactive materials, diffusion equation, mathematical modeling, Jordan atmosphere INTRODUCTION The artificial creation of radionuclides ma result from phsical processes involving nuclear fission, nuclear fusion and neutron activation. The most important source of artificiall created radionuclides is neutron-induced nuclear fission. The chemical and phsical forms of the active species determine deposition, migration and uptake are radioactivit b living organisms. A variet of sstems and processes ma introduce radioactivit into the environment. Human activities involving nuclear weapons and the nuclear fuel ccle (including mining, milling, fuel enrichment and fabrication, reactor operation, spent fuel storage and reprocessing and waste storage), leading to significant creation and release of radioactivit. Human technolog also releases pre-eisting natural radionuclides, which would otherwise remain trapped in the earth s crust []. The phsical and chemical form of radionuclides ma var depending on the release and transport conditions in addition to the element properties. A general distinction can be made between gases, aerosols and particulate material. The most serious dispersion of radioactive materials in the environment is that related to escaping of noble gases, halogens and aerosols of non-volatile radioactive materials, from the reactor containment in the event of a sever reactor accident []. In this stud we tr to make a mathematical simulation of radionuclide dispersion in the environment b matching the mathematical tools which are the partial differential equations, mainl the diffusion equation and the technical data of the nuclear reactors. This simulation ma help in determination of radiation dose ma received b the public during a sever reactor accident. A demonstration, with a computer program, reflecting the Jordanian atmosphere, will be carried out; the results from this demonstration will be compared with real data taken from the field. The stud is arranged as follows. First, we introduce the main concepts of the Partial Differential Equations (PDEs) and their applications. Then, we introduce the main features of transport phenomena with emphasis on the diffusion equation and its application in dispersion of radioactive materials as atmospheric pollution. Finall, arithmetic calculations related to Jordan atmosphere is carried out. MATERIALS AND METHODS Partial differential equations: A PDE is an equation that contains partial derivatives, in which the unknown function depends on several variables, e.g., temperature depends both on location and time t. The variables and t are called independent variables, whereas the unknown variable which we differentiate, e.g., temperature, is called dependent variable [-4]. Most phsical phenomena, whether in the domain of fluid dnamics, electricit, magnetism, mechanics, optics, or heat flow, can be described in general b Corresponding Author: Abdul-Wali Ajlouni, Department of Applied Phsics, Tafila Technical Universit, Tafila, Jordan 3

Partial Differential Equations (PDEs). Simplifications can be made to reduce the equations in question to ordinar differential equations, but, nevertheless, the complete description of the sstems resides in the general area of PDEs. Most of the natural laws of phsics, such as Mawell's equations, Newton's laws of motion and Schrödinger equation, are stated, or can be, in terms of PDEs, that is, these laws describe phsical phenomena b relating space and time derivatives. Derivatives occur in these equations because the derivatives represent natural things, like velocit, acceleration, force, friction and current. Hence, we have equations relating partial derivatives of some unknown quantit that we would like to find [-4]. The most important PDEs in phsical sciences are Equation of continuit: For a fluid of densit ρ(r, t), which has an element velocit v(r, t) at the position r at the time t. The rate of increase of mass per unit volume equals the net rate of mass addition per unit volume b convection, or: ρ + ( ρ v) = 0 Am. J. Environ. Sci., (): 3-7, 009 () Equation of diffusion or heat flow: The basic process in the diffusion phenomenon is the flow of the fluid from a region of higher densit to one of lower densit. The flow vector J = ρv can be represented as: J a = ρ () where, a is the diffusion constant of the medium. With the use of Eq. and, we get the equation of diffusion: ρ ρ = a 0 Denoting the temperature function b u(r, t) (3), the flow of heat from regions of higher temperature to those of lower temperature ma be described b the equation of heat flow: u = a u 0 (4) where, a = K, with K, σ and p are the thermal σρ conductivit, the specific heat and densit of the material, through which the heat flows, respectivel. Schrödinger equation for a free particle: With a simple division, the Schrödinger equation for a single particle of mass m in the absence of an applied force field can be rewritten in the following wa: 4 Ψ i Ψ = m 0 Where: I = The unit imaginar number, = Planck's constant divided b and Ψ = The wave function of the particle () This equation is a mathematical analogue of the particle diffusion equation, which one obtains through eas transformation [-4]. In addition to the above mentioned equations, there are man equations like the vibrating string equation, Laplacee's equation, longitudinal vibrations of a beam, transverse vibrations of a beam and man other PDEs that represent phsical phenomena. To solve a PDE there are man methods, the most important of them are those that change PDEs into Ordinar Differential Equations (ODE). The method of separation of variables which reduces a PDE in n variables to n ODEs, almost is the first and famous technique to solve a PDE. Other methods are integral transforms, change of coordinates, transformation of dependent variables, numerical methods, perturbation methods, integral equations, calculus of variations methods and eigenfunction epansion [-4]. All phsical problems have boundaries of some kind, called boundar conditions, so we must describe mathematicall what goes on there in order to adequate the problem. These problems must start from some value of time, generall assumed t = 0, so we must specif the phsical apparatus at this time. B writing the PDE which represents the phsical problem and the equations of boundar and initial conditions, we have what is called an Initial-Boundar-Value Problem (IBVP). Transport phenomena: The subject of transport phenomena includes three closel related topics: fluid dnamics, heat transfer and mass transfer. Fluid dnamics involves the transport of momentum, heat transfer deals with the transport of energ and mass transfer is concerned with the transport of mass of various chemical species. These three transport phenomena should be studied together for the following reasons: The frequentl occur simultaneousl in industrial, biological, agricultural and meteorological problems The basic equations that describe the three transport phenomena are closel related. This

Am. J. Environ. Sci., (): 3-7, 009 similarit of the equations under simple conditions is the basis for solving problems "b analog" The mathematical tools needed for describing these phenomena are ver similar The molecular mechanisms underling the various transport phenomena are ver closel related. All materials are made up of molecules and the same molecular motion and interactions are responsible for viscosit, thermal conductivit and diffusion The description of our phsical problem requires three tpes of equations: N N N N = K + K + K Where: N(,,, t) (7) = The concentration of the nuclide in Becquerel per cubic meter K, K and K = The diffusion coefficients in the, and directions, respectivel The initial conditions could be written as: Ν (,,,t = o) = Qδ( ) δ() δ ( ) (8) The PDE describing the phsical phenomenon of diffusion The boundar conditions describing the phsical nature of our problems on the boundaries The initial conditions describing the phsical phenomenon at the start of the eperiment [] Diffusion equation: the basic equation of onedimensional diffusion is the relationship: ρ ρ a = 0 Which relates the quantities: ρ ρ (6) = The rate of change in densit with respect to time measured in particle per unit of volume per second = The concavit of the densit profile p(, t), which essentiall compares the densit at one point to the densit at neighboring points. Eq. 6 is etended to Eq. 3 in three dimensions [-] When nuclear pollutants released from the nuclear reactor to the atmosphere, atmospheric edd currents causes a turbulent diffusion, which makes a pure mathematical description of the phenomenon is not completel epressed b the diffusion equation. Hence, approimations should be made and predictions of some semi-empirical models like the Gaussian plume model. Or, to treat the problem as an ideal situation and then tr to introduce other parameters that affect the results as what we are going to do here. The atmospheric concentration of a nuclide is epressed b the diffusion equation: where, we have assumed that the instantaneous release of Q Becquerel (Bq) occurs at = = = t = 0 and δ are the Dirac delta functions. The δ ( ), δ () and ( ) boundar conditions are Ν (,,, t) = o and: Ν Ν Ν = = = o as,, (9) Using the method of separation of variables, Eq. 7 will have the three dimensional spatial solutions as: N( ) = e πσ ( ) N and = N() = π σ π σ σ e e σ σ (0) () () It is eas to find that, the standard deviation which represents the dispersion coefficient is: σ = Kt (3) This leads to the general solution: 4k 4k t t 4kt e e e Ν (,,,t) = Q 4 π k t 4 π k t 4 π k t (4) This solution could be modified for insertion of wind speed, u and the height of the release point from

Am. J. Environ. Sci., (): 3-7, 009 the ground surface h, which changes the main spatial variables as follows: = + ut = + h = () Substitution of these new variables in Eq. 4 ields: ( ut) ( ) ( h) 4kt 4k t 4kt) e e e Ν (,,,t) = Q 4 π k t 4 π k t 4 π k t RESULTS (6) B assuming the speed wind is highl effective than self-diffusion. Eq. 6 is reduced to: (h ) ( ) σ σ Q Ν (,,,t) = e δ( t) πuσ σ v (7) which describes a plume of polluting radionuclide moving in the direction as an entire unit, while spreading b diffusion in the two other directions. If the source is continuousl releases pollutant radionuclide, Q (t), the concentration on the surface of Earth, = 0, is given b: h + σ σ Q (t) Ν (,,t) = e δ t πu σ σ v (8) with an accumulative, time-independent, pollution concentration: h + σ σ Q N(, ) = e (9) π u σ σ B differentiating Eq. 9, the maimum surface concentration could be obtained if the release point height is: A. Etremel unstable conditions B. Moderatel unstable conditions C. Slightl unstable conditions D. Neutral conditions E. Slightl stable conditions F. Moderatel stable conditions σ, Horiontal dispersion coefficient (m) σ, Horiontal dispersion coefficient (m) 0 4 0 3 0 0 A-Etremel unstable B-Moderatel unstable C-Slightl unstable D- Neutral E-Slightl stable F- Moderatel stable 4 0 º 0 0 3 0 4 0 3 0 3 0 3 0 0 Distance from source (m) A-Etremel unstable B-Moderatel unstable C-Slightl unstable D- Neutral E-Slightl stable F- Moderatel stable 0 0 Distance from source (m) 0 0 3 0 4 0 h = σ (0) In reference to Eq. 9, it is noticed that the surface pollutant concentration depends inversel on σ and σ. So, in order to get surface pollutant concentration at an point (, ), we should get σ and σ using the Fig.. In Fig., there are si tpes of weather conditions [6] : 6 Fig. : Horiontal dispersion coefficient σ (left) and vertical dispersion coefficient σ (right), versus downwind distance from source for different turbulence categories [6] In the last 30 ears, the average wind speed in Jordan was about m s, hence the best situation to

Am. J. Environ. Sci., (): 3-7, 009 describe Jordan weather conditions is C-Slightl unstable conditions. DISCUSSION In Jordan, there are no nuclear reactors, but, if an nuclear pollution occurs, it must come out of boarders, where the nearest nuclear reactor is Dimona 6 MW rector located in the southern part of Israel. B choosing Cs 37 nuclide which has a long half life, about 30 ears, so the calculations will not be affected b the radioactive deca of this radionuclide. Part of Cs 37 produced inside the reactor will be ordinaril released at a rate of 3.X0 6 Becquerel's (Bq) per MW per Year, thus for a period of ears, 4.X0 8 Bq are assumed to be released from the 6 MW reactor. Using the aforementioned numbers in application of Eq. 9 in calculating surface concentration of Cs 37 nuclide at a point located at km east of the reactor, with km width, =, leads to: (,) 36Bq / m Ν = () If the width is to be more than km, >, the surface concentration is: Ν (, ) = Bq / m () To give a general application o Eq. 9, a computer C++ program is introduced below. Results are represented in Table. Results in the Table showed that the surface concentration is increasing as we go from 00 m-4000 m and then start increasing. The maimum surface concentration is transformed from the distance h, as represented in Eq., to 4000 m due to effects of wind speed. This will make the pollutant distribution go far from the normal distribution. Also, the increase pattern of concentration on the left side of the line, that represent the maimum, is faster than the decrease pattern on the right side of the line, i.e., the distribution curve is not smmetric about the line 4000 m. CONCLUSION Using the diffusion equation, a mathematical model to calculate surface concentration of nuclear pollutants is constructed. Solving the equation b separation of variables and using some approimations leaded to a distribution represents a model for plume of radioactive pollutants dispersed in two directions normal to the wind direction in which the plume moves as an entire non-dispersible unit. Atmospheric edd currents causes a turbulent diffusion, which makes the surface concentration results go far awa from those of the pure mathematical description of the phenomenon as epressed b the diffusion equation. The result obtained here are ver near to those achieved eperimentall. REFERENCES. Fredrick Warner and Ro Harrison, 993. Radioecolog after Chernobl: Biogeochemical Pathwas of Artificial Radionuclides. st Edn., John Wile and Sons, Chichester, UK., ISBN: 0-47-9368-3, pp: 367.. Farlow, S.J., 98. Partial Differential Equations for Scientists and Engineers, st Edn., John Wile and Sons, ISBN: 0: 04866760X, pp: 44. 3. Dass, T. and Sharma, S., 998. Mathematical Methods in Classical and Quantum Phsics. st Edn., Universit Press, Haderabad, India, ISBN: 978-8-737-089-6, pp: 76. 4. Arfken, G., 98. Mathematical Methods for Phsicists. 3rd Edn., Academic Press, Orlando, Florida, ISBN: 00980, pp: 98.. Bird, R.B., W.E. Stewart and E.N. Lightfoot, 00. Transport Phenomena. nd Edn., John Wile and Sons, New York, USA., ISBN: 0474077, pp: 338. 6. Cember, H., 996. Introduction to Health Phsics. 3rd Edn., McGraw-Hill, New York, USA., ISBN: 0: 0070468, pp: 73. Table : General application: Result of C++ program X Y SY SZ Result/(Bq m ) 00 0 9 4 4.00 3 000 0 0 3 3.9 000 0 00 37.00 4000 0 90 4 8.3 8000 0 340 78 70.00 400 0 0 60.0 Q = 4 0 7 Bq sec. u = 4 m sec. h = 7 7