Question 30.2: Laboratory filtrations conducted at constant pressure drop o......

The first step is to prepare plots, for each of the five constant-pressure experiments, of t/V vs. V. The data are given in Table 30.2 and the plots are shown in Fig. 30.15.
The slope of each line is K_c/2, in seconds per liter per liter. To convert to seconds per cubic foot per cubic foot, the conversion factor is 28.31² = 801. The intercept of each line on the axis of ordinates is 1/q_0 , in seconds per liter. The conversion factor to convert this to seconds per cubic foot is 28.31. The slopes and intercepts, in the observed and converted units, are given in Table 30.3.

The viscosity of water is, from Appendix 14,0.886\,\mathrm{cP,or}\,0.886\times6.72\times10^{-4}=5.95\times10^{-4} \ \rm{lb/ft-s}. The filter area is 440/30.48² = 0.474 ft². The concentration c is (23.5 × 28.31)/454 = 1.47 lb/ft³ .

From the values of K_c/2 and 1/q_0 in Table 30.3, corresponding values of R_m and α are found from Eqs. (30.22) and (30.24). Thus

{\frac{\mu R_{m}}{A\,\Delta p\,g_{c}}}=\left({\frac{d t}{d V}}\right)_{0}={\frac{1}{q_{0}}}   (30.22)

K_{c}=\frac{\mu c\alpha}{A^{2}\;\Delta p\;g_{c}}   (30.24)

R_{m}={\frac{A\,\Delta p\,g_{c}(1/q_{0})}{\mu}}={\frac{0.474\times32.17\,\Delta p\,(1/q_{0})}{5.95\times10^{-4}}}

=2.56\times10^{4}\,\Delta p{\Bigg(}{\frac{1}{q_{0}}}{\Bigg)}

α={\frac{A^{2}\,\Delta p\,g_{c}K_{c}}{c\mu}}={\frac{0.474^{2}\times32.17\,\Delta p\,K_{c}}{5.95\times10^{-4}\times1.47}}

=8.26\times10^{3}\,\Delta p\,K_{c}

Table 30.3 shows the values of K_c/2 and 1/q_0 for each test, calculated by the method of least squares. In all but test I the first point, which does not fall on the linear graph. was omitted. Also in Table 30.3 are the values of α and R_m. Figure 30.16 is a plot of R_m vs. Δp.

Figure 30.17 is a logarithmic plot of α vs. Δp. The points closely define a straight line, so Eq. (30.26) is suitable as an equation for α as a function of Δp. The slope of the line, which is the value of s for this cake, is 0.26. The cake is only slightly compressible.

\alpha=\alpha_{0}(\Delta p)^{s}    (30.26)

Constant α_0 can be calculated by reading the coordinates of any convenient point on the line of Fig. 30.17 and calculating α_0 by Eq. (30.26). For example, when \Delta p=1000,\,\alpha=1.75\times10^{11},\,\mathrm{and}\,

\alpha_{0}={\frac{1.75\times10^{11}}{1000^{0.26}}}=2.90\times10^{10}\,{\mathrm{ft}}/{\mathrm{lb}}\ (1.95\times10^{10}\,{\mathrm{m/k}}g)

Equation (30.26) becomes for this cake

\alpha=2.90\times10^{10}\,\Delta p^{0.26}