Appendix A
Derivation of the Thin-Film Solution

A.1 Derivation of the Thin-Film Solution

The diffusion equation is a linear one, and a solution can, therefore, be obtained by adding several other solutions. An elementary solution ("building block") that is particularly useful is the solution to an instantaneous, localised release in an infinite domain initially free of the substance.

Mathematically, the problem is stated as follows:

Environment domain is assumed infite ( $- \infty < x < \infty$ )
Diffusion is assumed to be constant ( $\partial D ∕ \partial t = 0$ )
Localised release ( $c (x, t_{0}) = M_{0} δ (x)$ )

Since the substance will take an infinite time to reach the infinitely far ends of the domain, the following limits are imposed.

\lim_{x \to + \infty} c (x, t) = \lim_{x \to - \infty} c (x, t) = 0 .

(A.1)

In the above, $M_{0}$ is the total mass of the substance released per unit cross-sectional area, and $δ (x)$ is the Dirac function. Physically, the behaviour as displayed in Figure A.1 is anticipated. The pollutant patch gradually spreads on both sides of the release location, with a commensurate decrease in the maximum center value. Curves at later times appear similar to those at earlier times, only being flatter and wider. Anticipating such similarity in the solution, the following equation and the transform parameter is written;

Figure A.1: Diffusion process in time of a localized mass injection. As the mass concentration distributes along the

x

-axis, the area under the curve is preserved

c (x, t) = t^{- α_{D}} F (γ) w i t h γ = \frac{x^{2}}{4 D t},

(A.2)

where $t^{- α_{D}}$ represents the decay of the maximum concentration value attained at $x$ = 0. Since decay has to decrease the concentration $α_{D}$ parameter is a positive value ( $α_{D} \geq 0$ ). $F (γ)$ is the shape factor of the solution, to give similar curve profile given in Figure A.1.

The exponent present in the distance ( $x^{2}$ ) is chosen to represent the second order derivative of the diffusion PDE. The same rule applies to the $t$ parameter present in the $F (γ)$ . The $D$ factor is the make the function dimensionless. The factor 4 is introduced to simplify the mathematical process of derivation.

By introducing the proposed solution in Eq. (A.2) to the PDE shown in Eq. (2.11a) .

\begin{align} \frac{\partial c}{\partial t} & = - α_{D} t^{- α_{D} - 1} F (γ) + t^{- α_{D}} \frac{\partial F}{\partial γ} \frac{\partial γ}{\partial t} = - α_{D} t^{- α_{D} - 1} F (γ) - γ t^{- α_{D} - 1} \frac{\partial F}{\partial γ}, & (A.3a) \\ \frac{\partial c}{\partial x} & = t^{- α_{D}} \frac{\partial F}{\partial γ} \frac{\partial γ}{\partial x} = \frac{x t^{- α_{D} - 1}}{2 D} \frac{\partial F}{\partial γ}, & (A.3b) \\ \frac{\partial^{2} c}{\partial x^{2}} & = \frac{t^{- α_{D} - 1}}{2 D} \frac{\partial F}{\partial γ} + \frac{x t^{- α_{D} - 1}}{2 D} \frac{\partial^{2} F}{\partial γ^{2}} \frac{\partial γ}{\partial x} = \frac{t^{- α_{D} - 1}}{2 D} \frac{\partial F}{\partial γ} + \frac{t^{- α_{D} - 1}}{D} γ \frac{\partial^{2} F}{\partial γ^{2}} . & (A.3c) \end{align}

After deriving the new element, substitution of $\partial c ∕ \partial t$ and $\partial^{2} c ∕ \partial x^{2}$ in the diffusion equation yields:

- α_{D} t^{- α_{D} - 1} F (γ) - γ t^{- α_{D} - 1} \frac{\partial F}{\partial γ} = \frac{t^{- α_{D} - 1}}{2} \frac{\partial F}{\partial γ} + γ t^{- α_{D} - 1} \frac{\partial^{2} F}{\partial γ^{2}} .

(A.4)

The time factors cancel out due to the definition of $γ$ , and the partial-differential equation is reduced to an ordinary differential equation, with variable $γ$ :

γ \frac{\partial}{\partial γ} (\frac{\partial F}{\partial γ} + F) + \frac{1}{2} (\frac{\partial F}{\partial γ} + 2 α_{D} F) = 0 .

(A.5)

To decrease the complexity of the solution and thereby making the group in the parenthesis identical, $α_{D}$ is chosen as 0.5.

\frac{\partial F}{\partial γ} + F (γ) = 0,

(A.6)

which the solution is,

F (γ) = G \exp (- γ),

(A.7)

where $G$ is an arbitrary constant of integration. By putting Eq. (A.2) to Eq. (A.7) the solution can be written as;

c (x, t) = \frac{G}{\sqrt{t}} \exp (- \frac{x^{2}}{4 D t}) .

(A.8)

While the initial boundary condition are met with this equation, the $G$ needs to be defined. This can be achieved by imposing a conservation of mass to the equation.

\int_{- \infty}^{+ \infty} c (x, t) d x = \int_{- \infty}^{+ \infty} c (x, t_{0}) d x = M_{0} .

(A.9)

This condition defines the constant $G$ as;

G = \frac{M_{0}}{\sqrt{4 π D}} .

(A.10)

The solution can be finally written as;

C (x, t) = \frac{M_{0}}{\sqrt{4 π D t}} \exp (- \frac{x^{2}}{4 D t}) ■ .

(A.11)

Appendix ADerivation of the Thin-Film Solution

A.1 Derivation of the Thin-Film Solution

Appendix A
Derivation of the Thin-Film Solution