1 Introduction

An analytical solution of the advection-diffusion equation was calculated using the eddy diffusivity coefficients and wind speed profiles, which were assumed constant throughout the whole atmospheric boundary layer (ABL) or as following a power law (Pasquill & Smith, 1983; Seinfeld, 1986; Sharan et al., 1996; Tirabassi et al., 1986; Van Ulden, 1978). Moreira et al. (2005) presented a solution for the advection-diffusion equation based on the Laplace transform considering the ABL as a multilayer system.

Essa et al. (2011) studied the technique of two types of eddy diffusivities used to solve the advection-diffusion equation in two-dimensions analytically. Marrouf et al. (2015) studied the effect of eddy diffusivity on the advection-diffusion equation. By using the Hankel transform, the advection-diffusion equation with variable vertical eddy diffusivity and wind speed was evaluated Essa et al. (2020).

Recently, analytical solutions for the advection-dispersion reaction equation with first decay under constant and time-dependent boundary conditions were found. Mass transfer shape factor effects were studied by Abbasi et al. (2021). Also, an analytical solution for the diffusion equation under chemical reaction and wet deposition from the line source was investigated by Essa et al. (2022).

In this work, the effect of two analytical model solutions on the steady-state advection-diffusion equation for neutral, stable and unstable conditions was examined. The first model was solved using the variable separation technique, advection diffusion multilayer method (ADMM), Fourier transform and square complement method. The second model was solved using the advection diffusion multilayer method (ADMM), general integrated transport technique (GITT) and Laplace transform. The analytical solutions of both models were calculated by assuming that the wind speed u, crosswind eddy diffusivity \({k}_{y}\) and vertical eddy diffusivity \({k}_{z}\) are functions of power law in vertical distance "\(z\)." The analytical models were evaluated and used to calculate the predicted concentrations for neutral, stable and unstable conditions. The predicted concentrations were validated with already existing experimental data from the Egyptian Atomic Energy Authority test of radioactive iodine-135 (I135) in unstable condition. Also, they were validated with already existing experimental data of iodine I-131 (I131) released from the research reactor under neutral and stable conditions. The values of proposed concentrations were compared with the previous works. The second predicted model in three stabilities gave better agreement with the observed concentration data than the first predicted model.

2 The First Mathematical Model

The steady-state advection-diffusion equation in three dimensions (Essa et al., 2021) is written as follows:

$$u\frac{{\partial C\left( {x,y,z} \right)}}{\partial x} = \frac{\partial }{\partial y}\left( {k_{y} \frac{{\partial C\left( {x,y,z} \right)}}{\partial y}} \right) + \frac{\partial }{\partial z}\left( {k_{z} \frac{{\partial C\left( {x,y,z} \right)}}{\partial z}} \right),$$
(1)

where \(C\left( {x,y,z} \right)\) is the pollutant concentration (g/m3), and (Bq/m3), \(k_{y}\) and \(k_{z}\) are the crosswind and vertical eddy diffusivities, respectively; \(u\) is the wind speed (m/s), and \(x\) is downwind distance (m). Equation (1) is solved under the boundary conditions as follows:

(a) The flux vanished at the mixing height “h.”

$$k_{z} { }\frac{\partial C}{{\partial z}} = 0{ }\;{\text{at }}z = h{ }{\text{.}}$$
(2a)

(b) Flux at crosswind direction vanished at small distance “y0” and at large distance in crosswind direction Ly, i.e.,

$$k_{y} \frac{\partial C}{{\partial y}} = 0,\quad {\text{at }}y = y_{0} ,L_{y} .$$
(2b)

(c) The condition of null flux is applied on the ground surface

$$k_{z} \frac{\partial C}{{\partial z}} = 0\, {\text{at }} z = 0$$
(2c)

(d) Mass continuity is applied as follows:

$$uC\left( {0,y,z} \right) = Q\delta \left( {y - y_{0} } \right)\delta \left( {z - h} \right) \;{\text{at }}x = 0,$$
(2d)

where h is the top height of the atmospheric boundary layer (ABL) (m), "\(Q\)" is the emission release (g/s) or (Bq), \(\delta\) is a Dirac delta function,\(y_{0}\) is a small distance, and Ly is the biggest distance in the crosswind direction (m).

(e) The concentration tends to zero as z tends to ∞

$$C\left( {x,{\text{y}},{ }z} \right) \to 0{\text{ as }}z \to \infty .$$
(2e)

(f) The concentration vanished at the mixing height

$$C\left( {x,{\text{y}},{ }z} \right) = 0{\text{ at }}z = h.$$
(2f)

Assuming that u, \(k_{y}\) and \(k_{z}\) are functions of power law in vertical distance "\(z\)" it follows:

$${\text{u}} = \alpha z^{p} $$
(3)
$$k_{y} = \beta u$$
(4)
$$k_{z} = \gamma z^{n} ,$$
(5)

where \(\alpha ,\) \(\beta \;{\text{and}}\) \(\gamma\) are constants, which equal \(0.31\left( {\frac{{w_{*} }}{u}} \right)^{2}\), \(0.31\left( {\frac{{w_{*} }}{u}} \right)^{2}\) and \(0.16\left( {\frac{{w_{*} }}{u}} \right)^{2}\), respectively; w* is a convective vertical velocity (Essa & El-Otaify, 2007). n and p depend on stability conditions (Irwin 1979). Then, Eq. (1) is solved using variable separation technique as follows:

$$C\left( {x,y,z} \right) = \varphi \left( {x,y} \right)\psi \left( {x,z} \right).$$
(6)

Substituting from Eq. (6) in Eq. (1) and dividing by \(\varphi \left( {x,y} \right)\psi \left( {x,z} \right),\) it is shown that: -

$$\frac{u}{\varphi }\frac{\partial \varphi }{{\partial x}} + \frac{u}{\psi }\frac{\partial \psi }{{\partial x}} = \frac{\beta u}{\varphi }\frac{{\partial^{2} \varphi }}{{\partial y^{2} }} + \frac{\gamma }{\psi }\frac{\partial }{\partial z}\left( {z^{n} \frac{\partial \psi }{{\partial z}}} \right) .$$
(7)

Equation (7) is divided into the following two equations as follows:

$$\frac{1}{\varphi }\frac{\partial \varphi }{{\partial x}} = \frac{\beta }{\varphi }\frac{{\partial^{2} \varphi }}{{\partial y^{2} }}$$
(8a)
$$\frac{{\alpha z^{p} }}{\psi }\frac{\partial \psi }{{\partial x}} = \frac{\gamma }{\psi }\frac{\partial }{\partial z}\left( {z^{n} \frac{\partial \psi }{{\partial z}}} \right) .$$
(8b)

First Description Model: Equation (8a) is solved by assuming that

$$\varphi(x,y) = \mu_{l}(x) \eta_{l} (y)$$
(9)

Substituting Eq. (9) into Eq. (8a) then, one gets:

$$\frac{1}{{\mu_{l} \left( x \right)}}\frac{{\partial \mu_{l} \left( x \right)}}{\partial x} = \frac{\beta }{{\eta_{l} \left( y \right)}}\frac{{\partial^{2} \eta_{l} \left( y \right)}}{{\partial y^{2} }} = - \lambda_{l}^{2}$$
(10)

where \(\lambda_{l}\) is a constant of separation. Then, we have two equations as follows:

$$\frac{{\partial \mu_{l} }}{\partial x} = - \lambda_{l}^{2} \mu_{l}$$
(11a)
$$\frac{{\partial^{2} \eta_{l} }}{{\partial y^{2} }} = \frac{{ - \lambda_{l}^{2} }}{\beta }\eta_{l} .$$
(11b)

The solution of Eqs. (11a) and (11b) has the form:

$$\mu_{l} \left( x \right) = c_{1} e^{{ - \lambda_{l}^{2} x}}$$
(12a)
$$\eta_{l} \left( y \right) = c_{2} \cos \left( {\frac{{\lambda_{l} }}{\sqrt \beta }y} \right) + c_{3} \sin \left( {\frac{{\lambda_{l} }}{\sqrt \beta }y} \right),$$
(12b)

where \(c_{1} , c_{2},\) and \(c_{3}\) are constants. The condition in Eq. (2b) is applied at y = y0 = 0 and y = \(L_{y}\) into Eq. (12b), which gives \(c_{3} = 0\) and \(\lambda_{l} = \frac{l\pi \sqrt \beta }{{L_{y} }} ,l = 0,1,2,...\)

Then, the solution of Eq. (8a) is as follows:

$$\varphi \left( {x,y} \right) = \mathop \sum \limits_{l = 0}^{\infty } B_{l} e^{{ - \lambda_{l}^{2} x}} \cos \left( {\frac{l\pi }{{L_{y} }}y} \right),$$
(13a)

where \(B_{l}\) = \(c_{1} c_{2} ,\) Using the boundary condition \(C\left( {0,y} \right) = \delta \left( {y - y_{0} } \right)\) and Eq. (9), one gets: \(B_{0} = \frac{2}{{L_{y} }}, B_{l} = \frac{2}{{L_{y} }}\cos \left( {\frac{l\pi }{{L_{y} }}y_{0} } \right), l = 1,2,3, \ldots\); then, Eq. (13a) becomes:

$$\varphi \left( {x,y} \right) = \frac{2}{{L_{y} }}\left( {1 + \mathop \sum \limits_{l = 1}^{\infty } e^{{ - \lambda_{l}^{2} x}} \cos \left( {\frac{l\pi }{{L_{y} }}y} \right)\cos \left( {\frac{l\pi }{{L_{y} }}y_{0} } \right)} \right) .$$
(13b)

Second description model: Equation (8b) is solved by the advection diffusion multilayer method (ADMM) where the height of ABL (h) is discretized into N subinterval layers. Therefore, at each interval both \(k_{z} { }\) and \({\text{u}}\) are considered as average values. Then, Eq. (8b) is reduced to the solutions of N equations of the following type:

$$u_{i} \frac{{\partial {\Psi }\left( {x,{ }z} \right)}}{\partial x} = k_{i} \frac{{\partial^{2} {\Psi }\left( {x,{ }z} \right)}}{{\partial z^{2} }},$$
(14)

where

$$k_{i} = { }\frac{1}{{z_{i + 1} - { }z_{i} }}\mathop \int \limits_{{z_{i} }}^{{z_{i + 1} }} k_{i} \left( z \right){\text{ d}}z$$
$$u_{i} = { }\frac{1}{{z_{i + 1} - { }z_{i} }}\mathop \int \limits_{{z_{i} }}^{{z_{i + 1} }} u_{i} \left( z \right){\text{ d}}z,$$

for, \(z_{i} \le z{ } \le { }z_{i + 1} { }\), i = 1: N

By using separation of variables, the general solution of Eq. (14) is in the form:

$${\Psi }\left( {x,z,{ }h} \right) = X\left( x \right)Z\left( {z,h} \right).$$
(15)

Substituting from Eq. (15) into Eq. (14) and dividing by \(X\left( x \right)Z\left( {z,h} \right),{ }\) it is shown that:

$$\frac{1}{X\left( x \right)}\frac{{{\text{d}}X\left( x \right)}}{{{\text{d}}x}} = \frac{{k_{i} { }}}{{u_{i} { }Z\left( {z,h } \right)}}\frac{{{\text{d}}^{2} Z\left( {z,h} \right)}}{{{\text{d}}z^{2} }}{ } = { } - \xi^{2} ,$$
(16)

where \(\xi^{2} { }\) is a constant. Equation (16) is divided into the following two equations as follows:

$$\frac{{{\text{d}}X\left( x \right)}}{{{\text{d}}x}}{ } = - { }\xi^{2} { }X\left( x \right)$$
(17a)
$$\frac{{{\text{d}}^{2} Z\left( {z,h} \right)}}{{{\text{d}}z^{2} }} = { } - { }\frac{{u_{i} \xi^{2} }}{{k_{i} }}{ }Z\left( {z,h} \right).$$
(17b)

The solutions of Eq. (17a) and Eq. (17b) have the following forms:

$$X\left( x \right) = c\left( h \right)e^{{ - \xi_{l}^{2} x}}$$
(18)
$$Z\left( {z,h} \right) = A_{1} \left( h \right)e^{{i\xi_{l} z\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} { } + A_{2} \left( h \right)e^{{ - i\xi_{l} z\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} ,$$
(19)

where c \(\left( h \right)\), \(A_{1} \left( h \right) \;{\text{and}} A_{2} \left( h \right)\) depends on mixing height \(\left( h \right)\). Then, the general solution of Eq. (14) can be obtained as follows:

$${\Psi }_{i} \left( {x,z, h} \right) = c\left( h \right)A_{1} \left( h \right)e^{{ - \xi_{l}^{2} x + i\xi_{l} z\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} + c\left( h \right) A_{2} \left( h \right)e^{{ - \xi_{l}^{2} x - i\xi_{l} z\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} .$$
(20)

Since \(0 < \xi_{l} < \infty\), where \(l\) = 0, 1, 2,….., varies continuously as integer values; the sum of all these solutions depends on the integration of \(\xi_{l}\) so the general solution is as follows:

$${\Psi }_{i} \left( {x,z, h} \right) = \mathop \int \limits_{0}^{\infty } \left[ {c\left( {\xi_{l} ,h} \right) A_{1} \left( {\xi_{l} ,h} \right)e^{{ - \xi_{l}^{2} x + i\xi_{l} z\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} + c\left( {\xi_{l} ,h} \right) A_{2} \left( {\xi_{l} ,h} \right)e^{{ - \xi_{l}^{2} x - i\xi_{l} z\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} } \right] {\text{d}}\xi_{l} .$$
(21)

Also, we can write Eq. (21) in the form

$${\Psi }_{i} \left( {x,z, h} \right) = \mathop \int \limits_{ - \infty }^{\infty } \left[ {c\left( {\xi_{l} ,h} \right) A_{1} \left( {\xi_{l} ,h} \right) + c\left( { - \xi_{l} ,h} \right)A_{2} \left( { - \xi_{l} ,h} \right)} \right]e^{{ - \xi_{l}^{2} x + i\xi_{l} z\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} {\text{d}}\xi_{l} .$$
(22)

Let \(R\left( {\xi_{l} ,h} \right) = \left[ {c\left( {\xi_{l} ,h} \right) A_{1} \left( {\xi_{l} ,h} \right) + c\left( { - \xi_{l} ,h} \right)A_{2} \left( { - \xi_{l} ,h} \right)} \right]\) such that

$$R\left( {\xi_{l} ,h} \right) = \left[ {c\left( {\xi_{l} ,h} \right)A_{1} \left( {\xi_{l} ,h} \right)} \right] {\text{if}} \xi_{l} > 0$$
$$R\left( {\xi_{l} ,h} \right) = \left[ { c\left( { - \xi_{l} ,h} \right)A_{2} \left( { - \xi_{l} ,h} \right)} \right] {\text{ if}} \xi_{l} < 0$$

Then, Eq. (22) becomes

$${\Psi }_{i} \left( {x,z, h} \right) = \mathop \int \limits_{ - \infty }^{\infty } R\left( {\xi_{l} ,h} \right)e^{{ - \xi_{l}^{2} x + i\xi_{l} z\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} {\text{d}}\xi_{l} .$$
(23)

To get the value of \(R\left( {\xi_{l} ,h} \right),\) one can use the Fourier transform of \(\delta \left( {z - h} \right)\) as follows:

$$\delta \left( {z - h} \right) = \frac{1}{2\pi }\mathop \int \limits_{ - \infty }^{\infty } e^{{ i\xi_{l } \left( {z - h} \right)\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} {\text{d}}\xi_{l} .$$
(24)

By using the boundary condition \(uC\left( {0,y,z} \right) = Q\delta \left( y - y_{0} \right)\delta \left( {z - h} \right)\; {\text{at }}x = 0\) where h is the mixing height, the value of \(R\left( {\xi_{l} ,h} \right)\) can be written as follows

$$R\left( {\xi_{l} ,h} \right){ } = \frac{Q}{{2\pi u_{i} }}e^{{ - i\xi_{l } h\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} .$$
(25)

Then, Eq. (23) can be written as follows:

$${\Psi }_{i} \left( {x,z, h} \right) = \frac{Q}{{2\pi u_{i} }}\mathop \int \limits_{ - \infty }^{\infty } e^{{ - \xi_{l}^{2} x + i\xi_{l} \left( {z - h} \right)\sqrt {\frac{{u_{i} }}{{k_{i} }}} }} {\text{d}}\xi_{l} .$$
(26)

The square compliment method is considered to solve the above integration from Essa et al. (2011); then, the solution of Eq. (14) can be written as follows:

$${\Psi }_{i} \left( {x,z, h} \right) = \frac{Q}{{2 u_{i} \sqrt {\pi x} }}e^{{ - \frac{{\left( {z - h} \right)^{2} u_{i} }}{{4k_{i} x }}}} .$$
(27)

Substitute Eq. (13b) and Eq. (27) into Eq. (6), the concentration in three dimensions becomes:

$$C\left( {x,y,z,h} \right) = \frac{Q}{{2\sqrt {\pi x} u_{i} }}\frac{2}{{L_{y} }}\left( {1 + \mathop \sum \limits_{l = 1}^{\infty } e^{{ - \lambda_{l}^{2} x}} \cos \left( {\frac{l\pi }{{L_{y} }}y} \right)\cos \left( {\frac{l\pi }{{L_{y} }}y_{0} } \right)} \right) e^{{ - \frac{{\left( {z - h} \right)^{2} u_{i} }}{{4k_{i} x}} - \frac{\upsilon x}{u} }} ,$$
(28)

where ui and ki are presented in the two Eqs. (3) and (5), respectively, and \(e^{{ - \frac{\upsilon x}{u}}}\) is the radioactive decay for the specified nuclide, and \(\upsilon\) is the decay constant of iodine-135 or iodine-131 based on the experiment.

3 The Second Mathematical Model

Equation (1) is solved by the advection diffusion multilayer method (ADMM). We considered that the eddy diffusivities and wind speed depend on the vertical height, which this time resubmits Eq. (1) with constant parameters as a set of sub-layers like:

$$u_{n} \frac{{\partial C_{n} }}{\partial x} = k_{yn} \frac{{\partial^{2} C_{n} }}{{\partial y^{2} }} + k_{zn} \frac{{\partial^{2} C_{n} }}{{\partial z^{2} }},\;z_{n} \le z \le z_{n + 1} .$$
(29)

For n = 1:N, then, the general integrated transport technique (GITT) is used in crosswind direction (Moreira et al. 2005d). The problem began with eigenvalues with respect to boundary conditions.

$${\Psi }_{i}^{^{\prime\prime}} \left( y \right) + \lambda_{i}^{2} {\Psi }_{i} \left( y \right) = 0 {\text{ at}} 0 < y < L_{y}$$
(30a)
$${\Psi }_{i}^{^{\prime}} \left( y \right) = 0 \,{\text{at}}\, y = 0,L_{y} .$$
(30b)

Then, \({\Psi }_{i} \left( y \right) = \cos \left( {\lambda_{i} y} \right),\) where \(\lambda_{i} = \frac{i\pi }{{L_{y} }}, i = 0,1,2, \ldots \ldots \ldots\), since \({\Psi }_{i} \left( y \right)\) and \(\lambda_{i}\) are eigenfunctions and eigenvector, respectively, accompanying the problem of Sturm-Liouville, which satisfies the orthogonal condition:

$$\frac{1}{{N_{m}^{\frac{1}{2}} N_{n}^{\frac{1}{2}} }}\mathop \int \limits_{v}^{{}} {\Psi }_{m} \left( z \right){\Psi }_{n} \left( z \right){\text{d}}v = \left\{ {\begin{array}{*{20}c} {0, m \ne n} \\ {1, m = n} \\ \end{array} } \right.$$

where Nm is given by:

$$N_{m} = \mathop \int \limits_{v}^{{}} {\Psi }_{m}^{2} \left( z \right){\text{d}}v.$$
(31)

In the first step, one expands the variable concentration Cn(x,y,z) using GITT as follows:

$$C_{n} \left( {x,y,z} \right) = \mathop \sum \limits_{i = 0}^{\infty } \frac{{\overline{C}_{ni} \left( {x,z} \right){\Psi }_{i} \left( y \right)}}{{N_{i}^{1/2} }}.$$
(32)

Substituting from Eq. (32) into Eq. (29), one gets:

$$u_{n} \mathop \sum \limits_{i = 0}^{\infty } \frac{{\partial \overline{C}_{ni} \left( {x,z} \right)}}{\partial x}\frac{{{\Psi }_{i} \left( y \right)}}{{N_{i}^{1/2} }} = K_{yn} \mathop \sum \limits_{i = 0}^{\infty } \overline{C}_{ni} \left( {x,z} \right)\frac{{{\Psi }_{i}^{^{\prime\prime}} \left( y \right)}}{{N_{i}^{\frac{1}{2}} }} + K_{zn} \mathop \sum \limits_{i = 0}^{\infty } \frac{{\partial^{2} \overline{C}_{ni} \left( {x,z} \right)}}{{\partial z^{2} }}\frac{{{\Psi }_{i} \left( y \right)}}{{N_{i}^{1/2} }},$$
(33)

where \({\Psi }_{i}^{^{\prime\prime}} \left( y \right)\) is indicated to the second derivatives with respect to y.

By using the condition (30a) one gets \({\Psi }_{i}^{^{\prime\prime}} \left( y \right) = - \lambda_{i}^{2} {\Psi }_{i}\), and multiplying the above equation by \(\mathop \int \limits_{0}^{{L_{y} }} \frac{{{\Psi }_{j} \left( y \right)}}{{N_{j}^{1/2} }} {\text{d}}y\) and using the relation of orthogonally, then Eq. (33) can be written as:

$$\frac{{\partial^{2} \overline{C}_{ni} \left( {x,z} \right)}}{{\partial z^{2} }} - \frac{{u_{n} }}{{K_{zn} }}\frac{{\partial \overline{C}_{ni} \left( {x,z} \right)}}{\partial x} - \frac{{K_{yn} }}{{K_{zn} }}\lambda_{i}^{2} \overline{C}_{ni} \left( {x,z} \right) = 0.$$
(34)

Substituting from Eq. (32) in the source condition (2d), we have:

$$\mathop \sum \limits_{i = 0}^{\infty } u_{n} \overline{C}_{ni} \left( {0,z} \right)\mathop \int \limits_{0}^{{L_{y} }} \frac{{{\Psi }_{i} {\Psi }_{j} }}{{N_{i}^{1/2} N_{j}^{1/2} }}{\text{d}}y = \mathop \int \limits_{0}^{{L_{y} }} \frac{{Q \delta \left( {z - h_{s} } \right)\delta \left( {y - y_{0} } \right){\Psi }_{j} }}{{N_{j}^{1/2} }}{\text{d}}y.$$
(35)

After integration in Eq. (35), one obtains:

$$\overline{C}_{ni} \left( {0,z} \right) = \frac{{Q\delta \left( {z - h_{s} } \right){\Psi }_{i} \left( {y_{0} } \right)}}{{u_{n} N_{i}^{1/2} }}.$$
(36)

Applying Laplace transform on x is as follows:

$$L\left\{ {\overline{C}_{ni} \left( {x,z} \right)} \right\} = \tilde{C}_{ni} \left( {s,z} \right)$$
(37)

When Laplace transform technique is applied to Eq. (34), one obtains:

$$\frac{{\partial^{2} \tilde{C}_{ni} \left( {s,z} \right)}}{{\partial z^{2} }} - \frac{{su_{n} + \lambda_{i}^{2} K_{yn } }}{{K_{zn} }}\tilde{C}_{ni} \left( {s,z} \right) = - \frac{{Q\delta \left( {z - h_{s} } \right){\Psi }_{i} \left( {y_{0} } \right)}}{{K_{zn} N_{i}^{1/2} }}.$$
(38)

Equation (38) is solved in the form:

$$\tilde{C}_{ni} \left( {s,z} \right) = A_{1n} e^{{R_{n} z}} + A_{2n} e^{{ - R_{n} z}} + \frac{Q}{{2R_{a} }}\left( {e^{{R_{n} \left( {z - h_{s} } \right)}} - e^{{ - R_{n} \left( {z - h_{s} } \right)}} } \right)$$
(39)

where \(R_{n} = \sqrt {\frac{{\left( {su_{n} + K_{yn} \lambda_{i}^{2} } \right)}}{{K_{zn} }}}\) and \(R_{a} = \frac{{N_{i}^{1/2} }}{{{\Psi }_{i} \left( {y_{0} } \right)}}\sqrt {K_{zn} (su_{n} + K_{yn} \lambda_{i}^{2} }\)).

Using the boundary condition (2c) one gets \(A_{1n} = A_{2n}\), at z = 0; then, Eq. (39) becomes:

$$\tilde{C}_{ni} \left( {s,z} \right) = A_{1n} e^{{R_{n} z}} + A_{1n} e^{{ - R_{n} z}} + \frac{Q}{{2R_{a} }}\left( {e^{{R_{n} \left( {z - h_{s} } \right)}} - e^{{ - R_{n} \left( {z - h_{s} } \right)}} } \right).$$
(40)

Using the condition (2a), therefore, the value of \(A_{1n} = A_{2n}\) can be written as

$$A_{1n} = A_{2n} = \frac{{Q\left( {e^{{R_{n} \left( {h - h_{s} } \right)}} + e^{{ - R_{n} (h - h_{s} }} } \right)}}{{2R_{a} \left( {e^{{hR_{n} }} - e^{{ - hR_{n} }} } \right)}}.$$

Then, Eq. (40) becomes:

$$\tilde{C}_{ni} \left( {s,z} \right) = \frac{{Q\left( {e^{{R_{n} \left( {h - h_{s} } \right)}} + e^{{ - R_{n} (h - h_{s} }} } \right)}}{{2R_{a} \left( {e^{{hR_{n} }} - e^{{ - hR_{n} }} } \right)}}\left( {e^{{R_{n} z}} + e^{{ - R_{n} z}} } \right) + \frac{Q}{{2R_{a} }}\left( {e^{{R_{n} \left( {z - h_{s} } \right)}} - e^{{ - R_{n} \left( {z - h_{s} } \right)}} } \right){\text{H}}\left( {{\text{z}} - h_{s} } \right){ }{\text{.}}$$
(41)

The concentration \(\overline{C}_{n} \left( {x,z} \right)\) is obtained by inverting numerically using the Gaussian quadrature scheme:

$$\overline{C}_{ni} \left( {x,z} \right) = \mathop \sum \limits_{i = 1}^{8} a_{i} \frac{{p_{i} }}{x}\left[ {\frac{{Q\left( {e^{{G_{n} \left( {h - h_{s} } \right)}} + e^{{ - G_{n} (h - h_{s} }} } \right)}}{{2F_{a} \left( {e^{{hG_{n} }} - e^{{ - hG_{n} }} } \right)}}\left( {e^{{G_{n} z}} + e^{{ - G_{n} z}} } \right) + \frac{Q}{{2F_{a} }}\left( {e^{{G_{n} \left( {z - h_{s} } \right)}} - e^{{ - G_{n} \left( {z - h_{s} } \right)}} } \right){\text{H}}\left( {{\text{z}} - h_{s} } \right)} \right],$$
(42)

where \(G_{n} = \sqrt {\frac{{\left( {\frac{{p_{i} }}{x}u_{i} + K_{yi} \lambda_{i}^{2} } \right)}}{{K_{zi} }}}\) and \(F_{a} = \frac{{N_{i}^{1/2} }}{{{\Psi }_{i} \left( {y_{0} } \right)}}\sqrt {K_{zi} \left( {\frac{{p_{i} }}{x}u_{n} + K_{yi} \lambda_{i}^{2} } \right)}\) where H(z − hz) is the Heaviside function. Then, the final solution is obtained:

$$C_{i} \left( {x,y,z} \right) = \mathop \sum \limits_{i = 0}^{\infty } \frac{{{\Psi }_{i} \left( y \right)}}{{N_{i}^{1/2} }}e^{{ - \frac{\nu x}{u}}} \left\{ {\mathop \sum \limits_{i = 1}^{8} a_{i} \frac{{p_{i} }}{x}\left[ {\begin{array}{*{20}c} {\frac{{Q\left( {e^{{G_{n} \left( {h - h_{s} } \right)}} + e^{{ - G_{n} (h - h_{s} }} } \right)}}{{2F_{a} \left( {e^{{hG_{n} }} - e^{{ - hG_{n} }} } \right)}}\left( {e^{{G_{n} z}} + e^{{ - G_{n} z}} } \right) + \frac{Q}{{2F_{a} }}\left( {e^{{G_{n} \left( {z - h_{s} } \right)}} - e^{{ - G_{n} \left( {z - h_{s} } \right)}} } \right)} \\ {H\left( {z - h_{s} } \right) } \\ \end{array} } \right]{ }} \right\} ,$$
(43)

where \(e^{{ - \frac{\upsilon x}{u}}}\) is the radioactive decay for the specified nuclide (iodine-135 or iodine-131) based on the experiment. Therefore, \(\upsilon { }\) is the decay constant of iodine-135 or iodine-131 equals 2.9 × 10–5 and 9.95 × 10–7, respectively.

4 Results and Discussion

The proposed models in Eqs. (28) and (43) are used to calculate the predicted concentrations for neutral, stable and unstable conditions. Then, the predicted concentrations are compared with the already existing experimental data as follows.

4.1 First Experimental Data Measured on March and May 2006

The already existing observed data for I135 isotope concentration were obtained from dispersion experiments conducted in unstable condition air samples collected around the Egyptian Atomic Energy Authority, First Research Reactor, at Inshas, Cairo, Egypt. The samples were collected at a height of 0.7 m above ground from a stack of 43 m height. The reactor site was flat and had a roughness length of 0.6 cm; each run was made for 30 min. The values of power-law exponent ‘p’ and “n” of eddy diffusivity as a function of air stability are taken from Hanna Steven et al. (1982) and presented in Table 1). The meteorological data and the already existing observed concentrations of I135 isotope during the experiments were obtained from Essa and El-Otaify (2007) and are presented in Tables 2 and 3, respectively. Equations (28) and (43) are estimated using Eq. (3) and Eq. (4) below the plume center line. The comparison between two predicted concentrations was made using Mathematica program, previous work (Essa et al., 2022), and already existing observed concentration data of I135 from the Nuclear Research Reactor at the Egyptian Atomic Energy Authority are presented in Table 3).

Table 1 Exponents p and n for wind speed and eddy diffusivity for air stability in an urban area
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Table 2 Nine convective test runs of meteorological data at Inshas site on March and May 2006
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Table 3 Relation between observed and predicted concentrations for run 9 experiments
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Figure 1 shows the variation between two proposals, the previous work (Essa et al., 2022) and already existing observed concentrations of radioactive I135 via downwind distance in unstable conditions at Inshas. Also, the relation between the proposed, previous work (Essa et al., 2022) and already existing observed concentration is shown in Fig. 2.

Fig. 1
figure 1

Variation of concentrations (Bq/m3) of iodine-135 with downwind distance (m)

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Fig. 2
figure 2

Scattering diagram between predicted, previous Work (Essa et al., 2022) and already existing observed concentrations (Bq/m3) for iodine-135

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4.2 Second Experimental Data Measured in 2004

The already existing observed concentrations of iodine I-131 (I131) (Essa, 2009; Essa et al., 2014) were obtained from the experiments performed to collect air samples around the Second Reactor under neutral and stable conditions. The samples were collected at a height of 0.7 m above ground from a stack of 27 m height. The meteorological data of I131 isotope during the experiments under neutral and stable conditions were obtained and are presented in Table 4 (Essa, 2009, Essa et al., 2014).

Table 4 Meteorological data under stable and neutral conditions (Essa, 2009, Essa et al., 2014)
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The values of predicted concentration of iodine-131 (I131), the already existing observed data and previous Work (Essa et al., 2014) in stable cases with downwind distance are shown in Table 5.

Table 5 Observed and predicted concentrations for I131 in stable condition
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Figure 3 shows the variation between the proposed, previous Work (Essa et al., 2014) and already existing observed concentrations of radioactive I131 via downwind distance in stable conditions at Inshas. Also, the relations between the proposed, previous work (Essa et al., 2014) and already existing observed concentration are shown in Fig. 4.

Fig. 3
figure 3

Variation of concentrations (Bq/m3) of iodine-131 with downwind distance (m) in stable conditions

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Fig. 4
figure 4

Scattering diagram between predicted, previous Work (Essa et al., 2014) and already existing observed concentrations (Bq/m3) for iodine-131 in stable conditions

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The values of predicted concentration, already existing observed data and previous work (Essa et al., 2014) for iodine-131 (I131) in neutral case with downwind distances are shown in Table 6.

Table 6 Observed and predicted concentrations for I131 in neutral conditions
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Figure 5 shows the variation between two proposed models, already existing observed concentrations and previous work (Essa et al., 2014) on radioactive I131 via downwind distance in neutral conditions at Inshas. Also, the relation between the two proposed models, observed concentrations and previous work (Essa et al., 2014), are shown in Fig. 6.

Fig. 5
figure 5

Variation of concentrations (Bq/m3) of iodine-131 with downwind distance (m) in neutral conditions

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Fig. 6
figure 6

Scattering diagram between two predicted models, observed concentrations and previous work (Essa et al., 2014) (Bq/m3), for iodine-131 in neutral conditions

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One finds that the second proposed model of Eq. (43) is the closest to the observed concentration in most points via downwind distance in all conditions as shown in Figs. 1, 3 and 5. In unstable and neutral conditions, the second proposed model was followed the previous work and then the first proposed model of Eq. (28) as shown in Fig. 1 and Fig. 5, while in stable condition the second proposed model was followed by the first proposed model and then the previous work as shown in Fig. 3. Also, the most points of the second predicted model of Eq. (43) are located one to one with the observed concentration as shown in Figs. 2, 4 and 6 in all conditions, while most data are located within a factor of two but the first predicted model of Eq. (28) in unstable and the previous work in stable is outside a factor of 2.

5 Statistical Techniques

The statistical method is used to compare between predicted and already existing observed results (Hanna, 1989). The following standard statistical performance between predictions (\(C_{{\text{p}}} = C_{{{\text{pred}}}}\)) and already existing observations (\(C_{O} = C_{Obs}\)) is as follows:

$${\text{Fraction Bias }}\left( {{\text{FB}}} \right) = \frac{{\left( {\overline{{C_{{\text{o}}} }} - \overline{{C_{{\text{p}}} }} } \right)}}{{\left[ {0.5\left( {\overline{{C_{{\text{o}}} }} + \overline{{C_{{\text{p}}} }} } \right)} \right]}}$$
(44)
$${\text{Normalized Mean Square Error }}\left( {{\text{NMSE}}} \right) = \frac{{\overline{{\left( {C_{{\text{p}}} - C_{{\text{o}}} } \right)^{2} }} }}{{\overline{{\left( {C_{{\text{p}}} C_{{\text{o}}} } \right)}} }}$$
(45)
$${\text{Correlation Coefficient }}\left( {{\text{COR}}} \right) = \frac{1}{{N_{m} }}\mathop \sum \limits_{i = 1}^{{N_{m} }} \left( {C_{pi} - \overline{{C_{p} }} } \right) \times \frac{{\left( {C_{oi} - \overline{{C_{o} }} } \right)}}{{\left( {\sigma_{p} \sigma_{o} } \right)}}$$
(46)
$${\text{Factor of Two }}\left( {{\text{FAC2}}} \right) = 0.5 \le \frac{{C_{{\text{p}}} }}{{C_{{\text{o}}} }} \le 2.0.$$
(47)

where \(\sigma_{{\text{p}}}\) and \(\sigma_{{\text{o}}}\) are the standard deviations of predicted \(C_{{\text{p}}}\) and observed \(C_{{\text{o}}}\) concentrations, respectively. Over bars refer to the average overall measurements. A perfect model must have the following performance: NMSE = FB = 0 and COR = FAC2 = 1.0.

Table 7 shows that in unstable conditions the second proposed model of Eq. (43) is in very good agreement with already existing observed concentrations than both the first predicted model of Eq. (28) and the previous work (Essa et al., 2022). Also, the first proposed model, the second proposed model and the previous work (Essa et al., 2022) achieved approximately 39%, 97% and 78% from already existing observed data concentrations, respectively.

Table 7 Statistical calculations of present model in unstable condition
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Table 8 shows that the second proposed model of Eq. (43) is in very good agreement with already existing observed data concentrations compared with both the first predicted model of Eq. (28) and previous work (Essa et al., 2014). Also, the first proposed model, second proposed model and previous work (Essa et al., 2014) achieved approximately 63%, 100% and 28% from already existing observed data in stable conditions, respectively.

Table 8 Statistical calculations of present model in stable condition
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Table 9 shows that the second proposed model of Eq. (43) is in very good agreement with observed data concentrations compared with both the first predicted model of Eq. (28) and previous work (Essa et al., 2014) in neutral condition. Also, the first proposed model, second proposed model and previous work (Essa et al., 2014) achieved approximately 100% from already existing observed data concentrations, respectively.

Table 9 Statistical calculations of present model in neutral condition
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6 Conclusions

The effect of two analytical model solutions on the steady-state advection-diffusion equation for neutral, stable and unstable conditions has been studied. The first model was solved by using the variable separation technique, advection diffusion multilayer method (ADMM), Fourier transform and square complement method. The second model was solved by using the advection diffusion multilayer method (ADMM), general integrated transport technique (GITT) and Laplace transform. The assumption, for both analytical model solutions, is to consider the wind speed u, crosswind eddy diffusivity \({k}_{y}\) and vertical eddy diffusivity \({k}_{z}\) as functions of power law in vertical distance "\(z\)." The analytical models were evaluated and used to calculate the predicted concentrations for neutral, stable and unstable conditions. The predicted concentrations were validated with already existing experimental data of the Egyptian Atomic Energy Authority test of radioactive iodine-135 (I135) in unstable condition. Also, they were validated with already existing experimental data on iodine I-131 (I131) released from the research reactor under neutral and stable conditions. The values of proposed concentrations were compared with the previous works. The results showed that the second predicted model lies inside a factor of two with the already existing observed concentrations than both the first predicted model and the previous work. Also, while NMSE and FB are close to zero, COR and FAC2 are close to one. The second predicted model of Eq. (43) is in good agreement with the already existing observed concentrations in unstable, stable and neutral conditions compared with both the first predicted model of Eq. (28) and the previous work. The first proposed model, the second proposed model and the previous work (Essa et al., 2022) achieved approximately 39%, 97% and 78% from observed data in unstable conditions, respectively. The first proposed model, the second proposed model and the previous work (Essa et al., 2014) achieved approximately 63%, 100% and 28% from observed data in stable conditions, respectively. The second predicted model of Eq. (43) in three conditions gives the best agreement with the already existing observed concentration data compared with both the first predicted model Eq. (28) and the previous work.

Future work will include deriving analytical solutions of the fractional advection-diffusion equation with time-dependent pulses on the boundary as well as dispersion from an area source in an unstable surface layer.