We would like to assess the importance of dispersion in laminar pipe flow. Consider the steady$-$state dispersion of a solute $($A$)$ that decomposes according to a first$-$order chemical reaction. The convective$$dispersion equation is

d $=$ D d$2$ k zdzedz$2$ A

Appropriate boundary conditions are

$=$ cA$0,$z $=0$

$=0,$z $->\backslash$infty

with the effective diffusivity, De$,$ given by Equation $7\mathrm{.}46.$

a$.$ Write the problem in dimensionless form using the following

definitions:

u $=$ cA$0$

$\backslash$xi $=$z R

Pe $=$ R DA

$\backslash$phi $=$ kR$2$ DA

Find u $=$ u$(\backslash$xi $).$

b$.$ Plot u vs$.\backslash$xi for the case $\backslash$phi $=100,$ Pe $=500.$ In the same graph, plot u vs$.$

$\backslash$xi for the same conditions, but assuming that De $=$ DA $($this would be the

hypothetical case in which there is no dispersion$)$