Temperature gradient induced double stabilization of the evaporation front within a drying porous medium

Drying of porous media very often occurs in the presence of significant temperature gradients because heat fluxes are imposed in many situations in order to decrease the drying time or to facilitate the moisture removal at a higher humidity of the surrounding gas phase. Here we consider the situation where the temperature increases with depth. We show from experiments with a micromodel that the temperature gradient induces the stabilization of the evaporation front within the model porous medium according to two different mechanisms occurring consecutively. The first mechanism occurs in the liquid phase and is explained by the dependence of surface tension upon temperature. This results in the preferential invasion of the warmer zones. The second mechanism occurs gas-sided due to the dependence of saturation vapor pressure upon temperature. We show that the time scales of both mechanisms are different leading to the temporary formation of distinctive phase patterns from which different periods of drying can be discriminated.

Figure 1

(a) Temperature field with positive temperature gradient dT/dz imposed on the micromodel in the $z$ direction. ${\overline{T}}_{z=0}=24{}^{\circ }\mathrm{C}$ (top edge) and ${\overline{T}}_{z=49L}=50{}^{\circ }\mathrm{C}$ (bottom edge). (b) The temperature contour lines indicate nonlinearity of the temperature gradient in the $z$ direction.

Figure 2

Experimental phase distributions obtained from drying with $dT/dz=0.44\mathrm{K}\mathrm{m}{\mathrm{m}}^{-1}$ and ${\overline{T}}_{z=0}\cong 24{}^{\circ }\mathrm{C}$. Liquid and solid in black, gas phase in light gray. Vapor escapes from top edge. Shown are the snapshots for overall network saturation $S=\left[0.91,0.75,0.57,0.42,0.38,0.36,0.34,0.04\right]$, from (a) to (h), respectively.

Figure 3

PNM computed phase distributions obtained from drying with the same temperature setting as in Fig. 2. Liquid and solid in black, gas phase in white. Vapor escapes from top edge. $S=\left[0.91,0.75,0.57,0.42,0.38,0.36,0.34,0.04\right]$, from (a) to (h), respectively.

Figure 4

Initial phase distributions observed in different experiments. (a) Experiment with temperature field as specified in Fig. 1. (b) and (c) show two experiments with reversed flow direction of water along the top edge in the heat conducting plate (the temperature field is roughly flipped from right to left).

Figure 5

Graphical representation of capillary interconnectivity. Evaporation from the top edge. The blue arrows schematically illustrate the capillary pumping effect, i.e., the invasion of the liquid cluster pinned along the evaporative top edge of the micromodel from the bottom side.

Figure 6

Drying experiments: (a) drying curves and (b) drying rates in a semilogarithmic plot. The curves corresponding to the experiment discussed above and presented in Fig. 2 are shown in black color; experiments from Figs. 4 and 4 are shown in gray.

Figure 7

PNM simulations: (a) drying curves and (b) drying kinetics. The simulated curve is shown in black and compared to the experimental data from Fig. 6, shown in gray.

Figure 8

Drying experiments: horizontal slice averaged saturation profiles computed from the phase distributions shown in Fig. 2. The top row corresponds to $z=0\mathrm{mm}$. (a) Periods 1 and 2 up to the full fragmentation of the liquid phase. (b) Periods 3 and 4, i.e., shrinking two-phase zone. Notice the increase in slice saturation between $z\cong 8$ and 38 mm.

Figure 9

PNM simulations: total through-plane vapor flow rates from the simulation corresponding to Fig. 3. The top row is located at $z=0\mathrm{mm}$. Curves computed for various overall network saturation (a) $S=\left[0.91,0.75,0.57,0.42\right]$ and (b) $S=\left[0.38,0.36,0.34,0.04\right]$.

Figure 10

PNM simulations: through-plane vapor flow rates related to overall evaporation rate. Curves highlighted in the legend of the figure correspond to period 3 and period 4 (corresponding to Fig. 3). Note that vapor flow rates are constant (with $\sum {\dot{M}}_v={\dot{M}}_{\mathrm{ev}}$) in the dry zone above the two-phase zone ($z\le 15\mathrm{mm}$). Curves for $S=\left[0.91,0.75,0.57\right]$ partly overlap; they are shown as solid black lines and are not distinguished by the legend.

Figure 11

(a) Upper STPZ position, (b) lower STPZ position, (c) width, and (d) mean position of STPZ obtained from the experimental patterns. Results of the experiment presented in Fig. 2 are shown in black; results from the other two experiments (Fig. 4) are shown in gray.

Figure 12

(a) Saturation of the STPZ (experiment from Fig. 2 in black; additional experimental curves in gray). (b) Percentage ratio of shrinking (in red) and growing (in blue) clusters related to the current total number of clusters determined by image processing of experimental patterns (see text); the inset shows the same type of data from the PNM simulations (see text).

Figure 13

PNM computation of the variation of $M_{\mathrm{ev}}\left(t\right)$ (net evaporation), $M_{\mathrm{ev},\mathrm{int}}\left(t\right)$, and $M_{\mathrm{co}}\left(t\right)$ [Eq. (21)] related to the initial mass of water in the PN $\left[M\left(t=0\right)=0.0284\mathrm{g}\right]$.

Figure 14

Two mechanisms leading to a fragmented liquid phase spanning the network. Illustration from PNM simulations: (a) IPDG pattern with upward moving gas invasion front sweeping the network from below and (b) IP pattern with downward moving gas invasion front sweeping the system from the top and isolated liquid clusters stabilized by condensation ($S=0.75$).

Figure 15

Impact of the temperature gradient and the initial evaporation rate (a) on the overall drying time and (b) the total amount of condensed liquid volume from PN simulations with PN identical to the microfluidic network and different imposed thermal gradients. The initial evaporation rates vary in the range of ${\dot{M}}_{\mathrm{ev},0}=\left[15.7,3.15,1.6,0.8\right]\mathrm{mg}{\mathrm{h}}^{-1}$ according to a variation in the thickness of the diffusive boundary layer with which the mass transfer from the PN surface is simulated (refer to [48] for more details).