Numerical Investigation of Modifying the Design Parameters in Vortex Tube Separators for Enhanced Cooling Performance

Thermodynamic analysis of the separator: Figure 2 shows the different areas within the vortex separation tube.

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Figure 2. Conservation lows on vortex tube

Mass conservation equation: The mass entering the device per unit time equals the mass leaving it. This relationship is expressed mathematically during steady state as [9]:

$\dot{m}_{i n}=\dot{m}_c+\dot{m}_h$          (1)

where, $\dot{m}_{i n}$  is the mass flow entering the vortex chamber, $\dot{m}_c$ the flow leaving the cold outlet, and $\dot{m}_h$ the flow leaving the hot outlet [9].

The ratio $\xi=\frac{\dot{m}_c}{\dot{m}_{i n}}$  represents what is called the cold fraction, which represents the ratio of the time rate of the mass leaving the cold outlet concerning the time rate of the total mass entering the vortex chamber [10], and thus the equation becomes:

$\dot{m}_{i n}=\underbrace{(1-\zeta) \dot{m}_{i n}}_{\dot{m}_h}+\underbrace{\xi \dot{m}_{i n}}_{\dot{m}_c}$          (2)

Energy conservation equation: The total energy conservation equation for the vortex separator system, which does not perform any work and does not exchange heat with the surrounding environment in a steady state, can be expressed through the following relationship.

The term $h_o=\frac{1}{2} V^2+h$  represents the stagnation enthalpy or total enthalpy of the flow at the studied location. In contrast, h represents the static enthalpy of flow at the location or point studied. the subscribe (o) refers to the stagnation or total properties [11, 12].

Substituting $\xi=\frac{\dot{m}_c}{\dot{m}_{i n}}$  into the previous equation, we have:

$h_{o, i n}=\xi h \quad o, h_{o, c}$          (3)

But we have $h_o=c_p T_o$ , where $T_o$  represents the stagnation temperature or the total temperature, so we have:

$T_{o, i n}=\xi T_{o, c}+(1-\xi) T_{o, h}$           (4)

Hot temperature difference: It indicates the temperature difference between the hot outlet and the device inlet, represented by the relationship [13]:

$\Delta T_h=T_h-T_{i n}$           (5)

Cold temperature difference: It represents the static temperature difference between the device inlet and the cold outlet and is given by the relationship [13]:

$\Delta T_c=T_{i n}-T_c$          (6)

Vortex separator efficiency: The vortex separator works as a cooling machine and heat pump, which must be considered when studying the vortex separator. The performance factor of the vortex separator as a cooling machine is defined as the thermal cooling capacity $\dot{Q}_c$ with respect to the power expended on its operation and is given by the relationship [14]:

$\operatorname{COP}_c=\frac{\dot{Q}_c}{P}$          (7)

Refrigeration capacity $\dot{Q}_c$ is defined as the rate of heat withdrawn from the cooled gas from the inlet temperature to the cold outlet temperature and is given by the relationship [14]:

$\dot{Q}_c=\dot{m}_c c_p\left(T_{i n}-T_c\right)=\xi \dot{m}_{i n} c_p \Delta T_c$          (8)

P represents the power supplied to operate the compressor for the traditional refrigeration cycle. Still, there is no compressor in refrigeration systems using the vortex separator. The compressed air is supplied from a tank of air or compressed gas, so the power supplied to the separator is calculated as if it were a compressor compressing the gas by an isothermal process from the cold outlet pressure to the inlet temperature and pressure. Then the power supplied to the gas is [15]:

$P=\dot{m}_{i n} T_{i n} R \ln \left(\frac{P_{i n}}{P_c}\right)$          (9)

$\begin{gathered}C O P_c=\frac{\dot{Q}_c}{P}=\frac{\xi \dot{m}_{i n} c_p\left(T_{i n}-T_c\right)}{\dot{m}_{i n} R T_{i n} \ln \left(\frac{P_{i n}}{P_c}\right)}=\frac{\xi\left(T_{i n}-T_c\right)}{\alpha T_{i n} \ln \left(\frac{P_{i n}}{P_c}\right)} \\ =\frac{\xi \Delta T_c}{\alpha T_{i n} \ln \left(\frac{P_{i n}}{P_c}\right)}\end{gathered}$          (10)

When the machine operates as a heat pump, its performance is measured by the thermal power gained by the gas from the moment it enters until it exits through the hot outlet. This power is compared to the energy consumed during the compression process, which has been calculated previously [16].

$C O P_{h p}=\frac{\dot{Q}_{h p}}{P}$          (11)

Whereas $\dot{Q}_{h p}$  is the heat capacity acquired by the gas and given by:

$\dot{Q}_{h p}=\dot{m}_h c_p\left(T_h-T_{i n}\right)=(1-\xi) \dot{m}_{i n} c_p \Delta T_h$         (12)

Therefore, the coefficient of performance for the device as a heat pump is calculated according to the relationship [16]:

$\begin{array}{r}C O P_{h p}=\frac{\dot{Q}_{h p}}{P}=\frac{(1-\xi) \dot{m}_{i n} c_p\left(T_h-T_{i n}\right)}{\dot{m}_{i n} T_{i n} R \ln \left(\frac{P_{i n}}{P_c}\right)} \\ =\frac{(1-\xi)\left(T_h-T_{i n}\right)}{\alpha T_{i n} \ln \left(\frac{P_{i n}}{P_c}\right)} \\ =\frac{(1-\xi) \Delta T_h}{\alpha T_{i n} \ln \left(\frac{P_{i n}}{P_c}\right)}\end{array}$          (13)

$P_c$  was chosen because it is the smallest pressure from which it is compressed [17].

Physical model: The physical model consists of differential equations that govern flow, focusing on the analysis of the flow field within the separator and the mechanisms of energy transfer involved [18].

Motion and energy equations: The movement of fluids is governed by four differential equations: The continuity equation and the Navier-Stokes equation.

Continuity equation: The continuity equation expresses the principle of conservation of mass and states that the time rate of change of the mass of the control volume over time interval is equal to the difference between the rate of mass entering and exiting it. The relationship mathematically formulates it:

$\frac{\partial \rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0$         (14)

where, $\rho$  represents density. u, v, w  and w components of the velocity in the x, y, and z directions, respectively. Or in its compact form:

$\frac{\partial \rho}{\partial t}+\vec{\nabla} \cdot(\rho \vec{V})=0$          (15)

The first term represents the time rate, while the second term represents the convective term [19, 20].

Momentum equations: Newton's second law states that when a fluid's momentum changes, a force is acting on it. These equations are given as follows:

Momentum equation on the x-axis:

$\begin{aligned} \rho\left(\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\right. & \left.+w \frac{\partial u}{\partial z}\right) \\ & =-\frac{\partial p}{\partial x}+\rho g_x \\ & +\mu\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}\right)\end{aligned}$          (16)

Momentum equation on the y axis:

$\begin{aligned} \rho\left(\frac{\partial v}{\partial t}+u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}\right. & \left.+w \frac{\partial v}{\partial z}\right) \\ & =-\frac{\partial p}{\partial y}+\rho g_y \\ & +\mu\left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}+\frac{\partial^2 v}{\partial z^2}\right)\end{aligned}$           (17)

Momentum equation on the z-axis:

$\begin{aligned} \rho\left(\frac{\partial w}{\partial t}+u \frac{\partial w}{\partial x}+v\right. & \left.v \frac{\partial w}{\partial y}+w \frac{\partial w}{\partial z}\right) \\ & =-\frac{\partial p}{\partial z}+\rho g_z \\ & +\mu\left(\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}+\frac{\partial^2 w}{\partial z^2}\right)\end{aligned}$          (18)

where, P represents pressure, g represents the acceleration of gravity, and $\mu$  represents viscosity, and is given according to the viscous heating model according to the Sutherland relationship [21]. This law results from the kinetic theory of gases using the forces inherent between molecules and relates viscosity changes to temperature changes and is given for two parameters by:

$\mu=\frac{c_1 T^{3 / 2}}{T+c_2}$           (19)

where, v represents the kinematic viscosity (Pa. s). T is the static temperature K. $c_1, c_2$  represents the coefficients of this law. For air at moderate pressure and temperature, the values of these coefficients are $c_1=1.458 \times 10^{-6} \frac{\mathrm{Kg}}{\mathrm{m} . \mathrm{s} . \mathrm{k}^{1 / 2}}$  and $c_2=110.4 K$  [22].

Energy conservation equation:

$\underbrace{\frac{\partial(\rho e)}{\partial t}}_1+\underbrace{\vec{\nabla}\left(\rho \vec{V} h_o\right)}_2=\underbrace{\vec{\nabla}(\vec{K} \vec{\nabla} T)}_3+\underbrace{\vec{\nabla}(\tau \vec{V})}_4+\underbrace{\vec{F}_b \cdot \vec{V}}_5+\underbrace{\dot{Q}}_6$         (20)

$e$ expresses the internal energy and is given by:

$e=h-P / \rho+1 / 2 V^2$           (21)

h is system enthalpy. $h_o$  is total enthalpy or stagnation enthalpy. $\tau$ is the shear stress resulting from viscous forces and is given by the relationship [21].

$\tau=\mu\left[\vec{\nabla} \vec{V}+(\vec{\nabla} \vec{V})^T-\frac{2}{3}(\vec{\nabla} \vec{V}) \delta\right]$           (22)

The first term represents the rate of change of the mass's energy over time. The second term accounts for the energy transported to and from the system by the movement of the fluid flow. The third term indicates the energy transferred into the system through conduction. The fourth term represents the energy dissipated within the bulk due to viscous forces. The fifth term reflects the power exchanged between the system and external volumetric forces acting upon it. Lastly, the sixth term describes the heat generated within the system due to an internal heat source [21].

Ideal gas law: The ideal gas law relating the state variables to the value of density changes as a function of absolute pressure and absolute temperature and given by $\rho=P_{a b} / R T$ [22].

Turbulence equations: Turbulence is defined as a random and chaotic fluctuation movement of molecules, where the speed and pressure components at any point in the flow suffer continuous changes over time. The cause of this turbulence is attributed to a source of change in the face of the flow that leads to the formation of a turbulent signal transmitted throughout the entire flow. The flow is classified into: Turbulent and laminar according to the Reynolds number criterion ($R e=\rho V L / \mu$), where L is a characteristic length and V is a characteristic velocity. In most cases, the flow can be classified as turbulent at a high Reynolds number, and at relatively low Reynolds numbers, the flow is classified as laminar [21].

The turbulence equations contain six additional unknowns and therefore need six additional equations or need turbulence models that reduce the number of equations.

Reynolds Stress Model (RSM): This model is based on solving seven equations which give the value of Reynolds stresses, and it is considered one of the types that simulate flows with complex strain fields or with clear physical forces, these equations in tensor form are [21]:

$\frac{D R_{i j}}{D t}=\frac{\partial R_{i j}}{\partial t}+C_{i j}=P_{i j}+D_{i j}-\varepsilon_{i j}+\Pi_{i j}+\Omega_{i j}$          (23)

where, $R_{i j}=\overline{u_i^{\prime} u_j^{\prime}}$ are Reynolds stresses. $C_{i j}=\frac{\partial}{\partial x_k}\left(\rho \bar{u}_k \overline{u_i^{\prime} u_j^{\prime}}\right)$  are R transmission by convection. $P_{i j}=-\left(R_{i m} \frac{\partial \bar{u}_j}{\partial x_m}+R_{j m} \frac{\partial \bar{u}_i}{\partial x_m}\right)$  are the rate of production of R. $D_{i j}=\frac{\partial}{\partial x_m}\left(\frac{\nu_t}{\sigma_k} \frac{\partial R_{i j}}{\partial x_m}\right)$  Transmission of R by diffusion. $\Pi_{i j}=-C_1 \frac{\varepsilon}{k}\left(R_{i j}-\frac{2}{3} k \delta_{i j}\right)-C_2\left(P_{i j}-\frac{2}{3} P \delta_{i j}\right)$  rate of transmission of R due to the mutual effect of stress and strain. $\Omega_{i j}=-2 \omega_k\left(\overline{u_j^{\prime} u_m^{\prime}} e_{i k m}+\overline{u_i^{\prime} u_m^{\prime}} e_{j k m}\right)$  rate of transmission R due to rotation. Here, K is the turbulence energy calculated from the relationship:

$k=\frac{1}{2}\left(\overline{u_1^{\prime 2}}+\overline{u_2^{\prime 2}}+\overline{u_3^{\prime 2}}\right)=\frac{1}{2}\left(R_{11}+R_{22}+R_{33}\right)$          (24)

$\varepsilon$  is energy dissipation and the constant $C_1=1.8 \& C_2=0.6$.

Grid independence tests were conducted to ensure the accuracy of the numerical results. The computational domain was discretized using a pure hexahedral structural mesh generated in Ansys Meshing software. The mesh was refined iteratively to balance computational efficiency and result accuracy.

Initially, three different grid sizes were used: a coarse grid, a medium grid, and a fine grid. The solution converged when the residuals for all flow equations fell below a specified threshold. The temperatures at the cold and hot outlets were chosen as the key performance indicators for the grid independence analysis.

The results showed that the difference in temperature values between the medium and fine grids was minimal (less than 1%), indicating that the medium grid was sufficiently refined for accurate results. Further refinement did not significantly affect the performance, confirming the solution's grid independence.

The final mesh used in the simulations had an element number of 701,565 and a node number of 760,760. Its orthogonal quality was 0.88, and its average aspect ratio was 18.22, considered optimal for ensuring both computational efficiency and solution accuracy.

6.1 Model A truncated cone valve

Numerical results indicate that the temperatures of the hot outlet increase with the increase in the value of the cold fraction and with the rise in the value of the entry pressure into the device, while the temperatures of the cold part decrease with the increase in the entry pressure. It is noted that there is a minimum value for the temperature of the cold outlet at a specific value of the cold fraction, after which the temperatures begin to rise, as they appear in Figure 8.

Figure 8 compares the experimental values obtained from the research paper [6] with those obtained by numerical modeling using the RSM turbulence model. The results show agreement between the numerical modeling and experimental results, confirming the CFD results' validity.

Temperature change charts with cold fractions can be interpreted as follows: The temperature changes between the hot and cold outlets are influenced by a series of processes, the most significant of which is direct adiabatic expansion between the inlet and the cold outlet. As theoretical analysis indicates, this process leads to a decrease in temperature, particularly near the cold outlet.

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Figure 8. Temperatures at the cold (right) and hot (left) outlets numerically and experimentally in Model A

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Figure 9. Constant entropy contour

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Figure 10. Coefficient of performance as cooling (left) and heating (right) machine in Model A

Furthermore, the drop in temperature observed in the center area results from heat exchange between the hot central region and the cooler peripheral area near the walls. Additionally, viscous heating, caused by friction, contributes to an increase in temperature, mainly due to irreversible processes within the device. As a result, the hot region continues to heat up due to the predominance of temperature increases linked to viscous heating. Figure 9 shows the numerical modeling results of the magnitude of constant $T / P^{\frac{k-1}{k}}$  contour, representing the adiabatic process along this line, which dominates near the cold outlet.

Figure 10 illustrates the device's performance coefficient when functioning as a refrigeration machine and a heat pump. From the figure, we can observe that there are optimal values for the cold fraction at which the performance coefficient reaches its maximum. Additionally, as the pressure of the gas entering the separator increases, the cold fraction required to achieve this optimal performance decreases.

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Figure 11. Cold (right) and hot (left) outlet temperatures in the two models

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Figure 12. Coefficient of performance as cooling (left) and heating (right) machines in the two models

6.2 Model B full cone valve

In this part, we will compare the results of Model B to the results of the first Model A to compare the two models studied.

The numerical results obtained indicate that the behavior of the performance curves of the tubular separator with a complete cone valve is similar to the behavior of the performance curves of the tubular separator with a truncated cone valve in terms of improved performance with increasing inlet pressure and terms of the existence of optimal values for these curves at a specific cold fraction. However, a comparison of the performance between the separator in Model A and the separator in Model B indicates that the performance of the separator in Model B is better than the performance of the separator in Model A, as Figure 11 shows the hot and cold outlet temperature charts in Model B compared to these curves in Model A. We note that the hot outlet temperatures became higher in Model B compared to Model A, while the cold outlet temperature range became lower in Model B compared to Model A.

We observe that in Model B, the lowest temperature at the cold outlet occurs at a lower cold fraction compared to Model A. This phenomenon can be attributed to the complete cone design, which increases the pressure at hot and cold outlets. This increase in pressure reduces the speed of the axial flow of the return stream within the vortex chamber, thereby enhancing the mixing effectiveness. As a result, the optimum conditions begin to manifest at a lower cold fraction. Consequently, this leads to higher performance coefficients for the gas when using the full cone relative to the truncated cone, as illustrated in Figure 12.

We note that the optimal values occur at the highest inlet pressure value of 6.2 Bar. The performance coefficient of the device as a cooling machine increases in the second model by 57%. When the cold fraction in the second model is 18% higher than in the first model, the performance coefficient also increases as a heat pump in the second model by 122%, i.e., almost double what is in the first model at a cold fracture higher by 4%.