New Adapted One-Dimensional Mathematical and Regression Model to Predict Ejector Performance

It is proposed that an ejector, as a thermo-compressor, can be used to improve the thermal efficiency of refrigeration systems. The ejector consists of six parts, see Figure 1, primary nozzle, suction port, suction chamber, mixing chamber, constant mixing area, and diffuser. The design of the ejector provides the primary high-pressure flow, which entrains the low-pressure secondary flow and mixes the two streams. The resultant pressure at the outlet port is called the backpressure and, of course, will be an intermediate value between the high and low input pressures. The ejector generates a drive effect creating a so-called suction pressure due to the expansion at the primary nozzle exit, with the primary flow accelerating through the converging mixing chamber, the constant area mixing tube, and leaves from the diffuser. Figure 1 shows the main parts of the ejector.

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Figure 1. Ejector configuration

There are two main types of ejectors: Constant Area Mixing. (CAM) and Constant Pressure Mixing (CPM). CAM was proposed by Keenan and Neumann [1], their design positioned the primary nozzle discharge at the entry to the constant area section. CPM, suggested by Keenan et al. [2] located the primary nozzle discharge downstream in the suction chamber. Both proposals have been studied experimentally, with the CAM ejector providing a higher mass flow rate than the CPM ejector, but with the CPM more effective against the back pressure, which provided more stable operating conditions [3]. The advantages of CAM and CPM ejectors need to be combined in one ejector to maximize ejector entrainment. An ejector design technique is proposed by using a constant rate of momentum change (CRMC). The main concept is based on the mitigation of the shockwave by changing the area of the nozzle, thus enhancing performance (i.e., the pressure and entrainment ratios) [4].

There are three modes of ejector operating process shown in Figure 2. The ejector operation can be summarized as; the primary flow with high pressure and temperature enters the convergent-divergent nozzle to expand, and the flow exits the primary nozzle at supersonic speed, generating a low-pressure region, this will entrain the secondary flow from the second port and, by momentum exchange, shear effects, and low-pressure region suction effects, the primary flow is mixed with the secondary flow and forced towards the outlet port.

In the best conditions, the secondary flow should reach sonic speed to enhance the mixing process and the mixed flow will be compressed and its pressure increased due to the shock wave formed at the diffuser inlet. However, when the flow speed drops from supersonic to subsonic there are sudden increases in pressure and temperature. As the flow proceeds into the diffuser section, its velocity will decrease further as the cross-sectional area increases, with a consequent increment in pressure until the exit pressure equalizes the backpressure. This explanation is the ideal and preferred case of ejector operation. In the critical mode, the backpressure is equal to or lower than the critical pressure, and, therefore, the secondary flow is choked, and the velocity of its stream will reach the sonic condition with a maximum mass flow rate. Because the primary nozzle is also choked, this mode is termed the double choking mode. In the subcritical mode, the back pressure is higher than the critical pressure and the secondary flow is not choked, and the velocity of the stream will not reach the sonic condition. However, the primary nozzle will be choked, and this mode is referred to as the single choked mode. If the back pressure is higher than the cut-off point the secondary flow will be reversed, this mode is the backflow mode, no entrainment takes place, and the ejector is deemed to have malfunctioned as shown in Figure 2.

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Figure 2. Ejector operational modes

3.1 General governing equations

The study defined the ejector flow using the following equations.

The Continuity Equation:

$\frac{\partial \rho}{\partial t}+\frac{\partial\left(\rho u_i\right)}{\partial x_i}=0$                   (1)

Conservation of Momentum:

$\frac{\partial\left(\rho u_i\right)}{\partial t}+\frac{\partial}{\partial x_i} \cdot\left(\rho u_i u_j\right)=-\frac{\partial P}{\partial x_i}+\frac{\partial \tau_{i j}}{\partial x_j}$              (2)

Conservation of Energy:

$\frac{\partial(\rho E)}{\partial t}+\frac{\partial}{\partial x_i}\left(\rho u_i E+u_i P\right)=\frac{\partial P}{\partial t}+\frac{\partial}{\partial x_i}\left(k_{e f f} \frac{\partial T}{\partial x_i}\right)+\frac{\partial}{\partial x_i}\left(u_i \tau_{i j}\right)$                  (3)

where, $\tau_{i j}=\mu_{e f f}\left(\frac{\partial u_i}{\partial x_i}+\frac{\partial u_j}{\partial x_i}\right)-\frac{2}{3} \mu_{e f f} \frac{\partial u_k}{\partial x_k} \delta_{i j}$                  (4)

3.2 The proposed mathematical model assumptions

• The model works under steady-state circumstances and one-dimensional flow.
• Isentropic efficiencies are utilized to account for process losses and non-ideality.
• Section y-y is shown in Figures 3 and 4, it represents the effective area region and exists in both modes.
• At section (y-y), the primary and secondary streams begin to mix with a uniform pressure (i.e., P_py = P_sy = P_m).
• Section m-m is depicted in Figures 3 and 4, and it showed the complete mixing zone.
• The walls of the ejector are adiabatic.
• There is a stagnation condition at the inlet and outlet.
• The saturated vapor is the secondary and primary flow (no mixture or two-phase included).
• Heat capacity under constant pressure (Cp) and heat capacity ratio (γ) depend on the state.
• The isentropic efficiency of the primary nozzle, secondary flow inlet, diffuser, and the mixing chamber was considered in the study.

Figures 3 and 4 depict the theoretical pressure and velocity distribution at the ejector centerline of critical and subcritical modes. The flow description on the h-s diagram is shown in Figure 5, There are four major pressures to consider (primary, secondary, mixing, and condenser). The primary flow enters the primary nozzle at primary pressure, then expands and reduces in pressure as it passes down the nozzle's throat at point (t_s), but due to the irreversibility and the entropy increment the flow departs at point (t) while the flow continues to section (y-y), and the pressure of the flow dropped to pressure mixing (P_m), if the primary flow expands ideally it should be at point (py_s) but for the irreversibility, it deviates and becomes at point (py), on the other hand, the secondary flow enters from the secondary port with pressure (P_s) and due to the expansion, the pressure slumped to the mixing pressure (P_m) and start the mixing process with the primary flow at section (y-y). The mixed flow completes the process at section (m-m) at the pressure level (P_m). The mixed flow moves to the diffuser inlet while the pressure is augmented excessively from (P_m) level to condenser pressure (P_c), by the same concept of irreversibility the diffuser exit properties swerve from (de_s) to (de).

Double Choking Case (Critical Mode)

$\mathrm{M}_{\mathrm{t}}$ and $\mathrm{M}_{\mathrm{sy}}=1$

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Figure 3. Pressure and velocity distribution (Critical mode)

Single Choking Case (Subcritical Mode)

$\mathrm{M}_{\mathrm{t}}=1$ and $\mathrm{M}_{\mathrm{sy}}<1$

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Figure 4. Pressure and velocity distribution (sub-critical mode)

The flow description on the h-s diagram is shown in Figure 5.

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Figure 5. (h-s) diagram for ejector theoretical operational procedure

3.3 The model parameters

• Operating conditions:

$P_p, T_p, P_s, T_s, P_c, T_c$

• Isentropic efficiencies:

$\eta_p, \eta_{p y}, \eta_s, \eta_d, \psi_m$

• Ejector geometry dimensions:

$d_{p i}, d_{s c i}, d_t, d_{p n e}, d_{c a m}, d_{d i}, d_{d e}, L_{m c}, L_m, L_d$

Cross-section areas of the ejectors

$\left(A_{\mathrm{t}}, A_{\text {pne }}, A_{\mathrm{mc}}\right.$, and $\left.A_{\text {cam }}\right)=\frac{\pi}{4} \mathrm{~d}^2$            (5)

Area ratio $(\mathrm{AR})=\frac{A_{\text {cam }}}{A_t}$            (6)

From P_p & x=1, it is obtained (s_p, h_p & ρ_p)

From P_s & x=1, it is obtained (s_s, h_p & ρ_s)

All gas dynamics equations were extracted and adopted to build the model [16].

From the primary nozzle inlet to the throat

If the ejector operates in subcritical mode or critical mode, M_t=1, applying the gas dynamics law gives an expression for the relation between pressures as Eq. (7):

$\frac{P_p}{P_t}=\left[1+\frac{\gamma-1}{2} M_t^2\right]^{\frac{\gamma}{\gamma-1}}$               (7)

For the isentropic process between the primary inlet and nozzle throat, the entropy is obtained from the following relation:

$s_p=s_t$

The isentropic enthalpy at the throat is provided by the following function:

$h_{t_s}=f\left(P_t, s_t\right)$

The isentropic efficiency of the primary nozzle is calculated from Eq. (8):

$\eta_p=\frac{h_p-h_t}{h_p-h_{t_s}}$            (8)

The primary flow velocity at the throat is obtained by applying the conservation of energy, with the velocity calculated from Eq. (9):

$u_t=\sqrt{2\left(h_p-h_t\right)}$            (9)

The primary mass flowrate is calculated from Eq. (10):

$\dot{m}_p=\rho_t A_t u_t$            (10)

From primary nozzle throat to section y-y

For the isentropic expansion process between the primary inlet and section y-y, it is obtained from the following relation:

$s_p=s_{p y}$

The isentropic enthalpy for primary flow at section y-y is given by:

$h_{p y_s}=f\left(s_{p y}, P_{p y}\right)$

The isentropic efficiency of the primary flow at section y-y is calculated from Eq. (11):

$\eta_{p y}=\frac{h_t-h_{p y}}{h_t-h_{p y_s}}$            (11)

From secondary inlet to section y-y

The isentropic entropy for secondary flow at section y-y is obtained from the following relation:

$s_s=s_{s y}$

The isentropic enthalpy for secondary flow at section y-y is obtained from the following relation:

$h_{s y s}=f\left(s_{s y}, P_{s y}\right)$

The isentropic efficiency of secondary flow is calculated from Eq. (12):

$\eta_s=\frac{h_s-h_{s y s}}{h_s-h_{s y}}$             (12)

The density of the flow at the secondary flow is extracted as:

$\rho_{s y}=f\left(h_{s y}, P_{s y}\right)$

The velocity of the secondary flow at section y-y is obtained by applying the conservation of energy, with the velocity calculated from Eq. (13):

$u_{s y}=\sqrt{2\left(h_s-h_{s y}\right)}$           (13)

From the primary nozzle exit continues to section y-y

By applying the gas dynamics relation between the area ratio and Mach number we obtain Eq. (14):

$\left(\frac{A_{p n e}}{A_t}\right)^2=\left(\frac{1}{M_{p n e}^2}\right)\left[\left(\frac{2}{\gamma+1}\right)\left(1+\left(\frac{\gamma-1}{2}\right) M_{p n e}^2\right)\right]^{\frac{\gamma+1}{\gamma-1}}$             (14)

By applying the relationship between pressure ratio and Mach number, we obtain Eq. (15):

$\frac{P_p}{P_{p n e}}=\left[1+\frac{\gamma-1}{2} M_{p n e}^2\right]^{\frac{\gamma}{\gamma-1}}$               (15)

Similarly, considering the relation between pressure ratio and Mach number at the primary nozzle exit and section y-y, it is obtained in Eq. (16):

$\frac{P_{p y}}{P_{p n e}}=\frac{\left[1+\frac{\gamma-1}{2} M_{p n e}^2\right]^{\frac{\gamma}{\gamma-1}}}{\left[1+\frac{\gamma-1}{2} M_{p y}^2\right]^{\frac{\gamma}{\gamma-1}}}$              (16)

By applying the gas dynamics relation between the area ratio and Mach number at the primary nozzle exit and section y-y, it is obtained Eq. (17):

$\frac{A_{p y}}{A_{p n e}}=\frac{\frac{\eta_{p y}}{M_{p y}}\left[\left(\frac{2}{\gamma+1}\right)\left(1+\frac{\gamma-1}{2} M_{p y}^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}}{\frac{1}{M_{p n e}}\left[\left(\frac{2}{\gamma+1}\right)\left(1+\frac{\gamma-1}{2} M_{p n e}^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}}$            (17)

The calculation for the entrainment ratio

If the ejector is in critical mode, it is assumed M_sy=1, and if the ejector is in sub-critical mode, it is assumed M_sy<1.

By applying the relationship between pressure ratio and Mach number to the secondary inlet and critical pressure at section y-y, it is obtained Eq. (18):

$P_s=P_{s y_{\text {critical }}}\left[1+\left[\frac{\gamma-1}{2}\right] M_{s y}^2\right]^{\frac{\gamma}{\gamma-1}}$              (18)

where, A_sy and A_py can be found in Eq. (19) below, given A_cam:

$A_{c a m}=A_{p y}+A_{s y}$             (19)

The secondary mass flow rate is given by Eq. (20):

$\dot{m}_s=\rho_{s y} u_{s y} A_{s y}$                 (20)

The entrainment ratio is given by Eq. (21):

$\omega_{t h}=\frac{\dot{m}_s}{\dot{m}_p}$            (21)

From mixed flow at section y-y to section m-m

Applying the principles of conservation of momentum and energy to the flow in the mixing chamber at section y-y and section m-m.

The mixing efficiency is a non-dimensional parameter to measure the quality of the mixing process as shown in Eq. (22):

$\psi=\frac{u_m^2}{u_{m_{\text {ideal }}^2}}$            (22)

By applying the conservation of momentum, it is obtained Eqn. (23):

$\sqrt{\psi_m}\left(\dot{m}_p u_{p y}+\dot{m}_s u_{s y}\right)=\left(\dot{m}_s+\dot{m}_p\right) u_m$             (23)

To simplify, the secondary flow velocity is neglected at section y-y compared to that of the primary flow and divide Eq. (24) by $\dot{m}_p$:

$\sqrt{\psi_m}\left(u_{p y}\right)=(\omega+1) u_m$           (24)

The conservation of energy at constant area mixing region obtained as Eq. (25):

$\dot{m}_p\left[h_{p y}+\frac{u_{p y}^2}{2}\right]+\dot{m}_S\left[h_{s y}+\frac{u_{s y}^2}{2}\right]=\left(\dot{m}_s+\dot{m}_p\right)\left[h_m+\frac{u_m^2}{2}\right]$             (25)

Similarly, simplify Eq. (24) by neglecting the secondary flow velocity at section y-y compared to the primary flow and dividing Eq. (25) by $\dot{m}_p$:

$\left[h_{p y}+\frac{u_{p y}^2}{2}\right]+\omega\left[h_{s y}\right]=(\omega+1)\left[h_m+\frac{u_m^2}{2}\right]$           (26)

The mixing point entropy, s_m, is given as the following function:

$s_m=f\left(h_m, P_m\right)$

Mixed flow from section m-m to the diffuser inlet

Neglecting any change in enthalpy between section m-m and the diffuser inlet, the enthalpy at the inlet to the diffuser is expressed as the following:

$h_{d i}=h_m$

The isentropic efficiency of the diffuser was obtained as Eq. (27):

$\eta_d=\frac{h_{d e_s}-h_{d i}}{h_{d e}-h_{d i}}$           (27)

where, $h_{d e_s}=f\left(P_c, s_{d i}\right)$.

The local sound speed is obtained from the properties of the refrigerant as the following function:

$a_m=f\left(P_m, x=1\right)$

The Mach number at the mixing point is obtained from Eq. (28):

$M_m=\frac{u_m}{a_m}$               (28)

The Mach number at the diffuser inlet is obtained from Eq. (29):

$M_{d i}^2=\frac{1+\left[\frac{\gamma-1}{2}\right] M_m^2}{\gamma M_m^2-\left[\frac{\gamma-1}{2}\right]}$              (29)

By applying the relationship between pressure ratio and Mach number between section m-m and diffuser inlet, a length of the constant area, we obtain Eq. (30):

$\frac{P_{d i}}{P_m}=\left(1+2\left(M_m^2-1\right)\left[\frac{\gamma}{\gamma+1}\right]\right)$              (30)

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Figure 6. Flowchart of the mathematical model procedure

Diffuser inlet to (ejector outlet/condenser inlet)

Applying the relationship between pressure ratio and Mach number at the diffuser inlet and exit it is obtained Eq. (31):

$\frac{P_{d e}^*}{P_{d i}}=\left[1+\left[\frac{\gamma-1}{2}\right] M_{d i}^2\right]^{\frac{\gamma}{\gamma-1}}$            (31)

where, $P_{d e}^*$ compared to condenser pressure P_c.

The mathematical model procedure is summarized in the flow chart in Figure 6.

• The proposed and modified model to predict the ejector performance via a Mach number iterative procedure shows a good agreement with the published experimental data, with most predicted results located in a range of error of ± 10% concerning Huang et al. [6] and ± 15% for Yan et al. [15]. ANOVA test shows that the proposed model has no significant difference from the published experimental results, which increases the reliability of the model.
• Varying the Mach Number of the flow gives the procedure a more systemic and logical method to predict the ejector performance and specify the ejector mode of operation.
• A regression model implemented on the experimental results of Huang et al. [6] showed good agreement with R²=93.83%, the equation combined the working conditions parameters (i.e., expansion and compression ratios) and the geometrical parameter (i.e., area ratio).
• Pearson correlation has shown that the pressure ratio and temperature ratio are highly correlated, therefore, the regression model can be related to the pressure ratio, allowing the temperature to be omitted and so decreasing the number of independent variables in the regression equation.
• A second Pearson correlation test conducted between the independent variables (compression, expansion, and area ratios) and the dependent variable (entrainment ratio), showed a highly positive correlation for the area ratio and a negative for the compression and expansion ratio,
• The increment of the area ratio facilitates the flow of the streams with lower resistance; therefore, the entrainment ratio is enhanced. In the contrast, the compression ratio creates more resistance to the exit flow due to the increment of the condenser pressure or reduction of secondary pressure.
• The increment of expansion ratio deteriorates the entrainment ratio, and the entrained area region is reduced because of the expansion’s waves, and these occurred for both cases either the primary pressure increased or the secondary pressure decreases.
• The case study showed that the compression ratio and expansion ratio have an inverse effect on the refrigeration system’s COP while increasing the area ratio could increase COP due to the increment of the entrainment ratio.
• The COP and entrainment ratio shows a directly proportional relationship; hence they are influenced in the same way by compression, expansion, and area ratio.
• The mathematical and regression model, supported here by statistical tests, is an effective method that can give the designer a rapid estimation of the predicted likely performance without the complexity and expense of experimental tests and CFD methods.