Understanding the development of the flow and thermal processes in the VT

As previously discussed in Chapter 2.2, the TSE is mainly regarded as the combined effect of the adiabatic expansion and the internal friction; the former accounts primarily for the cooling effect and the latter for the heating effect at the opposite end of the VT. Three main stages which lead to the temperature separation can be identified based on the published works[120-123],involving the flow through the nozzle, the primary flow towards the hot end and the secondary flow towards the cold end, as shown in Figure 1.1. Using the current CFD model, some preliminary results are obtained that allow us to gain an appreciation in the associated thermal processes. 1) The flow through the nozzle

A convergent nozzle is always employed in VT. The fluid undergoes an isentropic expansion[118] when passing through this kind of nozzle, resulting in a considerable temperature drop[118]. As shown in Figure 4.7, the fluid static (actual) temperature at the nozzle outlet (or chamber inlet) area is around 245 K; with a nozzle inlet temperature set at 295 K (not shown), this represents a drop of around 50 K through the nozzle.

Figure 4.7 Static temperature distribution of the VT (air, μc = 0.3, Tin = 295K, pin = 0.17 MPa, pc = 0.10 MPa)

11 _{The interruption/stopper is applied to stop the flow from rotating before exiting the hot end}[17]

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2) The primary flow towards the hot end

Along the axial direction of the vortex chamber and the hot tube, the fluid expands and moves, from higher pressures at inlet towards a lower pressure at the hot end (as shown in Figure 4.8). At any cross-section along the hot tube, the primary flow can be seen as consisting of two movements, an outward and an inward movement, as shown in Figure 4.9 and Figure 4.10 (also shown in [81]). In the former the flow rotates around the axis and leaves at the hot end, and in the latter, the flow spirals inwardly towards the throttle. Part of this inward movement in fact turns around towards the cold end before even reaching the throttle and the rest gets rebounded by the throttle. Figure 4.10 presents the combined tangential and radial velocity vectors at cross- sections along 4 axial locations as indicated in Figure 4.9, and Figure 4.11a-c presents respectively the tangential, axial and radial velocity components at the same locations. As seen, the rotational strengths gradually decrease as the flow moves from positions 1 to 4. The radial velocity components are much weaker than the tangential ones, especially in the middle part of the hot tube; small positive radial velocities at location 4 indicate the flow are moving away from the axis existing the hot end, whereas negative radial velocities at location 1 suggest the flow is moving towards the axis joining the secondary flow.

Figure 4.8 Static pressure (gauge) distribution in the VT (air, μc = 0.3, Tin = 295K, pin = 0.17 MPa, pc = 0.101 MPa)

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Figure 4.9 Streamlines generated from CFD simulation, (air, μc = 0.3, Tin = 295K, pin= 0.17 MPa, pc = 0.101 MPa), the dotted line shows the boundary between the primary and the secondary flows

Figure 4.10 Combined tangential and radial velocity vectors at four positions corresponding to that shown in Figure 4.9

Position 1 Position 2 Position 3 Position 4

Inlet flow

Hot end Cold end

Axis

Outward movement Inward movement

Flow recirculation Primary flow Secondary flow y x Velocity scale 100 m/s

0 50 100 150 200 250 300 0 2 4 6 8 10 r ( mm) Tangential velocity (m/s) Position 1 Position 2 Position 3 Position 4 -60 -40 -20 0 20 40 60 80 0 2 4 6 8 10 r (mm) Axial velocity (m/s) Position 1 Position 2 Position 3 Position 4 -4 -3 -2 -1 0 1 2 3 4 0 2 4 6 8 10 r (mm) Radial velocity (m/s) Position 1 Position 2 Position 3 Position 4 a b c

Figure 4.11 Tangential (a), axial (b), radial (c) velocities of 4 positions corresponding to Figure 4.9

Figure 4.12 Flow vector (x-y plane) within the VT (air, μc = 0.3, Tin = 295K, pin = 0.17 MPa, pc = 0.101 MPa) Velocity scale (m/s) y x Flow recirculation Outward movement Inward movement Primary flow Secondary flow Primary flow Secondary flow

As illustrated in Figure 4.13, when considering the two adjacent fluid elements 1 and 2 in a y-z plane, entering the VT chamber, say, belonging respectively to an outward movement and an inward movement. The inertia force attempts to maintain their angular momentum when entering the rotating flow. At the same time, the shear force between these two elements slow down their rotating speed, as indicated by the decreasing tangential velocities in the radial direction, as presented in Figure 4.11a. Therefore, there must be a transfer of kinetic energy to the internal energy by some kind of internal friction, leading to a temperature rise of these fluid elements in the primary flow towards the hot end, as shown in Figure 4.7.

Figure 4.13 Sketch of the flow process within the VT (red and yellow line representing the primary flow, blue line the secondary flow)

In experimental research for VT, the total temperatures instead of static temperatures were often measured[1] and presented, and the TSE is also defined based on the difference of the total temperature[72]. Figure 4.14 presents the CFD results of the total temperature.

Figure 4.14 Total temperature distribution of the VT (air, μc = 0.3, Tin = 295K, pin = 0.17 MPa, pc = 0.101 MPa)

Before reaching the hot throttle, in the radial direction, element 2 (as illustrated in Figure 4.13 and represented by the inward movement in Figure 4.12) moves inwards and undergoes an adiabatic expansion due to the pressure differential, as seen in Figure 4.8, between the peripheral and the central parts of the VT. This is similar to the

1 2

o 

Exit (hot throttle)

Turning position

y

angular propulsion process for a rotating flow[63], in which the element both rotates around and moves toward the rotating centre. Within the rotating frame of the primary flow, element 2 which has the same angular velocity as the frame, will overcome the Coriolis force and the centrifugal force to move inward[63]. On its journey to the hot throttle, work must have been done by element, transferring its internal energy and the rotational (kinetic) energy to the rotational (kinetic) energy of the rotating frame. As a result, the total temperature of this fluid element drops[63]_{. This increased rotational} energy of the frame is transferred to the outer part of the primary flow by friction from shear force, resulting in an increase of total temperature towards the wall, as seem in Figure 4.14.

3) The secondary flow towards the cold end

Near the hot end, the turning around of the inner layers of the primary flow become the secondary flow moving towards the cold end (Figure 4.11b) due to pressure differential, while still maintaining the same rotating direction, as shown in Figure 4.11a. However, despite a pressure drop between the chamber inlet and the cold end, there is no direct flow between these two locations, though the experimental work by Xue et. al[123] using water shows a direct flow is possible.

As shown in Figure 4.11b, the axial velocities keep increasing when the secondary flow moves towards the cold end, suggesting there may be further transfer of internal energy to the kinetic energy, leading to a temperature decrease of the secondary flow from the hot to cold end, as shown in Figure 4.7. When the secondary flow approaches the cold orifice, the outer part of the secondary flow recirculates and mixes with the primary flow (as shown in Figure 4.9 and Figure 4.12) due to the centrifugal force.

This could further convey the energy to the primary flow[73].

In summary, the cooling effect appears mainly due to the expansion through the nozzle, and the adiabatic expansion in the spiral motion (angular propulsion effect) in the primary and secondary flows. The heating effect mainly comes from the internal friction due to shear friction.