Chapter 8 – magnetic Resonance Compatible Vortex Ring based Comple
8.4 Magnetic Resonance compatible Vortex Ring based Complex Flow Phantom Design
8.4.1 Introduction
Design limitations rapidly became apparent in this prototype. Specifically, the motor struggles to deliver enough energy to generate fast vortex rings (i.e. translational velocities higher than 30 cm/s). The long and narrow pipe also has a tendency to introduce turbulence into the flow that is visibly transferred into the vortex ring. Pressures generated by the system are insufficient to create water leakage at the piston cylinder interface. Limitations and potential alternatives are discussed in the following sections.
8.4.2 Design Limitations – Motor energy
A simple energetic analysis clarifies why the motor struggles to generate fast vortex rings. The motor moves the piston and the water contained within the “MR Cylinder” (Figure 8.7), the “Tank Cylinder” (Figure 8.8) and the pipe. The kinetic energy needed to generate a vortex ring with energy of ~0.3 mJ (Section 7.5.3, Chapter 7) is:
𝐸 =
1 2𝑚𝑣
2=
1 2𝑚
𝑝𝑖𝑠𝑡𝑜𝑛𝑉
𝑝𝑖𝑠𝑡𝑜𝑛 2+ 𝑚
𝑤𝑎𝑡𝑒𝑟/𝑀𝑅 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟𝑉
𝑤𝑎𝑡𝑒𝑟/𝑀𝑅 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟2+
𝑚
𝑤𝑎𝑡𝑒𝑟/𝑝𝑖𝑝𝑒𝑉
𝑤𝑎𝑡𝑒𝑟/𝑝𝑖𝑝𝑒2+ 𝑚
𝑤𝑎𝑡𝑒𝑟/𝑡𝑎𝑛𝑘 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟𝑉
𝑤𝑎𝑡𝑒𝑟/𝑡𝑎𝑛𝑘 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟2+ 0.3 mJ
(1)
195 Calculations are summarised in Table 8.2:
Radius (m) Length (m) Volume (m3) Mass (kg)
pipe 0.006 10 0.0011304 1.1304
tank cylinder 0.035 0.095 0.000365418 0.3654175
mr cylinder 0.035 0.1 0.00038465 0.38465
piston 0.33716*
Energy = 0.068 + 0.077 + 261.776 + 0.731 + 0.3 = 262.95 mJ = 0.263 J
Table 8.2: summary of kinetic energy calculations to generate a vortex ring with 0.3 mJ with the MRI compatible
complex flow phantom.
* value calculated in Section 7.5.2, Chapter 7.
When the flow is pushed through a pipe there is a resistance (analogue to electrical resistance V = IR) between the fluid and the vessel wall that generates negative work. For this case the resistance is assumed constant and it is defined as the ratio between the pressure difference (ΔP) and the rate of change of mass or volume (ΔQ).
𝑅 =
∆𝑃
∆𝑄 (2)
The flow through the pipe is affected by tube diameter, tube length and fluid viscosity (Equation 3). Flow is directly proportional to the fourth power of the tube radius (r4) and inversely proportional to
the length of the tube. Viscosity is a measure of frictional forces within the fluid layers, therefore, the flow is also inversely proportional to viscosity. All these variables are brought together by the Hagan- Poiseuille equation:
𝑄 =
𝑃𝜋𝑟
48𝜂𝐿
(3)
Where Q is the flow rate (Litres/seconds), η is the viscosity (Pascals x seconds), P is the pressure (Pascals), r is the radius of the tube (meters), L is the length of the tube (meters). From equation (3), it is possible to calculate the pressure difference between the two ends of the pipe.
∆P =
8ηLQ
πr
4(4)
Consequently, the pressure difference needed to push a column of fluid of ~3.07 x 10-3 litres, in 0.04 seconds, within a tube of 10 m length and 12 mm diameter, is 1.34 x 106 Pa. This calculation is based on assumptions of incompressible flow, laminar flow within the pipe, no deformation of the pipe walls, no acceleration of flow through the pipe and constant circular pipe cross-section. Under these assumptions, Bernoulli’s equation stated that (block scheme in Figure 8.12):
𝑃1
+ 1
2ρ𝑣1
2
+ ρgℎ
1
=
𝑃2
+
1
2ρ𝑣2
2
+ ρgℎ
2 (5)
Where P1 is the pressure within the MR cylinder, ρ the density of the water, v1 the fluid velocity within
the MR cylinder, g the gravitational constant, h1 the height of the fluid within the MR cylinder. P2, v2
196 and “Tank cylinder” (Figure 8.8) at the same level h1 = h2 , consequently, there is no variation of
potential energy.
𝑃1 =
𝑃2
+ 1
2ρ(𝑣2
2
− 𝑣
12) (6)
Assuming that no further work (J) is needed to push the fluid from the tube-end to the orifice (it is less than 1 mJ, Table 7.7, Chapter 7), 1.34 x 106 Pa. The work (Joule = Pascal/m3) needed to push a column of ~3.07 x 10-3 litres through the pipe in 0.04 s is:
𝑊 =
𝑃
1𝑚
3=
𝑃
2+
1 2ρ(𝑣
2 2−𝑣
12)
𝑚
3= 4.12 𝐽𝑜𝑢𝑙𝑒 (7)
The total kinetic energy needed to generate a vortex ring with 0.3 mJ energy is 4.12 J plus 0.263 J (Etot ~ 4.38 J). In the real case, there are losses within the tube, the fluid is turbulent at that speed (Reynolds number ~ 6000) and the tube walls are deformable. In a circular pipe with smooth internal surface and uniform diameter the pressure loss due to viscous forces can be calculated with the Darcy- Weisbach equation:
∆𝑝
𝐿
= 𝑓𝐷
𝜌
2
〈𝑣〉
2𝐷
(8)
Where ∆p/L is the pressure loss per unit of length, р the density of the fluid, D the hydraulic diameter of the pipe, <v>2 the mean flow velocity and fD the Darcy-friction factor. Consequently, the work (J)
needed is higher than the motor can supply in 0.04 seconds (Table 7.6, Chapter 7). Selecting the maximum input power of 96.84W, the motor is capable of delivering 4.38 J in 0.045s. The suggestion is to significantly increase the pipe diameter and reduce the length (where possible) or to control two motors with the same Arduino Board.
Figure 8.12: Schematic block of the MRI compatible Vortex Ring based Complex Flow Phantom design –
quantities involved in Bernoulli’s equation (Equation 3). Please note that this is a schematic representation and elements are not to scale.
197
8.4.3 Design Limitations – Turbulence
The flow within the pipe has a Reynolds number of ~6000. Consequently, the flow is turbulent and this is reflected in the generation of turbulent vortex rings. This effect is clearly visible from visual inspection. Turbulent flow is not convenient for a test object because it is characterised by chaotic fluid motion with random changes in pressure and velocity. Three potential alternatives were identified to overcome this problem. The “Tank Cylinder” might be manufactured big enough to avoid turbulence without compromise of vortex ring generation (option 1). However, a bigger “Tank Cylinder” implies a heavier weight needs to be sustained by the screw coupling and a compromise should be chosen. A flow straightener might be added into the “Tank Cylinder” to stabilise the flow before vortex ring generation (option 2). A flow straightener is a component, often based on a honeycomb or a circular structure, which minimises the lateral velocity components (caused by swirling motion and turbulence) of the flow. By way of an example, a simple flow straightener prototype was built using two laser-cut PMMA (Perspex) bases and (about) two hundred plastic mini cocktail straws of 3 mm diameter (Figure 8.13). Number of straws and dimensions should be chosen to create minimal resistance to the flow.
Figure 8.13: MRI Compatible Vortex Ring based Complex Flow Phantom – Flow Straightener (simple prototype).
Alternatively, the hydraulic piston (option 3) could be designed as shown in schematic block in Figure 8.14. A cylindrical block of Perspex can be manufactured with low tolerances following the method described in Section 6.2.2 (Chapter 6). Sucking fluid from the “Tank Cylinder” cylinder allows the piston to retract while routing pressurised fluid into the “Tank Cylinder” allows the piston to extend.
198 Figure 8.14: Schematic block MRI Compatible Vortex Ring based Complex Flow Phantom – hydraulic piston.
The piston connected to the pump hydraulically displaces the piston at the tank. Please note that this is a schematic representation and elements are not to scale.
Obviously, the introduction of a flow straightener or of a further Perspex block (hydraulic piston) implies further negative work (i.e. viscous losses of the fluid through the straightener, friction of the hydraulic piston) that needs to be considered for the choice of the actuator system (motor).
8.4.4 Design Limitations – Piston leakage
A negligible leakage (about 30 ml in 2 hours) was noticed from the piston cylinder composing the actuator system. The piston cap (Figure 6.1, Chapter 6) manufactured from Perspex with low (+/- 0.10 mm) tolerance offered optimal sealing properties and smooth coupling. However, the pressure difference exerted by the new system pushes the water around the piston cap, provoking a tiny leakage. Since the leakage is negligible, PTFE thread seal tape could be placed on the “Piston Guide” (Figure 8.9) internal screw to avoid any discharge. However, the best option is probably to design a hydraulic piston, as discussed in Section 8.4.3. The piston cap contained within the “Tank Cylinder” separates the hydraulic fluid that drives the actuator system (“Actuator System”, Figure 8.14) from the fluid needed for the vortex ring generation (“Imaging System”, Figure 8.14). Lubricants can be used because they do not interact with components contained in blood mimicking fluids that are compatible with medical imaging. Consequently, the piston design proposed in Chapter 4 (“Plunger”, Figure 4.4, Chapter 4) can be used. The rubber O-ring delivers optimal sealing properties while the lubricant facilitates smooth coupling during dynamic applications.
8.5 Summary
Hazards and safety regulations for Magnetic Resonance Units have been described in detail in Section 6.3. Ferromagnetic materials cannot be used because, beyond certain limits (3 mT), they experience a projectile effect. In addition, MR scanners are manufactured with very homogeneous magnetic fields and the introduction of electrical motors produces distortion of the magnetic field and artefacts in the
199 images. MRI compatible piezo linear actuators are available on the market but are expensive and limited in terms of power, velocity and displacement extent.
Slight modifications were made to the Vortex Ring based Complex Flow Phantom (described in Chapter 4) to adapt the design for the MR Environment. The component “Tank Cylinder” (Figure 8.8) was manufactured to be compatible with the “Imaging Tank” (Figure 4.3, Chapter 4) and with a BSPT connector. The piston cylinder system was manufactured following the method described in Chapter 6. Distinct from the previous design, the output of the cylinder provides coupling with a BSPT connector. A new base and further Perspex blocks were manufactured to secure an aligned connection between stepper motor, piston and cylinder. The “Imaging Tank” (Figure 4.3, Chapter 4), manufactured exclusively from plastic components, can be placed in the scanner while the piston cylinder unit remains outside the MR Environment (Zone IV). The two systems are connected through a PVC pipe hose which is secured with Nylon 6.6 plastic hose clips. Functionality and application of the system were demonstrated by generating vortex rings at Leeds Test Objects Ltd (Section 8.3.6). As expected from the manufacturing of a first prototype, limitations were identified and potential alternatives were proposed. Particularly, the pipe that connects the two systems is narrow (12 cm diameter) and very long (10 m), causing noticeable pressure difference between the extremities. The energy (J) needed to overcome this pressure difference is very close to the maximum energy the motor can deliver. The chosen stepper motor was one of the most powerful on the market (36 V, 3 A) at the time of writing (2018). Unless other technologies are released in the future, larger pipes (pressure inversely proportional to the forth power of the pipe radius) or the connection of two identical motors to the same Arduino Board are possible solutions. The narrow pipe also introduces other limitations. The flow within it is turbulent (Re ~ 6000) and affects the generation of laminar vortex rings. From experimental observations it was noted that most of the vortex rings produced exhibited a turbulent core. Potential alternatives are to increase the pipe diameter, increase the “Tank Cylinder” length, or introduce a flow straightener into the “Tank Cylinder”, or the manufacture of a hydraulic piston system (as shown in Figure 8.14). Fortunately, the pressure difference caused by the pipe generates negligible piston leakage. The manufacturing of a hydraulic piston system guarantees separation between the two systems (actuator system and imaging system, Figure 8.14), consequently, the piston design proposed in Chapter 4 could be used. The O-ring delivers waterproof sealing and a lubricant can be introduced to guarantee smooth piston displacement profiles.
Overall, the system has demonstrated reliability, and generated vortex rings as expected. Despite its limitations it is certainly capable of challenging clinical MRI Units. The instrumentation pack can be applied to the system and new reference flow values can be easily calculated following the methods described in Chapter 6.
8.6 Conclusion
A number of restrictions apply when working with MR Units. Notably, electrical motors and ferromagnetic materials cannot be included within the MRI Environment (Zone IV). New components were manufactured from Perspex and a cost-effective Magnetic Resonance Compatible Vortex Ring based Complex Flow Phantom design is proposed. Being just a first prototype, some limitations were identified but these have been discussed in detail and potential cost-effective alternatives were suggested. Overall, the phantom performed adequately and is in a form that can be scanned in a clinical environment.
200