Diffusion out of spherical bead, Anthony (Tony) Butterfield, Chemical Engineering, University of Utah

This simulation is meant to demonstrate aspects of mass transfer in radial coordinates A summary of this system and its governing equations may be found below the simulation. Set up your initial conditions at the bottom of the applet and press play to run the reaction simulation in "real-time". Press fast forward to run as quickly as your computer is capable. Mouse-over each input to see a description of that option or variable.

Governing Equations:

Concentration profile across the bead:

$\frac { \partial C\left( r,t \right) }{ \partial t } =D\left( \frac { { \partial }^{ 2 }C }{ \partial { r }^{ 2 } } +\frac { 2 }{ r } \frac { \partial C }{ \partial r } \right) \quad ,\quad \left\{ \begin{matrix} C\left( r,t=0 \right) \quad ={ \quad C }_{ b }^{ avg }\left( t=0 \right) , \\ C\left( r={ r }_{ C },t \right) \quad =\quad { C }_{ bc }\left( t \right) ,\quad C\left( r={ r }_{ B },t \right) \quad =\quad { C }_{ L }\left( t \right) \end{matrix} \right\}$

Total number of moles, and the concentration of the system at infinite time:

${ N }_{ \infty }={n}\frac { 4 }{ 3 } \pi { r }_{ c }^{ 3 }{ C }_{ cb }^{ 0 }+{n}\frac { 4 }{ 3 } \pi \left( { r }_{ b }^{ 3 }-{ r }_{ c }^{ 3 } \right) { C }_{ b }^{ 0 }+{ C }_{ L }^{ 0 }{ V }\quad ,\quad { C }_{ \infty }=\frac { { N }_{ \infty } }{ V+{n}\frac { 4 }{ 3 } \pi { r }_{ b }^{ 3 } }$

Number of moles in the bead centers, bead shells, and liquid (determined by a mole balance):

${ N }_{ cb }={n}\frac { 4 }{ 3 } \pi { r }_{ c }^{ 3 }{ C }_{ bc }\quad ,\quad { N }_{ b }={n}\int _{ r={ r }_{ c } }^{ { r }_{ b } }{ 4\pi { r }^{ 2 }Cdr } \quad ,\quad { N }_{ L }={ N }_{ \infty }-{ N }_{ b }-{ N }_{ cb }$

Average concentration in the beads:

${ C }_{ b }^{ avg }=\frac { { N }_{ cb }+{ N }_{ b } }{ \frac { 4 }{ 3 } \pi { r }_{ b }^{ 3 } }$

Concentration change in the center of the bead depends on the molar flux at the bead center's radius:

$\frac { \partial { C }_{ cb } }{ \partial t } =\frac { { A }_{ r={ r }_{ c } }{ J }_{ r={ r }_{ c } } }{ { V }_{ cb } } =\frac { \left( 4\pi { r }_{ c }^{ 2 } \right) D\frac { \partial { C } }{ \partial r } _{ r={ r }_{ c } } }{ \frac { 4 }{ 3 } \pi { r }_{ c }^{ 3 } } =\frac { 3D }{ r } { \frac { \partial { C } }{ \partial r } }_{ r={ r }_{ c } }$

Fraction of total solute released:

$Y(t)=\frac { { C }_{ b }^{ avg }(t=0)-{ C }_{ b }^{ avg }(t) }{ { C }_{ b }^{ avg }(t=0)-{ C }_{ \infty } }$

This simulation is also capable of determining the analytical solution for the concentration profile across the bead in the case where the liquid volume is infinite (it does not change with time) and there is no void in the center of the bead (r_c = 0). The analytical solution is:

$an$

Peppas approximation may be plotted in the simulation. This equation is an empirical relationship that is most valid when Y < 0.60:

$Y(t)\approx k{ t }^{ m }$

While this simulation models a more complex system with diffusion into a finite media with changing concentration, the traditional 1-D radial diffusion problem may be simulated by making the volume of water to beads very large. For example, one could reduce the number of beads to 1 and raise the liquid volume to a kiloliter. When this is done and dt is small enough to avoid numerical error, the concentration profile across the bead should be the same as that given by the analytical solution, C^a(r), which may be plotted using the appropriate check-box below the plot.

Plot What:	C_L	C_bc	C_b	Y	Y^f	C(r)	C^a(r)
Fit to f(Y)	t₀	t₁			m	b

Diffusion From Spheres

Associated Hands-On Teaching Module: http://www.che.utah.edu/module/?p_id=35