Experimental observation and pressure drop modeling of plug formation in horizontal millifluidic hydraulic conveying

The hydraulic conveying of glass beads is studied in a horizontal tube. At low flow rates, plugs can be observed moving across the tube, whereas pseudoplugs can be seen at higher flow rates. A statistical analysis of the plugs' and pseudoplugs' velocities and the plugs' lengths observed is conducted. A transition of the propagation speed distribution is established when the crossing over from a plug to a pseudoplug regime is reached, where the peaked plug velocity distribution turns into a uniform pseudoplug velocity distribution. On the other hand, the statistical distribution of plug lengths exhibits a log-normal mathematical shape. The interpretation of the measured pressure drop evolution with the imposed flow rate by means of an effective viscosity shows an apparent shear-thinning effect coming from the dilution of the granular material. This approach provides a predictive tool for pressure drop calculation in the pseudoplug regime.

Figure 1

Sketch of the experimental setup. A horizontal glass tube of inner diameter $D_{\text{tube}}$ contains a mixture of water and grains. A syringe pump imposes a constant flow rate of water $I_0$ and a second syringe filled with glass beads introduces the granular material into the stream at the tube's inlet, where a pressure sensor is returning the measurement $\mathrm{\Delta }P\left(t\right)$.

Figure 2

Pressure time series measured with imposed flow rates (a) $I_0=10\mathrm{mL}{\min }^{-1}$ and (b) $I_0=2\mathrm{mL}{\min }^{-1}$. The solid and dashed lines mark the steady-state pressure plateaus obtained without any grains and with grains, respectively.

Figure 3

Image of the treatment procedure applied to the original picture for $I_0=1{\mathrm{mL}\min }^{-1}$.

Figure 4

Spatiotemporal diagrams obtained from a section of the tube of length 234 mm located left of the tube outlet for the following flow rate values: (a) $I_0=1{\mathrm{mL}\min }^{-1}$, (b) $I_0=2{\mathrm{mL}\min }^{-1}$, (c) $I_0=3{\mathrm{mL}\min }^{-1}$, (d) $I_0=4{\mathrm{mL}\min }^{-1}$, and (e) $I_0=5{\mathrm{mL}\min }^{-1}$. Time goes downward for 231 s.

Figure 5

Details of the spatiotemporal diagram obtained for a flow rate $I_0=1{\mathrm{mL}\min }^{-1}$ for 85 s (left) and $I_0=5{\mathrm{mL}\min }^{-1}$ for 79 s (right). Thin red solid lines show the boundaries of the central white stripe after thresholding and contour detection for plugs traveling (left). The two upper and lower slopes in dotted lines indicate the front and the rear velocities, respectively, of the traveling plug. Only the front velocity $v_{\text{front}}$ is considered in this work. The plug's length $\lambda$ is obtained as the distance between the two front- and rear-velocity slopes. For pseudoplug propagation (right), only front velocities $v_{\text{front}}$ can be defined.

Figure 6

(a) Empirical cumulative functions for plug (solid lines) and pseudoplug (dashed lines) velocities normalized by the median velocity of each distribution. The shade of gray indicates the imposed flow rate ranging from $1{\mathrm{mL}\min }^{-1}$ (dark gray) to $5{\mathrm{mL}\min }^{-1}$ (light gray). (b) Median velocity values of plugs (closed squares) and pseudoplugs (open squares) obtained for different imposed flow rate values. Error bars associated with pseudoplug median velocities correspond to the observed minimal and maximal velocity values. The lines are a guide for the eye, giving the expected fluid velocity averaged over the cross-sectional full aperture (bottom dashed line), $50\%$ aperture (solid line), and $25\%$ aperture (top dashed line).

Figure 7

(a) Normalized empirical cumulative functions of the plug length ${[\lambda /\mathcal{M}\left(L\right)]}^{1/\sigma }$ (solid lines) and normalized cumulative function for a log-normal distribution (dashed line). The shades of gray gives the flow rate values ranging from $1{\mathrm{mL}\min }^{-1}$ (dark gray) to $3{\mathrm{mL}\min }^{-1}$ (lighter gray). (b) Median and variance values used for normalization, represented by closed squares and error bars, respectively.

Figure 8

Pressure drop measurements for pure water (green circles) and for hydraulic conveying of glass beads (purple squares) as a function of the imposed flow rate $I_0$. Each experiment was executed two to three times and the error bars show the minimum and maximum measurements. The purple solid line gives the expected pressure drop for a suspension with the Maron-Pierce correlation and a flow-rate-dependent solid filling fraction. The green dash-dotted line gives the expected pressure drop obtained with the Hagen-Poiseuille formula for pure water.

Figure 9

(a) Reduced pressure drop ${(\mathrm{\Delta }P/L)}^{*}$ as a function of the dimensionless velocity $v_G^{*}$ for pure water (green circles) and granular conveying (purple squares). The solid line shows the zero-ordinate baseline Hagen-Poiseuille prediction. (b) Effective viscosity ratio, calculated from the pressure drop measurement ${\mu }_{\text{eff}}$ with pure fluid (green circles) and with granular conveying (purple squares), to the nominal fluid viscosity ${\mu }_w$ as a function of the imposed flow rate $I_0$. The solid line shows the effective viscosity ratio given by the Maron-Pierce correlation with our flow-rate-dependent filling fraction model.