Submerged Gas Jet into a Liquid Bath: A Review - Industrial

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Submerged Gas Jet into a Liquid Bath: A Review S. S. Gulawani, S. S. Deshpande, and J. B. Joshi* Institute of Chemical Technology, UniVersity of Mumbai, Matunga, Mumbai - 400 019, India

M. S. Shah, C. S. R. Prasad, and D. S. Shukla Chemical Engineering & Technology Group, Bhabha Atomic Research Center, Trombay, Mumbai - 400 085, India

Gas-liquid jet reactors are widely used in commercial applications such as condensing jets for direct contact feedwater heaters and steam jet pumps, because of their efficient heat- and mass-transfer characteristics. These are also used for the blowdown of primary nuclear boiler systems into a water bath, without releasing fissionable materials into the atmosphere. Reacting jets are of major interest in metal processing and thermal energy sources that involve submerged injection of an oxidizer into a liquid metal bath. The design of gasliquid jet reactors is strongly dependent on the plume dimensions and the flow pattern in the liquid phase. In the present review paper, a critical analysis of the published literature on the fluid dynamics and heat transfer for gas-liquid jet reactors has been performed. The analysis has been extended for the empirical, semiempirical, and analytical attempts for the correlations of experimental observations. The published works on the computational fluid dynamics (CFD) simulations have also been critically analyzed. A comprehensive discussion has been presented and an attempt has been made to arrive at a coherent theme that clearly describes the present status of the published literature. Furthermore, recommendations have been made that are expected to be useful for the design engineers as well as researchers, to improve the reliability in the design of this important class of reactors. 1. Introduction Submerged gas jets into a liquid bath are widely used in thermal energy sources. In this system, thermal energy is produced in a reactor when an oxidizer gas jet is injected into a liquid bath. The energy produced is transferred through working fluid (water) through the boiler tubes. Superheated steam exits from the boiler reactor and is used to power a prime mover. As the gas is injected into a liquid, it reacts with liquid metal and produces a large amount of heat. This heat will evaporate the local liquid bath. The effect of fuel vaporization is important, because it influences the reaction mechanism as well as mixing of the oxidizer jet and fuel. Analysis of such system by visualization is practically impossible. Thus, the study of processes with similar attributes, such as condensation and evaporation of a bath, direct contact heat transfer must be a focal point. Typically, these processes involve condensation of a pure component (steam-water), dissolution of a gas into a bulk liquid phase (ammonia into water), or an absorption accompanied by instantaneous chemical reaction (HCl-aqueous NH3 solution, H2O/LiH-Al, O2/Al-Mg, F2/Li/Al-KClO4, etc.). The gas-liquid jet reactor consists of a nozzle inserted into a liquid bath. The gas is sparged through a nozzle into a liquid bath. The sparged gas phase takes the shape of a jet (Figure 1) when the gas-liquid reaction is instantaneous or the gas phase is a condensing fluid. This type of reactor is the subject of the present review paper and, for the sake of convenience, it henceforth will be called a jet reactor. The jet reactor has found a wide range of commercial applications, such as use as condensing jets for direct contact feed water heaters and steam jet pumps. These are also used for the blowdown of primary nuclear boiler systems into a water bath, without releasing * To whom all the correspondence should be addressed. Phone: 0091-22-2414 0865. Fax: 00-91-22-2414 5614. E-mail address: jbj@ udct.org.

Figure 1. Schematic of the gas-liquid jet reactor.

fissionable materials into the atmosphere. Reacting jets are of major interest in metal processing and thermal energy sources that involve the submerged injection of an oxidizer gas into a liquid-metal bath. For the design of these equipments, the principal requirement is knowledge of the jet shape and dimensions, which are primarily dependent on the nozzle type and size, the volumetric

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flow rate, and the rate of reaction/rate of condensation. This knowledge is a prerequisite for the selection of vessel dimensions. If reactor is to be provided with heat-transfer elements and some other internals, then the dimensions of the reactor are dependent on the jet dimensions. One of the major reasons of utilizing jet reactor in the industry is their efficient heat- and mass-transfer characteristics. Flow dynamics in a jet reactor leads to a faster renewal of the gas-liquid interface, resulting in more exposure of the surrounding bath to the incoming gas, which enhances further heat- and mass-transfer characteristics. In jet reactors, the gas-liquid reaction is usually instantaneous and, therefore, the mass transfer is a controlling mechanism. Thus, the mass-transfer coefficient is dependent on the local flow structures at the gas/liquid interface. Similarly, in the case of condensing vapors, the heat-transfer coefficient also is dependent on the local flow structures. Furthermore, the selection and design of the immersed heat-transfer area are also dictated by the flow pattern. Thus, the design of jet reactor is strongly dependent on the jet dimensions and the flow pattern in the liquid phase. Given this information, a large number of experimental and theoretical investigations have been undertaken during the past 30 years. Practically all types of flow visualization techniques have been used, and the results on jet dimension have been reported in the form of semiempirical correlations. For the description/estimation of jet dimensions, flow pattern, and heat- and mass-transfer characteristics, several empirical, semiempirical, and analytical, as well as numerical, techniques have been applied. In all these cases, simplifying assumptions have been made. It is important to assess the quantitative role of these assumptions on the reliability of the predictions and, hence, on the design. Although computational fluid dynamics (CFD) can reduce the number of simplifying assumptions, the progress that has been made so far is still preliminary and substantial improvements are needed in the description of interface momentum and mass and heat transfer. In the present review paper, experimental results that have been published so far, as well as all mathematical models, have been critically analyzed. A comprehensive discussion has been presented and an attempt has been made to arrive at a coherent theme that clearly describes the present status of the published literature. Furthermore, the recommendations that have been made are expected to be useful for the design engineers, as well as researchers. The knowledge gaps have been identified and suggestions have been made for further research, which, in turn, will improve the reliability in the design of this important class of gas-liquid jet reactors. 2. Jet Dimensions: Experimental Investigation and Semiempirical Correlations Whenever a gas is injected with very high velocity into a sub-cooled bath, the momentum and energy of the gas are transferred to the surrounding liquid, leading to threedimensional complex flow structures, which are generally turbulent. The turbulent flow pattern enhances the heat-transfer coefficient at the gas/liquid interface, as well as at the immersed surfaces. The injected gas also generates a small vapor cavity, called a plume, just next to the nozzle exit. Two types of gasliquid jets have been reported: (1) condensation jet and (2) reactive jet. A typical example, for the condensation jet, is steam injected into subcooled water; whereas, for the reactive jet, HCl gas injected into aqueous ammonia bath or F2 injected into lithium bath are typical examples. In the case of the steam jet, the entire plume length is due to condensation only, whereas in the case of reactive jets, the plume length consists of a reaction

zone and condensation zone.1 Thus, the condensation has a major role in deciding the plume length in these systems. Most of the authors had performed the experiments and developed semiempirical correlations for dimensionless plume length, in terms of various operating conditions. The dimensionless plume length can be defined as the ratio of the jet length to the nozzle internal diameter.2 The various measurement techniques used so far include photographic measurement, measurements of temperature profile along the nozzle axis, and pressure fluctuation measurements. Furthermore, the experimental as well as mathematical analysis is dependent on the modes of condensation. 2.1. Modes of Condensation and Regime Map. The pioneering work of developing a regime map, which classifies the complex condensation modes encountered in low-flow steam injection, has been obtained by Chan and Lee.3 They studied direct-contact condensation by injecting steam into a pool of subcooled water. The motion of the steam/water interface was recorded by high-speed movies, and the photographs were systematically analyzed to classify the modes of condensation, on the basis of injection rate and the extent of the pool subcooling. The schematics of various modes in the directcontact steam condensation (DCSC) have been shown in Figure 2. When the steam injection rate was high (>125 kg/(m2 s)) and ∆T was high, an oscillatory cone jet (Figure 2A) was observed, whereas as ∆T decreased, an oscillatory elliptical jet was observed. At intermediate steam injection rates and moderate ∆T, an oscillatory bubble mode (Figure 2B) was observed at the pipe exit, whereas at low ∆T, the shape of the plume reduces to an elliptical oscillatory bubble (Figure 2C). At a low rate of steam injection ( F2-Li > HCl-aqueous NH3 > steam-water Furthermore, the comparison of proportionality constants suggest that the constant c decreases in the following order:

SF6-Li < F2-Li < HCl-aqueous NH3 < steam-water

Figure 5. Plume length calculations by various correlations: (]) Kerney et al.,7 (O) Wiemer et al.,8 (2) Chen and Faeth,11 (0) Chun et al.,4 and (/) Kim et al.2 Table 3. Variation in Proportionality Constant of Dimensionless Correlations for Plume Length Prediction

reference

dimensionless correlation

Kim et al.2 Chun et al.4 Kerney et al.7 Weimer et al.8

L/d0 ) 0.503B-0.70127(G0/Gm)0.47688 L/d0 ) 0.5923B-0.66(G0/Gm)0.3444 L/d0 ) 0.26(1/B)(G0/Gm)0.5 1/2 L 1 (G0/Gm) ) 17.75 1/2 d0 B (F /F ) ∞ 0

()

variation in the proportionality constanta 0.18-0.24 0.14-0.24 0.26 0.11-0.28

a For the modified form of the dimensionless correlation: L/d ) 0 cB-1(G0/Gm)0.5.

For this purpose, all the semiempirical equations proposed were fitted with the following correlation proposed by Kerney et al.: 7

( )

G0 L ) c(B-1) d0 Gm

0.5

(8)

The value of the constant c for different authors is reported in Table 3. It can be observed that the overall variation in the constant c ranges from 0.11 to 0.28. In the case of reactive jets, the plume length was defined as

( )( )

∆Hrea F0 Lheat )c d0 Cp∆T F∞

0.5

(9)

The studies with various reactive systems such as HCl-aqueous NH3 (from Cho et al.1), Cl2-Na (from Avery and Faeth12), and F2-Li (from Chan et al.13) were reported in the literature. The proposed semiempirical correlations devised by different authors have been given in Table 4, and the correlations show much similarity, in regard to representing the condensation jets. The definition of driving potential must be developed properly for both types of jet systems. In the case of the condensation jet, the driving potential was based on the latent heat of steam and subcooling of the bath temperature, whereas in the case of the reactive jets, it is dependent on the amount of heat generated by the reaction and subcooling of the liquid-bath temperature. The comparison of predictions of the plume length for condens-

(see Table 4). The variation in the prediction of the plume length is due to differences in the vapor flow condition. The major reason for the greater plume length for a halogenated pair (SF6Li, F2-Li) is the higher heat of reaction, which leads to higher temperature in the plume zone. This evaporates more liquid from the surrounding bath, thus increasing the rate of entrainment of liquid, which gives the greater plume length. Furthermore, the data for the condensing jets were obtained under a well-defined condition where the vapor flow was fully choked and relatively steady at the injector nozzle. For reactive jets, oxidizer gas flow from the nozzle was always sonic. The fuel vapor produced by reaction heat arises some distance away from the nozzle, and it is in the subsonic condition that shows unsteady behavior. This unsteadiness enhances the mixing of vapor with the surrounding bath liquid, which leads to an increase in the plume length. 2.3.2. Effect of Mass Flow Rate on Plume Length. The gas-liquid reaction/ condensation phenomenon and the shape of the plume are mainly dependent on the velocity with which the gas enters the liquid bath. The regime map indicates the various modes of condensation, based on the shape of the plume formed under different steam mass flow rate conditions (section 2.1). The mass flow rate of gas decides the amount of gas available for the phenomenon of condensation/reaction and the amount of heat given to the liquid bath, which subsequently affects the shape and size of the plume. Figure 6 shows the variation in the plume length for condensation jets because of changes in the mass flow rates for the experimental data of Kerney et al.7 with flat and conical head nozzles with ID ) 6.35 mm. Here, the bath temperature varies from 336 K to 342 K, which is a relatively narrow range. It can be observed that the plume length increases as the mass flow rate increases for both types of injectors. Also, note that there is not much of a difference in the estimated plume length for both types of injectors. In the case of condensation jets, many authors2-4,7,8 developed semiempirical correlations to predict the plume length, which shows the dependency on the ratio of mass flow rate where the exponent varies from 0.3 to 0.5 (see Table 1). In the case of reactive jets, while developing the semiempirical correlation, Cho et al.1 assumed that the amount of entrainment of the bath liquid was directly proportional to the gas mass flow rate and that the reaction or heat transfer is controlled by the entrainment. Thus, we cannot have any explicit terms that involve the gas mass flow rate in the resulting semiempirical correlation (eq 9). 2.3.3. Effect of Bath Temperature on Plume Length. The heat-transfer characteristics have a significant role in regard to plume dimension. These characteristics are represented in terms of the driving potential (B), which can be defined as the ratio of heat generated due to the reaction or the latent heat of condensation, relative to the heat gained by the liquid bath. The rate of subcooling (the saturation temperature minus the bath temperature) dominates the driving potential B. The heat supplied by the gas due to condensation/reaction is being used to reach the saturation conditions, where the liquid bath becomes vaporized. At high bath temperatures (low subcooling rates),

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Table 4. Variation in the Proportionality Coefficient of Dimensionless Correlations for Condensation and Reaction Jets system

authors

steam-water

Chen and Faeth11

HCl-aqueous NH3

Cho et al.1

semiempirical correlation Condensation Jet (hs - hf) F0 L ) 17.8 d0 (hf - h∞) F∞ Lheat d0

F2-Li

Chan et al.13

Reactive Jet Lheat d0

SF6-Li

]( )

[

Chan et al.17

Lheat d0

gas travels a longer distance to reach the cut-off conditions, where the jets lose their energy to increase the local bath temperature to the saturation level, to convert them into vapor. Thus, the higher the bath temperature, the higher the plume length, because of the evaporation phenomena at the gas/liquid interface. In the case of reactive jets, bath vapors are lighter than the aqueous solution and the injected gas. As more vapors are evaporated from the jet interface boundary, the jet mixture density is reduced near the boundary. This leads to a higher penetration of fuel gas into the liquid bath, which increases the plume length. Figure 7 shows the variation of plume length with bath temperature for condensation4 and reaction jets.1 The heat generated in the reaction jets is very high, in comparison to the condensation jets, which leads to greater evaporation of the liquid bath at the gas/liquid interface. Thus, the plume lengths for the reaction jets are high, in comparison to the condensation jets. 2.3.4. Effect of Nozzle Diameter on Plume Length. The nozzle diameter determines the amount of gas that penetrates inside the liquid bath. Thus, the reaction/condensation phenomenon is dependent on the nozzle diameter, which affects the shape and size of the plume. Different types of nozzlessviz., flat head,2,7 conical head,2,4,7,8 and convergent divergent nozzles were used in the literature. The variation in the nozzle diameter changes the modes of condensation, which leads to the different shapes and sizes of the plume. Also, the nozzle submergence in the liquid bath decides the flow pattern and the extent of

Figure 6. Effect of mass flow rate on the plume length for Kerney et al.7 data for a nozzle inner diameter of ID ) 6.35 mm: ([) flat head and (0) conical head.

) 8.9

1/2

( )( ) ∆Hrea F0 Cp∆T F∞

) 7.29 ) 4.08

1/2

( )( ) ( )( )

density ratio variation 0.002-0.005 0.002-0.008

∆Hrea F0 Cp∆T F∞

1/2

2.37

∆Hrea F0 Cp∆T F∞

1/2

2.87

mixing inside the reactor. This significantly alters the rate of entrainment of liquid from the gas/liquid interface and the heattransfer characteristics of the liquid bath, which affects the plume length. 2.3.5. Effect of Bath Liquid Composition on Plume Length. In the case of reactive jets, the concentration of the liquid bath has a major role in the determination of the plume length. As the concentration of the liquid bath increases, more reactant is available for the reaction. Thus, a large amount of energy is released in the liquid bath, which leads to an increase in the bath temperature. Furthermore, as the temperature of the bath increases, the same explanation as that given in Section 2.3.3 can be given for the effect of bath temperature. Figure 8 shows the variation in the plume length for the HCl-aqueous NH3 system studied by Cho et al.1 It can be observed that, as the concentration of aqueous ammonia increases from 0.106 to 0.256, the plume length also increases. 2.3.6. Effect of Bath Pressure on Plume Length. The major application of gas-liquid jet reactors is in the thermal energy sources, where a large amount of heat is generated due to reaction. One of the major requirements for such a reaction is that the liquid bath must be kept in the molten condition and, hence, both temperatures are usually high. As the bath pressure is reduced, the boiling point of the liquid bath is reduced, which subsequently reduces the rate of subcooling. As the rate of subcooling decreases, the plume length increases. (Relevant discussion on the effect of rate of subcooling on plume length has been presented in section 2.3.3.) Figure 9 shows the variation in plume length for the HCl-aqueous NH3 system studied by

Figure 7. Effect of bath temperature on plume length: ([) Cho et al.1 and (0) Chun et al.4


transfer coefficient accurately predicts the plume dimensions (section 4.6.1), as well as flow and temperature (section 4.6.3), variations inside the reactor. 3. Heat Transfer in Jet Reactors: Experimental Investigation

Figure 8. Effect of ammonia concentration on the plume length (from Cho et al.1 data).

Figure 9. Effect of bath pressure on the plume length (from Cho et al.1 data).

Cho et al.,1 which clearly indicates that, as the bath pressure decreases, the plume length increases. 2.4. Conclusions. For condensation and reactive jets, various authors have conducted photographic measurements. The observed plume lengths have been correlated with system parameters such as mass flux, density ratio, nozzle diameter, and driving potential through semiempirical equations. Although there is fairly good similarity between the proposed correlations for condensation and reaction jets, the definition of driving potential for the condensing and reactive jets are different. Because most of the predictions of plume length are available in the semiempirical form, a strategy must be developed that reduces the empiricism for the design of jet reactors. Extensive work is still needed for the measurements of plume dimensions using the techniques of flow visualization, such as hot film anemometry (HFA), ultrasound velocity profiling (UVP), and tomography. Furthermore, an emerging numerical techniques namely, CFDscan be applied with appropriate governing equations, along with proper turbulence and reaction models, which will improve the understanding of flows in jet reactors. Recently, Gulawani et al.14 developed a new model for DCC phenomena, where interface mass transfer was modeled using a thermal phase change model (details are given in section 4.6.3). The approach of a phase-change model with the proper heat-

The jet reactors are associated with large heat effects that are either due to condensation (latent heat) or exothermic chemical reaction. In both cases, the heat must be removed from the reactor by providing a heat-transfer area in the form of a coil- or finger-type arrangement. The heat transfer occurs in two steps: (i) across the gas/liquid interface of the jet (this is the region where most of the heat is generated) and (ii) across the liquid/solid interface (where heat is removed from the reactor). Most authors (e.g., Kim et al.,2 Chun et al.,4 Kim et al.,9 Simpson et al.,15 and Aya and Nariai16) have restricted their studies to the calculation of the average condensation heattransfer coefficient near the vicinity of the gas/liquid interface. 3.1. Condensation Jets. Simpson et al.15 studied the dynamic behavior of subsonic jets by studying the condensation process and hydrodynamic pressure oscillations, when the steam was discharged into a subcooled water pool. The periodic interfacial motion was subdivided into three intervals: bubble growth, bubble translation, and bubble separation (necking). They found that the condensation rate during the bubble growth and the bubble translation was governed by the heat transfer in the liquid region, whereas the condensation rate during necking was governed by the heat transfer in the steam region. Furthermore, the heat-transfer coefficient was observed to be the highest in the necking interval and was unaffected by subcooling, because of the droplet stripping. The heat-transfer coefficient during bubble growth was observed to increase as the subcooling rate increased. The average heat transfer for subsonic jets was reported to be 1/5 to 1/10 of the sonic jet values. They also studied the pressure transience of a subsonic jet, which was marked by periodic impulses, originated by the necking process. The extent of pool subcooling was determined to exhibit the largest influence on the dynamic behavior. The pulse frequency and the intensity were correlated with the Jacob number (Ja) and the Reynolds number (Re). Kim et al.2 investigated DCSC phenomena experimentally, using five different sizes of nozzles with horizontal orientation, over a wide range of steam mass flux and pool temperature conditions. They reported that the average heat-transfer coefficient was in the range of 1.24-2.05 MW/(m2 K) and was observed to increase as (i) the pool temperature decreased, (ii) the nozzle size decreased, and (iii) the mass flux increased. Chun et al.4 studied the DCSC phenomena and proposed regime maps for the jet shapes. They proposed semiempirical correlations for the average heat-transfer coefficient, which was observed to be in the range of the experimental data ((30%). The average heat-transfer coefficient for steam injection in a stagnant pool of water was observed to increase as the steam flow rate and the degree of subcooling each increased. The average heat-transfer coefficient of condensation for the high steam mass velocities was determined to be in the range of 1.03.5 MW/(m2 K). Condensation was observed to be more efficient with smaller diameter nozzles. Aya and Nariai16 proposed correlations for the heat-transfer coefficient (assuming a simple interface shape) for different condensation regimes, with a steam jet in a pool of water and a water jet in a steam pool, using a numerical analysis technique. The correlations for different regions are given in Table 5. At chugging, the heat-transfer coefficient was calculated using

3200


Table 5. Proposed Correlations for Heat-Transfer Coefficient authors

model configurations

correlation h) h)

al.2

Kim et Chun et al.4 Kim et al.9 Kim et

al.9

Kim et al.9

(a) interfacial transport due to turbulence intensity (b) interfacial transport due to surface renewal (c) interfacial transport due to shear stress

Simpson and Chan15 Aya and Nairai16 Aya and Nairai16

Aya and

Nairai16

h)

numerical analysis techniques and it was observed to vary up to 2 × 106 W/(m2 K). At condensation oscillation, it ranged between 105-106 W/(m2 K). Kim et al.9 investigated DCSC phenomena that was occurring around a stable steam plume in subcooled water. The condensation heat-transfer coefficient at the steam/water interface was determined using three different models: interfacial transport model due to the turbulent intensity, the surface renewal model, and the shear stress model. They reported that the size of eddies developed on the liquid-side thermal boundary layer was important in determining the heat-transfer coefficient, using the turbulent intensity model. Furthermore, they introduced the new concept of a plume shape factor for the measurements of plume dimensions, which was used efficiently to determine the condensation heat-transfer coefficient. For the surface renewal model and the shear stress model, the measured plume shape was used to determine the heat-transfer coefficient. Furthermore, the model was corrected using the plume shape factor, which shows good agreement with the experimental data. Thus, the overall studies reveal that the size of eddies and the steam plume shape must be thoroughly investigated to determine the DCSC condensation heat-transfer coefficient. All the proposed semiempirical correlations for the heat-transfer coefficients have been tabulated in Table 5. 3.2. Reaction Jets. In the case of reaction jets, a large amount of heat is generated due to reaction. This leads to an increase in the bath temperature, from 1500 K to 4500 K. Measurement of the temperature profiles at such higher temperatures is practically impossible. Thus, no information was available for the temperature measurements inside the reactor. However, with low-temperature reaction pairs (such as the HCl-aqueous NH3 system), it is possible to measure the temperature variation inside the bath. Cho et al.1 have mentioned the various operating bath temperature conditions; however, the variation in the axial temperature was not reported. 3.3. Conclusions. Most authors have studied the heat transfer across the gas/liquid interface.2,4,9,15,16 The semiempirical correlations that have been proposed for the heat-transfer coefficient are based on the operating conditions. It was observed that, for the sonic condensation jet, the heat-transfer coefficient was in the range of 1-3.5 MW/(m2 K), whereas for the subsonic jet,

) )( )(

KLCp 1 νLFG 4 - 2ηsf

h ) 0.1819

KLCp 1 νLFG 4 - 2ηsf

hbtb + hntn , tb + t n

hi ) a

(b) steam jet into water chugging (low steam mass flux) condensation oscillation (high steam mass flux) (c) water jet into steam space

( )( [( [(

h ) 0.1409

h) (a) interfacial resistance

1.4453CpGmB0.003587(G0/Gm)0.13315 1.3583CpGmB0.0405(G0/Gm)0.3714 1 1/3 FL 2/3 CpG0 Stt 2ηsf FG

where h )

)] )]

0.5

0.5

B0.33Gm0.1722G00.8278 B0.33Gm0.1722G00.8278

m˘ hfgA0 ∆TAint pha

Fa

L2

x2πRTs Ts

h ) 6.5FLCpLuG0.6(VL/d0)0.4 λL dGG 0.9CPL∆T h ) 43.78 d0 FLVL L

( )

h)

K(Ts - T)W (1/4)πd2n

it was 1/5 to 1/10 that of the sonic jet.4,15 In the case of the reacting jet, a large amount of heat is generated due to reaction. Furthermore, this energy is transferred to the liquid bath through the gas/liquid interface, which results into an increase in the bath temperature. To remove the heat, a suitable geometry, along with a coolant, must be selected, so that the maximum amount of heat can be removed from the reactor. Water is the most suitable coolant for the removal of heat. Typically, subcooled water is used as a coolant and superheated steam is collected at the outlet. The water goes from various states of flow conditions, such as subcooled water to saturated water to saturated steam and finally to the superheated steam. Thus, the heat transfer from the liquid bath to the reactor wall and from the reactor wall to the coolant, at different states of flow conditions, must be explored further. 4. Computational Fluid Dynamics 4.1. Introduction. Computational fluid dynamics (CFD) is a new emerging technique for better understanding of the flow pattern and temperature distribution in the reactor. The main objective of the CFD simulation is to predict plume dimensions, flow patterns, and temperature distributions in the reactor for both types of jets. To predict the flow and heat variations in the system, nonlinear equations of mean flow and turbulence must be solved simultaneously. Along with these equations, heat-transfer coefficients (from the gas/liquid interface to the bath liquid and from the liquid interface to the solid wall), masstransfer coefficients, rates of reaction, and boundary conditions are also important. CFD model formulation mainly involves the following steps: (a) selection of flow model; (b) selection of turbulence model; (c) definition of rate of reaction; (d) individual species properties, which involves development of equilibrium state relationship and choosing of the probability density function (PDF) function; (e) heat- and mass-transfer coefficients; and (f) provision of the correct boundary conditions. To solve flow with condensation/reaction, the correct fraction of each species in the mixture after condensation/reaction must be predicted. This can be achieved via two methods: (1) by providing an equilibrium state relationship with the proper PDF function and (2) by providing reaction kinetics, with the required heat- and mass-transfer coefficients. In the first method,


local value of mixture fraction variance and the value of maximum probability. Furthermore, the mean scalar properties, such as temperature, density, and void fraction, were determined at a specified point in the flow using the probability function, in conjunction with the local equilibrium calculations, as

Table 6. Governing Equations for LHF Model ∂Fu 1 ∂ + (rFV) ) 0 ∂x r ∂r ∂φ ∂φ ∂φ 1 ∂ rµ + Sφ + FV ) ∂x ∂r r ∂r eff,φ ∂r

(

Fu φ uj hf k

µeff,φ µ + µtur (µ/Sc) + (µtur/σtur) µ + (µtur/σk)

ε

µ + (µtur/σε)

g

(µ/Sc) + (µtur/σg)

)

Sφ a(F∞ - Fj) 0 ∂uj 2 µt - Fjε ∂r ∂uj 2 ε Cε1µtur - Cε2Fjε ∂r k

φ)

( )

[

]

( )

Cg1µtur

(∂f∂rh) - C Fj(εgk) 2

g2

minimization of the Gibbs free-energy of formation was used and the equilibrium state function was developed. The state function, along with PDF and two-dimensional (2D) NavierStokes equations were solved to obtain the flow pattern. Thus, various investigators11-13,17-25 have undertaken CFD simulations to predict flow and turbulence parameters closer to reality. Although the state function was determined to predict the values of concentration of various species at equilibrium accurately, the prediction of the flow pattern was not closer to acceptable accuracy. To predict the flow pattern accurately, the second method is useful, wherein the reaction kinetics, along with heatand mass-transfer coefficients, are provided. 4.2. Flow Model Formulations. Most of the CFD analyses use the thin-shear-layer approximation for a steady, axisymmetric turbulent jet in stagnant surroundings that have uniform properties. Practically all the CFD analysis has been performed using LHF approximation, along with the PDF approach.11,13,17-25 It was observed that, for highly dispersed multiphase flow, where bubbles have small momentum, thermal, and mass capabilities, LHF approximation has shown reasonably good results. Furthermore, attempts have been made to introduce various multiphase submerged flow models, such as the two-fluid model (TFM) and the multifluid model (MFM), along with different shapes of PDF, to improve the predictions from CFD.13 In the case of reaction flow, the density must be decomposed to a time-average component and a fluctuating component. Favre26 suggested the density-weighted-average procedure, which gives the mass-averaged velocity as

u)

1 F

∫t T F(x,τ)uj(x,τ) dτ

(10)

4.2.1. Locally Homogeneous Flow (LHF). The LHF model implies that both phases have the same velocity and local thermodynamic equilibrium was maintained at each point in the flow. The properties (density, temperature, species, and void fraction) at each point in the flow were determined from adiabatic thermodynamic equilibrium calculation. The Favreaveraged approximation was applied to the transport equations of continuity, momentum, and mixture fraction (f) and the mean flow was solved. In addition, transport equations for the turbulence kinetic energy (k), the rate of turbulent dissipation (ε), and square of the mixture fraction fluctuations (g) were solved and utilized to model the turbulent flow. Table 6 shows the transport equations of mean velocities, mean mixture fraction (f), turbulent kinetic energy (k), rate of dissipation of turbulent kinetic energy (ε) and square of mixture fraction fluctuations (g). The solution of transport equations for f and g gives the

∫01 φ(f)p(f) df

(11)

4.2.2. Two-Fluid Model (TFM). TFM models use separate transport equations for each phase and the interaction between the two phases is taken into consideration by interfacial drag and mass exchange terms. The velocity and void fraction fluctuations were considered, but pressure and density fluctuations were ignored. The two-fluid model is free from the restrictions of the turbulent viscosity concept. Furthermore, it can adequately portray the interactions of pressure gradients and density fluctuations, which are major sources of generation of turbulent motion. Two-fluid turbulence model allows proper account of the large differences and steep gradients of temperature and concentration, which are present within turbulent reacting gases. Thus, the predictions made from this model were closer to the experimental measurements.13 4.2.3. Multifluid Model (MFM). The MFM model considers each phase to be a separate fluid entity. If the phase has different species (such as droplets), then it can be broken down into several size groups and each size group can be treated as a “fluid”. The mass and momentum can be solved based on timeaveraged analysis or by density-averaged analysis. In a timeaveraged MFM model, mass and momentum equations for all liquids have been written in terms of time-averaged quantities while pressure and density fluctuation were ignored, whereas in the density-averaged MFM model, density fluctuations were considered while the mass and momentum equations were being developed.13 4.3. Modeling of Condensation and Reaction Mechanism. Estimation of the plume length from the CFD simulation was performed based on concentration profiles of the species. Thus, the CFD simulation must predict accurate concentration profiles of each species. The modeling of the condensation and reaction mechanism has a vital role in the prediction of concentration profiles. The modeling of these processes can be achieved using two methods: (1) an equilibrium state relationship with proper PDF function and (2) reaction kinetics, with heat- and masstransfer coefficients. 4.3.1. Equilibrium State Relationship. The relationship between the various properties such as density, velocity, temperature, and void fraction for individual species at equilibrium, based on the mixture fraction, is called an equilibrium state relationship. The oldest method for equilibrium computation uses the concept of equilibrium constants27-29 to express the abundance of certain chosen species in terms of the abundance of other arbitrarily chosen species. However, the method requires several distinct reactions that occur under certain given thermodynamic conditions and the associated equilibrium constants to start solving for the chemical composition of species in the reacting system. Alternatively, one may use the element-potential method30 for equilibrium computations. It has a similarity with the elimination method and, consequently, becomes extremely complicated and difficult to use when nonideality of various chemical species is included in the analysis. In the case of liquid-metal solutions, more than two solutes are dissolved into their solution. Thus, the mathematical model must include nonidealities, in terms of the activity

3202


equation, to describe the interaction and solubility effects among the solutes and the solvent. This nonideality problem, which exists in a real chemical reaction, has made both the equilibrium constant method and the element potential method unusable for general and efficient programming of chemical equilibrium calculations that involve complicated chemical and physical interactions among the species.13 For nonideal mixtures, the best alternative seems to be the direct free-energy minimization method.20 The equilibrium state of a system, with the reaction occurring at constant temperature and pressure, is the state in which the Gibbs potential is minimized. The equilibrium calculation based on the principle of minimization of the Gibbs free energy was subjected to the following constraints: (1) the conservation of elements and (2) balance of charges. The nonlinear equations of imperfect gases, solutes, and solvent of nonideal electrolytic as well as nonelectrolytic solutions were solved by a modified Newton-Raphson iteration scheme using partial molal (or molar) enthalpy, activity coefficient expression, thermodynamic data and chemical potential expressions.21 The instantaneous scalar thermodynamic properties at each point in the flow correspond to the state reached when injected and ambient fluid at their initial states, having proportions given by the mixture fraction, are adiabatically mixed and brought to thermodynamic equilibrium at the ambient pressure of the flow. Therefore, the scalar thermodynamic properties of the flow were determined, as a function of the mixture fraction (termed as state relationships), by simple adiabatic mixing calculations for given injector exit and ambient conditions and pressure. Chan13 had given the equilibrium state relationships for various liquidmetal fuel reaction systems at different mass fractions of oxidizer and various bath temperatures. The reaction pairs involve F2Li, SF6-Li, Cl2-Na, H2O-Li, HCl-aqueous NH3, and H2ONa. Several computer programs are available for the solution of general chemical equilibria. For ideal, multiphase systems, the CEC-72 code (reported by Gordon and Mcbride31) and BNR and VCS programs (reported by Smith and Missen32) are wellknown, whereas, for nonideal, multiphase systems, the SOLGASMIX program (reported by Ericksson33) can be used.13 4.3.2. Probability Density Function (PDF). PDF methods are statistical approaches for treating turbulent flows. They can be used as an approach to the closure or as an enhancement to them for treating reactive flows. The PDF allows treatment of many distinct configurations (velocities, temperature, density, etc.) contributing to the mean, although phase and subgrid localization information was not available. Thus, unlike most turbulence closure models, PDFs can include intermittency effects.34 The functional forms assumed for PDFs were obtained from physical consideration of the system. Thus, the PDF can have a delta function at their extremities, representing the pure liquid that exists on either side of the convoluted surface, and a continuum portion that may be flat, Gaussian, or some other shape in between. The researchers have solved the problems of turbulent diffusion flames and turbulent reactions using various shapes of PDFs with different limits. Hawthorne et al.35 studied the turbulent diffusion flame and proposed a Gaussian PDF for the fluctuating concentrations of jet fluid bounded by the artificial limits of -∞ and +∞. Richardson et al.36 proposed a beta function probability distribution for the concentration fluctuations between the limits of 0 and 1 without taking into account the time spent at these limits. Bilger and Kent37 have used a Gaussian PDF for atomic hydrogen, truncated at the artificial limit of -2 and +2, to make predictions of a turbulent hydrogen flame. Bush and Fendell38 proposed a triangular wave

form, which corresponds to a uniform PDF, bounded by delta functions at the limit of 0 and 1. Becker39 has used a Gaussian distribution, extending from -∞ to +∞, to examine the behavior of turbulent diffusion flames in the region of the flow where the significance of fictitious concentration values is negligible. Lockwood and Naguib18 proposed a Gaussian PDF for the mixture fraction. The distribution in the ranges of -∞ < f < 0 and 1 < f < +∞ were lumped into Dirac delta funtions at f ) 0 and f ) 1.

p(f) )

1 1f-µ2 [u(f) - u(f - 1)] + exp 2 σ σx2π Aδd(0) + Bδd(1) (12)

[ ( )]

A)

∫-∞0

B)

∫1∞

1 1f-µ2 df exp 2 σ σx2π

[ ( )] 1 1f-µ exp[- ( df 2 σ )] σx2π 2

(13) (14)

Abou-Ellail and Salem22 developed a skewed PDF reaction model for the diffusion flames. They developed a transport equation for the skewness of the mixture fraction. The computed values of the mean mixture fraction and their variance and skewness were used to obtain the shape of skewed PDF. The skewed PDF was split into a turbulent component (the beta function) and a nonturbulent component (the delta function) at f ) 0. They reported that the inclusion of intermittency in the skewed PDF appreciably improves the numerical predictions for temperature and velocities obtained for turbulent jet diffusion methane flame. The major drawback of the PDF proposed by Abou-Ellail and Salem22 was that it requires a large number of nonlinear equations and model constants to be handled to get the solution, which takes greater computation time for CFD simulations. 4.3.3. Rates of Heat and Mass Transfer. The equilibrium state function method with PDF predicts the values of concentration of various species at equilibrium accurately. However, the prediction of the flow pattern was not accurate.11,13,17-25 Also, the prediction of the equilibrium state relationship with the PDF requires a large number of nonlinear equations with various model constants to be solved simultaneously. This leads to a large number of assumptions while solving the equations iteratively. Thus, the methods overestimate the flow and temperature variations. The method of rate kinetics with appropriate heat- and mass-transfer coefficients defines the numerical solution closer to reality. Also, it reduces the large number of nonlinear equations and their model constants. Furthermore, the closure of mass and momentum equations brings accurate predictions of the species concentration and flow and temperature variations. For the case of condensation jets, the phase change between the species must be expressed in terms of the heat-transfer coefficient of two phases from the interface. The mass balance can be done by adding the generated source due to phase change, which is dependent on the heat-transfer coefficient across the gas/liquid interface for both of the phases. In the case of reaction jets, most of the reactions are instantaneous. Thus, the mass transfer is the controlling mechanism. The rate of reaction can be expressed in terms of the masstransfer coefficient. Furthermore, the mass transfer can be expressed in terms of local kinetic energy and dissipation energy values. Based on stoichiometry and mass-transfer coefficients, the rate of conversion of species can be easily determined. To predict the heat-transfer properties in the jet systems, two types

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3203 Table 7. Numerical Constants Used To Solve Governing Equations of the LHF Model author Faeth11

Chen and Chan13 LHF model two-fluid and multifluid model Chan et al.17 Lockwood and Naguib18 isothermal inert flow nonisothermal inert flow reacting flow Chen and Faeth19 Chan et al.20 Chan and Shen21 Abou-Ellail and Salem22 Abou-Ellail and Chan23 Chan and Chern24 Ouellette and Hill25 a du du ∆r 0.2 F) . dx dx 1.6∆u

|( | |) |

code used/grid size (H × V)

Cµ

C2

Cg1

Cg2

σk

se

σf

σg

Sc

GENEMIX (90 × 1000)

0.09

1.44

1.84

2.8

1.92

1.0

1.3

0.7

0.7

0.7

GENENIX (-) GENENIX (-)

0.09 0.09 0.09

1.44 1.44 1.44

1.84 1.84 1.84

2.8 2.8 2.8

1.84 1.84 1.84

1.0 1.0 1.0

1.3 1.3 1.3

0.7 1.1 0.7

0.7 0.7 0.7

0.7 0.7 0.7

GENENIX (-) GENENIX (-) GENENIX (-) GENEMIX (93 × 2500) GENEMIX (93 × 1000) GENEMIX (93 × 1000) GENEMIX (26 × 250) GENEMIX (40 × 600)

0.09 0.09 0.09 0.09 0.09 0.09 0.09-0.04Fa 0.09 0.09 0.09

1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.50, 1.52

0.170 0.166 0.161 1.84 1.84 1.84 1.92-0.0667Fa 1.84 1.84 1.92

2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8

0.17 0.166 0.161 1.84 1.84 1.84 2.0 1.84 1.84

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

0.7 0.7 0.7 0.7 0.7 0.7

KIVA II (30 × 100)

of heat-transfer coefficient are required: (1) from the gas phase to the bulk liquid and (2) from the bulk liquid to the solid wall. The heat-transfer coefficient values determine the amount of heat transferred to the bulk liquid and the amount of heat removed from the system. The temperature of the system is controlled by the heat-transfer coefficients. The condensation/ reaction processes are highly temperature-dependent; therefore, appropriate heat-transfer coefficient values are required for the correct prediction of flow, temperature, and species concentration. 4.4. Boundary Conditions. In the case of the LHF model, the authors assumed that all the properties of the gas were constant at the injector exit, except for a shear layer that had a thickness equal to 0.1% of the injector radius.11,13,17-24 In the case of the condensation jet (the steam-water jet system),11 the constant property portion of the flow was specified as

x ) 0;

C1

2r m j e 0.999; uj0 ) ; hf 0 ) 1; k0 ) (0.02uj0)2; d0 Fj0 0.07(uj0) ; g0 ) 0 (15) d 3

ε0 )

In the shear layer (x ) 0, 0.999 < 2r/d0 e 1), the velocity and the mixture fraction were assumed to vary linearly, whereas k0 and 0 were obtained by solving the governing equations, assuming that the convective and diffusive terms were negligible. The ambient values of uj, hf, k, , and g were all assumed to be zero. The boundary conditions along the inner edge of the shear layer were specified until the shear layer reached the jet axis; all radial gradients at the axis then were set equal to zero. In the case of reacting jets, (HCl-aqueous NH3, Cl2-molten Na),13,20,21 choked flow conditions at the nozzle exit were assumed. The density, velocity, and temperature at the nozzle exit were determined from the stagnation temperature (293.15 K) and pressure (0.377 MPa), assuming an adiabatic, isentropic nozzle. The injected gas was then assumed to underexpand isentropically to the ambient pressure as it left the nozzle exit. The length of this external expansion region was assumed to be much less than the total penetration length of the jet. The calculation domain was started at the end of the expansion region where the density, velocity, and jet radius were computed by performing mass and energy balances on the expansion region. A pressure of 1 atm was used throughout the calculation domain.

All the initial conditions were assumed to be uniform across the entire jet, as

x ) 0; uj0 ) 382.5 m/s; hf 0 ) 1; F0 ) 2.208 kg/m2; k0 ) (0.03uj0)2; ε0 )

0.03(uj0)3 ; g0 ) 0 (16) d

Because of symmetry, the radial gradients of uj, hf, k, ε, and g at the plume axis was set to zero. At the outer, free boundary, g ) 0 and very small values of uj and hf (uj ) 0.0005uj0 and hf ) 0.0005fh0) were assumed, whereas k and ε were estimated as shown in the equation at the first step and by solving the governing equations with the convective and diffusive terms neglected at later steps.

∆x uj0

kd ) k0 - 0 and

εd ) ε0 - Cε2ε02∆x(k0uj0)

(17)

4.5. Method of Solution. Most authors11,13,17-25 have used the LHF, TFM, and MFM models, along with the PDF function. The turbulence was solved using the k-ε-g model. GENEMIX computer code was used for the CFD simulations. The grid size, constants of the flow model, and turbulence parameters for various authors have been mentioned in Table 7. In the case of the condensation jet, flow was initialized using eq 15, whereas, in the case of reaction jets, eqs 16 and 17 were used to initialize the flow. The flow and turbulence were solved using flow and turbulence model equations. Finally, the equations of the mixture fraction and variance were solved. The maxima of the PDF function was obtained from the calculated values of the mixture fraction and variance. Using this value of the PDF function, the properties of each species were calculated from eq 11. The detailed flow algorithm has been given in Figure 10, which elaborates the various steps in the CFD simulation. 4.6. Analysis of CFD Models for Condensation and Reaction Jets. 4.6.1. Plume Dimensions. Chen and Faeth11 conducted a theoretical investigation of turbulent condensing vapor jets submerged in subcooled liquids. They developed a CFD model using an LHF approximation of the two-phase flow, in conjunction with a k-ε-g model of turbulence. GENEMIX computer code was used with a grid size of 90 × 1000 (H ×

3204



Figure 10. Flow algorithm for CFD simulation with flow models (LHF, TFM, and MFM) with the PDF approach.

V). The model was capable of treating near-injection phenomena as well as the dispersed flow, where a long penetration length was observed. The model was evaluated using the existing data for turbulent condensing water, ethylene glycol, and iso-octane

vapor jets with liquid-to-vapor density ratios in the range of 324-31 200 and negligible effects of buoyancy. For injector exit pressures equal to the bath pressure, they obtained the semiempirical correlation for the penetration length. However,

3206


the model does not incorporate the underexpansion of vapor jets, with the exit plane pressure greater than the ambient pressure. The model was also not sufficiently comprehensive for the case of oscillatory subsonic flow, where the injector passage periodically fills with liquid. Chan et al.17 investigated the complex flow structure of choked, turbulent, reacting gaseous fluorine jets submerged in molten lithium. The LHF model was used for the two-phase flow, while the k-ε-g model was used for the turbulence modeling. The Li-LiF phase diagram was constructed, which was further used to develop the equilibrium state relationship as a function of the mixture fraction, over a wide range of temperature. They reported that the reaction was completed within a short distance from the injector but the plume jets penetrate further downstream, because of lithium fuel evaporation in the hot-temperature plume and subsequent condensation of the evaporated lithium and reaction products. Finally, the lithium subcooling effect on the flow structure was examined and a simple correlation of the jet penetration length was presented. Chen and Faeth19 developed a model for the prediction of turbulent, multiphase, and diffusion-flame structures of a gaseous oxidizer jet submerged in a molten metal fuel. An axisymmetric flow was considered using an LHF approximation of multiphase flow in conjunction with the k-ε-g model of turbulence. The model was evaluated using the existing data for the vertical chlorine jet submerged in the dilute Na-NaCl molten bath at atmospheric pressure. They reported that the portion of the flow containing a gas had a finite penetration length, because of the condensable products of the reaction. The complex flow structure, which contained several solid and liquid phases, was observed because of the large temperature range of the process and miscibility properties of the reaction mixture. Simulations for the injection of chlorine into pure sodium showed that the reaction was completed near the injector. A large amount of heat generated due to reaction vaporizes the surrounding liquid bath. Because the vapors formed in the hightemperature portions of the flow recondenses far away from the nozzle exit, the gas phase persists for a considerable penetration length. Furthermore, the effect of bath pressure on overall penetration length was also studied. The model provides reasonably good predictions of void penetration of the chlorine jet. Chan et al.20 investigated the structure of turbulent, gaseous, reacting, dissolving hydrogen chloride jets submerged in an aqueous ammonia bath. They developed a computer code for the computation of chemical equilibrium of complex reacting, nonideal, multiphase, and electrolytic mixtures. An extensive equilibrium state relationship as a function of mixture fraction was generated for the aqueous ammonia solution over a wide range of ammonia concentrations. A simplified method, called the boiling diagram technique, was developed and used to construct the state function. The LHF model, together with k-ε-g turbulence model was used to describe the two-phase flow. The Favre-averaged parabolic forms of the transport equations for continuity, momentum, and mixture fraction of mean flow were solved. A clipped Gaussian probability function for the square of the mixture fraction was used for the predictions of plume penetration length, temperature, void fraction, and concentration profiles of the products. The numerical computations were performed using a modified version of GENEMIX code with a nonuniform grid of 93 × 1000 (H × V). The predictions of penetration length, temperature, and void fraction were determined to be in good

Figure 11. Variation in plume length due to the addition of a radiation term with the turbulence equation reported by Chan:13 (1) with a radiation term and (2) without a radiation term.

agreement with the experimental data of Cho et al.1 They concluded that the penetration length was controlled by the extent of vaporization of the bath liquid, which was largely determined by the concentration of ammonia, resulting in the increase in the plume lengths with the concentration of ammonia solution. Chan and Shen21 extended the simulation studies by Chan et 17 al. for the dissolving hydrogen chloride jet submerged in an aqueous ammonia bath. They reported that the predictions of plume length from simulations show good agreement with the wide range of experimental conditions with varying ammonia concentration (10.6%, 12%, 19%, and 25.6%), bath temperature (14, 20, 25, 45, and 50 °C), and bath pressure (50.5, 84.4, and 101.3 kPa). They concluded that, with either increases in the fuel temperature or reductions in the ambient pressure, the total jet penetration length increases. Chan and Chern24 proposed a method to consider the turbulence-radiation interaction in the analysis of a multiphase diffusion flame. The study was conducted on a chlorine vapor jet submerged in molten NaCl bath. The radiation model compatible with mixture fraction fluctuation was incorporated in the LHF model, with k-ε-g as a turbulence model and chemical equilibrium with the PDF approach. The heat removed by radiation was incorporated as a source term in the energy equation and was solved simultaneously with Favre-averaged transport equations. The computation was performed for adiabatic conditions, as well as with turbulence/radiation interaction. They reported that, by coupling the radiation losses with turbulence, the plume length reduces by 18%, relative to that of the adiabatic condition (Figure 11). Chan13 has conducted an extensive review of the modeling approach for the various liquid-metal-fuel combustion system, viz., F2-Li, SF6-Li, Cl2-Na, H2O-Li, HCl-aqueous NH3, and H2O-Na. To predict the flow structure of submerged reacting jets, the LHF, TFM, and MFM models were proposed. Furthermore, the detailed equilibrium state relationships (pertinent to liquid-metal combustion) were reported, which were calculated by the latest version of the Chemical Equilibrium Calculation of Nonideal Multiphase System (CEC-NMS) computer code.13 Chan13 reported that the plume penetration length increased in the following order:

H2O-Na > Cl2-Na > SF6-Li > F2-Li > H2O-Li Furthermore, to overcome the drawbacks of the LHF model, the reacting TFM model, the reacting time-averaged MFM model, and the reacting density-averaged MFM models were discussed for the air-n-pentane, kerosene-air, and SF6-Li


Figure 12. Comparison between the predictions of the plume length using TFM and MFM models for SF6 injected into lithium, as reported by Chan: 13 (1) TFM model and (2) MFM model.

systems, respectively. The comparison was made for the predictions of void fraction profiles for the air-n-pentane system with the LHF and reacting TFM models. Chan13 also compared the predictions of void fraction for the system of SF6 injected into a molten lithium bath by LHF approximation and MFM, using the experimental data of Parnell et al.41 Figure 12 shows that the predicted plume length determined using the timeaveraged MFM model is longer than the LHF approximation. Gulawani et al.14 investigated DCSC phenomenon using the thermal phase-change model. The experimental data of Kim et al.2 and Kerney et al.7 were simulated for the prediction of plume dimensions and temperature profiles. Two prominent shapes of plume were observed: conical and elliptical. A conical shape was observed at smaller mass flux and lower pool temperatures with large nozzle diameters, whereas the elliptical shape of jet was observed at a relatively higher range of mass flux and pool temperatures with lower nozzle diameters. The plume length was measured by considering only the steam region and steam, as well as some part of the two-phase regions. For the identification of plume length, two criteria were developed, based on the hold-up profile of steam. In the first criterion, for the steam-only region, the volume fraction of steam was assumed to be 99%, whereas in the second criterion, a point was identified where there was a sharp change in the hold-up profile of steam, which was assumed to be the complete plume length. Here, the volume fraction of steam varies from 92.55% to 95.5%. Thus, the plume length consists of pure steam as well as a fraction of the condensation region. Figure 13 shows that good agreement prevails between the CFD predictions and the experimental data of Kim et al.2 and Kerney et al.7 4.6.2. Centerline Axial Velocity. In submerged reaction jets, as a gaseous oxidant is injected into a liquid fuel, the flow regime begins with a single-phase gaseous flow pattern at the nozzle exit. As the liquid fuel is entrained to react with the oxidant, a droplet flow pattern may evolve. Further downstream, as the mixture temperature decreases, a droplet flow or bubbly flow pattern may appear, because of condensation of the vapor. Eventually, it becomes a single fluid at the tip of the plume.13 Because of the complexity of undergoing several changes in the flow pattern, attempts have been made to understand the flow dynamics using CFD with various flow models. LHF approximation along with PDF was applied for most of the

Figure 13. Parity plot for condensation jets for CFD simulations using the thermal phase-change model devised by Gulawani et al.:14 ([) from Kerney et al.,7 flat, d0 ) 0.0004 m; (0) from Kerney et al.,7 flat, d0 ) 0.0007 m; (2) from Kerney et al.,7 flat, d0 ) 0.00635 m; (/) from Kerney et al.,7 flat, d0 ) 0.0095 m; (+) from Kim et al.,2 G0 ) 600 kg/s; (O) from Kim et al.,2 G0 ) 1045 kg/s; and (b) Chun et al.4

Figure 14. Variation in the prediction of the centerline axial velocity using the LHF model for various reaction jets with a gas temperature of 298 K and a liquid bath temperature of 1130 K (reported by Chan13): (1) F2-Li, (2) SF6-Li, (3) Cl2-Na, (4) H2O-Li, and (5) H2O-Na.

condensation/reaction flow. The approach was purely mechanical, wherein equilibrium state relationships between the properties of various species at different mixture fractions were developed along with the PDF function to solve the flow. Thus, the LHF model treats a two-phase flow like a single fluid flow with the mixture properties. Chan13 studied various reaction systems (F2-Li, SF6-Li, Cl2-Na, H2O-Li, and H2O-Na) using LHF approximation. Figure 14 shows the comparison of the prediction of centerline axial velocity profiles for all the systems with an inlet gas temperature of 298 K and a bath temperature of 1157 K. It can be observed that, for the same operating conditions, the profiles of axial velocity are almost identical. A slight difference is attributed to the different effective viscosity (µeff,φ) in the equations for f and uj. Furthermore, they have neglected the buoyancy effect in the source term of the momentum equation. The LHF model also does not accommodate for the effects of bubble/droplet size, slip velocity between phases, interfacial momentum, or energy and mass exchange between the phases. To overcome these draw-

3208


Figure 15. Comparison between the predictions of the mixture fraction and axial velocity using the LHF and TFM models (reported by Chan13): (1) hf, determined by LHF; (2) U/U0, determined by LHF; (3) hf determined by TFM; and (4) U/U0, determined by TFM.

Figure 16. Variation in the prediction of the centerline axial temperature using the LHF model for various reaction jets with a gas temperature of 298 K and a liquid bath temperature of 1130 K (reported by Chan13): (1) F2-Li, (2) SF6-Li, (3) Cl2-Na, (4) H2O-Li, and (5) H2O-Na.

backs, a reacting TFM model was developed.23 Figure 15 shows the variation in the prediction of the mean quantities of mixture fraction f and axial velocity. It can be clearly observed that the estimation of axial flow velocities were higher at low mixture fractions. Thus, the LHF model overestimates the flow velocities because of simplifying assumptions in the governing equations of the model. 4.6.3. Temperature Field. CFD analysis was conducted using various flow models with the PDF approach. Here, the equilibrium values were calculated based on the thermodynamic data under adiabatic conditions. Furthermore, the heat transfer was solved using enthalpy equations as

Chan13 studied the temperature variation in the various reaction systems (F2-Li, SF6-Li, Cl2-Na, H2O-Li, and H2ONa) using the LHF model with PDF. Figure 16 shows the temperature predictions for these reactive systems. The reaction of F2 gas with molten lithium liberates the highest heat of reaction (23 594 kJ/kg). Thus, the maximum centerline axial temperature of 4500 K was observed. However, the dissolution of water vapors into the molten sodium liberates only 9158 kJ/ kg. The centerline temperature was reported as 1500 K in this case. The centerline axial temperature decreases from 4500 K down to 1500 K as follows:

Hp(T,p,N1)p ) Hr(T,p,N1)r + ∆H

(18)

where Hr is the total enthalpy of the reactants, Hp the total enthalpy of the equilibrium products, and ∆H the specified enthalpy change. In all the cases, the enthalpy changes were due to heat transfer; however, for the adiabatic case, ∆H )0. The total enthalpy of a mixture (whether it be a mixture of products or a mixture of reactants) was calculated from n

H)

hiNi ∑ i)1

(19)

where hi is the partial molal (or molar) enthalpy of species i. The partial molar enthalpies of the species were calculated as

hi ) RT2

( ) ∂ ln γi ∂T

(20)

whereas the standard molar enthalpy (h/i ) of species i can be calculated from the specific heat data as

() () () () ()

a2 a3 2 a4 3 a5 4 a6 h/i (T) ) a11 + T+ T + T + T + RT 2 3 4 5 T (21) For the measurement of standard molar enthalpy, CEC data files were used. The two sets of coefficients (a11, a2, a3, a4, a5, and a6) were given for the two adjacent temperature intervals, namely, 300-1000 K and 1000-5000 K.13

F2-Li > SF6-Li > Cl2-Na > H2O-Li > H2O-Na Furthermore, Chan13 investigated the flow and temperature variations for a n-pentane spray flame with an air atomizing injector, using the TFM model with PDF. The predicted axial variation (Figure 17A) and radial variation (Figure 17B) of the mean centerline gas temperature were depicted in Figure 17, along with the corresponding experimental data and the predictions from LHF approximation. Figure 17 shows that the predictions from the TFM approach are more accurate than the LHF approximation. Furthermore, it can be seen that the radial spread of the temperature profiles was greater in the case of LHF predictions than that for the TFM model. This can be attributed to the proper modeling of transport equations of f and g in the TFM approach. The predictions from the TFM model further encouraged the researchers to develop the reacting MFM model. Chan13 compared the predictions of time-averaged axial temperature for the system of SF6 injected into a molten lithiun bath by LHF approximation and the reacting time- and density-averaged multifluid model, using the data of Parnell et al.41 Figure 18 shows the variation in the prediction of plume temperature by LHF approximation and the time-averaged MFM model. Although the predictions of centerline temperature profiles for the submerged reactive jets using the flow model (TFM and MFM) with a PDF approach were reasonably good, the enthalpy calculation (eq 21) was done using the model coefficients available in CEC. Most of the model coefficient data were given only up to 1000 K or up to the melting point of the solid species. As a result, the equilibrium calculation must be performed to avoid the extrapolation of coefficients of the


describes the phase change induced by the interphase heat transfer of intrinsic flows. The model was only applicable to a change of phase in pure substances and considers the heattransfer processes on each side of the phase interface. Hence, the two-resistance model for interphase heat transfer was used. The sensible heat flux was given by

qL ) hL(Ts - TL) (from the interface to the liquid phase) (22) qG ) hG(Ts - TG) (from the interface to the gas phase) (23) where hL and hG are the heat-transfer coefficients for the fluid and gas phases, respectively. The heat-transfer coefficient across the interface and gas phase (hG) was modeled using the zero equation model. Furthermore, the model assumes that the latent heat of the steam was completely transferred across the interface. Thus, the heat transfer was calculated by simple heat balance. The heat-transfer coefficient (hL) across the interface and liquid phase was modeled using the Ranz Marshal model (eq 24):

Nu ) 2 + 0.6Re0.6Pr0.3

Figure 17. Comparison of predictions from the LHF and TFM models for a n-pentane spray flame (reported by Chan13): (A) axial mean temperature profiles and (B) radial mean gas temperature. Legend for panel A: (O) experimental data, (curve 1) LHF, and (curve 2) TFM. Legend for panel B: (O) experimental data for L/d0 ) 74.5, (curve 1) calculated LHF curve for L/d0 ) 74.5, (curve 2) calculated TFM for L/d0 ) 74.5; (0) experimental data for L/d0 ) 170, (curve 3) calculated LHF curve for L/d0 ) 170, (curve 4) calculated TFM curve for L/d0 ) 170; (4) experimental data for L/d0 ) 340, (curve 5) calculated LHF curve for L/d0 ) 340, (curve 6) calculated TFM curve for L/d0 ) 340; and (4) experimental data for L/d0 ) 510, (curve 7) calculated LHF curve for L/d0 ) 510, (curve 8) calculated TFM curve for L/d0 ) 510.

Ts is the interfacial temperature determined from considerations of thermodynamic equilibrium, and it was assumed to be the same (the saturation temperature) for both phases. The interphase mass transfer was determined from the total heat balance. The total heat flux balance was given by eqs 25 and 26, respectively.

QG ) qG - m˘ GLHGs (from the interface to the gas phase) (25) QL ) qL + m˘ GLHLs (from the interface to the liquid phase) (26) Here, m˘ GL denotes the mass flux into the liquid phase from the gas phase and HGs and HLs respectively represent the interfacial values of enthalpy carried into and out of the phases due to phase change. The total heat balance determines the interphase mass flux (eq 27) as follows:

m˘ GL )

Figure 18. Comparison of predictions for temperature from the LHF model (curve 1) and the MFM model (curve 2) for the SF6-Li system (as reported by Chan13).

species. This brings uncertainty into the calculation of the temperature profiles. Recently, Gulawani et al.14 investigated the phenomenon of DCSC using a phase-change model. The thermal phase-change model was applied across the gas/liquid interface, which

(24)

qL + q G HGs - HLs

(27)

Thus, the model considers the interface mass transfer based on the heat-transfer coefficient. The validation of the model was done by predicting axial and radial temperature profiles for the experimental data of Kim et al.2 4.7. Conclusions. Most authors have modeled the condensation/reaction jets using LHF, TFM, and MFM models with the approach of an equilibrium state relationship and PDF formulation. The Gibbs free-energy minimization approach was used to develop an equilibrium state relationship. The approach requires chemical potential and activity coefficient models for the species in miscible and immiscible solutions. This formulation leads to a set of nonlinear equations, which were solved iteratively. The solution of these nonlinear equations encounters the problem of divergence in the iterative procedure. Thus, a new solution technique based on the simplex method was developed to enhance the convergence.13 Some authors13,24 have reported that the plume dimension predictions, using the LHF model with the PDF approach and the k-ε-g turbulence model, were good for both condensation and reaction jets. Although the LHF model predicts the plume dimension accurately, the axial centerline velocity and temperature were overestimated.13

3210


Figure 19. Schematic diagram for a reacting gas jet submerged in liquid (as reported by Chawla44).

To overcome the aforementioned drawback, the TFM and MFM models were developed. Comparison of the axial velocity and temperature using the TFM and MFM models with the LHF model reveals that the TFM and MFM models are better predictors than the LHF models. Although the phenomena of a submerged combustion jet are addressed efficiently using reaction models (LHF, TFM, and MFM) with PDF formulation, the prediction of the correct equilibrium state relationship for the species under specific operating conditions is a major drawback. The experimental determination of the mass-averaged quantities is one of the difficulties, which makes the PDF approach difficult to calibrate the condensation/reaction models. Recently, Gulawani et al.14 investigated the DCSC phenomenon using a thermal phase-change model. The model also considers buoyancy effects on the flow, along with mass and momentum exchange between the phases. The predictions of the plume dimensions, and the axial and radial temperatures, were in good agreement with the experimental data of Kim et al.2 and Kerney et al.7 The uncertainty in developing an equilibrium state function and a PDF formulation was reduced and a new model was developed based on the heat-transfer characteristics. This model is highly recommended for the condensation jets. To predict the heat-transfer coefficient, a generalized correlation was proposed for the condensation jets based on the nozzle diameter criteria:

( ) ( )

( ) ( )

G0 h ) 1.12 CpG0 Gm

0.31

G0 h ) 1.54 CpG0 Gm

0.12

(B)0.06 (B)0.04

Criteria: d0 < 2 mm (28) Criteria: d0 > 6 mm (29)

In the case of submerged reaction jets, the reaction is instantaneous. The rate of the reaction is controlled by the mass transfer across the gas/liquid interface. Thus, the gas/liquid interface has a major role in controlling the reaction, because the mass- and heat-transfer coefficients at the interface decide the overall system behavior. Furthermore, the turbulence generated in the reactor dominates the behavior of the gas/liquid interface, which decides the rate of reaction. Based on the local values of kinetic energy (k) and dissipation energy (ε), the rate of reaction can be estimated. This new approach of modeling reaction flow will reduce the formulation of equilibrium state relationship and PDF. The exact mass-transfer coefficient, stiochiometry, and proper turbulence modeling ensures the correct predictions of the plume dimension, velocity, and temperature in the system. Because such a type of modeling

approach for the submerged reaction jets brings better understanding of flow, along with the proper physics, it is highly recommended. 5. Jet Stability Analysis In the case of reactive flows, the local energy release causes a highly nonlinear feedback process between the reacting species and fluid dynamics. A change in temperature can increase the viscosity, which then reduces the Reynolds number and changes the dissipation scale. Turbulence alters the mixing and reaction times and the heat- and mass-transfer rates, which, in turn, modify the local and global dynamic properties of the system. The energy release in turbulent flame also causes the fluid to expand, which can generate appreciable vorticity and trigger fluid instabilities, which further complicates the flow. Several flow instabilities can contribute to the onset of turbulence. The two instabilities commonly discussed in the evolution of turbulence in variable density flows are the Rayleigh-Taylor and the Kelvin-Helmholtz instabilities. The Rayleigh-Taylor instability is caused by the acceleration of a heavy fluid through a light fluid; in this instability, vortex pairs of opposite sign are generated. The Kelvin-Helmholtz instability occurs at the interface between flows of different velocities. For the case of reactive jets with HCl gas injected into the aqueous NH3 solution, most authors (Chawla42-44 and Chan et al.45) have performed the Kelvin-Helmholtz instability analysis. 5.1. Kelvin-Helmholtz Instability Analysis. Chawla42-44 and Chan et al.45 have investigated the instability of the gas/ liquid interface due to injection of a gas into the liquid pool using Kelvin-Helmholtz instability analysis. Depending on the reacting and nonreacting systems, the appropriate terms were incorporated in the perturbation equation. The change in perturbation potential was expressed as a function of length of jet, time, gas velocity, rate of evaporation, and wavelength of disturbance. The resulting equation was solved to obtain the stability criterion. Chawla44 investigated the stability of the gas/liquid interface of a sonic gas jet submerged in an infinite mass of liquid under the action of a pressure perturbation, liquid viscosity, and surface tension (see Figure 18). The nonlinear wave equation for the gas jet was subjected to kinematic boundary conditions at the gas/liquid interface. Furthermore, the nonlinear equations were literalized by addition of a perturbation equation and the stability criteria that were applicable for different flows (such as subsonic, sonic and supersonic) were determined. Figure 19 shows the gas jet issuing from a jet orifice of radius r. The flow at the


orifice was assumed to be uniform, with the gas jet having a constant mean radius equal to the orifice radius. The expansion of the mean jet boundary was not considered, because short wave approximation had been applied to solve the nonlinear equations, which reduces the axisymmetric two-dimensional gas-liquid system to a two-dimensional planar configuration. It was further assumed that the thin shear layer at the jet boundary does not significantly alter the phase of the gas pressure perturbation, with respect to the disturbance wave; therefore, it was neglected. The various flow conditions, nozzle positions, and assumptions made in developing the stability analysis by various authors has been summarized in Table 8. The hydrodynamic equations that govern gas motion in axisymmetric coordinate systems are given by

[

]

∂F 1 ∂(rVr) ∂Vx ∂F ∂F +F + + V x + Vr ) 0 ∂t r ∂r ∂x ∂x ∂r

(30)

∂Vx ∂Vx ∂Vx 1 ∂P + Vx + Vr )∂t ∂x ∂r F ∂x

(31)

∂Vr ∂Vr ∂Vr 1 ∂P + Vx + Vr )∂t ∂x ∂r F ∂r

(32)

The sound speed and the velocity potential equations are given as follows:

Vx )

( )

φ , 1, φ

δλ , 1, φM02 , 1, φM02ω* , 1, a φM0δλ 2 , 1 (40) a

(34)

The nonlinear wave equation for the gas jet can be written by substituting eqs 33 and 34 in eqs 30, 31, and 32:

( )( ) ( )( ) (

)

The linearized equations of the motion of the liquid surrounding the gas jet, consistent with the short wave approximation, are given as

∂u ∂V + )0 ∂x ∂r

[ [

(41)

] ]

∂2u ∂2u 1 ∂P ∂u )+ν 2+ 2 ∂t F ∂x ∂x ∂r

(42)

∂2V ∂2V ∂V 1 ∂P )+ν 2+ 2 ∂t F ∂r ∂x ∂r

(43)

V)

∂η ∂t

(44)

The dynamic boundary conditions for (i) the continuity of tangential stresses at the interface is given as

(∂u∂r + ∂V∂x) ) 0

µ

(33)

∂φg ∂φg + Ug and Vr ) ∂x ∂r

(

The linearized kinetic boundary condition that satisfies the aforementioned equation is given as

and

dP ) cs2 dF

pressure be small. Thus, the following conditions were obtained:

(45)

and for (ii) the continuity of normal stresses at the interface is given as

∂ 2η ∂V σ -Pl + 2µ + Pg ) - σ 2 ∂r a ∂x

(46)

∂φg ∂2φg ∂φg 2 ∂2φg ∂φg ∂2φg + 2 + + 2 + ∂x ∂x ∂t ∂r ∂r ∂t ∂x ∂t2 ∂x2 ∂φg ∂φg ∂2φg ∂φg 2 ∂2φg ∂2φg ∂2φg 1 ∂φg 2 2 ) cs + 2 + + ∂x ∂r ∂x ∂z ∂r r ∂r ∂r2 ∂x2 ∂r (35)

The aforementioned equations of motion for the gas phase, as well as for the liquid phase, were solved using a stream function and the gas and liquid pressure equations at the gas/ liquid interface were obtained. Using the pressure terms on the gas and liquid side, the solution for the dispersion equation was also obtained, by taking into account the low- and high-viscosity liquids. The low-viscosity fluid was defined as one for which the following condition holds:

Without considering the expansion or contraction of the gas boundary, because of the vaporization or condensation between the gas jet and the surrounding reactive liquid, the kinetic boundary condition at the gas/liquid interface was given by

|R| . νκ2

∂2φg

( )

∂φg ∂η ∂η ∂η dx ) + ) ∂r i ∂t ∂t ∂x dt

)

(36)

Linearization of the equations of motion for the gas phase had been done using the following dimensionless variables:

( )

tUg (a - r)δ x x* ) , y* ) , t* ) w* λ a λ

(37)

φg(r,x,t) ) Ugx + φUgλφ(r*,x*,t*)

(38)

η(x,t) ) aη*(x*,t*)

(39)

where φ is the dimensionless perturbation potential and η is the dimensionless displacement. The theory of small perturbation requires that the deviations of the velocity components and

(47)

whereas, for high-viscosity fluid, the condition was as follows:

|R| , νκ2

(48)

R ) Rr - iRi

(49)

Here, R is defined as

where Rr is the time amplification factor and Ri is the angular frequency factor. It has been shown that the condition of linearization takes the form of the Weber numbersWe . 26(F/Fg)0.2 for the case of low-viscosity liquids and We . 36(µUg/σ) for high-viscosity liquidsssurrounding the gas jet. Chawla44 concluded that the instability of the gas/liquid interface of a gas jet submerged in a liquid was predominantly governed by the transfer of energy from the gas phase to the liquid layer, both through “wavedrag” and “lift” components of the pressure perturbation. He

3212


Table 8. Literature Survey for Stability Analysisa for the Reacting Jet (HCl-Aqueous NH3) System author/ system Chawla43

Chawla44

Chan et al.45

position of nozzle

flow condition

assumptionsb

remarks

vertical upward

sonic or transonic a, b, c, d, e, f, g, h, i (1) Described the phenomenon of liquid entrainment due to instability at the gas-liquid interface of a sonic gas jet submerged in liquid bath. (2) The initial non-entrainment length of jet was calculated as L0 ) (C0Ug/Rrm)[CL + ln(λm/a)iωj] vertical sonic or transonic a, b, c, d, e, f, g, h, i (1) The disturbance on the gas-liquid interface due to unsteadiness introduced upward into the flow by oscillation of the gas-liquid interface due to the presence of Kelvin-Helmholtz instability was significantly large. (2) The wave equation for submerged sonic gas jet into liquid bath with a disturbed gas-liquid interface had been linearized. (3) The condition of linearization for subsonic and supersonic jet was not as stringent as for the sonic jet. vertical sonic a, b, c, d, e (1) Analyzed the effect of mass transfer on the Kelvin-Helmholtz instability of the downward gas-liquid interface for submerged gas jet into liquid bath. (2) Developed a correlation for a plume length based on a dimensionless amplification factor, angular frequency and wave length corresponding to the maximum instability of the interfacial wave. (L0Rrm/Ug) ) -106 ln[2.282(λm/a)0.1533ω0.0761]

a All analyses were numerical analyses performed using Kelvin-Helmholtz instability. b Assumptions are coded as follows: (a) Gas jet with constant mean radius equal to the orifice radius. (b) Two-dimensional axisymmetric jet was reduced to two-dimensional planner jets by assuming gas is an ideal frictionless fluid. (c) Shear layer does not significantly alter the phase of the gas pressure perturbation, with respect to the disturbance wave. (d) Gas flow as an isentropic flow, and liquid flow field with potential flow theory was considered. (e) Suction and blowing was simulated by condensation and evaporation processes at the interface. (f) Thin shear layer approximation was applied for a steady, axisymmetric turbulent jet in an infinite stagnant environment having uniform properties. (g) Kinetic energy and viscous dissipation were neglected. (h) LHF approximation. (i) Exchange coefficient of mass and heat were identical. (j) Flows were dominated by momentum where the effects of flow stratification and buoyancy were small.

showed that, for the low-viscosity liquids, the phase angle between the pressure perturbation exerted by the gas phase on the liquid at the gas/liquid interface and the wave amplitude, which was the measure of the relative effectiveness of the lift and wave-drag components of the pressure perturbation, was a function of the density ratio (the ratio of gas density under throat conditions to the liquid density). At low density ratios, both of these components were operative; however, at high density ratios, the wave-drag component becomes dominant. The analysis further showed that the cut-off wave number and the wave number at maximum instability decrease as the density ratio increases. For highly viscous liquids and liquids that have a finite viscosity, the pressure perturbation was always out of phase with the wave amplitude, and no cut-off wave number exists; i.e., the gas/liquid interface was always unstable, despite the stabilizing effect of viscosity and surface tension. He also compared the sonic gas jet with the subsonic and supersonic jets and concluded that the conditions of linearization for these jets were not as stringent as those for the sonic jet. When in the transient motion, because of a disturbance at the gas/liquid interface that had a wave velocity much smaller than the gas velocity, the transient terms, in comparison to perturbation terms of basic flow, were of second order. In contrast to the supersonic and subsonic jets, the transient motion of the gas cannot be neglected for a sonic gas jet, even at very slow oscillations of the gas/liquid interface. Chawla43 developed a model that described the phenomenon of liquid entrainment resulting from the presence of KelvinHelmholtz instability at the gas/liquid interface of a sonic gas jet submerged in a mass of liquid. The initial non-entrainment length of jet was given as

[

() ]

C0Ug λm j L0 ) C + ln ω Rrm L a

The rate of liquid entrainment was given as

() ( ) ( )

Fga λm m n L - L0 RrmL0 Q/L ) CL1 ω exp FgCd a L0 C0Ug

The model has been validated with the experimental data presented by Bell et al.46 and Kennedy and Collier.47 Chan et al.45 analyzed the instability phenomenon of nonreacting and reacting stratified gas flows injected sonically into a liquid, using the theory of Kelvin-Helmholtz instability. They investigated the effect of mass transfer at the gas/liquid interface on the instability. According to them, the mass suction and blowing at the gas/liquid interface was due to condensation and evaporation processes. For nonreacting evaporating or condensing fluids, a blowing velocity (see V1 in Figure 20) into the gas side, relative to the interface velocity (-V1 ) Vg,r - Vi,r), was introduced such that eq 36 becomes

Vg,r )

( )

∂φg ∂η ∂η ∂x ) -V1 + + ∂r i ∂t ∂x ∂t

(50)

and, thus, the effective length of the jet over which liquid entrainment occurs is given by L - L0, where L is the actual exposed jet length.

(52)

where a positive value of V1 represents evaporation and a negative value means condensation. For linearization of the equation of motion for the gas phase, they introduced the term (aV1 ln r), which contributes to the evaporation at the gas/liquid interface. Thus, the dimensionless perturbation potential (eq 38) takes the following form:

φg(r,x,t) ) Ugx - aV1 ln r + φUgλφ(r*,x*,t*)

(53)

Also, the liberalized kinematic boundary condition at the gas/ liquid interface (eq 44) changes to

V)-

i

(51)

Fvap ∂η V + Ff 1 ∂t

(54)

and the dynamic boundary condition (eq 46) changes to

∂Vf,r σ ∂2η + Pg ) - σ 2 ∂r a ∂x (55)

m˘ vap(Vg - Vf) + (Pg - Pf) + 2µ


Figure 20. Schematic diagram for a reacting gas jet submerged in liquid (as reported by Chan et al.45).

Furthermore, the values of Pg and Pf were obtained from the gas-phase and liquid-phase equations and the dispersion equation was obtained from eq 55.

R2(κ2 + l2) κ -l 2

2

( ) ( ) ()

+ 2υRκ2

Fgκ κ2 + l2 +i (iκUg + R)2 + 2 2 FKc κ -l Fg σκ3 (56) V1κ(iκUg + R) ) F F

The roots of eq 56 were separated in real and imaginary parts. Applying low-viscosity approximation to these roots, a linear solution was obtained, in terms of the dispersion parameters P1 and P2. Furthermore, these parameters were expressed in terms of the dimensionless amplification factor (R), the dimensionless angular frequency (R/r ), and the dimensionless wavelength form (κ*):

Rr(µσ) Ri(µσ) κσ3/2 ; R/r ) ; R/i ) 1/2 5/2 5/2 µ FgUg FgUg FgUg5/2 1/2

κ* )

1/2

and

FgUg5/2σ-5/2µ5/2 FgUgσ-2µ2V1 ; P2 ) P1 ) F F

(57)

The instability occurs when the amplification factor is greater than zero. The effect of P1 and P2 on the dimensionless amplification factor R, dimensionless angular frequency R/r , and the dimensionless wavelength κ* has been shown in Figures 21A, 21B, and 21C, respectively, which shows that, at P2 ) 0, Chan et al.45 could reproduce the results presented by Chawla43,44 for the case without mass transfer at the gas/liquid interface. For P2 > 0, blowing at the interface (evaporation) increases the mean axial velocity, enhancing the time amplification factor of the disturbance, which leads to an increase in the instability of the interface, whereas, for P2 < 0, suction at the interface (condensation) reduces the instability. Furthermore,

they developed a correlation for the prediction of the break-off plume length observed in the submerged reacting jets and correlated with the experimental data of Cho et al.1

[ ()

L0Rrm λm ) -106 ln 2.282 Ug a

0.1533

ω0.0761

]

(58)

5.2. Conclusions. In the case of reactive jets, the localized heat release can cause a strong, transient expansion of gases. The resulting low-density region has a gradient, and when these gradients interact with the existing pressure gradient, they passively generate vorticity on the characteristics scales. Vorticity on these scales results in efficient mixing in the bath. Kelvin-Helmoltz instability analysis has been applied by various authors to study the gas/liquid instabilities. The analysis shows that the pressure perturbation due to the gas phase affects the liquid layer at the gas/liquid interface, both through a wavedrag component and a lift component, acting against the forces due to surface tension and liquid velocity. If the destabilizing effect of the former forces exceeds the stabilizing effect of the latter forces, the gas/liquid interface becomes unstable and the disturbance will grow over time until the amplitude has grown large enough for liquid to be torn off at those locations where it protrudes sufficiently far into the gas jet. For low-viscosity liquids (eq 47), the wave drag component becomes very small, in comparison to the lift component. For high-viscosity liquids (eq 48), the wave drag is always effective in the transfer of energy to the liquid layer, despite the stabilizing effect of surface tension and viscosity. Furthermore, in the case of reactive jets, the mass transfer at the gas/liquid interface contributes to the interfacial instability. Chan et al.45 applied the theory of KelvinHelmoltz instability analysis for such systems. They showed that the mass transfer affects the pressure perturbation, which acts to transfer energy from the gas phase to the liquid layer through their evaporation and condensation behavior, against the forces due to surface tension and liquid viscosity. The dimensionless wave frequency, amplification, and wavelength

3214


Figure 21. Effect of the dispersion parameter on (A) dimensionless amplification, (B) dimensionless frequency, and (C) dimensionless wavelength factor; each is plotted versus 1/P1 at different values of diffusion parameter P2 (as reported by Chan et al.45). Legend for all panels: (curve 1) P2 ) 0.005, (curve 2) P2 ) 0, and (curve 3) P2 ) -0.005.

at the maximum instability are presented as a function of the dimensionless surface tension/viscous parameter and a blowing parameter due to the interfacial mass transfer. The interfacial evaporation was observed to enhance the instability, whereas the interfacial condensation was determined to reduce the instability. Although the effect of pressure perturbation, surface tension, and viscosity on the gas/liquid interface has been studied using Kelvin-Helmoltz analysis, transient acceleration in the surrounding liquid due to expansion must be investigated, depending on the density variation in the liquid. This can be understood using Rayleigh-Taylor instability analysis. 6. Conclusions (1) In the case of condensation and reaction jets, the plume shape and size have been measured by analyzing high-speed photography images.1-9 Three plume shapessnamely, conical, ellipsoidal, and divergentshave been observed, based on the mass flux of gas and bath temperature.2 Plume length was measured based on two criteria: (a) the region up to which only gas vapors are present4,7 and (b) the region that contains steam vapor as well as that surrounding a two-phase mixture (a region of condensation and the evaporation zone of a bath mixture).2,8 In the case of reaction jets, the high heat of reaction leads to condensation and evaporation of the liquid bath, which results in a larger plume length than that of the condensation jets.

Furthermore, semiempirical equations have been developed based on the driving potential (B), the density ratio (F0/F∞), and the mass flux ratio (G0/Gm) for the prediction of plume length.1-9 Although attempts have been made to measure plume dimensions by analyzing the photographic images, a unique criterion must be developed for image analysis. (2) Computational fluid dynamics (CFD) analysis has been performed for the predictions of flow pattern and temperature profiles in the liquid bath.11-25 Various combustion models, such as the local homogeneous flow (LHF),11,13,17-22,24,25 two-fluid model (TFM),13,23 and multifluid model (MFM)13,23 models, along with different shapes of the probability density function (PDF) have been utilized for the flow predictions. LHF with the k-ε-g turbulence model (where k is the turbulence kinetic energy, ε the rate of turbulent dissipation, and γ the square of the mixture fraction fluctuations) was applied for the most of the systems. Furthermore, to overcome the assumptions in the LHF model, the TFM and MFM models were developed. The predictions from the TFM and MFM models were much better than that of the LHF model. Although all the models use the PDF approach to solve reactive flow, the prediction of the correct equilibrium state relationship between the species under specific operating conditions was the major difficulty. The condensation and reaction systems are mass-transfer-controlled. Thus, the local turbulence values will determine the rate of mass transfer across the gas/liquid interface. CFD models must be developed via the incorporation of real kinetics, which will lead to an accurate energy balance for the system. (3) In the case of the condensation jet, studies have been conducted to measure the average condensation heat-transfer coefficient. The heat-transfer coefficient is dependent on mass flux, bath subcooling, and the physical properties of the gas/ liquid. For a subsonic steam jet,15 the heat-transfer coefficient value has been determined to be

Submerged Gas Jet into a Liquid Bath: A Review - Industrial

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