Transport and deposition kinetics of polymer-coated multiwalled carbon nanotubes in packed beds

Ngoc H. Pham1, Changlong Chen2, Benjamin Shiau2, Jeffrey H. Harwell1, Daniel E. Resasco1, and Dimitrios V. Papavassiliou1[*]

1School of Chemical, Biological, and Materials Engineering

2Mewbourne School of Petroleum and Geological Engineering

University of Oklahoma

Norman, Oklahoma 73019-1004

Supporting Information

7 Pages

3 Figures

Determination of hydraulic diffusivity using tracer experiments

The hydraulic diffusivity Dh was determined by fitting Eq. 1 to the chloride tracer breakthrough curve. This tracer experiment was conducted prior to the injection of the studied nanoparticles at 10% salt into the same sand column where the particle injection experiments were conducted. In the tracer experiment, the column was saturated with 2wt% salt solution, followed by the flushing of 10PV of 10wt% salt solution with no nanoparticles at a flow rate of 0.3mL/min (see the “Crushed Berea Sandstone Column” subsectionin the “Materials and Methods” section), and the conductivity of the effluents was measured (S70-K SevenMultiTM conductivity meter withInLab®731conductivity probe, Toledo) instead of the concentration of nanoparticles. The transport of chloride tracer in the column was conservative and free from sand surface attachment and detachment, leading to the application of Eq. 1, but without saturation terms, resulting in a linear equation. In fact, we used the following d form of Eq. 1 to fit the tracer breakthrough curve for Dh as shown below:

(S-I)

This truncated form of Eq. (1) is the classical convection-diffusion equation, whose numerical solution is known. Presented in Figure S-I the full tracer profile along the column length at different times of injection. It can be noted that the fitting presented in Fig. 3a in the main manuscript was obtained from the 3d profile at a fixed column length of 7.62 cm.

Figure S-I. 3d profile of chlorine tracer breakthrough along the crushed Berea sand column at different time of injection. Operating conditions were those described in the “Materials and Methods” section – “Crushed Berea Sandstone Column” subsectionof the work.

Concentration and saturation profile calculations

Once the value of the hydraulic diffusivity was obtained, the particle concentration, C(x,t), profile and the surface saturation profile, S(x,t), were calculated in order to present results for the particle breakthrough as C/C0at the column outlet as a function of pore volume injected. The equations solved were as follows:

(S-II)

where C is the concentration of particles, u is the pore velocity, ρb is the bulk density of the column, ϕ is the porosity of the column, S is the surface concentration, Smax is the maximum surface concentration, ka1 is the deposition rate constant in sites of type 1, kd is the detachment rate constant, Dh is the hydraulic dispersion coefficient in the column, η0 is the single collector efficiency in the absence of the repulsive interaction, and ξ is a fitting parameter, which is expected to depend on pore velocity, ionic strength of electrolyte solution, and on particle size. We then discretized the column into 50 thin slabs along the column length for use of the method of lines, in which the spatial derivatives are approximated by finite difference relationships and the time derivative is of ordinary differential. The second – order central differencing scheme was employed for differencing the first and second order derivative in X direction. The Equation S-II in discretized formwas used for solving in Polymath as follows:

, i = 2 to 50(S-III)

where Δx is the slab thickness and i is the slab index, starting from 1 (column inlet) to 51 (column outlet). The initial and boundary conditions took the form:

(S-IV)

Note that the second – order backward differencing scheme was used to handle the boundary condition at the column outlet, and that the 51th slab (C51) was an isolated slab, allowing no particle flux to represent the collection of effluents at the column outlet. Discretization of Equation S-II resulted in a system of 99 ordinary differential equations (ODE), whose forms were as of Equation S-III.In our modeling with nanoparticles, input parameters for solving the system of ODE were Dh, u, ρb, ϕ, and Smax, typically obtained from experiments. The two remained fitting parameters were ka and ξ.

We provide herein the concentration profile of nanoparticles (Figure S-II) along with the surface saturation profile (Figure S-III) along the column length at different injectioninstances. The operating conditions were those of the 10% salt solution injection case (and slow velocity) presented in Fig. 6a, and are described in detail in the subsection entitled “Crushed Berea Sandstone Column”in the “Materials and Methods” section of the main manuscript.

It can be clearly seen in Figure S-II that the particle concentration profile exhibits a rectangular shape at inlet during the injection period. Later on, due to surface attachment and the tortuous path of the fluid through the porous medium, the rectangular shape deforms as the particles pass through the column. Along the column length, the first particle breakthrough point is now shifted to later times, and so is the final breakthrough point. The reason for the shift is mainly due to the deposition of the particles on the sand surface as depicted in Figure S-III. It can be concluded from Figure S-III that the saturation rate is really fast at all the column slabs. The surface saturation reaches 80% of the maximum surface capacity within small pore volume (less than 1PV). The saturation rate is then reduced because of the shadow effect, which makes the particle deposition a more competitive process.

Figure II. 3d plot of the particle concentration profile of the nanoparticles at 10% salt and slow velocity condition. Operating conditions are those described in the subsection entitled “Crushed Berea Sandstone Column” in the “Materials and Methods” section of the main manuscript.

Figure III. Corresponding 3d plot of the surface saturation profile of the nanoparticles passing through the crush Berea sand column, whose concentration profile is presented in Figure S-II.

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