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22.8.1 The Plain-Orifice Atomizer Model

The plain-orifice is the most common type of atomizer and the most simply made. However there is nothing simple about the physics of the internal nozzle flow and the external atomization. In the plain-orifice atomizer model in FLUENT, the liquid is accelerated through a nozzle, forms a liquid jet and then breaks up to form droplets. This apparently simple process is dauntingly complex. The plain orifice may operate in three different regimes: single-phase, cavitating and flipped [ 348]. The transition between regimes is abrupt, producing dramatically different sprays. The internal regime determines the velocity at the orifice exit, as well as the initial droplet size and the angle of droplet dispersion. Diagrams of each case are shown in Figures  22.8.1, 22.8.2, and 22.8.3.

Figure 22.8.1: Single-Phase Nozzle Flow (Liquid Completely Fills the Orifice)

Figure 22.8.2: Cavitating Nozzle Flow (Vapor Pockets Form Just after the Inlet Corners)

Figure 22.8.3: Flipped Nozzle Flow (Downstream Gas Surrounds the Liquid Jet Inside the Nozzle)

Internal Nozzle State

To accurately predict the spray characteristics, the plain-orifice model in FLUENT must identify the correct state of the internal nozzle flow because the nozzle state has a tremendous effect on the external spray. Unfortunately, there is no established theory for determining the nozzle state. One must rely on empirical models obtained from experimental data. FLUENT uses several dimensionless parameters to determine the internal flow regime for the plain-orifice atomizer model. These parameters and the decision-making process are summarized below.

A list of the parameters that control internal nozzle flow is given in Table  22.8.1. These parameters may be combined to form nondimensional characteristic lengths such as $r/d$ and $L/d$, as well as nondimensional groups like the Reynolds number based on hydraulic "head'' ( ${\rm Re}_h$) and the cavitation parameter ( $K$).

Table 22.8.1: List of Governing Parameters for Internal Nozzle Flow
nozzle diameter $d$
nozzle length $L$
radius of curvature of the inlet corner $r$
upstream pressure $p_1$
downstream pressure $p_2$
viscosity $\mu$
liquid density $\rho_l$
vapor pressure $p_v$

 {\rm Re}_h = \frac{d \rho_l}{\mu} \sqrt{\frac{2(p_1 - p_2)}{\rho_l}} (22.8-1)

 K = \frac{p_1 - p_v}{p_1 - p_2} (22.8-2)

The liquid flow often contracts in the nozzle, as can be seen in Figures  22.8.2 and 22.8.3. Nurick [ 266] found it helpful to use a coefficient of contraction ( $C_c$) to represent the reduction in the cross-sectional area of the liquid jet. The coefficient of contraction is defined as the area of the stream of contracting liquid over the total cross-sectional area of the nozzle. FLUENT uses Nurick's fit for the coefficient of contraction:

 C_c = \frac{1}{\sqrt{\frac{1}{{C^2_{\rm ct}}} - \frac{11.4r}{d}}} (22.8-3)

Here, $C_{\rm ct}$ is a theoretical constant equal to 0.611, which comes from potential flow analysis of flipped nozzles.

Coefficient of Discharge

Another important parameter for describing the performance of nozzles is the coefficient of discharge ( $C_d$). The coefficient of discharge is the ratio of the mass flow rate through the nozzle to the theoretical maximum mass flow rate:

 C_d = \frac{\dot{m}_{\rm eff}}{A \sqrt{2 \rho_l (p_1 - p_2)}} (22.8-4)

where $\dot{m}_{\rm eff}$ is the effective mass flow rate of the nozzle, defined by

 \dot{m}_{\rm eff} = \frac{2 \pi \dot{m}}{\Delta \phi} (22.8-5)

Here, $\dot{m}$ is the mass flow rate specified in the user interface, and $\Delta \phi$ is the difference between the azimuthal stop angle and the azimuthal start angle

 \Delta \phi = \phi_{\rm stop} - \phi_{\rm start} (22.8-6)

as input by the user (see Section  22.12.1). Note that the mass flow rate input by the user should be for the appropriate start and stop angles, in other words the correct mass flow rate for the sector being modeled. Note also that for $\Delta \phi$ of $2 \pi$, the effective mass flow rate is identical to the mass flow rate in the interface.

The cavitation number ( $K$ in Equation  22.8-2) is an essential parameter for predicting the inception of cavitation. The inception of cavitation is known to occur at a value of $K_{\rm incep} \approx$ 1.9 for short, sharp-edged nozzles. However, to include the effects of inlet rounding and viscosity, an empirical relationship is used:

 K_{\rm incep} = 1.9 \left(1 - \frac{r}{d}\right)^2 - \frac{1000}{{\rm Re}_h} (22.8-7)

Similarly, a critical value of $K$ where flip occurs is given by

 K_{\rm crit} = 1 + \frac{1}{\left(1 + \frac{L}{4d}\right)\left(1 + \frac{2000} {{\rm Re}_h}\right) e^{70r/d}} (22.8-8)

If $r/d$ is greater than 0.05, then flip is deemed impossible and $K_{\rm crit}$ is set to 1.0.

The cavitation number, $K$, is compared to the values of $K_{\rm incep}$ and $K_{\rm crit}$ to identify the nozzle state. The decision tree is shown in Figure  22.8.4. Depending on the state of the nozzle, a unique closure is chosen for the above equations.

For a single-phase nozzle ( $K >K_{\rm incep}, K \geq K_{\rm crit}$)  [ 207], the coefficient of discharge is given by

 C_d = \frac{1}{\frac{1}{C_{\rm du}} + 20 \frac{(1 + 2.25 L/d)}{{\rm Re}_h}} (22.8-9)

where $C_{\rm du}$ is the ultimate discharge coefficient, and is defined as

 C_{\rm du} = 0.827 -0.0085\frac{L}{d} (22.8-10)

For a cavitating nozzle ( $K_{\rm crit}\leq K \leq K_{\rm incep}$)  [ 266] the coefficient of discharge is determined from

 C_d = C_c \sqrt{K} (22.8-11)

For a flipped nozzle ( $K < K_{\rm crit}$) [ 266],

 C_d = C_{\rm ct} = 0.611 (22.8-12)

Figure 22.8.4: Decision Tree for the State of the Cavitating Nozzle

All of the nozzle flow equations are solved iteratively, along with the appropriate relationship for coefficient of discharge as given by the nozzle state. The nozzle state may change as the upstream or downstream pressures change. Once the nozzle state is determined, the exit velocity is calculated, and appropriate correlations for spray angle and initial droplet size distribution are determined.

Exit Velocity

For a single-phase nozzle, the estimate of exit velocity ( $u$) comes from the conservation of mass and the assumption of a uniform exit velocity:

 u = \frac{\dot{m}_{\rm eff}}{\rho_l A} (22.8-13)

For the cavitating nozzle, Schmidt and Corradini [ 323] have shown that the uniform exit velocity is not accurate. Instead, they derived an expression for a higher velocity over a reduced area:

 u = \frac{2C_c p_1 - p_2 + (1 - 2C_c) p_v}{C_c \sqrt{2 \rho_l (p_1 - p_v)}} (22.8-14)

This analytical relation is used for cavitating nozzles in FLUENT. For the case of flipped nozzles, the exit velocity is found from the conservation of mass and the value of the reduced flow area:

 u = \frac{\dot{m}_{\rm eff}}{\rho_l C_{\rm ct} A} (22.8-15)

Spray Angle

The correlation for the spray angle ( $\theta$) comes from the work of Ranz [ 295]:

 \frac{\theta}{2} = \left\{ \begin{array}{cl} \tan^{-1} \lef... ...tating} \\ \; \\ 0.01 & {\rm flipped} \end{array} \right. (22.8-16)

The spray angle for both single-phase and cavitating nozzles depends on the ratio of the gas and liquid densities and also the parameter $C_A$. For flipped nozzles, the spray angle has a constant value.

The parameter $C_A$, which you must specify, is thought to be a constant for a given nozzle geometry. The larger the value, the narrower the spray. Reitz [ 300] suggests the following correlation for $C_A$:

 C_A = 3 + \frac{L}{3.6d} (22.8-17)

The spray angle is sensitive to the internal flow regime of the nozzle. Hence, you may wish to choose smaller values of $C_A$ for cavitating nozzles than for single-phase nozzles. Typical values range from 4.0 to 6.0. The spray angle for flipped nozzles is a small, arbitrary value that represents the lack of any turbulence or initial disturbance from the nozzle.

Droplet Diameter Distribution

One of the basic characteristics of an injection is the distribution of drop size. For an atomizer, the droplet diameter distribution is closely related to the nozzle state. FLUENT's spray models use a two-parameter Rosin-Rammler distribution, characterized by the most probable droplet size and a spread parameter. The most probable droplet size, $d_0$ is obtained in FLUENT from the Sauter mean diameter, $d_{32}$ [ 200]. For more information about the Rosin-Rammler size distribution, see Section  22.12.1.

For single-phase nozzle flows, the correlation of Wu et al. [ 407] is used to calculate $d_{32}$ and relate the initial drop size to the estimated turbulence quantities of the liquid jet:

 d_{32} = 133.0 \lambda {\rm We}^{-0.74}, (22.8-18)

where $\lambda = d/8$, $\lambda$ is the radial integral length scale at the jet exit based upon fully-developed turbulent pipe flow, and We is the Weber number, defined as

 {\rm We} \equiv \frac{\rho_l u^2 \lambda}{\sigma} . (22.8-19)

Here, $\sigma$ is the droplet surface tension. For a more detailed discussion of droplet surface tension and the Weber number, see Section  22.7.2. For more information about mean particle diameters, see Section  22.16.8.

For cavitating nozzles, FLUENT uses a slight modification of Equation  22.8-18. The initial jet diameter used in Wu's correlation, $d$, is calculated from the effective area of the cavitating orifice exit, and thus represents the effective diameter of the exiting liquid jet, $d_{\rm eff}$. For an explanation of effective area of cavitating nozzles, please see Schmidt and Corradini [ 323].

The length scale for a cavitating nozzle is $\lambda = d_{\rm eff}/8$, where

 d_{\rm eff} = \sqrt{\frac{4\dot{m}_{\rm eff}}{\pi \rho_l u}}. (22.8-20)

For the case of the flipped nozzle, the initial droplet diameter is set to the diameter of the liquid jet:

 d_0 = d \sqrt{C_{\rm ct}} (22.8-21)

where $d_0$ is defined as the most probable diameter.

The second parameter required to specify the droplet size distribution is the spread parameter, $s$. The values for the spread parameter are chosen from past modeling experience and from a review of experimental observations. Table  22.8.2 lists the values of $s$ for the three nozzle states. The larger the value of the spread parameter, the narrower the droplet size distribution.

Table 22.8.2: Values of Spread Parameter for Different Nozzle States
State Spread Parameter
single phase 3.5
cavitating 1.5
flipped $\infty$

Since the correlations of Wu et al. provide the Sauter mean diameter, $d_{32}$, these are converted to the most probable diameter, $d_0$. Lefebvre [ 200] gives the most general relationship between the Sauter mean diameter and most probable diameter for a Rosin-Rammler distribution. The simplified version for $s$=3.5 is as follows:

 d_0 = 1.2726 d_{32} \left(1 - \frac{1}{s}\right)^{1/s} (22.8-22)

At this point, the droplet size, velocity, and spray angle have been determined and the initialization of the injections is complete.

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