Ch 9.8 Diffusion

Introduction

  • Particle diffusion shares characteristics with heat conduction.
  • Similar to Fourier's Law for heat conduction, Fick's Law of Diffusion will apply to mass transport.
  • This will lead to equations that model particle diffusion.

Particle Diffusion

  • Particles flow from high concentration to low concentration, and towards a concentration that is uniform towards available space.
  • This is called diffusion and is one method for transporting mass.

Concentration and Mass Flux

Concentration is the key variable in mass transport, and can be defined in a number of ways, including the following:

  • The mass of particles divided by volume containing the particles, with units kg/liter or mg/liter.
  • The number of particles in a volume divided by the volume.
  • The volume of all the particles (the solute) as a fraction of the total volume containing the particles (the solution).

Variables

We identify the following variables:

  • \( C(x) \) equilibrium concentration at location \( x \).
  • \( C(x,t) \) is the time dependent concentration at some point \( x \) and time \( t \) (comparable to temperature).
  • \( J(x) \) is the mass flux, or rate of flow of particles through a cross section at location \( x \) per unit time per unit cross-sectional area (comparable to heat flux).

Fick's Law

  • Similar to Fourier's law of heat conduction, in mass transport there is Fick's law of diffusion for equilibrium concentrations:

\[ J(x) = -D \frac{dC}{dx} \]

  • For time dependent concentrations, Fick's law is

\[ J(x,t) = -D \frac {\partial C}{\partial x} \]

The Constant of Proportionality

  • Fick's law is

\[ J(x,t) = -D \frac {\partial C}{\partial x} \]

  • \( D \) is called the diffusion coefficient or the diffusivity, and has units of \( \frac{m^{2}}{s} \)
  • \( D \) takes on different values depending on the size of particles and the type of fluid in which the particles are diffusing.
  • Temperature also impacts the value of \( D \), as particles diffuse more easily at higher temperatures.
  • In general, diffusion as a mechanism for mass transport is a slow process (see values in table on next slide).

Table of Diffusivity Values

  • Table 9.4 depicts \( D \) for several different media.
  • The SI units of \( D \) are given by

\[ \left[ D \right] = \frac{m^{2}}{s} \]

Formulating a Differential Equation

  • Consider a pipe filled with fluid that is not moving, but which contains particles.
  • To establish the diffusion process of these particles within the fluid, we set up a mass balance in region \( x \) to \( x + \Delta x \), a slice across the cylinder with cross-sectional area \( A \).

Word Equation

  • The rate of change for the mass of particles inside the region, or slice, is determined by the net amount flowing into and out of the region:

\[ \begin{Bmatrix} \mathrm{rate \, of \, change} \\ \mathrm{of \, mass} \\ \mathrm{within \, \, region} \end{Bmatrix} = \begin{Bmatrix} \mathrm{rate \, of \, mass} \\ \mathrm{ flowing \, in \, at } \\ \mathrm{ \, x \, } \end{Bmatrix} - \begin{Bmatrix} \mathrm{rate \, of \, mass} \\ \mathrm{ flowing \, out \, at} \\ \mathrm{ \, x + \Delta x \, } \end{Bmatrix} \]

Word Equation for Equilibrium

  • At equilibrium the rate of change of mass with time will be zero, so the LHS is zero.

\[ \begin{Bmatrix} \mathrm{rate \, of \, change} \\ \mathrm{of \, mass} \\ \mathrm{within \, \, region} \end{Bmatrix} = \begin{Bmatrix} \mathrm{rate \, of \, mass} \\ \mathrm{ flowing \, in \, at } \\ \mathrm{ \, x \, } \end{Bmatrix} - \begin{Bmatrix} \mathrm{rate \, of \, mass} \\ \mathrm{ flowing \, out \, at} \\ \mathrm{ \, x + \Delta x \, } \end{Bmatrix} \]

Governing Equations

  • The rate at which mass flows in at x is given by the mass flux \( J(x) \) (mass per unit time per unit area) multiplied by the cross-sectional area, and similarly for the amount flowing out.
  • Our word equation can then be written as

\[ J(x)A-J(x+\Delta x)A=0 \]

Governing Equations

  • From the previous slide,

\[ J(x)A-J(x+\Delta x)A=0 \]

  • Dividing both sides by \( \left(-A\Delta x\right) \), we obtain

\[ \frac{J(x+\Delta x)-J(x)}{\Delta x} =0 \]

  • Letting \( \Delta x\rightarrow 0 \) yields

\[ \frac{dJ}{dx}=0 \]

Differential Equation

  • Our ODE for mass flux \( J(x) \) is:

\[ \frac{dJ}{dx}=0 \]

  • To get an ODE in terms of \( C(x) \), recall Fick's Law:

\[ J(x) = -D\frac{dC}{dx} \]

  • Thus the ODE for the concentration \( C(x) \) at equilibrium is

\[ \frac{d^2C}{dx^2}=0 \]

Linear Geometry

  • For linear diffusion inside cylinder, the ODE we derived is

\[ \frac{d^2C}{dx^2}=0 \]

Cylindrical Geometry

  • For radial diffusion inside the cylinder, the ODE is

\[ \frac{d}{dr} \left( r \frac{dC}{dr} \right) = 0 \]

  • This follows since the area through which diffusion occurs changes with distance \( r \) from the center.

Spherical Geometry

  • For spherical geometry, we have

\[ \frac{d}{dr} \left( r^2 \frac{dC}{dr} \right) = 0 \]

  • This follows since the area through which diffusion occurs changes with square of the distance \( r^2 \) from the center.

Volumetric Mass Source

  • If we add an additional source/sink of particles into the system at distance \( r \) from the center of the sphere, then the ODE gets more complicated; see next slide.

\[ D \frac{1}{r^2} \frac{d}{dr} \left( r \frac{dC}{dr} \right) + M(r) = 0 \]

Volumetric Mass Source

  • Assuming volumetric mass source at distance \( r \) from the center, we have

\[ D \frac{1}{r^2} \frac{d}{dr} \left( r \frac{dC}{dr} \right) + M(r) = 0 \]

  • \( D \) is the diffusion coefficient and \( M(r) \) is the rate of the 'production' of mass, in \( \frac{kg} {m^3 s} \).
  • Note that \( M(r) \gt 0 \) and \( M(r) \lt 0 \) with the addition and removal of mass, respectively.