Question 3.8: Evaluation of the Hydraulic Conductivity As discussed, a pos......
Evaluation of the Hydraulic Conductivity
As discussed, a possible porous-medium model is a structure with n parallel tubes of diameter d, representing straight pores or capillaries. Assuming steady laminar fully-developed flow in horizontal tubes, find an expression for K= æ /μ.
| Sketch | Concept |
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• Equate volumetric flow rate obtained from Poiseuille flow with Darcy’s law, where dp / dx ≈ ¢ and \left\langle\mathrm{u}\right\rangle={\frac{Q}{\mathrm{A}}}=-\mathrm{K}\,{\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}\mathrm{x}}} |
Using Eq. (E.3.4.7a) to obtain the flow rate per pore or tube, we have:
{\rm Q=\frac{\pi r_0^4}{8\mu}} \left\lgroup\frac{\Delta {\rm p}}{l} \right\rgroup (E.3.4.7a)
\frac{\mathrm{Q}}{\mathrm{n}}=\mathrm{v_{{}_{av}}A_{{}_{\mathrm{tube}}}=-{\frac{\pi\mathrm{d^{4}}}{128\mathrm{\mu}}}}\left\lgroup\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}\mathrm{x}}\right\rgroup (E.3.8.1a)
As indicated in Fig. 3.3, \rm \left\langle u\right\rangle =Q/A:=\varepsilon v_{av}, , so that \rm A=\frac{n}{\varepsilon } A_{tube}.
Hence,
\frac{\mathrm{Q}}{\mathrm{n}}\!=\!\left\langle\mathrm{u}\right\rangle\frac{1}{\varepsilon}\,{\mathrm{A}}_{\mathrm{tube}}=\!-\!{\mathrm{K}}\!\left\lgroup\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}\mathrm{x}}\right\rgroup\frac{\mathrm{d}^{2}\pi}{4\varepsilon} (E.3.8.1b)
so that by inspection
\mathrm{K=}{\frac{\mathrm{d}^{2}\varepsilon}{32\mathrm{{\mu}}}} (E.3.8.2)
where 0 < ε <1.0 .
Comment: Once the average pore diameter has been estimated, setting ε = 0.5 and the fluid viscosity known, K and æ can be calculated and hence suitable porous medium flow analyses can be carried out.
