A very long cylindrical pipe of radius R is carrying a hazardous compound X$.$ Species X has corroded the

pipe so that is now leaks out at a constant rate of MX $[=]$ amt$/($area $$ time$)$ into the infinite fluid reservoir

surrounding the pipe. A scavenging agent S is added to the reservoir in excess to consume X and convert

it to an inert compound Y $.$ The reaction X $->$ Y occurs throughout the entire volume outside the pipe,

consuming X at a rate of kCX$($r$),$ where CX$($r$)$ is the volumetric concentration of X in the external fluid and

k $[=]1/$time is the rate constant. Far away from the pipe, the concentration of CX$(\backslash$infty $)=0.$ The diffusivity

of X is D$.$

Figure $4$

$($a$)$ The ODE governing the steady$-$state CX$($r$)$ is

D

d

$2$CX

dr$2$

$+$

D

r

dCX

dr $=$ kCX$.(3)$

Use order$-$of$-$magnitude analysis on this ODE near the pipe surface to show that if the Damk$$ohler number

Da $=$ kR$2/$D is sufficiently large and most of X is consumed over a distance $$ away from the pipe surface,

the ODE can be approximated as

D

d

$2$CX

dr$2$

$$ kCX $(4)$

In other words, the pipe$$s curvature can be neglected.

$($b$)$ Solve this ODE $($eqn$.(4))$ to obtain either CX$($r$)$ or CeX$($re$).$ The following dimensionless variables may

be helpful

re $=$

r

R

$,$ CeX $=$

CX

MXR$/$D $,\backslash$lambda $=$ Da$1/2$