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Three-dimensions advection-diffusion equation in unstable condition
Khaled S. M. Essa, * Sawsan E. M. Elsaid
Mathematics and Theoretical Physics, NRC, Atomic Energy Authority, Cairo- Egypt
*Email: mohamedksm56@yahoo.com
Abstract
The three-dimensional advection-diffusion equation is solved by using Laplace transform with constant eddy diffusivities coefficients along horizontal and crosswind coordinates, while vertical eddy diffusivity in unstable condition depends on vertical height and Monin- Obukhov length (L) at different effective heights for two stacks for different emission rate for Iodine-131 (131I ) with different runs. Comparison between the observed data are taken on Enshas, Cairo, Egypt and predicated shows that there are some predicated data which are agreement with observed data (one to one) and others lie inside the factor of two. Also the statistical one finds that the predicted concentrations are agreement with the observed concentrations.
Keywords: Analytical Solution; Advection-Diffusion Equation; Atmospheric Dispersion.
Introduction
Atmospheric dispersion modeling for the release of radioactive gases and volatiles is an important contribution for various stages in the nuclear technology safety criteria. These stages are: Licensing requirements for the selection of nuclear reactor site, normal operating conditions stage, and finally, accidental release in case of reactor accident (IAEA, Vienna, 1979). The potential damage from a nuclear power reactor (or even research reactor) is initiated in case of accident occurring, and the radioactive cloud diffuse and transport in the atmosphere and finally, the dramatic exposure of public to radiation. The area on the ground surface or in the atmosphere - which is affected by the radioactive release, is determined by various factors (Davidson Moreira and et. al, 2005). Amount of radiation released (source strength), wind direction and speed, weather conditions (particularly, atmospheric thermal stability states) and the physical characteristics of radioactive material released (half-life time and its deposition velocity). The advection-diffusion equation has long been used to describe the dispersion of contaminants in the atmosphere (Seinfeld and et al. 1998). Time dependent one-dimensional linear advection–diffusion equation with Dirichlet homogeneous boundary conditions and an initial sine function is solved analytically by separation of variables and numerically by the finite element method. It is observed that when the advection becomes dominant, the analytical solution becomes ill-behaved and harder to evaluate. Therefore another approach is designed where the solution is decomposed in a simple wave solution and a viscous perturbation. It is shown that an exponential layer builds up close to the downstream boundary. Discussion and comparison of both solutions are carried out extensively offering the numeric’s a new test model for the numerical integration of the Navier–Stokes equation, (Abdelkader Mojtabi and et al. 2015).
In this work, The three-dimensional advection-diffusion equation is solved by using Laplace transform with constant eddy diffusivities coefficients along horizontal, and crosswind while eddy diffusivity in unstable condition depends on vertical height and Morin- Obukhov length (L) at different effective heights for two stacks for different emission rate for Iodine-131 (131I ) with different runs. Comparison between the observed data are taken on Enshas, Cairo, Egyptand predicated shows that there are some predicated data which are agreement with observed data (one to one) and others lie inside the factor of two.
Mathematical solutions
The basic gradient transport model can be written (Hanna and et al. 1982):
where:
C is the average concentration of diffusing point (x, y, and z) (Bq/m3).
u is mean wind velocity along the x-axis (m/s).
Kx, ky and kz are the eddy diffusivities coefficients along x, y and z axes respectively (m2/s).
x is along –winds coordinate measured in wind direction from the source (m).
y is cross-wind coordinate direction (m).
z is vertical coordinate measured from the ground (m).
S is source/ sinks term (Bq/m3-s).
is time rate of change and advection of the cloud by the mean wind.
and represent turbulent diffusion of material relative to the center of the pollutant cloud. (The cloud will expand over time due to these terms).
S source term which represents net production (or destruction) of pollutant due to sources (or removal).
The mean wind components (u, v and w) and mean concentration (C) represent average over a time scale (Ta) and space scale (xa).
One simplified Eq. (1), one gets that:
(2)
Let (Shir and Shieh (1974))
where
K_(z )=(0.4 u_* Z)/(1-15Z/L)^(-1/4) ,
(K_z ) ́=(-1.5〖 u〗_* Z)/(L(1-(15 Z)/L)^0.75 )+0.4〖 u〗_* (1-(15 Z)/L)^.0.25 and (K_z^´)/( K_(z ) )=(4L-45Z)/(4Z(1-15L)) in unstable condition
L is the Monin-Obukhov length; L is positive for stable conditions, negative for unstable conditions and approaches infinity for neutral conditions (Hanna and et. al, (1982)) and u* is friction velocity..
Equation (2) is subjected to the following boundary condition
1-The flux at ground and the top of the mixing layer can be given by: -
at z= 0, h (i)
2-The mass continuity is written in the form:-
uC(x,y,z,t)=Qδ(z-h) at x=0 (ii)
3-The concentration of the pollutant tends to zero at large distance of the source, i.e.
C (x, y, z, t) =0 at x, y, z→∞, t→ 0 (iii)
4-The flux at crosswind direction can be given by: -
at y=0 ,Ls (iv)
where Ls is constant large value in crosswind direction.
5-The flux at horizontal direction can be given by: -
at x=0 (v)
Let the solution of the equation (2) in the form
C(x, y, z, t) = X (x) Y(y) Z (z) T (t) (3)
Substituting from equation (3) in equation (2) and dividing on X (x) Y(y) Z (z) T (t) and equal to λ2,one gets that:-
T^'/T(t) =(K_x X^″/(X(x))-u X^'/(X(x)))+(K_y Y^″/(Y(y)))+(K_z Z^″/(Z(z))+K_z^' Z^'/(Z(z)))=-λ^2 (4)
This equation are divided into four equations as follows
(I1)
(I2)
(I3)
(I4) Applying the Laplace transform on equation (I1) respect to t, one gets that:
where Lp is the operator of the Laplace transform.
(I11)
Applying the Laplace inverse transform on equation (I11) and Substituting from equation (ii), one gets that:
(I12)
Applying the Laplace transform on equation (I2) respect to x, we get that:
X ̃(p)=((p-u/K_x )X(0)+u/K_x X ́(0))/(p^2-(u/K_x ) p+λ^2/K_x )=Q/u δ(Z-h)[(p-u/〖2K〗_x )/((p-u/〖2K〗_x )^2+(λ^2/K_x +(u/〖2K〗_x )^2 ) )-(u/〖2K〗_x )/((p-u/〖2K〗_x )^2+(λ^2/K_x +(u/〖2K〗_x )^2 ) )]
(I21)
Applying the Laplace inverse transform on equation (I22) and Substituting from equations (ii) and (iv), one gets that:
X(x)=Q/u 〖e^(-ph) e〗^(u/〖2K〗_x )x [cos(√((λ^2/K_x +(u/〖2K〗_x )^2 ) ))x-(u sin(√(√((λ^2/K_x +(u/〖2K〗_x )^2 ) )) )x)/(〖2K〗_x √((λ^2/K_x +(u/〖2K〗_x )^2 ) ))] (I22)
where
L^(-1) ((p-b)/((p-b)^2+a^2 ))=e^bx cos(ax) and L^(-1) (1/((p-b)^2+a^2 ))=(e^bx sin(ax))/a
L^(-1) (δ(Z-h_s ))=e^(-ph)
Applying the Laplace transform on equation (I3) respect to y, one gets that:
(I31)
Applying the Laplace inverse transform on equation (I31) and Substituting from equations (ii) and (v), we get that:
(I32)
Differentiating this equation respect to y, one gets that:
Y^' (y)=(-λ)/√(K_y ) (sin(λ/√(K_y ))y )=0
"λ" /√(K_y ) L_s=nπ→"λ=" nπ/L_s √(K_y )
where:
L^(-1) (q/(q^2+a^2 ))=cos(ay)
Applying the Laplace transform on equation (I4) respect to z, we get that:
(Z ) ̃(β)=((β+(K_z^')/K_z ) Z(0)+Z^' (0))/(β+(K_z^')/K_z β+λ^2/K_z )=Z(0)[((β-((〖-K〗_z^')/〖2K〗_z )))/((β-((〖-K〗_z^')/〖2K〗_z ))^2+(λ^2/K_z -(K_z^'2)/(4 K_z^2 )) )+((K_z^')/〖2K〗_z )/((β-((〖-K〗_z^')/〖2K〗_z ))^2+(λ^2/K_z -(K_z^'2)/(4 K_z^2 )) )]+(Z^' (0))/((β-((〖-K〗_z^')/〖2K〗_z ))^2+(λ^2/K_z -(K_z^'2)/(4 K_z^2 )) ) (I41)
Put:
Applying the Laplace inverse transform on equation (I41) and Substituting from equations (ii) and (i), one gets that:
Z(z)=e^(((〖-K〗_z^')/〖2K〗_z ) z) [cos(√((λ^2/K_z -(K_z^'2)/(4 K_z^2 )) ))z+(K_(z )^' sin(√((λ^2/K_z -(K_z^'2)/(4 K_z^2 )) ))z)/(〖2K〗_z √((λ^2/K_z -(K_z^'2)/(4 K_z^2 )) ))] (I42)
Substituting from equations (I12), (I22), (I32) and (I42) in equation (3) one gets the general solution as follows
C(x, y, z, t) = Q/u (cos(nπ/L_s )y) e^(-(nπ/L_s √(K_y ))^2 t) e^(-(h-u/〖2K〗_x )x) e^(((〖-K〗_z^')/〖2K〗_z ) z) [cos(√(((nπ/L_s √(K_y ))^2/K_z -(K_z^'2)/(4 K_z^2 )) ))z-(K_(z )^' sin(√(((nπ/L_s √(K_y ))^2/K_z -(K_z^'2)/(4 K_z^2 )) ))z)/(〖2K〗_z √(((nπ/L_s √(K_y ))^2/K_z -(K_z^'2)/(4 K_z^2 )) ))][cos(√(((nπ/L_s √(K_y ))^2/K_x +(u/〖2K〗_x )^2 ) ))x-(u sin(√(√(((nπ/L_s √(K_y ))^2/K_x +(u/〖2K〗_x )^2 ) )) )x)/(〖2K〗_x √(((nπ/L_s √(K_y ))^2/K_x +(u/〖2K〗_x )^2 ) ))] (4)
Results and Discussion
Table (1): shows that the meteorological data (downwind distance ‘x’, wind speed ‘u’, stability classes and different effective heights for two stacks for different emission rate for 131I (AEA, 2006). Comparison between the predicated and observed concentration different downwind distance for the different runs (Ledina and et al. 2011). Fig (1) shows that the Comparison between the predicated and observed concentration. Fig (2) shows that the Comparison between downwind distance and Concentration. Fig (3) shows that the comparison between concentration and effective height. From the figures one finds that there are some predicated data which are agreement with observed data (one to one) and others lie inside the factor of two.
Table (1) Meteorological data (downwind distance ‘x’, wind speed ‘u’, stability classes and different effective heights for two stacks for different emission rate for 131I) (AEA,2006), observed and predicated concentration.
Experiment Stability
classes Distance
‘x’(m)
U(m/s) L H
(m) Concentration (Bq /m3)
Observed present predicted
1 A 92 4 -198 49 0.025 0.069
2 A 96 4 -198 48 0.037 0.044
3 B 97 6 -192 45 0.090 0.069
4 C 98 4 -192 46 0.200 0.240
5 A 99 4 -112 45 0.270 0.251
6 D 100 4 -112 45 0.190 0.171
7 E 115 4 -112 47 0.450 0.373
8 C 132 4 -82 46 0.120 0.176
9 A 134 4 -82 47 0.030 0.080
10 D 165 3 -82 28 0.420 0.185
11 B 184 2 -130 28.3 0.420 0.291
12 A 200 3 -130 30.8 0.670 0.399
13 A 300 3 -130 30.6 0.670 0.110
Fig (1) comparison between observed and predicated concentrations
Fig (2) comparison between concentrations and downwind distance
Fig (3) comparison between concentrations and effective height
Statistical method
Now, the statistical method is presented and comparison among analytical, statically and ob¬served results will be made (Hanna 1989).The fol¬lowing standard statistical performance measures characterizes the agreement between model predic-tion
(Cp=C pred / Q) and observations (Co=Cobs/Q):
where σp and σo are the standard deviations of Cp and Co respectively. The subscript o and p refer to observed and predicted quantities, respectively. Here the over bars indicate the average over all measurements (Nm). The statistical index FB says if the predicted quantities underestimate or overestimate the observed ones. The statistical index NMSE represents the model values dispersion in respect dispersion. The results are expected to have values near to zero for the NMSE and FB, and near to one in indices COR and FA2.
A perfect model would have the following idealized performance:
NMSE = FB = 0 and COR = FAC2 = 1.0.
According to standard statistical performance measure in this work the value of NMSE is 0.14, FB is 0.19, COR is 0.97 and FAC2 is1.09.
From the statistical one finds that the predicted concentrations are agreement with the observed concentrations.
Conclusion
The three-dimensional advection-diffusion equation is solved by using Laplace transform with constant eddy diffusivities coefficients along horizontal, and crosswind coordinates, while eddy diffusivity in unstable condition depends on vertical height and Monin- Obukhov length (L) at different effective heights for two stacks for different emission rate for Iodine (131I ) with different runs. One finds that the predicated data are agreements (one to one) and others lie inside the factor of two with observed data which are taken on Enshas, Cairo, Egypt. Also the statistical one finds that the predicted concentrations are agreement with the observed concentrations.
Reference
Abdelkader Mojtabi, Michel O. Deville (2015). “One-dimensional linear advection–diffusion equation: Analytical and finite element solutions. Computer& Fluids 107.189-195.
AEA (2006).”Safety analysis report”.0797-5325-3IBLI-001-10: ETRR 2.AEA.Egypt.
Davidson Moreira M., Tiziano Tirabassi, Marco Vilhena, T., Jonas Carvalho, C. A. (2005). “Semi- analytical model for the tritium dispersion simulation in the PBL from the Angra I nuclear power plant, Ecological Modeling, 189.
Hanna Steven R., Gary A., Briggs and Rayford P. Hosker. Jr. (1982). “Handbook on atmospheric diffusion”. Technical Information Center U.S. Department of Energy.
Hanna, S. R., (1989). “Confidence limit for air quality models as estimated by bootstrap and Jackknife resembling methods”, Atom. Environ. 23,1385-1395.
Ledina Lentz Pereira, Camilla Pinto da Costa, Marco Tullio Vilhena. Tiziano Tirabassi (2011). “Puff models for simulation of fugitive hazardous emissions in atmosphere”. Journal of Environment, protection, 2,154-161.
IAEA, Vienna, (1979). ”Information to be submitted in support of licensing application for nuclear power plants”, A safety guide, technical report series No. 50-SG-G2.
Shir and Shieh (1974).”Prediction and Regulation of Air Pollution". Atmospheric Sciences library.
Seinfeld J. H. and Pandis S. N. (1998), “Atmospheric chemistry and physics,” John Wiley & Sons, New York.

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Nuclear research

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