Pressure-swirl atomiser modelling T.L. Chan & W.T.V. Leung" Department of Mechanical and Marine Engineering, Faculty of Engineering, Hong Kong Polytechnic, Hong Kong * Graduated student ABSTRACT The pressure-swirl atomiser, probably is one of the most widely used atomisation devices, having applications in broad areas of industrial, chemical, and agricultural engineering. Hence, successful simulation will be valuable in terms of time and cost-effectiveness for experimental work and will also simplify the atomiser design process. The Ballistic Modelling (BM)TDiscrete Droplet Modelling (DDM) method has been employed for modelling a pressure-swirl atomiser (Dyna-Coin nozzle) with solid spray cone. The characteristic of its liquid spray is of considerable importance to the operation and performance of combustion systems. A twodimensional spray model has been developed to simulate a continuous spray under steady-state condition. This computer model can simulate the resultant drop size of atomisation and reveal the effect of the important physical variables such as fuel injection pressure, air pressure (or density), co-axial air flow and fuel properties on the result of atomisation process. Dimensional analysis is used to simulate the dropsize immediately after jet breakup and further breakup of the droplets is determined by testifying the critical condition of aerodynamics breakup i.e., (We<j)c = 8/Co. INTRODUCTION Atomisation is the process through which a continuous liquid jet is broken up into a large number of droplets with sizes ranging approximately from 10-500)1. This increases greatly the total surface area of the liquid in preparation for its subsequent evaporation and combustion in many liquid fuel combustion systems. Two commonly employed categories are the pressure-swirl atomiser and twin-fluid atomiser. A lot of research work has been done and is still proceeding in order to elucidate the effect of physical variables on the final product of atomisation, and to lead to a better understanding of the whole phenomena. Computer modelling has always been applied in the prediction of the atomisation process because of its cost-effectiveness; however, the success rate and its popularity are not so high. The Ballistic Modelling (BM)TDiscrete-Drop Modelling (DDM) and Computational Fluid Dynamics (CFD) methods are commonly employed in the modelling of fuel spray systems. The BM/DDM method involves the consideration of fuel spray as individual droplets. Equations of motion
540 Heat Transfer and energy balance of these droplets are set up and by solving these equations, the position and velocity of these droplets can be obtained, while the droplets breakup can be detected. This method is used by Chin et al. [1] to investigate the effect of acceleration or deceleration of the surrounding air on the travelling droplets. On the other hand, the CFD method approaches the problem by treating the liquid spray as continuous fluid flow. Inevitably, this will lead to the application of equations of fluid dynamics to elements of volume in the spray and the distributions of drop-size and velocities are considered. Thereby, solving these equations to obtain the change of these distributions with respect to the dimensional space. Sultanian and Mongia [2] illustrated this CFD method on the fuel nozzle air flow modelling. TWO-DIMENSIONAL PRESSURE SWIRL ATOMISER MODEL A two-dimensional Dyna-Coin nozzle with solid spray cone model using BM/DDM method has been developed to investigate the result of atomisation of a continuous spray under steady-state condition. This model is constituted of the primary spray breakup regime and the secondary spray breakup regime in accordance with real atomisation process. Primary spray breakup regime Three factors affect the disturbance of a liquid jet discharged from the orifice of an atomiser with a certain velocity described by Giffen and Muraszew [3]. These factors are: the conditions of fuel flow at exit from the atomiser; the properties of the medium into which the jet is discharged; and the physical properties of the discharged liquid. Two dimensional analysis equations have been established for the prediction of drop-size resulting from primary breakup of liquid jet as follows: SMD ( Y* = 0.l(Rej)~ '*(Wej)~ n, for stagnant air movement (1) Do IpaJ SMD v-02/ Wp TVuXjf' = 0.07(Rej) (Wej) ' -, for co-axial air movement (2) Do (,pj I Ua ) The above constants are obtained cautiously to incorporate into the developed Fortran program and compare with the similar trend of behaviour obtained by Rizk and Lefebvre [4] & [5], Weiss and Worsham [6], Ingebo [7] and DeCorse [8]. In the preponderance of them; fuel such as heavy fuel oil, kerosene, methyl alcohol and water are used in these trials. Their drop-size distribution resulting from primary breakup can then be passed to next regime for the determination of secondary breakup. Rosin-Rammler distribution function Accurate knowledge of the drop-size distribution is a prerequisite for fundamental analysis of atomisation, and the Rosin-Rammler and Nukiyama-Tanasawa distributions are well-known for correlating the drop-size in a fuel spray. More recently, Bhatia and Durst [9] have used the Log-hyperbolic distribution method to describe the liquid atomisation. However, the Rosin-Rammler distribution has been used frequently to represent drop-distribution resulting from that of swirl nozzles, as described by Fraser and Eisenklam [10], and Zhao et al. [11]. The Rosin-Rammler distribution has been developed as follows:
Heat Transfer 541 D = SMD 1/2.987 In(l-V)] 0.41 J (3) Since in the spray cone, the larger droplets are distributed along the peripheral and the smaller ones are distributed into the interior as described by DeCorse [8], Bachalo [12] and Asheim et al. [13], it is reasonable to assume that the cumulative volume fraction of the whole area will vary from the central axis as follows: (1 - V)i = 0.99, for the central axis, i = 0.99- (i-l)a 0.175 for the outer cone i = 2 & 3 and A = 0.0583mm. (4) (1-V)4 = 0.01, for the outer cone, i = 4. Based on the experimental results, the axial velocity of spray droplets near the nozzle along the central axis and the outer cone measured approximately 1.2 m/s and 3 m/s respectively. It is discovered that, if all the droplets' axial velocity immediately after breakup is around 80% of the calculated jet velocity, the above observation can be achieved at an axial displacement of 60 mm from the nozzle after several trials with the synthesis of the computer model as follows: UXd=0.8UXj (5) Secondary spray breakup regime Before any secondary break-up can be detected, the paths of the droplets must be established. For the atomisation in stagnant air condition, droplets are assumed to be travelling in cones inscribed in each other with the largest cone subtending an angle of 15 with the central axis. In order to simplify the model, only droplets travelling in three of these cones (subtending 5, 10, 15 to the central axis respectively) and the central axis are investigated as shown in Figure 1. x4,(vd)4 ri(xo) r4(xo) Parallel circle at 10 mm from nozzle Parallel circle at xl mm from nozzle Figure 1 The spray conical conformation for atomisation. In order to subtend a cone, the velocity components of a single droplet are assumed to have the following relationships:
542 Heat Transfer URd=Kr(UXd) (6) U6d = Ke(URd) = KrKe(UXd) (7) Dumouchel et al. [14] described the ratio of tangential and radial velocity components in the swirl chamber, Ke is approximately 5 and this ratio is assumed to be carried also by the spray droplets. However, the value of Kr will change with the cone angles of the droplets. For a cone angle, 6i = 0,5, 10 and 15, the corresponding value of the constant(kr)i is given by (-Lj- The radius of the parallel circle on the i-th cone at a particular displacement xi,, ri(xi), is given by ri(xi) = ^[ri(0) + xi(kr)if + [(Kr)iKexif (9) In accordance with the above conditions, the following equation of motion in stagnant air condition can be used to describe the trajectories of a drop, where V< = V, = UX, and the effects of virtual mass, Bassett forces, Magnus forces and gravitational forces etc., are negligible as described by Kuo [15]. Since the drag coefficient, CD is a function of the Reynolds number of droplet, Red as described by Asheim et al. [13], it is convenient to write into ( n H \-084 where K = 27 ^- & a = -0.84, for 0 < Red < 80, or vm-a ; / \ 0.2 17 K = 0.271 -^ & a = 0.217, for 80 < Rej < 10*' or Ua ) K = 2& a = 0, for Red > 10^ Solving equation (10) in stagnant air condition with respect to time using relation in equation (11), we have (11)
Heat Transfer 543 (12) where S = K(l + a)- (13) d PI and the value of 'K' and 'a' will change with the Reynolds number of droplet. Solving equation (10) with respect to x; using relation in equation (11), we have _ j \ where (Axi)j = (xi)j + i-(xi)j (14) For atomisation with co-axial air movement, it is usually a convenient practice to assume that these droplets are travelling only in the axial direction, as described by Asheim et al. [13] and Chin et al. [1]. This is inevitable as the motion of the droplets will be undefined as the drag force are not in the direction of the drop velocity. As a result, the relative velocity (Vr)x in the axial direction is given by (Vr)x= Ua-UXd (15) It is reasonable to assume that the spray cone will have a cylindrical conformation when the co-axial air velocity exceeds or equals to 20 m/s. In addition, it is stated that for Mach number, Ma < 0.3, i.e., Ua < 100 m/s, the density effects of air, the kinetic energy of the mean flow and viscous dissipation in the combustion chamber are negligible as described by Kuo [15], therefore the coaxial air movement inside the chamber can be considered as incompressible flow under this condition. At the same time, the application of atomiser at such a drastically high speed air movement is very limited. Hence the greatest co-axial air movement in this model is limited to 100 m/s for reason of simplicity. In accordance with the above conditions, the equation of motion of a droplet in co-axial air movement can be described as follows, Solving the equation (16) with respect to time using relation in equation (11) we have have (17) O Solving equation (16) with respect to xi using relation in equation (11), we -SA, = 0 (18)
544 Heat Transfer In order to detect the droplets breakup along their prescribed paths, readings of the relative velocities are taken every millimeter in the central axis, Ax, = 1 mm. For the conical conformation, the corresponding displacement for the path other than the central axis, we have for i = 2, 3 & 4. While for the cylindrical conformation, all the displacement will be constant. The relative velocities are used to test against the critical condition, (We<0c = S/CD as described by Faeth [16] for the secondary breakup of droplets. If the secondary breakup is detected, the breakup time will be calculated immediately. Where the drop-size along the path of droplet should be replaced by the new dropsize and velocity when the breakup time is reached as described by Andersen and Wolfe [17]. RESULTS AND DISCUSSION In this computer model, secondary breakup is found to occur only when the critical velocity of droplet is detected. Hence, the SMD of droplets resulting from primary break up will remain unchanged along the downstream when the relative velocity of droplet is less than the critical droplet velocity. A comparison of the effect of different fuel properties, air pressure (or density) and fuel injection pressure on the SMD under the primary breakup is shown in Figure 2. The increase in fuel injection pressure will lead to finer droplets when different fuels are used. In conjunction with increasing air pressure, it will lead to even finer droplets. A similar trend of experimental results have been reported by Rizk and Lefebvre [4]. The acceleration of droplets in the axial direction should be observed when the co-axial air velocity is greater than the initial fuel droplet velocity as shown in Figure 3. The increase in co-axial air velocity will lead to finer droplets under different fuels. However, no secondary breakup of droplets is detected yet in the radial direction under this condition. In response to different fuels, secondary breakup occurs under the specific co-axial air velocity at a distance very close to the breakup of ligaments as shown in Figure 4. The spray drop sizes at the peripheral decreased in the downstream direction as the corresponding number of droplets increased. CONCLUSIONS This computer model can simulate the resultant drop size of atomization and reveal the effect of the important physical variables such as fuel injection pressure, air pressure (or density), co-axial air flow and fuel properties on the result of atomization. It is eligible for good prediction of aerodynamic breakup of droplets in high air velocity. Satisfactory theoretical results have reconciled the rectification and modification of theories. Moreover, better solutions and explanations to the intricate problems and uncertainties in the atomisation process will be obtained through successful research work, so that modification and refinement of this model can be made. (19)
Heat Transfer 545 100 90 80 _ 70 I 60 I 50 Q Z on 40 30 20 10 Heavy Fuel Oil Fuel Injection Pressure Methyl Alcohol 1 1.5 2 1 1.5 2 Air Pressure (bar) Figure 2 Effect of different fuel properties on the SMD co-axial air velocity - 0 m/s. 1 1.5 2 versus air pressure when the 10 15 20 25 30 35 40 45 50 55 60 Axial displacement of droplet (mm) Figure 3 A typical result of the effect of different fuels on the mean axial velocity of droplet versus axial displacement of droplet when injection pressure = 10 bar, air pressure = 1 bar & co-axial air velocity = 20 m/s. Heavy Fuel Oil * Heavy Fuel Oil Droplet(s) o- Kerosene -a Methyl Alcohol -* Kerosene o Droplet(s) Methyl Alcohol Droplet(s)? 0.875 1.763 2.679 Radius of spray cone (mm) Fig. 4 A typical result of the effect of different fuels on the SMD & no. of fuel droplets versus radius of spray cone at the axial displacement = 20 mm, when fuel injection pressure = 10 bar and air pressure = 1 bar.
546 Heat Transfer NOMENCLATURE CD D, d = drag coefficient of droplet = fuel droplet diameter, m U6j U0j = axial velocity of droplet at spray angle 6, m/s = axial velocity of droplet at the i-th spray angle DO Kj- = orifice diameter of swirl atomiser nozzle, m = radial velocity constant V 9j, m/s = volume fraction in Rosin-Rammler distribution K0 = tangential velocity constant Vj = velocity of droplet, m/s Ma = Mach number V,- = relative velocity of droplet, m/s Rej = Reynolds number of liquid jet (We)<j = weber number of droplet at critical condition r, = radius of the i-th spray cone, m Wej = weber number of liquid jet SMD = Sauter Mean Diameter, Dg^ m xj = droplet displacement on the i-th spray cone, m tj = time, s Ax; = droplet displacement interval on the i-th spray Atj = time interval, s cone, m Ua = co-axial velocity of air, m/s p; = density of liquid fuel, kg/m URj = radial droplet velocity of spray cone, m/s p% = density of air, kg/m URj UXj = radial velocity of i-th drop = axial velocity of liquid jet, m/s \L% = viscosity of air, kg/m.s REFERENCES [1] Chin J.S., Nickolaus D. & Lefebvre A. H., "Influence of Downstream Distance on the Spray Characteristics of Pressure-Swirl Atomizers," ASMEJ. of Engineering for Gas Turbines and Power, Vol. 108, pp. 219-224, 1986. [2] Sultanian B. K. & Mongia H. C., "Fuel Nozzle Air Flow Modeling," AIAA/ASME/SAE/ASEE 22nd Joint Propulsion Conf., Alabama, 16-18 June 1986, AIAA-86-1667, pp. 1-14. [3] Giffen E. & Muraszew A., The Atomisation of Liquid Fuels, Chapman & Hall Ltd., London, 1953. [4] Rizk N. K. & Lefebvre A. H., "Spray Characteristics of Simplex Swirl Atomizers," Progress in Astronautics and Aeronautics, Vol. 95, pp. 563-580, 1984 [5] Rizk N. K. & Lefebvre A. H., "Influence of Downstream Distance on Simplex Atomizer Spray Characteristics," ASME Paper 84-WA/HT-25, 1984. [6] Weiss M. A. & Worsham C. H, "Atomization in High Velocity Airstreams," ARS Journal, April 1959, pp. 252-259. [7] Ingebo R. D., "Capillary and Acceleration Wave Breakup of Liquid Jets in Axial-Flow Airstreams," NASA Technical Paper 1791, 1981. [8] DeCorso S. M., "Effect of Ambient and Fuel Pressure on Spray Drop Size," ASME J. of Engineering for Power, pp. 10-18, Jan. 1960. [9] Bhatia J. C. & Durst F., "Description of Sprays Using Joint Hyperbolic Distribution in Particle Size and Velocity," Combustion and Flame 81, pp. 203-218, 1990. [10] Fraser R. P. & Eisenklam P., "Liquid Atomisation and the Drop Size of Spray," Trans. Instn Chem. Engrs, Vol 34, pp. 294-319, 1956. [11] Zhao Y. H., Hou M. H., Kong X. Z. & Chin J. S., "Investigation on Drop Size Distribution," AIAA/SAE/ASME 20th Joint Propulsion Conf., 11-13 June 1984, Ohio, AIAA-84-1314, pp. 1-7. [12] Bachalo W. D., Houser M. J. & Smith J. N., "Evolutionary Behavior of Sprays produced by Pressure Atomisers," AIAA 24th Aerospace Sciences Meeting, 6-9 Jan. 1986, Nevada, pp. 1-12. [13] Asheim J. P., Kirwan J. E. & Peters J. E., "Modeling of a Hollow-Cone Liquid Spray Including Droplet Collisions," J. of Propulsion, Vol. 4, no. 5, Sept.-Oct. 1988, pp. 391-398. [14] Dumouchel C., Ledoux M., Bloor M. I. G., Dombrowski N. & Ingham D B, "The Design of Pressure Swirl Atomizers," 23rd Symposium (Int.) on Combustion, The Combustion Institute, pp. 1461-1467, 1990. [15] Kuo K. K., Principles of Combustion, John Wiley & Sons, Inc., New York, 1986. [16] Faeth G. M., "Structure and Atomization Properties of Dense Turbulent Sprays," 23rd Symposium (Int.) on Combustion, Orleans, Paper No. 23-536, pp. 1-23, 1990. [17] Wolfe H. E. and Andersen W. H., "Kinetics, Mechanism, and Resultant Droplet Sizes of the Aerodynamic Breakup of Liquid Drops," Report No. 0395-04(18)SP, Aerojet-General Corp., Downey, April 1964.