Appendix
Appendix A: novel solution with a different set of dimensionless variables
The transport equations, Eqs. 1 and 2, may be written in terms of a different set of dimensionless variables. Let’s define the following new set of dimensionless variables.
$$ \begin{array}{cc}\hfill {t}_D=\frac{\phi_m^2{R}_m{D}_mt}{R^2{b}^2}\hfill & \hfill\;\lambda =\frac{k{R}^2{b}^2}{\phi_m^2{R}_m{D}_m}\hfill \end{array} $$
(A1)
$$ \begin{array}{cc}\hfill {x}_D=\frac{R_m{\phi}_m^2{D}_mx}{Ru{b}^2}\hfill & \hfill {z}_D=\frac{R_m{\phi}_mz}{Rb}\hfill \end{array} $$
(A2)
Based on the above dimensionless variables, Eqs. 1 and 2 reduce to
$$ \frac{\partial {C}_D}{\partial {t}_D}+\frac{\partial {C}_D}{\partial {x}_D}+\lambda {C}_D-{\left.\frac{\partial {C}_{mD}}{\partial {z}_D}\right]}_{z_D=0}=0 $$
(A3)
$$ \frac{\partial {C}_{mD}}{\partial {t}_D}+\lambda {C}_{mD}-\frac{\partial^2{C}_{mD}}{\partial {z}_D^2}=0 $$
(A4)
In terms of the dimensionless variables, the related initial and boundary conditions given in Eqs. 13 and 15 remain unchanged but Eqs. 14 and 6 becomes the following:
$$ {C}_D= \exp \left(-\lambda {t}_D\right)\;\mathrm{at}\kern0.62em {x}_D=0 $$
(A5)
$$ \frac{\partial {C}_{mD}}{\partial {z}_D}=0\kern0.5em \mathrm{at}{z}_D={a}_D $$
(A6)
Closely following the same method of solution, namely Laplace transform and binomial series expansion employed to solve Eqs. 11 and 12, we obtain the following complete solution in terms of the new variables:
$$ {\overline{C}}_D=\left[\begin{array}{l}{\displaystyle \sum_{n=0}^{\infty }{\left(-1\right)}^nF(q)}+\\ {}{\displaystyle \sum_{n=1}^{\infty }{\left(-1\right)}^{n+1}n{x}_D{G}_1(q)}+\\ {}{\displaystyle \sum_{n=2}^{\infty }}{\displaystyle \sum_{m=2}^n{\left(-1\right)}^{n+m}}\left(\begin{array}{l}n\\ {}m\end{array}\right)\frac{x_D^m}{m!}G(q)\end{array}\right] $$
(A7)
where
$$ F(q)= \exp \left(-q{x}_D\right)\left\{\begin{array}{l}\frac{ \exp \left(-{\alpha}_1\sqrt{q}\right)}{q}+\\ {}\frac{ \exp \left(-{\alpha}_2\sqrt{q}\right)}{q}\end{array}\right\} $$
(A9)
$$ {G}_1(q)= \exp \left(-q{x}_D\right)\frac{2}{\sqrt{q}}\left(\begin{array}{l} \exp \left(-{\alpha}_1\sqrt{q}\right)+\\ {} \exp \left(-{\alpha}_1\sqrt{q}\right)\end{array}\right) $$
(A10)
$$ G(q)=\left\{\begin{array}{l}{2}^m{q}^{\left(m-1\right)/2-1/2}\\ {} \exp \left(-q{x}_D\right)\left(\begin{array}{l} \exp \left(-{\alpha}_1\sqrt{q}\right)+\\ {} \exp \left(-{\alpha}_2\sqrt{q}\right)\end{array}\right)\end{array}\right\} $$
(A11)
$$ {\alpha}_1=2n{a}_D+{x}_D\kern0.5em \mathrm{and}\kern0.5em {\alpha}_2=2\left(n+1\right){a}_D+{x}_D $$
(A12)
Note that Eqs. A9–A11 differ from their counterparts Eqs. 32, 33, and 34, only with exp(−qx
D
) which is absent in Eqs. 32, 33, and 34. In the Laplace inversion process, this additional term introduces a Heaviside step function and shift in the time variable to yield:
$$ \begin{array}{l}F\left({t}_D\right)=H\left({t}_D-{x}_D\right)\kern0.36em \exp \left(-\lambda {t}_D\right)\\ {}\kern2.88em \left\{\begin{array}{l} erfc\left(\frac{\alpha_1}{2\sqrt{t_D-{x}_D}}\right)+\\ {} erfc\left(\frac{\alpha_2}{2\sqrt{t_D-{x}_D}}\right)\end{array}\right\}\end{array} $$
(A13)
$$ \begin{array}{l}{G}_1\left({t}_D\right)=H\left({t}_D-{x}_D\right)\kern0.36em \exp \left(-\lambda {t}_D\right)\\ {}\kern3em \left\{\begin{array}{l}\frac{2 \exp \left(-\frac{\alpha_1^2}{4\left({t}_D-{x}_D\right)}\right)}{\sqrt{\pi \left({t}_D-{x}_D\right)}}+\\ {}\frac{2 \exp \left(-\frac{\alpha_2^2}{4\left({t}_D-{x}_D\right)}\right)}{\sqrt{\pi \left({t}_D-{x}_D\right)}\kern0.24em }\end{array}\right\}\end{array} $$
(A14)
$$ \begin{array}{l}G\left({t}_D\right)=H\left({t}_D-{x}_D\right)\kern0.96em \\ {}\kern1.92em \exp \left(-\lambda {t}_D\right)\kern0.36em \frac{2^{\left(m+1\right)/2}}{\sqrt{\pi}\;{\left({t}_D-{x}_D\right)}^{m/2}\;}\\ {}\kern1.8em \left\{\begin{array}{l}{e}^{-\frac{\alpha_1^2}{4\left({t}_D-{x}_D\right)}}H{e}_{m-1}\left(\frac{\alpha_1}{\sqrt{2\left({t}_D-{x}_D\right)}}\right)+\\ {}{e}^{-\frac{\alpha_2^2}{4\left({t}_D-{x}_D\right)}}H{e}_{m-1}\left(\frac{\alpha_2}{\sqrt{2\left({t}_D-{x}_D\right)}}\right)\end{array}\right\}\end{array} $$
(A15))
Writing both Eqs. 35, 36, and 37 multiplied by the term exp(-λ2
x
D) and Eqs. A13–A15 in terms of dimensional variables, one can see that they have the same arguments. The only difference the Heaviside step function, H(t
D
− x
D
), appearing in Eqs. A3–A15 is implicitly included in Eqs. 35, 36, and 37 as Eqs. 35, 36, and 37 are defined only for positive t
D values.
This verifies that dimensionless variables are correctly employed and Eq. 11 is indeed an unsteady state equation. Only algebra is minimized as an advantage.
Appendix B: Laplace inversion relation by Skopp and Warrick
Skopp and Warrick (1974) provide the following Laplace transform relation in their notable work.
$$ {F}_1(s)= \exp \left[-{x}_D\sqrt{q} \tanh \left(\sqrt{q}\right)\right] $$
(B1)
First, they derive the inverse transform of Eq. B1 given by Eq. B2.
$$ {F}_1=\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\lambda \exp \left(-{\lambda}_R\right)\;Cos\left(\frac{\lambda^2\tau }{2}-{\lambda}_I\right)}\kern0.24em d\lambda $$
(B2)
where
$$ {\lambda}_R=\frac{x_D\omega }{2}\frac{Sinh\left(\omega \right)- Sin\left(\omega \right)}{Cosh\left(\omega \right)+Cos\left(\omega \right)} $$
(B3)
$$ {\lambda}_I=\frac{x_D\omega }{2}\frac{Sinh\left(\omega \right)+ Sin\left(\omega \right)}{Cosh\left(\omega \right)+Cos\left(\omega \right)} $$
(B4)
Let us rewrite Eq. 23 as follows:
$$ {\overline{C}}_D={e}^{-{\lambda}_2{x}_D}\frac{ \exp \left[-{x}_D\sqrt{q} \tanh \left(\sqrt{q}\right)\right]}{q} $$
(B5)
and its Laplace space variable q,
$$ q=s+{\lambda}_1 $$
(B6)
Based on the above solution given by Eq. B2 and using the shifting and convolution properties of Laplace transform (Abramowitz and Stegun 1972; Bateman and Erdelyi 1954) the inverse of Eq. B5 can be written as follows:
$$ \begin{array}{l}{C}_D={e}^{-{\lambda}_2{x}_D}\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\lambda \exp \left(-{\lambda}_R\right)d\lambda}\;\\ {}\kern1.44em {\displaystyle \underset{0}{\overset{t_D}{\int }} \exp \left(-{\lambda}_1\tau \right)Cos\left(\frac{\lambda^2\tau }{2}-{\lambda}_I\right)\;}d\tau\;\end{array} $$
(B7)
carrying out the integration with respect to τ, we obtain the following:
$$ {C}_D=\frac{2{e}^{-{\lambda}_2{x}_D}}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-{\lambda}_R}\;\frac{\left[g\left({t}_D\right)+g(0)\right]}{\lambda_1^2+\frac{\lambda^2}{4}}}d\lambda $$
(B8)
where
$$ g\left({t}_D\right)={e}^{-{\lambda}_1{t}_D}\left(\begin{array}{l}\frac{\lambda^2}{2}\mathrm{Sin}\left(\frac{\lambda^2{t}_D}{2}-{\lambda}_I\right)-\\ {}\;{\lambda}_1Cos\left(\frac{\lambda^2{t}_D}{2}-{\lambda}_I\right)\end{array}\right) $$
(B9)
$$ g(0)=\left(\frac{\lambda^2}{2} Sin\left({\lambda}_I\right)+{\lambda}_1Cos\left({\lambda}_I\right)\right) $$
(B10)