Chapter 3: Results and Discussions
3.2 Optimization of The Vortex Tube Energy Separation
3.2.1 Effect of Tube Internal Tapering Angle
The current section investigates the connection between flow behavior and energy separation in straight, diverging, and converging vortex tubes. To this end, we examine the velocity, pressure, and temperature profile at a specified axial point of z/L = 0.45. It is worth noting that the energy separation in the vortex tube occurs solely due to the high-pressure flow that enters it, as there are no external sources of work or heat.
Additionally, the vortex tube's unique shape facilitates a sudden increase in the flow velocity while most of the flow pressure dissipates through the input nozzle. For more details regarding operation conditions, see Table 3.
3.2.1.1 Velocity Distribution Total Velocity
The total fluid velocity is the sum of the axial, tangential, and radial velocity components. The axial component is parallel to the flow axis, the tangential component is perpendicular to the flow axis, and the radial component is directed towards or away from the flow axis. Figure 13 (a) shows the radial distribution of total velocity for the convergent vortex tube, while Figure 13 (b) shows the radial distribution of the total velocity for the divergent vortex tube. These graphs demonstrate that the total velocity increases with increasing angle of convergence and decreases with increasing angle of divergence.
The study found that the maximum total velocity, 238.7 m/s, occurred at a 2o convergent vortex tube, while the minimum total velocity, 149.5 m/s, occurred at a 6o divergent vortex tube angle. The total velocity increases with an increase in the tapering angle of convergent or decreases the tapering angle of divergent, and vice versa; this is due to the effect of increasing the angle of divergent or reducing the angle of convergent on the vortex's form (which is a Rankin vortex, as reported in the literature by Hamdan et al. [46]) and angular momentum distribution. The diameter of the vortex tube increases with increasing tapering angle of the divergent or decreases the tapering angle of the convergent, it found that the vortex's shape expands as the radius increases, and the angular momentum per unit length grows, So the velocity decreases to conserve angular
31 momentum. as shown in Figures 13 (a) and (b). These results are consistent with the findings of Hamdan et al. [46].
Figure 13: Radial Profile of Total Velocity for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT
0 50 100 150 200 250 300
0 0.2 0.4 0.6 0.8 1
Vtotal (m/s)
r/Rt
0° vortex tube 0.5° convergent vortex tube
0.75° convergent vortex tube 1° convergent vortex tube 1.25° convergent vortex tube 1.5° convergent vortex tube 1.75° convergent vortex tube 2° convergent vortex tube
(a)
0 50 100 150 200 250 300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Vtotal (m/s)
r/Rt
0° vortex tube 0.5° divergent vortex tube
1° divergent vortex tube 2° divergent vortex tube 4° divergent vortex tube 6° divergent vortex tube
(b)
32
Tangential Velocity
Figures 14 (a) and (b) illustrate the radial distribution of tangential velocity at the axial position of z/L = 0.45 for various internal tapering angles of convergent and divergent vortex tubes, respectively. The vortex that forms in the tube is a Rankin vortex, which consists of three distinct regions: forced vortex, transition, and free vortex. In the forced vortex region, the fluid velocity increases linearly with the radius until it reaches a critical value, after which the relationship between the tangential velocity and the tube radius becomes nonlinear as the forced vortex transitions into the free vortex. The free vortex then dominates the flow, and the tangential velocity drops abruptly to zero at the wall due to the no-slip condition.
The study examined the impact of the internal tapering angle on the tangential velocity. The results indicated that the tangential velocity decreases as the convergence angle decreases and the divergence angle increases, and vice versa. Figures 14 (a), (b), and (c) show that the maximum tangential velocity is observed at a 2° convergent vortex tube angle with a velocity of 236.8 m/s. In contrast, the minimum tangential velocity is observed at a 6° divergent vortex tube angle with a velocity of 121.5 m/s. As the tapering angle of the divergent increases, the transition region expands, and the 6° divergent vortex has a more extensive free vortex region than the other angles. Changing the internal tapering angle affects the angular momentum distribution and vortex form, causing an increase in tube diameter as the convergent angle decreases or the divergent angle increases, resulting in an increase in angular momentum. The conservation of momentum then causes a decrease in tangential velocity.
33 Figure 14: Radial Profile of Tangential Velocity for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT (c) Maximum Tangential Velocity
0 50 100 150 200 250
0 0.2 0.4 0.6 0.8 1
Vtangential (m/s)
r/Rt
0° Vortex Tube 0.5° convergent vortex tube
0.75° convergent vortex tube 1° convergent vortex tube 1.25° convergent vortex tube 1.5° convergent vortex tube 1.75° convergent vortex tube 2° convergent vortex tube
(a)
0 50 100 150 200 250
0 0.2 0.4 0.6 0.8 1
Vtangential (m/s)
r/Rt
0° Vortex Tube 0.5° divergent vortex tube
1° divergent vortex tube 2° divergent vortex tube 4° divergent vortex tube 6° divergent vortex tube
(b)
34
Figure 14: Radial Profile of Tangential Velocity for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT (c) Maximum Tangential Velocity
(Continued) Axial Velocity
Various configurations, including convergent, straight, and divergent angles, were analyzed to determine the effect of internal tapering angle on the radial distribution of axial velocity. Figure 15 (a) illustrates the radial distribution of axial velocity for the convergent internal tapering angle. In contrast, Figure 15 (b) shows the same for the divergent vortex tube at the axial location of z/L = 0.45. The findings indicate that the velocity reaches zero at a particular radius in the main line, known as the stagnation point, after which the flow direction reverses towards the cold end. The negative axial velocity zones in Figures 15 (a) and 15 (b) correspond to the forced vortex or cold flow regime. In contrast, the positive axial velocity zones correspond to the free vortex or hot flow regime. The free vortex region exhibits a decrease in flow velocity in the r/R >
0.95. The results indicate that the straight vortex tube, with a 0° angle, shows the highest axial velocity at 71.2 m/s. The vortex intensifies as the convergence angle increases, increasing particle movement restrictions in the axial direction. In contrast, as the angle of divergence increases, the vortex expands, causing a reduction in axial velocity.
0 50 100 150 200 250
-3 -2 -1 0 1 2 3 4 5 6 7
Vtangential (m/s)
ϴo
(c)
35 The Figures show that the cold flow region increases as the convergent angle decreases and the divergent angle increases the vortex core becomes bigger, growing cold flow regions.
Figure 15: Radial Profile of Axial Velocity for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT
Radial Velocity
Radial velocity is a fluid velocity component that characterizes the fluid's velocity concerning the flow axis. This velocity component is perpendicular to the fluid flow's axial and tangential directions. To investigate the influence of the internal tapering angle on the radial velocity, Figure 16 (a) illustrates the radial distribution of radial velocity for
-80 -60 -40 -20 0 20 40 60 80
0 0.2 0.4 0.6 0.8 1
Vaxial (m/s)
r/Rt
0° vortex tube 0.5° convergent vortex tube
0.75° convergent vortex tube 1° convergent vortex tube 1.25° convergent vortex tube 1.5° convergent vortex tube 1.75° convergent vortex tube 2° convergent vortex tube
(a)
-80 -60 -40 -20 0 20 40 60 80
0 0.2 0.4 0.6 0.8 1
Vaxial (m/s)
r/Rt
0° vortex tube 0.5° divergent vortex tube
1° divergent vortex tube 2° divergent vortex tube 4° divergent vortex tube 6° divergent vortex tube
(b)
36
convergent vortex tubes. In contrast, Figure 16 (b) displays the radial distribution of radial velocity for the divergent vortex tubes at the axial location of z/L = 0.45. The depicted figures show that the radial velocity in the vortex tube is at its minimum in the core flow and increases radially until it reaches its maximum in the annular region.
However, the trend of radial velocity is complex and peculiar. Moreover, results show that the complex velocity trend in convergent angles is more than the divergent angles due to the high turbulence flow within the convergent vortex tube. The results indicate that the vortex tube's maximum radial velocity occurs at an angle of 1o convergent. The prof radial velocity trend profile is compatible with Oberti [78] trend.
Figure 16: Radial Profile of Radial Velocity for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT
-10 -8 -6 -4 -2 0 2 4 6 8 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Vradial (m/s)
r/Rt
0° vortex tube 0.5° convergent vortex tube
0.75° convergent vortex tube 1° convergent vortex tube 1.25° convergent vortex tube 1.5° convergent vortex tube 1.75° convergent vortex tube 2° convergent vortex tube
(a)
-10 -8 -6 -4 -2 0 2 4 6 8 10
0 0.2 0.4 0.6 0.8 1
Vradial (m/s)
r/Rt
0° vortex tube 0.5° divergent vortex tube
1° divergent vortex tube 2° divergent vortex tube 4° divergent vortex tube 6° divergent vortex tube (b)
37 3.2.1.2 Pressure Distribution
Static Pressure
Figure 17 (a) illustrates the radial distribution of static pressure for a convergent vortex tube at an axial location of z/L = 0.45. In contrast, Figure 17 (b) shows the radial distribution of static pressure for a divergent vortex tube at the exact axial location.
These figures demonstrate that static pressure increases radially, with maximum static pressure occurring at the tube's wall and minimum static pressure occurring at its center.
Moreover, the internal tapering angle influences the static pressure values. As depicted in Figure 17 (a), increasing the convergence angle increases static pressure. Conversely, in Figure 17 (b), increasing the divergence angle causes a decrease in static pressure.
Therefore, the maximum static pressure value in the annular region corresponds to a 2°
convergent angle. However, for a 1.75° convergent vortex tube, the annular and central areas exhibit the maximum difference in static pressure. Increasing the tapering angle of the convergent vortex intense the fluid particles, and the centrifugal forces increase, increasing static pressure. On the other hand, increasing the internal tapering angle of the divergent vortex expands due to a decrease in centrifugal force, causing the static pressure to decrease.
Figure 17: Radial Profile of Static Pressure for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT
100000 120000 140000 160000 180000 200000 220000 240000 260000
0 0.2 0.4 0.6 0.8 1
Pstatic (pa)
r/Rt
0° vortex tube 0.5° convergent vortex tube
0.75° convergent vortex tube 1° convergent vortex tube 1.25° convergent vortex tube 1.5° convergent vortex tube 1.75° convergent vortex tube 2° convergent vortex tube
(a)
38
Figure 17: Radial Profile of Static Pressure for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT (Continued)
Total Pressure
The total pressure of a fluid in motion is the sum of its static and dynamic pressures. At an axial location of z/L = 0.45, Figure 18 displays the radial profile of total pressure for different internal tapering angles. Figure 18 (a) shows the effect of varying convergent tapering angles on the total pressure, while Figure 18 (b) represents the effect of divergent tapering angles on the total pressure. The results indicate that the total pressure profile increases radially in the tube, where the maximum total pressure is near the wall, and the minimum is in the central region.
The static pressure and flow velocity significantly influence the value of total pressure. The results indicate that an increase in the angle of a convergent tube increases total pressure. In contrast, an increase in the divergent angle causes a decrease in total pressure, similar to a static pressure trend. Figures also demonstrate that the highest total pressure corresponds to a 2° convergent angle, with the most significant difference between the annular and central regions occurring with a 1.75° convergent vortex tube.
Furthermore, the figure illustrates that the static and total pressure at the wall is equal, as the dynamic pressure is zero at the wall.
100000 120000 140000 160000 180000 200000 220000 240000 260000
0 0.2 0.4 0.6 0.8 1
Pstatic (pa)
r/Rt
0° vortex tube 0.5° divergent vortex tube
1° divergent vortex tube 2° divergent vortex tube 4° divergent vortex tube 6° divergent vortex tube
(b)
39 Figure 18: Radial Profile of Total Pressure for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT
3.2.1.3 Thermal Distribution Static Temperature
Figure 19 depicts the radial profile of static temperature at an axial position of z/L 0.45 for various internal tapering angles, including convergent, straight, and divergent.
The results demonstrate that the static temperature remains almost constant radially within the forced vortex, due to the heat conduction between the fluid particles, region
100000 120000 140000 160000 180000 200000 220000 240000 260000
0 0.2 0.4 0.6 0.8 1
Ptotal (pa)
r/Rt
0° vortex tube 0.5° convergent vortex tube
0.75° convergent vortex tube 1° convergent vortex tube 1.25° convergent vortex tube 1.5° convergent vortex tube 1.75° convergent vortex tube 2° convergent vortex tube
(a)
100000 120000 140000 160000 180000 200000 220000 240000 260000
0 0.2 0.4 0.6 0.8 1
Ptotal (pa)
r/Rt
0° vortex tube 0.5° divergent vortex tube
1° divergent vortex tube 2° divergent vortex tube 4° divergent vortex tube 6° divergent vortex tube
(b)
40
until reaching the transition region, where it starts to drop to a particular value and then rises sharply to be significantly higher in the annular region. In addition, the radial profile of static temperature can be utilized to classify various regions of a vortex tube as forced, transition, and free vortex.
According to Alsaghir et al. [66], the static temperature profile suddenly drops in the transition region due to the flow's attempt to compensate for the decreasing density due to the pressure not being sufficient, so the temperature falls, followed by a sudden temperature increase, indicating the presence of a free vortex. As the fluid moves radially towards the periphery, the temperature rises due to the conversion of kinetic energy to thermal energy. As shown in Figure 19, the results indicate that the tapering angle of the vortex tube significantly affects the static temperature. With an increase in the convergent tapering angle, the temperature separation increases However, when the angle reaches 2° convergent vortex tube, the relationship between temperature separation and the convergent tapering angle is destroyed, making an angle of 1.75° optimal for temperature differences. Conversely, increasing the divergent angle slightly improves the separation until a specific angle, beyond which the temperature separation deteriorates, where the worst temperature separation occurs at 6° divergent.
Figure 19: Radial Profile of Static Temperature for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT
280 285 290 295 300 305 310 315 320 325
0 0.2 0.4 0.6 0.8 1
Tstatic (K)
r/Rt
0° vortex tube 0.5° convergent vortex tube
1° convergent vortex tube 0.75° convergent vortex tube 1.25° convergent vortex tube 1.5° convergent vortex tube 1.75° convergent vortex tube 2° convergent vortex tube
(a)
41 Figure 19: Radial Profile of Static Temperature for Different Internal Tapering Angles (a) Convergent RHVT (b) Divergent RHVT (Continued)
Total Temperature
Figure 20 represents the impact of internal tapering angles, including divergent, straight, and convergent angles, on thermal separation performance at an axial location of z/L = 0.45. The total temperature at the tube's periphery was higher than at its center, indicating the occurrence of thermal separation among the vortex tube particles.
Furthermore, the results demonstrate a correlation between the internal tapering angle and the total temperature, with an increase in the divergent angle leading to the destruction of temperature separation. Conversely, increasing the convergent angle enhances the temperature separation between the annular and core regions until the angle of 1.75o of convergence, after which the temperature separation decreases. Consequently, a 1.75o convergent vortex tube provides the greatest temperature separation.
280 285 290 295 300 305 310 315 320 325
0 0.2 0.4 0.6 0.8 1
Tstatic (K)
r/Rt
0° vortex tube 0.5° divergent vortex tube
1° divergent vortex tube 2° divergent vortex tube 4° divergent vortex tube 6° divergent vortex tube
(b)
42
Figure 20: Radial Profile of Total Temperature for Different Internal Tapering Angles (a) Convergent (b) Divergent RHVT
Performance of the Energy Separation
Figure 21 compares the heating coefficient of performance, the refrigeration coefficient of performance, and the temperature differences between the cold and hot outlet temperatures for different internal tapering angles of the tube, all at the same cold mass fraction (α) of 0.317. Simes-Moreira [79] provided the heating and refrigeration performance coefficient of the vortex tube. As expected, the optimal heating and refrigeration coefficient of performance and temperature separation were achieved with
280 285 290 295 300 305 310 315 320 325
0 0.2 0.4 0.6 0.8 1
Ttotal (K)
r/Rt
0° vortex tube 0.5° convergent vortex tube
1° convergent vortex tube 1.5° convergent vortex tube 2° divergent vortex tube 0.75° convergent vortex tube 1.25° convergent vortex tube 1.75° convergent vortex tube
(a)
280 285 290 295 300 305 310 315 320 325
0 0.2 0.4 0.6 0.8 1
Ttotal (K)
r/Rt
0° vortex tube 0.5° divergent vortex tube
1° divergent vortex tube 2° divergent vortex tube 4° divergent vortex tube 6° divergent vortex tube
(b)
43 an internal tapering angle of 1.75 degrees for the convergent vortex tube, resulting in an optimal temperature separation of approximately 38.17 K.
Coefficient of Performance of Heating (COPHP)
COPHP= (1 − α) k
k−1 (TH
Tin−1) ln(Pin
po) (12) Coefficient of Performance of Refrigeration (COPR)
COPR= α k
k−1 (1−Tc
Tin) ln(Pin
po) (13) Where K = 1.4 is the ideal gas heat capacity ratio.
Figure 21: Energy Separation Performance for Different Internal Tapering Angle (a) Coefficient of Performance of Heating and Refrigeration (b) Temperature Differences between the Hot and Cold Outlets for Different Internal Tapering Angles
0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065
-2 0 2 4 6
COP
ϴo
COP_R COP_HP
(a)
0 5 10 15 20 25 30 35 40 45
-2 0 2 4 6
T=TH -TC (K)
ϴo
(b)
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.