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22.8.2 The Pressure-Swirl Atomizer Model

Another important type of atomizer is the pressure-swirl atomizer, sometimes referred to by the gas-turbine community as a simplex atomizer. This type of atomizer accelerates the liquid through nozzles known as swirl ports into a central swirl chamber. The swirling liquid pushes against the walls of the swirl chamber and develops a hollow air core. It then emerges from the orifice as a thinning sheet, which is unstable, breaking up into ligaments and droplets. The pressure-swirl atomizer is very widely used for liquid-fuel combustion in gas turbines, oil furnaces, and direct-injection spark-ignited automobile engines. The transition from internal injector flow to fully-developed spray can be divided into three steps: film formation, sheet breakup, and atomization. A sketch of how this process is thought to occur is shown in Figure  22.8.5.

Figure 22.8.5: Theoretical Progression from the Internal Atomizer Flow to the External Spray

The interaction between the air and the sheet is not well understood. It is generally accepted that an aerodynamic instability causes the sheet to break up. The mathematical analysis below assumes that Kelvin-Helmholtz waves grow on the sheet and eventually break the liquid into ligaments. It is then assumed that the ligaments break up into droplets due to varicose instability. Once the liquid droplets are formed, the spray evolution is determined by drag, collision, coalescence, and secondary breakup.

The pressure-swirl atomizer model used in FLUENT is called the Linearized Instability Sheet Atomization (LISA) model of Schmidt et al. [ 325]. The LISA model is divided into two stages:

1.   film formation

2.   sheet breakup and atomization

Both parts of the model are described below.

Film Formation

The centrifugal motion of the liquid within the injector creates an air core surrounded by a liquid film. The thickness of this film, $t$, is related to the mass flow rate by

 \dot{m}_{\rm eff} = \pi \rho ut(d_{\rm inj} - t) (22.8-23)

where $d_{\rm inj}$ is the injector exit diameter, and $\dot{m}_{\rm eff}$ is the effective mass flow rate, which is defined by equation Equation  22.8-5 . The other unknown in Equation  22.8-23 is $u$, the axial component of velocity at the injector exit. This quantity depends on internal details of the injector and is difficult to calculate from first principles. Instead, the approach of Han et al. [ 134] is used. The total velocity is assumed to be related to the injector pressure by

 U = k_v \sqrt{\frac{2 \Delta p}{\rho_l}} (22.8-24)

where $k_v$ is the velocity coefficient. Lefebvre [ 200] has noted that $k_v$ is a function of the injector design and injection pressure. If the swirl ports are treated as nozzles and if it is assumed that the dominant portion of the pressure drop occurs at those ports, $k_v$ is the expression for the discharge coefficient ( $C_d$). For single-phase nozzles with sharp inlet corners and $L/d$ ratios of 4, a typical $C_d$ value is 0.78 or less [ 207]. If the nozzles are cavitating, the value of $C_d$ may be as low as 0.61. Hence, 0.78 should be a practical upper bound for $k_v$. Reducing $k_v$ by 10% to 0.7 approximates the effect of other momentum losses on the discharge coefficient.

Physical limits on $k_v$ require that it be less than unity from conservation of energy, yet be large enough to permit sufficient mass flow. To guarantee that the size of the air core is non-negative, the following expression is used for $k_v$:

 k_v = \max \left[0.7, \frac{4 \dot{m}_{\rm eff}}{d_0^2 \rho_l \cos \theta}\sqrt{\frac{\rho_l}{2 \Delta p}}\right] (22.8-25)

Assuming that $\Delta p$ is known, Equation  22.8-24 can be used to find $U$. Once $U$ is determined, $u$ is found from

 u = U \cos \theta (22.8-26)

where $\theta$ is the spray angle, which is assumed to be known. At this point, the thickness and axial component of the liquid film are known at the injector exit. The tangential component of velocity ( $w = U \sin \theta$) is assumed to be equal to the radial velocity component of the liquid sheet downstream of the nozzle exit. The axial component of velocity is assumed to remain constant.

Sheet Breakup and Atomization

The pressure-swirl atomizer model includes the effects of the surrounding gas, liquid viscosity, and surface tension on the breakup of the liquid sheet. Details of the theoretical development of the model are given in Senecal et al. [ 326] and are only briefly presented here. For a more robust implementation, the gas-phase velocity is neglected in calculating the relative liquid-gas velocity and is instead set by the user. This avoids having the injector parameters depend too heavily on the usually under-resolved gas-phase velocity field very near the injection location.

The model assumes that a two-dimensional, viscous, incompressible liquid sheet of thickness $2h$ moves with velocity $U$ through a quiescent, inviscid, incompressible gas medium. The liquid and gas have densities of $\rho_l$ and $\rho_g$, respectively, and the viscosity of the liquid is $\mu_l$. A coordinate system is used that moves with the sheet, and a spectrum of infinitesimal wavy disturbances of the form

 \eta = \eta_0 e^{ikx + \omega t} (22.8-27)

is imposed on the initially steady motion. The spectrum of disturbances results in fluctuating velocities and pressures for both the liquid and the gas. In Equation  22.8-27, $\eta_0$ is the initial wave amplitude, $k = 2 \pi/\lambda$ is the wave number, and $\omega = \omega_r + i \omega_i$ is the complex growth rate. The most unstable disturbance has the largest value of $\omega_r$, denoted here by $\Omega$, and is assumed to be responsible for sheet breakup. Thus, it is desired to obtain a dispersion relation $\omega =\omega(k)$ from which the most unstable disturbance can be calculated as a function of wave number.

Squire [ 353], Li and Tankin [ 206], and Hagerty and Shea [ 131] have shown that two solutions, or modes, exist that satisfy the governing equations subject to the boundary conditions at the upper and lower interfaces. The first solution, called the sinuous mode, has waves at the upper and lower interfaces in phase. The second solution is called the varicose mode which has the waves at the upper and lower interfaces $\pi$ radians out of phase. It has been shown by numerous authors (e.g., Senecal et. al. [ 326]) that the sinuous mode dominates the growth of varicose waves for low velocities and low gas-to-liquid density ratios. In addition, it can be shown that the sinuous and varicose modes become indistinguishable for high-velocity flows. As a result, the atomization model in FLUENT is based upon the growth of sinuous waves on the liquid sheet.

As derived in Senecal et al. [ 326], the dispersion relation for the sinuous mode is given by

\omega^2[\tanh(kh) + Q] + [4\nu_l k^2 \tanh(kh) + 2iQkU] \; +

 4\nu_l k^4 \tanh(kh) - 4\nu_l^2 k^3 \ell \tanh(\ell h) - QU^2 k^2 + \frac{\sigma k^3}{\rho_l} = 0 (22.8-28)

where $Q = \rho_g/\rho_l$ and $\ell^2 = k^2 + \omega/\nu_l$.

Above a critical Weber number of ${\rm We}_g$ = 27/16 (based on the liquid velocity, gas density, and sheet half-thickness), the fastest-growing waves are short. For ${\rm We}_g < 27/16$, the wavelengths are long compared to the sheet thickness. The speed of modern high pressure fuel injection systems is high enough to ensure that the film Weber number is well above this critical limit.

An order-of-magnitude analysis using typical values shows that the terms of second order in viscosity can be neglected in comparison to the other terms in Equation  22.8-28. Using this assumption, Equation  22.8-28 reduces to

\omega_r = \frac{1}{\tanh(kh) + Q} \left\{-2 \nu_l k^2 \tanh(kh) + \phantom{\sqrt{\frac{1}{2}}} \right.

 \left. \sqrt{4 \nu_l^2 k^4 \tanh^2(kh) - Q^2 U^2 k^2 - [\tan... ...] \left[-QU^2 k^2 + \frac{\sigma k^3}{\rho_l}\right]} \right\} (22.8-29)

For waves that are long compared with the sheet thickness, a mechanism of sheet disintegration proposed by Dombrowski and Johns [ 84] is adopted. For long waves, ligaments are assumed to form from the sheet breakup process once the unstable waves reach a critical amplitude. If the surface disturbance has reached a value of $\eta_b$ at breakup, a breakup time, $\tau$, can be evaluated:

 \eta_b = \eta_0 e^{\Omega \tau} \Rightarrow \frac{1}{\Omega} \ln \left(\frac{\eta_b}{\eta_0}\right) (22.8-30)

where $\Omega$, the maximum growth rate, is found by numerically maximizing Equation  22.8-29 as a function of $k$. The maximum is found using a binary search that checks the sign of the derivative. The sheet breaks up and ligaments will be formed at a length given by

 L_b = U \tau = \frac{U}{\Omega} \ln \left(\frac{\eta_b}{\eta_0}\right) (22.8-31)

where the quantity $\ln (\frac{\eta_b}{\eta_0})$ is an empirical sheet constant that you must specify. The default value of 12 was obtained theoretically by Weber [ 392] for liquid jets. Dombrowski and Hooper [ 83] showed that a value of 12 for the sheet constant agreed favorably with experimental sheet breakup lengths over a range of Weber numbers from 2 to 200.

The diameter of the ligaments formed at the point of breakup can be obtained from a mass balance. If it is assumed that the ligaments are formed from tears in the sheet twice per wavelength, the resulting diameter is given by

 d_L = \sqrt{\frac{8h}{K_s}} (22.8-32)

where $K_s$ is the wave number corresponding to the maximum growth rate, $\Omega$. The ligament diameter depends on the sheet thickness, which is a function of the breakup length. The film thickness is calculated from the breakup length and the radial distance from the center line to the mid-line of the sheet at the atomizer exit, $r_0$:

 h_{\rm end} = \frac{r_0 h_0}{r_0 + L_b \sin \left(\frac{\theta}{2}\right)} (22.8-33)

This mechanism is not used for waves that are short compared to the sheet thickness. For short waves, the ligament diameter is assumed to be linearly proportional to the wavelength that breaks up the sheet,

 d_L = \frac{2 \pi C_L}{K_s} (22.8-34)

where $C_L$, or the ligament constant, is equal to 0.5 by default.

In either the long-wave or the short-wave case, the breakup from ligaments to droplets is assumed to behave according to Weber's [ 392] analysis for capillary instability.

 d_0 = 1.88d_L (1 + 3 {\rm Oh})^{1/6} (22.8-35)

Here, $Oh$ is the Ohnesorge number which is a combination of the Reynolds number and the Weber number (see Section  22.7.2 for more details about Oh). Once $d_0$ has been determined from Equation  22.8-35, it is assumed that this droplet diameter is the most probable droplet size of a Rosin-Rammler distribution with a spread parameter of 3.5 and a default dispersion angle of 6 $^{\circ}$ (which can be modified in the GUI). The choice of spread parameter and dispersion angle is based on past modeling experience [ 324]. It is important to note that the spray cone angle must be specified by the user when using this model.

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