Impact of clogging on the transfer dynamics of hydraulic networks

We investigate how clogging affects the transfer properties of a generic class of materials featuring a hydraulic network embedded in a matrix. We consider the flow of a liquid through fully saturated hydraulic networks which transfer heat (or mass) by advection and diffusion, and a matrix in which only diffusion operates. Networks are subjected to different clogging scenarios (or attacks), changing their microstructure and flow field. A series of canonical cooling tests are simulated using a tracer method prior to and following attack. Results quantify the threat posed by different types and degrees of attack to a system's transfer properties. An analytical framework is introduced to predict this vulnerability to attacks and rationalize it in terms of permeability evolution and population dynamics of island size.

Figure 1

Simulated example. (a) Hydraulic network comprising approximately 500 channels embedded in a diffusive matrix. The system is bidimensional and periodic in both directions. (b) The local flow field. (c) Typical tracer motions at a given time step, including random walk (orange arrow) and advection (blue arrow). Tracers carry a scalar indicator of temperatures $T_0$ (red) and $T_{in}$ (blue). Temperature in a domain is the average scalar of all the tracers it contains. Only a few tracers are represented here for clarity; the typical density of tracer used is $\rho =\frac{2}{R^2}$ (tracers/$m^2$).

Figure 2

Cooling dynamics of undamaged Gabriel networks. (a) Scaling of the cooling efficiency with the Péclet number: each symbol corresponds to cooling test simulated with a different triplet Pe, $\varphi$, and $\lambda$. Dashed horizontal lines denotes the maximum efficiency obtained by using the ${\text{Pe}}_{\infty }$ method (see text). Solid lines represent the predictions of the model in Eq. ((9) and (9) cooling dynamics of systems in regimes II and III, respectively, simulated with the tracer method $[\varphi =0.5, \lambda =22.8$, (b) $\text{Pe}=62$ and (d) $2.6\times {10}^6]$. In panel (d) the dashed line corresponds to prediction of the analytical model (6), where ${\ell }_e=0.52\overline{l}$, truncating the sum to $n,m<=19$. (c), (e) Snapshots of the temperature field during the cooling test.

Figure 3

Transfer efficiency ($E$) versus Péclet number (Pe) for systems featuring (a) lattice (b) $K$-nearest ($K=3$), and (c) Epsilon ($\epsilon =11$) hydraulic network topologies. For each type of topology, the following is presented: series of results from tracer method simulations (markers), the transfer efficiency limits obtained from ${\text{Pe}}_{\infty }$ simulations (dashed lines), predictions of the analytical model (9) (solid lines), and an example schematic of each type of topology (right inset).

Figure 4

Flow in a Gabriel network before (undamaged) and after examples of each of the four types of attack: major iterative (MI), major block (MB), minor iterative (mi), and minor block (mb). The proportion of channels clogged, $\mathcal{P}$ in all but the MI attack is $20\%$. $\mathcal{P}=2.8\%$ for the MI attack and corresponds to the state just before the first loss in connectivity when one or several nodes becomes disconnected from the network. (a) Cumulative probability density function of normalized channel flow velocity, $\tilde{v}=\frac{\left|{\overline{v}}_{ij}\right|}{v_0}$ where $v_0=\frac{R^2}{8\eta }\mathbf{\nabla }P$ is the normalization velocity scale. (b), (c) Polar plots of the angular distribution of channel flow velocity. The angular coordinate corresponds to the channel orientation with respect to the flow direction, and the radial coordinate represents the magnitude of the average normalized velocity of channels at this orientation. (d), (e) The flow velocity field after the MI and mb attacks, respectively. Channels are colored as a function of their normalized flow velocity. All but the fastest channel velocity for the MI attack $[\tilde{v}=5.4$, identified as ${\tilde{v}}_{\max }$ in gray in (d)] are presented in (a) and the color bar of (d).

Figure 5

Thermal response of an attacked network in regime II. The example results are from a cooling experiment performed using the tracer method after a major block attack ($\mathcal{P}=20\%, \varphi =0.41, T=2.27, \text{Pe}=18$). (a) Normalized average temperature variation of the matrix (red) and liquid in the channel network (blue) versus time normalized by $t_{adv}$. (b) Snapshot of the system's temperature field at $t_{50}$ where $\mathrm{\Delta }{\overline{T}}_s(t=t_{50})=0.5$. (c) Normalized transfer efficiency results for regime II (markers) versus proportion of clogged channels for each network attack. Dashed lines represent predictions of Eq. (13) for MB ($\xi =2.45$), mi and mb attacks ($\xi =-0.58$), and Eq. (14) for MI attacks (${\mathcal{P}}_c=0.04$).

Figure 6

Thermal responses of attacked Gabriel networks in regime III following the same attacks illustrated in Fig. 4. (a) Example normalized temperature variation of the matrix (red) and liquid in the channels (blue) versus dimensionless time for a MI attack ($\mathcal{P}=20\%, \varphi =0.48, \text{Pe}=8.84\times {10}^5$). The cooling dynamics for the matrix obtained using the tracer method (red) are compared with the prediction (red dashed) of the analytical model [Eq. (6)]. Best fit was achieved with $\ell =0.58\overline{l}$, truncating the sum to $n,m<=19$. Snapshots of the temperature field at $t_{50}$ (where $\mathrm{\Delta }{\overline{T}}_s(t=t_{50})=0.5$) for systems subjected to a (b) MI, (c) mi, (d) MB, and (e) mb attack.

Figure 7

Effects of attacking the network on the transfer in regime III. (a) Markers represent normalized effective length of matrix islands (determined from the thermal responses in regime III) versus proportion of attacked channels for each attack type. Dashed line represents the semiempirical model (15) with $\eta =1.8$. (b) Normalized transfer efficiency results for regime III versus proportion of clogged channels for each network attack (markers) compared to model prediction [Eq. (16) with $\eta =1.8]$ (dashed line).

Figure 8

Increase in effective length $\ell$ induced by different attacks on a $K$-nearest ($K=3$) (a) and an Epsilon (b) network. Symbols represents the measured effective length by fitting the cooing dynamics with Eq. (6). Dashed lines represents the prediction of the model (15), using the values of the parameter $\eta$ given in Table 2. Snapshots represents temperature fields taken at $t_{50}$ for the undamaged and attacked networks, as in Fig. 5.