Combination of Fix Law and Continuous Equation

Homogeneous reactor and neutron diffusion equation

K. Devan , Abhitab Bachchan , in Physics of Nuclear Reactors, 2021

4.5.2 Fick's law limitations

Fick's law is derived based on various assumptions. Therefore, it has certain limitations to apply for various systems. In such cases, suitable modifications are necessary. Let us discuss each of the assumptions one by one.

1)

Finite medium

Fick's law is derived for an infinite homogeneous medium. The integrand in Eq. (4.16) has an exponential function which dies off quickly with distance. So, the neutrons coming from a point of few mean free paths away will have negligible contribution to the integral. For finite media, this law is valid up to the point which is away from the edges of the medium by few mean free paths.

2)

Nonuniform medium

For a nonuniform medium, a re-evaluation of the derivation of Fick's law is needed. The collision rate Σ _s ϕ is space dependant because of both Σ _s and ϕ vary in the medium More detailed calculations have shown that contributions to neutron current from localized neutron scattering collisions exactly cancels the effects due to higher attenuation, provided that Σ _s   ≫   Σ _a . At the boundary between two media of different scattering properties, Fick's law is still valid, provided that the sharp change in scattering properties does not lead to the rapid variation of flux.

3)

Proximity to sources or sinks

In the case of neutron sources present at few mean free paths away from the considered volume, the Fick's law is still valid. It is because of the attenuation factor, a very small number of source neutrons will contribute to the flux and Fick's law is still valid. Similarly, Fick's law is valid at a few mean free paths away from sinks like vacuum boundary.

4)

Anisotropic scattering in LAB system

The assumption of isotropic scattering in the LAB system, in general, is not true. This is true only if scattering is from heavy nuclei at lower energies. With moderate scattering anisotropy, Fick's law can still be used if the diffusion coefficient is modified based on transport theory as:

(4.22b) $D=\frac{1}{3{\mathrm{\Sigma }}_t\left(1-\overline{\mu }\right)\left(1-4{\mathrm{\Sigma }}_a/5{\mathrm{\Sigma }}_t+\cdots \right)}$

If Σ _a   ≪ Σ _t , then above expression becomes

(4.22c) $D=\frac{1}{3{\mathrm{\Sigma }}_t\left(1-\overline{\mu }\right)}=\frac{1}{3{\mathrm{\Sigma }}_{\mathit{tr}}}=\frac{{\lambda }_{\mathit{tr}}}{3}$

5)

Highly absorbing medium

Derivation of Fick's law assumes that the neutron flux, $\varphi \left(\overrightarrow{\mathrm{r}}\right)$ , is slowly varying. In case of large spatial variation of $\varphi \left(\overrightarrow{\mathrm{r}}\right)$ , higher-order terms have to be included in Taylor's series expansion of neutron flux. But the contribution from second-order terms cancels out and contribution from third-order terms are small beyond a few mean free paths. So, Fick's law is valid if the medium is not too absorbing in nature. In the case of highly absorbing media, it is advisable to use the exact neutron transport theory.

6)

Time-dependent flux

The time-independent neutron flux was assumed for the derivation of Fick's law. If ϕ is not constant, the flux in the integral used to compute the neutron current should be evaluated at an earlier time period. It is because of the finite time needed for the neutron to travel from the collision site to the point where current is evaluated. As already indicated, neutrons from the regions of few scattering mean free paths away do not contribute significantly to the neutron current. So the requirement of time independence is possible to be relaxed provided the fractional change in ϕ is small during the time interval of neutron to travel about three scattering mean free paths (λ _s ). The time needed for a slow neutron to travel three mean free paths of distance by assuming λ _s = 1 cm is

$\mathrm{\Delta }t\approx \frac{3{\lambda }_s}{v}\approx \frac{3\times 1\mathrm{cm}}{2\times {10}^5\mathrm{cm}/\mathrm{s}}=1.5\times {10}^{-5}s$

Even assuming a high rate of fractional flux change, say 10%/s, then

$\frac{\mathrm{\Delta }\varphi }{\varphi }\approx \frac{\mathrm{\Delta }\varphi /\varphi }{\mathrm{\Delta }t}\times \mathrm{\Delta }t\approx 0.1\times \mathrm{\Delta }t=1.5\times {10}^{-6}$

This is a very small fractional change of neutron flux amplitude during the time of travel for three scattering mean free paths of distance.

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Transport phenomena in fuel cells

Bengt Sundén , in Hydrogen, Batteries and Fuel Cells, 2019

9.4.1 Charge transport by diffusion

Fick's law gives the mass transport of charge flux due to a concentration gradient, i.e.,

(9.38) ${\text{J}}_{\text{i}}=-{\text{D}}_{\text{i}}·\nabla {\text{C}}_{\text{i}}$

where D_i is the diffusion coefficient of an ion in the electrolyte and C_i is the charge concentration.

Using molar charge flux instead, Eq. (9.38) is written as

(9.39) ${\text{J}}_{\text{i}}=-\left({\text{z}}_{\text{i}}\text{F}\right){\text{D}}_{\text{i}}·\nabla {\text{C}}_{\text{i}}$

where z_i is the charge number of the charge carrier and F is the Faraday constant. The charge number z_i is −1 for an electron (e⁻) charge and for a proton or hydrogen ion it is +1.

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Compartmental Modeling

John D. Enderle PhD , in Introduction to Biomedical Engineering (Third Edition), 2012

7.3.2 Fick's Law of Diffusion

Fick's Law of diffusion describes the time course of the transfer of a solute between two compartments that are separated by a thin membrane, given by

(7.1) $\frac{\mathit{dq}}{\mathit{dt}}=-\mathit{DA}\frac{\mathit{dc}}{\mathit{dx}}$

where

q = quantity of solute

A = membrane surface area

c = concentration

D = diffusion coefficient

dx = membrane thickness

$\frac{\mathit{dc}}{\mathit{dx}}$ = concentration gradient

Consider the system of two compartments shown in Figure 7.4, where

Figure 7.4. Two-compartment model with a membrane of width Δx = dx.

V ₁ and V ₂ are the volumes of compartments 1 and 2

q ₁ and q ₂ are the quantities of solute in compartments 1 and 2

c ₁ and c ₂ are the concentrations of solute in compartments 1 and 2

and an initial amount of solute, Q ₁₀, is dumped into compartment 1. After approximating the derivative $\frac{\mathit{dc}}{\mathit{dx}}$ as $\frac{c_1-c_2}{\mathrm{\Delta }x}$ , the rate of change of solute in compartment 1 is given by

(7.2) ${\dot{q}}_1=-\mathit{DA}\frac{\mathit{dc}}{\mathit{dx}}=-\mathit{DA}\frac{\left(c_1-c_2\right)}{\mathrm{\Delta }x}$

Next, the quantity is converted into a concentration by

(7.3) $q_1=V_1{c}_1$

and after differentiating Eq. (7.3), gives

(7.4) ${\dot{q}}_1=V_1{\dot{c}}_1$

Substituting Eq. (7.4) into Eq. (7.2) yields

(7.5) $V_1{\dot{c}}_1=\frac{-\mathit{DA}}{\mathrm{\Delta }x}\left(c_1-c_2\right)$

With the transfer rate K defined as

$K=\frac{\mathit{DA}}{\mathrm{\Delta }x}$

when substituted into Eq. (7.5) yields

(7.6) ${\dot{c}}_1=-\frac{K}{V_1}\left(c_1-c_2\right)$

From conservation of mass, we have

$Q_{10}=q_1+q_2$

which after converting to a concentration gives

(7.7) $V_1{C}_{10}=V_1{c}_1+V_2{c}_2$

where $C_{10}=\frac{Q_{10}}{V_1}$ is the initial concentration in compartment 1 due to the initial amount of solute dumped into the compartment.

The concentration in compartment 2 is found from Eq. (7.7) as

(7.8) $c_2=\frac{C_{10}{V}_1-V_1{c}_1}{V_2}$

which when substituted into Eq. (7.6) gives

${\dot{c}}_1=\frac{-K}{V_1{V}_2}\left(V_2{c}_1-V_1{C}_{10}+V_1{c}_1\right)=\frac{K{C}_{10}}{V_2}-\frac{K{c}_1}{V_1{V}_2}\left(V_1+V_2\right)$

or

(7.9) ${\dot{c}}_1+K\left(\frac{V_1+V_2}{V_1{V}_2}\right)c_1=\frac{K{C}_{10}}{V_2}$

This is a first-order linear differential equation with forcing function

(7.10) $f\left(t\right)=\frac{K{C}_{10}}{V_2}$

and initial condition c ₁(0) = C ₁₀.

Assume for simplicity that V ₁ = V ₂. Then Eq. (7.9) becomes

(7.11) ${\dot{c}}_1+\frac{2K}{V_1}{c}_1=\frac{K{C}_{10}}{V_1}$

To solve Eq. (7.11), note that the root is $-\frac{2K}{V_1}$ and the natural solution is

(7.12) $c_{1_n}=B_1{e}^{\frac{-2\mathit{Kt}}{V_1}}$

where B ₁ is a constant to be determined from the initial condition. The forced response has the same form as the forcing function in Eq. (7.9), $c_{1_f}=B_2$ , which when substituted into Eq. (7.11) yields

$\frac{2K}{V_1}{B}_2=\frac{K{C}_{10}}{V_1}$

or

$B_2=\frac{C_{10}}{2}$

Thus, the complete response is

$c_1=c_{1_n}+c_{1_f}=B_1{e}^{-\frac{2\mathit{Kt}}{V_1}}+\frac{C_{10}}{2}$

To find B ₁, the initial condition is used

$c_1(0)=C_{10}={B_1{e}^{-\frac{2\mathit{Kt}}{V_1}}|}_{t=0}+\frac{C_{10}}{2}=B_1+\frac{C_{10}}{2}$

or

$B_1=\frac{C_{10}}{2}$

The complete solution is

$c_1=\frac{C_{10}}{2}\left(e^{-\frac{2\mathit{Kt}}{V_1}}+1\right)$

for t ≥ 0. Note that the concentration in compartment 2 is found using Eq. (7.8) as

$c_2=\frac{V_1{C}_{10}-V_1{c}_1}{V_2}=\frac{C_{10}}{2}\left(1-e^{-\frac{2\mathit{Kt}}{V_1}}\right)$

If $V_1\ne V_2,$ then

$c_1=\frac{C_{10}}{\left(V_1+V_2\right)}\left(V_2{e}^{-K\frac{\left(V_1+V_2\right)}{V_1{V}_2}t}+V_1\right)u(t)$

and

$c_2=\frac{V_1{C}_{10}}{(V_1+V_2)}\left(1-e^{-K\frac{(V_1+V_2)}{V_1{V}_2}t}\right)u(t)$

At steady state, the concentrations on either side of the membrane are equal. In fact, if the volumes of the compartments are not equal, the concentrations at steady state are still equal. This should also be clear using Eq. (7.6); setting the derivative term equal to zero gives c ₁ (∞) = c ₂ (∞). Note, however, that the number of moles of solute will be greater in the larger compartment.

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Review of Diode Physics and the Ideal (and Later, Nonideal) Diode

Marc T. Thompson Ph.D. , in Intuitive Analog Circuit Design (Second Edition), 2014

Diffusion

Fick's law of diffusion describes how particles under random thermal motion tend to spread ¹⁸ from a region of higher concentration to a region of lower concentration. This principle is illustrated by opening a perfume bottle in the corner of a closed room. If you wait long enough, the perfume odor will permeate the room because the perfume molecules have diffused from one side of the room to the other, from a region of high concentration to a region of low concentration. Mathematically, three-dimensional diffusion is characterized by Fick's diffusion law, which states that the diffusion flux is proportional to the concentration gradient, as:

(3.9) $F=-D\nabla C$

where C is the concentration of the diffusing particles, F is the diffusion flux (particles per square meter per second), and D is the diffusion constant, which has units of cm² per second. For a one-dimensional problem, Fick's law reduces to:

(3.10) $F=-D\frac{dC}{dx}$

Therefore, charged particles tend to flow down a concentration gradient. This diffusion process also occurs in PN junctions whenever there are gradients of free charged carriers.

We can work out the form (but not the detail) of Fick's law by considering a thought experiment. Consider a region of space where there is a changing concentration of free charges, in this case, holes (Figure 3.6). The holes are undergoing random thermal motion. For instance, at x  =   −x ₀, on an average, one half of the holes are traveling to the left and one half are traveling to the right. The same is true at x  =   +x ₀. In order to find the net current at x  =   0, we recognize that the current at x  =   0 is the sum of the current from the left plus the current from the right, or:

FIGURE 3.6. Hole concentration gradient, resulting in a net hole flux from left to right.

(3.11) $J(x=0)=k\left[p(x=-x_{\text{0}})-p(x=+x_{\text{0}})\right]=-k\left(2x_{\text{0}}\right)\left[\frac{dp}{dx}\right]$

where k is some constant that makes the units work out. Note that the net current at x  =   0 is proportional to the difference in the concentrations at x  =   −x ₀ and x  =   +x ₀. Through a mathematical manipulation ¹⁹ above we see that the current is also proportional to dp/dx or the gradient of the concentration. We now recognize the familiar form for diffusion current for holes:

(3.12) $J_{\text{h},\text{diff}}=-q{D}_{\text{p}}\frac{dp}{dx}$

where q is the electronic charge and D _p is the diffusion constant ²⁰ for holes. Using a similar derivation, we can find the electron diffusion current:

(3.13) $J_{\text{e},\text{diff}}=q{D}_{\text{n}}\frac{dn}{dx}$

Let us do a diffusion thought experiment, illustrated in Figure 3.7. At time t  =   0, a high concentration of particles (in this case, electrons) exists at x  =   0. These particles can be created by illuminating a piece of semiconductor, or by other mechanisms. The particles are in random thermal motion; some diffuse to the left and to the right. The concentration of particles n(x, t) at various times is shown. At t  = t ₁, the maximum concentration at x  =   0 has dropped, and the particles have spread to the left and the right. Further smearing of the particle concentration occurs at t ₂ and t ₃. As time reaches infinity, the concentration is the same everywhere, and diffusion ends. A closed-form solution for this diffusion problem exists; ²¹ the electron concentration everywhere is:

FIGURE 3.7. Illustration of a diffusion thought experiment. At time t  =   0 there is a very high concentration of particles at x  =   0. The particle concentration spreads out and varies with time as shown, with t ₃  > t ₂  > t ₁, etc.

(3.14) $n(x,t)=\frac{A}{\sqrt{4\mathrm{\pi }{D}_{\text{n}}t}}{\mathrm{e}}^{\frac{x^2}{4D_{\text{n}}t}}+n_{\text{0}}$

where A is a constant, D _n is the diffusion constant for electrons, and n ₀ is the equilibrium concentration of electrons.

In a famous experiment devised in 1949 called the Shockley–Haynes experiment ²² (Figure 3.8(a)), an area in a piece of semiconductor material was illuminated while an electric field E _x was applied. The electric field is proportional to the voltage applied to the left side of the sample, and the electric field points from left to right.

FIGURE 3.8. (a) Cartoon illustration of the Haynes–Shockley experiment. (b) The concentration of extra holes (p ′(x, t)) moves to the right with speed μ _p ग _x where μ _p is the mobility of the holes.

At time t  =   0, a light source is turned on and creates a high concentration P ₀ of holes at x  =   0. The light is turned off and the concentration of holes diffuses due to the concentration gradient and drifts to the right due to the applied electric field. A detector is used to measure the hole concentration at x  = L. The solution for the hole concentration is given by:

(3.15) $p(x,t)=\frac{P_{\text{0}}}{2\sqrt{\mathrm{\pi }{D}_{\text{p}}t}}{\mathrm{e}}^{-\left[\frac{{(x-{\mu }_{\text{p}}{E}_{\text{x}}t)}^2}{4D_{\text{p}}t}+\frac{t}{{\tau }_{\text{p}}}\right]}+p_{\text{no}}$

The evolution of the concentration of holes in the sample is shown in Figure 3.8(b). Using this result we can find the drift coefficient for holes μ _p by viewing how fast the pulse center moves from left to right. The drift coefficient is:

(3.16) ${\mu }_{\text{p}}=\frac{v_{\text{x}}}{E_{\text{x}}}$

where v _x is the velocity of the center of the pulse from left to right. The diffusion coefficient D _p can be found by viewing how fast the pulse spreads.

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SURFACE AND INTERFACE ANALYSIS AND PROPERTIES

Kathleen J. Stebe , Shi-Yow Lin , in Handbook of Surfaces and Interfaces of Materials, 2001

APPENDIX A DERIVATION FOR EQUATION (3.5)

Fick's law in spherical coordinate is

(A.1) $\frac{D}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial C}{\partial r}\right)=\frac{\partial C}{\partial t}(r>b,t>0)$

with initial conditions

(A.2a) $C(r,t)=C_{\infty }(r>b,t=0)$

(A.2b) $\Gamma (t)={\Gamma }_b(t=0)$

and boundary conditions

(A.3a) $C(r,t)=C_{\infty }(r\to \alpha ,t>0)$

(A.3b) $d\Gamma /dt=D\left(\partial C/\partial r\right)\left(r=b,t>0\right)$

(A.3c) $C(r,t)=C_s(t)(r=b,t)$

Define dimensionless variables:

$\begin{array}{c}\theta =\frac{C_{\infty }-C}{C_{\infty }},\tau =\frac{t}{b^2/D},z=r/b,\\ \text{and}{\Gamma }^{*}=\frac{\Gamma -{\Gamma }_b}{{\Gamma }_{\infty }}\end{array}$

Here, Γ_∞ is the maximum surface concentration. Equation (A.1) becomes

(A.4) $\frac{\partial \theta }{\partial \tau }=\frac{1}{z^2}\frac{\partial }{\partial z}\left(z^2\frac{\partial \theta }{\partial z}\right)$

Initial and boundary conditions become

(A.5a) $\theta (z,\tau )=0(z>1,\tau =0)$

(A.5b) ${\Gamma }^{*}(z,\tau )=0(z=1,\tau =0)$

(A.6a) $\theta (z,t)=0(z\to \infty ,\tau >0)$

(A.6b) $\frac{{\Gamma }_{\infty }}{C_{\infty }b}\frac{d{\Gamma }^{*}}{d\tau }=-\frac{\partial \theta }{\partial z}(z=1,\tau >0)$

(A.6c) $\theta (z,t)=\frac{C_{\infty }-C_s}{C_{\infty }}={\theta }_1(\tau )(z=1,\tau )$

Taking Laplace transform with respect to time, Eqs. (A.4), (A.6b) and (A.6c) become

(A.7) $s\overline{\theta }(z,s)-\theta (z,\tau =0)=\frac{1}{z^2}\frac{d}{d_z}\left(z^2\frac{d\overline{\theta }}{dz}\right)$

(A.8a) ${\frac{{\Gamma }_{\infty }}{C_{\infty }b}\left[s{\overline{\Gamma }}^{*}(s)-{\Gamma }^{*}(0)\right]=-\frac{d\overline{\theta }(z,s)}{dz}|}_{z=1}$

(A.8b) $\overline{\theta }(z,s)={\overline{\theta }}_1(s)(z=1,s)$

According to Eq. (A.5b), Γ*(0) = 0; therefore, Eq. (A.8a) becomes

(A.8c) ${\frac{{\Gamma }_{\infty }s}{C_{\infty }b}{\overline{\Gamma }}^{*}(s)=-\frac{d\overline{\theta }(z,s)}{dz}|}_{z=1}$

Apply initial condition equation (A.5a), θ(z, τ = 0) = 0 and Eq. (A.7) then becomes

(A.9) $s\overline{\theta }(z,s)\frac{1}{z^2}\frac{d}{dz}\left(z^2\frac{d\overline{\theta }}{dz}\right)$

Equation (A.9) is a second-order ordinary differential equation of $\overline{\theta }\left(z\right)$ with variable coefficient. We can solve Eq. (A.9) by assigning a new variable f(z, s).

$\overline{\theta }(z)=\frac{f(z)}{z}$

The left- and right-hand sides of Eq. (A.9) become sf(z)/z and 1/z(d ² f/dz ²), respectively. Therefore, Eq. (A.9) becomes

(A. 10) $\frac{d^{2}f}{d{z}^2}=sf$

Equation (A. 10) can be solved easily by assuming

(A.11) $f=e^{\lambda z}$

Substitute Eq. (A.11) into Eq. (A.10) and solve for eigenvalue λ

$\lambda =\pm s^{1/2}$

Therefore,

(A.12) $f=A{e}^{-\sqrt{sz}+Be\sqrt{sz}}$

(A.13) $\overline{\theta }(z,s)=A\frac{e^{-\sqrt{sz}}}{z}+B\frac{e^{-\sqrt{sz}}}{z}$

By applying boundary condition equation (A.6a), coefficient B must be equal to zero.

Therefore, Eq. (A.13) becomes

(A.14) $\overline{\theta }(z,s)=A\frac{e^{-\sqrt{sz}}}{z}$

Apply boundary condition equation (A.8b) $\overline{\theta }$ (z = l, s) = ${\overline{\theta }}_1$ (s), and Eq. (A.14) becomes

${\overline{\theta }}_1(s)=A{e}^{-\sqrt{s}}$

$\therefore A={\overline{\theta }}_1(s)e^{\sqrt{s}}$

Therefore,

(A.15) $\overline{\theta }\left(z,s\right)=\frac{{\overline{\theta }}_1(s)}{z}{e}^{\sqrt{s}(1-z)}$

(a)Obtain the bulk concentration distribution C(z, τ):

By taking an inverse Laplace transform on Eq. (A. 15),

(A.16) $L^{-1}\left[F(s)G\left(s\right)\right]=f(t)*g(t)={\int }_0^{t}f(t-\omega )g(\omega )$

we get

(A.17) $\begin{array}{c}\theta (z,\tau )=L^{-1}\left[\overline{\theta }\left(z,s\right)\right]=L^{-1}\left[\frac{{\overline{\theta }}_1(s)}{z}{e}^{\sqrt{s}(1-z)}\right]\\ =L^{-1}\left[\frac{{\theta }_1(s)}{z}\right]*L^{-1}\left[e^{-\sqrt{s}(z-1)}\right]\\ ={\int }_0^1\frac{1}{z}{\theta }_1(t-\omega )\frac{(z-1)}{2\sqrt{\pi {\omega }^3}}{e}^{-\frac{{\left(z-1\right)}^2}{4\omega }d\omega }\end{array}$

Note that

(A.18) $L^{-1}\left[e^{-a\sqrt{s}}\right]=\frac{a}{2\sqrt{\pi {\omega }^3}}{e}^{-a^2/4t}$

Therefore,

(A.19) $\frac{C_{\infty }-C(z,\tau )}{C_{\infty }}={\int }_0^1\frac{1}{z}{\theta }_1\left(t-w\right)\frac{\left(z-1\right)}{2\sqrt{\pi {\omega }^3}}{e}^{-{(z-1)}^2/4wdw}$

(b)To obtain the surface concentration distribution Γ*(τ), recall Eq. (A.8c)

(A.20a) $\begin{array}{c}{\frac{{\Gamma }_{\infty }s}{C_{\infty }b}{\overline{\Gamma }}^{*}(s)=-\frac{d\overline{\theta }(z,s)}{dz}|}_{z=1}\\ =\frac{d}{dz}{\left[\frac{{\overline{\theta }}_1(s)}{z}{e}^{\sqrt{s}(1-z)}\right]}_{z=1}\end{array}$

(A.20b) $={\overline{\theta }}_1(s){\left[\frac{e^{\sqrt{s}(1-z)}}{z^2}+\frac{\sqrt{s}}{z}{e}^{\sqrt{s}(1-z)}\right]}_{z=1}$

(A.20c) $={\overline{\theta }}_1(s)\left[1+\sqrt{s}\right]$

Therefore,

(A.21) ${\overline{\Gamma }}^{*}(s)=\frac{C_{\infty }b}{{\Gamma }_{\infty }}\frac{{\overline{\theta }}_1(s)\left(1+\sqrt{s}\right)}{s}$

Now we obtain

(A.22) $\Gamma (\tau )=L^{-1}\left[{\overline{\Gamma }}^{*}(s)\right]=\frac{C_{\infty }b}{{\Gamma }_{\infty }}{L}^{-1}\left[\frac{{\overline{\theta }}_1(s)}{s}+\frac{{\overline{\theta }}_1(s)}{\sqrt{s}}\right]$

Note that

(A.23) $L^{-1}\left[\frac{F\left(s\right)}{s^n}\right]={\int }_0^{t}f(w)\frac{{(t-\omega )}^{n-1}}{\left(n-1\right)!}d\omega$

(A.24) $\therefore L^{-1}\left[\frac{{\overline{\theta }}_1\left(s\right)}{s^n}\right]={\int }_0^t{\theta }_1(\omega )d\omega$

and

(A.25) $L^{-1}\left[\frac{{\overline{\theta }}_1\left(s\right)}{\sqrt{s}}\right]=L^{-1}\left[\frac{1}{\sqrt{s}}\right]*L^{-1}\left[{\overline{\theta }}_1(s)\right]$

Note that

(A.26) $L^{-1}\left[\frac{1}{\sqrt{s}}\right]=\frac{1}{\sqrt{\pi t}}$

Therefore

(B.27) $\Gamma (\tau )=\frac{C_{\infty }b}{{\Gamma }_{\infty }}\left[{\int }_0^{\tau }{\theta }_1(\omega )d\omega +{\int }_0^{\tau }\frac{1}{\sqrt{\pi t}}{\theta }_1(\tau -\omega )d\omega \right]$

According to Eq. (A.6a),

(B.28) $\begin{array}{ccc}{\theta }_1(\tau )& =& \frac{C_{\infty }-C_s(\tau )}{C_{\infty }}\hfill \\ \Gamma (\tau )& =& \frac{b}{{\Gamma }_{\infty }}{\int }_0^{\tau }\left[C_{\infty }-C_s(\omega )\right]d\omega \hfill \\ & & +\frac{b}{{\Gamma }_{\infty }}{\int }_0^{\tau }\frac{1}{\sqrt{\pi \tau }}\left[C_{\infty }(\omega )-C_s(\tau -\omega )\right]d\omega \end{array}$

Integrate Eq. (A.28) and apply the following equation to it:

(B.29) ${\int }_0^{\tau }\frac{C_s\left(\tau -\omega \right)}{\sqrt{\tau }}d\omega =2{\int }_0^{\sqrt{\tau }}{C}_s\left(\tau -\omega \right)d\sqrt{\omega }$

Then change the dimensionless variables back to the dimensional variables. An equation relating the surface concentration, subsurface concentration and time is obtained:

(B.30) $\begin{array}{c}\Gamma (t)={\Gamma }_b+\frac{D}{b}\left[C_{\infty }t-{\int }_0^t{C}_s(\omega )d\omega \right]\\ +2\sqrt{\frac{D}{\pi }}\left[C_{\infty }{t}^{1/2}-{\int }_0^{\sqrt{t}}{C}_s(t-\omega )d\sqrt{\omega }\right]\end{array}$

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Coalification, Gasification, and Gas Storage

Romeo M. Flores , in Coal and Coalbed Gas, 2014

Gas Diffusion and Flow

Gas diffusion and flow in coal beds may be described as a two-phase or binary transfer of gas diffused from the pore system (intramolecular nanopores and micropores–macropores) in the coal matrix to the cleat (fracture) system followed by flow of the gas through the reservoir to boreholes. This gas diffusion-flow system is known as "dual porosity system" and "primary and secondary porosity system" (Mavor & Gunter, 2004; Shi & Durucan, 2005). These workers detailed representation of the order of diffusion within the multiscale pore system in the coal matrix is called the bidisperse diffusion model. The model is a two-step gas diffusion process in the coal matrix occurring as (1) surface gas diffusion in the micropores (e.g. <2 nm) and (2) pore diffusion in the mesopores (2–50 nm) and macropores (>50 nm).

Simply put, gas diffusion in coal occurs in the matrix following Fick's law in contrast to the cleat (fracture) system in which gas is transported in laminar flow obeying Darcy's law. These two laws explain how the sorbed gas is transported from the pores in the coal matrix to the cleat (fracture) system and eventually to the open borehole. The Fick's diffusion law postulates that the diffusive flux goes from a high-concentration area to a low-concentration area proportional to the concentration gradient. The Darcy's law hypothesizes that the apparent velocity of a flowing fluid in a permeable medium is directly proportional to the applied pressure gradient. Thus, the concentration gradient drives the gas diffusion across the porous coal matrix and once the gas enters the cleat system, gas flow is driven by pressure gradient. This two-phase mechanism of gas flow is what separates the unconventional coal from the conventional reservoir (e.g. sandstone).

Although Fick's law is applicable to fluids, in this book, the principle focuses on its application to gas diffusion in coal. Several publications have described Fick's law as applied to coal; however, the reader is specially referred to a recent article by Moore (2012) for an additional excellent explanation. Moore (2012), and Zarrouk (2008) explained Fick's law as

$F=-D\text{d}C/\text{d}x$

where:

F = diffusion flux (kg/[m² s]),

D = effective diffusivity (m² of coal control surface area]/[s]),

dC/dx = concentration gradient ([m³ of gas]/[m³ of coal]/[m length along gradient]),

C is in [m³ of gas]/[m³ of coal], and

x is in meters

The diffusion coefficient is the most critical part of the equation because it is in part a function of the properties of the gas. For example, a gas mixture of light and heavy molecules when passed through a porous–permeable medium is assumed to transport the light gas faster than the heavy gas at the end of the porous–permeable medium. This phenomenon is comparable to the coal bed reservoir, which consists of a binary gas system consisting of methane and carbon dioxide with proportional concentrations of about 9:1. The gases, however, are mainly transported in water-saturated coal reservoirs that behave as aquifers.

The study of Cui et al. (2004) on the differential transport of methane and carbon dioxide in the coal bed aquifers demonstrates some revealing results. Their work indicates that the differential transport of methane and carbon dioxide through the coal aquifer is a function of adsorption equilibrium and water solubility. They argue that although carbon dioxide is 20 times more soluble at lower temperature (50 °C), it is only several times more efficiently transported than methane. They account for the difference with the stronger adsorption of carbon dioxide than methane. This leads to enrichment of carbon dioxide such as seen in the San Juan and Powder River Basins. However, in the Powder River Basin, the enrichment may be complicated by nitrogen concentration probably derived from meteoric water as a result of groundwater recharge (Stricker et al. 2006).

Coal reservoirs are considered aquifers, which contain and transmit quantities of water under normal field condition. Therefore Darcy's law, which governs gas flow in the cleat (fracture) system of water-saturated coal reservoirs, has been unanimously suggested in the literature as an appropriate model to apply in the gas flow transport beyond the porous coal matrix (see Figure 4.29). However, the most recommended model of Darcy's law to apply in the coalbed gas flow is the one-dimensional, single-phase or linear dimensional model (Adeboye, 2011; Pakham, Cinar, & Moreby, 2009). Thus, once the diffused gas from the porous coal matrix enters the cleat (fracture) system, the gas flow or transport is governed by the following Darcy's law equation (Adeboye, 2011):

$\nu =-\kappa /{\mu }_{\text{f}}(\nabla \text{P}-{\rho }_{\text{f}}g)$

where:

ν = velocity flux (volume per time, e.g. ft³/s or m³/s),

κ = function of coal bed permeability (darcy/millidarcy),

μ_f = fluid viscosity (kg/(m s),

$\nabla \text{P}$ = pressure gradient,

ρ _f = fluid density, and

g = acceleration due to gravity.

Darcy laminar flow is valid if cleat or fracture walls are smooth and the openings are unimpeded. However, in natural conditions and more often than not, they are closed by mineralization and/or pinched. Gamson, Beamish, and Johnson (1993) observed that cleats are partly blocked by diagenetic minerals (see Chapter 5). These workers suggested that combined diffusion and flow probably transport the gas under this condition depending on the size of the unfilled channels.

In general, the mechanism of gas flow in the porous and permeable coal is complex in that about 10% of methane in the cleat systems is as occluded gas and about 90% of the methane in the matrix pore system is as adsorbed gas (Basu & Singh, 1994). Presumably the large volume of gas from the microporous coal matrix must be diffused and flow out through the cleat system. However, there is still some debate about the mechanics of gas transport and the idea that gas flow through the cleat systems is the last stage. It has been demonstrated through computer and numerical modeling and field application that when a borehole is drilled through the coal reservoir, the pressure (e.g. confining stress, hydrostatic head) on the hole sidewall drops with respect to the in situ pressure causing radial or linear flow to the borehole (Basu & Singh, 1994). The pressure distribution is a function of time. Flow through the porous coal reservoir is treated at steady state if conditions are unchanged with time. However, depletion of reservoir pressure (e.g. dewatering) causing gas desorption is an unsteady state. The initial gas desorption produces the "free" gas from the cleat (fracture) system, which is at the early stage and not late stage as proposed. When flow becomes stabilized, reservoir conditions return to a steady state. Thus, Darcy's law applies when the initial gas pressure is high; then once the pressure decreases and stabilizes, Fick's law of diffusion applies.

During this later stage, radial gas flow or diffusion following Fick's law is applied within the blocks of coal matrices toward the cleat system (Figure 4.29). Because the face cleats are more continuous than the butt cleats, the ratio of permeability in face cleats over butt cleats is about 1:1 to 1:17.

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Skin: Physiology and Penetration Pathways

Bozena B. Michniak-Kohn , ... Victor M. Meidan , in Delivery System Handbook for Personal Care and Cosmetic Products, 2005

3.3.6 Supersaturation of the Drug Solution

Since Fick's laws of diffusion state that the flux of a molecule is directly proportional to its thermodynamic activity, drug delivery can be optimized by using saturated solutions or suspensions of drugs. Higuchi ^[201] and later Coldman, et al., ^[202] addressed the importance of chemical potential on diffusion and proposed the use of supersaturation to further augment cutaneous transport rates. This concept seems attractive since it does not involve modifying the barrier properties of the stratum corneum by adding potentially irritating chemicals.

In order to test this hypothesis, supersaturated systems have been prepared by three techniques: heating and cooling, ^[203] use of cosolvent mixtures in which the drug has a very low solubility in one component, ^[204] and solvent evaporation methods employing a range of volatile:nonvolatile solvent mixtures. ^[205] Unfortunately, supersaturated formulations commonly exhibit instability and both drugs and salts may precipitate out during manufacturing, storage, or application.

Stability of such supersaturated systems can be promoted by incorporating polymers that act as antinucleant crystal growth inhibitors such as hydroxpropyl methylcellulose (HPMC) and methylcellulose (MC). ^[206] Raghavan, et al., employed supersaturation to maximize the transport of hydrocortisone acetate across a model silicone polymer membrane. ^[206] By adjusting concentrations of the antinucleant crystal growth inhibitors and supersaturation levels, the workers could modulate the crystallization process. This approach was capable of maintaining high flux rates throughout the duration of the experiment. Polymer concentrations up to 1% yielded the maximum molecular flux, while higher concentrations increased vehicle viscosity, therefore reducing flux. Furthermore, by using inhibitor polymer concentrations greater than 2%, it was possible to maintain supersaturation for one week. Kondo, et al., ^[207] adopted a supersaturation approach in order to enhance transdermal nifedipine delivery in rats. Even though the bulk vehicle was stabilized with inhibitor polymers, the formulation formed an appreciable undesirable mass on the skin surface. Overall, the supersaturation approach still has problems associated with stability during manufacture and storage. These problems remain to be resolved. ^[208]

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Physical Background

Zeev Zalevsky , Ibrahim Abdulhalim , in Integrated Nanophotonic Devices, 2010

1.2.3 Currents

According to Fick's law the flux is proportional to the gradient of the concentration:

(1.207) $F=-D\frac{\text{d}n}{\text{d}x}$

where D is the diffusion constant, F is the flux and n is the concentration. The current in the semiconductor can be divided into two types: current due to diffusion and current due to drift. Drift is movement of free carriers due to electrical field. The drift current equals:

(1.208) $I_{drift}=qA{\mu }_{n}nE+qA{\mu }_{p}pE$

where q is the charge of an electron (1.6×10^-19[cb]), A is the cross-section area, μ_n and μ_p are the mobility of electrons and holes respectively. n and p are the free concentrations of electrons and holes and E is the applied electrical field.

The current due to diffusion is equal to:

(1.209) $I_{diffusion}=qAD\frac{\text{d}n}{\text{d}x}-qAD\frac{\text{d}p}{\text{d}x}$

Following the Einstein equation there is a relation between mobility and diffusion constant D:

(1.210) $\frac{D_p}{{\mu }_p}=\frac{D_n}{{\mu }_n}=\frac{k_{B}T}{q}$

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Vapour permeation modelling

M. Giacinti Baschetti , M.G. De Angelis , in Pervaporation, Vapour Permeation and Membrane Distillation, 2015

8.3.1 Diffusivity and mobility

According to Fick's law, governing the majority of membrane separation processes, the diffusive flux through the membrane, J, is proportional to the concentration gradient as reported in the first of Eqn (8.4).

In reality, the driving force for diffusive flux is the chemical potential gradient rather than the concentration gradient, used by Fick because it is a very intuitive and easily measurable quantity. Use of concentration instead of chemical potential leads to some drawbacks, such as the lack of continuity of concentration profile at the interface between fluid and membrane, which requires the introduction of a partition factor. Another drawback is that the diffusivity is not a purely kinetic parameter but is somehow also affected by a thermodynamic (solubility) contribution. Such contribution can give rise to uncommon trends of diffusivity with concentration which cannot be explained by considering only kinetic aspects. An example is given by the behaviour of alcohols diffusivity in poly[1-(trimethylsilyl)-1-propyne] (PTMSP), which has a very peculiar shape, shown in Figure 8.2(a), with maxima and minima that are difficult to explain. Such behaviour is due to the fact that the diffusivity is not only related to mobility but also to solubility behaviour (Doghieri & Sarti, 1997).

Figure 8.2. Data relative to fluid sorption and diffusion in PTMSP at 300   K: □ n-C₅, high-density samples; ■ n-C₅, low-density samples; ○ C₂H₅OH, high-density samples; ● C₂H₅OH, low-density samples; (a) diffusivity; (b) solubility; (c) thermodynamic factor τ; (d) mobility. L: high density: 0.80   g/cm³; low density: 0.75   g/cm³ (Doghieri &amp; Sarti, 1997).

Copyright © 1997 John Wiley &amp; Sons, Inc.

To separate the two contributions, one can define a mobility L that relates the diffusive flux to the chemical potential gradient and is a purely kinetic quantity, in the following way:

(8.7) $J=-\rho \frac{L}{RT}\omega \frac{d\mu }{dx}=-\rho \frac{L}{RT}\omega \left(\frac{\partial \mu }{\partial \omega }\frac{d\omega }{dx}\right)=-\rho \frac{L}{RT}\left(\frac{\partial \mu }{\partial \mathit{ln}\omega }\frac{d\omega }{dx}\right)$

Combining Fick's law and Eqn (8.7), one obtains:

(8.8) $J=-\rho D\frac{d\omega }{dx}=-\rho \frac{L}{RT}\left(\frac{\partial \mu }{\partial \mathit{ln}\omega }\frac{d\omega }{dx}\right)$

then the purely thermodynamic contribution in the previous expression, defined as τ, thermodynamic factor, can be isolated:

(8.9) $\tau \equiv \frac{L}{RT}\left(\frac{\partial \mu }{\partial \mathit{ln}\omega }\right)$

To obtain a clear expression of diffusivity as the product of a purely kinetic factor (mobility L) and a purely thermodynamic factor (τ):

(8.10) $D=L\tau$

The thermodynamic factor τ can be retrieved immediately from a solubility isotherm. Indeed, one can remember that the chemical potential μ can be expressed with respect to a reference state as follows:

(8.11) $\mu ={\mu }^{\text{ref}}+RT\ln a$

where a is the activity of the fluid with respect to a reference state. If the reference state, as often done for vapours, is the pure liquid at the same temperature, one has:

(8.12) $a_i=\frac{f_i(T,p,x_i)}{f_i^0(T,p_{i,\text{SAT}}(T))}\underset{\text{pressure}}{\overset{\text{low}}{\to }}\frac{p_i}{p_{i,\text{SAT}}(T)}$

The activity is therefore the ratio between the partial pressure of the fluid, p _i , and its saturation pressure, p _i,SAT, at the same temperature in the pure state. The thermodynamic factor, then, becomes:

(8.13) $\tau \equiv \frac{L}{RT}\left(\frac{\partial \mu }{\partial \ln \omega }\right)=\frac{L}{RT}\left(\frac{\partial \ln a}{\partial \ln \omega }\right)\cong \frac{L}{RT}\left(\frac{\Delta \ln a}{\Delta \ln \omega }\right)=\frac{L}{RT}\left(\frac{\ln a_1/a_2}{\ln {\omega }_1/{\omega }_2}\right)$

where 1 and 2 are two adjacent equilibrium points on the solubility isotherm. Now the data in Figure 8.2(a) relative to ethanol diffusivity in PTMSP can be treated considering those contributions. In particular, the solubility isotherm for this system (Figure 8.2(b)) can be used to estimate the thermodynamic factor (Figure 8.2(c)) and the mobility (Figure 8.2(d)), that is a smooth decreasing function of concentration, as it is consistent with the nature of fluid diffusion within high FV glassy polymers like PTMSP (Doghieri & Sarti, 1997). The corresponding behaviour of alkanes in PTMSP is also shown in the plots: because of the different chemical affinity between such fluids and the polymer, their thermodynamic behaviour is different from that of alcohols as well as their thermodynamic factor. However, when the mobility is considered, the nature of the two different types of fluids (alkanes and alcohols) plays no role and their behaviours are very similar.

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Diffusive Mass Transfer

Nikolaos D. Katopodes , in Free-Surface Flow, 2019

3.12 Inertia-Moderated Diffusion

The diffusion model based on Fick's law postulates that a point load will spread infinitely fast albeit with an exponential attenuation away from the source. This is a consequence of neglecting the inertia of the solute particles, which are allowed to move in a purely random walk. This may be unrealistic, but leads to satisfactory results in many environmental applications. There are, however, instances where the assumption that the mass flux adjusts instantaneously to the gradient of the concentration may lead to poor results.

Cattaneo (1948) argued that a small relaxation time, ${\tau }_r$ , must elapse before a concentration gradient can have an impact on the mass flux. Therefore, Fick's law should be replaced by the following expression

(3.219) $(1+{\tau }_r\frac{\partial }{\partial t})\mathbf{q}=-D\mathrm{\nabla }C$

This is known as Cattaneo's equation. The flux vector in Eq. (3.219) needs to be considered in conjunction with the conservation of solute mass equation, i.e.

(3.220) $\frac{\partial C}{\partial t}+\mathrm{\nabla }\cdot \mathbf{q}=kC$

where k is the velocity of a typical first-order reaction. This is a first-order hyperbolic system describing the propagation of concentration wave fronts in three-dimensional space and time. To simplify the analysis, let us restrict the problem to one space dimension. Then, differentiation of Eq. (3.219) with respect to time, and Eq. (3.220) with respect to distance, and elimination of the mixed derivatives leads to the following inertia-moderated, diffusion-reaction equation

(3.221) ${\tau }_r\frac{{\partial }^{2}C}{\partial t^2}+(1-{\tau }_{r}k)\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial x^2}+kC$

For a conservative solute, $k=0$ , Eq. (3.221) reduces to the well-known telegraph equation, as follows

(3.222) $\frac{1}{c^2}\frac{{\partial }^{2}C}{\partial t^2}+\frac{1}{D}\frac{\partial C}{\partial t}=\frac{{\partial }^{2}C}{\partial x^2}$

where $c=\sqrt{D/{\tau }_r}$ is the propagation speed of a small disturbance in the concentration. Eq. (3.222) is named following its original derivation for propagation of a voltage signal along a transmission line. Compared to the standard wave equation, i.e. Eq. (II-5.26), the telegraph equation represents a damped wave equation with a damping constant equal to $1/D$ . In general, Eq. (3.222) represents a wave-like dissipative process, in which solute mass travels with a finite speed while the concentration attenuates. In the limit, as ${\tau }_r\to 0$ , the propagation speed becomes infinite, and we recover the classical diffusion equation. If $D\to 0$ , while $t_r$ remains finite, the telegraph equation approaches the linear wave equation, in which the disturbance propagates without attenuation.

Physically, Eq. (3.222) describes a heat wave, which is a rather uncommon phenomenon, as heat conduction ordinarily resembles diffusion, and is accurately described by Fourier's law. A diffusion wave is also an improbable phenomenon, as actual measurements of mass transfer are accurately described by the classical diffusion equation. Nevertheless, the Cattaneo model allows us to consider the impact of inertia on mass transfer.

Mathematically, the telegraph equation poses an increased demand on the initial data due to the second derivative in time. Thus, not only the concentration must be known at the starting time of any problem, but also the mass flux. This further complicates the acceptance of Eq. (3.222) as a valid model for solute mass transport. Furthermore, in addition to the increased demands on initial data, we must also ensure that the damping coefficient, i.e. the term in parenthesis in Eq. (3.221) remains positive. For otherwise the concentration may become negative, which is unrealistic. This implies that ${\tau }_{r}k<1$ , i.e. the relaxation time may not be selected arbitrarily in order to prevent non-physical solutions.

Let us consider a unit impulse load placed at the origin of an infinitely long channel at $t=0$ . If both $C(x,0)$ and $\frac{\partial C}{\partial t}$ are equal to zero initially, the telegraph equation describes a bi-directional wave propagating with speed $c=\pm \sqrt{D/{\tau }_r}$ , with the dissipation of the impulse load limited in the wake of the wave. The analytical solution is complicated, but it can be shown that the concentration profile resulting from Eq. (3.222) is given by (Zauderer, 2006, p. 467)

(3.223) $C(x,t)=\frac{1}{\sqrt{4D{\tau }_r}}{e}^{-t/2{\tau }_r}{I}_0\left(\sqrt{\frac{D{t}^2-{\tau }_r{x}^2}{4D{\tau }_r^2}}\right)$

where $I_0$ is the modified Bessel function. The concentration is plotted in Fig. 3.27 together with the unit impulse response function of the diffusion equation, i.e. Eq. (3.118). The results shown correspond to ${\tau }_r=0.08s$ and $D=0.2m^2/s$ .

Figure 3.27. Unit impulse load response; t = 2 s

It appears at first that there is no discernible difference between the two approaches, however, Fig. 3.28 shows some interesting details of the solution. While the diffusion equation predicts that the concentration decays exponentially, and becomes zero at infinity, the telegraph equation shows a concentration front at $x=\pm 3.16m$ . Near the front, the concentration decays as $t^{-1/2}$ , but in the core of the wake the profile behaves exactly like that of the diffusion equation. Thus, the Cattaneo model eliminates the unrealistic property of infinite signal speed of the diffusion equation, albeit with the introduction of an additional empirical parameter, ${\tau }_r$ .

Figure 3.28. Unit impulse load response; detail of telegraph wave front

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