Mechanical Stirring System

The mechanical stirring system is intended to increase the gas velocity near the walls of the reactor tube. Since plasma formation is more difficult in high-velocity regions, formation near the walls is discouraged and the plasma will instead favour the centre of the tube. The system is shown in Figure 3.5a. The paddles of the stirring system are located upstream of the plasma, meaning that the gas must maintain its circumferential velocity until it reaches the plasma region for the design to be effective and successful. A simplified scenario takes the velocity profile of the gas as it leaves the stirring system and models the velocity decay over time. Using the volumetric flow rate, this decay time can then be converted into an axial travel distance through the reactor tube. As a starting point, the circumferential component of the Navier-Stokes equation in cylindrical coordinates can be used, and several simplifying assumptions can be applied. These assumptions are that (1) the system is at steady-state, (2) the gas is nitrogen at 600 K, (3) the initial gas velocity profile matches that of the paddles, (4) there is zero velocity at the tube wall, (5) there is zero circumferential velocity on the tube axis, (6) the system is circumferentially symmetric, (7) there is no radial velocity, and (8) there are no circumferential acceleration or pressure terms. With these assumptions, the Navier-Stokes equation reduces to:

ρ∂ uθ ∂ t = µ  1 r ∂ ∂ r  r∂ uθ ∂ r  −uθ r  . (3.6)

Here, uθ is circumferential velocity, t is time, ρ is density, and r is radius. This

equation was then input into a partial differential equation (PDE) solver in Matlab. The boundary conditions are uθ = 0 at r = 0 and at r = rout (the inner radius of the

3.5 Plasma Stabilization 43

reactor tube). Additionally, the initial condition is that the velocity profile matches that of the paddles (i.e. increases linearly with radius) between the tube axis and the paddle radius, and is a decreasing linear function between the paddles and tube wall:

u_θ = ωpadr, for 0 < r < rpad, (3.7)

u_θ = ωpadrpad

r− rout

r_pad− rout

, for rpad< r < rout. (3.8)

Here, ωpad is the angular velocity of the paddles and rpad is the outer radius of the

paddles. The circumferential velocity profile from the solution to this PDE as a function of time and radius is shown in Figure 3.5b. Similarly, the velocity profile as a function of radius at multiple time steps is shown in Figure 3.5c. As expected, the peak of the velocity profile reduces in magnitude over time and moves towards the radius equidistant from the zero-velocity boundary conditions. From the volumetric flow rate and the inner diameter of the tube, the velocity profile as a function of time can be converted to a profile as a function of axial distance from the paddles (z direction), as long as the scenario is simplified so as not to include the parabolic flow profile of fully-developed flow: L= uzt= Q At= Q π r2_out t. (3.9)

Here, L is axial distance from the paddles, Q is volumetric flow rate, and A is cross- sectional area. These distances are also shown in Figure 3.5c. This information can be used to determine whether the mechanical stirring mechanism is viable and if so, the proximity of the stirring paddles to the plasma can also be determined to maintain sufficient gas swirl. Under these conditions, the peak velocity is approximately 0.3 m s−1and quickly decays below 0.1 m s−1 in less than one second (corresponding to an axial travel of approximately 10 cm at the example flow rate of 5 SLPM). Importantly, the Reynolds number in the tube at this flow rate is approximately 120, meaning the flow should be laminar and any swirl imparted to the gas from the paddles will not be immediately disrupted by turbulence. Experimentally, this amount of vorticity produced from the paddles was still not able to stabilize the plasma and unfortunately the rotational speed of the paddles presents a limitation. It was not possible to fit a motor which was able to spin the paddles any faster, while still maintaining an appropriate gas seal. Instead, other solutions to increase gas velocity and vorticity have been explored and are discussed below.

Part of the issue with the mechanical stirring mechanism is that the paddles must be located relatively far from the plasma, so preservation of the vorticity is difficult. The shaft and paddles themselves are fabricated from stainless steel which has a reasonably high operating temperature window. However, since the vapour pressure of metals

b) c) a) (m s -1) t = 0.00 s, L = 0.00 cm t = 0.10 s, L = 1.12 cm t = 0.20 s, L = 2.24 cm t = 0.40 s, L = 4.49 cm t = 0.80 s, L = 8.98 cm t = 1.20 s, L = 13.47 cm t = 1.60 s, L = 17.96 cm t = 2.00 s, L = 22.45 cm (m s -1)

Fig. 3.5 (a) Mechanical stirring mechanism schematic, (b) surface plot of circumferen- tial velocity (u_θ) as a function of both time and radius, and (c) circumferential velocity (u_θ) as a function of radius at various time steps, which have also been converted to axial distance past paddles given a flow rate of 5 SLPM

is higher than most ceramics, they tend to produce nanoparticles as they are heated which can contaminate the process. This is particularly troubling for the applications presented here since the primary component of stainless steel is iron which is an excellent catalyst for CNT growth, thus bringing into question whether the resultant CNTs would have been grown from contaminating particles or intentional catalyst particles. A second but more immediate concern is that closer to the plasma, very high microwave field strengths are present. These fields cause additional heating of materials with high or infinite dielectric strengths, and can often result in arcing as well as melting of metal components. It is conceivable that a ceramic shaft and paddles could be constructed but this would be challenging to fabricate and as mentioned above, the maximum velocity that can be produced is still rather low since it is limited by the motor speed and torque. Therefore, inducing vorticity using fluid flow rather than mechanical means has been pursued.

3.5 Plasma Stabilization 45