Design Enhancement of Eductor for Active Vapor Transport and Condensation during Two-Phase Single-Species Flow

Abstract

This study is focused on enhancing secondary vapor entrainment and direct-contact condensation in a water jet eductor for the purpose of developing a compact, medium-scale desalination system. It encompasses an extended investigation of an eductor as a condenser, or heat exchanger, for the entrained phase. Exergy study, experimental measurement, and computational analysis are the primary methodologies employed in this work. The target parameters of the optimization work were set through exergetic analysis to identify the region of maximum exergy destruction. In the case of water and water vapor as primary and secondary fluids, mixing and condensation initiates in the mixing chamber of the eductor and is where the maximum exergy destruction was calculated. Therefore, multi-jet primary nozzle eductors were studied to determine the effect of increased interphase interaction area on the exergy destruction and the maximum suction and cooling capacities. Increases in the entrainment ratio, condensation rate and heat transfer coefficient were noted for increasing numbers of nozzles when comparing one-, two- and three-jet eductors.

1. Introduction

In a thermal membrane desalination process, adequate vapor generation and condensation are essential. The characteristics of eductors as a combined vacuum generator and condenser make them a unique option for such applications. The vacuum generation ensures sub-atmospheric vapor generation (suitable for low-temperature applications), and direct-contact condensation accelerates mass transfer. Investigation of the proposed application would aid in developing a desalination method considering the direct-contact condensation feature of DCMD and the pressure gradient due to the vacuum feature in VMD systems. The performance of eductor-based membrane distillation units has been studied in the past, and showed an enhanced performance and reduced footprint of the membrane distillation unit when coupled with an eductor. This study is focused on enhancing the performance of an eductor based on the earlier findings on limiting factors for eductor performance (e.g., thermal resistance due to limited mass exchange).

Eductors (Figure 1) are simple mechanical devices for converting primary flow pressure energy to velocity energy, which in turn creates a low-pressure zone for suction of a secondary fluid. With the primary fluid as liquid water at normal temperature, and the secondary fluid as water vapor, the entrained vapor also condenses. Different physical parameters affect each of the potential functions of an eductor differently. Hence, particular consideration of the specific application is essential when designing an optimized eductor. The use of analytical methods for this purpose has significant limitations, and experimental analysis may not always be practical. Therefore, the strength of computational tools must be increased to ease the design analysis process. The use of Fluent-based Eulerian flow analysis with two resistance models has been the most robust technique for these kinds of applications.

The performance of an eductor is given by the equation

$$E_r=\frac{{\dot{m}}_s}{{\dot{m}}_p}$$

(1)

which is the ratio of the suction mass flow rate to the primary mass flow rate.

The entrainment ratio in current eductors is meager compared to their cooling capacity. There are several sources in the literature on design modifications to eductors that are focused on the comparative study of geometric variations. Chang et al. [1] proposed a petal nozzle design for maximizing back pressure in refrigeration applications and found it gave some improvement relative to a standard nozzle. The installation of a pressure exchange device at the primary nozzle was proposed by Garris et al. [2], Hong et al. [3] and Pietrowicz et al. [4] to enhance mixing compared to a conventional static jet. The use of a lobbed nozzle improved the entrainment ratio but harmed compression ratios [5]. The use of sinusoidal pulsed jets for enhancing mixing was found to be comparatively efficient [6].

The nozzle position is a prime driving factor in an ejector/eductor; with the nozzle positioned outside the mixing chamber, more significant entrainment was achieved [7]. Lin et al. [8], Ariarfar et al. [9] and Ruangtrakoon et al. performed a computational and experimental study for different area ratios. Small variations in entrainment values and a significant improvement in back pressures were achieved. Yapici et al. [10], Wang et al. [11], Sriveerakul et al. [12] and Mohamed et al. [13] evaluated the influence of φ (the ratio of the diameters of the mixing section to the throat). The entrainment ratio increases until a critical value of φ, at the cost of a reduction in back pressure. The use of a larger throat length was found to have better back pressure but will not influence the entrainment ratio [14].

The conventional design method suggests starting by sizing the diameter of the throat [15,16]. Other significant practices involve the use of an adjustable primary nozzle [17], spear valve and swirling vanes at the primary nozzle, mixing vanes at the throat, etc., to increase the system’s robustness under off-design conditions.

In contrast to the existing literature, this work focuses on enhancing the condensing capacity of the eductor by increasing the interphase interaction area. The modification is based on identification of the region of maximum exergy destruction. It makes a foundation for proof of concept to enhance the mechanical and thermal performance of eductors.

2. Methodology

This study involves experimental, computational and analytical methods to strengthen the outcomes. The experimental studies were performed to verify the numerical process, computational study was performed for detailed study of flow phenomenon and exergetic analysis was performed to identify the key parameters for optimization.

2.1. Experimental Method

The objective of this experimental study is to verify the computational and numerical processes for analyzing design modifications. The experiment considers single-jet eductors to correlate to the computational method. Figure 2 shows the experimental setup developed at RMIT University to perform this study. Two sets of experiments were conducted to compare with computational results: axial pressure measurement and flow visualization. The experimental rig loop consists of a large sump of 315 L, a centrifugal pump and a pressure tank, to ensure thermodynamic and hydraulic stability. The centrifugal pump was used to provide pressure energy to the eductor for suction and thermal property exchange. The suction vapor was introduced using a plate heater to saturate liquid water contained within a sealed flask. A 3D-printed eductor with the physical dimensions shown in Figure 2a was installed in the setup. At the suction pipe, suction chamber, mixing chamber and throat, four piezo-resistive pressure transducers with a measurement range of 0–200 kPa absolute were installed. In addition, inlet and outlet pressures and temperatures were also monitored. The flow rate of the inlet water was measured with a vortex flow meter (Grundfos VFS 2-40). All the data monitoring systems were linked to a data acquisition system (dataTaker DT80) and measurements were recorded at an interval of 25 ms through the LabView interface. All the sensors were calibrated, and the uncertainties of individual signals were checked to minimized signal noise. The random uncertainty of individual sensors was within 0.8%. Since the evaporation mass flow rate is very small, it was physically estimated based on the initial and final weight of the flask. A viewing piece was installed at the outlet of the eductor to observe the presence of bubbles in the water. It was also used to ensure that no inward air leakage occurred during the process. Another set of experiment was also performed for qualitative observation of the eductor mechanism. A 3D-printed transparent cascade, analogous to the eductor, was developed to visually capture the flow and phase-change process within the system. The video was captured with a mobile device at 480 FPS, and compared with the phenomenon observed during computational analysis. The visual study does not include any measurements.

An uncertainty analysis of the measured data was performed for all the experiments. The random uncertainty of the data was calculated using the following process, where M_i is the calculated value, $\overline{M}$ is its mean, N is the number of data, i is the position of data, $\sigma$ is the standard deviation of the mean, $\delta$ is the standard error of the mean and $\epsilon$ is the percentage random uncertainty of mean.

Mean value of measured data

$$\overline{M}=\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N{M}_i$$

(2)

Standard deviation of measured data

$$\sigma =\sqrt{\frac{1}{N-1}{{\displaystyle \sum }}_{i=1}^N{\left(M_i-\overline{M}\right)}^2}$$

(3)

Standard error of mean

$$\delta =\pm \left(1.96\times \frac{\sigma }{\sqrt{N}}\right)$$

(4)

Percentage random uncertainty of mean

$$\epsilon =\pm \frac{\delta }{M_i}\times 100$$

(5)

2.2. Computational Method

2.2.1. Solver Theory

The primary and secondary flow modelling was performed using an Eulerian Multiphase Model. The model solves individual momentum and continuity equations for each of the phases. The instantaneous pressure at the point of study is considered the same for all the phases present at the computing point for the considered instance. This multiphase model has been considered to be a suitable option for evaporation and condensation modelling.

Conservation of Mass:

$$\frac{1}{{\rho }_{rq}}\left(\frac{\partial \left(a_q{\rho }_q\right)}{\partial t}+\nabla \times \left(a_q{\rho }_q{\overrightarrow{v}}_q\right)\right)={{\displaystyle \sum }}_{p=1}^n\left({\dot{m}}_{pq}-{\dot{m}}_{qp}\right)$$

(6)

where ρ_rq is the volume average density of the qth phase in the domain.

Conservation of momentum:

Two fluid momentum equations were solved with following equation.

$$\begin{array}{c}\frac{\partial \left(a_q{\rho }_q{\overrightarrow{v}}_q\right)}{\partial t}+\nabla .\left(a_q{\rho }_q{\overrightarrow{v}}_q{\overrightarrow{v}}_q\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-a_q{\nabla }_p+\nabla .{\overline{\tau }}_q+a_q{\rho }_q\overrightarrow{g}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \sum }}_{p=1}^n(K_{pq}({\overrightarrow{v}}_p-{\overrightarrow{v}}_q)+{\dot{m}}_{pq}{\overrightarrow{v}}_{pq}-{\dot{m}}_{qp}{\overrightarrow{v}}_{qp})+({\overrightarrow{F}}_q+{\overrightarrow{F}}_{lift, q}+{\overrightarrow{F}}_{wl,q}+{\overrightarrow{F}}_{vm,q}+{\overrightarrow{F}}_{td,q})\hfill \end{array}$$

(7)

Conservation of Energy:

$$\frac{\partial \left(a_q{\rho }_q{h}_q\right)}{\partial t}+\nabla .(a_q{\rho }_q{\overrightarrow{u}}_q{h}_q)=a_q\frac{\partial \left(p_q\right)}{\partial t}+{\stackrel{=}{\tau }}_q:\nabla {\overrightarrow{ u}}_q-\nabla .{\overrightarrow{ q}}_q+S_q+{{\displaystyle \sum }}_{p=1}^n\left(Q_{pq}+{\dot{m}}_{pq}{h}_{pq}-{\dot{m}}_{qp}{h}_{qp}\right)$$

(8)

where h_q is the specific enthalpy of the qth phase, q_q is the heat flux, S_q is a source term that includes sources of enthalpy, Q_pq is the intensity of heat exchange between the pth and qth phases and h_pq is the interphase enthalpy. The process of exchange of heat agrees with the local energy balance; Q_pq = −Q_qp and Q_qq = 0.

The prediction of turbulence in the current multiphase model is a complex process. This study uses a k-ω mixture turbulence model to simulate the properties of the two phases to provide a convincing result.

The solver theory guide finds a better convergence of multiphase heat and mass transfer when heat transfer is predicted using the Two-resistance model and mass transfer is predicted using the Thermal phase-change model for this multiphase model.

$$\frac{\partial \left({\rho }_m\epsilon \right)}{\partial t}+\nabla .\left({\rho }_m{\overline{v}}_m\epsilon \right)=\nabla .\left(({\mu }_m+\frac{{\mu }_{t,m}}{{\sigma }_{\epsilon }}){\nabla }_{\epsilon }\right)+\frac{\epsilon }{k}\left(C_{1\epsilon }{G}_{k,m}-C_{2\epsilon }{\rho }_m\epsilon \right)+{\mathsf{\Pi }}_{{\epsilon }_m}$$

(9)

And

$$\frac{\partial \left({\rho }_{m}k\right)}{\partial t}+\nabla .\left({\rho }_m{\overline{v}}_{m}k\right)=\nabla .\left(({\mu }_m+\frac{{\mu }_{t,m}}{{\sigma }_k}){\nabla }_k\right)+G_{k,m}-{\rho }_m\epsilon +{\mathsf{\Pi }}_{k_m}$$

(10)

where ρ_m is the density of the mixture, µ_m is the molecular viscosity and ${\overline{v}}_m$ is the velocity of the mixture. The heat transfer process (Figure 3) between phases becomes complicated if an overall heat transfer coefficient is used; hence either side individually calculates phase transfer with the same or a different model. Heat transfer to liquid and vapor phases from the interface are defined with Equations (11) and (12), respectively.

$$Q_l=h_l{A}_i\left(T_s-T_l\right)-m_{lv}{H}_{ls}$$

(11)

$$Q_v=h_v{A}_i\left(T_s-T_v\right)\mp m_{lv}{H}_{vs}$$

(12)

T_s is the surface temperature, which is considered to be similar to the saturation temperature. At the interface neither mass nor heat is stored; hence the overall energy balance is

$$Q_l+Q_v=0$$

(13)

For heat transfer, considering no mass transfer at interphase, the interfacial temperature is calculated as

$$T_s=\frac{h_l{T}_l+h_v{T}_v}{h_l+h_v}$$

(14)

The interphase heat transfer is

$$Q_l=-Q_v=h_{lv}{A}_i\left(T_v-T_l\right)$$

(15)

$$\frac{1}{h_{lv}}=\frac{1}{h_l}+\frac{1}{h_v}$$

(16)

For mass transfer:

From Equations (15) and (16), for the mass transfer rate,

$${\dot{m}}_{lv}=-\frac{h_l{A}_i\left(T_{sat}-T_l\right)+h_v{A}_i\left(T_{sat}-T_v\right)}{H_{vs}-H_{ls}} \{\begin{array}{c}if {\dot{m}}_{lv}\ge 0; evaporation \\ if {\dot{m}}_{lv}<0; condensation\end{array}$$

(17)

The solver has flexibility of selecting from- and to-phases; hence, the value of m_lv and the related sign convention will change accordingly [18].

2.2.2. Computational Domain

A commercially available single-nozzle ejector design was chosen for the computational study. The physical eductors used are the result of reverse-engineering commercially available eductors into 3D-printed specimens. In addition, we investigated the effect of increasing the interaction area between phases by making two similar 3D-printed samples, but increasing the number of nozzles in each while keeping the total cross-sectional area constant. The combined perimeters of 2 and 3 nozzles are 1.4 and 1.7 times larger than in the reference case, respectively. This is expected to improve both the thermal and mechanical performance of the eductor due to the increased interactions between the phases.

A Mesh Independent Test was performed to select computational size. The convergence criteria of 1% was set for a mesh size interval of 1.5 times the former one. Pressures were observed at four different points (P1, P2, P3 and P4) in the flow channel, as shown in Figure 4. A mesh size of 1.8 million was selected for this study. Figure 5 shows the schematic of the eductor studied along with the arrangement of nozzles for each case.

2.2.3. Boundary Conditions

For the simulation, the commercial CFD software ANSYS Fluent 2019 R2 was used. Two-phase flow modeling is solved using Euler’s multiphase model, and fluid turbulence is solved using the k-ω turbulence model. The activation of the primary water pressure at the inlet, the back pressure at the outlet and the low-pressure steam entrainment at the suction port are used as boundary conditions. The eductor wall is treated as a no-slip adiabatic wall boundary. The water used as the main working fluid is the continuous phase, while the entrained steam is the discrete phase. The interaction of heat transfer is calculated using two resistance models (from: Ranz–Marshall, to: zero resistance), and mass transfer is calculated using a thermal phase-change model, where the saturation condition depends on the user-defined local absolute pressure.

The pressure-based double precision solver of the parallel processing system is used to solve the problem. The solution control setting for phase coupling is through a phase-coupled SIMPLE scheme, spatial discretization of gradient is through a least squares cell-based model, pressure is through the PRESTO! Model and the rest are with a first-order upwind scheme. To prevent the solution from diverging, it is very important to set appropriate sub-relaxation factors for pressure, turbulent kinetic energy and turbulent dissipation rate. This work uses 0.1, 0.2 and 0.4, respectively. Additionally, BCGSTAB and multi-grid F-cycle stabilization methods, and Gauss–Seidal smoothing are used to maintain advanced dissolution control. The initialization of the solution is conducted using a hybrid method, followed by a steady-state analysis, and finally a non-steady-state solution is solved. Condensation is an unsteady phenomenon, and the mass transfer rate is a time-dependent parameter, so non-steady-state simulations must be run to obtain reliable results. The boundary condition is described in

Table 1. Boundary conditions.

Inlet Pressure	250 kPa (absolute)
Outlet Pressure	100, 110, 120, 130, 140, 150 kPa
Suction pressure	82 kPa
Primary mass flow rate	6.2 LPM
Nozzle Reynold’s number	100,820.2
Outlet Reynold’s number	62,732.58

.

3. Anergy and Exergy in an Eductor

Exergy is the maximum amount of useful work a system can produce, while anergy is the amount of thermal energy left over (i.e., that cannot be converted to useful work). Anergy is also called exergy destruction; identification of its magnitude will quantify the irreversibility (which is a cause of performance deterioration). Identification of a region of maximum exergy destruction and modification based on it is a systematic approach. This second law concept aids in the identification of major anergy contributors in a system. In a thermo-fluid process specifically, this helps to identify the exact site of inefficiency, which supports implementation of more concrete design optimization for enhanced performance. For the eductor (Figure 6), the region with maximum modification potential is being identified based on regions of maximum exergy destruction and modification attempts have been performed based on it. Here, liquid water at normal temperature is the primary fluid and saturated water vapor from an infinite source is the secondary fluid.

The specific exergy equations for an eductor are described in

Table 2. Specific exergy at various point in the eductor.

Point	Eductor Region	Exergy Equation	Unit
1	Primary inlet	${}_x{}^{}e{}_{1,l}=(h_{1,l}-h_o)-T_o\left(s_{1,l}-s_o\right)+\frac{v_{1,l}{}^2}{2}+g{z}_{1,l}$	J/kg	(18)
2	Primary outlet	${}_x{}^{}e{}_{2,l}=(h_{2,l}-h_o)-T_o\left(s_{2,l}-s_o\right)+\frac{v_{2,l}{}^2}{2}+g{z}_{2,l}$	J/kg	(19)
3	Secondary inlet	${}_x{}^{}e{}_{3,v}=(h_{3,v}-h_o)-T_o\left(s_{3,v}-s_o\right)+\frac{v_{3,v}{}^2}{2}+g{z}_{3,v}$	J/kg	(20)
4	Secondary outlet	${}_x{}^{}e{}_{4,v}=(h_{4,v}-h_o)-T_o\left(s_{4,v}-s_o\right)+\frac{v_{4,v}{}^2}{2}+g{z}_{4,v}$	J/kg	(21)
5	Mixing chamber inlet	${}_x{}^{}e{}_{5,v}=(h_{5,v}-h_o)-T_o\left(s_{5,v}-s_o\right)+\frac{v_{5,v}{}^2}{2}+g{z}_{5,v}$	J/kg	(22)
		${}_x{}^{}e{}_{5,l}=(h_{5,l}-h_o)-T_o\left(s_{5,l}-s_o\right)+\frac{v_{5,l}{}^2}{2}+g{z}_{5,l}$		(23)
6	Mixing chamber outlet	${}_x{}^{}e{}_{6,v}=(h_{6,v}-h_o)-T_o\left(s_{6,v}-s_o\right)+\frac{v_{6,v}{}^2}{2}+g{z}_{6,v}$	J/kg	(24)
		${}_x{}^{}e{}_{6,l}=(h_{6,l}-h_o)-T_o\left(s_{6,l}-s_o\right)+\frac{v_{6,l}{}^2}{2}+g{z}_{6,l}$		(25)
7	Throat outlet	${}_x{}^{}e{}_{7,v}=(h_{7,v}-h_o)-T_o\left(s_{7,v}-s_o\right)+\frac{v_{7,v}{}^2}{2}+g{z}_{7,v}$	J/kg	(26)
		${}_x{}^{}e{}_{7,l}=(h_{7,l}-h_o)-T_o\left(s_{7,l}-s_o\right)+\frac{v_{7,l}{}^2}{2}+g{z}_{7,l}$		(27)
8	Outlet	${}_x{}^{}e{}_{8,l}=(h_{8,l}-h_o)-T_o\left(s_{8,l}-s_o\right)+\frac{v_{8,l}{}^2}{2}+g{z}_{8,l}$	J/kg	(28)

.

The general exergy equation is defined by

$${}_x{}^{}E{}_{r,p}={\dot{m}}_p\times {}_x{}^{}e{}_{r,p} \left[\mathrm{J}\right]$$

(29)

The localized exergy destruction in an eductor is given by the equations in

Table 3. Exergy destruction at various zones in Eductor.

Zone	Exergy Destruction	Unit
Primary nozzle	${}_x{}^{}E{}_{d,1-2}={}_x{}^{}E{}_1-{}_x{}^{}E{}_2$	J	(30)
Secondary nozzle	${}_x{}^{}E{}_{d,3-4}={}_x{}^{}E{}_3-{}_x{}^{}E{}_4$	J	(31)
Suction chamber	${}_x{}^{}E{}_{d,4-5}=({}_x{}^{}E{}_{4,v}+{}_x{}^{}E{}_{2,l})-({}_x{}^{}E{}_{5,v}+{}_x{}^{}E{}_{5,l})$	J	(32)
Mixing chamber	${}_x{}^{}E{}_{d,5-6}=\left({}_x{}^{}E{}_{5,v}+{}_x{}^{}E{}_{5,l}\right)-\left({}_x{}^{}E{}_{6,v}+{}_x{}^{}E{}_{6,l}\right)$	J	(33)
Throat	${}_x{}^{}E{}_{d,6-7}=\left({}_x{}^{}E{}_{6,v}+{}_x{}^{}E{}_{6,l}\right)-\left({}_x{}^{}E{}_{7,v}+{}_x{}^{}E{}_{7,l}\right)$	J	(34)
Diffuser	${}_x{}^{}E{}_{d,7-8}=\left({}_x{}^{}E{}_{7,v}+{}_x{}^{}E{}_{7,l}\right)-\left({}_x{}^{}E{}_{8,l}\right)$	J	(35)
Overall	${}_x{}^{}E{}_d={}_x{}^{}E{}_{1,l}+{}_x{}^{}E{}_{3,v}-{}_x{}^{}E{}_{8,l}$	J	(36)

.

4. Performance Comparison

Figure 7 is the description of the thermal and mechanical mechanism inside an eductor for single-species two-phase flow. The start-up, degassing, pumping and condensation are the major processes involved. In the start-up process, the primary fluid starts flowing into the system, while there is no work occurring in the system. The degassing process starts once the primary fluid starts flowing through its path. The primary fluid transfers surface energy to the stagnant atmospheric air within the eductor; because of this momentum transfer, the stagnant atmospheric air starts forming eddies. The inter-fluid surface friction pulls the air in the direction of primary fluid flow. The pressure inside the mixing chamber gradually lowers until it reaches a point lower than the pressure of the secondary fluid source, which allows movement of the secondary fluid (vapor) into the eductor. Here, secondary fluid vapor is pumped inside the eductor. This pumping process starts from the end stages of the degassing process. Once the vapor is pumped into the eductor, the interphase shear stress and momentum transfer (as in the degassing process) takes the vapor forward. During this process, there is also a thermal interaction between the two phases at the interphase interface, resulting in condensation of the vapor. When the condensing vapor molecules collapse into the liquid state, they form voids which are then filled by more of the surrounding vapor. The larger the rate of condensation, the larger the number of voids that are created and simultaneously filled. During pumping and forward action (in the primary flow path), the availability (secondary fluid in suction chamber) and shearing limit (of primary flow) create flow resistance; hence, the creation of voids during condensation enhances the incoming flow. For the case of non-condensing flow, there is no mass transfer between the primary and secondary fluids; hence the entire process is like the degassing process, with the existence of flow resistance in the forward path. The comparative performance of a two-phase single-species eductor is therefore best described if the mechanical work, thermal work and thermal capacity are evaluated independently. This study presents two primary methods: relative enhancement in entrainment (due to mechanical and thermal activities) and heat transfer coefficient. The relative entrainment describes the contribution of thermal and mechanical activities in the total entrainment value. Similarly, the heat transfer coefficient describes the thermal interaction axially.

4.1. Relative Enhancement Study

Physically, the driving mechanism of an eductor has two aspects. One is the interphase shear stress purely due to momentum transfer, and the second is the rushing of vapor into the spaces formed after condensation. There is an absence of a relative performance indicator suitable for comparing applications that utilize the combined pumping and condensing functions of an eductor with single-phase applications. Hence a relative performance indicator for design alternatives based on the entrainment ratio has been developed. The influence of this design modification was summarized based on the relative change in the entrainment ratio from the reference case. Koirala et al. [19] studied the comparative performance of an eductor in entraining condensing and non-condensing flow with otherwise equal properties. This was part of series of publications in developing an eductor as an effective alternative for active vapor transfer and condensation in the desalination process [20,21]. The entrainment ratio was much higher in the case of condensing flow compared to non-condensing flow. Hence, this study expects to have two different changes in performance, mechanical and thermal. These have been studied based on two indicators: the relative enhancement in thermal performance coefficient (${\varphi }_T$) and the relative enhancement in mechanical performance coefficient (${\varphi }_m$).

4.1.1. Relative Enhancement in Mechanical Performance Coefficient (${\varphi }_m$)

The pumping action of an eductor is due to the momentum transfer between the high-velocity primary fluid and its surroundings. At the start-up, when the primary fluid is just introduced and no work has been performed by the system, the high-speed and low-pressure primary flow causes the stagnant air to start circulating and form eddies. The circulating air is drawn out by the shear stress of the primary fluid jet. The improvement in mechanical performance is primarily related to the suction capacity and axial energy conversion. This is the ratio of the change in entrainment ratio of the new design with respect to the reference design, to the entrainment ratio of the reference design (all for non-condensing flow).

The relative enhancement in mechanical performance coefficient is

$${\varphi }_m=\frac{E_{r,nc,i}-E_{r,nc,1}}{E_{r,nc,1}}$$

(37)

where ϕ_m is the relative enhancement in mechanical performance coefficient, E_r,nc,i is the entrainment ratio of non-condensing flow of design number i and E_r,nc,1 is the entrainment ratio of non-condensing flow of the reference design.

4.1.2. Relative Enhancement in Thermal Performance Coefficient (${\varphi }_T$)

The relative enhancement in thermal performance is a dimensionless number defined to measure the relative entrainment performance of the eductor due to thermal effects. This describes the enhancement in performance because of the creation of voids resulting from condensation. It is the ratio of the change in entrainment ratio between condensing and non-condensing flow of the new design to the entrainment ratio for non-condensing flow of the reference design. Therefore, the contribution of thermal activity in the calculated total entrainment ratio and the influence of design changes on thermal enhancement is best described by Equation (38):

$${\varphi }_T=\frac{E_{r,c,i}-E_{r,nc,i}}{E_{r,nc,1}}$$

(38)

where ϕ_T is the relative enhancement in thermal performance coefficient, E_r,c,i is the entrainment ratio of condensing flow of design number i, E_r,nc,i is the entrainment ratio of non-condensing flow of design i and E_r,nc,1 is the entrainment ratio of non-condensing flow of the reference design.

4.2. Heat Transfer Coefficient

The direct-contact heat transfer coefficient inside the eductor describes the performance of the axial thermal process. This allows a more holistic and precise description of the influence of the geometry on the thermal outcome. It also standardizes the selection of eductors for condensation purposes. The direct-contact heat transfer coefficient h_DCC is given by:

$$h_{DCC}=\frac{{\dot{m}}_v h_{fg}}{\left(T_v-T_l\right) A_{iA}}$$

(39)

where m_v is the mass flow rate of vapor, h_fg is the latent heat of vaporization, T_v is the vapor temperature, T_l is the liquid jet temperature and A_iA is the interface area. The change in heat transfer coefficient between three designs, along with axial variation of this value, is discussed here.

5. Results and Discussion

5.1. Verification of the Computational Process

This simulation method has been adopted based on the two-phase condensing flow analysis performed for a single-nozzle case. The single-nozzle design of an eductor has been experimentally and computationally analyzed. The comparison of the respective results has been considered for verification of the computational method. The verification includes both qualitative and quantitative analysis. The quantitative comparison includes the pressure at three different points, entrainment ratio, and maximum back pressure limit (zero entraining back pressure). Similarly, the qualitative analysis includes comparison of the phase volume fraction and mechanism of condensation, which has been captured through both computational and experimental approaches. The experimental setup, as described in Figure 2, was used to verify the computational method. The prime objective of the visual comparison as to ensure whether the axial oscillation of flow is practically observed or is limited to a theoretical limit.

Figure 8a,b are the comparison plots for axial pressure and zero entraining back pressure, respectively, based on experimental and computational studies. The results are in fairly good agreement with each other. For the point P₁, the length of the suction port from the source is longer in the experimental setup compared to the computational case; hence, larger differences were observed. The value of zero entraining back pressure is the limiting factor for compressibility. Hence, it has been compared between the two studies and found to be within 3%. The reasons for the variation are primarily the experimental uncertainty, variation from ideal no-slip wall conditions and limitation in maintaining adiabatic walls. Figure 9 is the visual comparison of the mechanism within the eductor. The contour in the figure was generated through computational simulation and representing the vapor volume fraction within the eductor flow channel. The oscillating trend of the complete condensation zone was observed in both the computational study and the cascade flow experiment. Although the boundary conditions between the two studies are different, the mechanisms are identical. The magnitude of the boundary conditions might have caused the variation in frequency of the oscillation seen in the comparative study.

5.2. Exergy Analysis

The exergy of an eductor gives the maximum theoretical value of useful work it can perform, at the given boundary conditions, when the final state is in equilibrium with the environment. This allows the identification of the locations and magnitude of maximum exergy destruction so that any modifications can be focused on these priority areas. An analysis of two-phase single-species flow was performed using ANSYS Fluent. The simulation represents sub-atmospheric vapor pumping and condensation, which in turn represents sub-atmospheric vapor generation and condensation processes. Figure 10 shows the exergy at different boundary locations and the total exergy destruction within the system. The inlet exergies are 122.3 J and 341.4 J from inlet and suction, respectively. The outlet exergy is 34.7 J. About 429 J of exergy is destroyed within the eductor during heat and mass transfer and change in axial pressure conditions.

Further, a section-wise analysis of exergy destruction (_xE_d) was performed. Figure 11 shows the exergy destruction at different locations of the eductor. The locations are identified based on the nature of the flow: primary nozzle, mixing zone and throat and diffuser of the eductor. The maximum amount of exergy destruction was calculated to occur in the mixing zone, which includes the suction chamber and mixing chamber. Therefore, optimization of this part of the system is expected to give the greatest improvement in operational performance of the system.

The design modification activity carried out for this study is based on this calculation. The visual observations of unutilized cooling potential and the region of maximum exergy destruction introduced the idea of a multi-jet system, i.e., using a multi-nozzle system for the motive flow. The total cross-sectional area of the reference design’s nozzle outlet was divided by the required number of nozzles (to ensure the same inlet flow conditions), and the corresponding diameters were calculated for each case. This work involves the study of one-, two- and three-nozzle systems. The manufacturability and understanding of the requirement for even flow to all the nozzles are the primary reasons for the selection of these three nozzle designs.

5.3. Computational Study

The observations from the computational study are presented below for all three designs of eductor. The radial plane plots are with reference to Figure 5a. The explanation in each of the sections includes the location of the plane to clarify the results.

Performance Study

Five performance criteria were observed: the comparative entrainment ratio, relative enhancement in thermal performance coefficient, relative enhancement in mechanical performance coefficient, direct-contact heat transfer coefficient and relative improvement in exergetic efficiency.

Figure 12 shows the entrainment ratio (E_r) at different outlet conditions (P_b/P_in). At P_b/P_in = 0.41, the value of E_r is 0.0075, 0.0110 and 0.0168 for one, two and three nozzles, respectively. These values all decrease with increasing back pressure. The study considers five different back pressure conditions ranging from 0.41 to 0.485. The maximum entraining capacity is significantly higher for larger numbers of primary nozzles.

Figure 13 shows the relative enhancement in thermal performance coefficient and relative enhancement in mechanical performance coefficient with respect to the back pressure ratio. The increase in nozzle number from two to three resulted in an improvement of ϕ_T from 26.3 to 39.9 and ϕ_m from 0.38 to 0.91 for P_b/P_in = 0.41. The relative improvement was higher at P_b/P_in = 0.485 for both ϕ_T and ϕ_m. The value P_b/P_in = 0.41 is not the limiting condition for zero-entrainment; rather, it is simply one of the cases considered for study. The difference in the values of each of the coefficients plotted shows there is a larger contribution from thermal activity within the eductor to the enhanced E_r.

Table 4. Heat transfer coefficient for the geometry.

Design	Dn	Nn	m˙s	ATSA	hDCC
	m		kg/s	m2	kW/K·m2
Design 01	0.0028	1	0.00300	0.00048	210.53
Design 02	0.0019	2	0.00439	0.00027	532.50
Design 03	0.0016	3	0.00671	0.00031	711.61

shows the direct-contact heat transfer coefficient (h_DCC) calculated for each of the cases at P_b/P_in = 0.41. The table shows the diameter and number of nozzles along with h_DCC. The value of the direct-contact heat transfer coefficient was calculated to be 210.53, 532.50 and 711.61 kW/K·m² for the one-, two- and three-nozzle cases, respectively. For direct-contact condensation in the eductor, the value of the heat transfer coefficient is dependent on the temperature and flow rate of both primary and secondary flows. This h_DCC is the maximum achievable value for the considered design and operating conditions, as it considers the maximum secondary flow obtained from the computational analysis.

Figure 14 is the plot for the relative improvement in the exergetic efficiency of the eductor when the two- and three-nozzle designs are analyzed with respect to the reference case (one nozzle). This shows the enhancement in utilization of available thermal and mechanical resources within the eductor for each case. In comparison to one nozzle, the two-nozzle design has a relative improvement in exergetic efficiency of 36.94% and the three-nozzle design has an improvement of 51.20%. The boundary conditions and associated energy consumption for the process remain unchanged while the process shows enhanced performance.

Figure 15 shows the flow vectors at the 62 mm radial plane, which is at the suction chamber of the eductor. The large arrow represents the direction of secondary flow. The positioning of the nozzles and direction of flow around it are clearly shown. The flow of liquid water inside the nozzle is axial. The velocity vectors show the formation of eddies around the primary nozzle and the interaction at the interface between the flows. Since the secondary flow in contact with the primary jet is being taken away, additional secondary fluid is being introduced to the configuration. The effectiveness of having multiple jets can also be seen, as the vapors are seen flowing into the gaps between the jets.

The flow of vapor and liquid along the eductor is shown in Figure 16. The vapor volume fraction plot provides visual confirmation of the zone of complete condensation. The thermal interaction at the interphase interface and the flow interference in multi-jet cases are additional important observations from the plots. The entrained vapor forms an annular covering to the primary water jet. In the multi-jet cases, rapid mass transfer in between the jets can been seen.

Figure 17 is the radial temperature distribution at different axial positions in the eductor. The temperature value ranges from 300 K, which is the primary inlet temperature, to 368 K, which is the vapor suction temperature. For one jet, the core jet and annual vapor flow can be clearly observed. The region of core temperature is found to be slowly expanding as the vapor in the vicinity is being condensed. For the two-jet case, dual-core flow is initially seen up to 80 mm, and eventually mixes after 90 mm because of the complete condensation and change in pressure conditions. The thermal effect of the jets was dominant in the abscissa (the axis on which the nozzles are aligned) compared to the ordinate. Although there is effective heat transfer in the flow between the jets, the overall effect may be enhanced further with an elliptical mixing chamber having its major axis towards the abscissa. For the three-nozzle case, the core flow has good heat transfer and hence has uniform temperature around the region of the nozzles. The temperature distribution is more even compared to the one- and two-nozzle systems, indicating better utilization of the cooling potential.

Figure 18, Figure 19 and Figure 20 are the plots for the mass transfer rate for one, two and three nozzles, respectively. In the one-nozzle system, the mass transfer is observed only around the central core flow. There is an uneven distribution of mass transfer axially, with the maximum rate of 3845.3 kg m⁻³ s⁻¹ at around 96 mm (zone of complete condensation). The mass transfer area is also observed to be expanding as the flow moves forward towards the outlet. For the two-nozzle system, the largest mass transfer is observed in between the jets, and the region of mass transfer expands in the abscissa. The influence of the core reduces past 90 mm, and complete condensation occurs around 103 mm. The maximum rate of mass transfer is 3800 kg m⁻³ s⁻¹, which is consistent from the mixing chamber to the zone of complete condensation. If the nozzles were placed further apart, a more uniform utilization of potential could be achieved. For the case of the three-nozzle geometry, the rate of mass transfer is more uniform and significantly larger than one- and two-nozzle geometries. Although the interaction area was increased in the two-nozzle case, the energy transfer in the ordinate was not as effective. With three nozzles, both ordinate and abscissa have active heat and mass transfer within them. A uniform mass transfer of up to 4850 kg m⁻³ s⁻¹ was achieved. The larger thermal load requires a larger area for mass transfer; hence in this case the complete condensation point occurred at around 120 mm. As the flow moves forward, a growth in the thickness of the mixing zone is observed, which covers the entire flow area at the throat. The use of multiple nozzles might increase the manufacturing complexity but has also been shown to significantly enhance the two-phase single-species fluid handling for combined vacuum generation and condensation.

Zhang et al. [22] proposed an empirical model verified experimentally on a water-driven steam injector for waste heat recovery. The flow phenomena they modelled are similar to the current scope of this study. They performed theoretical modeling based on three considerations: mass exchange dominantly at the interphase interaction, a single-phase outlet from the eductor and annular flow of subcooled water jacketed by the vapor. The entire phenomenon obtained in this visual process is identical. Shah et al. [23] performed experimental and computational studies of flow in a steam jet ejector. The mass transfer was observed primarily at the interphase interaction area when the jet enters into the sub-cooled water pool. With primary liquid and secondary vapor in our case, the interphase interaction continues along the throat for complete condensation. This work extends the existing knowledge pool by introducing the impact of an increase in the interphase interaction area and flow mechanism inside an eductor. Moreover, this develops eductor as an active vapor-transfer and condensing device. In addition to the theoretical understanding of the process, in Figure 21, Koirala et al. compared the outcome with the current model [19].

6. Conclusions

This is a continued work from previous understanding which includes computational and experimental studies on the design modification of an eductor to use it as a condensing unit.

The exergy analysis helped in identifying the two-phase mixing zone as the region of maximum exergy destruction. The single-jet primary flow design was modified to use two and three nozzles, to increase the interphase interaction area. An experimentally verified computational model was adopted for further analysis. With increasing number of nozzles, the relative improvement in mechanical and thermal performance was calculated along with its heat transfer coefficient and exergetic efficiency. The relative enhancement coefficients were specifically developed to show the relative magnitudes of the effect of thermal and mechanical activity within an eductor. Furthermore, it was found that the entrainment ratio in a two-phase single-species eductor is affected more by the thermal activity than the mechanical activity, indicated by the values of ϕ_m and ϕ_T. The increase in mass transfer rate creates more voids for the surrounding vapor to fill, and hence the mechanical resistance is reduced, which supports a larger incoming mass flow. The addition of extra nozzles adds some manufacturing complexity in the design but results in a significant enhancement of performance. Since this is a novel application of the existing technology, any further similar application can make use of the findings of this work. The increase in the interphase interface area and its influence on thermal performance of two-phase single-species flow is the major outcome of this work.