Chapter 3: Results and Discussions
3.2 Optimization of The Vortex Tube Energy Separation
3.2.2 Effect of Tube Length-to-Diameter Ratio
44
Negative values in Figure 21 correspond to convergent angles, while zero represents a straight angle and positive values indicate divergent angles. The low Coefficient of Performance (COP) of a vortex tube can be attributed to the absence of an external energy source to power it, unlike conventional refrigeration systems. In traditional refrigeration systems, an external energy source, such as electricity or fuel, powers the compressor, compressing the refrigerant and creating a temperature difference. Conversely, in a vortex tube, the sole driving force behind energy separation is the high-pressure flow entering the tube. This limitation results in the energy efficiency of the vortex tube being constrained by the properties of the compressed air, including its pressure, temperature, and flow rate. Additionally, the design parameters of the vortex tube, such as its diameter, length, number, and size of tangential inlets, and the angle of the conical nozzle, can significantly influence its efficiency, leading to alterations in its COP.
45 Figure 22: Radial Distribution of Total Velocity for Different Lt/Dt
Tangential Velocity
The findings are presented in Figure 23, which reveals that an increase in the length-to-diameter ratio of a convergent vortex tube can reduce the tangential velocity.
The maximum tangential velocity is also observed at a Lt/Dt ratio of 2.5. Therefore, a shorter length-to-diameter ratio increases the turbulence and, consequently, the velocity within the convergent vortex tube. In contrast, reference [30] used a divergent vortex tube with a low Lt/Dt ratio of three.
Figure 23: Radial Distribution of Tangential Velocity for Different Lt/Dt -50
0 50 100 150 200 250 300
0 0.2 0.4 0.6 0.8 1
Vtotal (m/s)
r/Rt
Lt/Dt = 2.5 Lt/Dt= 3 Lt/Dt= 5
Lt/Dt= 7 Lt/Dt= 8.33 Lt/Dt= 11
-50 0 50 100 150 200 250 300
0 0.2 0.4 0.6 0.8 1
Vtangential (m/s)
r/Rt
Lt/Dt = 2.5 Lt/Dt = 3 Lt/Dt = 5
46
Axial Velocity
The results presented in Figure 24 demonstrate that increasing the length-to- diameter (Lt/Dt) ratio of a convergent vortex tube can lead to a decrease in axial velocity.
This can be attributed to the fact that the reduction in tube diameter as the length of the tube increases leads to an increase in friction between the gas particles due to the intense Rankin vortex, resulting in a decrease in the kinetic energy of the vortex. As a result, the vortex becomes weaker and less stable as being moved away from the inlet, reducing axial velocity. Consequently, the maximum axial velocity is attained when the Lt/Dt ratio equals Three.
Figure 24: Radial Distribution of Axial Velocity for Different Lt/Dt
Radial Velocity
As demonstrated in Figure 25, as the length-to-diameter ratio of a convergent vortex tube increases, the radial velocity exhibits a peculiar trend. Several variables, including the geometry of the vortex tube, the operating conditions, and the flow regime, influence radial velocity. Although the relationship between Lt/Dt ratio and radial velocity in the convergent vortex tube may not be straightforward, an increase in Lt/Dt
may result in a weaker and less stable vortex, exemplifying the flow's peculiar behavior as Lt/Dt increases.
-80 -60 -40 -20 0 20 40 60 80
0 0.2 0.4 0.6 0.8 1
Vaxial (m/s)
r/Rt
Lt/Dt = 2.5 Lt/Dt = 3 Lt/Dt = 5
Lt/Dt = 7 Lt/Dt = 8.3333 Lt/Dt = 11
47 Figure 25: Radial Distribution of Radial Velocity for Different Lt/Dt
3.2.2.2 Pressure Distribution Static Pressure
The radial profile of static pressure in RHVT is studied by varying the length-to- diameter ratio, as depicted in Figure 26. The results indicate that as Lt/Dt decreases, the static pressure drops in the cold region and rises in the annular region, indicating improved energy separation as the flow becomes more concentrated. Since the flow became highly turbulent, this occurred to the radial pressure profile. Moreover, the results indicate that the annular region has the highest static pressure for Lt/Dt of 2.5, while the Lt/Dt of 3 has the lowest static pressure in the central region.
Figure 26: Radial Distribution of Static Pressure for Different Lt/Dt -1
-0.5 0 0.5 1 1.5 2 2.5 3
0 0.2 0.4 0.6 0.8 1
Vradial (m/s)
r/Rt
Lt/Dt = 2.5 Lt/Dt = 3 Lt/Dt = 5
Lt/Dt = 7 Lt/Dt = 8.3333 Lt/Dt = 11
10000 60000 110000 160000 210000 260000 310000 360000 410000 460000
0 0.2 0.4 0.6 0.8 1
Pstatic (Pa)
r/Rt
Lt/Dt = 2.5 Lt/Dt = 3 Lt/Dt = 5
48
Total Pressure
The total pressure of a moving fluid equals the sum of its static and dynamic pressures. Figure 27 demonstrates the radial profile of total pressure for various Lt/Dt
vortex tubes at an axial location of z/L = 0.45. It has the same trend as static pressure but with a higher value due to the dynamic effect; additionally, the static and total pressure at the wall is identical due to zero dynamic pressure when the velocity is zero because of a no-slip condition.
Figure 27: Radial Distribution of Total Pressure for Different Lt/Dt
3.2.2.3 Thermal Distribution Static Temperature
Figure 28 depicts the relationship between Lt/Dt and static temperatures. The illustration shows the effect of Lt/Dt, with decreasing Lt/Dt resulting in a decrease in the cold temperature of the core. The optimal temperature separation for the convergent vortex tube occurs at a Lt/Dt ratio of 3. Since the flow becomes more turbulent with decreasing Lt/Dt, the viscous effect and pressure expansion increase, causing the central region to lose more heat. Moreover, if the Lt/Dt ratio is too low, the vortex becomes unstable, and temperature separation is less effective, as evidenced by a Lt/Dt ratio of 2.5. The results also indicate that as Lt/Dt increases, the temperature of the hot flow in the annular region increases.
10000 60000 110000 160000 210000 260000 310000 360000 410000 460000
0 0.2 0.4 0.6 0.8 1
Ptotal (Pa)
r/Rt
Lt/Dt = 2.5 Lt/Dt = 3 Lt/Dt = 5
Lt/Dt = 7 Lt/Dt = 8.3333 Lt/Dt = 11
49 Figure 28: Radial Distribution of Static Temperature for Different Lt/Dt
Total Temperature
Figure 29 depicts the effect of Lt/Dt on the total temperature, demonstrating that it follows the same trend as the static temperature, given that the total temperature depends on both the static temperature and the velocity. A Lt/Dt ratio of 3 is optimal for achieving the desired total temperature, while a ratio greater than 3 has a negative effect. Since the velocity at the nonslip wall is zero, the wall's static and total temperatures are equal.
Figure 29: Radial Distribution of Total Temperature for Different Lt/Dt 260
270 280 290 300 310 320 330
0 0.2 0.4 0.6 0.8 1
Tstatic (K)
r/Rt
Lt/Dt = 2.5 Lt/Dt = 3 Lt/Dt = 5
Lt/Dt = 7 Lt/Dt = 8.3333 Lt/Dt = 11
260 270 280 290 300 310 320 330
0 0.2 0.4 0.6 0.8 1
Ttotal (K)
r/Rt
Lt/Dt = 2.5 Lt/Dt = 3 Lt/Dt = 5
Lt/Dt = 7 Lt/Dt = 8.3333 Lt/Dt = 11
50
Performance of the Energy Separation
Figure 30 compares the heating coefficient of performance, refrigeration coefficient of performance, and temperature differences between the cold and hot outlet temperatures when the tube length-to-diameter ratio is varied between 2.5 and 13 with the same cold mass fraction of 0.56. Equations (12) and (13) are utilized to determine the heating and cooling coefficient of performance. As predicted, the optimal heating and refrigeration coefficient of performance and temperature separation are achieved with a length-to-diameter ratio of 3. The maximum outlet difference in temperature occurs at Lt
/Dt of 3 with a value of 43.59 K.
Figure 30: Energy Separation Performance for Different Lt/Dt (a) Outlet Temperature Differences between the Hot and Cold Exit (b) Coefficient of Performance of Heating and Refrigeration
40.0 40.5 41.0 41.5 42.0 42.5 43.0 43.5 44.0
0 2 4 6 8 10 12
T=TH -TC (K)
Lt/Dt (a)
0.04 0.05 0.06 0.07 0.08 0.09 0.1
0 2 4 6 8 10 12
COP
Lt/Dt
COP_R COP_HP
(b)
51 In Figure 30, The low COP of a vortex tube is the absence of an external energy source to power the vortex tube, unlike traditional refrigeration systems. In conventional refrigeration systems, the compressor is powered by an external energy source, such as electricity or fuel, which compresses the refrigerant and facilitates a temperature difference. In contrast, in a vortex tube, the high-pressure flow that enters the tube is the sole driving force behind the energy separation process. This restriction results in the energy efficiency of the vortex tube being limited by the properties of the compressed air that enters the tube, including its pressure, temperature, and flow rate. Moreover, the design parameters of the vortex tube, such as its diameter, length, number, and size of tangential inlets, and the angle of the conical nozzle, can also significantly impact the efficiency of the vortex tube, leading to changes in its COP.