Spray equation approach

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

Two numerical methods have been used for the solution of the spray equation. In the first method, i.e., the full spray equation method 543 544 the full distribution function / is found approximately by subdividing the domain of coordinates accessible to the droplets, including their physical positions, velocities, sizes, and temperatures, into computational cells and keeping a value of / in each cell. The computational cells are fixed in time as in an Eulerian fluid dynamics calculation, and derivatives off are approximated by taking finite differences of the cell values. This approach suffersfrom two principal drawbacks (a) large numerical diffusion and dispersion... 

Drop breakup enters the spray equation via the source term/bu in (19.45). There are various ways of accounting for drop breakup, most of which are also used for a rudimentary description of the atomization process. Some of these approaches are discussed in more detail in Chap. 9, and include the TAB model of O Rourke and Amsden [37], the Wave Breakup model of Reitz and coworkers [46, 40], the Unified Spray Breakup model of Chryssakis and Assanis [10], and the Cascade Atomization and Drop Breakup model of Tanner [54]. 

The practice of estabHshing empirical equations has provided useflil information, but also exhibits some deficiencies. Eor example, a single spray parameter, such as may not be the only parameter that characterizes the performance of a spray system. The effect of cross-correlations or interactions between variables has received scant attention. Using the approach of varying one parameter at a time to develop correlations cannot completely reveal the tme physics of compHcated spray phenomena. Hence, methods employing the statistical design of experiments must be utilized to investigate multiple factors simultaneously. 

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas 2-D equations of motion for a sphere 5791 or the simplified forms)154 1561 The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from... 

A typical form of flooding-point curve for spray towers is shown in Figure 13.35 where values of udf are plotted against ucf. The limiting values of each flowrate as the other approaches zero may be determined readily from equations 13.34 and 13.35. Thus, when... 

Flooding-point. Because the flooding-point is no longer synonymous with that for spray towers, equations 13.34 and 13.35 predict only the upper transition point. Dell and Pratt 30 1 adopted a semi-empirical approach for the flooding-point by consideration of the forces acting on the separate dispersed and continuous phase channels which form when coalescence sets in just below the flooding-point. The following expression correlates data to within 20 per cent ... 

More recent approaches [42], based on studies with isolated leaf cuticles in vitro with a wide range of formulations, considered transport through the cuticle to be a three stage diffusion mechanism, namely absorption into the cuticle, diffusion through the cuticle and finally desoiption from the cuticle into the internal leaf cells. This experimental method used a constant cuticular area, constantly covered by a volume of spray solution, so variable spreading was not an issue. An alternate version of this approach uses droplets applied to cuticular membranes which is more realistic [43]. The modified Pick s equation used in these in vitro studies [41,42] has the following form ... 

The formulation that has been given here is not the only approach to the description of two-phase flows with nonequilibrium processes. Many different viewpoints have been pursued textbooks are available on the subject [43], [44], and a reasonably thorough review recently has been published [45]. Combustion seldom has been considered in this extensive literature. Most of the work that has addressed combustion problems has not allowed for a continuous droplet distribution function but instead has employed a finite number of different, discrete droplet sizes in seeking computer solution sets of conservation equations [5]. The present formulation admits discrete sizes as special cases (through the introduction of delta functions in fj) but also enables influences of continuous distributions to be investigated. A formulation of the present type recently has been extended to encompass thick sprays [25]. Some other formulations of problems of multiphase reacting flows have been mentioned in Sections 7.6 and 7.7. 

Attempts have been made to develop design equations for preformed spray contactors on the basis of a more foudemental approach by considering muss transfer to individual drops. However, this requires a characterization of the drop size distribution, time of flight, and other factors. The approach is useful when test data are not available bat cannot be expected to provide accurate results for commercial equipment because the effects of drop agglomeration, entrainment, wall contact, gas recirculation, drop velocity changes, and other factors are very difficult to model analytically. 

Compared with Equation 4.6, Equation 4.12 contains the term -pUi Uj, the so-called Reynolds stress, which represents the effect of turbulence and must be modeled by the CFD code. Limited computational resources restrict the direct simulation of these fluctuations, at least for the moment. Therefore the transport equations are commonly modified to account for the averaged fluctuating velocity components. Three commonly applied turbulence modeling approaches have been used in the CFD model of spray drying system, i.e., k-Q model (Launder and Spalding 1972, 1974), RNG k-e model (Yakhot and Orszag 1986), and a Reynolds stress model (RSM) (Launder et al. 1975). 

Abstract Evaporation of multi-component liquid droplets is reviewed, and modeling approaches of various degrees of sophistication are discussed. First, the evaporation of a single droplet is considered from a general point of view by means of the conservation equations for mass, species and energy of the liquid and gas phases. Subsequently, additional assumptions and simplifications are discussed which lead to simpler evaporation models suitable for use in CFD spray calculations. In particular, the heat and mass transfer for forced and non-forced convection is expressed in terms of the Nusselt and Sherwood numbers. Finally, an evaporation model for sprays that is widely used in today s CFD codes is presented. 

One of the most widely used approaches for the simulation of sprays is the stochastic discrete droplet model introduced by Williams [30]. In this approach, the droplets are described by a probability density fxmction (PDF),/(t,X), which represents the probable number of droplets per unit volume at time t and in state X. The state of a droplet is described by its parameters that are the coordinates in the particle state space. Typically, the state parameters include the location x, the velocity v, the radius r, the temperature Td, the deformation parameter y, and the rate of deformation y. As discussed in more detail in Chapter 16, this spray PDF is the solution of a spray transport equation, which in component form is given by... 

Advances in computational technology have made continuum-based methods one of the most widely used approaches for the description of sprays. In this approach, the gas phase is formulated by conservation equations for mass, momentum, and energy, and the dispersed or liquid phase is described by a multidimensional distribution function. In most applications, the gas phase is turbulent and, therefore, plays an important role in almost every aspect of a spray. 

In contrast to the eddy viscosity approach, the Reynolds stress tensor, R, can be modeled directly by introducing one transport equation for each term. Such models are called Reynolds stress models, second-order closure models or second-moment closure models and have been pioneered by Rotta [47]. Since R is a symmetric second-order tensor, this requires six transport equations, and an additional three transport equations are needed to model T. Obviously, the predictive capabilities of the Reynolds stress models over the two-equation models are improved, but these models are significantly more complex and there is a considerable additional computational cost. A Reynolds stress model for spray applicatiOTis has been developed and tested by Yang et al. [63]. 

In the RANS approach, the gas phase is modeled with the Reynolds-Favre-averaged conservation equations for a compressible fluid. For a reacting spray. 

The drop deformation is modeled by Taylor s drop oscillator [56], as introduced by O Rourke and Amsden [37] into the context of sprays. In this approach, the drop distortion is described by a forced, damped, harmonic oscillator in which the forcing term is given by the aerodynamic drag, the damping is due to the liquid viscosity, and the restoring force is supplied by the surface tension. More specifically, the drop distortion is described by the deformation parameter, y = 2x/r, where x denotes the maximum radial distortion from the spherical equilibrium surface, and r is the drop radius. The deformation equation in terms of the normalized distortion parameter, y, is... 

In order to obtain a correlation, the outflow of the effervescent spray was simulated by a numerical model based on the Navier-Stokes equations and the particle tracking method. The external gas flow was considered turbulent. In droplet phase modeling, Lagrangian approach was followed. Droplet primary and secondary breakup were considered in their model. Secondary breakup consisted of cascade atomization, droplet collision, and coalescence. The droplet mean diameter under different operating conditions and liquid properties were calculated for the spray SMD using the curve fitting technique [43] ... 

---