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Different solutions of the diffusion equation and its applications
BeniSuef University Journal of Basic and Applied Sciences volume 10, Article number: 82 (2021)
Abstract
In this report, we solved the advection–diffusion equation under pollutants deposition on the ground surface, taking wind speed and vertical diffusion depend on the vertical height. Also, we estimated a simple diffusion model from point source in an urban atmosphere and the conservative material with downwind was evaluated. Then, we calculated the extreme groundlevel concentration as a function of stack height and plume rise in two cases. Comparison between the proposed models and the emission from the Egyptian Atomic Research Reactor at Inshas had been done. Lastly, we discussed the results in this report.
Background
The Gaussian plume equation was semianalytical solution, assuming that wind velocity and eddy diffusivities were constant. The nonGaussian models agree well with the observed data by Hinrichsen [17]. Wortmann et al. [35] and Essa et al. [10] calculated a new technique to get the concentration of contaminants in the planetary boundary layer.
Early solutions used only two dimensions [26, 28, 34]. The Green's function method was studied by Stakgold [30]. Yeh [37] calculated advection–diffusion equation in three dimensions under boundary conditions. Also, these equations have been studied in turbulent models for unbounded domain by [18, 38]. Also, [33, 32] utilized dispersion modeling. All techniques have been restricted to a single isolated point source located at the origin. The details of several solutions were investigated by Carslaw and Jaeger [7], Sutton [31], Yaglom [36], Pasquill [24], Berlyand [4] and Lin and Hildemann [21]. Also, the diffusion from a point source in an urban atmosphere was estimated by Essa and ElOtaify [9]. Heines and Peters [16], and Bennett [3] investigated the physical effect of model for the dry deposition of pollutants to a rough surface. Recently, Essa et al. [11, 12] estimated the solution of the advection–diffusion equation in two dimensions with variable vertical eddy diffusivity and wind speed using Hankel transform. Also, Essa et al. [11, 12] found the solution of advection–diffusion equation in three dimensions using Hankel transform.
In the first part: Pasquill and Smith [22, 23] introduced the advection–diffusion equation in steady state for a continuous point source as follows:
where u(z) and K(z) are the mean horizontal wind velocity and vertical eddy diffusivity as a function of vertical height. Assuming that u(z) = 0 at z = 0, as defined by [29] as follows:
where φ_{m}(z/L) is the nondimension wind gradient, L is the Monin–Obukhov length, u_{*} is the friction velocity, and k is the vonKarman's constant equals 0.4. One can get by integrating Eq. (2) w.r.t (z) the following velocity profiles [9]:
The vertical diffusion coefficient was taken from Hanna et al. [13, 14]as follows:
where K_{o} is a constant value. Equation (1) is solved using the boundary conditions as follows:

(1)The mixing layer was assumed not allowing fluid to pass through to the pollutants
$$C\left( {x, \, z} \right) = 0, \quad {\text{at}} \quad z = h$$(9a)And
$$\frac{{\partial {\kern 1pt} \,{C}\left( {{x,z}} \right)}}{{\partial {\kern 1pt} {z}}} = {0}\quad{at}\;z = h$$(9b)where h is the height of the planetary boundary layer (PBL).

(2)Assuming that there is a deposition at the ground surface as follows:
$${K}\left( {z} \right)\;\frac{{\partial {\kern 1pt} {\kern 1pt} {C}\left( {{x,z}} \right)}}{{\partial {\kern 1pt} {z}}} = \,{v}_{{d}} \;{C}\left( {{x,z}} \right)\quad{at}\;z = 0$$(10)where v_{d} is the deposition velocity.
The solution of Eq. (1) is solved theoretical under different stabilities [10] as follows:
Integrating Eq. (1) from zero to h with respect to z, and applying Eqs. (9b) and (10), one gets:
For neutral conditions
Substituting C(x, z) and u (z) by Eqs. (11) and (3) in Eq. (12), one obtains:
Equation (13) becomes:
where
Equation (14) becomes:
where F_{0} is a constant. Then, the solution in the neutral case was obtained as follows:
Taking
where \({x}_{{{dn}}}\) is the decay distance of pollutant in neutral condition. Equation (17) becomes:
Now, F_{0} can be determined by using the formula:
where Q is the emission rate which after substitution of Eqs. (3) and (19), one gets:
Then, the concentration of pollutants in neutral condition becomes:
Similarly, C(x, z) in stable condition takes form [10]:
where X_{ds} is given in the stable case as follows:
where M = N – (5.2 h^{2}/12L).
Also, C(x,z) in unstable condition is given [10] as follows:
where X_{du} is given in unstable case as follows:
where D is taken from [10].
In the second part: We assumed twodimensional structure with homogeneity in the lateral coordinate, where ū(z) is described in Fig. 1. The ground surface is treated with no deposition of matter exist.
The conservation of mass can be taken from [9]:
where
\(\overline{u}(z)\) is the average wind velocity, C(z) is the concentration of pollutants, and H is the effective stack height of the plume.
The Effective stack Height (H)
We estimated the height of the plume as follows:
where w_{o} is the exit velocity of the pollutants (m/s) and D_{1} is the inside stack diameter (m). Then, H equals:
The concentration profile will be assumed in form [9]:
where C_{o} is the concentration value at the edge of the plume. C(z) is the concentration at vertical height; α_{1} and α_{2}, etc., are constants.
The fitting in Fig. 2 is obtained by taking the first two terms as follows:
If α_{1} is a r percentage of the concentration as follows:
if r = 0, then
After substituting, Eq. (26) becomes using different stabilities as follows:
Neutral case
The integration over "z" yields:
Stable case
After integration, C_{o} takes the form:
Unstable case
where f(z) stands for: \(f(z) = \frac{{16(z + z_{0} )}}{L}\).
By substituting from the above equation into Eq. (35) and integrating w.r.t “z,” one gets:
where \(\Omega\) was taking from [9]:
The time dosage integral of the concentration “D” is defined as:
The mass balance condition takes the form:
Substituting from Eq. (36) in Eq. (37), one gets:
The second research reactor in E.A.E.A., (ETRR2) has stack height H_{1} equals 27 m which is a point source of (I^{131}), (H) equals 31.29 m, (Q) equals 35 Bq, the wind velocity (u) equals 2.8 m/s, and the lapse rate (ΔT/ΔZ) (^{o}c/100 m) is 0.36, which is a stable case. Using Eq. (34), one gets (C_{o}) equals 1.208 Bq/m^{3}. Then, the concentration at ground level becomes:
The observed concentration at x = 300 m and H = 31.29 m was 0.16 Bq/m^{3}. The corrected source strength becomes:
Then, the corrected C (ground) = 0.18 Bq/m^{3}.
In the third part: The maximum groundlevel concentration with two cases was estimated [9]. The concentration at the ground reaches a maximum and thereafter decreases as given by Hans et al.[15] with the formula:
σ_{i} (i = y, z): are dispersion parameters of concentration in the lateral and vertical directions (m).
The extreme concentration at ground occurs along the plume centerline (y = 0) at ground (z = 0) [25]).
σ_{y} and σ_{z} can be estimated as follows:
where a, b, c, d values are taken from Curtiss [8].
The groundlevel concentration along centerline was estimated from Eq. (40) taking σ_{y} and σ_{z} from Eq. (41) one gets:
The maximum concentration value can be obtained after solving the following equation:
The maximum concentration and effective height were obtained as follows:
First: the effective height as a function of x
A—Unstable or neutral conditions:
The buoyancy flux parameter, F_{b,} was written [5] as follows:
For (h_{s} < 305 m), g is the acceleration (ms^{−2}), v_{s} is the exit vertical speed (ms^{−1}), r_{s} is the exit radius (m), T_{s} is the gas exit temperature (K), and T_{a} is the ambient temperature (K) at h_{s} [27].
The critical x* is given by:
For x ≤ x*, we have:
Taking the constant equals 1.6 [6], one gets:
This equation is used if T_{s} > T_{a} (2/3 law) [27].
u_{h} can be calculated at 10 m as follows:
where the parameter p is given from [20].
Substituting from (47) into (43), one gets:
where H* is the maximum value.
In unstable condition, taking d = 1.17 ([8] H* becomes:
Then,
The maximum value for \(\chi\)* has the form:
In neutral condition, taking d = 0.95 [8] \(\chi\)* becomes:
In stable stability, there are two methods for maximum concentration at z = 0 as follows:
(First)—In stable, where d = 0.67 [8] the maximum value for \(\chi\)* is:
(Second)—In stable (E and F), the stability parameter is written as follows:
where \(\left( {\frac{\Delta \theta }{{\Delta Z}}} \right)\) = 0.02 K/m for E and \(\left( {\frac{\Delta \theta }{{\Delta Z}}} \right)\) = 0.035 K/m for F ([27].
\(\Delta h\) is written as follows:
H* becomes:
Also, χ* has the form:
Second case: H is a constant of downwind distance:
If H is constant, then,\(\frac{\partial H}{{\partial x}} = 0\), assuming H_{c} = h_{s} + 3w_{o}D/u, where H_{c} is constant. H* becomes:
The maximum value for \(\chi\)* has the form:
First case study
The decay distance and the concentration of pollutant at Inshas, Nuclear Research Center, Egyptian Atomic Energy Authority, were calculating. The stability was found in stable condition by using these data. The value of L was taken from Gifford work [13]. One uses Eq. (4) to calculate the values of u_{*} in terms of the wind velocity u(z) at 10 m height; hence, the values of the mixing height could be estimated from [2] for stable condition:
where f is the Coriolis parameter. v_{d} equals 0.04 m/s for Iodine [19]. Hence, the decay distance of iodine has been estimated using Eq. (23). The friction velocity and the mixing height with downwind distance are illustrated in Fig. 3. One gets the change of the friction velocity and mixing height from 0.25 to 0.3 m/s and 180 to 190 m, respectively [10].
The values of mixing height proportional with the decay distance are shown in Fig. 4; the normalized concentration (C/Q) of iodine at heights z = 0, 10, 50 m as a function of downwind distance has been evaluated by Eq. (22). One finds that the maximum normalized groundlevel concentration equals 18*10^{–6} at 100 m and then decreases to reach near minimum value at 5000 m as shown in Fig. 5.
Second case study
C_{o}/Q is applied on the first research reactor at Inshas, Nuclear Research Center, Egyptian Atomic Energy Authority. h_{s} equals 43 m, D equals 1 m, w_{o} equals 4 m/s, and the total ventilation equals 39,965 m^{3}/hr. Also, one takes α = 1. Figure 6 shows that a straightline fit well to these data in three conditions between Co/Q and H.
Third case study
One gets the maximum concentration on the second research reactor at Inshas, Nuclear Research Center, Egyptian Atomic Energy Authority. h_{s} equals 27 m, D equals 1 m, w_{o} equals 0.3 m/s, and the total ventilation equals 3400 m^{3}/hr [1]. One finds that the maximum concentrations are 6.2*10^{–4}, 4.8* 10^{–4} and 2*10^{–4} Bq/m^{3} at 40 m in unstable, neutral and stable conditions, respectively; when plume rise depends on downwind distance as shown in Fig. 7, these values are less than the maximum system activity (MSA).
One finds that the effective height is quadratic with downwind distance in unstable case and the correlation coefficient (R) equals 0.99, (x_{max}) equals 1000 m, and (H_{max}) reaches 978 m. In neutral case, the curve is still quadratic and smaller than that in the unstable case, R = 1. Also, in the stable case, the relation is linear, R = 0.984, x_{max} reaches to maximum value, and H_{max} will reach 59 m as shown in Fig. 8.
We find that the maximum concentrations are 2.5* 10^{–4}, 2.1* 10^{–4} and 2* 10^{–4} Bq/m^{3} in neutral, unstable and stable conditions, respectively, at 40 m when H is a constant as shown in Fig. 9. Also, the relation between "H" and "x" is linear in unstable condition, H_{max} = 163 m when x_{max} = 1000 m and R = 0.997. In neutral case, the relation is quadratic and R = 1 and H_{max} = 83 m at x_{max} = 1000 m. Also, in stable case the relation is still quadratic and R = 0.998, H_{max} = 58.5 m at x_{max} = 1000 m as shown in Fig. 10. We must take H that depends on x because one gets the values of "H" and the concentration near the true. This is the best result to reach it.
Conclusions
The concentration released from an elevated point source in the presence of both deposition and elevated mixed layer is estimated. The decay distance in the downwind distance is calculated. The maximum normalized concentration equals 18*10^{–6} (s/m^{3}) at 100 m and then decreases to reach minimum value at 5000 m at the ground level.
The logarithmic law of wind velocity in different conditions is used to get the plume rise, effective stack height and the normalized concentration at the axis of the plume from the reactor release through different stability classes. We find that the groundlevel concentration of iodine (I^{131}) agrees with the observed concentrations.
The values of extreme concentration are 6.2*10^{–4}, 4.8* 10^{–4} and 2*10^{–4} Bq/m^{3} at 40 m in unstable, neutral and stable conditions, respectively; when the plume rise depends on downwind distance and then decreases to values less than MSA, after that the values are close to zero at 400 m from the stack. In stable case, the maximum values of the groundlevel concentration of air pollutants were similar in the two cases. One obtains a good model when the plume rise depends on downwind distance and it is the best result.
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Abbreviations
 PBL:

Planetary boundary layer is (PBL).
 \({x}_{{{dn}}}\) :

Decay distance of pollutant in neutral condition.
 X_{ds} :

Decay distance of pollutant in the stable condition.
 X_{du} :

Decay distance of pollutant in the unstable condition.
 D:

Time dosage integral of the concentration.
 H:

Effective stack height of the plume.
 C_{o} :

Concentration value at the edge of the plume.
 MSA:

Maximum system activity
References
AEA (2003) 0767–5325–3IBLI001–10: ETRR2, Safety Analysis Report. AEA, Egypt
Arya SP (1999) Air pollution meteorology and dispersion. OxfordUniversity Press, New York, p P98
Bennett M (1988) Atmos Environ 22(12):2701–2705
Berly MY (1975) Contemporary problems of atmospheric diffusion and pollution of the atmosphere. Gidrometeoizdat, Leningrad, 444 pp, translated into English by NERC, USEPA Raleigh, N.C.
Briggs GA (1969) Plume rise. USAEC Critical Review Series, TID 25075 National Technical Information Service, Springfield, Virginia
Briggs GA (1971) Some recent analysis of plume rise observations. In: Proceedings of the 2nd international clean air congress, England. Academic Press, New York
Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Oxford University Press, London, p 510
Curtiss PS, Rabl A (1996) Impacts of air pollution: general relationships and site dependence. Atmos Environ 30(19):3331–3347
Essa KSM, ElOtaify MS (2006) Diffusion from a point source in an urban atmosphere. Meteorol Atmos Phys 92:95–101
Essa KSM, Soad M, Etman ME (2007) New analytical solution of the dispersion equation. Atmos Res 85:337–343
Essa KSM, Shalaby AS, Ibrahim MAE, Mosallem AM (2020) Analytical solutions of the advection–diffusion equation with variable vertical eddy diffusivity and wind speed using hankel transform. Pure Appl Geophys. https://doi.org/10.1007/s0002402002496y
Essa KSM, Mosallem AM, Shalaby AS (2020) Evaluation of analytical solution of advection diffusion equation in three dimensions using Hankel transform. Under publication in Izvestiya Atmospheric and Oceanic Physics
Hanna SR (1982) Review of atmospheric diffusion models for regulatory applications. WMO Tech. Note No. 177, WMO, Geneva, 37
Hanna Steven R, Briggs GA, Hosker RP, Jr (1982) Handbook on atmospheric diffusion of energy. (DoEl / TIC11223 (DE 82002045))
Hans AP, Jhon AD (1984) Atmospheric turbulence, models and methods for engineering applications. Willy, New York
Heines TS, Peters LK (1974) The effect of ground level absorption on the dispersion of pollutants in the atmosphere. Atmos Environ 8:1143–1153
Hinrichsen K (1986) Comparison of four analytical dispersion models for nearsurface releases above a grass surface. Atmos Environ 20:29–40
Huang CH (1979) A theory of dispersion in turbulent shear flow. Atmos Environ 13:141–144
IAEA (1982) Generic models and parameters for assessing the environmental transfer of radio nuclides from routine release".
Irwin JS (1979) A theoretical variation of the wind profile power law exponent as a function of surface roughness and stability. Atmos Environ 13:191–194
Lin JS, Hildemann LM (1997) A generalized mathematical scheme to analytically solve the atmospheric diffusion equation with dry deposition. Atmos Environ 31:59–71
Pasquill F, Smith FB (1983) Atmospheric diffusion, 3rd edn. D. Van Nortrand, London
Pasquill F, Smith FB (1983) Atmospheric diffusion. Wiley, New York
Pasquill F (1974) Atmospheric diffusion. 2nd ed., Wiley, New York
Richard EF, Shultis KJ (1993) Radiological assessment, sources and exposures. PTR PrenticeHall, Englewood Cliffs
Rounds W (1955) Solution of the twodimensional diffusion equation. Trans. Am. Geophys. 36:395–405
Karl BS Jr., Dey PR (2000) Atmospheric dispersion modeling compliance guide. McGraw Hill Companies, New York
Smith FB (1957) The diffusion of smoke from a continuous elevated pointsource into a turbulent atmosphere. J Fluid Mech 2:49–76
Smith FB (1990) College on atmospheric boundary layer physics “air pollution modelling for environmental impact assessment” international center for theoretical physics (4–15 June). SMR/ 462:1
Stakgold I (1968) Boundary value problems of mathematical physics. Vol. 1. MacMillan, New York
Sutton OG (1953) Micrometeorology. McGrawHill, New York
Tirabassi T (1989) Analytical air pollution advection and diffusion models. Water Air Soil Pollut 47:19–24
Tirabassi T, Tagliazacca M, Zannetti P (1986) KAPPAG, a nonGaussian plume dispersion model description and evaluation against tracer measurements. JAPCA 36:592–596
Walters TS (1957) Diffusion from an infinite line source lying perpendicular to the mean wind velocity of a turbulent flow. Quart J Mech Appl Math 5:214–219
Wortmann S, Turbulence VM, Moreira DM, Buske D (2005) Anew approach to simulate the pollutant dispersion in the PBL. Atmos. Env. 39:2171–2178
Yaglom AM (1972) Turbulent diffusion in the surface layer of the atmosphere. IZV Atmos And Ocean Phys 8(6):333–340
Yeh GT (1975) Green’s function of a diffusion equation. Geophys Res Lett 2:293–296
Yeh GT, Huang CH (1975) Threedimensional air pollution modeling in the lower atmosphere. Boundary Layer Met 9:381–390
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All sources of funding for the research reported should be paid by BeniSuef University, Egypt.
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KE, SE, MS, ME, AM and AS contributed in all sections. All authors read and approved the final manuscript.
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Essa, K.S.M., Etman, S.M., ElOtaify, M.S. et al. Different solutions of the diffusion equation and its applications. BeniSuef Univ J Basic Appl Sci 10, 82 (2021). https://doi.org/10.1186/s43088021001534
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DOI: https://doi.org/10.1186/s43088021001534
Keywords
 Advection–diffusion equation
 Pollutants deposition
 Extreme groundlevel concentration
 Egyptian atomic research reactor
 Urban atmosphere
 The conservative material
 Continuous point source
 Maximum concentration value
 The logarithmic law of wind velocity
 Contaminants in the planetary boundary layer