Seeding a precipitative softening reactor with calcite

can sometimes increase the efficiency of the process.

Precipitation on the pre$-$existing solids requires transport

of calcium to the particle surface $($carbonate is

also required, but calcium supply seems to limit the

reaction$),$ followed by the precipitation reaction, in

which the ions are incorporated into the solid. A

number of researchers $($e$.$g$.,$ Chao and Westerhoff,

$2002)$ have modeled the overall reaction as irreversible and second order with respect to the

calcium concentration $$rcalcite $$ k$2$Ca$22.$

In a system containing a bulk Ca$2$ concentration

of $2103$ mol$=$L$,$ the overall reaction rate normalized

to the calcite surface area is $4106$ mol$=$m$2$

min. The diffusion coefficient for Ca$2$ in the solution

is $7\mathrm{.}9106$ cm$2=$s$,$ the diameter of the seed particles

is $3$ mm$,$ and the water temperature is $20$C$.$ For the

experimental conditions, the mass transfer coefficient

for Ca$2$ transport through the liquid boundary layer

around the particles can be estimated based on the

following correlation for mass transfer during flow

around a sphere:

where Dr$($rp$$rL$),$ and the other parameters have

their usual meanings. The density of calcite is

$2\mathrm{.}71$ g$=$cm$3.$

$($a$)$ What is the surface$-$normalized rate constant for

the precipitation reaction, in L$2=($m$2$ min mol$)?$

$($b$)$ What is the mass transfer coefficient $($m$=$s$)$ for

Ca$2$ transport through the boundary layer in

this system?

$($c$)$ What is the concentration $($mol$=$L$)$ of calcium at

the particle$=$water interface? What do you infer

about the resistance associated with mass transfer

compared to that associated with the incorporation

step? Briefly explain how you reached

your conclusion.$k_{mt}=\frac{D_i}{d_p}[2+0\mathrm{.}31(\frac{d_p}{{}_L})(\frac{D_i{}_L}{{}_L}{)}^{0\mathrm{.}67}(\frac{}{{}_L}\frac{{}_{L}g}{{}_L}{)}^{0\mathrm{.}33}]$