Supersonic Nozzle Location in Steam Ejector Effect on the Mass Fraction and Vacuum of Second Fluid

To achieve the research objectives of knowing the location of the nozzle inside the steam ejector on the necessary injection pressure values at the identical dimensions of the steam ejector. The first stage will include designing the steam ejector, using one-dimensional gas dynamics equations to calculate all diameters of the steam ejector [2]. The second stage includes the flow inside the steam ejector simulation based on the dimensions obtained from the first stage, which is carried through the CFD system using the ANSYS program [12].

2.1 One-dimensional gas dynamics calculation

For normal and stable operation of the steam jet see Figure 1, it must be designed under critical conditions, the design of the continuous pressure ejector was based on several assumptions, which are summarized as follows [13]:

The ejector flow is one-dimensional and at steady state conditions.

a)   The flow is adiabatic, so that there is no heat transfer between the fluids inside ejector and the surrounding.

b)   Steam follows ideal gas behavior.

c)   Friction losses in the nozzle, diffuser, and mixing chamber are taken into account within the efficiency of each part.

d)   The velocities of the motive steam and the entrained vapor are negligible because it comes from relatively large tanks.

e)   Constant isentropic expansion exponent and the ideal gas behavior.

f)   The motive steam and the entrained vapor have the same physical properties.

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Figure 1. Steam ejector scheme

The steam ejector is designed according to the following steps see Figure 2.

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Figure 2. Scheme of the algorithm for the calculation process of steam ejector

The following is an explanation of the steam ejector calculation equations according to the above model:

  1) Design parameters, and calculation flow rates of motive steam and entrained vapor:

• Material balance:

$m_p+m_e=m_c$           (1)

where,

m_p: Primary stream mass flow rate $\left(\frac{\mathrm{kg}}{\mathrm{s}}\right)$.

m_e: Entrained vapor mass flow rate (kg/s).

m_c: Compressed vapor mixture mass flow rate (kg/s).

• Entrainment ratio:

$w=\frac{m_e}{m_p}$          (2)

  2) Calculation the Mach numbers:

The Mach number of the primary fluid at the nozzle outlet level. This will be calculated by assuming the initial value of the pressure at the Converging–Diverging Nozzle outlet.

$M_{P 2}=\sqrt{\frac{2 \eta_n}{(\gamma-1)}\left[\left(\frac{P_p}{P_2}\right)^{\frac{(\gamma-1)}{\gamma}}-1\right]}$             (3)

$M_{P 2}{ }^*=\sqrt{\frac{M_{P 2}{ }^2(\gamma+1)}{M_{P 2}{ }^2(\gamma-1)+2}}$                (4)

where,

γ: is the isentropic expansion coefficient.

η_n: is the nozzle efficiency.

M^*: is the ratio between the local fluid velocity to the velocity of sound at critical conditions.

The same applies to the secondary fluid:

$M_{e 2}=\sqrt{\frac{2}{(\gamma-1)}\left[\left(\frac{P_e}{P_2}\right)^{\frac{(\gamma-1)}{\gamma}}-1\right]}$           (5)

$M_{e 2}{ }^*=\sqrt{\frac{M_{e 2}{ }^2(\gamma+1)}{{M_{e 2}}^2(\gamma-1)+2}}$                      (6)

Using the one-dimensional continuity equation and momentum-energy equations for the compressible flow. The critical Mach number of the mixture at point (5) is calculated using:

• The critical Mach number for the primary and Mach fluids at point (2).

• The Mach number at point (4) (at the site of the shock).

$M_4{ }^*=\frac{M_{P 2}{ }^*+w M_{e 2}{ }^* \sqrt{T_e / T_p}}{\sqrt{(1+w)\left(1+w T_e / T_p\right)}}$               (7)

$M_4=\sqrt{\frac{2 M_4{ }^{* 2}}{(\gamma+1)-M_4{ }^{* 2}(\gamma-1)}}$                    (8)

$M_5^2=\frac{M_4^2+\frac{2}{(\gamma-1)}}{\frac{2}{(\gamma-1)} M_4^2-1}$                 (9)

  3) Check the pressure value at point (2):

• Calculate the pressure value before the shock wave at point (4):

$P_5 / P_4=\frac{1+\gamma M_4^2}{1+\gamma M_5^2}$        (10)

•   Assuming that the pressure remains constant at points (2, 3 and 4). So, the pressure at the exit of the steam ejector (at point c) is:

$P_c / P_5=\left[\frac{\eta_d(\gamma-1)}{2} M_5^2+1\right]^{\frac{\gamma}{\gamma-1}}$           (11)

where,

η_d: is the diffuser efficiency.

Through the results obtained, it is checked whether the pressure value at point c matches what was obtained. The calculations are repeated for the pressure values according to equations from (3) to (11) until a match in the values is reached at point c.

4) Calculate the cross-sectional areas of the steam ejector.

•   Calculate the cross-sectional area at the throat of Converging–Diverging Nozzle:

$A_1=\frac{m_p}{P_p} \sqrt{\frac{R T_p}{\gamma \eta_n}\left(\frac{\gamma+1}{2}\right)^{\frac{(\gamma+1)}{(\gamma-1)}}}$         (12)

•   Calculate the cross-sectional area at the exit of the Converging–Diverging Nozzle:

$A_2 / A_1=\frac{1}{M_{P 2}}\left(\frac{1+\frac{(\gamma-1) M_{P 2}{ }^2}{2}}{\frac{(\gamma+1)}{2}}\right)^{\frac{(\gamma+1)}{2(\gamma-1)}}$                    (13)

•   The area ratio of the nozzle throat (point 1) and diffuser constant area (point 3):

$\begin{aligned} & A_1 / A_3=\frac{P_c}{P_p}\left(\frac{1}{(1+w)\left(1+w\left(\frac{P_e}{P_p}\right)\right)}\right)^{1 / 2} \frac{\left(\frac{P_2}{P_c}\right)^{1 / \gamma}\left(1-\left(\frac{P_2}{P_c}\right)^{\frac{(\gamma-1)}{\gamma}}\right)^{1 / 2}}{\left(\frac{2}{\gamma+1}\right)^{1 /(\gamma-1)}\left(1-\frac{2}{\gamma+1}\right)^{1 / 2}}\end{aligned}$                    (14)

Through the calculations that were made using the above equations, all the cross-sectional areas of each part of the steam ejector were obtained. As for the lengths of the parts. The correctness inclination angle of the diffuser is less than 10 degrees was adopted for the publisher in each of the primary Converging-Diverging Nozzle and the region Mixing As for the suffocation area [5, 6, 14], certain lengths were adopted commensurate with the mixing stability (for the mixing area) [6], and accordingly all the dimensions of the steam ejector were obtained and were as in Table 1.

The current model's primary flow is superheated steam at total pressure (7.5 bar), temperature (175℃) and mass flow rate of superheated steam (0.02 kg/s). The steam condition in the suction area at pressure (0.65 bar) and temperature (84℃). The amount of steam entering was about (0.01). In the discharge condition, the vapour is superheated steam at pressure (0.15 bar) and temperature (96℃).

The isentropic expansion coefficient for steam flowing through the ejector approximately is (γ=1.3) [15], as well as the nozzle efficiency (η_n=0.9) and the diffuser efficiency (η_d=0.8) [13].

Table 1. Geometrical parameters of the steam ejector

2.2 CFD modeling

The flow pattern within a steady-state steam ejector and a compressible fluid can be represented by the continuity equations with Navier-Stokes in addition to Reynolds averaging and the energy equation and as follows [12]:

a) Continuity equation:

$\frac{\partial}{\partial x_i}\left(\rho u_i\right)=0$              (15)

b) Navier-Stokes equation:

$\begin{aligned} \frac{\partial}{\partial x_i}\left(\rho u_i u_i\right)=- & \frac{\partial p}{\partial x_i}+\frac{\partial}{\partial x_j}\left[\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right.\right.\left.\left.-\frac{2}{3} \delta_{i j} \frac{\partial u_k}{\partial x_k}\right)\right]+\frac{\partial}{\partial x_j}\left(-\rho \overline{u_i^{\prime} u_j^{\prime}}\right)\end{aligned}$                      (16)

c) Energy equation:

$\frac{\partial}{\partial x_i}\left(\rho c_p u_i T\right)=\frac{\partial}{\partial x_i}\left(\lambda \frac{\partial T}{\partial x_i}-\rho c_p \overline{u_l^{\prime} T^{\prime}}\right)+\mu \phi$                   (17)

where,

$\rho \overline{u_l^{\prime} u_l^{\prime}}$ : Reynolds stresses.

$\rho c_p \overline{u_i^{\prime} T^{\prime}}$ : Turbulent heat fluxe.

λ: The heat conductivity.

μϕ: The viscous dissipation.

In this simulation, the realizable (k-ε) turbulence model is used to represent the turbulence features, based on previous literature, Hanafi used a turbulence model to represent the flow inside the steam ejector, which demonstrated better agreement with experimental data and more accuracy in forecasting the spreading rate of both planar and round jets. In their compact Cartesian form, the transport equations for (k-ε) are as follows [2].

$\begin{aligned} & \frac{\partial}{\partial t}(\rho K)+\frac{\partial}{\partial x_i}\left(\rho K u_i\right) =\frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_k}\right) \frac{\partial k}{\partial x_j}\right]+G_k-\rho \varepsilon-Y_M\end{aligned}$           (18)

$\begin{aligned} \frac{\partial(\rho \varepsilon)}{\partial t}+\frac{\partial\left(\rho \varepsilon u_i\right)}{\partial x_i}= & \frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_{\varepsilon}}\right) \frac{\partial \varepsilon}{\partial x_j}\right]+\rho C_1 S_{\varepsilon}-\rho C_2 \frac{\varepsilon^2}{K+\sqrt{\nu \varepsilon}}+C_{1 \varepsilon} \frac{\varepsilon}{K} C_{e \varepsilon} C G_b\end{aligned}$                           (19)

where,

$C_1=\max \left[0.43 \frac{\eta}{\eta+5}\right]$                  (20)

$\eta=S K \varepsilon$                      (21)

$\mu_t=\rho C_\mu \frac{K^2}{\varepsilon}$                   (22)

Note:

C₁=1.44 (constant).

C₂=1.9 (constant).

σ_k=1.0 (the turbulent Prandtl numbers for k).

σ_ε=1.2 (the turbulent Prandtl numbers for ε).

By representing the flow using the ANSYS program and with the exact flow specifications mentioned in the previous paragraph, the turbulent flow type (k-ε) was adopted, and accordingly, the amount of pressure and velocity at the central axis of the steam ejector. In this work, we relied on representing the flow using the ANSYS program only to determine whether the flow inside the steam ejector works within the design conditions, represented in Figure 3 and Figure 4, were obtained, respectively, in addition to the shape of the flow inside the steam ejector (represented by the velocity vector) see Figure 5, which shows the location of the shock wave at the entrance of the diffuser. The steam velocity at the outlet of the internal nozzle reaches about (440 m/s), while in the vacuum chamber, the velocity is less than (45m/s) and at the steam ejector outlet, it is at a rate of (50 m/s).

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Figure 3. Pressure inside the steam ejector

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Figure 4. Mach number inside the steam ejector

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Figure 5. Flow inside the steam ejector (represented by the Mach number)

The current steam ejector is part of a water desalination system with a predetermined throughput, and the steam ejector is designed based on the amount of steam as well as the operating conditions, so it is challenging to compare the results of the dimensions of the steam ejector in this work with previous literature.

2.3 The primary nozzle location effect calculation

After completing the design of the steam ejector and obtaining all the required dimensions, as mentioned in the previous two paragraphs, it is now the turn to find out the best location for the primary nozzle in relation to the mixing area shown in Figure 6 and its effect on the pressure required for injection (primary stream pressure P_p) in addition to its effect on the amount of steam withdrawn from the area suction. The previous design of the steam ejector was tested at the same pressure in the suction area (entrained vapor pressure P_e) as well as the pressure at the end of the steam injector (compressed vapor pressure P_c).

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Figure 6. The distance between the primary nozzle and the mixing chamber

Using the values of the initial pressure and through equation No. 12, the mass of vapor passing through the primary nozzle can be calculated according to the equation:

$m_p=P_p A_1 \sqrt{\frac{\gamma}{R T_p}\left(\frac{2}{\gamma+1}\right)^{\frac{(\gamma+1)}{(\gamma-1)}}}$          (23)

Since the design of the steam ejector relied on its best performance, which requires that the location of the shock wave at the start of the diffuser be at point (5) and immediately after the constant area, through the values obtained when representing the flow using CFD at each of the primary nozzle locations inside the vapor ejector.

Depending on the pressure values at point 2, the amount of steam entering from the suction area is calculated, then:

$m_e=\rho_{e 2} V_{e 2} A_{e 2}$                (24)

where:

$\rho_{e 2}=\frac{P_{e z}}{R T_{e 2}}$                   (25)

$V_{e 2}=M_{e 2} a_{e 2}$                 (26)

$a_{e 2}=\sqrt{\gamma R T_{e 2}}$                   (27)

From Eq. (7)

$M_{e 2}=\sqrt{\frac{2}{(\gamma-1)}\left[\left(\frac{P_e}{P_2}\right)^{\frac{(\gamma-1)}{\gamma}}-1\right]}$               (28)

Also:

$T_e / T_{e 2}=\left\lceil\frac{(\gamma-1)}{2} M_{e 2}^2+1\right\rceil$                   （29）

So

$m_e=P_{e 2} M_{e 2} A_{e 2} \sqrt{\frac{\gamma}{R T_{e 2}}}$              （30）

Note: The values of the area (A_e2) were calculated from the internal area of the steam ejector minus the area of the external nozzle at point 2, and Figure 7 shows the area that the secondary fluid passes through in the suction area, where:

$A_{e 2}=A_{l-E j}-A_{O-N o z} C$                    （31）

Or

$A_{e 2}=\frac{\pi}{4}\left(D_{l-E j}^2-D_{O-N o z}^2\right)$                   （32）

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Figure 7. Cross Section Area (A_e2) for secondary fluid at point (2)

In order be able to know the extent of the effect the nozzle location on the efficiency of the steam ejector, which is the ratio of the compressed secondary fluid one of the most important measures of performance efficiency, the percentage of the secondary fluid quantity to the total fluid amount must be calculated through:

$M F_e=\frac{m_e}{m_c}=\frac{m_e}{m_p+m_e}$               （33）

This study included the result of flow analysis for steam ejector using the CFD model of the flow. The effect of the location of the initial nozzle inside the steam ejector is evident in the suction pressure, As well as the amount of steam removed from the container of vacuum pressure. The following conclusions are drawn based on the calculation:

• Regarding the quantity of steam produced in the vacuum container, the best location for the nozzle is at the distance (5 mm) from the beginning of the mixing zone.
• Regarding the effect of the nozzle location when determining the exit pressure in the steam ejector, we note that the smaller the distance between the nozzle (the outer opening of the primary nozzle) and the beginning of the mixing zone (the area that has a constant area), the better the performance of the steam ejector. In other words, the distance of the nozzle from the mixing zone is inversely proportional to the performance of the steam ejector.

The use of the steam ejector determines the best location for the primary nozzle has been located. The purpose of a steam ejector is to increase steam generation in a multi-effect evaporation desalination system. So, Prefer the primary nozzle placed at a distance of (5 mm).

Many factors affect the performance of the steam ejector, including what was discussed in previous research and the proposal in the current paper. Examining the effect of more than one factor at the same time on the performance of the steam ejector would play a role in obtaining better results. Therefore, it was suggested that the following effects be studied:

• Studying the possibility of the effect of the location of the nozzle and the length of the mixing zone (the region with a constant area) on the performance of the steam ejector.
• Study the possibility of the influence of the nozzle location and diffuser angle beyond the mixing zone at constant back pressure.