Question 29.2: A batch grinding mill is charged with material of the compos......

(a) For the 4/6-mesh material there is no input from coarser material and Eq. (29.11) applies. At the end of time t_T, x_1 \text { will be } 0.0251 \times 0.9=0.02259 \text {. Thus }

\frac{d x_u}{d t}=-S_u x_u    (29.11)

-S_u \int_0^{t T} d t=\int_{0.0251}^{0.02259} \frac{d\left(x_1\right)}{x_1}

or

t_T=\frac{1}{S_u} \ln \frac{0.0251}{0.02259}=\frac{1}{0.001} \ln 1.111=105.3~ s

(b) Assume S_u \text { varies with } D_p^3 \text {. Let } S_1 \text { and } S_2 be the values for 4/6- and 6/8-mesh material, respectively. Then S_1=10 \times 10^{-4} ~s ^{-1},

S_2=S_1\left(\frac{D_2}{D_1}\right)^3=10^{-3}\left(\frac{2.362}{3.327}\right)^3=3.578 \times 10^{-4}~ s ^{-1}

Values of S_3 to S_7 are calculated similarly; the results are given in Table 29.2.
The breakage function ΔB_{n,u} is found as follows. When n and u are equal, or whenever n<u, \Delta B_{n, u}=0. The total mass fraction smaller than 6/8-mesh resulting from breakage of 4/6-mesh particles, B_{2,1} from Eq. (29.13), is

B_{2,1}=\left(\frac{2.362}{3.327}\right)^{1.3}=0.6407

Then ΔB_{2,1}, the fraction of broken material retained on the 8-mesh screen, is 1 – 0.6407, or 0.3593.

The total mass fraction smaller than 8/10-mesh resulting from breakage of 4/6-mesh material,B_{3,1}, is

B_{3,1}=\left(\frac{1.651}{3.327}\right)^{1.3}=0.4021

In general, the individual breakage functions are found from the relation

\Delta B_{n, u}=B_{n-1, u }-B_{n, u }    (29.14)

Thus the mass fraction of the broken 4/6-mesh material retained on the 10-mesh screen, \Delta B_{3,1} \text {, is } 0.6407-0.4021=0.2386 . \text { Other values of } B_{n, u} \text { and } \Delta B_{n, u} are found in the same way, to give the results shown in Table 29.3. Note that when n=u, B_{n, u} is unity, by definition. When u = 1, as shown in Table 29.3, 0.6407 of the broken particles from the 4/6-mesh material is smaller than 8-mesh, 0.4021 smaller than 10-mesh, 0.2564 smaller than 14-mesh, and only 0.0672 smaller than 35-mesh.

(c) Let x_{n,t} be the mass fractions retained on the various screens at the end of t time increments \Delta t \text {. Then } x_{1,0}, x_{2,0} \text {, } etc., are the initial mass fractions given in Table 29.2.

The left-hand side of Eq. (29.12) is approximated by \Delta x_n / \Delta t, \text { where } \Delta t in this example is 30 s and \Delta x_n=x_{n, t+1}-x_{n, t} Successive values of x on the screens can then be calculated from the following form of Eq. (29.12):

\frac{d x_n}{d t}=-S_n x_n+\sum\limits_{u=1}^{n-1} x_u S_u \Delta B_{n, u}    (29.12)

\begin{aligned}x_{n, t+1} & =x_{n, t}-S_n \Delta t x_{n, t}+\Delta t \sum_{u=1}^{n-1} x_{u, t} S_u \Delta B_{n, u} \\ & =x_{n, t}\left(1-S_n \Delta t\right)+\Delta t \sum_{u=1}^{n-1} x_{u, t} S_u \Delta B_{n, u}\end{aligned}    (29.15)

For the top screen n = 1 and ΔB = 0. Hence Eq. (29.15) becomes

After 30 s, then, the mass fraction on the top screen is

x_{1,1}=0.970 \times 0.0251=0.02434

After 30 s more,

x_{1,2}=0.970 \times 0.02434=0.02360

and so forth. On the 8-mesh screen (n = 2), the mass fractions are, from Eq. (29.15),

x_{2,1}=x_{2,0}\left(1-S_2~ \Delta t\right)+\Delta t ~x_{1,0} S_1 ~\Delta B_{2,1}

Substituting the values of S_1, S_2 \text {, and } x_{1,0} \text { from Table } 29.2 \text { and } \Delta B_{2,1} from Table 29.3 gives

Thus

x_{2,1}=(0.98926 \times 0.1250)+0.00027=0.12393

Similarly,

The values of x_ 3 ~to~ x_7 are found in the same way. The results are given in Table 29.4†and illustrated in. Fig. 29.1. Initially x_ 1 , x_2 , ~and ~x _3 all decrease with time and the other mass fractions increase. At the end of 1 h, 99.95 percent of the 4/6-mesh material ( x_1 ) has disappeared and x_7 has more than doubled. The fraction of material finer than 35-mesh has increased from 0.0384 to 0.0931. During the first hour the changes in x_3 and x_7 are almost linear with time. At about 70 min x_4 reaches a maximum and then diminishes with time. If the grinding were continued, the still finer fractions would eventually do the same.