Refer to Example 17.12. In Run 15, Fitchett and Tarbell also made measurements of number density of crystals at $200 \mathrm{rpm}$, for which the data can be fitted well by the equation
$$
\text { In } n=26.3-0.407 L
$$
where
$n=$ number density of crystals
$L=$ crystal size, $\mu \mathrm{m}$
Using the MSMPR model, determine in the same units as for Example 17.12:
(a) $n^{\circ}$
(b) $G$
(c) $B^{\circ}$
(d) mean crystal length
(e) $n_{c}$
(f) Are your results consistent with the trends found in Example 17.12?
(g) Using your results and those in Example 17.12, predict the growth rate and mean-crystal length if no agitation is used.
Figure 17.37 Population density of $\mathrm{CaCO}_{3}$ for Exercise 17.28.
$17.28$ Tai and Chen $[A I C h E J .41,68-77(1995)]$ studied the precipitation of calcium carbonate by mixing aqueous solutions of sodium carbonate and calcium chloride in an MSMPR crystallizer with $\mathrm{pH}$ control, such that the form of $\mathrm{CaCO}_{3}$ was calcite rather than aragonite or vaterite. In Run S-2, which was conducted at $30^{\circ} \mathrm{C}$, a pH of $8.65$, and $800 \mathrm{rpm}$, with a residence time of $100 \mathrm{~min}$, the crystal population density data were as shown in Figure 17.37. Because the data do not plot as a straight line, they do not fit (17-38).
(a) Develop an empirical equation that will fit the data and determine, by regression, the constants.
(b) Can nucleation rate and growth rate be determined from the data? If so, how?
17.29 Tsuge and Matsuo ["Crystallization as a Separation Process," ACS Symposium Series 438 , edited by Myerson and Toyokura, ACS, Washington, DC (1990), pp. 344-354] studied the precipitation of $\mathrm{Mg}(\mathrm{OH})_{2}$ by reacting aqueous solutions of $\mathrm{MgCl}_{2}$