For very small values of 5, the steady and oscillating parts of the fluid motion make nearly equal contributions to the overall heat transfer. The transfer rate at low Pe is always higher for very low f than for very high f. The periodic part of the fluid motion tends to enhance the transfer rate. The high Pe limit is treated in a manner similar to that in Morrison []. At high Pe, the periodic part of the fluid motion may either enhance or detract from the overall time-averaged Nu.

Series expressions for the Nu have been provided in this study. The analysis considers a liquid-liquid system. On the basis that the mass diffusivities of such systems may be usually small, rendering the Peclet number to be large, thin diffusional boundary layers are assumed to exist on both sides of the drop interface. The mass transfer process is regarded as quasisteady.

The results include a closed form expression for the Sherwood number, Sh. It is found that only when W , [Eq. The conjugate problem formulation has to be solved when the transport resistance of the continuous phase is comparable to that of the dispersed phase. Chang et al. The stream functions reflect the effects of gravity and electric field on either side of the interface. Unsteady and quasisteady transfer rates have been obtained for the limit of high Pe.

The thin thermal boundary layer approximation and similarity transformation similar to those employed by Chao [41] are adopted in this study. The shortcomings of the developments described in Chao [41] also apply here see discussions in section III. The results illustrate the enhancement in heat transfer due to the presence of the electric field.

The overall steady-state Nu is noted to be independent of the direction of flow. It is found that only when the absolute value of W [Eq. The low Pe limit of the conjugate problem has been investigated by Nguyen and Chung [ by treating the continuous phase as quasisteady and the transport in the drop as a transient process.

The paper appears to have a different definition for W [see Eq. The singular perturbation method is used to obtain the temperature profile for the continuous phase, whereas a regular perturbation procedure along with the method of weighted residuals is employed for the dispersed phase. The temperature has been computed up to and including the first order in Pe; however, higher order effects are also examined in order to ascertain the influence of an external field on the transport rates. In the first-order solution, the electric field presence is found to alter the temperature profiles but the net heat transfer rate remains unchanged.

The role played by electroconvection has been delineated by examining the variation of the ratio of heat transfer rates with and without the electric field as a function of time for two different values of Pe. Spherically Symmetric Condensation and Evaporation of a Drop in the Presence of Pure Vapor Plesset and Prosperetti [ have estimated the spherically symmetric flow rate of pure vapor in the region surrounding a drop experiencing evaporation or condensation from kinetic theory considerations as:.

For drops in the size range of 1 mm, a radial Peclet number of the order Pe would be the theoretical upper limit. The temperature field for a drop in an infinite vapor medium is given in Sadhal et al. For detailed kinetic theory treatments, see Shankar [] and Loyalka []. The parameter W is a function of the thermodynamic conditions p,, T,, and T,. For drop sizes of, say, 0. In a spray of drops, the interaction between drops might have to be considered in ascertaining transport rates.

The droplet is assumed to have fully developed internal. Geometry and coordinate system. Droplet undergoing slow translation and experiencing condensation. Here, u,,,,, is the maximum of the normal velocity at the interface u,. The drop is initially cold at a temperature To,whereas ' ".

Chung et al. Flow fields in both phases are treated as quasisteady. The energy and species equations for the continuous phase have been solved through a singular matched-asymptotic technique due to the existence of a region of nonuniformity in the neighborhood of the point at infinity [ In the singular perturbation procedure, the leading-order description for the far-field temperature and mass fraction includes the perturbed velocity field. A transient solution of the dispersed phase has been developed using the semianalytical series truncation method to provide the surface temperature, and the continuous and dispersed phases are connected through the energy flux continuity equation:.

The is the average heat flux corresponding to a Nusselt-type heat flux iNu condensation heat transfer to an isothermal solid sphere situated in an atmosphere of quiescent, pure, saturated stream. The temperature of the solid sphere is set equal to the initial temperature of the drop. To evaluate the transport quantities, the fall-velocity of the drop has been established through a balance between drag and gravity forces.

The drag force has been evaluated using results provided in Sadhal and Ayyaswamy []. For illustration, the droplet initial temperature has been taken to be. Reprinted with permission from Chung et al. The droplet environment has been assumed to consist of air saturated with steam, with the air-mass fraction ranging from 0. The ambient temperatures range between to C.

The reason for the reduction in heat transfer due to the presence of air is that a buildup of air film at the interface causes a reduction in the partial pressure of the vapor at the interface.

## Advances in Heat Transfer, Volume 33

In turn, this reduces the saturation temperature at which condensation takes place. The net effect is to lower the effective thermal driving force, thereby reducing the heat transfer. It is also evident from Fig. In Chung et al. The flow field, fluid temperature, and mass function of the noncondensable gas are expanded as complete series of Legendre polynomials, and in the actual implementation, a six-term expansion series has been employed.

The numerical results are shown to compare very well with those of Chung et al. The presence of a large noncondensable mass fraction in the bulk causes the radial flow of vapor to be weakened. As a consequence, the internal circulation is weaker, and the corresponding liquidside Peclet number, E, is smaller. The asymptotic solution should be capable of representing this situation very well, and it is seen to do so.

The quasisteady assumption for transport in the continuous phase and for flow in the phases has been invoked in the first three publications, and the other papers address the fully transient formulation. The equations for the flow fields and the transport in the continuous phase have been solved by a hybrid finite-difference scheme see Sundararajan and Ayyaswamy [ for details. The transport in the drop interior is solved by the CrankNicolson procedure. These studies have considered a cold water droplet of radius R , and initial bulk temperature To that is introduced into an environment consisting of a mixture of steam and air.

The droplet is assumed to be projected with an initial velocity U, and at an angle Po with respect to the vertical direction as shown in Fig. The formulation considers a coordinate frame that coincides with the drop center and moves with the instanta,. The instantaneous value of Re is taken to neous translational velocity U be O up to Drop deformation due both to inertial effects Weber number We and to hydrostatic pressure variation Eotvos number Eo are assumed to be small, and are neglected. For the range of Re considered here, based on Us the maximum circulation velocity at the drop surface is of U [18, The droplet trajectory has been determined by a gravity-drag force balance.

In addition to the initial conditions for velocity, temperature, and mass fraction, and the interface conditions requiring the continuity of tangential velocity and shear stress, the continuity of mass flux and the. In Sundararajan and Ayyaswamy [], it is assumed that the instantaneous surface temperature of the drop, T,, is uniform over the drop surface, and this assumption is useful in decoupling the quasisteady equations from the formulation for the drop heat-up [84].

In the liquid phase, the stream surfaces are taken to be isothermal. The changes in the internal and external flow structure due to condensation are shown in Fig. Within this wake, fluid particles recirculate. In the drop interior, a primary liquid vortex generated by the positive shear. The secondary vortex strength is very small compared to that of the primary and the sense of circulation in the secondary vortex is opposite to that of the primary vortex. In the presence of condensation, the wake length and volume are reduced.

A dividing stream surface b ' exists, inside which fluid particles either condense or recirculate. In the drop interior, the strength of the primary vortex is higher. This is due to the increased shear stress at the interface caused by the bending of the gaseous phase stream surfaces toward the drop.

The internal secondary vortex does not exist at high rates of condensation. As W is increased, the primary vortex center shifts toward the drop equatorial plane, and there is a reduction in the asymmetry in the circulation. The comparisons between numerical results and some experimental 1 for the dimensionless, instantaneous data from Kulif. The drop bulk temperatures are predicted very well. Note also that, for a larger noncondensable fraction in the bulk, the drop heat-up rate is slower. This illustrates the effect of gas-phase resistance. The rate of drop growth is examined in Fig.

The dimensionless growth rate is given by:.

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The growth rate increases , - f,. Effect of condensation on flow pattern. Reprinted with permission from Sundararajan and Ayyaswamy The variation of drop bulk temperature with time. Reprinted with permission from Huang and Ayyaswamy [94]. The numerical calculations are in agreement with this. Because the drop growth rate is intima! Improved correlations may be available in the future with greater computational accuracy, which could cause slight changes in the numerical coefficients in the correlations. But the structure is likely to stay because it has been developed with reasonable scaling arguments.

The numerical calculations also show that the dimensionless average heat flux i j defined by. In the front portion of the drop, the condensation velocity has an approximate cosine variation with angle. The maximum condensate rate that occurs at the front stagnation point is approximately twice the average condensation rate on the droplet. Thus, analyses developed for the stagnation region [ 1 1 can serve to obtain useful bounds for the transport to the droplet. The transport to the drop in the rear region is enhanced due to recirculation in the wake, and this enhancement increases with Re.

The shear stress and the friction drag coefficient increase with condensation. The pressure drag coefficient, on the other hand, rapidly decreases with condensation due to a large pressure recovery in the rear portion of the drop. The relative importance of pressure drag over the friction drag increases with drop size or temperature differential. The contribution to the total drag from the linear momentum of the inward flow is significant only for very small drops and at very high rates of condensation. At high levels of condensation, the drag on the drop is very much reduced, and if the drop size is very large, the drag force cannot balance the weight of the drop.

Such an imbalance might lead to excessive drop acceleration and eventual breakup. Because ti, depends directly on the average concentration gradient of the noncondensable at the interface, this square-root dependence may be attributed to an overall boundary layer type variation for the noncondensable concentration. At high Re, the regions of steep radial gradients of velocity, temperature, and noncondensable concentration in the gas-vapor phase near the drop surface can be represented by thin boundary layers.

This feature is discussed in Sundararajan and Ayyaswamy H i g h Reynolds Number Condensation: Solution by Boundary Layer Formulation Outside of the immediate transient period following the introduction of the drop, it is reasonable to consider established, quasisteady boundary. However, due to flow separation and wake recirculation, the theory would be inapplicable in the rear region. Because transport is maximum in the front region, bounds for transport rates and approximate results for the overall transport may be derived. In Sundararajan and Ayyaswamy [], the quasisteady continuous phase equations are decoupled from those for the liquid phase by prescribing both a suitable surface circulation velocity and the surface temperature.

In the front stagnation region of the drop, a similarity transformation has been introduced to obtain a system of ordinary differential equations and a similarity solution. For the entire boundary layer region, a series-truncation procedure with expansions in terms of Legendre polynomials has been employed to develop ordinary differential equations.

These are solved by a numerical scheme. A Crank-Nicolson procedure is used for the drop interior. Tables summarizing results applicable for a wide range of condensation rates have been provided. A boundary layer analysis of the effects of an insoluble monolayer surfactant on high Re condensation on a drop is presented in Chang and Chung [40]. The strength of internal motion ranges from an order of magnitude smaller than the free stream velocity for slight surfactant contamination to almost a complete stop for high concentrations.

It is found that surfactants with lower surface diffusion coefficients are more effective in weakening the strength of internal circulation in condensing drops. Huang and Ayyaswamy have carried out a numerical study of the effect of insoluble surfactants in condensation on a drop moving in the intermediate Re regime. The droplet environment is taken to consist of its own vapor and a noncondensable, and the droplet is assumed to be initially contaminated with an insoluble monolayer of surfactant.

The ambient pressure is 1 atm. The surfactant induced force F, 8 has been evaluated on the basis of a steady convection-diffusion balance model [28, This force modifies the shear stress at the drop surface. Results have been provided for the interface velocity, drag, surface vorticity, external and internal flow structures, surfactant concentration along the droplet surface, and the Nusslet and Sherwood numbers.

Some conclusions from the study of Huang and. The gradient of the surface concentration increases with increasing Re. With increasing Re, the angular location of the maximum surfactant concentration moves toward the front. The surface mobility and the strength of internal circulation both decrease with increasing amounts of surfactant. The covering angle of the surfactant increases with increasing difisivity of the surfactant. During early stages of the condensation, Nu and Sh are relatively lower for a contaminated drop. At higher Re numbers, the contribution from the inertial forces to the normal stress balance increases, and a spherical water droplet translating in a stream-air mixture progressively deforms into an approximately prolate spheroid.

Deformation of a droplet experiencing condensation would be smaller compared to that of a noncondensing drop under otherwise identical conditions as a consequence of the higher pressure gain in the rear. The shapes of water droplets falling through air and the effect of the pressure profiles on droplet deformation are discussed in LeClair et al. Hijikata et al. Droplet oscillation and shape deformation are shown to enhance heat transfer to the droplet. The oscillation is shown to mix the droplet interior.

The experimental results yield a heat transfer coefficient about 10 times higher than that for a solid sphere and about 4 times higher than the theoretical result for a spherical droplet. The predictions of the model with oscillation and deformation taken into account compare well with experimental observations. At present, no systematic, careful, and rigorous analysis exists for reporting on this important area.

Earlier experimental studies mostly related to nuclear reactor emergency core cooling spray. Some of the earliest attempts to study condensing sprays include the contributions by Tanaka [], Tanaka et al. Significant contributions have been made in the recent experimental studies on various aspects of direct-contact condensation as evident in the report by Cum0 [58]. In particular, these include a study by Celata et af. The injection has been varied from 0. The following equation for the average nondimensional droplet temperature Om, see Pasamehmetoglu and Nelson [, and Carra and Morbidelli [36] for details is stated to fit the experimental data:.

Huang and Ayyaswamy [96] have numerically investigated condensation on a spray of equal-sized drops moving in the Re O regime by using a unit cylinder cell model [, with a body fitted coordinate system [63, The distance between neighboring droplet centers is assumed to remain the same during the entire process of condensation, although this distance in the flow direction front to back is allowed to be different from that in the lateral direction side to side. The drop environment in each cell is taken to consist of its own vapor and a noncondensable.

The ambient pressure is taken to be 1 atm. The results for the interface velocities, drag, surface vorticity, external and internal flow structure, far-stream velocity, and the variations in Reynolds, Nusselt, and Sherwood numbers for various drop arrangements have been provided. Here A is the lead drop, and the follower drops, B , C, and D, are traveling in tandem; h l and h2, are half-distances between neighboring drops in the plane of motion of the drops, and w l is the half-distance between neighboring drops in the plane.

The variation in surface shear stress with 0 for drops A , B , C, and D. The variation in Nusselt number with 0 for the first, second, third, and fourth drops. At a given value of Re and for any particular drop, the shear stress increases to a maximum value on the front surface and then decreases. This region of negative shear stress corresponds to recirculation on the rear of the drop beyond the point of separation. For the follower drops, the shear stress profiles are similar, except that the magnitudes are lower in view of the weaker flow fields experienced by them.

The local variation of the Nusselt number with angular position for a row of drops is shown in Fig. The Nu attains its highest value at the front stagnation point of the lead drop where the temperature gradient is the highest. Away from the stagnation point, with increasing angle, Nu decreases up to the separation ring, following an approximate cos6 variation.

For the follower drops, the steepest temperature gradient does not necessarily occur at the front stagnation points due to the wake effects of the leader drops. The dynamics and transport associated with a row of drops are also strongly influenced by the presence of drops on lateral boundaries. The numerical calculations also show that drag coefficients increase with the proximity of the side drops. A compound drop is formed when a liquid drop undergoes evaporation or when a vapor bubble experiences condensation while passing through another immiscible liquid.

With a compound drop, the liquid and its vapor are both present. Condensation of a Bubble in an Immiscible Liquid. The articles by Sideman and Moalem-Maron [], Johnson and Sadhal [, and Jacobs [98] contain critical reviews of related information and must be consulted for details. The review by Jacobs [98] provides discussions of the studies by Lerner and Letan [] vapor condensation with a thin condensate film and Jacobs and Major [99] bubble collapse taking into consideration the heat and mass transfer in the continuous phase. A Freon vapor-liquid water compound drop has been investigated in Lerner et al.

Water is lighter than condensate Freon. The experimental aspect of the study is based on visualizations of temperature shadowgraphy , flow pattern color entrainment , and condensate shape dye injection together with screen tracing of the videotaped bubble shape and path. As the spherical bubble accelerates away from the nozzle, its shape progressively deforms into an ellipsoid, and hydrodynamic and. Viscous and thermal wakes form in the rear. This is followed by deceleration.

Vortices in the wake advance forward and a thermal cloud covers the bubble. Vapor material inside the collapsing bubble is noted to be spherical, eccentrically positioned, and adhering to the top front stagnation point of the bubble. The condensate film is progressively thicker away from the front stagnation point. On completion of condensation, the droplet which may contain noncondensables moves away from the vortices. In the analytical model, the noncondensable gases in the bubble are assumed to be uniformly distributed in the vapor.

The bubble velocity is taken to be time dependent, and the drag coefficient is allowed to vary with the size and shape. Heat transfer in the condensate is assumed to be quasisteady conduction along radial flow paths emanating from the center of the inner sphere. The continuous phase transport is assumed to be quasisteady, and the thermal resistance is evaluated using correlations described in Lee and Barrow [].

The results show that the rate of collapse is high during acceleration and is significantly reduced during deceleration. The predictions of the model are shown to compare favorably with the experimental measurements. Motion of a gas bubble completely engulfed by another liquid and moving in a third immiscible fluid is analytically examined using the bipolar coordinate system for low Reynolds number flow in Sadhal and Oguz []. The surface tension at both the interfaces is assumed large enough to preserve sphericity of the bubble and the drop.

The transient convective heat transfer associated with a collapsing bubble moving at low Reynolds number under the influence of buoyant forces is evaluated using finite-difference methods in Oguz and Sadhal []. The drag component induced by radial velocity contributes to the total drag on the bubble in eccentric configuration. The time-dependent Nusselt number depends on the compound drop configuration and on the conductivities of the participating liquids.

When the heat transfer rate is high enough, the bubble away stays inside the drop due to rapid shrinkage until it finally disappears. The velocity of the drop and the relative velocity of the bubble decrease as the bubble gets smaller due to the changes in the buoyant forces. Radial convection decreases the Nusselt number. Evaporation of a Drop in an Immiscible Liquid The review articles by Johnson and Sadhal [lo71 and Avedisian [16] contain related information about evaporation in an immiscible liquid and must be consulted for details.

Vaporization of liquid drops in an immiscible and low volatility liquid medium has been studied extensively in view of the numerous applications of this process. The classic contribution of Sideman and Taitel [] has been followed by a very large number of studies with varying degrees of success. The evaporation of a liquid drop into a vapor bubble can occur in three different configurations: nonengulfing, completely engulfing, and partially engulfing based on the balance of surface tension forces [17]. The partially engulfed configuration is most common in direct-contact heat transfer.

Note that in single drops, significant superheating can occur before evaporation commences. We discuss only situations where evaporation is occurring. Sideman and Taitel [] have experimentally studied pentane and butane drops evaporating in distilled water and seawater. An analytical expression for Nu has been developed by solving the external heat transfer problem assuming a spherical shape for the drop valid only to a limited type of fluid system and potential flow in the continuous phase. The predictions match the experimental data only at the stage when the drop has almost completely evaporated and the residual liquid is small and thin.

Evaporating compound drops may experience oscillation, causing the dispersed phase liquid to slosh from side to side, forming a thin film of liquid over the top of the inside bubble surface. This feature has been observed by Simpson et al. The oscillation is ascribable to periodic vortex shedding of the wake at high Reynolds number. The drop changes shape from spherical through ellipsoidal to a cap-shaped bubble. A compound drop rising in low- and moderate-viscosity fluids follows a zigzag trajectory.

A pentane drop evaporating in highly viscous glycerol has been studied by Tochitani et ai. In the analytical model, the Stokes solution for a sphere has been assumed for the flow field, and the heat transport is treated as a quasisteady process. The latter assumption is questionable is O The surface forces are evaluated for the limits of potential flow and Stokes flow; and, for finite values of Re, correction factors are employed.

The drag coefficients for air bubbles moving in water from Haberman and Morton [ have been generally employed in the force balance. The RayleighPlesset equation is used to determine the growth rate of the bubble. A semiempirical expression for the instantaneous velocity of a compound drop is given by Raina et al. The energy conservation equation in these studies is usually a heat balance.

This heat transfer coefficient is obtained from semiempirical expressions for the instantaneous Nusselt number such as the one developed by Battya et al. In Raina and Grover [], a model for determining the liquid-liquid area taking into account the effect of viscous shear on the spreading of dispersed liquid over the bubble surface has been described. A theoretical model for evaporation that takes into account the effects of temperature difference between the phases and the bubble growth rate is developed in Battya et al.

The predictions of this model agree with Eq. The mechanical equilibrium of the bubble droplet has been further explored in the second part of Tadrist et al. The limitations of the Raina-Grover viscous shear model have been discussed in this study. Experimental results, correlation for volumetric heat transfer coefficient, and a model extending the single-drop study to multidroplet systems have all been included.

A partially engulfed drop has been analyzed in Vuong and Sadhal[, In the first of this two-part study [], the motion is analyzed in the limit of Stokes flow, and the bubble growth rate has been prescribed arbitrarily. An exact analytical solution for the axisymmetric flow field has been developed in a toroidal coordinate system by combining the solutions separately obtained for a flow field resulting from drop translation and a flow field resulting from the moving boundaries of the drop due to the growth.

The drag force on the compound drop is shown to depend on many parameters besides the viscosity of the continuous phase, the drop size, and the free stream velocity. Among the important parameters are 4p,the drop geometry, that is, the liquid-to-vapor volume ratio together with contact angles that depend on the nature of the fluid systems, and the ratio of the growth to translational velocities.

During the vaporization process, for a compound drop, the drag force has been shown to even exceed the value for a solid sphere whose volume is equal to the total liquid-vapor dispersed phase volume. In part two of the study [], the heat transfer problem is solved under the assumption that the liquid-vapor interface and the vapor phase are at the equiIibrium temperature corresponding to the hydrostatic pressure.

The vapor is effectively decoupled from the liquid []. The energy equations, for both the continuous phase and the liquid portion of the. Reproduced with permission from Vuong and Sadhal []. The time history of evaporation of a pentane drop immersed in a bath of glycerol has been provided, and the predictions compare favorably with the experimental results of Tochitani et al. Note that the measured heat transfer coefficient in the Tochitani et af.

This contradicts the well-established theory that the heat transfer coefficient is inversely proportional to the size of the drop. Here, U, is the initial velocity of the compound drop and R , is the initial radius of the vapor bubble. The results reveal that when the drop is first introduced into the high-temperature fluid, the average Nusselt number for the liquid-liquid interface denoted by Nu, is very high.

Soon after, a very thin conduction layer in the region of the liquid-liquid interface begins to form resulting in a steep drop of Nu,. Simultaneously, the temperature at the liquid-vapor interface begins to increase due to the energy absorbed from the external fluid, and a rapid rise in the Nusselt number for the liquid-vapor interface denoted by Nu; is noted. Both Nu, and Nu, approach asymptotic limits at a large time since the transient effect is no longer important.

It may be observed that,. This is explained by the transient oscillations in Nusselt number and the role of internal circulation, both of which depend on the level of AT compare with section III. An approximate analytical model applicable for preagglomerative and postagglomerative stages of compound drop motion in multidroplet evaporation situations is described in Smith et al. During the preagglomeration stage, it is assumed that the droplets are relatively small and do not interfere with one another. For the postagglomeration stage, it is assumed that the rate of coalescence is such that the dispersed phase volume fraction is constant.

Experimental measurements related to the evaporation of cyclopentane droplets in stagnant water are reported. Results for volumetric heat transfer coefficients in terms of travel distance that compare favorably with experimental observations are provided in this study. Seetharamu and Battya [ have experimentally investigated the evaporation of R and n-pentane droplets in a stagnant column of distilled water. The variation of volumetric heat transfer coefficient with column height and dispersed flow rate has been established. The volumetric heat transfer coefficient increases with a a decrease in column height and b with an increase in the flow rate of the dispersed phase.

A lower column height is associated with larger temperature differences, greater acceleration of the drops, and increased turbulence. With distilled water for the continuous phase, the volumetric heat transfer coefficient is lower for R compared to n-pentane. Based on their experimental observations and a multiple-linear regression analysis, Seetharamu and Battya propose the following expression for calculating the initial drop diameter: 0. Modifications to the theoretical model of Smith et al. Although the compound droplet problem is among the more difficult class of problems involving phase change, considerable progress on the problem can be expected in the coming years due to the recent availability of advanced computational methods and supercomputers.

The properties are evaluated at the film conditions. The Ranz and Marshall correlation is based on certain quasisteady, constant radius, porous wetted sphere experiments and is only valid for small values of transfer number. The other commonly used correlation is the Spalding's correlation []:. The correlation of Eq. Neither of the two correlations accounts for transient heating, regressing interface, and internal circulation.

Experimental correlations for the evaporation of water droplets in air have also been reported in Beard and Pruppacher [27], Yao and Schrock [], Yuen and Chen [, , and Pruppacher and Rasmussen []. The evaporation rate of small water droplets moving in air has been numerically predicted by Woo and Hamielec [ Recent studies of droplet vaporization, particularly as related to fuel droplets, have contributed to a deeper understanding of the mechanisms involved, and many new and comprehensive correlations are available.

Fuel sprays consist of a spectrum of droplets of various sizes and velocities. The major objective of fuel drop vaporization studies has been to develop suficiently accurate yet simple enough models for use in a spray analysis. A vaporizing droplet can experience a range of Reynolds numbers during its lifetime. Typically, Re will decrease with time as the droplet diameter and the relative velocity decrease.

There are exceptions to the monotonic behavior, e. They penetrate deep into the spray core. The bigger droplets determine the overall behavior of the spray combustion process, because they constitute most of the spray mass. The smaller. From an energy transport point of view, in typical combustion situations, the duration of the transient droplet heating is comparable with the droplet vaporization time. A complete investigation of the various flow regimes governing droplet motion and the evaluation of the associated transient effects are prohibitively difficult.

This is true in spite of the availability of advanced experimental and numerical techniques and supercomputers. However, with intelligent modeling of combustion spray systems, a wealth of information can be obtained, and optima1 systems can be designed. In regard to combustion modeling, two major categories are distinguished by Faeth [68]. A spray model consisting of very small droplets is termed the locally homogeneous flow LHF model; here, the gas and liquid phases are assumed to be in dynamic and thermodynamic equilibrium at each point in the flow. Although this model cannot describe a real spray, it provides the lower bound for the size of spray process.

The second category, called the separated flow SF model, considers the effect of finite rates of transport between the phases and, hence, can model practical sprays with finite size droplets. It is noted in Faeth that the behauior of individual drops in a spray must be examined in order to assess the validity of LHF models and to undertake SF models. This involues drop-life-history computations to yield the size, velocity, temperature, and composition of individual drops as a function of position in the pow.

Small Mach number flow is almost always considered so that kinetic energy and viscous dissipation are negligible, Gravity effects, droplet deformation, radiation, Defour energy flux, and mass diffusion due to pressure and temperature gradients are all usually neglected. The multicomponent gas-phase mixture is assumed to behave as an ideal gas. Phase equilibrium usuaIly described by the Clausius-Clapeyron equation is assumed at the single-component dropletgas interface.

Gas-phase density and thermophysical parameters are generally considered variable. Liquid-phase viscosity is generally taken as variable but density and other properties are typically taken to be constant. Spherically Symmetric Vaporization of a Fuel Drop.

For droplet evaporation in a stagnant medium, the spherically symmetric continuous phase motion is Stefan convection in the radial direction. The liquid phase motion is also spherically symmetric. The droplet surface regresses into the liquid. At moderate pressures, the gas phase may be. Hubbard et al. The results are developed by solving transient one-dimensional problems for the continuous and dispersed phases.

Spherically symmetric vaporization in the presence of an exothermic reaction in the continuous phase between the fuel vapor and a suitable reactant is discussed in a number of articles [, These articles contain details in regard to the following. At the liquid-vapor interface, the energy balance is:. The position of the flame zone for a thin flame may be determined from:. The solution to the transient energy equation for the liquid phase will provide the necessary relation between the interface temperature and the liquid heating rate. The liquid temperature would continue to rise until the wet bulb temperature Twbis reached heat flux entering the liquid, G, ceases.

With an underestimated surface temperature, however, the corresponding heat flux to the drop during the early period is overestimated, and the droplet is ultimately predicted to vaporize at a faster rate compared to the solution of the diffusion limit model. In this case, if the thermal diffusion across the thin layer is regarded as negligible, the vaporization rate is predicted by yet another d 2 law but with a lower absolute value of the slope on a time plot []. During spray combustion in devices such as liquid-fueled rocket engines and diesel engines, the evaporation of droplets may occur at pressures near or above the critical pressure of the fuel.

Under these conditions, many effects that are assumed negligible at low and moderate ambient pressures need t o be reevaluated. Such effects include the transient character of the gas phase, nonideal gas phase behavior, the real gas effect on the heat of vaporization, the vapor-liquid equilibrium condition at the droplet surface, and the solubility of the ambient gas into the liquid phase.

Jia and Gogos [ l O l , have numerically investigated the spherico-symmetric vaporization of an n-hexane droplet into nitrogen environment for ambient pressures of 1 to atm and ambient temperatures of to K. The papers include discussions of important earlier work. In the. Dimensionless droplet lifetime with ambient pressure: a Low and moderate ambient temperatures and b high ambient temperatures.

Reproduced with permission from Jia and Gogos [loll. The second paper focuses on gas solubility. The droplet lifetime dependence on ambient pressure and temperature has been predicted. At the lowest temperature considered K , the droplet lifetime increases monotonically with ambient pressure in the pressure range investigated. At a higher ambient temperature, it exhibits a maximum with ambient pressure. At supercritical pressures, the droplet surface temperature keeps rising throughout its lifetime.

At high temperatures K , droplet lifetime decreases monotonically with pressure. In a fuel-rich environment, at relatively low ambient temperatures and high ambient pressures, condensation may occur during the early part of the droplet lifetime. At high pressures, a model that neglects gas solubility will either underestimate the droplet lifetime low ambient temperature or break down by failing to predict the correct criteria for vapor-liquid equilibrium high ambient temperature.

Figures 19 a and b show the lifetime histories as a function of pressure for various temperatures. Reproduced with permission from Jia and Gogos [ Convective Droplet Vaporization With a moving droplet, the vaporization problem is very much more complicated. The nonuniform blowing effect due to evaporation will affect the drag force components and the heat transfer to the drop in complicated ways.

The composition of the gas phase is modified, resulting in very significant changes in thermophysical properties. Evaporation and combustion of a slowly moving droplet may be analyzed by a method similar to the one used for studying condensation associated with a drop undergoing slow translation. For a vaporizing droplet of finite size, the Reynolds number based on relative gas-droplet velocity, droplet radius, and gas properties may be large [Re O 10O l during a significant part of the droplet lifetime and it may decrease with time.

At high Re, we may envision thin boundary layers for the efficient transport of momentum, heat, and mass, both on the outside and on the inside of the droplet surface. Separated near wakes on the outside and an internal wake and core flow inside the drop complete the picture. But again, the R e may decrease with time and boundary layer theory may be inapplicable beyond a certain portion of the drop lifetime. Approximate boundary layer analyses and fully numerical solutions have been attempted to predict the. Axisymmetry is usually assumed in most analyses based on the observation that the characteristic time for a change in direction of the relative velocity is long compared to the residence time for an element of gas to flow past the droplet even when the droplet changes its direction of motion.

Evaporation at Low Translational Reynolds Number The evaporation of a slowly moving n-hexane droplet initially at K and in a K, atm, fuel-free environment has been analytically examined by Gogos and Ayyaswamy [ Whereas the flow and the transport processes in the gaseous phase and the droplet internal circulation are treated as quasisteady, the droplet heating is treated as a transient process.

The transport properties of the gaseous phase have been considered variable. The gaseous-phase flow solution has been developed in a manner similar to the treatment in Sadhal and Ayyaswamy []. The transport equations of the gaseous phase require analysis by a singular perturbation technique and this is treated in a manner similar to that in Chung et al.

The transient heating of the drop interior is solved by a series-truncation method.

## Advances in Heat Transfer, Volume 26 - 1st Edition

The solution to the total problem is obtained by coupling the results for the gaseous and dispersed phases. The enhancement in the evaporation rate due to convective motion has been predicted. The friction drag coefficient is much lower compared to the Stokes value. The reduction in the pressure drag coefficient is smaller. The evaporation drag coefficient, which is due to momentum flux at the interface, becomes increasingly important as the nonuniform radial flow due to evaporation becomes vigorous.

The combustion of a fuel droplet that is slowly moving in an oxidant environment is investigated in Gogos et al. Variable density effects have been taken into account. The combustion process is modeled by an indefinitely fast chemical reaction rate. The convective flow considered is such that an envelope diffusion flame is established. The energy and species equations of the gaseous phase in terms of Shvab-Zeldovich variables are solved through a singular perturbation matched-asymptotic technique. The transient heat-up of the drop interior is solved by a series-truncation numerical method.

A single effective binary diffusion coefficient is used for all pairs of species. A one-step, irreversible chemical reaction has been assumed. The Burke-Schumann flame front approximation is used to represent the thin reaction zone. The reactants. Nondimensional time, T FIG. Time history of the drag coefficients. Reprinted with permission from Gogos and Ayyaswamy The analysis by Gogos et al. The motivation for developing this higher order theory has been that it can be subsequently extended to study the ignition and extinction of evaporating and burning droplets.

The reasoning is as follows. Consider, for example, droplet ignition. The onset of droplet ignition is governed by two characteristic time scales: the diffusion time and the chemical reaction time. The ratio of these time scales is the Damkohler number CQ. This is the frozen limit. This is the fast chemistry limit. Ignition and extinction phenomena are characterized by 0 O 1 and represent the transition between the frozen and the fast chemistry limits.

Ignition may occur when the diffusion time is of the same order of magnitude as the reaction time. The presence of convective flow introduces competition between the characteristic time scales, varying the time scales along the periphery of the droplet. Thus, the location along the droplet surface where the ignition initiates is decided in part by the effects of convection. This location is important because it determines both the nature of the flame and the extent of burning. The presence of convection also affects the ignition delay time by influencing the rates of droplet heating and vaporization.

Similar discussions apply for extinction. Ignition and extinction phenomena can be studied by allowing for finite-rate kinetics and resolving the flame structure. These may be accomplished by the method of activation energy asymptotics AEA , AEA assumes that the chemical reaction rate has an Arrhenius temperature dependence with a large activation energy. A distinguished limit is taken in which the activation energy 0 and the Damkohler number 0 2 go to infinity [loo]. When the temperature is less than q. When the temperature is greater than T,, the exponential term cannot be balanced unless the product Y,Y, is zero.

Therefore, the chemistry is confined to a thin but finite reaction zone where T is close to T,. Variations of the perturbed temperature with a range of reduced Damkohler numbers reveal the conditions for ignition. To ascertain the effect of convective flow on ignition, the development of solutions for flow field and transport to order 6 is the logical first step.

Note that in practical applications involving liquid hydrocarbon fuels, S is 0 1 O p 2 and, hence, it is of O E. It is therefore necessary to obtain the flow and transport to 0 e 2. The inclusion of higher order perturbations greatly increases the mathematical complexities. In particular, the determination of the flow field itself. However, the solution for u 2 does not go to zero as r 0. Hence, it cannot be made to satisfy boundary conditions at infinity. Thus a regular perturbation scheme is not adequate to obtain uniformly valid l. In Jog et a temperature and species distribution have been solved up to and including terms of O e2.

Evaporation has been addressed first, and subsequently combustion has been examined. Evaporation results will assist ignition studies, whereas combustion results are useful for exploring extinction. Properties have been evaluated using the rule. The development in Jog et al. To satisfy the velocity variation at the droplet interface and the uniform flow at infinity, the stream function is expanded in terms of Gegenbauer polynomials as m.

Only the terms required to satisfy the boundary conditions are retained in this expansion. In the outer region, it is inappropriate to scale velocity by the. The liquid phase solutions to O E are obtained in a manner similar to that in Gogos et al. The drag force on the liquid drop nondimensionalized by puU,R is calculated up to and including terms of. With Shvab-Zeldovich variables,. Similar to the expansion for the temperature in the gaseous phase, a perturbation scheme in terms of E and Legendre polynomials is used:.

With this expansion, the governing differential equations are solved numerically using a finite-difference method. An implicit algorithm is used to solve for the transient temperature field inside the droplet. At each time step, several iterations are required to obtain consistent convergent solutions. The dimensionless heat transfer from the gaseous phase to the droplet is. The interfacial heat transfer variation with time is shown in Fig. For the heat from the a droplet introduced at its wet bulb temperature, Twh, gaseous phase, q , is completely utilized for fuel evaporation, q,.

For a droplet whose initial temperature is less than the wet bulb temperature, for about a third of the droplet lifetime, part of the heat input is used for fuel evaporation and a substantial portion is used for liquid heating. As the surface temperature of the droplet increases, q decreases while qe increases.

At higher T,, the mass fraction of the fuel at the interface is higher. For droplet initial temperatures higher than the wet bulb value, the interface may receive heat both from the droplet interior and the gas phase. This will result in very high values for the initial evaporation rates. High Reynolds Number Vaporization: Solution by Boundary Layer Formulation At high Re, the evaporation rate can be sufficiently accurately determined by solving the coupled boundary layer equations of motion, energy, and species concentration in the gas and liquid phases. Harpole [84] has examined water droplet evaporation into a high-temperature dry air or pure steam environment by solving the laminar boundary layer equations at an axisymmetric stagnation point.

It is claimed that transport to the whole droplet may be estimated with suitable corrections to the stagnation-point solution. A number of correlations are given to demonstrate and justify this claim, but the validity has been seriously questioned by Renksizbulut and Yuen Prakash and Sirignano [, have used boundary layer theory for a single-component drop. Law et al.

The droplet is assumed to vaporize in a hot inert environment. It is also assumed that the specific heats ci and binary diffusion coefficients Dj of all the gas phase components are equal, and Le is unity. In the high Re regime, gas flow is essentially unsteady due to the temporal change in the size of the droplet. This time is much smaller than the droplet lifetime, which is typically of O 5 msec for a droplet of this size vaporizing in a convective field. In the time scale of the drop heating, the gas phase is, therefore, taken to consist of quasisteady momentum and thermal boundary layers near the surface.

The velocity and pressure distribution outside the momentum boundary layer are assumed to be those for potential flow over a sphere. These assumptions preclude the determination of the correct total drag coefficient. In many circumstances, the drag force effects are such as to rapidly decrease the flow Reynolds number with time. This book comprises heat transfer fundamental concepts and modes specifically conduction, convection and radiation , bioheat, entransy theory development, micro heat transfer, high temperature applications, turbulent shear flows, mass transfer, heat pipes, design optimization, medical therapies, fiber-optics, heat transfer in surfactant solutions, landmine detection, heat exchangers, radiant floo This book comprises heat transfer fundamental concepts and modes specifically conduction, convection and radiation , bioheat, entransy theory development, micro heat transfer, high temperature applications, turbulent shear flows, mass transfer, heat pipes, design optimization, medical therapies, fiber-optics, heat transfer in surfactant solutions, landmine detection, heat exchangers, radiant floor, packed bed thermal storage systems, inverse space marching method, heat transfer in short slot ducts, freezing an drying mechanisms, variable property effects in heat transfer, heat transfer in electronics and process industries, fission-track thermochronology, combustion, heat transfer in liquid metal flows, human comfort in underground mining, heat transfer on electrical discharge machining and mixing convection.

The experimental and theoretical investigations, assessment and enhancement techniques illustrated here aspire to be useful for many researchers, scientists, engineers and graduate students. Rao and Hari Sankar. By Ryuji Yamada and Kazuo Mizoguchi. By Alireza Zolfaghari and Mehdi Maerefat.

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