Case setup

The three-dimensional geometry consists of two reservoirs of water, held at different pressures, separated by a (15,15) carbon nanotube (CNT) of length 50 nm and diameter approximately 2 nm, see Figure 5.2(a); since the domain is periodic in the y- and z-directions this setup represents a regularly repeated array of CNTs. To save computational time of the CFD only a 6° wedge of the domain is simulated, with symmetry boundary conditions are used so that the mesh represents the full cylinder, see Figure 5.2(b).

We again compare our enhanced CFD predictions against full MD results and against the standard CFD models outlined in §4.3.1. The pressure difference between the reservoirs is set to be 200 MPa because it is very challenging to obtain useful information from MD using only low pressure differences due to the extended sampling times required to filter low-velocity signals from the thermal noise [104]. These high pressure (and consequently density) differences make the CFD predic- tions even more challenging. The full MD simulation was performed in parallel on 48 CPUs, with the majority of the computational effort attributable to the two reservoir regions. These have dimensions 4.4×10.6×10.3 nm and are chosen to be large enough to avoid any effects on the CNT flow due to reservoir boundaries. The intermolecular potentials used are the same as those in the pre-simulation, as given in Section 2.3.3.

5.5

Mesh dependency study

As in the previous chapter, we must perform a dependency study on the number of cells required to obtain results that are independent of the mesh. We measure the mass flow rate at the center of the channel for meshes with between 10,000 and

650,000 cells. The results are plotted in Figure 5.3 and show that the results become effectively mesh independent when 160,000 cells are used. Therefore, the remainder of the simulations presented in this chapter have been performed with 160,000 cells.

0.03 0.035 0.04 0 2·105 4·105 6·105 Mass Flo w Rate, ˙ m [ng/s] Number of Cells

Figure 5.3: Mass flow rate (measured in the middle of the CNT) versus the total number of cells used in the simulation.

5.6

Results

Figure 5.4 shows pressure and density plots along the centreline of the CNT; in the MD this data is collected within a cylinder of radius 0.1577 nm about the centerline. Due to the substantial molecular layering within the CNT (see Figure 5.5(b)) a bulk density effectively does not exist, and the choice of the size of this sampling region can substantially affect the ‘bulk’ density measured.

The no-slip CFD model does not exhibit large pressure drops at the inlet and the outlet (shown in Figure 5.4 at the grey dashed lines) due to the much lower velocity in the tube (and therefore there are lower accelerations at the inlet and outlet) than in the slip cases; this is shown in equation (5.1) by the dependence on mass flow rate. The classical continuum pressure drop along a cylinder ∆pHP can be predicted using equation (5.2); using this the pressure drop is predicted to be ∼_{50 MPa, in good agreement with our CFD results. The MD results, on the other} hand, show a channel pressure drop much lower than this. This might suggest that the slip length is larger in the MD than was used in the CFD (asξ _{→ ∞},∆pHP →0), possibly due to a geometry dependence, e.g. curvature effects, which is not captured by the pre-simulation methodology.

Despite this the cross-sectional velocity profiles in the centre of the CNT, plotted in Figure 5.5 (a), shows good agreement. The mass flow rate in the full

MD simulation is measured to be 4.3_×10−14_{kg/s, which is 23 % greater than that}

predicted by our enhanced CFD. That this is a significant improvement on conven- tional CFD model predictions is indicated in Table 5.1. The under prediction for the enhanced CFD could be explained partly by a lower slip length used than was exhibited in the Full MD and partly by the high variation in the fluid density as shown in Figure 5.5 (b). 0 100 200 0 10 20 30 40 50 Pressure, p [MP a] Streamwise Direction,x[nm]

Full MD Enhanced CFD No slip CFD Incomp. Slip CFD

(a) 800 900 1000 1100 1200 0 10 20 30 40 50 Densit y, ρ [kg/m 3] Streamwise Direction,x[nm] (b)

Figure 5.4: Water density and pressure along the centerline of the CNT. The noise in the MD data is due to the relatively small number of molecules being sampled in the absence of a well defined ‘bulk-flow’ region. The vertical lines atx= 4.4 nm and

x= 54.8 nm represent the inlet and outlet, respectively. Full MD (solid), enhanced CFD (dashed), incomp. slip (dotted) and no-slip CFD (dash-dotted).

Given the _∼2 nm diameter of the (15,15) CNT and despite the molecular layering that actually occurs within the flow field, as evidenced in Figure 5.5 (b), our enhanced CFD approach can be considered reasonably robust in predicting important averaged fluid properties to the correct order of magnitude. These CFD results are obtained with negligible cost in comparison to full MD simulations.

0 10 20 30 0 0.25 0.5 0.75 1 V elo cit y, v [m/s]

Normal distance to wall, n[nm]

Full MD Enhanced CFD No slip CFD Incomp. Slip CFD

(a) 0 1000 2000 3000 0 0.25 0.5 0.75 1 Densit y, ρ [kg/m 3]

Normal distance to wall, n[nm]

Full MD Enhanced CFD Pre-Simulation

(b)

Figure 5.5: (a) Streamwise velocity cross-sectional profile in the longitudinal center of the CNT, and (b) the radial water density profile within the CNT. The vertical lines at x = 0.266 nm indicate the position of the CFD surface; the CNT surface atom centres are atx= 0.

5.7

Summary

We have shown in this chapter that it is possible to obtain reasonable predictions for three-dimensional flows through nano channels at the continuum limit. The likely reason that our enhanced CFD estimates are reasonable is that the flow in the CNT is dictated by the liquid interaction with the smooth graphitic surface (which is adequately modelled), despite the non-continuum conditions within the fluid. However, there is possibly a curvature dependence on the slip length, which was not included in the MD pre-simulations. This could be rectified by performing pre-simulations of a section of the CNT to capture any geometry dependence on the slip length or surface displacement. As mentioned in Section 3.3.3, if too many pre-simulations are required to obtain data for the parameter space, a concurrent

Table 5.1: Water mass flow rate predictions for each model of the CNT. The per- centage difference (error) between the mass flow rates predicted by the CFD models and the full MD results are presented in parentheses.

full MD enhanced CFD no-slip CFD incomp. slip CFD ( ˙mmd×1014kg/s) ( ˙mA×1014kg/s) ( ˙mB×1014kg/s) ( ˙mC×1014kg/s) CNT 4.3 3.3 (-23%) 0.15 (-97%) 9.0 (+109%)

hybrid methodology, such as using the IMM [125], could be more appropriate to accurately model CNTs of this size.